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NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 



'■l-^ . 



Space Programs Summary 37-51, Vol. Ill 

Supporting Research and Advanced Development 



For the Period April 1 to May 31, 1968 



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ff653 July 65 



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N68-3'?^9'rJ N68-3742 



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%, I (NASA CK OR TMX OR AD NUMBER) 



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JE1 PROPULSION LABORATORY 

CALIFOSNIA INSTITUTE OF TECHNOLOGY 
PASADENA, CALIFORNIA 



June3C, 1968 



NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 






Space Programs Summary 37-51, Vol. Ill 

Supporting Research and Advanced Development 

For the Period April 1 to May 31, 1968 



JET PROPULSION LABORATORY 

CALIFORNIA INSTITUTE OF TECHMOLOOY 
PASADENA, CALIFORNIA 

June 30, 1968 



SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



Copyright® 1968 

Jet Propulsion Laboratory 

California Institute of Technology 

Prepared Under Contract No. NAS 7-100 
National Aeronautics & Space Administration 



Preface 

The Space Programs Summary is a bimonthly publication that presents a review 
of engineering and scientific work performed, or managed, by the Jet Propulsion 
Laboratory for the National Aeronautics and Space Administration during a two- 
month period. Beginning with the 37-47 series, the Space Programs Summary is 
composed of four volumes: 

Vol. I. Flight Projects (Unclassified) 

Vol. II. The Deep Space Network (Unclassified) 

Vol. HI. Supporting Research and Advanced Development (Unclassified) 

Vol. IV. Flight Projects and Supporting Research and Advanced 
Development (Confidential) 



Approved by: 



W. H. PickiKing, Director ^ 

Jet Propulsion Laboratory 



in SPACE PROGRAMS SUMMARY 37-51, VOL. Ill iii 






\UAi^T>. 



Contents 



SYSTEMS DIVISION 

I. Systems Analysis Research 1 *^ 

A. Shadow Equation for a Satellite 

NASA Code 8 M-) 2-02-0 1,, iore/l 1 

B. A Consistent Ephemeris of the Major Planets 
in the Solar System 

NASA Code :29-04-O4-O2.W.G.Melbayrnc and D.A.O'Handley 4 

C. Correction of the Lunar Orbit Using Analytic Partial 
Derivatives 

NASA Code 129-04-04-02, J. 0. Mulholhnd 13 

D. Bayesian Estimation Based on the Gram-Charlier Expansion 

NASA Code I29-04-0I-0I,W.Ki2ner 15 

II. Systems Analysis 19 i' 

A. A Proposed Venus Coordinate System 

NASA Code 684-30-0?-)0,F.M.Sfurms,Jr IV 

III. Computation and Analysis 1A y/ 

A. OrthogonnI Transformations for Linear Algebraic 
Computations 

NASA Code 129-04-04-01, C. I. lowson 24 

GUIDANCE AND CONTROL DIVISION 

IV. Spacecraft Power 29 '^ 

A. Solar Cell Standardization 

NASA Code 120-33-01-03, R.f. Greenwood 29 

B. Solar Power System Definition Studies 

NASA Code I20-33-O5-0J,H./M. Wick 30 

C. Development of Improved Solar Cell Contacts 

NASA Code 120 J3-0I-)2, p. Berman 31 

D. CapsuleSystem Advanced Development: Power Subsystem 

NASACodeI20-33-05-03andI20-33-0S-07. ft. G./vanoff and O.J. Hopper ... 32 

E. Computer-Aided Circuit Analysis 

NASA Code I20.33-08-02, D.J. Hopper 35 

F. Electric Propulsion Power Conditioning 

NASA Code 120-26-04-05, E.N.Costogue 35 

G. Mars Spaciscraft Power System Development 

NASA Code 120-33-05-04, H. W. W/ck 36 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. \\\ 



Contents (contd) 

H. Planetary Solar Array Development 

NASA Code 120-33-01-08, W. A. Hasboch 37 

I. Thermionic Research and Development 

NASA Code 120-33-02-06, O. S. Merrill 41 

J. Thermionic Converter Development 

NASACode I20-33-02-0I,P. Rouklove 44 

V. Guidance and Control Analysis and Integration 48 V 

A. Automation of Variational Techniques for the Solution of 
Optimum Control Problems 

NASA Code I25-I7-05-0I,H. Mock, Jr 48 

B. Optical Approach-Guidance Flight Feasibility Demonstration 

NASA Code I25-)7-02.0I,T C.Duxbury 50 

C. Development of Computer-Oriented Operational Support Equipment 

NASA Code 186-68-02-27, J. P. Perrid 51 



y 



/ 



VI. Spacecraft Control 53 

A Sterilizable Inertial Sensors-. Gas Bearing Gyros 

NASA Code 186-58-02-03, P.J. Hond 53 

B. Analysis of Ion Thruster Control Loops 

NASA Code 120-26-08-0), p. A. Mueller and E.V.Powlik 55 

C. Pov*/ered Flight Control Systems 

NASA Code 186-68-02-33, R.J. Monkov/fz 58 

D. Spacecraft Antenna Pointing for a Multiple-Planet Mission 

NASA Code l25-I9-04-0I,G.£. Fleischer 63 

E. Extended Mission Control Systems Development 

NASA Code 186-68-02-31,1. McGlinchey 65 

VII. Guidance and Control Research . . 72 

A. Josephson Junction Memory Elements 

NASACode)29-02-05-02, p. V.Mason 72 

B. Frequency Response of Thin-Film Thermal Detectors 

HM/^ Code 129-02-05-09, J. Maser(ian 75 

C. GoSe Schottky Barrier Gate 

NASACodeI29-02-05-09,S.KurfinondC. A. Mead 77 

D. Metal Contacts to Photoconductors 

NASA Code 129-02-05-07, R. J. Sfirn 78 

E. Pre-ignition Characteristics of Cesium Thermionic Diodes: Part II 

NASA Code 129-02-01-07, K.Shimodo 82 

F. Thermionic Diode Switch 

NASA Code 129-02-01.07, S. Luebbers 86 

vi JPL SPACE PROGRAMS SUMMARY 37-5?, VOL. Ill 



Contants (contd) 

ENGINEERING MECHANICS DIVISION 
VIII. MQtviiaU 91 i^ 

A. Effect of Notch Severity on Cross-Rolled 
Beryllium Sheet 

NASA Code I24-09-0I-02, R. Moss 91 

ENVIRONMENTAL SCIENCES DIVISION 

IX. Aerodynamic Facilities 95 ''^^ 

A. Heat Transfer Stuay of 60-cleg Half-Angle Cones 

NASA Code 124-07.01-04, M. F. B/oir 95 

X. Environmental and Dynamic Testing 97 '^'"^ 

A. Low-Frequency Plane-Wave Sound Generator and 
Impedance-Measuring Device 

NASA Code 124-08-05-04, C. D. Hayes 97 

PROPULSION DIVISION 

XI. Solid Propellant Engineering 101 i^ 

A. Molecular Momentum Transfer From Regressing Solid 
Propellant Surfaces 

NASA Code 128-32-06-01, O. K. Heiney 101 

B. T-Burner Studies 

NA5A Code ) 28-32-06-0 1, E. H. Perry 105 

XII. Polymer Research 109 ''' 

A. Investigation of the Transport Characteristics of an 
lonene Memb'one 

NASA Code 12044-01-03, H. y. Tom and J. Mooconin 109 

XIII. Research and Advanced Concepts 116 '""''^ 

A. Laminarization in Nozzle Flow 

NASA Code 128-31-06-08,1. K Bock, R.F.CuWe/, and P. f. Mossier . . . .116 

B. Liquid-Metal MHD Power Conversion 

NASA Code 120-27-06-03, D. G. E/i-ott, L. G. Hayes, and D. J. Cerini . . . . 1 20 

C. Evaluation of the SE-20C Thruster Design 

NASA Code 120-26-08-0), T. D. Mosek 1 24 

D. Radial Distribution of Enthalpy in a High-Temperature 
Swirling Flow 

NASA Code t29-OI-05-J0, P. f. Mossier 128 

E. Some Effects of an Applied, Transverse Magnetic Field 

on Heat Transfer With Swirling and Nonswirling Gas Flow 

NASA Code ) 29-0 1-05- M, E. J. Roschke 130 

jn SPACE PROGRAMS SUMMARY 37-51, VOL. 1(1 vii 



^ 



^ 



Contents (contd) 

F. Some Effects of on Applied, Transverse Magnetic Field on 
Wal! Pressure in a Square Channel 
NASACode I29-0I-0S-IJ, E. J. Roschke 134 

XIV. Liquid Propulsion 137 

A. Heat-Sterilization Compatibility of Ethylene-Propylene 
Rubber in WM, 

NASA Code 731.12-03-03, O. F. Ke/ler 137 

SPACE SCIENCES DIVISION 

XV. Lunar and Planetary Instruments 143 

A. Atmospheric Entry Sampling System 

NASA Code 185-37-34-01, S. Rich 143 

B. Prototype Moss Spectrometer for Planetary Atmosphere Analysis 

NASA Code 185-37-34-01. H.R.AAerti 146 

XVI. Space Instruments 152 

A. A Pulse-Height Analyzer for Space Application 

NASA Code 166-68-06-06, W. J. Schneider 152 

B. Quantitative Use of Imaging Systems: An Electronic 
Camera System 

NASA Code 125-24-01-09, A. T. Young and F. p. tondouer 159 

C. On the Slow-Scan Characteristics of the WX30691 SECVidicon 

NASA Code 125-?4-01-03, K.J. Ando 162 

XVII. Science Data Systems 169 

A. Digital Techniques for Generating a Time-Dependent 
Acceleration Voltage for a Mass Spectrometer 

fMSA Code ; "5-23-02 ?2,M. Perlmcn 169 

B. Capsule System Advanced Development Woven Plated-Wire 
Memory 
NASA Code 1 86-5^-03-02, P. B. Whifeheod . . 175 

XVIII. Lunar and Planetary Sciences 182 

A. Scattering in the Twilight Atmosphere of Venus 

NASA Code 185-47-33-01, K. D. Abhyonkor 182 

B. WoterVapor Variations on Venus 

NASA Code )8S-4l-2)-0), R. A. Schorn, I. D. Gray, I. S. Soricer, 

ondR. C.Moore 184 

XIX. Physics 187 \.'^' 

A. Auroral Arcs -. Result of the Interaction of a Dynamic 
Magnetosphere With the Ionosphere 

NASA Code 129.02-07.02, G. A>lcinion 187 

B. Rates and Mechanisms of the Gas Phase Ozonation of Ethylene 
and Acetylene 

NASA Code 129-02-01-04, W. 0. OeMore 189 

viii JPL SPACE PROGRAMS iMtAIAkW 37-51, VOL. W 



Contents (contd) 



C. Prediction of OH Radical Microwave Lambda Doubling 
Transitions Below 1 20 GHz 

NASA Code 129-02-06-01, P. L. Poynttr and R. A. Beaudef , 

D. An Ion Cyclotron Resonance Study of the Escape of Helium 
From the Earth's Atmosphere 

NASA Code 129-02-01-09, J. King, Jr. ond D. D. £/(eman, 

E. Shape of the Magnetosphere 

NASA Code 129-02-07-02, G. Afkinion ond T. Unti . . ^ , . 



193 

198 
200 



TELECOMMUNICATIONS DIVISION 

XX. ''ommunicaHont SyttOMS Research 203 

A. Coding and Synchronization Studies-. A General Formulation 
of Linear Feedbcck Communications Systems With Solutions 

NASA Code 125-21-02-03, S. Bulmon 203 

B. Combinatorial Communication : The Maximum Indices of Comma 
Freedom for the high-Data-Rate Te'emfc'fy Codes 

NASA Code I2S-21-0I-0M. 0. Boumert ond H. C. Rumse>',Jf 215 

C. Propagation Studies : AMopof the Venus Feature /J 

NAf A Code 125-21-02-04, S. Zoiior ond R. Goldsfein 217 

D. Propagation Studies : The Variance of Scattering-Law Estimates 

NASA Code 150-22-11-08, O.G.Ke/ly 219 

E. Communications Systems Development: Design of One- and Two-Vi/oy 
High-Rote Block-Coded Telemetry Systems 

NASA Code 150-22-1 1-08, W. C.tindsey 225 

F. Communications Systems Development: A Digito' Demonstration 
of Sequential Decoding and Comparison With Block-Coded Systems 

NASA Code 150-22-11-08, p. Stonek 232 

G. Communications Systems Development: The Optimum Cross-Correlation 
Function for a First-Order Tracking Loop Under Unit Power 
Constraint 

NASA Code 150-22-11-08, J. W. toylond 240 

H. Information Processing : Disjoint Cycles From the de Bruijn Graph 

NASA Code 150-22-11-09, H.fredrickien 244 

I. Information Processing: Estimating the Correlation Between 
Two Normal Distributions When Only the Means are Known 
NASA Code 150-22-11-09, 1. Eiienberger 250 

J. Infoi motion Procassing : The Distribution of the Ratio 
of Two Jointly Normal Random Variables 

NASA Code 150-22-11-09, /.Eiienberger 254 

K. Astrometrics : Pulsar Observations 

NASA Code 150-22-M-IO, R. M. Gofdtrein 256 

L. Astrometrics : Optimum Range Gates 

NASA Code 150-22-1 1 -10, A. Gorjio, E. Rodemlch, ond H. Rumiey, ir 258 



^ 



jn SMCE PROGRAMS SUMMARY 37-51, VOL. Ill 



ix 



y 



Contents (contd) 

M. Data Compression Techniques : Product Entropy of Gaussian 
Distributions 

NASA Code )50-22-?7-08, C. C. Poiner, i. R. Rot/emich, und H. Rumiey, it. . . . 266 

N. Data Compression Techniques: Estimators of the Parameters 
of an Extreme- Value Distribution Using Quantiles 

NASA Code 150.22.17-08, /. fisenberoer 277 

O. Data Compression Techniques: Mass Spectrogram Data Compression 
by the Slope Threshold Melhod 

NASA Code 150-22-17^)8, t.K/einrotk 285 

P. Data Compression Techniques: Estimating the Correlation Between 
Two Normal Populations Using Quantiles of Conditional Distributions 

NASA Code ?50-22.I7O8, /. Eijenberger 289 

XXI. Communications Elements Research 295 

A. RF Techniques: Sv itching Frequency Determination for the 
Nodding Subdish System 

NASA Code 125-21-03-0^, T. Sofo, W. V. T. Rujch, C. T. Stedried, S. 0. S/obin, 

and O. B. Porhom 295 

B. Precision Calibration Techniques: Microwave Thermal Noise 
Standards 

NASA Code 150-22-11-07, C.Sfelzried 299 

C. RF Breakdown Studies: RF Breakdown in Coaxial Transmission 
Lines 

NASA Code 125-22-01-02, R. Woo 302 

D. Spacecroft Antenna Research: 400-MHz Coaxial Cavity Radiator, 
Part II 

NASA C.de 184-68-04-02, K. Woo 307 

XXII. Spacecraft Telemetry and Command 310 

A. Multiple-Mission Telemetry System: Bit-Sync Lock Detector 
Evoluation 

' NASA Code I50-22-I7-I3,N. Borrow and A. Voijnyj 310 

B. Re'ay Telemetry Modulation System Development 
NASA Code I84-68-04-J9, C. Cor/ 311 

5 

XXIII. Spacecraft Radio 314 

A. Lunar Orbiter V Side-Looking R'-.dor Experiment 
NASA Code )25-l/.O3-03,R. I. Horftor 314 

B. Power Spectral Densities for Binary Frequency-Shift-Keyed 
Waveforms 

NASA Code 186-^-04-11,0. W.Boyd 328 

f ADVANCED STUDIES 

I XXiV. Future Prelects 335 ^ 

R A. Science Utility of Automated Roving Vehicles 

I NASA Code 68440<)l-I0,R.G.Brer«fon 335 

X Jn SMCE UtOQRAMS SUMMAHY 37-51, VOL. Ill 



X 



N 68-37398 



i. Systems Analysis Research 

SYSTEMS DIVISION 



A. Shadow Equation for a Satellite, j. Lore/f 

1. Introduction 

This article discusses computation procedures for find- 
ing the shadow entry and exit angles for an artificial satel- 
lite of the moon. The results are also applicable to 
satellites of earth or the planets. To determine whether 
a given position in space is in simlight or in shadow is 
relatively simple; however, the edges of the shadow, i.e., 
intersection points of an elliptic orbit with a cylindrical 
shadow, are not so directly computable. 

The difficulty lies in the fact that a fourth-degree 
algebraic equation must be solved. The roots of such 
equations may be written down immediately using 
Ferrari s (Cardan's) formula, but the result involves the 
cube roots of ccnnplex numbers — even when the solutions 
are real. 

R. P. Yeremenko (Ref. 1) solves the problem using 
Ferrari's formula, in spite of the inconvenience of the 
complex numbers. Another approach, taken by A. A. 
Karytevn (Ref. 2), first solves the problem for a circular 
orbit, and then treats the low eccentricity orbit as a 
perturbation. 

In this article, a third approach is presented, viz., the 
use of an iterative, or search procedure. This method is 
particulariy useful when shadow conditions are required 



for each of a sequence of orbits. The fact that orbit 
precession and shadow rotation produce only slowly 
changing values of the entry and exit angles is used to 
advantage. 

2. Shadow Geometry 

Consider the geometry associated with a lunar satellite 
and its intersections with the moon's shadow. In Fig. 1, 
the x-y plane is the plane of the satellite orbit, labelled 
SAT, which is assumed to be an ellipse with one focus 
at the center of the moon, 0. 



/ 










1 








SAT 


> 


^ If 




SHAD 




Nvl** y 


/ 


f If 
1 1 1 


^ 




m 


^X 











w 


ik. 















Fig. 1 . Configuration in plane of satellite orbit 



JPL SPACE PROGRAMS SUMMARY 37-5?, VOL. I» 



The orbit plane must intersect the ihoon's shadow 
(assumed to be bounded by a haU-circular-cylinder ema- 
nating from the moon) in a semi-ellipse with center at 
and major axis along x. This shadow ellipse (labelled 
SHAD) is also shown in Fig. 1. Only the shaded portion 
represents shadow. Note that point is simultaneously 
the center of the shadow ellipse and the focus of the 
orbit ellipse. 

The SAT may intersect SHAD at as many as four 
points, although only two of these, at most, can be on 
the shadow side. Let these be labelled E, and Ej, such 
that the satellite exits the shadow at E, and enters at Eo. 
Of jourse, E, and Ej may either coincide (langency of 
ellipses) or there may be no intersection on the shadow 
side (satellite always in the sun). The latter case is of no 
concern to the present discussion. 

If we let Eo be the point of orbit crossing the shadow 
side of the ar-axis, there are several possibilities which 
may be listed as follows: 

(1) Eo is in shadow. In this case, E, and Eo exist and 
are on opposite sides of Eo. 

(2) Eo is in sun and there are no intersection points, Ei 
and E2. In this case, the satellite is always in the 
sun. 

(3) Eo is in sun and there is one intersection point, 
E2 = Ej. Here, also, the satellite is always in sun. 

(4) Eo is on the shadow-sun boundary, and hence coin- 
cides with either E, or Ea or both. 

(5) Eo is in the sun and there are two intersection 
points. El and E,. This is the case illustrated in 
Fig. 1. Here, both £1 and E^ are on the same side 
of Eo. 

The problem is to specify an algorithm, appropriate 
for computer use, to determine E, and Ej. 

3. Dvrivotion of Shadow Equation 

R. W. Bryant (Ref . 3) introduces the shadow equation 
in terms of eccentric anomaly, E 

F(E) = P • u (cos E - e) + Q • u (1 - e^)^ sin E 

+ [(1 - e cos Ey - pVo^]^ = (I) 

The values of E satisfying Eq. (1) correspond to positions 
of the satellite on the shadow-sun boundary. When 
F(£) > 0, the satellite is in sun; when F(£) < 0, the 
satellite is in shade. 



As shown in Fig. 2, the derivation of the shadow equa- 
tion is straightforward. In Fig. 2, coordinate axes are 
shown in the plane parallel to the moon-sun line. The 
unit vector in the direction of pericenter is P, Q is a unit 
vector in the direction with 90 deg true anomaly, and R 
(not shown) is an out-of-plane vector completing a right- 
handed system. The unit vector u is in the moon-sun 
direction, while r is the radius vector from the moon- 
center to the satellite. 



SAT 




Fig. 2. Coordinate system for shadow equation 

Consider an arbitrary satellite position r, on the shadow 
border. The projection of r in the shadow direction 
(along u) is given by (u • i")u. On the other hand, this 
same quantity may be obtained geometrically as 
— (r- - p-)^" u (see Fig. 2). The negative sign is required 
since we have specified shadow side. Hence, it follows 
that 



u • r -I- (r= - p')^ = 



(2) 



Then Eq. (1) follows from Eq. (2) and the standard 
relations for an elliptic orbit 



and 



r/a= (cosE- e)P -l-(l-e')'*sin£Q (3) 



r/a— 1 — e cos £ (4) 



The significance of F(£) is easily inferred from Fig. 3, 
which is an edge-on view of the orbit. Consider any 
point $ on the orbit in the sun, and pass a circle, center 0, 
through S intersecting the shadow at S'. Then 



(r» - p")^ = b8' > - u • r 



(5) 



JPL SPACE fItOGRAMS SUMMAkY 37-51, VOL. Ill 



a' ' 




Fig.3. Configuration in plane >or|<endicular to satellito 
orbit and containing moon— sun line 

Hence, F(E) > when the satellite is in the sun [see 
Eq. (2)] and F(E) < when if is in the shade. 

4. Shadow Computation Algorithm 

The compulation starts with Eo [the value of the 
eccentric anomaly on the +i-axis crossing (Fig. 1)] and 
proceeds as a search using small increments in £. The 
search may require many trials for the 'arst orbit; how- 
ever, for successive orbits, the nui.it)er of trials is minimal 
since the search can start with the previous value of Ei 
or Ea instead of with Eo. 

It is convenient to consider three regimes as follows: 

(1) F(Eo) < 0. 

(2) < F(£„) < K. 

(3) K < F(£o). 

where K is a constant to be computed by Eq. (6). 

In regime (1), the satellite is in shadow at Eo, and the 
search procedurij is followed as given. In regime (3), 
the satellite is always in the sun, and no search is needed. 
If regime (2) occurs, the satellite rray or may not be 
always in the sun. In either oase, a search for Ei and Ej 
must be followed. However, it need only proceed in one 
direction from Eo since, if they f^xist, both £i and Ea 
are on the same side of Eo. To Jetennine the direction, 
note whether Eo is less than or greater tlian 180 deg and: 

(1) If < Eo < 180 deg, then both < E, < £„ and 
< Ea < £„. 

(2) If 180 deg < £o < 360 deg, then both Eo < £, < 
360 deg and £„ < Ej < 360 deg. 



(3) If Eo -^ deg or Eo = 180 deg, then the satellite 
is always in sun.' 

The search can be limited by noting (1) that it need not 
be pursued past pericenter and (2), since the change in 
true anomaly from Eo to Ei or £2 can not exceed 90 deg, 
the search on £ can be limited to a span of slightly -nore 
than 90 deg (say 100 deg) for practical purposes. 

It remains only to determine a value for K. 

5. Value of the Constant K 

We shall show that when K is appropriately defined 
[see Eq. (12)], then K may be computed by the formula 



K^^\ 



(1 - K\lr 



t\ 



(6) 



where 



x»» 



= fl„/ I u • R I 

= a (1 - e j sin £0 1) 



Rj, = radius of moon 

This value of K corresponds to tangency of SHAD with 
the line connecting £0 and the closest point of intersec- 
tion of the orbit and the t/-axis. In Fig. 4, the points 
r^, x»)., and the distance aK between xs* and Eo, are 
identified. To derive Eq. (6), it is sufficient to solve 



'Implicit in our argument is the assumption that only one shadow 
region can occur. This is intuitively obvious, but not too easily 
shown mathematically. 




Fig. 4. Satellite orbit satisfying sufficiency criterion 
for satellite always in sun 



JPL SPACE PROGRAMS SUAftMARY 37-51, VOL. \\\ 



algebraically for the intersection of the line r^ Eo and 
SHAD, and then require tangency. Thus* 



_X y_ ^1 



X,\ 






SHAD: 
Lliminate x to obtain the quadratic in y 

whose discriminant is 



(7) 
(8) 



(9) 



which must vanish for tangency. Solving for Eor (value 
Eo for tangency) 



iioT 



x»» 



(1 - Rl/rlY 



(11) 



Then Eq. (6) follows from Eq. (11), and the definition 
ofKis 



K - — (Eor — X«») 



(12) 



'Using the symbol E, to represent the length of OEo. 

References 

1. Yeremenko, R. P., "Exact Solution of the Shadow Equation," 
Inst. TeoT. Astron., Vol. 10, No. 6, pp. 446-^49, 1965 (in Russian). 

2. Karytov, A. A., "Determination of the Time in Which an Artificial 
Earth Satellite is Illuminated by the Sun," Kosm. Issled., Vol. 5, 
No. 2, pp. 298-301, 1967. 

.1. Bryant, R. W., "The Effect of Solar Radiation Fremire," NASA 
TN D-1063. National Aeronautics and Space Administration, 
Washington, Sept. 1961. 



B. A Consistent Ephemeris of the Major Planets 
in the Solar System, 

W. G. Ale/bourne and 0. A. O'Handhy 

1. Introduction 

The system of computer programs known as the solar 
system data-processing system (SSDPS) has been used 



to compute a consistent ephemeris of the major planets 
that has been fit in a weighted least-squares sense to both 
optical and radar-time-delay observations of the planets. 
The SSDPS has been described fully in SPS 37-49, 
Vol. Ill, pp. 1-14. This ephemeris has been adopted for 
the planetary ephemerides contained in developmental 
ephemeris (DE) 40. Although the developmental ephe- 
merides are continually being updated by the pro- 
cessing of new or refined data, or by the improvement of 
the mathematical model used in the data processing, 
DE 40, nevertheless, represents something of a milestone 
in the ephemeris development activity. For this reason, 
a brief summary of the data processing, and the resulting 
ephemeris, is presented here. 

Until 1967, the planetary ephemeris tape system at 
JPL was obtained from least-squares fits to source ephe- 
merides based on planetary theories fit to meridian circle 
observations of the sun and the planets (Refs. 1 and 2). 
In early 1967, ephemerides of Venus and the earth-moon 
barycenter were produced that had been fit to both 
1950-1966 U.S. Naval Observatory meridian-circle obser- 
vations and planetary radar range and doppler observa- 
tions of Venus taken over the period 1961 to 1966. The 
best example of this series is DE 24' which was used in 
the Mariner V operations. These ephemerides were ob- 
tained with the "phase I" system of programs. These 
included an orbit determination system used in early 
work on the determination of the astronomical unit (AU) 
and the radius of Venus (Ref. 3), but modified to include 
optical data. The path generation for the phase I system 
was the PLOD II system (Ref. 4). Although intended to 
be valid only over a relatively short arc, DE 24, never- 
theless, represented an improvement of between one and 
two orders of magnitude in accuracy over previous 
ephemerides. The phase II program development activ- 
ity, begun in late 1966, has led to the current version of 
the SSDPS. 

2. Data Set 

The optical data set used in DE 40 is presented in 
Table 1. These are all the meridian observations from 
the 6-in. transit circle of the U.S. Naval Observatory over 
the interval 1949-1967. This set of observations di£Eers 
from those reported in SPS 37-48, Vol. Ill, pp. &-9 
primarily by the data taken between 1966-1967. 

The planetary radar data have been taken since 1961. 
Initially, the data type was doppler, and, beginning in 



•Lawson, C. L., Announcement of JPL Developmental Ephemeris 
No. 24, Apr. 1967 (JPL internal document). 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. HI 



Tabu 1 . DE 40 optical data stt obsarvatient 



Ptonal 


No. of eb»rvcrttont 


Sun 


2136 


Mwcury 


353 


Vanut 


116S 


Mart 


243* 


jupitar 


34t 


Sotiirn 


338 


Uranui 


330 


Ntplun* 


325 


hit Iktcy e( fttan. 



1964, both doppler and range were obtained. The four 
sources of this planetary range data have been Arecibo 
Ionospheric Observatory in Puerto Rico, Haystack and 
Millstone Hill sitss, and the Venus DSS (SPS 37-48, 
Vol. ni, pp. 8-9). The usable data (in the sense of accu- 
racy), cover the period from 1964 onwards. This set of 
planetary range data is given in Table 2. 

Table 2. Planetary range data 



nanci 


Na. of 
oburvotienf 


Souk* 


Paried 


Marcury 


■ SI 


Arecibo 


Apr. 1964-Aug. 1967 


Venut 


■81 


Arecibo 


Mar. 1964-Oc». 1967 




35 


Haystack 


July 1967-Sept. 1967 




99 


Millilone 


Aug. 1967-Oc». 1967 




281 


Venus DSS 


May 1964-Ocf. 1967 


Man 


39 


Are-ibo 


Nov. 1964-June 1965 




10 


Hayslcck 


Apr. 1967-Jun« 1967 



An additional discussion of these data appears in 
SPS 37-48, Vol. Ill, pp. 8-9. The total of the Venus DSS 
set is radically changed from that of Table 1. The values 
given in Table I referred to the uncompressed data. The 
full discussion of these data appears in Ref. 5. In addi- 
tion, the total includes 15 time-delay measurements of 
Venus obtained by D. A. O'Handley* at the Venus DSS 
during July-October 1967 inferior conjunction. 

Because of advances in radar technology involving 
larger antennas, increased transmitter power levels, and 
improved data reduction techniques, the precision of the 
time-delay measurements has improved by an order of 
magnitude over the 1964-1967 period, i.e., a typical stand- 
ard deviation of a 1964 Venus time-delay measiurement 
is in the 20-50 /xs range, while a 1967 inferior conjunction 



•O'Handley, D. A., ReconitructUm of ]PL Radar-Bange of Venus— 
29 Jtdy, 1967 to 27 October, 1967. ( JPL internal documeat). 



measurement lies in the 3-5 /ts range. Current Mercury 
observations are precise to about 10 us and the 10 normal 
points for Mars, based on the 1967 Haystack observa- 
tions, are of similar quality. On the other hand, the 
precision of a radar doppler ineas'irement is about 1 Hz. 
A simple calculation will show that for a typi;:al orbital 
parameter, a precision in doppler of 1 Hz is equivalent 
to a precision of about 10' /as in a time-delay measure- 
ment. Further, doppler does not provide information 
about planetary radii. For these reasons, doppler infor- 
mation, although extremely valuable in the radar data 
reduction process and in the study of planetary topog- 
raphy and surface characteristics, is not presently used in 
ephemeris development. 

Special mention should be made of the 10 high- 
precision, time-delay normal points of Mars taken during 
the April-June, 1967 period at the Haystack facility. Each 
point corresponds to the observations taken in one night. 
The 10 observation nights are spread over the 2-month 
period at weekly intervals. During an observation session, 
the planet rotates under the radar beam, and the half- 
power width of the return beam covers about 200 km on 
the Martian surface. Consequently, topographic features 
on Mars are observed to move through the return radar 
beam giving variations in time delay with a magnitude 
of up to 100 fis. The regions on Mars observed on suc- 
cessive nights partially overlap, and, during the 2-month 
period, a strip covering the entire 360 deg of longitude 
was observed. Because of this overlap, it is possible to 
determine the relative altitude, on every observation 
night, of any point on this strip. A reference point was 
chosen that was close to representing a mean altit.ade 
with respect to the topographic variations; it is the range 
to this reference point that is given in the data set for the 
10 observing sessions.' 

3. Parameter Set 

The conditional equations were formed from the resid- 
uals constructed from the observations and the pre- 
dicted observations (observed minus computed) based 
on DE 35. The DE 35 was generated from the N-body 
integrator in SSDPS using an up-to-date set of planetary 
masses (SPS 37-45, Vol. IV, p. 17) that incorporates the 
mass determinations by radio tracking data from space- 
craft. The initial conditions of DE 35 were based on a 
least-squares St to an earlier JPL ephemeris (DE 26) 
in order to minimize the secular effects resulting from 
adopting a new set of planetary masses significantly 
different, in some cases, from the lAU set used previously. 
The planetary masses in DE 40 are the same as in DE 35. 



'Private communication from G. H. Pettengill (Apr. 2, 1968). 



JPL SPACE PROGRAMS SUM/MARY 37-5?, VOL. Ill 



The orbital coefficients of the conditional equations con- 
structed from DE 35 are basically the osculating Set III 
elements of D. Brouwer and G. M. Cleinence (Ref. 6) at 
the epoch JD 2440800.5. 

The simultaneous incorporation of optical and range 
observations in a single solution for all the planets, with 
the except'on of Pluto, has not been accomplished pre- 
viously. It therefore became necessary to examine the 
parameters that could be solved for in light of the limited 
data set currently available. 

With range data alone, a 21-parameter solution for 
Mercury, Venus, and Mars gave a solution in which the 
parameters were reasonably determined (see Table 3). 
The first 6 rows of Table 3 correspond to the Set III 
orbital parameters in Ref. 6. 

Table 3. ParameUr determination using range dat; 



Marcury 


Vtnui 


Earth— Moon 


Mars 


Al 


At 





Al 


Ap 


Ap 


— 


— 


A, 


Aq 


— 


— 


tAr 


•Af 


•Ar 


•Ar 


A* 


A* 


A* 


Ae 


Aa/a 


— 


Ao/o 


— 


Radiui 


Rodiui 


— 


Radius 


AU 


— 


— 


— 



Several comments should be made with regard to this 
set of parameters. With radar data only, it is necessary 
to limit the parameter set to those parameters that are 
sensitive to time-delay measurements. Consequently, the 
parameters defining the orientation of the orbit of the 
earth relative to the astronomical iight ascension and 
declination coordinate system were not adjusted. Even 
with a well-distributed data set, solving for the semimajor 
axis of the orbits of these planets simultaneously leads, 
in the pure radar solution, to a near singular normal 
matrix. The dominant signature in the time-delay observ- 
able resulting from adjusting the semimajor axis is due 
to the change in the mean motion of the planet rather 
than the direct effect of the change in the semimajor axis 
itself. The orbits of Venus and the earth are nearly 
coplanar and circular; therefore a change in the mean 
motion of Venus is almost indistinguishable from a 
corresponding negative change in the mean motion of 
the earth. In the radar-only solution, the semimajor axis 
of the earth-moon barycenter is used because it gives a 
slightly better fit in the least-squares sense, and because 
it has the we'3ht of all the range observations. 



The radar data for Mars are too scant and not well 
enough distributed to give good determination of the 
quantities Ap, \q, and a/a. The quantities Ap and Aq 
are rotations of the orbit plane about orthogonal axes 
embedded in tht orbit plane, and cause displacements 
of the planet perpendiculai to its orbit plane. Since the 
inclination of the orbit plane of Mars to the ecliptic is 
only 1.9 deg, these two out-of -plane quantities are in 
excess of an order of magnitude more difficult to deter- 
mine than the in-plane quantities, even with an optimally 
distributed data set. The high-precision Haystack points, 
jouplrd with the relatively low-precision 1964 Arecibo 
data, are not sufficnnt to obtain a definitive value for the 
mean motion quantity Ao/a. 

With the inclusion of optical data for all the major 
planets, with the exception of Pluto, an expanded param- 
eter set is used. This set consists of 56 unknowns as 
follows: 

(1) Six elements of 7 planets. 

(2) Six elements of the earth-moon barycenter. 

(3) Four limb corrections, right ascension, and declina- 
tion of Mercury and Venus. 

(4) Three radii (Mercury, Venus, and Mars). 

(5) One AU. 

The 18-yr span of this data does not permit a definitive 
set of corrections for the outer planets. Including the 
Aa/a parameters of tbe outer planets is somewhat ambi- 
tious for this data set; however, this parameter set gave 
corrections for each planet that diminished or removed 
the secular trends in the residuals published by the 
U.S. Naval Observatory from transit circle observations. 

4. Solutior; 

Two solutions were made in order to arrive at the 
current ephemeris. Initially, a solution that utilized both 
the optical and radar-range conditional equations was 
made. The motivation here was to allow the optical data 
set to determine those quantities that are sensitive only 
to the optical data, but simultaneously using the range 
data to anchor the range sensitive parameters. The rank 
52 solution from an Eigenvalue-Eigenvector analysis" of 
the 56-parameter set was chosen because the correction 
to Ap and \q of the earth stabilized at a value which is 
in agreement with the known error in these quantities. 
For solutions of rank greater than 52, the normal matrix 



"Lawson, C. L., Eigenvalue-Eigenvector Analysis for SSDPS, 
Jan. 17, 1968 (JPL internal document). 



jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



is too near singular and causes significant instability in 
these and other parameters. The resulting ephemeris is 
DE39. 

At this point, the optical data have provided the ref- 
erence frame to which the relative measurements of range 
can be evaluated. The radar data was felt to be a much 
more accurate source of information for those parameters 
best solved for by this type of data. It was suspected that 
this data type would be degraded when used simul- 
taneously with optical data. For this reason, an iteration 
was made on this solution using the range observations 
alone. The range data were compared against DE 39 and 
corrections to this ephemeris were calculated based upon 
the 21-parameter radar set described in Subsection 3. 
None of the 21 corrections obtained was statistically 
significant when cc^ipared to its formal standard devia- 
tion; nevertheless, they were applied f"r reasons of con- 
sistency. The ephemeris generated by applying these 
corrections is called DE 40. 



deviations of the estimated parameters and the cor- 
relations among them. Table 4 gives the formal standard 
deviations of the 24 orbital parameters of the inner 
planets, the three planetary radii, and the astronomical 
unit. The units of the standard deviations are arc seconds 
except for the radii and the AU which are in kilometers. 

Table 4. Standard deviotions of orbital parameters, 
planetary radii, and AU 



Data lyp* 


MercHry 


Venut 


Earlh-Moen 


Man 


M 


0.031 


0.031 


0.031 


0.032 


Ap 


0.023 


0.019 


0.018 


0.030 


Aq 


0.022 


0.019 


0.019 


0.031 


eAr 


0.005 


0.0006 


0.0007 


0.004 


A* 


0.002 


0.0005 


0.0004 


0.007 


Aa/a 


0.00007 


0.0001 


0.0002 


0.0006 


•■ 'iot 


1.0 


0.2 


— 


7.0 


AU 


0.27 


— 


— 


— 



The values of the constants to be used with DE 40 are 
as follows^: 

(1) AU = 149,597,895.8. 

(2) Radius of Mercury = 2437.3. 

(3) Radius of Venus - 6055.8. 

(4) Radius of Mars = 3375.3. 

The \alues of the AU and the radii of Mercury and Venus 
given here are essentially in agreement with those found 
by the MIT group (Ref. 7). The value of the radius of 
Mars, however, is weakly determined (see Subsection 5) 
because of the poor distribution of radar points; the best 
value available at this time is the Mariner IV occultation 
experiment value of 3393 ± 4 (SPS 37-43, Vol. IV, p. 7). 

5. Standard Deviations 

The subject of the relative sigmas of each data type 
present in the solution was considered. The optical data 
were given the following sigmas: 

(1) Right ascension = l'.'0/cos 8. 

(2) Declination = I'.'O. 

The range data were given the standard deviation 
assigned by the respective observers. 

The covariance matrix resulting from the optical and 
range data may be used to obtain both formal standard 



'In converting from "light-seconds" to kilometers, the velocity of 
light is taken to be exactly the lAU value of 299,792.5 km/s. 



The formal standard deviations exhibit thcr usual 
degree of optimism. The reader, therefore, should be 
aware that they do not account for either possible sys- 
tematic error factors in the data or unmodelled param- 
eters in the mathematical model. The correlation matrix, 
although not shown here, verifies that high correlations 
exist among the mean longitude parameters (aL), and the 
mean motion parameters (Aa/o). With this exception, the 
problem is well-conditioned. 

In spite of known biases in the optical data related to 
limb corrections, there is some encouraging evidence of 
consistency between the optical and radar data. For 
example, the corrections from an optical solution alone 
to the orientation of the orbit plane of Venus relative to 
the ecliptic are found to agree with the values obtained 
in a pure radar solution. The radar data also exhibit a 
degree of internal consistency. For example, the cor- 
rections to eAr and Ag of the earth from processing 
Mercury range data alone are the same as those obtained 
when only Venus ranging data are processed. 

The standard deviations for aL, Ap, Aqf, and Aa/fl are 
an order of magnitude smaller in the 21-parameter pure 
radar solution. This is due to the precision of the radar 
measurements and the fact that these parameters become 
relative quantities for which radar obtains extremely 
powerful solutions. The reader can easily verify with a 
simple model consisting of circular coplanar orbits, that 
a set of one-hundred 10-/is quality range points, well 
distributed, enables one to determine the longitude of 



JPL SMCE PROGRAMS SUMMAPY 37-57, VOL HI 



Ve- s relative to the longitude of the earth to about O'/002. 
Furthermore, additional error analyses show that with 
only 3 years of ranging to Venus, the mean motion of 
Venus relative to the mean motion of the earth is deter- 
mined with a precision (formal) of O'.'1/IOO yr. It has 
been known for several years that the relative longitude 
of Venus required a correction ranging between +0'.'5 
and +1''0. This is, most likely, an accumulated effect 
due to an error in relative mean motion; the current 



analysis gives a correction to the relative mean motion 
of Venus of + I'.'2/IOO yr. 

6. Rctidunit 

The raiige residuals for Mercury, Venus, and Mars are 
shown in Figs 5-9. The residuals of Mercury are shown 
in Fig. 5 based on an ephemeris (contained in DE 35) 
which closely matches the Newcomb ephemeris (Ref. 2). 



4000 


































9 






























* 
























































































































• 




















































4 






















* 












3000 




























r 


















* 






























* 
















« 














































* 






























* 














• 




















































^ 
























* 




































" - 9 




2000 












1 
















* 












« 








** 





































t 





































































1 


4 


« 
































r 


































* 






















































f 






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1000 


If 








tt 
























































»■ 








* * 


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9 


% * 






\ « 








* 
















* 


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« 




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m 


i 




f 






V 
















• 






































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* ^ 








* 


































** 
























* 




































*• 
















ft 














t 
























* 






















* 








^■f 






















i 








•i 






































































-1000 






































































* 






























4 






























♦ 






























* 
















































! 


















































































-2000 




_* 










L , 













200 



1000 



400 600 800 

(toys AFTER JO 243 8400 
Fig. 5. Mercury residuals based on DE 35 ephemeris and OE 40 M and radius 



1200 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. 11/ 



< 

a 
</) 

UJ 
(T 

UJ 

o 

z 
< 



1000 


















































































































































































































































« 


* 


• 








* 




















* 


«: 




* 




* 
1 




i, ♦ ♦ 


Im*-j 






^mM 


f 






* 






f't 


* iJ 


1 


4 

1 


k 

• 


^r 






*7^ 








* 






























































« 




































































• 












































































* 


























1000 































zoo 



400 600 SOO 

days AFTER JD 243 8400 



1000 



1200 



1400 



Fig. 6. Improvemtnt in Mercury residuals resulting from DE 40 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



coo 



CODE LOCATION : 

I ARECIBO IONOSPHERIC - 

OBSERVATORY DATA : 

; 2 LINCOLN LABORATORY.: 

HAYSTACK DATA : 

3 . -OLN LABORATORY, - 

MILLSTONE DATA ; 

; 4 VENUS DSS DATA : 



u> 200 

a. 



< 

o 
(/) 

LU 

q: 

UJ 

o 

z 
< 



an: 



3 




•-++ 



xr 



li-c 



-F^V 



;^ 



ri ; 



-200 



::pz 



200 



«00 



600 aoo 

doyi AKTER JD 243 8400 



1000 



1200 



Fig. 7. Rctidvcit of all availabi* Vcnut ran9ing data obtained from DE 40 



10 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL Hi 



3000 



2000 



in 1000 
4. 

in 

-I 
< 

a 

en 
ui 
q: 

u 
o 
z 
< 



-1000 



-2000 



1 
















































1 


























































































































', 








































1 
























t 








i 
























1 
















s 
















































t 






























1 


















f 






1 


























' 
























« 
















































i 
























f 






















. ' ' 


I 






























































i 










t 














































« 


1 














z 










1 
























t 
















































{ 






























































































I 




















































































































































































CO 


DE LOCATION 




















1 ARrrinn >r)NO!>Pi-iFRr- 


















caSERVATORY DATA 




1 
































c LINCOLN LAnOnATO 
HAYSTACK DATA 
1 1 



































































zoo 



400 



600 



•00 



1000 



I ZOO 



doy« AFTER JO 243 8600 
Fig. 8. Mart residuals comporvd to DE 35 



JPL SPACE PROGRAMS SUMMARY 37.51, VOL. IN 



11 



< 
a 

CA 

Ul 



o 



< 
q: 



4000 
































































































































































i 










































































3000 


































































































































































































































































2000 




•l 
































































































































1 


























I 


























t 


















^ 1 
















































































































I 
























t 































































































































































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lliki'i' 
















"■r 










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"* 






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"* 






















1 














CO 


DE LOCATION 

ARECIBO lONOSPHER 
OBSERVATORY DbJl 

! ..'n:oln laboratof 

-lAYSTACK DATA 






I 














t 


c 




















1 
























' "1 












I 






t 








































1000 

























zoo 



400 600 800 

days AFTER JD 243 8600 



1000 



1200 



Fig. 9. DE 40 residuals for Mars 



12 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



In this figure, the AU and radius from DE 40 were used. 
The improvement in residuals resulting from DE 40 is 
shown in Fig, 6. All of the range data shown here were 
taken at the Arecibo Ionospheric Observatory, The 
residuals of all available Venus ranging data obtained 
from DE 40 arc shown in Fig. 7. 

The tremendous improvement of radar techniques over 
the perod 1964-1967 is shown in all of the figures after 
solution. An as yet unexplained anomaly in the residuals 
of the 1965-1966 ranging period is shown in Fig. 7. 
The fact that the radar-range residuals from both JPL 
and Millstone show this anomaly independently estab- 
lishes that it is not due to an instrumentation efiFect. 
Current conjecture is that it is due to second-order effects 
of fixed parameters. 

The residuals from ranging Mars, when compared to 
DE 35, are shown (Fig. 8) to have very large trends. The 
Mars ephemcris in DE 35 closely fits Clemence's second- 
order theory of Mars used as a source ephemeris for 
DE 19 (Ref. 2). The DE 40 residuals for Mars are 
shown in Fig. 9. 

DE 40 should not be considered the final "best" ephe- 
meris. The lunar ephemeris incorporated into this 
ephemeris is LE 4. There is a new version DE 43 which 
has LE 6 on it. Certain problems with the 1967 Venus 
radar-range data, from the Arecibo Ionospheric Observ- 
atory, lead to the conclusion that another solution should 
be made. There is, at present, new data on Mercury and 
Venus, and some revised data over other periods to be 
added. A few data points should be edited. Finally, the 
SSDPS is a rather complex and evolving system contain- 
ing over 150 subroutines and about 200,000 words of 
machine-level instructions. The possibility of subtle 
errors in this system is not unlikely, and efforts are con- 
tinuing to validate the current working version. 

References 

1. Peabody, P. R., Scott, J. F., and Orozco, E. G., Users' Description 
of JPL Ephemeris Tapes, Technical Report 32-580. Jet Propulsion 
Laboratory, Pasadena, Calif., Max. 2, 1964. 

2. Devine, C. J., JPL Development Ephemeris Number 19, Tech- 
nical Report 32-1181. Jet Propulsion Laboratory, Pasadena, Calif., 
Nov. 15, 1967. 

3. Muhleman, D. O., and Holdridge, D. A., and Block, N., "The 
Astronomical Unit Determined by Radar Reflections from Venus," 
Astron. /., Vol. 67, p. 191, 1962, 

4. Devine, C. J., PLOD II: Planetary Orbit Determination Program 
for the IBM 7094 Computer, Technical Memorandum 33-188. 
Jet Propulsion Laboratory, Pasadena, Calif., Apr. 15, 1965. 

5. Muhleman, D. O., O'Handley, D. A., Lawson, C. L., and 
Holdridge, D. B., JPL Radar Range and Doppler Observations of 



Venus 1961-1966, Technical Report 32-1123. Jet Propulsion Lab- 
oratory, Pasadena, Calif., 1967. 

6. Brouwer, D., and Clemence, G. M., Methods of Celestial Me- 
clmnics, p. 241, Academic Press, New York, 1961. 

7. Ash, M. E., Shapiro, 1. 1., and Smith, W. B., "Astronomical Con- 
stants and Planetary Ephemerides Deduced from Radar and 
Optical Observations," Astrmi. /., Vol. 72, p. 338, 1967. 



C. Correction of the Lunar Orbit Using Analytic 
Partial Derivatives, J. 0. Mulholhnd 

As reported in an earlier article (SPS "^7-49, Vol. Ill, 
pp. 21-23), work is underway on the numerical inte- 
gration of the lunar ephemeris. The primary difficulty 
in such an undertaking lies in the formulation of the 
differential correction process — not a trivial process for 
such a highly perturbed object. 

In order for a differential correction process to work, 
it is necessary that the vector p (ij,), whose first variation 
is represented by the left-hand side of the conditional 
equation (SPS 37-50, Vol. Ill, pp. 50-53), be a reasonably 
close approximation to the real motion over the correction 
arc.** As a result of recent efforts, it is now known thr.t 
Keplerian or Hansen-type approximations are not ade- 
quate for the correction of the lunar orbit for arcs of 
5 years. What is required is some formulation of the 
conditional equations that conforms rather closely with 
the motion that is being used as the "observations," in 
this case lunar ephemeris (LE) 6. Three ways in which 
this might be accomplished are as follows: 

(1) Integration of the variational equations. 

(2) Construction of finite difference quotients. 

(3) Derivation of high-accuracy analytic partial de- 
rivatives. 

All three means are being investigated and compared. 

Integration of the variational equations repre.sents the 
most accurate and the most rigorously correct of the 
possible approaches. If done properly, the conditional 
equations would represent the correct first variation of 
the computed state vector. This process, however, re- 
quires large amounts of computer time and, for this 
reason, does not seem promising. 

Finite difference quotients a.-^ approximations of the 
integrals of the variational equations. They are formed 

This is an intentionally ambiguous statement, becau.e this quali- 
tatively true statement can only be given a quantitative meaning in 
terms of the specific problem of interest. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



13 



by making a series of computations, varymg one element 
at a time, and computing the differential effects Ap/Aiji. 
Thus, 13 orbit integrations are required, rather than one. 
Again, this is an expensive process. 

The use of analytic partials would appear to be very 
desirable if they can be made to provide an adequate 
representation of the perturbed motion. This will be 
assured if thev are derived directly from the Lunar 
Theory; they will then represent the correct first variation 
of the observed motion — the Lunar Theory itself. Unfor- 
tunately, there is no simple correspondence between the 
parameters of the Lunar Theory and the set of elements 
to be corrected, the Brouwer and Clemence Set III pa- 
rameters (Ref. 1). Define the following sets of parameters: 

8(c : { A€, At, An, A(u, Aa/o, Ae} 
8III: {Ago + Afx, Ap, A(7, eAfj., Aa/a, Ae} 



?: {A,/3,r,X,/J,f} 

f: [x, y, z, X, y, z} (ecliptic) 



where e = n + 
usual meanings. 



+ go, and all other symbols have their 



The difficulty lies in the circumstance that the PLOD II 
differential correction treats the orbit elements k, osculat- 
ing at the epoch, while the Lunar Theory is developed 
in terms of the mean elements kq. Thus, it is necessary to 
form the conditional equations according to the matrix 
relation 

[ dsi ^ [js_-\ vdKoi r g/c.i 
LsiiiJ L3k„j1_5*,JL3"U 

The factor I3s/5ko] is obtainable directly from the theory, 
while the factor [Oki/SIII] is strictly geometric and is 
readily shown to be the matrix 



SUlo) ,, .. cos U) ,, ., 

— (1— cosi) — : — :- ^^l— cost) 



smt 



smi 



cos (1) 



— sm b> 



sm<tf 
sin t 



COS(i> 

sint 



sm^ 
tani 



COS «i> 

tant 



14 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. /(/ 



What, howe\'er, is the relationship between the mean 
elements and the elements osculating at epoch? Recalling 
that the [9s/3k„] are available, one may write 






where s evaluated at the epoch is denoted Si. Since ko and 
K, are each sets of 6 linearly independent Darameters, then 
the inverse exists and 






To find the formulation of [8ki/?Si], one ma\' write 






The matrix [3r,/?s;] is readily found from geometric re- 
lations and will not be given here. The problem finally 
comes down to the computation of [dKi/dr\] = [dri/dx,]-^. 
A relatively simple approach to this is to define the rec- 
tangular state vector p in orbit-fixed coordinates 



a(cos E — c) 
a(l-e-')''sinE 


— rta sill E/(l — e cos E) 
+ na{l- e")^ cos E/(l - e cos E) 



(pi = 



If the matrix A\i(i) is used to effect a rotation about the 
fc-axis through the angle a, then 

{r} - A3(n)A,(i)A3M{p} 

Define the matrix 

B^.{a) = d[A,(a)]/da 

Then the columns of [3r/9Ki] are given by 

|i^|=A3(n)B.(t)A3W(^} 



|^|=B3(n)A,(f)A3(<o){^} 

where H and K are as defined previously (SPS 37-50, 
Vol. Ill, pp. 50-53). 

The application of these relations would be as follows: 
At the beginning of the differential correction process, 
it is necessary to form the matrix 

t^'M[ll]'[t][t]}-[lfr] 

At every subsequent time point at which conditional 
equations are required, one need only form the matrix 
product 



As = 



L9«oJ 



[C] {8111} 



where {SIII} is the vector of unknown increments that 
the solution is expected to determine. 

It is expected that this approach will be mors eco- 
nomical of computer time by a factor of 3 to 6 over the 
other methods of computing accurate partial derivatives. 

Reference 

1. Brouwer, D., and Clemence, G. M., Methods of Celestial Me- 
chanics, Academic Press, New Vork, 1961. 



D. Bayesian Estimation Based on the 
Gram-Charlier Expansion, W. Kizner 

1. Introduction 

In previous articles (SPS 37-49, Vol. Ill, pp. 23-31, and 
SPS 37-50, Vol. Ill, pp. 20-22), the author discussed a 
method that uses a numerical approximation to find the 
coefficients of a Hermite series expansion (used in non- 
linear estimation). This article shows that an approxi- 
mation based on the Gram-Charlier expansion may be 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. \\\ 



15 



optimal in cases vhere !;il that is desired is the condi- 
tional mean of the distribution, and the distribution is, 
approximately, gaussian. As is well known, this is desirable 
with quadratic loss functions. 

2. A Numerical Approximation of the Gram-Charlier 
Expansion 

Let X be a scalar and assume that all the moments of 
the probability density function p(x) exist. Then p(x) may 
be represented by the Gram-Charlier expansion 

1 f" 
fln = — j- / p{x) He„{x) dx (lb) 

Here He„{x) is the Hormite polynomial of nth degree 
and can be defined by 



Then 



Heo{x) = 1, Hei(x} ^ i 
He„^,{x) = xHe„(x) - nHe„.i (x) 



(2) 



It is known that these polynomials are mutually orthog- 
onal with respect to the weight function exp(— A-V2). The 
fact that the area of p(x) is one implies that Co is one. 

To find a numerical approximation for a„ without 
having to evaluate the integral in Eq. (lb) analytically, 
one proceeds as follows: 

Let ir„(|) be a polynomial of degree n or less. Applying 
the theorem of Gauss and Jacobi 



/ ,r„(g)exp(-|^)d^=£W7^„(^»'j 

J-<o i = i 



(3) 



where 2m — 1 < n, and the Wv* and |?* are weights and 
nodes for gaussian quadrature. They are tabulated for 
the weight function exp (—!■); the nodes correspond to 
the zeroes of //^(x). Let i - x/2'^. Tfien Eq. (3) becomes 
equal to 

Define a new nth degree polynomial by 



j^ Mx) exp(-f-^ dx = 2^ £ Wr <^ (2^ C 

If we approximate p(x) by 

p(x) s exp (-J-j ^x i^) 



(4) 



(5) 



Then substituting Eq. (5) into Eq. (lb), and using 
Eq. (4), we arrive at 



9V^ m 



a„ = fl" = -rZ^^r V (2'' C) exp [(^r)1 We„ (2^ ^;») 
1 = 1 

(6) 
Let 



p"'^'^^ " 72^ ^^p (~r) [ % < ^^" ("^^J 



(7) 



Then it can be shown (as previously) that p'"(x) coincides 
with p(x) at the m points 2''^ ||^, i = 1,2, •• •, m which are 
the zeroes of He„(x). Also as before, we are led to believe 
that whenever Eq. (lb) exists as a Riemann integral 



lim a"' = a„ 



(8) 



For a k dimensional distribution^ the procedure is 
similar to the case for the Hermite functions. 



3. The Convergence of the Gram-Charlier Expansion 

The reason for employing this expansion is as follows: 

Theorem. Assume that p(x) is given exactly as a com- 
bination 

Then the approximation given in Eq. (7), using the values 
of p(x) at n points, is exact as far as the area and moments 
up to the (n — l)th order. 

Vroof. Since this method is an interpolation using the 
zeroes of He„(x), the result will be exact as far as the first 
n coeflBcients go (a,,, Oj, ■••, a„-i). These determine the 



16 



in SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



area (if the distribution is not nonnalized) and the 
first n — 1 moments. 

Thus, if the distribution can be accurately approxi- 
mated by an expression of the form Eq. (5), then this 
method should allow one to calculate the moments with 
great accuracy. 



Checks on the convergence of this method are given 
in Table 5. These may be compared with the results in 
SPS 37-49, Vol. Ill, pp. 23-31, using the Hermite expan- 
sion. It will be seen that the first 2 moments (when they 
exist) are given more accurately by this procedure, but 
the approximation does not generally converge uni- 
formly or in the mean-square sense. 



Table 5. Convergence of Gram-Charlier approximation 







No. of 












Namo of 
dlitribulion 


S<ale 
factor 


inter- 
polation 
points 


Area 


Mean 


Variance 


1: norm 
of error 


l^ norm 
of orror 


Unknown phase 


1 


72 


0.89464 


0.03834 


1.00607 


0.00000 


0.00000 


angle 




2 


0.89371 


0.01270 


not defined 


0.00637 


0.00434 






3 


0.89420 


0.03808 


1.00046 


0.00926 


0.00724 






4 


0.89464 


0.03818 


1.00405 


0.00073 


0.00052 






5 


0.89464 


0.O383C 


1.00600 


0.00026 


0.0OO20 






6 


0.89464 


0.03834 


1.00604 


0.00056 


0.00049 






7 


0.89464 


0.03834 


1.00606 


O.00O12 


0.00O10 






8 


0.89464 


0.03834 


1.00607 


0.00003 


0.00003 






9 


0.89464 


0.03834 


1.00607 


0.00005 


0.00005 






10 


0.89464 


0.03834 


1.00607 


0.00002 


0.00002 






12 


0.89464 


0.03834 


1.00607 


0.00001 


0.00001 






14 


0.89464 


0.03834 


1.00607 


0.00000 


0.00000 


Cauchy distribution 


1 


2 


0.65774 








0.15407 


0.09913 






3 


0.82991 


— 





0.14425 


0.09250 






4 


0.78861 


— 


— 


0.13492 


0.09066 






5 


0.85464 


— 


— 


0.08738 


0.055S5 






6 


0.84093 








0.12698 


0.11324 






7 


0.87276 


— 





0.10223 


0.06921 






8 


0.86864 


— 


— 


0.31294 


0.23133 






9 


0.88631 


— 


— 


0.12124 


0.09026 






10 


0.88590 








0.86881 


0.76146 






12 


0.89780 


— 





3.06090 


2.59333 






14 


0.90661 


— 





11.80041 


10.21137 






16 


0.91347 


— 


— 


49.40664 


42.85682 






20 


0.92359 








0.102X10' 


0.895 Xltf 






40 


0.94733 


— 





0.204X10"" 


0.183X10" 






48 


0.95215 


— 


— 


0,250X10" 


0.22 X10" 


Norinatized student 


jM. 


2 


0.65774 





not defined 


0.20799 


0.23768 


t distribution, v = 3 




3 


1.21284 





0.36854 


0.12301 


0.11172 






4 


0.84759 





0.86082 


0.18303 


0.20991 






5 


1.09702 





0.55811 


0.08198 


0.06813 






6 


0.92305 





0.82206 


0.11766 


0.14354 






7 


1.04953 





0.66315 


0.06892 


0.05695 






8 


0.95777 





0.81389 


0.12469 


0.14404 






9 


1.02707 





0.72590 


0.04485 


0.03975 






10 


0.97536 





0.81655 


0.03654 


0.04593 






12 


0.98491 





0.82311 


0.22953 


0.22499 






14 


0.99038 





0.83084 


0.50863 


0.41463 






16 


0.99365 





0.83861 


2.02287 


1.77571 






48 


0.99979 





0.90469 


0.294X10" 


0.263X10" 


Normalized student 


1.0540 


2 


0.97285 





not defined 


0.01803 


0.01636 


f distribution, v = 20 




3 


1.00220 





0.92257 


0.00974 


0.00657 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



17 



Table 5 (contd) 







No. of 












Namo of 
distribution 


Scale 
factor 


Inter- 
polation 
points 


Area 


Mean 


Variance 


li norm 
of error 


1„ norm 
of error 


Normalized itudeni 


1.0540 


4 


0.99810 





0.99545 


0,01714 


0.01545 


t distribution, v = 20 




5 


1.00014 





099057 


0.00117 


0.00091 






6 


0.99979 





0.99850 


0.00123 


0.00135 






7 


1.000CO 





0.99838 


0.00126 


0.00090 






8 


0.99997 





0.99955 


0.00352 


0.00317 






9 


1.00000 





0.99963 


0.00015 


0.0001 1 






10 


0.99999 





0.99985 


0.001 30 


0.00104 






12 


1.00000 





0.99995 


0.00244 


0.0Q213 






U 


1.00000 





0.99998 


0.00285 


0.00245 






20 


1.00000 





l.OOOOO 


0.01690 


0.01478 






36 


1.00000 





1.00000 


0.597X10' 


0.533X10= 






56 


1.00000 





l.OOOOO 


0.533X10' 


0.479X10' 


Exlrtir.e value 


1 


2 


0.81 657 


0.36418 


not defined 


0.10223 


0.09394 






3 


1.02431 


0.39141 


0.53758 


0.08334 


0.07552 






4 


0.98532 


0.35252 


0.95907 


0.10672 


0.09395 






5 


0.98253 


0.47214 


0.77955 


0.04019 


0.03835 






6 


1.01088 


0.39468 


0.96166 


0.04312 


0.04719 






7 


0.98518 


0.46418 


0.91741 


0.07504 


0.07114 






8 


1.00778 


0.42780 


0.95980 


0.05766 


0.05532 






9 


0.99343 


0.45136 


0.97471 


0.02371 


0.02370 






10 


1 .00239 


0.44489 


0.96709 


0.06557 


0.06054 






12 


0.99969 


0.45080 


0.97845 


0.13009 


0.10453 






20 


0.99991 


0.44975 


0.99928 


6.16576 


5.34279 






32 


0.99999 


0.45007 


0.99980 


0.922X10* 


0.818X10' 






40 


1.00000 


0.45005 


0.99998 


0.212X10' 


0.190X10' 



18 



JPl SPACE PROGRAMS SUMMARY 37-51, VOL /// 



N 68-37399 



II. Systems Analysis 

SYSTEMS DIVISION 



A. A Proposed Venus Coordinate System, 

F. M. Sturms, Jr. 

1 . Radar Studies of Venus 

During 1964 and 1966, radar studies of Venus (Hefs. 
1-3) have produced solutions for the radius, axis and 
rotation period, and also identified several siurface fea- 
tures. This knowledge permits, for the first time, speci- 
fication of coordinate systems associated with the 
equatorial plane of Venus. Selection of such a coordinate 
system is complicated somewhat by the fact that Venus 
rotation is retrograde. 

From Ref. 2, the best solutions for thv rotation (or 
angular momentum) vector and period are as follows: 



(1) Right ascension (ao) ~ 

(2) Declination (8o) = 

(3) Period = 



98 ±5 deg. 
-S9±2deg. 
242.6 ±0.6 days. 



From Refs. 1 and 2, the prime meridian, or zero aphro- 
diographic longitude, is chosen to pass through a promi- 
nent narrow feature denoted as F or a. However, the 
coordinate system proposed in this report is based on a 
choice of north pole opposite that used in Refs. 1 and 2. 
This article discusses the reasons for this choice. 



2. Coordinate System Geometry 

In 1964, R. Richard^ presented arguments for stan- 
dardizing the method of choosing the north pole and 
the direction for measuring positive longitude. The ad- 
vantages described include a reduced possibility of con- 
fusion, due to the proposed analogy to terrestrial 
conventions, and a single set of formulas for expressing 
rotations, angles, and oblateness perturbations. Accord- 
ingly, the following conventions are adopted for Venus: 

(1) The north pole is that end of the rotational axis in 
the direction of the angular momentum vector 
(right-hand rule). 

(2) Body-fixed longitude is measured positive in the 
direction of rotation, i.e., with convention (1), to 
the east. 

Convention (1) is opposite to that given in Refs. 1 and 2, 
and convention (2) is opposite to that generally used in 
Refs. 4 and 5. 

a. Adopted pole and rates. By the above convention, 
the north pole of Venus has the right ascension and 



'Richaid, R. J., On a StandartUxed Method of Beckoning Longitude 
on the Various Celettial Bodies, June 16, 1964 ( JPL internal docu- 
ment). 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. (If 



19 



declination given in Subsection 1. Because of the fairly 
large uncertainty in the values, the epoch associated with 
these values is not tightly constrained and shaM be taken 
as 1964.5 (a convenient value near the Ve: as conjunc- 
tion of that year). Also, the values shall be taken as being 
with respect to the mean equator and equinox of date. 

The values of the pole location will change with time 
due to the precession of both the earth and Venus equa- 
tors. At the present time, since no estimate of the oblate- 
ness of Venus is available, the precession of Venus is 
taken as zero. Therefore, due to the precession of earth 



dt 

dSp 
dt 



— m + n sin ao tan 80 



= n cos a,, 



Using values of the annual general precession in right 
ascension, m, and the annual general precession in decli- 
nation, n (Ref. 5, p. 38), the resulting pole location is 

tto = 98 -0,0015551 {t - 1964.5) deg 
8„ = -69 -0.0007748 (t - 1964.5) deg 

where {t — 1964.5) is in tropical years. 

b. Equator and orbit angles. Given the location of the 
pole of Venus, several useful angles describing the 
orientation of the equator and orbit of Venus may be 
computed. Using the formulas on p. 332 of Ref. 5, and 
the values of the mean orbital elements of Venus from 
p. 113 of Ref. 5, the results for the epoch 1964.5 are a.s 
follows: 

n = angle froin mean equinox along ecliptic to ascend- 
ing node of the Venus mean orbit = 76.360 deg 

t = inclination ot the Venus mean orbit to ecliptic 
= 3.394 deg 

n = angle from node, ft, along the Venus orbit to 
descending node of orbit on equator (Venus au- 
tumnal equinox) = 290.878 deg 

I = inclination of Venus orbit to Venus equator 
(Venus obliquity) = 176.545 deg 

A = angle from ascending node of Venus equator on 
earth mean equator along Venus equator to au- 
tumnal equinox = 180.075 deg 



Note that for Venus, the vernal equinox is analogous to 
that of earth, i.e., the point where the sun crosses from 
the southern hemisphere to the northern hemisphere 
(beginning of northern spring). Because of the porth-pole 
convention used, the Venus obliquity is greater than 
90 deg. The proper quadrants for these angles follow 
unambiguously from the equations in Ref. 5. 

The obliquity of the Venus equator is very nearly 
180 deg and, consequently, the seasons are not very dif- 
ferent from one another in terms of the incidence of the 
sun's rays and the maximum elevation of the sun at 
noon. Coupled with the nearly circular orbit of Venus, 
this results in a day-night cycle that is near' constant. 

Finally, it is interesting to note that the Venus equi- 
noxes lie very nearly in a plane parallel to the earth 
equatorial plane. 

3. Venus Rotation 

a. The Venus day. The Venus sidereal day is, as given 
in Subsection 1, 242.6 ephemeris days, and the sidereal 
rotation rate is, correspondingly, 1.484 deg/day. With 
the adopted north-pole convention, the apparent star 
motion is from east to west. 

The mean orbital motion is 1.602 deg/day, and the 
sun appears to move from east to west in right ascension 
against the star background, which is opposite to that 
seen from earth. 

These motions combine to form a solar day that is 
shorter than the sidereal day, contrary to that of earth. 
The mean rotation rate, with respect to the sun, is the 
sum of the above rates (3.086 deg/day), and the Venus 
mean solar day is, therefore, 116.7 ephemeris days. The 
apparent solar motion is from east to west. 

b. The prime meridian and central meridian. In Refs. 
1 and 2, the Venus prime meridian is chosen to pass 
through a narrow feature identified as F for a. This 
choice is also made here. The method for establishing the 
prime meridian is to define the sub-earth longitude or 
central meridian^ at some epoch. Thus, following Ref. 1, 
the apparent aphrodiographic longitude of the earth at 
0^ ephemeris time (ET) on June 20, 1964 (JD 243 8566.5) 
is +40 deg. (Note that the epoch has been arbitrarily 
changed to 0^ ET, rather than 0" UT, in order to simplify 
the computations below.) Because of the reversed pole, 
the values in Table 1 of Refs. 1 and 2 should be changed 



niie longitude at the apparent center of Venus as seen from earth. 



20 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. HI 



Toble 1 . Phenomtna for propestd V»nut 
coordlnatt systam 



Sam* ai tarlh 


Oppotlta to aanh 


D«pand*nt en narth-pol* can«*nHen, i.a., ravanci If (onvmHon 
li ravartad 


1. Apparantitar motion aait to wait 


1 . Vanui obliquity graatar than 
90 dag. Sun movai aait to 
wait In RA 


2. Right otcantton (RA) potiliva In 
diraetion of rotation 




3. Sun rtiat In aait, tatt In watt 




4. Hour ongla of aquinox oppotlta 
rotation, Incraoiai willi tlma 




5. Effad of Vanut pracatilon i> to 
Incraata Vanui RA 




Indapandani of nei1h-p«l* canvantien 


1 . longituda potltlva ''Ott 

(longltuda of cantrol maridion ra- 
varsat with convantion) 


t. Sun movai oppoiita rotation 


2. Daflnition of varnol aquinox 
(Idantity of givan inlarteclion re- 
variai with convantion) 


2. Solar day ihortar than lida- 
raol day 


3. Hour ongla posltlva wail 

4. RA poiitiva aoit 


3. Eftact of Vanui pracatilon ii 
to dacraoia Vanut calattlal 
longituda 



by reversing the signs on the latitudes and longitudes of 
the features. 

From the discussion on pp. 335 and 336 of Ref. 5, the 
longitude of the central meridian, A, is given by 

. ,, , Rttt 
A = A, - V + 



(note reversed signs to account for reversed convention 
for positive longitude) 

where 

A;{ = Venus right ascension of apparent earth 

V = hour angle of Venus vernal equinox from prime 
meridian 

and the third term is the rotation during the light time, 
where 

R = earth-Venus distance (AU) 

T = light time for 1 AU - 499.012 s 

w = sidereal rotation rate 



From the equations on p. 334 of Ref 3, Ag is computed 
in terms of the right ascension and declination of the 
Venus pole (oo, 8o) and the apparent Venus coordinates 
(a, fi). Note that a, «, Oo, So and A must be consistently 
given with respect to either the mean or true earth 
equator and equinox. The quantities Ag and Dg are inde- 
pendent of the choicp, and P will be measured from a 
mean or true declination circle, respectively (Dg = aphro- 
diocentric latitude of earth; P = position angle of Venus' 
north pole from earth declination circle). 

From Ref. 4 for 0^ ET on June 20, 196i with respect 
to the true equator and equinox of date 

a = 5^ 53" 59r71 = 88.4988 deg 
8 = 21° 3& 28'.'3 = 21.6079 deg 
R = 0.2895 AU 

Converting to mean equinox and equator 



a = 88.5040 deg 
8 ^ 21.6081 deg 

As - 278.75 deg 

De = 0.87 deg 

P = 176.61 deg 



and 



Then, the reference value of V is (light time correction is 
negligible to significance retained) 

V„ = 278.75 - 40 + 0.002 = 238.75 deg 

and subsequently 

V = 238.75 + 1.483924 (JD - 243 8566.5) deg 
\= As- V + 0.0086R 

Finally, it should be noted that the Venus right ascen- 
sion and declination of the earth, A» and D«, are mea- 
sured positive east (in direction of rotation) and north, 
from the Venus vernal equinox and equator, respectively, 
and the hour angle of the equinox, V, is measured posi- 
tive west from the prime meridian to the equinox. These 
are analogous to the measurement conventions on earth. 

4. Coordinate Trantformotion* 

The mean earth equator and equinox of 1950.0 is a 
standard non-rotating coordinate system in common use. 



jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



21 



The cartesian position transfonnations to Venus coordi- 
nate systems involve the following rotations: 

(1) Rotate to mean earth equator and equinox of date 
(precession matrix A). 

(2) Kdtate Oo + 90 deg about Z axis (matrix S,). 

(3) Rotate 90 - o„ deg about K axis (matrix Sj). 

(4) Rotate A + 180 deg about Z axis (matrix Sj). 

At this point, the coordinates are with respwt to the 
Venus equator and equinox of date. Two options are as 
follows: 

(1) Rotate I about X axis (matrix £). This yieldi co- 
ordinates with respect to Venus mean orbit and 
equinox of date. 

(2) Rotate V about Z axis (matrix H). This yields co- 
ordinates with respect to Venus equator and prime 
meridian (aphrodiographic). 

Then, in summary 
Venus equator and equinox: (X) = S., So S, A (X), 950.0 
Venus orbit and equinox: (X) ~ E S3S2S1 A (X),95„.o 
Aphrodiographic: (X) = H S^S^Sx A (X)i9!,o.o 



cosV 


sinV 





-siiiV 


cosV 











1 



where 



A = precession matrix (Ref. 5) 



S. = 



E = 



— sin Oo 


COSOo 





-cos do 


— sin Oq 











1 


1 











sin So 


COsfio 





-cos 80 


sin So 


—cos A 


— sin A 





sin A 


— cos A 











1 


1 











cos/ 


sini 





—sin I 


COS I 



H 



The velocity rotations are obtained from differentiation 
of the above matrix equations. 

5. Discussion 

The proposed Venus coordinate system is based on 
conventions for defining the north pole and the direction 
of positive longitude. The convention for measuring 
longitude positive east has been adopted by the Inter- 
national Astronomical Union (lAU) (Ref. 6, p. 174) in 
conjunction with gravitational potential expressions, and 
is undoubtedly the best choice. The choice of north-pole 
convention is not so clear, however. In making the choice, 
it was desired to retain as much analogy and consistency 
with earth as possible. Accordingly, Taole 1 presents a 
list of items pertainjjg to the proposed Venus coordinate 
system. The table is a useful aid in visualizing phe- 
nomena as they appear relative to a Venus observer. 

The adoption of the proposed coordinate system leads 
to the question of how improved values are ir .jrporated. 
The following procedures are based on historical 
precedent. 

Improved values of the pole location should be stated 
in term-: of the 1964.5 values. This can be done by map- 
ping a solution for a current epoch backward, or by 
solving directly in terms of the 1964.5 value and mapping 
forward to compute current observations. The improved 
location should be included in the rates due to the 
earth precessit 1. 

vVhen information on the figure of Venus has been 
obtained, the precession of the Venus equator can be 
detennined (Ref. 5, p. 327). This can then be incor- 
porated into the computation of the pole location rates. 
If the rate of precession of the Venus equator on the 
Venus orbit is denoted by /*, the contribution to the 
rates is (Ref. 7). 

— T^ — 11 sin I cos A sec So 

dSo ■ J „ 

(Note: for Venus, /i is positive, i.e., in the direction o^ 
orbital motion, whereas it is normally negative for other 
planets.) 



22 



JPl SPACE PROGRAMS SUMMAttf 37-51, VOL. \\\ 



' a 



A more precise value for the rotation period of Venus 
will directly update the rate term in the expression for V. 
The lengths of the solar and sidereal days are easily 
corrected. 

The leading term in the expression for V must be re- 
derived for improved values of ao and So- Changes will 
enter through a di£Ferent value of Ag and also the 
inclusion of the light time term, if it is significant. In this 
step, the longitude of the centi-al meridian at the refer- 
ence epoch is unchanged, i.e., it is fixed at a value of 
40 deg. This procedure is similar to that followed for the 
physical ephemeris of Mars, where initially the longitude 
of the central meridian is computed to place the prime 
meridian through a prominent feature. Subsequently, 
hovever, the longitude of the central meridian at the 
reference epoch is held constant, and the longitudes of 
the prominent feature, as well as all other features, will 
vary slightly. 



References 

1. Carpenter, R. L., "Study of Venus by CW Radar-1964 Results." 
AstTon. J., Vol. 71. No. 2, Mar. 1966. Also available as Technical 
Report 32-963, Jet Propulsion Laboratory, Pasadena, Calif. 

2. Goldstein, R. M., "Radar Studies of Venus," Moon and Planets, 
North-Holland Publishing Co., Amsterdam, 1967. Also available 
as Technical Report 32-1081, Jet Propulsion Laboratory, Pasa- 
dena, Calif. 

3. Ash, M. E., Shapiro, 1. 1., and Smith, W. B., "Astronomical Con- 
stants and Planetary Ephemerides Deduced from Radar and 
Optical Observations," Ajtron. /., Vol. 72, No. 3, Apr. 1967. 

4. American Ephemeris and Nautical Almanac, 1964. United States 
Government Printing Office, Washington, 1962. 

5. ExiMmatory Supplement to the Ephemeris. Her Majesty's Sta- 
tionery Office, London, 1961. 

6. "Proceedings of the Eleventh General Assembly, Berkeley, Calif., 
1961," Tron*. lAU, Vol. XIB. Academic Press, New York, 1962. 

7. de Vaucouleurs, C, '"The Physical Ephemeris of Mais," Icarus, 
Vol. 3, 1964. 



JPL SPACE PROGRAMS SUMMARY 37-5?, VOL. HI 



23 



^N6S-S74Ga 



III. Computation and Analysis 

SYSTEMS DIVISION 



A. Orthonormal Transformations for Linear 
Algebraic Computations, C. L Lawson 

1. Introduction 

The basic step in many methods for solving systems of 
linear equations, or computing eigenvalues or singular 
values of a matrix, may be interpreted as premultipli- 
cation of a matrix A by a matrix T, where T is chosen 
so that certain elements of TA are zero. Methods in which 
r is orthonormal are stable with respect to growth of 
rounding errors and are particularly appropriate in least- 
squares computations because of their property of pre- 
serving the euclidean length of vectors. 

In this article, we review the properties of two ortho- 
normal trapsformations that are well known in numerical 
analysis, and introduce a third orthonormal transfor- 
mation that combines certain features of the first two. 

We denote the transpose of an m-vector v by v'' and 
its euclidean norm by 



: v'V 



Zv^(0 



All of these transformations can be discussed in the 
following setting: 

Problem. Given an m-vector v, find an m X m ortho- 
normal matrix Q such that components 2 through m of 
Q\ are zero. 

Since only the first element of Q\ is permitted to be non- 
zero, and since ] | Qv | | = j | v 1 1, it follows that the first 
component of Q\ must be either | j v ] 



or - 



The identity matrix of order m is denoted by Im- 



2. The Jacob! Transformation 

A single Jacobi transformation alters only two com- 
ponents of a vector, one of which will be transformed to 
zero if the transformation matrix is appropriately chosen. 
Thus, the Problem, above, can be solved by a sequence 
of m— 1 Jacobi transformations. 

A Jacobi transformation matrix can be denoted by Bij,, 
and is identical with the m X m identity matrix /„ with 
the exception of the four elements 

bii = bjj — c = cosfl 
bij = — faj, = s = sin^ 



24 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



Let V — Bj, j, s V, and suppose is to be chosen so that 
V ,^.) = 0. This is accomplished by computing 



d - Ko+ v=,,)- 



c = 



V(,/rf 



V,^)/d 



ifd^O 
ifd = 

ifd^O 
ifd = 



Then 






A geometric interpretation of the Jacobi transformation 
is given in Fig. 1. 




Fig. 1 . Geometric interpretation of a Jacobi 
transformation 



If I I w I I = 1, the matrix H^ is a reflection matrix 
characterized by the fact that it transforms w to — w and 
acts as an identity on the (m — l)-dimensional subspace, 
S, orthogonal to w. These properties completely charac- 
terize the eigenvalue-eigenvector structure of H„ and 
thus permit an explicit construction of H„ as follows: 



Let p2, •■■, Pm be an orthonormal basis for S and let 
P = [w>P2, ■ ■. Pm]- Then P is an m X m orthonormal 
matrix and 

H^P - PD 
where D = diag ( — 1, 1, 1, • • •, 1). 

Let Ell denote an m X m matrix whose only nonzero 
element is a one in the (1, 1) position. Then 

H^ = POr = P(I - 2£i,)P'' 

= 1 - 2PEii F = / - 2ym'' 

Now consider the Problem presented in Subsection 1. 
Let the /n-vector v be given. Define 



a-sgn(vj,,)= _^ ,f 



+ 1 



'fv,i,>0 



Ev,i,<0 



Define the m-vector Ci by Ci = (1,0, • • • , 0)' , Let w be the 
unit vector bisecting the angle between v and a 1 1 v 1 1 Ci; 
explicitly 



The multiplication x = Bx, where x is an arbitr?.ry 
m-vector, can be done as 



"{t) CX(i) + ^U) 



-U) 



\k} ■*{)£) 



'(ij-1- cxj^j 



f or fc ^ i and kj^j 



3. The Householder Transformation 

An m X m Householder transformation matrix, H„, may 
be parameterized by an m-vector w, where either w = 
or I I w I I = 1. If w = 0, we define H„ = /„. 



u = V + <T V Ci 



w 



«/||u| 




ifu^O 
ifu = 



The matrix H„ = l^ — 2ww^ solves the FrobleTn since 



v= H^v= -<r||v| Ici 



A geometric interpretation of a Householder transfor- 
madon is given in Fig. 2. 



JPL SPACE PROGRAAtS SUMMARY 37-57, VOL. Ill 



25 






\J 




•rllvlle. o-*i 



\ 



Fig. 2. Geometric interpretation of a Householder 
transformation 



In a computer program, this computation is commonly 
organized so that the vector w is not explicitly com- 
puted. We may write 



where 



-b=||ulj72 

= (v + cr||v||eOMv + a]|v]|e,)/2 

= !!v|p + !|vl|-lv,„| 

= -||v|l(-||v||+v,.,) 



(1) 



(2) 



Note that since u differs from v only in the first component, 
the constructioti of Hw, as given iti Eq. (1), requires only 
the computation of Uj,jand h. Furthermore, the only non- 
zero element of v is v^,,. The computation of U(,j, b, 
and Vj,j can be organized as 



h^r -(|:/(i))'*sgn(V(,)) 



■{!) 



^{1) ^{1) 



^ = "^W "{!) 



(3) 

(4) 
(S) 



If the matrix H, is to be saved, it suflBces to save the 
m-vector u and the scalar h. If v p. is also being saved, 
then one need not save h as it can be recomputed when 
needed using Eq. (5). 



26 



The multiplication x = i/«x for an arbitrary m-vector 
X proceeds as 



= (|:-.,^n)/^ 



*{i} =^ *{i) + C"(i) 



4. The RSP Transformation 



i = l,--,m 



The RSP (Rotation in a Selected Plane) transformation 
will combine the Householder-like ability to transform 
m— I elements of an m-vector to zero in a single trans- 
formation with the Jacobi-like property of using a plane 
rotation instead of a reflection. 

Let S denote a two-dimensional subspace of m-space 
with orthonormal basis vectors Wj and Wj. We wish to 
construct an orthonormal matrix R that will act as a 
rotation in S, rotating Wj through an angle toward 
Wi, and act as an identity on S^, the orthogonal comple- 
ment of S- 

Let W3, • • •, w„ be an orthonormal basis for S^. Define 

c — cos 6 
s = sinO 
c 



B 



:] 



Then 



RW = W 



R = W 



B 


- 





/m-.. 


B 


■ 





I.-.. 



W^ = 7„ 



(6) 



+ [Wi.Wa] 



'"-'-'[<] 



Consider the Problem given in Subsection I. Let v be 
a given m-vector, and again let 



tr = sgnv,,, = 



+ 1 



«V, >0 



1 ifV(,, <0 

«! = [1, 0, ■ • •, 0]'' (m-dimensional) 

JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



We seek orthonoimal vectors w, and Wj and an angle $ 
such that the matrix R, defined by Eq. (6), satisfies 



v = Rv = <T|lv||ei 



(7) 



Define 



Wi — oei 

it'/llull ifu^j^O 



w, = 



c = 



[0,1,0, ••,0]'^ ifu = 

|v,„|/||v|l ifv^O 



(8) 



1 



l«l|/| 







ifv = 

ifv^O 
ifv=0 



It can be verified by substitution into Eq. (6) that these 
values of Wi, Wa, c, and s provide a matrix R that satis- 
fies Eq. (7). 

A geometric interpretation of these quantities is pro- 
vided in Fig. 3. 




{=ae^) 



Fig. 3. Geomefric interpretation of an RSP 
transformation 



We now consider computational details. When u = 0, 
we have R = /«, and this case can be given special 
treatment. We thus consider only the case of Ut-^O, 
which of course implies v 7^ 0. 

JPL SPACE PROGRAMS SUMMARY 37-51, VOL. HI 



It is possible to rewrite Eq. (6) as 



».•»!"['„;] 



(9) 



where the elements of the 2X2 matrix F are 

f,. = .*||u||-> = <r|lv||- = y-, 
til ~ /12 

h, = {c- l)||u||-=-sMc + l)-Mlu|h 

fii = C - 1 = j I U I (= fss 

The computation of these quantities could proceed as 

\H\'= t ^h (10) 

i = 2 

til — ~/12 

Saving the matrix R would require space for the m — 1 
nonzero elements of u plus f^i, fiu and fu, i.e., a total of 
m + 2 locations. If Vj^j is also being saved, then /„ need 
not be saved. 

To compute x = Rx for an arbitrary m-vector x 

i=2 

h = -/i2X,,j + fiig 

*(1) = *(1) + (/""{l} +/l2g) 



"{«) 



= Xjj, -f ftuji, i = 2, ■••, 



m 



17 



■ » > !■ 



5. Conclusion 

The Jacobi (ransformation is used primarily in cases 
in which the pattern of elements to be zeroed is some- 
what irregular. When a number of elements in one col- 
umn are to be zeroed, it is more economical and more 
accurate to use the Householder transformation. The 
RSP transformation is nearly as economical as the 
Householder transformation and could reasonably be 
used in the same circumstances. 



The relative roundoff error, ««, m computing 
X(„j = X -f Gx using arithmetic having relative precision 
a, is bounded by 



e«||< 



"O-'IT^)^"^'"'!!''' 



with a similar bound of 



) = 3a 

(14) 



Although the Householder transformation is very stable 
with regard to roundoff error propagation (see Ref. 1, 
p. 101), the RSP transformation may be even slightly 
more stable. The Householder transformation is applied 
in the form H = I + G where, using the spectral matrix 
norm, 1 1 G | | = 2 for all w except for the special case of 
w = (which we will henceforth exclude). Similarly, the 
RSP transformation is applied in the form R = I + K, bu' 



iX|| = [2(l-c)]^ 



(11) 



where c is defined by Eq. (8). In particular < c < 1, 
and thus 



and, consequently 



< I I K I I < 2" 



K I I < 0.71 • I I G I 



(12) 
(12) 



lh«ll<«(i + tfllT)^"^^'"*"l''^N) 



oil + [2(1 - c)]^} < 2.42a 



(15) 



for the computation oix^^y = x + Kx. Since G and K are 
of i.mk 1 and 2, respectively, the ratio, j | Gx | |/| | x | | 
and I I Kx I |/ ! I X I I are usually not close to their respec- 
tive upner bounds, | | G | | and ] ] K | | (say, averaging 
over iit, i i V 1). Thus, comparison of average be- 

havior cannot be based on Eqs. (14) and (15). Furtlier 
investigation will be made of the relative merits of the 
Householder and the RSP transformations. 

Reference 

1. Ralston, A., and Wilf, H., Matheinatica', Methods for Digital 
Computers: Volume U, John Wiley & Sons, Inc., New York, 1967, 



28 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



N68-37401 



IV. spacecraft Power 

GUIDANCE AND CONTROL DIVISION 



A. Solar Cell Standardizafion, ft. F. Greenwood 

1. Introduction 

A project was initiated by ]PL in 1962 to improve the 
accuiacy of predicting solar array performance in space. 
High-altitudc balloon flights have been used to achieve 
the near-zero air-mass conditions required for calibrating 
the solar cell standards. 

Balloon-calibrated solar cells in modular form were 
recovered an<l mounted on a temperature-controlled 
housing {Fig, 1) and used as intensity reference stand; rds 
during performance testing of solar arrays under ter- 
restrial suiilight conditions. It has been shown by Ritchie 
(Ref. I, pp. 6 and 7) that, if the standard solar cell and 
the cells used for array fabrication have the same spectral 
response, the space short-circuit current output of the 
solar array can be predicted with an accuracy of better 
than 2%. Since 1962, high-altitude balloon (lights for 
so'ar cell standardization have been conducted at the 
rate of three or four flights per year. Cooperative efforts 
between JPL and other NASA and government agencies 
have provided standard solar cells at minimum expense 
for c variety of space projects and advanced development 
work. 

2. 1968 Bolloen Flight Project 

Three 80,000-ft balloon flights are scheduled for July 
and August 1968. Fabrication and testing of standard 




Rg. 1. Balleofi-calibraltd itondard folar c*H modul* 
on t«mp«rolur«-controli*d heuitng 



JPL SPACE PROGHAm SUMMARY 37-51, VOL lit 



29 



solar coll modules art' in progress. The cooperative cHort 
between JPL and other govcmmtMit agencies is con- 
tinuing this year with the Air Force Aero Propulsion 
Laborator>\ the Johns Hopkins University, tho NASA 
Langley Research Center, and the NASA Goddard Space 
Flight Center supplying standard solar cell modules for 
calibration. 

Improvements to the ballooii flight system are currently 
in progress. Design modifications of the solar tracker 
having an inrrcascd payload capacity have been com- 
pleted, and actual modification has begun. Figure 2 shows 
the old solar tracker configuration The modified solar 
tracker will provide for 36 solar cell calibration channels, 
an increase of 12 channels. This will be accomplished by 
replacing the old 24-posi*ion stepping ■witch with a new 
36-positfon steppmg switch. At the same time, the solar 
cell module mounting area will be increased to accom- 
modate the added module capability. 

Due to the increased amount of data returned per flight 

as a result of increased payload capacity, improved 
methods of data reduction are required. To meet this 



P 




Fig. 2. Pf«»nt bollean ap»x-moun(«d solar tracker 



problem, flight data will be supplied by the balloon flight 
project contractor on IBM punched cards. A JPL com- 
puter program is in the process of being updated, which 
will be compatible with the contractor-supplied data. 
The computer program will reduce, average, and correct 
tlie solar cell data for intensity and temperature. A sum- 
mary sheet will give solar cell descriptive information 
along with calibration data at a standard intensity and 
temperature. It is expected that, through improved data 
handling and processing mcthod.s, calibr.ition data will 
be available within a few days following a balloon flight 
series. 

R«fer«nc* 

I. Ritchie, D. W., Decchpmcnt of PhotovoUoic Slanttard Cdh for 
XASA, Tt-clinital Report 32-634. Jet Propubion Lijbo.alory, 
I'iisiidtiia. Calif., June I, 1964. 



B. Solar Power System Definition Studies, 

H. M. VVjcJt 

1 . Introduction 

The overall objective of this effort is to investigate the 
problems :issoeiated with developing spacecraft power 
systems for unmanned planetary niission.s. Tlie effort 
stresses development of the technologj' retiuired to sohe 
system de.sigii j>roble]iis a.ssociated with meeting JPL 
mission rcquirertients. One task which is presently being 
under taki-n is the investigation and development of 
eomputi'r programs for power system design, int<'gratioii 
,ukI analysis. 

2 Power Profile Computer Program for o 
Marx 1971 Million Study 

The successful development of a spacecraft power 
system requires a compromise between user i>o\vtr re- 
quirements and available po^er limitations. Due to the 
mu'titude of changes in the user s>'stcm power require- 
ments during the spacecraft design phases, continuous 
monitoring by the spacecraft power design team becomes 
an absolute necessity. Data processing methods provide 
both in effective and .iccurate method for maintaining an 
up-to-date status of th<? spacecraft power requirements. 

A computer program was recently developed to assist 
in determining the spacecraft electrical power require- 
ments for power system sizing and spacecraft power 
managemeut for a Mars 1971 mission. 

A functional block diagram of tl e power system with 
the spacecraft loads is shown in Fig. 3, which represents 



jn SPXC£ PKOORAMS SUMMARY 37-51, VOL. (If 



mtm^m^m 



SOLAR 
PANELS 



POWER 

SWITCHING 

AND 

LOGIC 



n 



BATTERY 



BATTERY 
CHARGER 



LINE 
REGULATOR 



MAIN 



r*- SCIENCE 



INVERTER L^ 



THREE- 
PHASE 
INVERTER 



SINGLE- 
*) PHASE 
INVERTER 



• ENGINEERING 



■ GYROS 



SCAN 
PLATFORM 



_^ COMMUNICATIONS __ .,i._., - 

,\Xy CONVERTER [^ REGULATED _p'°'WBALS 



L^ TEMPERATURE 
CONTROL 



CONVERTER 



ru 



J 



ACTUATORS 



Fig. 3. Power system functional block diagram 

the power system model used by the power profile com- 
puter program. Power is derived from photovoltaic solar 
panels and a secondary battery. The power switch and 
logic (PS&L) distributes raw power to the line regulator, 
battery chaiger, communications converter, and tempera- 
ture control system. 

The first two pages of the computer printout lists all 
input data for reference. The spacecraft systems and their 
power requirements for each of the mission flight phases 
are tabulated on the first page. On the second page is a 
listing of power and efficiency data points for each of 
the inverters and the line regulator. These data are used 
by an interpolation subroutine to define the operating 
efiBciency as a function of power output. The program 
then calculates and prints out the power output, effi- 
ciency, and power input for each of the inverters of the 
power system along with their respective user system 
requirements. The line regulator power output, efficiency, 
and power input is then determined and listed. The total 
power demand of the spacecraft power system is obtained 
by summing the PS&L loads and dividing by the PS&L 
efficiency. This process continues until all power system 
operating modes and mission flight phases have been 
considered. This computer program is an extension of 
the programming work done to support Mariner Mars 
1969. The program has been written in Fortran IV for 
the IBM 7094. 

3. Battery Cell Data Reduction Program 

A Fortran II program was written for the IBM 1620 
data processing system. The program appropriately re- 
duces raw battery cell data in addition to providing 
plots of the cell discharge curves. 



4. Shepherd's Equation Battery Discharge Programs 

The BATT3 and BATT4 battery discharge programs 
have been verified. These programs are now considered 
operational; however, additional checkout runs are 
planned as soon as more detailed battery discharge data 
become available. 



C. Development of Improved Solar Cell 

Contacts, P. Sermon 

1. Introduction 

Silicon solar cells are presently the most reliable direct 
energy converters for space applications, and it appears 
that this will continue to be the case for some time to 
come. Over the past years, there have been significant 
increases in cell conversion efficiency, as well as reduc- 
tion in cell size and manufacturing costs; however, im- 
provements of the same magnitude have not been made 
in the area of solar cell contacts and solar cell intercon- 
nection techniques. The environmental limitations im- 
posed on the solar cell contacts to avoid mechanical and 
electrical degradation have remained the same for many 
years, and in some cases have even become more restric- 
tive. Therefore, solar panels are environmentally limited 
in many cases as a result of solar cell contact restrictions, 
and it can be expected that signfficant improvements 
on solar panel reliability will result from improvements in 
solar cell contacts and interconnection techniques. 

The objective of this study is the development of 
silicon solar cell electrical contacts and interconnection 
techniques which are less susceptible to mechanical and 
electrical degradation resulting from exposure to extremes 
of earth- and space-type environments. A major objective 
is the development of cell electrical contacts and inter- 
connection techniques which do not require the use of 
solder. There should be less degradation of contact 
strength and electrical characteristics after exposure to 
thermal shock, humidity-temperature, vacuum- 
temperature, high-temperature, and low-temperature en- 
vironments. The solar cell contact and interconnection 
techniques are also to be optimized with respect to the 

(1) eflfects on solar cell current-voltage characteristics, 

(2) series and/or cont<ict resistance, (3) stresses due to 
fabrication procedure, (4) compatibility with require- 
ments for fabrication into submodules, (5) reliability, (6) 
handling and manufacturing characteristics and re- 
straints, (7) repair or rework capability, (8) reproduci- 
bility, (9) production cost, (10) ease of production, (11) 
weight, (12) compatibility with large-area cells, (13) 



JfL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



31 



requirements for special equipment and toolmg, and (14) 
compatibility with inorganic, integral protective coatings. 

2. Otvclopmant Activities 

Development contracts have been awarded to Ion 
Physics Corporation and the Librascope Division of Gen- 
eral Precision Systems, Inc., and work was initiated in 
January 1968. Ion Physics is presently utilizing its high- 
vacuum sputtering technique, and Librascope is utilizing 
its cold-substrate deposition process to deposit the con- 
tact materials onto the silicon. 

The tirst material to be investigated by both organiza- 
tions is aluminum. Ion Physics has produced and deliv- 
ered to JPL sample solar cells having aluminum contacts. 
Cells have been fabricated having an efficiency of 9-10% 
at air mass zero (28 "C). This compares quite favorably 
with the 11-11.5% efficiencies characteristic of state-of- 
the-art solar cells when one considers the developmental 
status of the former cells. Thus far, only adhesive tape- 
peel tests (utilizing Scotch tape) have been used to 
evaluate contact adherence. Several cells exhibited con- 
tact peeling as a result of these tests; however, most of 
the cells were capable of passing with no apparent peel- 
ing of the contacts. Two cells which did not exhibit 
peeling were placed in a humidity chamber at 60°C and 
95% relative humidity for a period of 1 week. The cells 
showed no apparent deterioration in contact adhesion, 
r reliminary attempts to utilize parallel gap welding have 
not been successful, due to the oxide layer on the alumi- 
num which inhibits constant curr*^ »it How. The technique 
is still under investigation, 

Librascope has deposited aluminum on low resistivity 
(approximately 0.001 O-cm) n-type silicon, which is rep- 
resentative of the diffused layer of an n/p solar cell. The 
first attempts gave rise to rectifying (non-ohmic) contacts 
which exhibited nonlinear current-voltage characteristics. 
Through a series of experiments it was found that the 
glow-discharge operation, which was utilized as a clean- 
ing procedure prior to aluminum deposition, was a major 
reason for the non-ohmic behavioi of the contact. Use of 
a field to ionize the aluminum and yield an average ion 
energy of 112 eV (200 eV maximwn), in conjunction with 
tlie elimination of the glow-discharge operation, pro- 
duced contacts which exhibited ohmic behavior. The 
same technique was then utilized on ptype wafers that 
are representative of the base region of an n/p solar cell, 
and ohmic contacts were also obtained. The contact re- 
sistances, especially in the latter "\'i, s^ppear to be 
extremely high, and will probably not result in high- 
e£Bciency solar cells. 



3. Condutiont 

Significant progress has been made in the use of alumi- 
num as a contact material for silicon solar cells. It has 
been demonstrated that the high-vacuum sputtering pro- 
cess is capable of producing aluminum-contact cells with 
reasonable etfic'encies. The series resistance of these cells 
was found to be of the order of 0.5 O for 2 X 2-cm 
cells, in comparison with state-of-the-art titanium-silver 
contact cells which exhibit series resistance of the order 
of 0.3 n for 2 X 2-cm cells. 

At the present, problems scm to exist with the cold- 
substrate deposition process in achieving contact lesist- 
ances low enough to yield high-efficiency solar cells, 
although it was possible to obtain contacts to the n and p 
Ipyers which are ohmic in nature. 

D. Capsule System Advanced Development: 
Power Subsystem, K. G. /vonoff and D. J. Hopper 

1. Introduction 

The primary purpose of the Capsule Syste a Advanced 
Development (CSAD) project is to obtain an improved 
understanding of planetary entry lander capsule system 
design and integra'^'on problems and to obtain experience 
in several critvcu. and new technologies that relate to 
planetary capsule missions. To accomplish this objective, 
a specific Mars entry and hard-landing capsule system 
is being designed to obtain scientific information on the 
Martian atmosphere and surface conditions during cap- 
sule system entry and subsequent landing. 

To support the CSAC activity, pfjwer subsystems capa- 
ble of supplying electrical energy at discrete levels wt^ ; 
developed tor the entry and lander capsules. These sub- 
systems were designed to survive the sterilization require- 
ments of the capsule system, and for the lander capsule, 
impact survival was required. 

Power subsystems to be incorporated into an entry 
and lander capsule, as part of the CSAD, have been 
fabricated, integrated in»-o the capsule system, and are 
now underjToing syitem-level tests.. 

2. Power Subsystem Design 

Each power subsystem consists of a sterili"able silver- 
zinc battery and a pow'^r control unit to provide switch- 
ing, conditioning and distribution of several regulated 
voltages. The entr> cpsule power subsystem functional 
block diagrcm (Fig. 4) illustrates the power subsystem 
design and inetho<J of electrical power distribution. 



32 



jn SPACE PROGRAM* SUMMAItY 37-51, VOL. til 



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JPl SrMCE PROGRAMS SUMMARY 37.57, VOL. 11/ 



33 



I'pon comni;)nd from tlu- Piitry sitinciioiT and timer, 
till' powtT subsystom baUcry is swilchfd tm-lint- provid- 
ing power to llic major c.ipsulo subsystiuiu. Th"' bn'.tt'ry 
voltage of 15 to 25 V is boosttnl and rfgulatcd to 28 V 
by the boost reguhtor und used by llu- radio sub:>ysti'in. 
This output is also distributed to a dt-dc convcttor. The 
con\'erter, using v'oltage and current feedback, pro^'idcs 
six regulated voltage outputs tliat are used by all sub- 
systems V} .pt entry-capsule radi(» and ei>try timer. The 
entry timer is lunied on by command well In advance of 
other subsystems and is. therefore, supplied power from 
a separate regulator to reduir losses inherent in the 
power conversion etjuipnu'nt. The re^julator consists of 
a shunt ;^ener control. The eiitry-c;ipsule power control 
unit (Fig. 5^ is capable of providing a maximum power 
of ^) W. Figuri' 6 illustrates lu»w the power control unit 
and battery are comliined prior to assendily in the entry 
eapsuK', The battery consists of 1-1 cells, eacli having a 
capacity of 5 A-ii. 

Thi' lander-capstilc power subsystem is similar in de- 
sign to the entry-capsule power subsystem and performs 
identical functions. The major difference is the reqiiire- 
ment for high-impact .sunival of (lie lander ca siilo and 
the .ihility lit turn tiie lander capsule radio on and off 
independent of the other power loads. 

3. Devetoofnenl Status 

The i)roci'diire for the development of the power su!i- 
.systcnis consists of design, fabrication, and testing of thi 
individual units. The power subsystems are then inte- 





Fig. 5. Entry capswte power tufatysfem control unit 



Pig. 6. Entry capsute power subsystem 

grated into the ca[)sule system and functionally tested 
after being subjected to the selected environmental 
requirement s. 

Prototypes of the entry- and landcr-capsule power sub- 
systems have been tested at the subsystem level: in each 
ease, results wi^rt- within design limiis. Afur inteijration 
of the power .iubsjstem into the lander capsule, addi- 
tional system-level tests were performed, including steri- 
li/atioii. Till' power sub.system was also integrated into 
the entr>- capsule. A\\ s>stem tests indicate nominal per- 
formance of the power subsysteni. 

The entry and lander capsnles were combined and 
tested as a complete capsule system. Both entry- 
ai.d lander-capsule power suhsyst-.^ms performed as 
expect'd. 

The capsule system has undergone .sterilizat'on at 
12.5°C. Poy'-sterilization tests u.s'ng external power indi- 
cate no loss in performance in the entry- or the lander- 
capsule power control unit, which l.as now undergone 
iwo sterilization cycles and one impact test. Entn'- and 
liin'' r-capsule batteries art now heing charged. 

On May 4, :*S, the lander capsule was dropped ^rom 
an altitud,' of 2-jO ft onto the dry lake bed at Goidrionc, 
California. The wp.sule impacted after reaching a ter- 
minal veliK-ity of ] 15 ft/s. The capsule then cycled 
through the nO!r.,.i;J mission profile with no anomalies. 
Subsequent .sy.stem tests indicated all siiisy stems were 
operational within design limits. .\t the conclusion of the 



34 



JPL SPACe PROGRAMS SUMMARY 37-SI. VOL. IK 



drop test, the battery was monitored and found to have 
an open-circuit voltage of 17.5 Y. TiTis voltage indicates 
the battery was operating within design limits and had 
not been discharged more than expected. For the nominal 
mission profile, no more than 50% of the total battery 
capacity would be used. 

A second drop test of the lander capsule was per- 
formed on May 28, 1968. The unit was dropped from an 
altitude of 250 ft onto a macadamized road to achieve a 
higher impact force than experienced on the Goldstone 
dry lake bed. The power subsystem performed all sched- 
uled functions, with no apparent loss in capability. 

E. Computer-Aided Circuit Analysis, D. J. Hopper 

1. Introduction 

The objective of this eflFort is to provide a generalized 
system of computer programs for analyzing electronic 
ciicuits. Computer programs are presently available to 
simulate circuits and to perform a steady-state, transient, 
or cyclic (AC) analysis on these simulated circuits 
depending upon which computer program is used. One 
of the advantages of being able to simulate a circuit is 
that an engineer can use components having "worst- 
case" values. Construction of an actual worst-oase bread- 
board in the laboratory is a very difiBcult, if not an 
impossible, task. The major difficulty lies in obtaining 
components that have worst-case properties. 

2. Simulation Problem 

The computer simulation is accomplished by describing 
the circuit to the computer in an engineeiing-oriented 
computer language. Most of the programs can work with 
the simpler elements of a circuit, i.e., resistors, capacitors, 
inductors, mutual inductance, and ideal diodes. The rest 
of the circuit components must be described using these 
basic elements. This is where the difficulty lies. For ex- 
ample, one of the most common circuit elements, the 
transistor, has a small-signal model, a large-signal mode), 
and a saturated model. The result obtained from the 
cow^Duter could be radically different from the expected 
resuK if the wrong model is used. The modeling of a 
transformer is another compk. problem. In observing 
any magnetic induction versus field intensity (B-H) loop 
for magnetic materials, it is evident that B is a complex 
function of H, also the loop is dependent upon frequency. 

3. Survey of Existing Computer Programs 

A survey of the existing computer piograms shows 
the number of programs available is extremely large, but 



most of the programs were written to solve specific prob- 
lems instead of being general analysis programs. Several 
programs were studied, and it was found that a few 
programs could satisfy the total requirements. 

With SCEPTRE, a program developed by IBM, one 
can perform both transient and steady-state analysis. 
Another feature of SCEPTRE is that it has a component- 
model library tape. Once a model has been derived, it 
can be stored on U.e hbrary tape and used repeatedly. 
A copy of SCEPTRE was obtained on tape, and several 
sample circuits were analyzed. A few problems were 
uncovered, but the recent runs on SCEPTRE have been 
satisfactory. 

ECAP is another useful prograih. With this program 
one can perform AC analysis. Unfortunately, it does not 
have the library tape feature possessed by SCEPTRE. 
This is an inconvenience, but the AC analysis feature is 
well worth the extra work involved. 

4. Model Development 

During this reporting period, several semiconductor 
models have been developed for transistors and diodes, 
but the major effort has been to develop a model of a 
transformer. Two methods of core modeling are cur- 
rently under investigation. A piecewise linear model was 
obtained from IBM for use with SCEPTRE. It allows 
the entry of coordinate points corresponding to the 
magnetic induction versus field intensity {B-H) loop and 
provides a means of computing and storing values of B 
obtained as the transient solution proceeds. Operation 
within the B-H loop is simulated by taking the last value 
of 6 and a slope corresponding to that of the elastic 
region. Operation is otherwise constrained to points on 
the B-H loop. The other method being studied is the use 
of an exponential model. This model relates B and H 
with the use of exponential functions. The major difficulty 
with this model is that it is hard to simulate hysteresis. 



F. Electric Propulsion Power Conditioning, 

£. N Cosfogue 

1. Introduction 

The electric propulsion power conditioning project has 
two principal tasks. The first task, which is scheduled 
for completion in the early part of 1969, is to test a power 
conditioning unit with two ion engines. A switching mod- 
ule will be utilized to switch power to the engines by 
command. The second task, which is scheduled for com- 
pletion in 1970, is to design, fabricate and test the 



JPL SPACE PROGRAMS SUMMARY 37.5*, VOL. »f 



35 



complete power conditioning portion of an electric pro- 
pulsion system. " 

The power conditioning unit will consist of (1) four 
or five units that will power five ion engines, (2) a 
switching unit that will switch power conditioning units 
to available ion engines as required, and (3) a maximum 
power point seeker unit that will examine the solar panel 
characteristics and verify the available maximum power 
of the source. Item (1), the power conditioning units, will 
be developed under contract. Items (2) and (3) will be 
developed at J PL. 

2. Task 1 Power Conditioning Unit 

Power conditioning hardware built for the SERT II 
program will be modified for the first task. The modified 
units are scheduled to be received from the contractor 
by November 1968. The power switching unit for the 
first li^sk, which will switch power from one ion engine 
to the other by command, has been designed, and fabri- 
cation of the unit will be completed by August 1968. 

The block diagram of the power switching unit is 
shown in Fig. 7. The major blocks of the unit are (1) the 
switch-position sense-logic circuit, (2) the switch driver, 
or stepper, and (3) the switch. The switch-position sense- 
logic circuit accepts th>? command for switching to the 
position requested and compares the position requesfed 
to the pre3ent position of the switch. When the signal 
received i.i satisfactoiy, indicating that the switch can 
move to the ne.xt position, the sense logic issues a signal 
drive to the switch driver. After the switch has moved to 



POWER 
CONOITtONING 



POWER 

conditioning 
"on" inhibit 

SWITCH Jbo-IhJ swncH 

POSITION |co-4* POSnON 
POWER I 

CONDITIONS Id 

TO 3E 
SWITCHED 



SWITCHING 
COMPLETE 
INDICATOR 



'jDo-^* LOGIC 



SE^JSE 



28 Vdc 8A 



POWER "ON" I 

COMMAND ^"^ 

no Vac 60 Hz 



T 



SWITCH 
STEPPER 



1 



T 



!♦ SWITCH -tr^NGINEl 



-|-»»-ENGINE 2 



POWER 
SUPPLY 



_J 



the position requested, a signal is generated to indicate the 
completion of the switching. 

The switch driver (stepper) receives (1) the input for 
the switch-position sense-logic circuit, (2) verification 
that the power conditioner is functioning, ai.d (3) verifi- 
cation that the switch is ready to switch. When all the 
signals received are satisfactory, the driver circuit gener- 
ates the drive to move the switch. The switch is a heavy- 
duty, multiple-deck unit with high breakdown voltage. 

3. Task 2 Design Studies 

A study has been initiated to evaluate the merits of 
switching ion engines to power conditioning units versus 
providing a power conditioning unit per engine. The 
study will evaluate the reliability, weight, and cost of one 
system over the other. 

Another study has been initiated to estabUsh an effi- 
cient and acceptable method of determining the maxi- 
mum power point of the solar panel source. The study 
will recommend (1) a design that will ensure safe oper- 
ation of the engines throughout the mission, and (2) a 
means of identifying the available maximum power out- 
put of the panel. 

G. Mars Spacecraft Power System Development, 

H. M. Wick 

1. Introduction 

A two-phase study was initiated to design an improved 
Mariner spacecraft power system for possible future 
Mars missions. The latest system design techniques and 
component technology are being employed to develop 
optimum power systems for both Mars orbiter and flyby 
spacecraft. 

In Phase I, General Electric Missile and Space Division 
and TRW Systems were selected to investigate and 
analyze various baseline power system configurations. 
In Phase T^. (FY 1969). JPL will select the best power 
system design and award a contract for the detail design 
and construction of a power system feasibility model. 

2. General Electric Missile and Space Division 

A contract' was awarded to the General Electric 
Missile and Space Division on Januaiy 28, 1Q68, for the 
Mars spacecraft power system development program. 
A detailed analysis of the load power profile and its efiPect 



Fig. 7. Power switching unit block diagram 



'JPL Contract 9521,50. 



36 



JPL SPACE PROGMMS SUMMARY 37-51, VOL. Ill 



on poMrer system sizing was perfonned: a partial-shunt 
regolatton system was selected for analysis. The sdar 
anay/partial-shunt system integration investigation and 
the battery/battery charger interface study are continuing. 

A distributton frequency optimization study indicated 
that a change from the presently used frequency of 2.4 
kHz would not provide su£Bcient weight savings to war- 
rant its amsideration. 

Reliability sensitivity studies indicated that fault- 
sensicg and switchover to redui^ant devices should be 
considered onlv if dieir net reliability is equal to or 
greater than Jte reliability of the functions being {nto- 
tected. No distinct reliability advantage was determined 
for &ult-sensing the regulatcHr and inverter separately or 
as a pair. Fault criteria vere identified for the principal 
power-conditi(Miing units. 

Power system reliability modeling was perfonned oa 
the Mariner Man 1969 system. A similar model is being 
pn^rammed for the shunt system and a reliability 
comparison will be completed. 

3. TtW Syttcm* 

TRW Systems began their investi^tion and analysis 
eftnt* for the Mars spacecraft power system develop- 



'JFLCcHrtract 952151. 



ment program on March 4, 1968. Mission and spacecraft 
requirements were reviewed and load power profile and 
I>ower distribution/control requirements defined. 

Five power system coi^gurations were selected and 
are to be subjected to further detailed analysis to 
determine die optimum system. For these selections, a 
computer program was used to examine 70 baseline 
configuratiims. The selection criteria included weij^t and 
reliability assessments, maximization <rf solar array power 
margin, and minimum bus voltage excursion. 



H. PlaiMlary Solar Amiy D«v*lopnMnt, 

W. A. Hotboch 

1. bUroducHon 

A report of the objectives and environmental design 
considerati(»s wei% reptwted in SPS ?7-49, Vol. Ill, 
pp. 112-114. Effcnrts to date have been in the ctmceptual 
design and analysis of three feasibility models capable of 
producing not less tiian 200 W of electrical power (nt 
die Martian surfaice (Refe. 1-3). Trade-off studies of wei^t 
and structural integrity versus eqmsure to the Martian 
environment have been conducted. Selectkm d materials, 
mechanisms, and solar ceO panel conjurations has 
confirmed three approaches tfiat have the potential of 
meeting tiie goals of the program. The characteristics of 
each solar orray syst<%m are summarized in Table 1. 



Tobl* 1 . CharactorisHcs of Hi* toUu onuf syttwns 



, — 


S«l»iio» 












TyiMOf can 


O.OIOin.lhkfc.2 X 2 OR 


0.010 ir. Ihidi, 2 X 2 ca 


0.010in.lWcii,2 X2aii 


C*« ovtpwl, <rt 485 aV. aW 


5S 


sa 


53 




307«0 


14,400 


— 


Hmht of (wb/dfToy 


— 


— 


10300 




2 


— 


— 




— 


3 


2 




— 


— 


2 






— 


— 


Oriwitolion itod* 













wiin MNOi ccn mmi svnsors 


wMi aolar call mni Mmon 












33X)7 


1M3 


20.94 




33.37 


10.W 


3.24 


Total 


56.44 


30.) • 


2y.it 



m »ACi ptooitAMS sumMAitr 37.51, vol. in 



37 



jSjte =¥-'i?i(* '■'■-■- 



TRANSMiTTlNG ANTENNA ASSEMBLY 



SOLAR ARRAY HALF- SECTION 
IN HALF-DEPLOYED CONDITION 



TRUE CENTER 
OF GRAVITY 




SOLAR ARRAY HALF- 
SECTION IN FULLY 
DEPLOYED CONDITION 



Fig. 8. Conical nontracking array 



2. Nontracking, Oopleyabio, Conical Solar Array 

TIm objective of the coniral nontracking array (Fig. 8) 
is to produce power win a minimum of deployment 
medianisms. Tbe goal is to avoid complex medianisms 
fm latitude, slqpe, and position corrections and eliminate 
tibe need for a continuous tracking capability. This sys- 
t«n, once released from its locked, launched, aitd flight 
positions, will not require power from tfie lander system 
for deployment or continuous operation for the mission 
life (rf 1 yr. 

As recognized initially in its conception, this array will 
not meet the desired goal of 20 W/lb (1 AU) and under 
'ATOfst-case omditicnis will be under die minimum power 
requirements of 200 W of electrical power at solar no(m. 
In l3ae majority of the cases, the power output exceeds 
die minimum requirement of 200 W. The minimum 
power output at the worst-case condition of 46 mW/cm* 
(sumnwr) is 5% low or 190 W, while the best-case 
condition is 35% high or 256 W. The average noon power 
ou^t <rf dw limiting conditions is 17% high (^3 W). 
At U^^ber solar inteositfes occurring in tiw spring and 
fall teasom, the power level is above 200 W for all 
conditimis. 

The power -to-wei^t r&tio varies with die power out- 
put <rf the array at noon at a specific Martian location. 



llie specific power output is based on the equivalent 
power at 1 AU. Taking the power output at die wwst- 
case condition of 46 mW/cm* and converting to 1 AU 
by the ratio of 46/140 mW/cm» = 0.328, 

256W/0.328 = 780W 
190W/0.328 = 580W 

Thus, the specific power would lie between die range of 

780 W/56.44 lb = 13.8 W/lb 
580 W/56.44 lb = 10.3 W/lb 

3. Two Solar ^anolt Having Sun Tracking CapabiliNot 

The objective of tbe two-panel-oriented solar array 
(Fig. 9) is to provide a diree-axis tracking capability. 
In this design, die solar panels are mounted on opposite 
sides of the spacecraft so that the other two spacecraft 
sides are always unobstructed, and there is no interfer- 
ence with die vehicle antenna system. The design is a 
trade-off against the antenna shadow problem in whk^ 
die total array was sized at 10 circuits over the minimum 
of 30 circuits required. 

This system will meet the desired goal of 20 W/lb at 
1 AU aiMi exceed the minimum powor of 200 W at solar 
noon fat worst-case ocmditimis. Hie po^/er output wiB 
vary, depending on die number of circuits diat may 
possibly be shadowed at nocm. For die kmest solar 



M 



jn SMCf ^iOOIlAMS WHmAltY 37-51,, VOL. Ill 



TRANSMITTING ANTENNA ASSEMBLY 




Fig. 9. Two solar panels having sun tracking capabilities 



intensity (46 mW/cm^) occurring at the first day of 
Maitian summer, the power output limits are: 

Maximum shadow (30 circuits) = 205.6 W 
No ^dow (40 circuits) = Z74.2 W 

For the higher solar intensities occurring in the spring 
and fall seasons, die power levels range from 234.4 to 
312.5 W, considerably over the minimum requirement. 

The power-to-weight ratio, ba^d on 1 AU, varies vriA 
the power output of the panels as a function of shadow- 
ing. Taking the lowest output condition of noon at the 
sununer solstice with a solar intensity on the Martian 
siuface of 46 mW/cm" and converting to 1 AU by the 
ratio of 46/140 mW/cm^ = 0.328, 

205.6 W/0.328 = 626 W 
274.2W/0.328 = 837W 

Thus, die specific power at 1 AU would be 

626 W/30.185 lb = 20.8 W/lb 
837 W/30.185 lb = 28.6 W/lb 



4. Solar Panel and Integrated Antenna System 

The objective of the single-panel-oriented solar array 
(Fig. 10) is to provide a three-axis tracking capability. 
The deployment of the solar panel and antenna on a 



vertical boom eliminates the possibility of shadowing 
from the spacecraft body and antenna. This allows the 
minimum number of circuits (30) to be used to achieve 
the required power output of 200 W under worst-case 
conditions. 

Combining the antenna and solar array mounting 
presents a problem in maintaining the point accuracy of 
the antenna when the system is buffeted by wind gusts. 
The vertical boom has been sized to minimize the 
deflection due to wind loads; however, other factors are 
present. The drive mechanisms will have to be designed 
to eliminate, as much as possible, any backlash in the 
gearing, and the latching mechanism of the vertical boom 
will have to be of a self-tightening design. Other factors, 
such as the stability nf the spacecraft body and legs and 
the soil condition of the vehicle landing area, will affect he 
antenna point accuracy, but these are unanswerable at 
the present time. 

Tlie single-panel-oriented array of 30 circuits will meet 
the desired goal of 20 W/lb at 1 AU, and exceed the 
minimum powe^ requirement of 200 W at solar noon for 
worstcase seasonal conditions. The power outputs for the 
limiting seasoiial conditions are: 

Summer noon = 205.6 W 
Spiing/fall noon = 234.4 W 



jn SMCf PftOGKAMS 'MMMARY 37-51, VOL. Ill 



39 



B 



ANTENNA ASSEMBLY AND SOLAR ARRAY 
ROTATED 90 d«g TO SHOW KULL 
FRONT VIEW 



SOLAR PANEL 




Fig. 10. Solar panel and integrated antenna system 



The power to weight ratio, based on 1 AU, for the 
output condition at noon at the sununer solstice with a 
solar intensity on the Martian surface of 46 mW/cm" is 
22.2 W/lb. 



5. Solar Cell Covering 

A solar cell power supply operating on the Martian 
surface has a problem that is unique and not found in 
space applicatious. Mars has a dusl condition that is 
considered severe; the dust is assumed to be iron oxide 
and electrically conductive. The electrical shorting caused 



by the dust would be catastrophic to the solar array. 
One solution that was considered was to coat all elec- 
trically exposed areas of the solar cell circuit with filter 
adhesive. However, this method is tedious, virtually 
impossible to guarantee complete protection, and will 
add considerable weight to be solar array. 

A second consideration was to eliminate the cover glass 
and use a semiorganic resin.^ This c'>ating has been 



'Developed by B. Mark;:, I.ockheed Missiles and Space Po., Palo Alto, 
Calif. 



40 



jn SPACE PIIOGRAMS SUMMARY 37-51, VOL ill 



developed specifically as a solar cell coating and can be 
applied by spraying. Principal drawback to this coating 
technique is the inability to completely insulate all 
electrically conducting suriFaces. Sprt.)' application of the 
coating would not insulate the back side cf the connector 
tab, and dipping the total array is highly unpractical. 

The selected method for insulating the solar array is 
by encapsulating the cells with a continuous sheet of 
Tedlar fibn.* Tedlar film is essentially transparent to, and 
unaffected by, solar radiation in the near ultraviolet, 
visible, and near infrared regions of the spectrum. 

The advantages of tlie film coating for the solar cell 
circuit are: 

(1) Total insulation is obtained of all electrical con- 
ducting surfaces. 

(2) Electrical loss is low due to coating. 

(3) Installation is easily accomplished using a space- 
proven aclhesive system. 

(4) Finished coating eliminates gaps between cells, 
which would form a trap for Just accumulation. 

(5) The coating protects the cells from low-energy 
proton radiation, as the cells will have no exposed 
areas common to typically filtered cells. 

(6) The film has little weight. 

(7) The flexible film, bonded with a resilient adhesive, 
should have better abrasion resistance to the "sand- 
blasting effect" of the dust than the hard surface 
of glass or quartz filters. 

References 

1. Quarterly Report 7254-Q-l, Electro-Optical Systems, Inc., Pasa- 
dena, Calif., Oct. 13, 1967. 

2. Quarterly Report 7254-Q-2, Electro-Optical Systems, Inc., Pasa- 
dena, Calif., Jan. 6, 1968. 

3. Quarterly Report 7254-Q-3, Vols. I, II, and III, Electro-Optical 
Systems, Inc., Pasadena, Calif., Apr. 15. 1968. 

I. Thermionic Research and Development, 

O. S. Mtrrin 

1. Introduction 

A program to improve the output performance of 
cesium-vapor thermionic converters has been in progress 
for several years. The work reported in this article is a 
continuation of this program and was performed (Ref. 1) 



under contract to NASA, but with JPL technical direc- 
tion, by Electro-Optical Systems, Inc. The first of th:ee 
tasks of this effort was reported in SPS 37-50, Vol. Ill, 
pp. 82-92. The second and third tasks are reported here. 

2. Variable-Spacing Test Vehicle* 

a. Desjgt.. Two variable-spacing test vehicles of the 
same basic design but incorporating different sets of 
electrode materials were fabricated. The test vehicles, 
drive mechanism, and supporting structure are of the 
same design as reported in SPS 37-39, Vol. IV, pp. 15-19. 
The first test vehicle had a polycrystalline rhenium emit- 
ter and a polycrystalline molybdenum collector; the sec- 
ond had a vapor-deposited rhenium emitter and collector. 

b. Teat reauUs, Data typical of that taken in this project 
is shovm in Fig. 11, where the voltage output at a con- 
stant current of 38 A is shown versus interelectrode 
spacing for both the rhenium-molybdenum and a previ- 
ously tested polycrystalline rhenium-rhenium test vehicle 



> 
< 

00 

I- 

0. 

K 

o 

UJ 

i 

o 

> 




'Polyvinylfluoride film manufactured by Film Dept., E. I. du Pont 
de Nemours and Co., Vernon, Calif. 



2 3 4 5 

INTERELECTRODE SPACING, mils 

Fig. 1 1 . Comparison of performance of rheniiim- 

riienium and rhenium-molybdenum 

variable-spacing test vehicles 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



41 



(SPS 37-39, Vol. IV). The quantitative difference between 
these electrode systems is of the order of 60 inV at the 
optimum spacing of 3 to 4 mils. This figure demon- 
strates the central thesis of ^Jiis program, namely, that 
increased thermionic converter performance results when 
a higher bare work fimction (lower cesiated work func- 
tion) collector is used. The bare work function of vapor- 
deposited rhenium at ITSS^C (2008''K) is 5.08 eV, as 
reported in SPS 37-50, Vol. Ill, while that of poly- 
crystallipe molybdenum is 4.2 eV. 

Extensive data were taken with both test vehicles. 
When operated at high temperatures, the performance 
of the rhenium-molybdenum test vehicle approached 
and eventually reached that of the rhenium-rhenium 
test vehicle and remained comparable thereafter at both 
high and low temperatures. This was attributed to the 
deposition of rhenium onto the molybdenum collector, 
thus essentially changing this vehicle to a rhenium- 
rhenium test vehicle; however, subsequent long term, 
low emitter temperature operation of the device resulted 
in a return to the rhenium-molybdenum performance. It 
is postulated that the deposited rhenium diffused into 
the molybdenum collector substrate, although the device 
has not been disassembled and metallurgical tests per- 
formed to determine if this is indeed the explanation. 
To change the rhenium-molybdenum vehicle into a 
rhenium-rhenium vehicle, and to maintain it as such 
over uesurable test periods, *he rhenium emitter was 
periodically operated at temperatures as high as 2200° C 
for « h (with cesium reservoir heater turned off) to 




ensure a sufficient and stable rhenium coverage on the 
molybdenum (x)Uector (believed to be at least two or 
tliree monolayers thick). Data from this vehicle match 
those from the rhenium-rhenium test vehicle to 
\vithin 2%. 

Some of the most significant and useful data obtained 
from the test vehicles are shown in Fig. 12, showing, for 
the rhenium-rhenium electrode system, voltage output 
versus interelectrode spacing for a constant load cur- 
rent, constant emitter temperature, constant collector 
temperature, and constant cesium vapor pressure (as 
indicated by constant cesium reservoir temperature). 
Whcu the current and temperatures are held constant 
to within experimental error and when the spacing be- 
tween electrodes is precisely determined, the product 
of the cesium pressure p and the interelectrode-spacing d 
(i.e., pd) at the point of optimum voltage (and power) 
output is observed to be constant at a value of 16.0 ±0.8 
mil-torr (Table 2). It can be noted from the curves and 
the tabulated data that the pd product is independent of 
emitter temperature. The optimum pd product al<io 
appears to be independent of the collector and emitter 
materials. It is further observed that at the lower emitter 
temperat'TTfcs the optimum voltage for a given current is 
lower, pnd the interelectrode spacing is considerably 
larger and less critical; i.e., the optimum voltage is less 
sensitve to variations in the spacing. 

The performance testing of the vapor-deposited 
rhenium-rhenium test vehicle was not successful due to a 
leak in one of the electron-beam welded flange joints. 
This was discovered after about 120 h of operation. 

TabI* 2. Summary of pr*ttur*-distanc« data taken 

from lnt«r«l«etrodt spacing vtrtus voltag* 

output curvot 



INTERELECTRODE SPACING, milt 

Fig. 1 2. N«ar-optimii«d voltagt output vc intoroltetrodo- 

spaclng for a rlianium-rhtnium oltctrodo syttom 

at various constant eporating conditions 



ImMMt 

•e 


CniwH 
"C 


CmImiu 
pmvm 


ipMlnt 


MU-Mtr 


1327 


2S9 


1.33 


12.5 


16.6 


1427 


291 


1.43 


11.0 


157 


U27 


303 


KM 


8.0 


157 


)527 


310 


2.35 


7.1 


16.a 


1527 


320 


3.01 


5.3 


15.9 


1527 


331 


4.02 


3.9 


157 


U27 


331 


4.02 


3.9 


157 


1 735 


331 


4.02 


3.9 


157 


1735 


344 


5.30 


3.0 


15.9 


1735 


350 


6J}i 


2.7 


U.3 



42 



JH SMCe nOOKAMS SUfAMAMY 97-51, VOL III 



By comparing data from this vehicle with those from the 
previous ^'^st vehicles, its perfonnance was observed to 
be {nconsistent and considerably lower. A second set of 
parts has since been assembled into another test vehicle" 
and early tests on the second unit show it to be perform- 
ing satisfactorily. Besults of the tests will be reported in 
a future article of SPS, Vol. III. 

3. Pix«d<Spacln9 Vaper-D«petlttd Rhtnium Cenv*rt«r< 

a. Oengn. Two thermionic converters of planar geom- 
etry employing vapor-deposited rhenium electrodes were 
fabricated. The two converters, designated SN-109 and 
SN-110, are identical to the SN-101 series converters 
(SPS 37-39, Vol. IV). The desigu criteria for these con- 
verters were to have bee.- based on the vapor-deposited 
rhenium-rheniimi test vehicle data. Since those data were 
lacking, the design criteria were chosen based on data 
from the polycrystalline rhenium-rhenium test vehicle. 

b, Tett remit*. The test data from the polycrystalline 
rhenium-rhenium test vehicle and &e data for the cor- 
responding fixed-spacing converters SN-109 and SN-110 
are compared in Table 3 and in the design temperature 
curves of Fig. 13. The agreement is very close and sug- 
gests that the electrode systems of vapor-deposited and 
ix>lycrystalline rhenium yield nearly equivalent ther- 
mionic performance. 

Converters SN-109 and SN-110 were also tested at a 
higher emitter temperature for additional comparison to 



•Under JPL Contract 952217. 

Tabic 3. Comparison of porformanco of cenvortors 

SN>109 and SN-IIO with polycrysfallino rhtniwm- 

rhonlum variablo-spacing tott vohicU data 

(of convtrtor design point) 



P-minitttr 


Tm>v«McI« 
SN-IO* 


SN-10* 
4Mi 


■wr vvnicw 
diMaM 
SN-IIO 


SN-IIO 
Ma 


Intwalaclred* 
•padng, mHi 


*±i 


6.2 


'0±J 


10.5 


Iwt.'C 


1325 


1524 


1427 


1423 


CmIwri raMTtoir 
<Mip«ralHr«, °C 


320 


321 


303 


302 


CeilMler MrfoM 


722 


720 


713 


706 


VohoB. o«»»l»«». V 


0.4 


0.4 


0.3 


0.3 


■•ITMt, A 


4S7 


45.2 


35.1 


35.1 



SN-IIO 

HIGHER TEMPERATURE 
5-»l7!5»C,J''38A 

;^'l735«C,i".60A" 




4 6 a 10 12 14 16 18 20 22 24 
INTERELECTRODE SPACING, mil* 

Fig. 13. Comparison of porformanco of fixod-spacing, 

vapor-dopotitod rhonium convortors SN-109 and 

SN-110 with polycnrttolllno rhonlum-rhonium 

voriablo-vpacing tost vohlcio 

polycrystalline rhenium-rhenium performance. Tlieir per- 
formance at' an emitter temperature of ITSS^C and at 
constant currents of 38 and 60 A (where optimum volt- 
ages were 0.8 and 0.7 V, respectively) was also found to 
be in excellent agreement with the variable-spicing test 
vehicle data for the same conditions. Their pf rformance 
was considerably o£F optimum, however, inasmuch as die 
optimum interelectro^ie spacing at this temperatiure is 
approximately 3.5 mils. The higher temperature cur/es 
of Fig. 13 also show this performance compaiisor. The 
interelectrode spacing is a few tenths of a mil larger at 
the higher temperature due to increased expansion of the 
emitter support sleeve and other converter components, 

4. Analysis of Tost Vohicio Data 

The primary objective of this task was to formulate a 
theoretical description of thermionic converter perform- 
ance and to correlate it with an analysis of die para- 
metric vehicle data The effort proceeded sequentially in 
three parts, each part covering one of the regions of 
parametric vehicle operation as defined by Fig. 14. 
Region I is the electror space charge region and extends 
from zero interelectrode spacing to the minimum voltage 
identified as the plasma onset point. Region II is the 
transtdon region and extends firom die plasma onset point 
to the (qptimum output point. Region III is the positive 
column region and exterds from the optimum output 
point to the rig^t margin ot the ttgure and beyond. 



Jn SMCC nOORAMS SUMMARY 97-51, VOL III 



43 



4 



3 
0. 
I- 
3 
O 

UJ 
(9 

3 

o 
> 



TgimiT ' I736»C 

/ • 38 A (CONSTANT) 

O COMPUTER SOLUTION 
FOR Re-Re SYSTEM 

DATA FROM Re-Re 
SYSTEM 




INTERELECTROOE SPACING, mile 

Fig. 14. Comparison of oxporimtntai rotults with 

computed roiults (rogion I otily) for a typical 

voltcgt output vs intoroloctrodo- 

spacing curvo 

This task was directed mainly toward the analysis of 
region I. The formulation of the problem and the analysis 
are given in Ref. 1, where the converter is viewed as a 
"double diode" described by Poisson's equation. A com- 
puter program was set up and solutions obtained. A 
comparison of the computer solution for region I and the 
test vehicle data is also shown in Fig. 14. The discrf ;)iincy 
in output voltage at low spacings is related to losses in 
lead resistance between the electrode surfaces and the 
point at which the potential was measured. 

Rofortnct 

1. Campbell, A. T., and Jacobson, D. L., Final Report, ThemUoPic 
Reieorcfc and Dev^opmant Program, NASA Confaract NA5 7-514, 
EOS Report 7118-Final Electro-Optical Systems, Im-.. Pasadeiw, 
Calif., Mar. 1. 1968. 



J. Thermionic Convtrtor D«v»lopmMif, P. Roukhv 

1. Introduction 

The development oi advanced technology thermionic 
converters is continuing at JPL. The series 9 planar 



converters, built by Thermo Electron Co., are s'Ul being 
used as test vehicles for technical improvements. The 
development of this type of converter was discussed 
previouiiy in SPS 37-48, Vol. IIL pp. 58-60. 

2. Convortor Dotigns and Tost Rosults 

Measurements performed on converter T-206 pointed 
out that any further improvement in power output was 
lii'.iited by the radiator geometry. This geometry, which 
was derived from the necessity to incorporate the con- 
verter into a 16-converter solar-heated generator, limited 
the cross-sectional area available for the collector- 
radiator heat flow and resulted in excessively hi^ col- 
lector temperatures. The advantages of the application 
of the heat pipe as a collector heat rejection medium 
weij presented in SPS 37-48, Vol III, pp. 60-ii3. 

Converter T-206 was assembled incorporating a 
niobium heat pipe as a collector-radiator structure 
(Fig. 15). The converter was assembled using a rheniimi 
emitter and a rhenium collector, the latter consisting of 
a sheet of rhenium vanadium-brazed to the niobiuci pipe. 
Prior to the assembly, the emitter surface was electro- 
etched and thermally stabilized in vacuum at approxi- 
mately 2050''C for 2 h. 

During the tests, it was observed that the performance 
of converter T-208 was inferior not only to that of 
thermal model T-3, which had a collector heat-pipe 
assembly, but also to that of T-206, which used a fiimed- 
type radiator. Both converters T-206 and T-208 used 
rhenium electrodes. The difference in the collector area 
(2.52 an' for T-206 versus 2.34 cm* for T-208) did not 
accoiujt for the performance reduction. However, the 
collector surface in converter T-208 was further reduced 
to a net electrode area of 2.16 cm' by a groove cut in the 
colleii^or for cesium vapor distribution and outclassing 
Tlie net ratio of collector areas of these converter was 
C.86, or a 14% smaller area for T-208. Figure 16 indicates 
by dashed line the performance of converter T-206 
reduced by 14% for comparison purposes. 

Examination of the results implied tliat the inter- 
electrode spacing in the two converters v^as different, 
being larger in the case of converter T-.208. Comparison 
of the ce.sium conduction was made from test data and 
the following empirical formula was used to calculate the 
interelectiode spa umg: 



[ 



0.0 001475A(r, - Yc) (T, -I- T,) 



T' _ 1 



006(r,+rc) 
p 



^PL SPACE PKOOPAMS SUMMAHY 37-51, VOL HI 









f: 



EMITTER 
TERMINAL 



EMITTER 
COLLECTOR 

CERAMIC SEAL 




COLLECTOR 
TERMINAL 



CAPILLARY 
STRUCTURE- 



-HEAT PIPE WALL 



-RADATIVE COATING 



CESIUM PIPE 




-FILL TUBE 



CESIUM RESERVOIR 



where 



Fig. 1 5. Converter with collector heat pipe 

d = interelectrode spacing, mils 
A = interelectrode area, cin^ 
Te = emitter temperature, °K 
Tc = collector temperature, °K 
^Q/^p = slope of cesium conduction curve 
p = preysure, torr 

JfL SPACE fROGRAMS SUMMARY 37-51, VOi. Ill 



70 



CO 



50 



40 



-? 



O 



30 



20 



» 





1 \ 










\ 












\\ 
\\ 
\\ 








2081 


v\ 

U206 
\\ 










\ \^ 










\ \ 


\\ 
\\ 
\\ 










NJ 


^ 



0.4 



L6 



2.0 



0.8 12 

VOLTAGE kb.V 
Fig. 16. Converter performance comparison 

The results of the calculations are presented in Table 4 
for various cesium pressures. These data indicate a 65% 
di£Ference in the interelectrode spacing between con- 
verters T-208 and T-206; the actual magnitude of the 
spacing should be larger because only the interelectrode 
areas were considered in the calculations, disregarding 
tne side effects. 

Table 4. Comparison of converters T-206 and T-208 
at various cesium pressures 





ConvMlw T-206 


Convwlar T-20S 






ol iiMHcotad praMM* 


tlocr 


12tafr 


SlOfT 


laiwr 


A.c-n' 


2.52 


2.52 


2.16 


2.16 


rt,°K 


19V0 


1990 


2000 


2000 


rc,°K 


861 


875 


880 


885 


nO/Hp, W/lofr 


1.60 


1.05 


0.90 


0.50 


d. mib 

d, mils (avaragal 


1.29 


1.38 


2.06 


2.33 


1.33 


2.20 



45 



-f 



The lower performance of converter T-208 was also 
tentatively related to an overheating of the collector 
surface. Although no direct measure of the collector 
surface temperatures could be obtained, the inability to 
reproduce the dynamic curves in steady state pointed to 
an overheated collector. It was tentatively attributed to 
an excessive restriction in the vapor channels in the heat 
pipe 4:t the heat receiving end near the collector. This 
could lead to an excessive temperature drop at the 
liquid-vapor interface and was estimated to be 60 to 
WC. Corrective measures have been taken in the 
assembly of converter T-2 



Converter T-207 was assembled using a rhenium emit- 
ter and a palladium-clad molybdenum collector. The 
configuration of converter T-207 was identical to that of 
converter T-206. This duplication was done to facilitate 
the comparison of experimental data and evaluate the 
performance of the palladium as a collector material. 
Tests were performed at emitter temperatures of 1800, 
1900, and 2000°C. The converter configuration and the 
use of a finned radiator again did not allow proper col- 
lector cooling. The cesium conduction data corresponded 
to an interelectrode spacing of 2.54 mils. Some un- 
certainty exists as to the exact comparison between 
emitter surface temperatures of the two converters, due 
to a possible influence of the electron bombardment 
filament shape and to variations in the location of the 
hohlraum. An approximate 13% difFerence in the required 
power input, for otherwise similar test conditions, was 
observed between the two converters. This would cor- 
respond to a possible difference in emitter surface 
temperature of approximately 80° C. 

The analysis of the test data indicated that the ap- 
parent cesiated work function of the palladium used as 
collector material in converter T-207 was higher by 
0.037 eV than that of the rhenium utilized in converter 
T-206. This corresponds to a reduction in output current 
of between 3.9 and 6.1 A or a voltage shift between 
0.030 and 0.044 V, with the current-voltage character- 
istics of converter T-207 to a lower output voltage 
(Fig. 17). 

3. Generator Design 

Because of the necessity of using converters with heat 
pipe collector-radiators, the original design of the multi- 
converter generator had to be modified. The new as- 
sumptions for the generator design are a converter output 
of 70 A at 0.80 V and 28 A at 1.0 V, corresponding to a 
maximum power point power density of 20 W/cm*. Two 
types of heating systems were considered for terrestrial 



50 



z 

u 30 

C 

3 















^, 










XI 


4'' 


\y 


^ 






08V 


-\y/ 


<^A 


f y^ 


_,-" 










4'A 


^fj^* 


Liov 








A^ 




1 '^ 


206 ( 

207 ( 


Pd) 






J" 


r 











100 ISO 200 ZSO 300 KO 400 4$0 SOO 

BOMBARDMENT POWER, W 

Fig 17. Converter work function comparison 



tests: solar, using an 11.5-ft-d:am mirror capable of a 
6770-W thermal input into a 1.60-in. cavity aperture, 
and electron-bombardment heating. A study was made 
for comparison purposes using a 9.5-ft-diam mirror for 
cislunar application. 

Parametric studies of optimum generator performance 
with varying converter complements as a function of 

Table 5. Predicted generator performance data using 
solar and electro-bombardment heating systems 







healing (yslem 


Ground 


Citluirar 








(1I.5-II 
diam 


(9.S-fl 


Ca$e 1 


Cose 2 




mirror) 


mirror) 






Cavity aperture 


1.61 


1.33 


1.33 


1.33 


diameter, in. 










Covity input, W 


6770 


7800 


9070 


8680 


Available converter 


255 


315 


394 


380 


input, W 










Converter output 


31.0 


54.5 


87.0 


81.2 


current, A 










Converter output 


0.90 


0.87 


07^0 


OJ'4 


voltage, V 










Converter output 


30 3 


47.5 


61.0 


60.0 


power, W 










Generator output 


485 


760 


975 


960 


povrer, W 










Abwrber-generator 


7.2 


9.7 


_ 





efficiency, % 










Generator effi- 








10.7 


10.8 


ciency, % 











46 



ifl SPACE PROGRAMS SUMMARV 37-51, VOL. \\\ 



cavity aperture diameter indicated the desirability of 
selecting a 16-converter configuration composed of axial 
rows of 8 radially mounted converters. Calculations of the 
individual converter total power input requirements, in- 
cluding emitter support conduction, electrical output, 
cesium ccduction, interelectrode re-radiation losses, 
etc., were performed. These calculations lead to the 
predicted generator performance shown in Table 5. 



In the case of the ground tests using the 11.5-ft-diam 
mirror, the following assumptions were made: incident 
flux 90 W/ft*, reflectivity 88%, shadow factor loss due to 
the vacuum housing of the generator, 5%, and window 
loss 11%. For the 9.5-ft-diam mirror for cislunar appli- 
cation, an incident flux of 130 W/ft* was assiuned, with 
a mirror reflectivity of 89% and a generator support 
shadow factor loss of 2%. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



47 



N68 



-3'?i^;^ 



V. Guidance and Control Analysis and Integration 



GUIDANCE AND CONTROL DIVISION 



A. Automation of Variational Techniques for 
the Solution of Optimum Control Problems, 

H. Mack, Jr. 

1. Introduction 

Computer programs have been written to completely 
automate the solution of optimal control problems where 
the computation scheme assumes small scale variations 
about a nominal solution. This automation of variational 
techniques will enable the user to solve small as well as 
large scale optimal control problems with a minimum 
amount of programming for each specific problem. Since 
all of the variational equations are derived by a com- 
puter and compiled by Fortran IV with no intervening 
human action, the most time-consuming part as well as 
the greatest source of errors in the solution of variational 
problems has been eliminated. 

2. Description of DEVNEC and QUASI Programs 

The most widely used variational techniques are auto- 
mated by two separate programs. The first program is 
called DEVNEC and is written in the IBM FORMAC 
language, which is currently available on the IBM 7094 
as an extension of the Fortran IV compiler. DEVNEC 
uses the system equations and the boundary conditions 
as inputs to derive all of the necessary conditions for an 



optimum solution by use of the maximum principle. The 
maximum principle is automated in DEVNEC because it 
is one of the best methods for obtaining the solution 
to two-boundary-value problems that result from the 
formulation of the optimal control problem. This method 
has a significant advantage over the classical calculus of 
variations method and the dynamic programming method 
in that the maximum principle can be applied to prob- 
lems where the control is constrained. The second pro- 
gram is called QUASI and automates a generalized 
Newton-Raphson method for the solution of two-point 
boundary-value problems. QUASI uses the variational 
equation derived by DEVNEC to obtain a numerical 
solution to the optimum control problem. 

The flow chart in Fig. 1 shows the functions of each 
program in obtaining a solution to the optimal control 
problem. The process starts with the input of the system 
equations 

x = F(x,u.f) 

and the boandary conditions at the initial and final times 
(to and tf) 

D{x,n,to) = 
G (x, u, t,) = 



48 



JPL SPACE PROGHAMS SUMMARY 37-51, VOL. Ill 



I 



«>. 



1 



rSTARTJ 

SYSTEM 
EQUATIONS 
AND BOUNDARY 
CONDITIONS I 



DEVNEC 
(FORMAC-7094) 



z 



SYSTEM 
(FORTRAN IV) 



ALL NECESSARY 
CONDITIONS AND 
SYSTEM EQUATIONS 



LIST OF ALL 
NECESSARY 
CONDITIONS 



FORTRAN IV 
COMPILER 



SUBROUTINE 
SYSTEM 



b 



QUASI 



USERS 
PROGRAMS 



LIST 

SOLUTION 

1 



INITIAL 
CONDITIONS 
AND 

PROCEDURES 
FOR "QUASI" 



f STOP ^ 

Fig. 1. Computer program flow chart showing functions 
performed in solving optimal control problems 

into the DEVNEC program. The numerical value for the 
dimensions of the state and control vectors (x, u) and 
the boundary conditions are also inputs to DEVNEC; 
these dimensions are as follows: 

dim (x) = n 
dim (u) = m 
dim (D) < n 
dim (G) < n 

The quantity or performance index that is to be maxi- 
mized or minimized is 

7(U)= <C,-K{t,)> 



Using these inputs and the mechanization of the maxi- 
mum principle, DEVNEC computes the adjoint system 
of equations 



X = 



-^<X,F(x,u,0> 



where X is the adjoint vector, and then computes all par- 
tials of the state and adjoint equations, the Hamiltonian, 
and the boundary conditions. The state and adjoint 
equations and partials are output as equations in a sub- 
routine that may be compiled directly by a Fortran IV 
compiler. 

The DEVNEC output equations are compiled in a sub- 
routine called SYSTEM. The SYSTEM subroutine is 
called by QUASI when the optimal control problem 
is solved numerically. QUASI is a mechanization of the 
quasilinearizatioi. or Newton-Raphson technique for 
solving two-point boundary-value problems. This tech- 
nique solves the state and adjoint equations, which are 
usually nonlinear, by solving a sequence of linearized 
state and costate equations. The boundary, transversality, 
and optimality conditions are satisfied by constraining 
their variations to be zero. The quasilinearization pro- 
cess is initiated by a call statement in the user's pro- 
gram that contains an initial approximation to boundary 
conditions and some logic variables that specify optional 
procedures to be taken by QUASI. 

3. Applications and Results 

The programs DEVNEC and QUASI have been 
checked by applying them to the solution of several 
optimal control problems where the dimensions of the 
systems have varied from 4 to 10. The computer object 
run time for DEVNEC varied from 1.5 min for the 
4-dimensional case ^o 5 min for the 10-dimensional case. 
The run time for QUASI was dependent on the linearity 
of the system equations and some of the options offered 
by QUASI. The run time for the 4-dimensional case 
was 25 s and for the 10-dimensional case it was 10 min. 



where c is a constant vector. The angular brackets < > 
denote the inner-product operation, where 



<c, x>=Y, 



CiXi 



and d and Xi are elements of the c and x vectors, re- 
spectively. 



The mechanization is being tested on trajectory prob- 
lems of dimensions higher than 10, where QUASI at- 
tempts automatically to make trade-offs between run 
time and the amount of storage required. Procedures are 
also included so that QUASI will change its procedure 
automatically for solving a new optimum trajectory. 
These procedures will drastically reduce the necessary 
computations as the solution converges. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



49 



B. Optica! Approach-Guidance Right Feasibility 
Demonstration, T. C. Duxbury 

1. Introduction 

Studies indic.itf th;it improvement can be obtained in 
a Mars-encounter c.irth-bascd orbit estimate if infor- 
mation defining the direction from the spacecraft to 
Mars to an accuracy better than 1 mrad ( 1 a) is included 
in the estimate. An optical approach-j;uidanec flight 
feasibihty demonstration (SPS 39-42, Vol. IV, pp. 4&-49) 
on the Mariner Mars 1969 mission was to use an on- 
board planet tracker. However, with the deletion of the 
planet tracker from the mis.sion (as a result of budget 
constraints), a study was initiated to evaluate existing 
on-board sources of optical data that could be used for 
orbit determination. 

2. Planet Tracker 

The planet tracker was to give pointing angles t<j 
Mars during the 10 days before encounter. These angles 
along with spacecraft attitude information were to be 



ground-processed to produce a spacecraft trajectory 
estimate in near-real time. 

A planet tracker was built, and data interfaces were 
established between the telemetry data stream in the 
Space Flight Operations Facilities and the optical data 
pre-processing software in the spacecraft performance 
analysis and command area. A draft description of a 
computer program for implementing each data interface 
was vcritten. Equations were derived and documented 
that related the planet tracker measurements to the 
spacecraft trajectory parameters and to measurement 
errors. Orbit determination accuracy studies were per- 
formed to define mea.surement sy.stem accuracy requiie- 
ments, A computer program simulating on-board system 
measurements was developed thrf'Ugh a contracted effort 
(SPS .37-50, Vol. HI, p. 1(M) to test and evaluate the 
ground processing software. 

3. Alternate Sources of Optical Data 

These sources include the fa r-t;n counter planet sensor, 
the scan platform, the attitude-control celestial sensors, 



V 

ill 




1 



J 



liKHES 

^y*^. ''■ 

Fig. 2. Two TV calibration target! ilmulotinQ Man and eight itctri ; (a) 25-d«{i angin, (b) 75-deg onglo 



90 



4n SPACE PfOGMMS SUMMAItY 37-51. VOL. Itt 



and the TV system. Sufficient accuracy can be obtained 
using data accumulated over a 24-h pen'od. 

Studies have also shown the usefulness ot TV data in 
the orbit determination process when Mars is in a TV 
frame. The value of the TV data is greatly increased if 
a star is visible in the TV frame al ng with Mars. A 
star {i Serpentis) of 3.64 magnitude has a high proba- 
bility of being in the TV field-of-view for trajectories 
having a launch data before March and an arrival date 
between July 31 and August 15. The TV was not spe- 
cifically designed to photograph stars; therefore, it may 
not have the capability of detecting stars as dim as 
i Serpentis. To aid in determining if this capability exists, 
two test targets (Fig. 2) simulating Mars and eight stars 
ranging in magnitude from 1 to 5 will be included in the 
TV calibration schedule. The large hole in each test 
target simulates Mars, the cross-hair mtersection desig- 
nates the center of the large hole, and the eight apertures 
about the perimeter of the large hole simulate the 
stars. The geometric relationships between the simulated 
stars and planet have been measured to ascertain the 
accuracy with which the angle between a star and planet 
can be reconstructed from the TV data. 

Selection of the set of optical data sources that pro- 
vide the best orbit determination capability is under 
study; results will be reported in future articles o*^ the 
SPS, Vol. III. 



C. Development of Computer-Oriented 
Operational Support Equipment, J. p. Pcrrili 

1. Objectives 

The long-range objective is to develop the guidance 
and control operational support equipment (OSE) tech- 
nology to meet the requirements of possible future plan- 
etary missions. Within this objective, the near-term goal 
is to develop an "OSE unified approach" concept. This 
concept is to be applied to the three guidance and con- 
trol flight subsystems (electrical power, guidance and 
control, and central computer and sequencer) to provide 
an integrated approach to subsystem testing in the lab- 
oratory, manufacturing area, system test complex, and 
launch compLx. 

2. OSE Unified Approach 

This concept will specify the use of the same basic 
OSE in all test areas where a flight subsystem exists as 
an assembled entity. Adaptors, buffers, or additional 



cabling \-ill be added in areas where more test points 
are available and where more detailed tests are required. 

The basic control element in all test areas is a small, 
commercially available general-purpose computer. A 



LINE PRINTER 



PAPER TAPE 
STATION 




COMPUTER 



KEYBOARD AND 
CATHODE RAY TUBE 
GRAPHIC SYSTEM 



RANDOM -ACCESS 
DISK (CAPACITY 
> 500,000 bits) 



INTERFACE UNIT 



:juumj[iijiiuiii iiL U uuuul: 

/ I go o[ fo] looQool 1. 1 > \ \ 




^^^m^^\ 



o o 



o o 



o o 



o o 



O G O O 

o o 





Fig. 3. Proposed hardware configuration for flight 
project operational support equipment 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



51 



versatile man-machine interface is provided by a cathode 
ray tube graphic system, permitting instant input/output 
access to the user. A test language is provided that re- 
quires relatively little training and is based on user 
requirements rather than computer characteristics. The 
language has on-line response and presents the user 
with the capability of direct control access to the unit 
under test, willi automatic checking of responses. Alter- 
ations from mission to mission are expected to be more 
economical than the present method of reworking exist- 
ing OSF hardware. 

The hardware interface unit provides special-purpose 
logic, signal conditioning, and buffering between the 
unit under test and the general-purpose input/output 
channels of the computer. The interface unit is designed 
as a "sample-and-hold" device in both directions, with 
control reserved and maintained by the computer. 
Figure 3 shows the envisioned OSE hardware that would 
form the computer-oriented test system. 

The software functions as an interpreter between the 
test engineer and the unit under test. A primary func- 
tion of the interpreter is to take a user-defined engineer- 
ing language test program and develop the subsystem 
test program while maintaining an error monitoring and 
display capability. 

The software is tailored to a hardware configuration 
of a central processing unit, display and/or typewriter, 
subsystem interface, and bulk storage. The software is 
modular in concept, real time in operation, and is classed 



as an interpreter. Among the attractive features planned 
for this interpreter are: 

(1) A basic set of frequently used, thoroughly checked 
elementary programs that may be .'selected and 
run by the user with minimal effort. 

(2) Orderly growth of programs obtained by user ex- 
perience and the sequencing of elementary subpro- 
grams. 

(3) A programming language consisting of engineering 
parameters rather than the mnemonics used by 
programmers. 

(4) The ability to change, update, clear, and insert 
into the existing program on line; i.e., dynamic 
change of the checkout program without the aid 
of an c^-line software assembly. 

3. Progrcis 

The preliminary design of the computer-oriented test 
system is essentially complete. A functional requirements 
document was generated in December 1967, describing 
a feasibility demonstrp«'on model to test a Mariner Mars 
1969 central computer and sequencer (CC&S) spacecraft 
subsystem. This CC&S is considered to be representative 
of future flight project hardware. Preliminary software 
flow charts that incorporate the CC&S as the subsystem 
to be tested have been prepared up to the point of 
machine dependency, and will be completed when the 
particular computer system is selected and the computer 
procured. 



52 



JPL SPACE nOGRAMS SUMMARY 37-5?, VOL. Ill 



N68-37403 



VI. Spacecraft Control 

GUIDANCE AND CONTROL DIVISION 



A. Slerilizabie Inertial Sensors: Gas Bearing 
Gyros, P. J. Hand 

1 . introduction 

Hie objective of this task is to perfect a complete family 
of miniature inertial sensois that will be capable of with- 
standing both thermal and g&s sterilizatiun without sig- 
nificant degradation of performance. Included in this 
family are long-Ufe rate-integrating gas bearing gyros, 
subminiature baU bearing gyros, and high-performance 
linear accelerometers. These sensors have potential appli- 
cations m both -A-a'icrd '^acecraft and entry capsule 
attitude control systems. 

The gas bearing gyrosc(^pes selected for this effort are 
the Honeywell, Inc. type CG159 and its wide-angle coun- 
teijKirt, GG-'VMS. The gas bearing gyro does not demon- 
strate any wearout conditions during operatioQ and can, 
therefore, be c-onsidered for application on very long 
missions. 

2. Dovolopmontal Baclcground 

Experience widi Ac GG159 gyro began at ]PL in j.982 
with the evaluati(Hi of a standard production version 
(GG159B1). Evaluation of the Bl version was followed in 



1963 with a development program at Honeywell to im- 
prove the g capabihty of the gas bearing mote-. The 
improvement program resulted in a motor design that 
was capable of passing the JPL shock requirement of 
200 g peak. This environment is required for (operation 
on all JPL-designed spacecraft. 

In 1964 the development program was broadened tu 
cover: (1) a gyro containing die 200-g motor (GG159C7), 
(2) a study to develop a gimbal suspension pump to 
operate at higher frequencies, and (3) a study to improve 
the gyro twquer efficiency as well as the first attempt at 
a thermally Si.jrilizable gyro (GG159D1). Knowledge and 
experience obtained from these study programs and from 
die Dl gyro wt re used in the redesigned GG159D2 gyro. 
This gyro succ-3ssfullv passed sev«i sterilization cycles at 
135°C without significant degradation of the iiiipoit.ant 
gyro drift parameters. Worst-case drifts were less than 
0.5 deg/h. 

During 1966, while JPL was evaluating die D2 gyro, 
Honeywell was devel(q>ing a wide-angle gas bearing gyro 
(GCJ34A). Initial <tttempts to develop a low-power 
(4.0 W) spin motor we:e also started. In mid-1966 a con- 
tract fur a sterilizablc version of this instrument, to be 
known as CG334S, was released. Later in 1967 work was 



jn SMCE MtOGKAMS SUMMAHY 37-51. VOL. Ill 



53 



started for JPL on the GG159E. This gyro will contain 
all the improvements developed since 1962 plus the low- 
power spin motor whidi was brought to an advanced 
state of development in the GG334S program. Salient 
characteristics of the GG159E and GG334S designs are 
compared in Table 1. 

Table 1 . Comparison of gas bearing gyros 





GyroKope 
Oei59E 


GyroKope 
GG334S 


Gyro gain (input to output), dcg/deg 


200 


0.40 


Diomatcr, in. 


2.2 


2.2 


Ungtii, in. 


3.1 


3.0 


Weiglit, lb 


I.l 


1.1 


Gimhal sufpension 


Pumped fluid 


Dithered pivot 
and jewel 


Gimbal freedom, deg 


±0.5 


±3.0 


Operating temperature, *F 


115 


180 


Motor power (at 26 V rmt. 


4.0 


4.0 


800 Hi), W 






Motor speed, rev/min 


24,000 


24,000 


Angular momenluni, g-cm'/t 


100,000 


100,000 


Drift rotes 






S-sensitive (spin axis), deg/li/g 


±0.50 


±0.50 


g-sensitive (input axis), deg/li/g 


±0.46 


±0.46 


g-insensitive, deg/h 


±0.30 


±0.46 


Kondom drift (1 v), deg/h 


0.008 


0.01 


Elastic restroint, deg/h/mrad 


0.06 


0.06 


Anisoelostic coefficient, deg/h/g' 


0.15 


0.15 



3. Gas Bearing Gyros Description 

The Honeywell type GG159 is a miniatiure, high-gain, 
single-axis, floated, rate-integrating gyroscope, utilizing 
a hydrodynamic spin-motor bearing. This gyro was se- 
lected for sterilization development because it had the 
greatest potential for surviving the thermal environment. 
(Tlie gas film bearing did not su£Fer from lubrication 
breakdown at the original sterilization temperature of 
145°C). 

The spin motor (Fig. 1) is designed to operate on 
26-V rms, 800-Hz power and rotates at 24,000 rev/min, 
producing an angular momentum of 100,000 g-cmVs, As 
with all hydroilynamic bearings, the rotor is in contact 
with the journal at the start of motor operation, but ^ifts 
off within a few revolutions. The rotor is then carried ou 
a gas film less than 100 /uin. thick. 

The spin-motor construction materials are largely 
ceramic except for the magnetic parts and the inertia 
ring. To prevent scuffing or abrasion between rotor and 




-THRUST PLATE 
(CERAMIC) 



JOURNAL BEARING 
WITH GROOVES 
(CERAMIC) 



-INERTIA RING 

(TUNGSTEN ALLOY) 



Fig. 1 . Miniature gas bearing spin motor 

journal during starts and stops, the ceramic materials are 
made very hard and are highly polished. 

The gimbal structure, which carries tlie spin motor, is 
also made of ceramic to provide matching thermal expan- 
sion characteristics. ITiis gimbal is floated at neutral buoy- 
ancy in a dense fluorolube fluid. The GG159 uses a very 
low viscosity fluid which allows a high input-to-output 
gain to be obtained at the gyro gimbal. A normal gain of 
200 at 115°F operating temperature is developed. The 
floated gimbal of the GG159 is also suspended by pump- 
ing the same fluid through controlled orifices between the 
gimbal and the outer case. 

The flotation fluid in the GG334S is very viscous and, 
therefore, supplies large damping forces to the gimbal. 
The GG334S gain is 0.40, allowing the gyro to store an 
input angle of ±7.5 deg. This flotation fluid is too viscous 
to allow pumping in the manner of the GG159. The gimbal 
suspension of the GG334 is more conventional in that 
pivots and jewels are used; however, the jewel is oscil- 
lated by a piezoelectric dither plate to eliminate static 
friction from the suspension. In both gyros, the outer case 
is made of conventional aluminum alloy with an integral 
heater and temperature sensor attached. 

The GG159E is the culmination of the effort to perfect 
a gas bearing gyro for spacecraft operation. The GG334S 
will contain the same imprc"-'ements as the GG159E but 
will be capable of storing up to ±7.5 deg of input prgle 
information direcdy on the gyro gimbal without requiring 
the large integrating capacitors which the present Mariner 
spacecraft uses. 



54 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL /// 



4. Statu* 

Final fabrication of both the GG159E and the GG334S 
has been delayed due to a moisture contaminaticHi prob- 
lem within the gimbal. This has been solved by redesign 
of the joiunal bearing to cause gas to flow through the 
bearing. Evaluation of both types of gyros will take place 
at both JPL and Honeywell during the latter half of 1968 
and early 1969. Performance data will be presented in 
future editions of the SPS, Vol. III. 

B. Analysis of Ion Thruster Control Loops, 

P. A. Mu9llw and E. V. Pawlik 

1. Introduction 

Data (HI electric propulsion systems indicate ion 
thrusters to have several nonhnear properties that make 
the use of computer simulation and analysis quite attrac- 
tive. Computer studies have been performed on controls 
for a thruster suitable for use as primary propulsion of a 
spacecraft for deep space missions (such as a Jupiter 
flyby»). 

Tliese computer studies have been performed for 
thruster controls and power matching that have been 
previously proposed for a thruster employing an oxide- 
coated cathode (Refs. 1 and 2). In the proposed control 
scheme, two control loops are utilized to maintain the 
thruster at a desired operating point (thrust) despite 
variations in cathode emission, vaporizer porosity, and 
thruster thermal emissivity during the thruster operating 
lifetime that may be as long as 10,000 h. In addition, the 
relationship between the two thruster control loops is used 
to indirectly specify the mercury propellant flow rate to 
the thruster. 

The computer simulation was performed with Digital 
Simulation Language 90 on the IBM 7094 computer. 

2. Computer Simulation Model Considerations 

The thrust is approximately proportional to the product 
of the ion beam current h and the net acceleration volt- 
age. Typical power conditioning and control loops for the 
20-cm-diam thruster being simulated are shown in Fig. 2. 
For the present systeip the net acceleration voltage, the 
output of supply V5 (the high-voltage screen supply), is 
held constant and the ion beam current is commanded to 
operate over a two-to-one range corresponding to 0.5 to 
1.0 A. Other fixed value supplies are VI, V4, and V6, which 
are the electromagnet supply, the arc or discharge voltage 



NB3PRENE r-VAPORIZER DISCMARGE MAGNETIC 



BLADDER 



CHAMBER 



"7 i-* ^ J / GRID 

/ I f- ^ 



-SCREEN GRID 
ACCELERATOR 



ERCURVV«WR^_^^^^^,5-;0NBEAM-^ 

1 C _ 




yi/B reference)! (<>^ 

I *I 



'1975 Jupiter Flyhy MissUm UHng a Solar Electric Propulsion 
Spacecraft, Mar. 1968 (JPL Internal document). 



^ reference /* 

spacecraft ground 

Fig. 2. Power conditioning and controls block diagram 

supply, and the high-voltage accelerator supply, respec- 
tively. These fixed values simplify the control loops to 
those presented in Fig. 3. Supplies V2 and V3 are the 
two controlled supplies, the vaporizer supply and the 
cathode supply, respectively. 

Figure 4 presents the thruster ion chamber nonlinearity 
for a typical thruster. The straight lines for constant ion 
chamber or arc current I^ approximate the curved lines 
obtained from a characteristic mapping of a thruster 
which is not operating at maximum efficiency (Ref. 2). 
Characteristics of an optimized thruster have constant I a 
lines that have negative slopes at low propellant utihza- 
tion values. (This condition is also being studied.) For 
the nonoptimized thruster, the straiglit-line approximation 
is accurate for points where propellant utilization (the 
fraction of mercury propellant ionized and accelerated as 
the ion beam) is 0.8 or less. For higher values of utilization 
the constant I a Unes have greater slopes. Operation at a 
constant utilization value implies a unique combination 
of beam current aad arc current (except where utilization 
is 1.0). This unique relationship is the function generated 
in the block, "Arc Current Reference." The particular 
function chosen depends on the propellant utilization 
value desired. All simulations to date have been at 0.80 
utilization which is indicated by the heavy line in Fig. 4. 

Without the controllers in the arc loop, the loop has 
one time lag of approximately 120 s due to the cathode. 
Nonlinearities cause the gain to vary by as much as a 
factor of 4, depending on the thruster conditions and the 



JPL SPACE PROGRAMS SUMIAARY 37-51, VOL III 



55 



PERTURBATIONS 



BEAM 



CURRENT 
REFERENCE 



ARC 

CURRENT 

REFERENCE 

GENERATOR 



^O 



ARC CURRENT 

CONTROLLER 

AND CATHODE 

HEATER SUPPLY 



THRUSTER 
CATHODE 



o 



BEAM CURRENT 

CONTROLLER 

AND VAPORIZER 

SUPPLY 



VAPORIZER 

AND 
MANIFOLD 



MERCURY MASS 
FLOW RATE 



ARC CURRENT 



THRUSTER 

ION CHAMBER 

NONLINEAR 

CHARACTERISTICS 



HIGH VOLTAGE SUPPLY PERTURBATIONS • 
ARC VOLTAGE SUPPLY PERTURBATIONS • 
MAGNET SUPPLY PERTURBATIONS 



JTT 



BEAM CURRENT 



Fig. 3. Block diagram of two ion thruster control loops 



degradation with use of the cathode. Without the con- 
troller in the beam loop, there are time lags of 0.02 s in 
the thruster manifold and from 120 to 600 s in the vapor- 
izer. Gain variations attributable to component nonlineari- 
ties are on the order of 2. 



3. Controllers 

Several controllers have been studied for the two arc: 
loops, including simple gain factors (proportional), type 1 
or integral, and integral with lead compensation. Propor- 
tional and integral >^ith lead appear most promising. 
Similar controllers are being considered for the beam 
loop. With ideal components used in the controllers, stable 
performance with 0.1* beam current error (difference be- 
tween the specified and the actual beam currents) and \% 
utilization error (difference between the desired and the 
actual utilizations) appears to be realisticaUy feasible. 
Drift, offset, and other errors associated with the con- 
trollers' electronics, are being considered as the compute r 
simulation becomes more complete. 



4. Ion Chamber Perturbation Studies 

The model of the thruster ion chamber nonlinear char* 
acteristics presented in Fig. 4 depends on other thruster 



parameters remaining constant. These parameters include 
the magnetic field, discharge voltage, screen voltage, and 
accelerator voltage. Deviations from these fixed values 
introduce variations in the ion chamber characteristics. 
The regulation of the power supplies, therefore, becomes 
an important consideration. Variations of the fixed out- 
puts of the electromagnet supply, the high-voltage sup- 
plies tor maintaining the voltage between the grids of the 
ion ( xtraction system (the sum of the screen and accel- 
er.^l.ir voltages), and the discharge of arc voltage supply 
(V. . V5, V6, and V4, respectively) are the important per- 
lurhations. The predominant effect of the variation in 
power supply output is a translation effect in the charac- 
teristics. The dashed lines in Fig. 4 indicate such a shift 
due to a very small perturbation and are explained in the 
folli wing paragraphs. 

'I Jie computer was again used in determining the per- 
turbation effects. A perturbation factor was calculated 
for each of the supplies using the experimental data 
obtained for large differences ia supply outputs. These 
factors were introduced into the simulation program and 
computer runs were made for ±10? errors on the outputs 
)f the power supplies. 

While perturbations were introduced, the closed-loop 
system maintained the set point values of J^ and !«. How- 



56 



JPH SPACE <>X06RAMS SUMMARY 37-51, VOL. Iff 




3 4 5 6 7 

MERCURY MASS FLOW RATE m, g/h 

Fig. 4. Typical ion chamber nonlinear characteristics 



ever, the set point values in this situation no longer de- 
fined the specified propellant utilization. Because of both 
the slight slopes in the ion chamber characteristics and 
the large perturbation factors, small variations in supply 
outputs resulted in large variations in propellant utiliza- 
tion. The worst-case variations obtained for the electro- 
magnetic supply, high-voltage, and arc-voltage errors were 
2, 1.5, and 0.5%, respectively, with propellant utilization 
errors of 0.05 in each case. All three supplies have the 
same e£Fect so that the error can be additive; i.e., if the 
three .supplies had the above errors, the utilization error 
would be 0.15. An error of 0.15 in the utilization when 
the set point is 0.80 is an error of 18.8%. The dashed lines 
in Fig. 4 are for this case. 

To minimize the susceptibility of propellant utilization 
to such perturbations, greater characteristic slopes are 
desirable. One method of achieving this is to run at a 



higher value of utilization since, as previously mentioned, 
the true thruster ion chamber characteristics have a 
greater slope by a factor of 3 or 4 in the utilization range 
of 0.9 to 0.95. This was not considered in the computer 
model since 0.80 utilization was the designated set point. 

5. Arc Reference Perturbations 

The arc current reference function generator is also a 
critical component in determining the tolerances in pro- 
pellant utilization. An error in the reference has the same 
effect as an error in the arc current itself. Computer simu- 
lation resulted in a worst-case error of 7.5% in utilization 
corresponding to a 1% error in the arc reference. Since the 
arc reference is a critical factor, it must be maintained 
with much better stability than 1% if the propellant utiliza- 
tion errors are to be kept within a realistic range of a few 
percent. 



JPl SPACE PROGRAMS SUMMARY 37-51, VOL. I» 



57 



6. Conclusions 

Several conclusions can be drawn from this study of 
throttled ion thruster controls. Stable ion thruster per- 
formance is feasible with a beam current error limit:.tion 
of 0.12 and a propellant utilizaticm error limitation of U. 
Errors of a few percent in the fixed output power supplies 
may cause utilization errors 10 times as large. Arc ref- 
erence errors of 1% may yield propellant utilization errors 
of 7 SI. Thruster performance in the utilization range of 
0.8 to 0.95 merits further investigation. 

References 

1. Pawlik, E. v., Power Matching of an Ion Thwiter to Solar Cell 
Power Output, Technical Memorandum 33-392. Jet Propulsion 
Laboratoiy, Pasadena, Calif, (in press). 

2. Maser, T. D., and Pawlik, E. V., "Thrust System Technology 
for Solar Propulsion," Sections III and IV, Paper 68-541, to be 
presented at the AIAA Fourth Propulsion Joint Specialists Con- 
ference, Cleveland, Ohio, June 1968. 

C. Powered Flight Control Systems, 

R. J. Mankovitz 

1 . Introduction 

The original objective of this task was to develop non- 
linear digital computer programs for various powered 
flight control systems, and to utilize these programs to 
perform parametric trade-off studies that could be used 
to select optimum systems for given requirements. After 
completion of a six-degree-of-freedom program for a 
gunballed-engine (chemical propulsion) autopilot system, 
the objective was revised, due to budgetary constraints, 
to the study of the attitude control of an electric- 
propulsion-powered (ion engine) vehicle, during the pow- 
ered flight phase. This work was directly applicable to 
an Advanced Technical Studies task related to a solar 
electric-powered spacecraft mission to Jupiter. 

2. Basic Considerations 

Trade-off studies have been conducted for the attitude 
control of an electric-propulsion-powered vehicle. A com- 
plete six-degree-of-freedom digital computer simulation 
has been developed and used to evaluate the foUowing 
basic concepts. 

(1) Three-axis cold gas control. 

(2) Two-axis engine translation with third-axis cold gas 
control. 

(3) Two-axis engine translation with engine gimballing 
for third-axis control. 



In addition to the basic concepts, a hot gas system (resisto- 
jets) was considered in place of the cold gas system. Solar 
pressmre control augmentation was also considered by 
rotating the solar panels (panel trim) to obtain solar 
torques. As a result, the tliird alternative (3) was chosen 
as the baseline configuration. This configuration reduces 
the gas usage to zero for powered flight control, and only 
requires a total of 20 lb of cold gas during the non- 
powered phase. The hot gas and panel trim alternatives 
were rejected on the basis that the significant increase in 
complexity does not result in a significant reduction in 
stored-gas weight. 

A basic control law has been developed and analyzed 
for the chosen configuration and has been demonstrated 
to p'' ide stable operation. 

3. Functional Description 

During the powered flight portion of a solar electric 
mission, the spacecraft must remain sun-Canopus-oriented 
and have the ability to point the ion engine thrust vector 
over a 180-deg angle in the ecliptic plane, even when out 
of the ecliptic plane by as much as ±3 deg. A total thrust 
vector pointing error (in celestial coordinates) of less than 
1 deg is desired. 

a. Three-axis attitude control. The method selected for 
providing three-axis attitude control during that portion of 
powered flight when more than one ion engine is operat- 
ing consists of a two-axis, bi-directional engine-translation 
system with third-axis control (thrust vector axis) pro- 
vided by gimballing the outermost ion thrusters on 
the engine array (Fig. 5). The ion thruster gimbals are 
single-degree-of-freedom with opposite thruster gimbals 
slaved to each other. If one of the outermost thrusters 
should fail, control is switched to the other opposite pair. 
The baseline control system requires ±10-deg gimbal ex- 
cursions and ±12-in. translator travel. Both the translator 
and gimbals are stepper motor-controlled, with resolu- 
tions of 0.005 in./step on the translator and 0.1 mrad/step 
on the gimbals. Maximum stepping rate for all systems is 
50 steps/second. All control loops are passively compen- 
sated and do not require gyro signals. 

b. Cold-gas/ion-engine stcitchocer.^ Upon completion 
of the Canopus acquisition phase, the ion engines are 
activated. A 5-min duration is allowed to permit the en- 
gines to achieve full thrust. During this period, the tians- 
lation and gimbal systems are inactive, and the cold gas 



'See also: Section E, "Extended Mission Control Systems Develop- 
ment." 



58 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



X TRANSLATION 



y TRANSLATION 




SINGLE-AXIS 
GIMBALS ON 
OUTER-FOUR 
THRUSTERS 



Fig. 5. Powered flight control using two-axis 
translation and third-axis gimballing 

system maintains attitude control in the presence of thrust 
vector misalignments. At the end of this 5-min period, 
the engine control systems are activated, and the cold 
gas control system deadbands are increased from ±0.5 
to ±3 deg. Under normal operating conditions, the engine 
translation and gimbal systems will be operating within 
the deadband of the gas system. If, however, a large dis- 
turbance should cause loss of acquisition, the ion engines 
will be deactivated and the celestial references reacquired 
using the cold gas system. 

In the limit cycle mode, the engine control system 
stepper motors will pulse at a maximum rate of 1 pulse/ 
32 s. The average deadband size is approximately d=I 
mrad. The control system can recover from a 3.0-ft-lb-s 
torque impulse without losing celestial lock. 

c. Orientation of the thrust vector. 

Full thrust mode. Nominally, the spacecraft x-z plane 
is co-planar with the ecliptic plane. To meet the require- 
ment of 180-deg angular freedom in pointing the thrust 
vector in the ecliptic plane, the ion engine array is 
mounted on a single-degree-of-freedcm platform which 
can rotate 180 deg in the spacecraft x-z plane. The 
position of this turret will be changed in increments 
(< 0.1 deg) determined by a central compiter and se- 
quencer (CC&S)-stored thrust poinung program. 

The ±:3-deg out-of-plane pointing capability is mecha- 
nized by appropriately biasing the pitch sun sensor and 
the Canopus tracker error signals. Thus, the spacecraft 
itself is rotated about the pitch and roll axes to point the 
dirust vector. Utilizing sensors w-ith ±8-deg linear fields 
<A view enables the spacecraft to perform these turns 



without losing the celestial references. As in the case of 
the in-plane pointing angle, the out-of-plane angle is 
suppUed by a stored CC&S program. This angle is 
resolved into sensor bias signals wdthout the use of 
trigonometric functions. 

A functional block diagram of the attitude control sys- 
tem during this phase is shown in Fig. 6. Since the engine 
control axes will, in general, not coincide with the space- 
craft axes, a coordinate transformation is required to 
convert error signals from the spacecraft axes to the con- 
trol axes. Since the engine pointing angle varies through- 
out the powered flight phase, the variable transformation 
mixing matrix is mechanized with resolvers The resolved 
error signals are sensed by the control systems, and trans- 
lation and gimbal deflections of the thrust vector produce 
the three spacecraft body control torques. The control 
torques act through the spacecraft structural dynamics to 
counteract the disturbing torques and produce the error 
signals flj., 6,/, S.- 

Reduced thrust mode. During the latter portion of the 
powered flight phase, only one ion engine is operating. 
Since this engine is centered about the spacecraft center 
of gravity by the translator, engine gimballing can no 
longer provide tl^'Td-axis control. During this phase, the 
gimbal control system is deactivated, and the cold gas 
system is used to control the torques about the engine 
axis. The only disturbance torque generated by the single 
engine is due to swirl of the ion stream (engine misalign- 
ments are removed by the translator), so that only a small 
amount of cold gas is required during this phase. 

4. Control System Analysis 

Some of the powered flight attitude control mechaniza- 
tions that were considered for this phase aie: 

(1) Three-axis cold ga3 control. 

(2) System (1) with solar panel trim. 

(3) Two-axis translator control plus cold gas third-axis 
control. 

(4) System (3) with solar panel trim. 

(5) Two-axis translator control plus gimbal third-axis 
conttol. 

(6) System (5) with solar panel trim. 

(7) Any of the above systems using heated Nj (resisto 
jets). 

The three basic systems are (1), (3), and (5). For those sys- 
tems requiring cold gas for control, the options of heating 



jn SPACE PROGRAMS SUMMARY 37-51, VOL Ul 



59 







B 


K 














T. 






















By 


STRUCTURAL 
DYNAMICS 




Ty 


TORQUE 

GAIN 
MATRIX 












e. 


" 


^ 






















i'c 










PITCH 
SU^l SEI.SOR 






VOLTAGE -CONTROLLED 
OSCILLATOR 










K. 


IrS 


t* 


COORDINATE 

TRANSFORMATION 

MIXING MATRIX 

SPACECRAFT BODY 

TO ENGINE 

CONTROL AXES 

^ 

RESOLVER 1 

.in y 1 

1 cos y 1 

1 1 




■NJ 


-- 




ENGINE 

TRANSLATION 

LOOP 






"R 






K 












YAW 
SUN SENSOR 




VOLTAGE-CONTROLLED 
OSCILLATOR 




8/c 






K, 






^ 


L 




ENGINE 

TRANSLATION 

LOOP 


















k 












CANOPUS 
TRACKER 




VOLTAGE-CONTROLLED 
OSCILLATOR 












Kc 


^ 


> 




^L 


- 




6IMBAL 
LOGIC 




GIMBAL 
ACTUATOR 




9 

+ 




ICP 




k 












r' 


ECLIPT 








1 








OUT-OF-ECUPTIC 

PLANE POINTING 

ANGLE 




LANE POINTING ANGLE 


ENGINE 
FAILURE 































Fig. 6. Attitude control system during thrust phase 



the gas to increase the Nj 7s^ and tilting the solar panels 
to balance the disturbance torques were considered. 

To permit an evaluation of these systems, a digital com- 
puter program was used to determine the attitude control 
gas storage requirements for each system during a 1200- 
day mission. 

The mission gas storage requirements for each system 
is presented in Table 2. The use of a gas system for three- 
axis control, assuming a 0.1-ft engine array center of 
gravity offset and a 1-deg engine angular misalignment, 
was eliminated immediately due to excessive gas weight. 

For the system using a two-axis translator plus gas sys- 
tem third-axis control (assuming a 1-deg engine angular 
misalignment), the only case that appeared feasible from 
a gas-weight standpoint required hot gas and solar panel 
trim capability. Considering the added weight, the de- 
crease in reliability attendant ^^dth rotation of 46-ft-long 
solar arrays (as well as structural dynamics problems), 
and the lack of long term flight experience with hot gas 
systems, this mechanization was eliminated. 



The third, baseline, mechanization, which requires a 
20-lb gas weight, uses a two-axis translator with gimballed 
engine third-axis control. Cold gas is only used for acquisi- 
tions, cniise (nonpowered flight), and for third-axis con- 
trol during that portion of the powered flight phase when 
only one engine is operating. Neither heated gas nor solar 
panel trim is required for the baseline mechanization. 

The basic control loop for either the translator or gim- 
bal system is shown in Fig. 7. The input is a position 
signal from a celestial sensor, referenced to spacecraft 
axes. An attitude bias, in the form of a do voltage simmied 
with the sensor output, may be present to orient the space- 
craft out of the ecliptic plane for thrust vector pointing. 
The sensor signals are mixed in a transformation matrix 
to go from spacecraft axes to engine axes. The matrix is a 
function of the angle y, which is the ecliptic plane engine 
pointing angle. The engine-referenced error signal is used 
to drive a voltage-controlled oscillator (VCO), yielding a 
variable frequency pulse train that is used to drive a 
stepper motor. Thus, the stepper motor rate is propor- 
tional to the error magnitude. The motor is used to drive 
either the translator platform or the engine gimbals to 



60 



in SPACE PROGRAMS SUMMARy 37-51, VOL. /» 



Tobl* 2. Attitude control gat storage roquiramonts 



Basic ty>l*m 


Cold gas, lb 


Hoi gas, lb 


No panol 
tHm 


ranol 
trim 


No ponol 
trim 


Ponol 
trim 


Thr«a-axii cold gai 


-1100 


~800 


~6S0 


~500 


Twc-axii translator 
plus old gas 
third axis 


85 


S6 


51.3 


35.2 


Two-axis translator 
plus gimbal third 
axis 


20.4 


20.4 


15 


15 



/ TORi 



DISTURBING 
TORQUE 



RESTORING 
TORQUE 



SPACECRAFT 

STRUCTURAL 

DYNAMICS 



SPACECRAFT 
POSITION 



ENGINE 
TRANSLATION 
OR ROTATION 



CELESTIAL 
SENSOR 



STEPPER 
MOTOR 



' + 



COMPENSATION 



VOLTAGE-CONTROLLED 
OSCILLATOR 



(Sh 



DEADSPACE 



^♦Hg^izK 



ATT ITUDE 
BIAS 



TRANSFORMATION 
MIXING 
MATRIX 



OTHER 
AXES 



Fig. 7. Basic translation or gimbal control loop 

produce restoring torques. These act on the structure, 
which includes the dynamics of the solar arrays. 

Compensation networks are required to stabilize the 
loops, and since gyros cannot be used for extended dura- 
tions, passive rate compensation must be employed. 

The electronics required for the control systems are 
mechanized with linear and digital integrated circuits, 
employing triple modular redundancy for increased reli- 
ability. Redundant sun sensors are employed, and it also 
appears desirable to employ dual Canopus h'ackers which 
can be switched by ground command. 

Since the translator and gimbal positions (and thus the 
restoring torques) are a discrete function of time, the 
steady-state behavior of the control loops will be a limit 
cycle of nominally ±:1 step about the balanced torque 
point. 

To analyze the control loops for the large signal mode, 
linear analysis methods were used to approximate the 



nonlinear loops. Considering the baseline conSgiuration, 
it can be shown that the sampling rate (VCO output) of 
the actual loop is sufficiently high, in all modes except the 
steady-state limit cycle, to permit the use of linear analy- 
sis for preliminary investigations. Digital computer simu- 
lation programs were constructed to verify the analysis. 
Some of the major problems in mechanizing these loops 
are: interai.tion with the solar panel structural dynamics, 
passive rate compensation alone, and sensor noise. 

The block diagram of a single-axis translator control 
system is shown in Fig. 8. Compensation (lag) is placed 
in the feedback loop, as opposed to lead compensation in 
the forward (sensor) loop, to minimize problems due to 
sensor noise. In addition, the sensor output is fed through 
a deadspace which is sufficiently wide to reject the am- 
bient tracker noise at null, thus preventing stepper motor 
dither. A combination of positive and negative feedback 
is used to minimize steady-state error in spacecraft posi- 
tion. The structure (solar array) dynamics are modeled as 
a fourth-order polynomial, with coefficients chosen as a 
function of the spacecraft configuration. A first-order lag 
is associated with the sensor signals to model the efiFects 
of noise filters. 

The gimbal control system block diagram is also shown 
in Fig. 8 and is analogous to the translator loop. The 
stepper motor output is proportional to gimbal angulai* 
position, which acts through the engine thrust Fa and 
moment arm Lg to produce restoring torque. 

To optimize the compensation networks and determine 
the operating point for the system, a digital computer 
root locus program was used to analyze the open-loop 
transfer function. 

The major parameters for the translator and gimbal 
loops are shown in Table 3. Many of the parameter values 
were dictated by hardware constraints, such as: 

(1) A high stepper motor rate (SLEW) is desired. To 
achieve good dynamic response, 50 steps/s is con- 
sidered a reasonable value for a magnetic detent 
stepper motor. 

(2) It is desirable to minimize the step size to achieve 
accurate attitude control. The values chosen for K„ 
are within hardware capability. 

(3) The translator and gimbal limits (8mu and 8mi„) 
were chosen as large as possible (to maximize 
restoring torque capability) within the structural 
limitations. 



jn SMCE PROGRAMS SUMMARY 37-51, VOL. Ill 



61 




E 



e 

c 
o 

w 

"5 

M 

E 

n 

■o 
e 
s 

e 
e 

e 



62 



jn SPACE PffOGMAU SUMAMffr 37-51, VOL. Ill 



Tabia 3. Trgnslotor and gimbal control 
tystom poramolors 



Param*l«r 


Tranilotar 


Olmbal 


Cclaitial iinic ;<itn K., V/rad 


134 


134 


Ctlailiol ttntor lag r>, > 


0.S 


0.5 


VCO gain Kv, iltpi/i/V 


2S7 


1914 


Maximum (tapping ral* SLEW, 


SO 


50 


tiapi/t 






Stapptr motor gear train gain Ka, 


4.167 X 10* 


— 


ft/itap 






$t*pp*r malor g*ar train gain K., 


— 


10* 


rad/it«p 






FMdback gain Kr, V/ft 


16.7 


— 


Ftrdback gain K,, V/rad 


— 


10.45 


Poiitive fatdbacli lag Tp, > 


1000 


1000 


Ntgaliv* fmdback lead ri, i 


50 


50 


Nagativ* ftadback lag r:, i 


500 


500 


Moximum translator txcurtlon 


1 


— 


«m..,fl 






Maximum gimbal oxcurtion Snm, dag 


— 


10 


Minimum tranilator axcurtion 


-1 


— 


«,„ln, ft 






Minimum gimbal axcurtion j„,in, dag 


— 


-10 


Ion angina tliruit F, lb 


0.01-0.04 


__ 


Ion angina ttiruit Fo (2 anginas), lb 


— 


0.02-0.03 


Spocacraft inartia J, «lug-ft' 


15,000-30,000 


15,000-30,000 


Coafficianti of structural dynamics 


6.34 


6.34 


modal M. 






Coafficinnts of structural dynamics 


0.08 


0.08 


modal M, 






Coafficiants of structural dynamics 


5.04 


5.04 


modal M, 






Coafficiants of structural dynomici 


0.032 


0.032 


mc-J*l M, 






Coafficiants cf structural dynamics 


0.73 


0.73 


modal N. 






Ceaftiriants of structural dynamics 


0.08 


0.08 


modal N, 






Coafficiants of structural dynamics 


2.8 


2.8 


modal Ni 






Coafficiants of structural dynamics 


0.032 


0.0S2 


modal N, 






Distanva from angina gimbal to array 


— 


1.25 


cantar of gravity to, ft 






Calaslial tansor daoiibond DB 


K. X 10' 


K. X 10' 


(oquivolant to 1 mrod), V 







(5) The sensor filter time con tant (t.) is chosen to 
achieve the noise figure indicated above. 

(6) The ion engine thrusf range for the translator ^F) 
covers the range from one to four engines operating 
from full to throttled-back thrust. The engine thrust 
range for the gimbal system (Fj) covers the throt- 
tling range of two engines. 

(7) Since the engine bank can rotate 180 deg about the 
spacecraft yaw axis, both the translator and gimbal 
systems must be able to operate over the full range 
of spacecraft inertias (/). 

(8) The torque moment arm (La) for the gimballed 
engines is determined by the engine diameter and 
engine mounting positions. 

(9) The coefiBcients of the linear structural dynamics 
model (Ms and Ns) are calculated by a computer 
program. A solar array natural frequency of 0.1 Hz 
and a damping ratio of 0.005 were used, represent- 
ing worst-case conditions. 

From a closed-loop Bode plot (at a dc gain of 4 X 10"'), 
the control bandwidth can be determined as 0.005 Hz. 
The effect of the panel dynamics occurs at 0.1 Hz. 

To verify the performance and stability of the systems, 
a six-degree-of-freedom digital computer simulation pro- 
gram was constructed. Simulation results indicate stable 
operation over the gain variations anticipated (12:1 gain 
change due to thrust and moment of inertia variaHons). 

The simulations also indicated that, with a ±5-deg 
sensor field of view and with minimum engine thrust, 
the control loops could maintain the spacecraft orienta- 
tion when subjected to a 3-ft-lb second-torque impulse 
about all axes (corresponding to a step anf^ular rate of 
'-0.1 mrad/s in all axes). 

Further discussion of the results of the six-riegreeof- 
f reedom simulation will be presented in a ft ■ edition 
of the SPS, Vol. III. 



D. Spacecraft Antenna Pointing for a 
Muitiple-Planet Mission, G. E. F/«itcher 



(4) The sensor deadband (DB) is chosen su£Bciently 
large, so that tracker noise will fall within its limits. 
Worst-K^ase tracker noise, when acquired to Cano- 
pus, is estimated at 0.53 mrad peak to peak. The 
deadband width is chosen as ±1 mrad. 



Current preliminary studies of a gravity-assist mission 
(Gi-and Tour) to the Jovian planets (Jupiter, Saturn, 
Uranus, and Neptune) have included a rather broad look 
at the spacecraft high-gain antenna pointing problem. 
Several different pointing systems are being compared on 



JPL SPACE PMOGR AMS SUMMAKY 37-51, VOL. Ill 



63 




-ANTENNA BORESIGHT 
VECTORS 



CANOPUS SENSOR 
CLOCK ANGLE 
BIAS 



Fig. 9. Singi«-axit anl»nna with CanepHt 
(•ntor clock bia« 



CONE ANGLE BIAS 



>6 



14 



I 2 



? I 
uj 06 

(9 



gO.6- 



04 



0.2 



T 



T 



T 



T 



200 



■CANOPUS UPDATE 
SEQUENCE (BIAS) 



CANOPUS UPDATE SEQUENCE 
TIME.days ANTENNA CLOCK ANGLE, deg 

100 90 

314 86 

507 80 

716 78 

914 76 

1114 70 

1321 75 

1514 77 

1708 78 

1893 80 

2274 82 

2465 83 

2654 89 

2846 86 



POINTING 
ERROR 




I 

V) 

< 
20 £ 

UJ 
10 (O 



- 



URANUS 
ENCOUNTER 



-10 



-20 



z 

4 
U 



400 



600 



1800 



BOO 1000 1200 1400 1600 

TIME FROM LAUNCH, days 

Fig. 10. Sirtgi«-degrto-of-fr*«dem antanna articulation 



2000 



2200 2400 



64 



in SPACi MOGMMS SUMMARY 37-51, VOL. Ml 



the basis of their contributions to total communications 
system weight, power, and performance. A cone-clock 
type pointing system is briefly described here. 

For the case in w!uch the high-gain antenna is assumed 
to be articulated with respect to the spacecraft in one 
degree of freedom, two-degree-of-freedom earth point- 
ing can efiFectively be achieved through the use of a 
spacecraft-fixed Canopus sensor whose field of view is 
electronically biased in clock angle (in addition to the 
present bias capability in cone angle). Thus, the clock 
angle degree of freedom is provided by the roll attitude 
control loop at relatively little cost in terms of weight and 
power. The antenna's degree of mechanical freedom quite 
naturally then becomes a rotation about an axis perpen- 
dicular to the spacecraft roll axis (Fig. 9). Antenna rota- 
tion is controlled by a stored program that generates the 
proper angular function of time (earth's cone angle). The 
result is a cone-clock type of pointing system with a 
limited capability (±15 deg) for biasing the Canopus 
sensor view in clock angle. 

Due to this hmitation in clock angle rotation of the 
spacecraft, an inherent pointing error occurs as the appar- 
ent earth track passes near the sun. Assuming all other 
pointing errors are neghgible, that portion of the earth 
track outside of the region in which clock angle freedom 
is available for antenna pointing cannot be seen with 
zero error. 

Assuming that very accurate pointing is not required 
during the earlier portions of the Grand Tour, the effect 
of a relatively coarse progr; m of Canopus sensor clock 
bias angW was investigated and the result plotted in 
Fig. 10 Fourteen bias-angle updates of the clock angle 
are provided for the entire Grand Tour mission. Of course, 
the effects of spacecraft attitude errors, mechanical mis- 
alignmoits, cone angle program errors, etc., on total point- 
ing error have not been included in Fig. 10; only the error 
resulting from a discrete and limited clock angle rota- 
tional capability is given. 

E. Extended Mission Control Systems 
Development, L. McGUnchey 

1 . Introduction 

The extended mission control systems development 
study is a new task for FY 1968. During the first quarter, 
a project was started to study the attitude control of 
vehicles utilizing electric propulsion sy iems. 'xhe scope 
of this work was directly applicable to an Advanced Tech- 



nical Studies task related to a solar electric-powered 
spacecraft mission to Jupiter. 

Providing attitude control fo' a solar electric spacecraft 
poses many new and unique configuration and design con- 
siderations not encountered previously. Tlie mass and 
inertial properties of solar electric spacecraft pose unique 
problems with regard to sizing the control capability of 
the attitude control system, due to the constraints posed 
by dynamic interaction. The Jupiter spacecraft has inertias 
on the order of 15 000 slug-ft* about the pitch and yaw 
axes and 30,000 slug-ft" about the roll axis. These large 
inertias require a much higher control torque level to pro- 
vide reasonable acquisition times and recovery from dis- 
turbances. In addition, the change in inertias (60:1) after 
solar array deployment requires that the attitude control 
system have a very large dynamic range. 

The deployment of large solar arrays (1500 ft^) can 
introduce disturbance torques that could cause such 
severe interaction with the attitude control system that 
the solar array structure and the deployment procedure 
would be adversely affected. Detailed structural analyses 
are required to evaluate this problem. At present, no 
detailed information is available regarding the structural 
properties of so\cr electric spacecraft. In this article the 
results of the attitude control system study are based on 
a linear lumped parameter model of the solar array struc- 
tural dynamics, with the remainder ot the spacecraft con- 
sidered as a rigid body. On this basis, the baseline attitude 
control system was designed to be compatible with the 
structure. However, considerable analysis must be done 
to fully investigate and model all possible adverse struc- 
tural resonance modes that can affect the attitude control 
of this type of spacecraft. 

In addition to structural interaction, incident solar radi- 
ation on the large solar arrays can reduce significant 
disturbing torques on the spacecraft. Similarly, gravity 
gradient disturbance torques can be significant in the 
vicinity of the planet, especially a planet the size of 
Jupiter. In the case of the Jupiter mission, the attitude 
control system was configured for worst-case solar pres- 
sure and gravity gradient unbalance torques. 

Several alternate attitude control configurations for the 
Jupitur spacecraft were examined during the course of 
the study. The following discussion describes the baseline 
attitude control system for the nonpowered flight portion 
of the mission. Attitude control during the powered flight 
phase is described in Section C. 



jn SPACE PROGRAMS SUMMARY 37-51, VOL HI 



65 



2. Baselins Configuration Functionol Dtscription 

a. General attitude control requirements. The basic 
requirements for the attitude control of the spacecraft are 
as follows: 

(1) ProWde initial rate removal and stabilization of 
the spacecraft following separation and solar panel 
deployment. 

(2) Acquire celestial references (sun and Canopus). 

(3) Provide thrust vector orientation and maintain a 
stable attitude during the thrust phase. 

(4) Maintain a stable attitude during the cruise phase. 

(5) Provide immediate reacquisition of the celestial ref- 
erences as required. 

(6) Provide antenna and science instrument orientation 
as re^iuired. 

The above requirements, with the exception of the 
third, do not pose serious constraints on the selection of 
an appropriate attitude control configuration. The third 
requirement presents unique problem areas because of 
(1) the duration of the thrust phase (470 days), and (2) a 
requirement for pointing the thrust vector out of the 
ecliptic plane (see Section C). 



h. Nonpowered jlight fttnctional sequence and attitude 
control modes. During all phases of the mission, except 
the powered flight phase, attitude control and stabiliza- 
tion of the spacecraft is obtained by control torques pro- 
vided by a Nj cold gas mass expulsion system. The Nz 
cold gas system was selected as the most feasible for the 
following reasons: 

(1) Simplicity and inherent reliability. 

(2) Space proven, particularly on Mariner TV where 
this type of system operated for over 1000 days. 

(3) Minimum weight consistent with the attitude con- 
trol requirements. 

The basic operation of the cold gas system is as follows: 
error signals are measured by position and rate sensors 
and svunmed in their respective channels to operate gas-jet 
valve-switching amplifiers, which provide an onr-off type 
control torque. A position limit cycle about each of tibe 
control axes is established by a switching amplifier dead- 
band. A rate feedback signal provides the proper rate 
dam:}ing. 



A description of the attitude control modes during each 
of the nonpowered flight phases of the mission sequence 
is given below. 

Initial rate reduction and stabilization. Following sepa- 
ration from the launch vehicle, the structures supporting 
the gas jets are deployed. In the present configuration, the 
yaw jets are located on and are deployed with the low- 
gain antenna; the pitch and roll jets are located on a 
deployable boom (Fig. 11). 



REDUNDANT 
PRIMARY 
SUN SENSORS 




REDUNDANT 
SECONDARY 
SUN SENSORS 



Fig. 1 1 . Gas jet and celestial sensor locations 

During this phase, the purpose of the attitude control 
system is to reduce the initial tumbling rates imparted to 
the spacecraft at separation to within a controlled rate 
deadband. The attitude control loop (single axis) during 
this mode is shown in Fig. 12. Three single-degree-of- 
freedom high-gain gyros operating in a caged configura- 
tion provide rate damping by sensing the components of 
spacecraft rate about each axis. After the initial rates have 
been removed, the solar arrays are deployed. 

Acquisition. After reduction of the initial spacecraft 
tumbling rates and deployment of the solar arrays, sun 
acquisition will begin automatically. The sun senson, 
which have a 47r-sr field of view, provide the pitch and 
yaw position error signals. Redundant sun sensors are 
employed to improve the reliability of this primary sys- 
tem. The controlled sun acquisition rate corresponds to 
the saturated output of the sun sensor. The pitch and yaw 
sun acquisition rates will nominally be 2.0 mrad/s. After 
acquiring the sun, Canopus acquisition will begin auto- 
matically. Upon receipt of the Sun gate (sun acquisition 
signal), a calibrated command current is fed into the roU- 



66 



jn SfACE PROGRAMS SUMMARY 37-51, VOL. (// 



SWITCHING 
AMPLIFIER 




SOLAR ARRAY 
STRUCTURAL DYNAMICS 



M^S* + M^S^ -t- MzS^ + 4/,5 +1 



US{fi/^S* + /Vj5' + /VzS* + A^,5 + 1) 



CONTROL LOOP FOR INITIAL 
STABILIZATION AND ACQUISITIONS 



CELESTIAL SENSOR 




v + 








CONTROL LOOP FOR CRUISE PHASE 


DERIVED 
RATE 


CELESTIAL SENSOP 




*> 



















Ife ~ (fl, + kre,) 




'/ 






CONTROL LOOP FOR 
INERTIAL HOLD 
DURING OCCULATION 


GYRO AND 

CAGING 

ELECTRONICS 











Fig. 1 2. Single-axis attitude control loops 



switching amplifier. This signal causes the roll gas jets to 
fire and accelerate the spacecraft to a rate proportional 
to the magnitude of the command current. When the gyro 
feedback signal exactly balances the command current 
signal, the spacecraft is at roll search rate which is nomi- 
nally 2 mrad/s. The basic control loop is the same as dur- 
ing separation rate reduction and is shown in Fig. 12. 
Nominally, acquisition of the sun and Canopus will re- 
quire no more than 1.5 h. 



Powered flight phase. Upon compIetitMi of Canopus 
acquisition, the {."owered flight phase begins. He attitude 
control system during this phase and its operation in 
conjunction with the N, cold gas system is described in 
Section C. 

Cruise phase. Attitude control during the cruise phase 
is provided by the cold gas system. Hie basic system is 
identical to the Mariner system. A block diagram of die 



in SMCE PROGRAMS lUMMAkY 37-51, VOL. H( 



67 



cruise attitude control system is shown In Fig. 12. Dur- 
ing this phase of the mission, rate dampitig is provided 
by derived rate feedback around th i switching amphfier. 
Passive derived rate compensation is used instead of the 
gyros, primarily to improve the reliability of the system. 
In addition to the derived rate circuitry, the switching 
amplifier will incorporate a minimum on-time circuit. 

The operation of the cinuitry is such that when the 
celestial sensor output reacnes the deadband level of the 
switching amplifier, the amplifier is switched on for a 
time v-qual to the set mininium on-time. At the instant 
the amplifier turns on, the derived rate output builds up 
as a ramp function. At the er ' of the minimum on-time, 
the derived rate output volt;.ge is large enough to turn the 
amplifier off and keep it tamed off. The result is a stable 
and known controlled limit rvcK'. Deadbands of ±0.5 deg 
are used in all axes. 

Periodically throughout the mission, the sun sensor 
scale factor will be updated (through central computer 
and sequencer update commands) to counteract the op- 
tical gain reduction caused by the decreasing solar energy. 



is identical to that used on previous Mariner spacecraft 
during occultation. The Manner gyros are high-gain, 
narrow-angle, rate-integrating gyros. Attitude control is 
accomplished by caging the gyro through a capacitor lead 
network. To ensure that the proper spacecraft attitude 
and antenna pointing accuracy is maintained, a drift com- 
pensation scheme may have to be incorporated in the 
control system. This, of course, depends on gyro drift and 
the length of time the spacecraft is in occultati )n. Upon 
completion of occultation, reacquisition of the celestial 
references, if required, is performed in the manner de- 
scribed previously. 

Postencounter. The spacecraft remains on gyro inertial 
control until a sufficient postoccultation time period has 
elapsed to permit reacquisition of the celestial references 
without interference (stray light) from Jupiter. At this 
time, reacquisition of the sun and Canopus occurs in the 
manner described previously. Since the cold gas storage 
has been si. sd for a 1200-day mission, and lominal en- 
counter time is 900 days, the attitude control system will 
continue operating in the cruise mode for an additional 
300 days. 



Reacquisition. If loss of acquisition should occur, reac- 
quisition of the celestial references will be performed by 
the cold gas attitude control system. The system configu- 
ration is the same as during ir.'tial acquisition and is 
shown in Fig. 12. When loss of acquisition is detected 
by (he celestial sensor logic, the gyros automatically turn 
on and the acquisition sequence is initiated. If loss of 
acquisition occurs during the powered flight phase, the 
ion engines are first shut down and then the control is 
switched to the cold gas system. 

Encounter. Approximately 10 days before closest ap- 
proach, the ecliptic plane engine pointing control system 
(which serves the dual purpose of science platform point- 
ing) is slewed to a nominal science platform pointing 
position to ensure that the planet will be within the 
planet tracker field of view. The platform tracks the planet 
in two axes until approximately 45 min before closest 
appr )ach. At this time, the gyros are turned on in prepa- 
ration for the sun occultation mode. The effects of the 
Jovian radiation belt on optical sensor performance were 
not evaluated in this study. 

Occultation. During occultation, position reference can- 
not be maintained using the sun and Canopus. Attitude 
control during this phase will be accomplished by using 
the gas systems with the gyros in inertial hold. A block 
diagram of this system is .shown in Fig. 12. This method 



3. Cold Gas Attitude Control System Analysis and 
Description 

During all phases of the mission, other than powered 
flight, attitude control is provided by <.\e three-axis cold 
gas system. Figure 11 shows the location of the gas jets 



TWO- STAGE 
REGULATOR 



TWO -STAGE 
REGULATOR 



^ ^ ^ ^ ^ 
M ^ ^ ^ 









v_ 



y- 

SAME FOR YAW 




Fig. 13. Quad-redundant valve and gas jets 



68 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



and their lever aims to the spacecraft center of gravity. 
The gas jets are located in these positions due to shroud 
packaging constraints and to eliminate gas impingement 
on the solar arrays. The gas valves for each control axis 
are connected in a quad-redundant fashion (Fig. 13). 
Connecting the valves in this manner provides high relia- 
bility and eliminates the requirement for storing addi- 
tional gas to allow for a valve open failure. In addition, 
redundant sun sensors are employed to increase optical 
system reliability, and techniques employing triple re- 
dundancy will be incorporated to increase circuit relia- 
bihty. 

The attitude control position deadband 9db, consistent 
with attitude pointing accuracy requirements, is set at 
±0.5 deg. The gas jet minimum on-time i^t„n, which 
assures . s'ible and predictable limit cycle, is set at 
100 ms. Selection of this value is based on previous experi- 
cace iili this type of attitude control system. Conserva- 
tive estimates of the spacecraft moments of inertia h 
were determined as 

L = Iy= 14,216 slug-ft"" 

h = 29,653 slug-ft ■ 

Determination of the gas system thrust level Fj is based 
on a trade-off between limit cycle behavior, acquisition 
time, recovery from disturbances, and interaction between 
the control system and the spacecraft structural dynamics. 
Ideally, the thrust level is set so that the minimum dis- 
turbing torque Tp wiU cause an ideal soft limit cycle 
resulting in lower gas consumption and less valve actu- 
ations. The minimum To is due to solar radiation pressure 
and will occur at the maximum distance from the sun. 
For zero rate at one side of the deadband, 



Tci — FiLi 



(5) 



m 



= 2aD (26 ob) 



* i 



(1) 



(2) 



where 



Adi = minimum rate increment about tth control axis 
ao = angular acceleration due to disturbing torque 



Also, 



TciATon 



h 



(3) 
(4) 



where 



«ci = gas jet angular acceleration constant (each axis), 
i = x,y,z 

Tci = gas jet control torque about ith axis, i = x,y,z 

Li = gas je* lever arm for ith control axis, i = x,y,z 



Substituting, 






(6) 



Solving for the thrust level yields 

iiTaenBhY'^ 



Fi- 



LiATo 



(7) 



The minimum disturbing torque due to solar radiation 
pressure is determined from 

Ta^^^ = A, (1 + f,) ^^) Lo - 1.72 X 10-« ft-lb (8) 

where 



Ap = area of one solar array = 380 ft^ 

f B = solar array reflectivity coefiicient — 0.2 

Po = solar radiation pressure at 1 AU =9.72X 10-« Ib/ff 

Lo = disturbance torque effective lever arm = 1 f t 

This disturbing torque would primarily influence the 
limit cycle behavior about the pitch and yaw axes since 
the solar arrays he in the spacecraft x-y plane. Substitut- 
ing ^".n.n into Eq. (7) yields 

F, = F, = 0.09 lb 

F, = 0.151b 

The control angular acceleration constants are 

F^Lr 



Ctcx ^cy 



Otcz 



ix 

F,L, 



4.3X10-»rad/s^ 



-2.9X10-''rad/s=' 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



69 



Reduction of the separation rates is done prior to solar 
panel deployment. Due to the much smaller spacecraft 
inertias, the effective inertias are approximately 60 times 
greater than after solar panel deployment. For this rea- 
son, reduction of worst-case separation rates (3 deg/s) is 
accomplished in less than 1 min. A suitable value for the 
search rate 9, from which sun and Canopus acquisition 
will occur is 2 mrad/s. For sun acquisition, this corre- 
sponds to the saturated output of the pitch and yaw sun 
sensors. Acquisition of the sun in pitch and yaw will 
occur simultaneously and will take at the most 



t = 



2.0 X 10-' 



= 1570 s 



For Canopus acquisition, 



< = 



2,r 



2.0 X 10- 



= 3140 s 



Control system damping during acquisition is determined 
by the rate to position gain fer. This gain establishes the 
proper scaling between the gyro gain and celestial sensor 
gain. The criterion for determining this gain is as follows; 
when tlie celestial reference (sun or Canopus) enters its 
sensor field of view ^„, the gyro rate signal must be suf- 
ficiently large, relative to the position error signal, to 
activate the jets having the polarity that will decelerate 
the spacecraft from the search rate. For example, if B, is 
negative, then the positive gas jets should fire when 
B = Or. Referring to Fig. 12, the gas jets fire when 



Bob ~ Br ~ Kt 6, 



fcr- 



Bv- 



e. 



(9) 



(10) 



where 

Kog = derived rate gain 
Tc = derived rate charge time constant 

The derived rate damping factor to rate disturbances is 
defined as 



(12) 



where 



Bo = initial rate disturbance 

Br = retium rate after first excursion out of deadband 

To provide good limit cycle performance and reacquisi- 
tion capability, the derived rate time constants are usu- 
ally different for the charge (tc) and discharge (t^) cycles, 
and the output is clamped at some level lower than the 
full scale output aKoR- The following relationships can 
be derived for determining the derived rate parameters. 
The relation between the limited derived rate output Bdrl 
and damping y can be shown to be 



6drl = {Bv- Bob) {2 -y)y 
The derived rate gain can be determined from 



Ocf'-DIC 



1 



-[(^)(^)] 



(13) 



(14) 



A lower limit on the discharge time constant can be 
determined from 



Substituting the nominal parameter values, fcr = 40 s. '■''n,in ~ "2 \q ) ( j _ ) Y (^»b/«c)''^ (15) 



Upon completion of the powered flight phase, control 
is switched back to the gas system in the manner de- 
scribed previously. During the cruise phase, the gyros 
are off and rate damping is provided by derived rate feed- 
back as shown in Fig. 12. Selection of appropriate de- 
rived rate parameters is based on (1) providing a high 
degree of damping and thereby good reacquisition capa- 
bility to rate disturbances, and (2) assuring stable mini- 
mum impulse limit cycle operation. The derived rate 
output, when the switching amplifier fires, is 



Bob ^KoRadl -€'/■'') 



(11) 



The above equations can be solved parametrically for 
different values of y and Ti/rc. This was done and the 
results verified through computer simulation. The selected 
values are 

y=0.5 

^oRL — ctc^nit = SOmrad 
Tc = 50s 
Ti = 100s 



70 



JPL SPACE PROGRAMS SUMMARY 37-5?, VOL. /// 



To verify the control system analysis and to investigate 
the dynamic interaction between the control system and 
the spacecraft structure, a six-degree-of-freedom com- 
puter simulation program was written to assess the over- 
all system performance. In addition, a digital computer 
program wa^ written to facilitate the analysis of the atti- 
^Jde control gas storage requirements due to the many 
system iterations that were made in the course of deter- 
mining the baseline system. The gas system for the base- 



line system is sized for a 1200-day mission. The required 
initial gas storage weight is 20 lb. 

A future SPS, Vol. Ill, article will present (1) typical 
results from the six-degree-of-freedom computer simula- 
tion program and from the gas storage analysis computer 
program, and (2) a description of the various alternate 
system configurations that were examined during the 
course of the study. 



JPL SPACE PROGaAMS SUMMARY 37-51, VOL. Ill 



71 



^ * 



^66 






/^O 



VII. Guidance and Control Research 

GUIDANCE AND CONTROL DIVISION 



A. Josephson Junction Memory Elements, 

p. V. Mason 

1. Introduction 

A previously reported study (SPS 37-44, Vol. IV, p. 57, 
and SPS 37-46, Vol. IV, p. 97) led to the conclusion that the 
most useful application of superconducting phenomena 
on board spacecraft is probably in the area of high-density 
and/or high-speed computer memory and logic devices. 

At present, the superconducting memory closest to 
actual application utilizes the cryotron, a device based on 
the superconducting-to-normal transition in a magnetic 
field. Several laboratories have devoted considerable effort 
to the development of such memories (Ref. 1), and it now 
appears that the obstacles to practical use are those ot 
economics and production rather than of fundamentals. 

Another type of cryogenic memory, which should 
provide extremely short cycle times, is based on the 
Josephson effect, as described by J. Matisoo (Ref. 2). The 
primary reason for interest in such a device is the very 
high switching speed. Matisoo has shown the switching 
time to be less (probably much less) «^han 0.8 ns. Since 
such junctions also lend themselves to batch fabrication 
by microcircuit techniques, they seem to be very attrac- 
tive devices for high-speed, high-density, low-cost-per-bit 
memory. 



72 



2. Functional Description 

The basic element of such a memory is a junction 
formed of two superconductors separated by an insulat- 
ing layer a few tens of angstroms thick. Such a junction 
can pass current in two different modes: (1) a zero voltage- 
drop superconducting tunneling mode, and (2) a finite 
voltage-drop normal tunneling mode. The current vs 
voltage (I vs V) characteristic of such a junction, taken 
from a junction made in the laboratory, is shown in 
Fig. 1. As the current increases from zero, the junction 
conducts without voltage drop until a critical junction cur- 
rent (Ij) is reached. There is an abrupt transition to the 
normal conduction mode, with voltage drop about equal 
to the superconducting energy gap (E,) cf the metal 
forming the junction. As the current is further increased, 
the voltage changes little until the line representing the 
ohmic drop of the normal junction is reached. If the cur- 
rent is now reduced, the junction follows a steep line of 
low resistance (about 1.4 n for the junction shown here) 
to a low current. A moderately abrupt transition to the 
superconducting mode then takes place. Reverse current 
yields identical behavior in the negative region. 



The current Ij is, in theory, given by the equation 



I. = 



4R„ 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. HI 




■A 



A- 



1^ 



■ si 



Fig. 1. Cun-«n^ vs V >ltage for Jotephson junction 72-1 

where R„ is the normal resistance of the junction at the 
operating temperature but a lower value is usually ob- 
tained in practice. Also, Ij depends on magnetic field as 
shovn in Fig. 2, which is taken for another of our diode^. 
The periodic minima occiu* at fields satisfying the condi- 
tion 

* = BA = ^ n - n 2.1 X 10"' G cmS n - 1, 2, 3, • • • , 

where 9 is *otal fiux, B is magnetic field, and A is the 
cross-sectional area of the function normal to the field. 



soo 






400 


- 


n;-- \ 


300 


- 


/ 


200 


- 




100 




..^W^V^ 


! 


'i VtV~N-'-t 



-zoo -ISO -wo -90 

0, mC 



SO 100 ISO 200 



Fig. 2. Dopondonco of If on applied mognotic 
fiold-iunction 28-1 



It is now observed that, if a bias current /o that is less 
than Ij is applied, the junction may be in one of two 
.stable states (indicated by 1 and 2 in Fig. 1). If we are 
at 1, we may switch to 2 by .pplying a switching pulse J, 
such that Jo + /* exceeds Ij, ana we may switch back to 1 
by reducing 7o to zevo as, for example, by applying a 
pulse /, > lo in the reverse direction. 

Another, and more uf eful, means exists to switch the 
junction from 1 to 2. By n ans ot an external field (which 
will usually be generated by a current in a nearby wire), 
1 1 may be reduced below its zero field value Ij max- If we 
reduce Ij below /„, the junction must switch to 2. Thus, 
we have the necessary elements of a three-tenninal, bi- 
stable device capable of serving as a memory element. 



3. Experimental Program 

a. Fundamental measurementa. Several fundamental 
questions thai arise are as follows: 

(P What is the actual switching speed? 

(2) What physical process determines it? 

(3) Are we process limited or circuit limited? 

(Measurements so far are circuit limited, but there are 
indications that the fundamenta] limitation is less than 
the present measurement limitation of 0.1 ns). 

In order to answer these questions, sevcal test samples 
were fabricated with the junctions in the center of a 
thin-film superconducting transmission line (see Fig. 3a) 
in order to reduce eflEects of the circuit on the measured 
rise times. However, considerable diflSculty was found in 
obtaining good losephson characteristics. Because the 
fabrication in transmission-line form is complicated and 
therefore slow, it was decided to experiment with fabri- 
cation techniques in a simpler crossed-film form (see 
Fig. 3b). As fabrication methods improve, it will become 
feasible to return to the measurement of switching speed. 

If switch ing speeds are slow enough to me»<!ur»: with 
present experimental techniques (about 0.1 lu), an at- 
tempt will be made to correlate them with theory. If they 
are faster, this, of coutse, will be impossible, but the 
device applications will be of even more interest. 

b. Selection of materials and fabrication techniqtie$. 
The selection of materials must be based on ease of fabri- 
cation in thin-film form, high superconducting transition 
temperature (in order to minimize cryogenic refrigerator 
power and weight), and reliability. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. HI 



73 



(a) 



GLASS SUBSTRATE 



(b) 



0.5 mm 




— JOSEPHSON JUNCTIO^ 



SILICON 
MONOXIDE 
INSULATOR 

LEAD BASE 
ELECTRODE 



LEAD COUNTER 
ELECTRODE 



LEAD BASE ELECTROD? 



0.5 mm ^ LEAD COUNTER ELECTRODES (6 EACH) 

Ffg. 3. Test sompU configurations: (a) transmission-lino 
tost sample; (b) ladder-form test sample 



A number of methods of forming thin insulating films 
are available. These include the following: 

(1) Chemical reactions with the metal (e.g., oxidation 
or nitridization, both with and without voltage- 
induced reactions). 

(2) Deposition of insulating materials (e.g., silicon 
monoxide by vacuum evaporation). 

(3) Polymerization of organic materials on the surface 
(e.g., silicone pump oil by electron bombardment). 

In general, the simplest method is the oxidation of the 
metal surface, which is the method most used by those 
investigating Josephson junctions. Therefore, for this 
study, the oxidation process was chosen to fonii the 
insulating layer. Since oxidation proceeds less rapidly as 
the oxide becomes thicker, the process tends to be self- 
healing and self-limiting with time. Thus, pinhole-free 
films of uniform thickness should be the end result of 
the process. It was also decided to begin with thermal 
oxidation, rather than anodization (i.e., voltage-induced 
oxidation) because of its relative simplicity, although it 
will probably be necessary to also experiment with 
anodization. 



We chose to begin our investigations using vacuum- 
evaporated lead. Lead has two major advantages. First, 
its critical temperature (T^) is 7.2°K, second among the 
elements only to niobium, whose T^ is 9.2°K. Second, 
being a low-melting-point material, it is far easier to 
deposit than niobium. Lead can be easily evaporated from 
a resistively heated boat at temperatures of SOO-SOCC, 
while niobium must be evaporated from a high-power 
electron gun at 2500-2600''C. Furthermore, the deposition 
of niobium must be done either in an ultra-high vacuum or 
onto a heated substrate in order to obtain good supercon- 
ducting properties. The high Tc of lead has another aQ^'an- 
tage; since the energy gap is directly proportional to r^, 
junctions made of lead have a relatively large output 
signal (about 2.6 mV) in the normal conducting mode. 



Tne crucial step in the fabrication process is the 
formation of the insulating film that separates t^e metal 
conductors. This film must be, uniformly, a few tens of 
angstroms thick, free of pinholes that would permit the 
formation of superconducting bridges between the metal 
electrodes, and must maintain constant electrical proper- 
ties in storai^e and operating environments. Furthermore, 
the electrical properties must be quite uniform over a large 
number of samples, since wide variations over the large 
number of elements would make the memory inoperable. 



The process variables under our control are tempera- 
ture, relative humidity, £;nd time. Pi^sumably, for rea- 
sonable ranges of time, the chemical and physical nature 
of the film should be fairly uniform, and the film thick- 
ness should simply increase monotonically. For the tem- 
perature range used (20 to 100°C), a logarithmic time 
dependence and a temiination of growth at a few tens 
of angstroms were exjiected (Ref. 3). 

Likewise, dependence on temperature might be ex- 
pected to be straightforward, that is, faster rates and 
thicker films would be produced as the temperature 
increases. There are possible complications, however, in 
that lead has a number of oxides, and it is entirely pos- 
sible that growth of one or the other could be favored 
at ditferent temperatures. 

The dependence on relative humidity is likewise com- 
plicated. The production of hydrates at high humidities 
could (and probably did) lead to p(X>r insi;lators. 

A number of films were made under various condi- 
tions of the parameters. Temperatures ranged from room 
temperature to 100°C with relative humidity from 10 to 
80% ; time ranged from 6 min to 24 h. It was found pos- 
sible to form excellent films under nearly all conditions 
of temperatiu-e and humidity by adjusting the time, but 



74 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



reproducibility was very poor. Some general conclusions 
were: (1) Humidities above 70% usually give poor re- 
sults, probably due to formation of lead-oxide hydrates. 
Humidities below 25% take an excessively long time to 
form an oxide. Between 30 and 50% gives highest yield. 
(2) Temperatures above 50*0 seem to lower the yield, per- 
haps because of diffusion of lead into the oxide. In this 
connection, it should be remarked that we have found 
that it is necessary to store good diodes at liquid-nitrogen 
temperature in order to preserve their characteristics. 
This strongly suggests that a temperature-induced dif- 
fusion process is at work, and also thai it would be very 
desirable to find an insulator with more stable properties. 
Anodization is known to produce stable films, especially 
on the harder materials such as niobium and tantalum. 
Thus far, room-temperature oxidation has not been 
explored except to note that it takes an inconveniently 
long time to form a film. 

Films with excellent characteristics, with Ij ranging 
from a few microamperes to 1.7 mA, have been made. 
Attempts to produce higher Ij have invariably resulted 
in poor characteristics, probably as a result of supercon- 
ducting bridges across the insulating film. 

Refersncci 

1. Sass, A. h., Stewart, W. C, and Cosentino, L. S., "Cryogenic 
Random-Access Memories," IEEE Sped., Vol. 4, p. 91, July 

1967. 

2. Matisoo, J., 'The Tunneling Cryotron-A Superconductive Logic 
Element," Proc. JEEE, Vol. 55, p. 172, 1967. 

3. Kubaschewski, O., and Hopkins, B. E., Oxidation of Metah and 
Alloys, Second Edition, p. 39, The Academic Press, New York, 
1962. 



B. Frequency Response of Thin-Film Thermal 

Detectors, J. Maserjian 

1. Introduction 

The response of a thermal detector to radiant energy 
is a two-fold process involving, first, the temperature rise 
resulting from the absorbed radiation and, second, the 
conversion of the temperature rise into a useful output 
signal by the active detector clement. A new kind of 
thin-fihn thermal detector, discussed previously (SPS 
37-41, Vol. IV, p. 115; SPS 37-47, Vol. Ill, p. 44), derives 
much of its sensitivity from the large temperatuio re- 
sponse possible in a thin-film structure. For a practical 
design, it is important to consider this temperature re- 
sponse in detail and, in particular, its dependence on the 



modulation frequencies of the incident radiation. This 
article summarises our analysis of this problem. 

2. Tho Thin-film Structuro 

The structure under consideration, which closely 
matches the experimental structure, is a thin film sus- 
pended across a hole in a supporting frame that is main- 
tained at a fixed temperature To (Fig. 4 inset). Radiant 
flux Qo + Q cos lot, containing a harmonic component 
Q (^ Qo) with angular frequency oi, is absorbed over a 
disc of radius a less than or equal to the radiu: h of the 
hole. Only thermal conduction along the film to the frame 
is considered. The Blm is assumed to he i» an evacuated 
chamber r>o that radiation is the only additional mecha- 
nism of heat loss; however, this becomes significant only 
in extreme cases which may then be considered sep- 
arately. If the film is composed of layers of diltercnt 
materials, one is still free to use effective values for the 
thermal parameters to describe the composite film. We 
start with the diffusion equations 



KV=1 - DC 






f or < r < fe 



(1) 



K V -T + Qo-r Q cos <at = pc 



dt 



for r < a 



10" 



^ 10" 



lo-i 

6 

4 

2 

io-» 



























































1 










b/o 


= 00 








^0.308-^ 

1 










10 


"*^ 




; ^. 1 


















5 






■*^ 














V^ 


^ 






-/9-* 








? 










^ 


V 


\u- 




















*k 


^V\ 










1 














^ 


k 




1 


















--3 


Oo 


+<; 


'COI 


\ uit 










\ 






^cc 










\ 






~-l*l---~ 










^ 


i 


/^^^^^^''X 










\ 




f ^— y-y.-- 


1 

i 










\ 


























\ 



I0-* 2 4 6 W-' 2 4 • to* 2 4 6 O' 

Fig. 4. Froquoncy retponso of lliin-film thomial dttoctor* 



m SMCE noGRAm summary 37-51, vol. III 



75 



with the boundary condition T{r = b) — To, where K, 
p, and c are the effective values of the thermal conduc- 
tivity, density, and specific heat, respectively. We seek 
the steady-state solution which is obtained at time -> oo 
and consists of only the particular solution to the dif- 
ferential equations. The temperature may be assumed 
uniform throughout the thickness of the film r, and with 
the radial symmetry, the Laplacian operator in Eq. (1) 
reduces to one dimension involving r in cylindrical co- 
ordinates. The general solution for arbitraiy a and b has 
been obtained, and the amplitude of the harmonic com- 
ponent j AT 1 at r = is plotted in dimensionless form 
for several ratios of b/a, where k is the effective diffusivity 
of the fihn (k = K/pc) and fi is the ratio of the radius a 
to the diffusicm length [(«/«)^]. 

The general solution, expressed in terms of the tabu- 
lated Kelvin functions, is rather cumbersome and will 
not be reproduced here. However, the solution reduces 
to a much simpler form for r = and b/a = «, the har- 
monic component being given by 

Ar(0) = -2-^^cos(u-«) 



vKt pcose 



= tan-' { — 



pkeT'fi \ 



P keV p y 



(2) 



where iei' and hef, the first derivatives of the Kelvin 
fimctions Jtet and \xr, are tabulated by H. B. Dwight 
(Ref. 1). The low- and high-frequency asymptotes are 
plotted as dashed curves in Fig. 4. The solutions for 
finite ratios of b/a are seen to follow a nearly constant 
plateau from their low-frequency limit until intersecting 
the above solution, after which they rapidly merge. The 
low-frequency limit may be readily calculated for arbi- 
trary r < « as follows: 



-«=:i^[i-(iy 



+ 2 



'"^] 



for< 



= 

(3) 



3. Obssrvations and Conclusions 



Some important observations can be made from these 
results. First of all, the solution is exact for the type of 
structure considered, and differs significantly from the 
approximation often made by assuming a fixed value for 
the thermal relaxation time. In such approximations, the 
relaxation time is calculated from the ratio of the value 
of the thermal capacitance o^ the irradiated region to the 
value of the low-frequency thermal conductance between 
this region and the heat sink — the calculation in this case 



giving a*(l 4- 2 In &/a)4K. T..e dependence <rf tiie ampli- 
tude, in terms of /3, then becomes 

which is a fair approximation onit/ for the particular case 
of b/a — 2. Thus, one cannot, in general, characterize 
the thermal response of such a structure by a relaxation 
time. 

Secondly, the curve in Fig. 4, given by Eq. (2) for 
b/a = 00, may be considered an envelope that essen- 
tially encompasses solutions for all finite values of b/a. 
Therefore, if one wishes to detect radiation modulated at 
a given frequency (or fi), there exists a minimum ratio, 
b/a, above which one obtains aproximately the same 
response at this frequency. This minimum ratio b/a cor- 
responds to that curve in which its low-frequency asymp- 
tote intersects the envelope at this response. Larger 
values of b/a add little to the detector s response at this 
frequency, but may increase the fragility and fabrication 
difficulties of the detector. 

The effects of the constants K, «, a, and t are also note- 
worthy. The response is seen to be inversely proportional 
to the thickness t independent of frequency. Thus, it is 
highly advantageous to make the composite film as thin 
as possible. The mechanical limitations th» increase the 
importance of using the smallest radius b consistent with 
the operating frequency, as discussed above. The re- 
sponse also appears to be inversely proportional to the 
thermal conductivity K; however, this is actually true 
only at low frequencies. At high frequencies, the response 
approaches the ^"^ asymptote which, when expressed in 
terms of the constants, gives 



ATI 



ita pc T<o 



for (u > > a-A 



which is independent of K and depends instead on the 
product pc. In this case, the harmonic component of heat 
is entirely contained in the irradiated region, and the 
temperature change is determined only by the thermal 
capacity of the region. Also, the response decreases ac- 
cording to a more rapid \/f dependence in this range; 
however, this may still be a useful range, particularly 
when the signal bandwidth is of primary importance, or 
when excess noise of a \/f dependence is present at low 
frequencies. The onset of this high-frequency limiting 
dependence is seen from Fig. 4 to occur at j8 «= 2. If an 
area of 10-* cm* is assumed, this value of p corresponds 
to frequencies ranging up to about 20 Hz for dielectrics 
and 200 Hz for metals. If the thermal response of the 



76 



in SPACE PK06RAMS SUMMARV 37-51, VOL. »> 



suspended thin-film detector in this high-frequency limit 
is compared wi^h that of a thin-film detector in direct 
contact with an insulating substrate, the advantage is 
still maintained up to much higher frequencies where 
the thermal difFusion length becomes comparable to the 
film thickness (> 10« Hz for 2000 A film). 

Reference 

1. Ehvight, H. B., Tables of Integrab and Other Mathematical. Dita, 
pp. 278-279, MacMillan Company, New York, 1957. 

C. GaSe Schottky Barrier Gate, S. Kurtin and 

C. A. Mead' 

The Schottky barrier gate (Ref. 1) is ideal for the con- 
struction of field-eflfect devices since it avoids the difiB- 
culties of p-n junction formation, particularly in 



'Performing work supported by JPL at the California Institute of 
Technology. 



wide-band-gap materials, and the Schottky barrier deple- 
tion layer is not affected by the presence of surface 
states. A properly-formed Schottky barrier has nearly 
theoretical reverse current and does not exhibit the drift 
and instability problems associated with metal-oxide 
semiconductor structures. Hence, the Schottky barrier- 
gate technique can be employed to construct active 
devices from materials which cannot be otherwise utilized. 

GaSe (Refs. 2 and 3) is a layer semiconductor having a 
2-eV band gap. A recent study of surface barriers on 
GaSe (Ref. 4) indicates that the advantages of the 
Schottky barrier-gate technique will allow the construc- 
tion of a field-effect device from this material. 

Experimental devices were constructed from approxi- 
mately 8-/im-thick cleaved layers of p-type (p ~ 10"/cm') 
GaSe. A schematic cross section appeal's in the inset of 
Fig. 5. The source and drain ohmic contacts were alloyed 



60 



50 



40 



iO 



< 
4. 



S^ 



20 — 



-10 



/ ALUMINUM GATE 






l^y, V = 


^^ — OHMIC \ 

CONTACTS—^ ^ 










/ 






^ 


/ 


'y^^ 








~//^ 




- — 






£^ 


— — -^ 






20 

















20 



30 



40 



50 



I'zj.v 



Rg. 5. Electrical characteritHcs of GaSe Schottky barrier gate 



ifl SMCE PROGRAMS SUM/MARr 37-51, VOL. Ill 



77 



Zn-Au spaced 0.5 mm apart; the width of the device was 
3 mm. An aluminum gate, 0.1 mm across, was evaporated 
directly onto the freshly cleaved surface. Open gate 
channel resistance was 300 kn. The /drain-Vrfrain curves 
are shown in Fig. 5. Observed transconductance and 
pinch-off voltage agree well with those calculated for the 
materials and geometry employed. Channel depth was 
measured optically, and carrier concentration determined 
from the capacitance-voltage characteristic of the gate- 
channel barrier. Note that the zero-bias transconductance 
is equal to the channel conductance at small drain voltage. 

References 

1. Mead, C. A., "Schottky Barrier Gate Field Effect Transistor," 
Proc. IEEE, Vol. 54, p. 307, 1966. 

2. Fisher, G., and Brebner, J. L., "Electrical Resistivity and Hall 
Effect of Single Crystals of GaTe and GaSe," /. Phys. Chem. 
Solids, Vol. 23, p. 1363, 1962. 

3. Leung, P. C., Andermann, G., Spitzer, W. G., and Mead, C. A., 
"Dielectric Constants and Infrared Absorption of GaSe," 7. Phyi. 
Chem. Solids, Vol. 27, p. 849, 1968. 

4. Kurtin, S., and Mead, C. A., "Surface Barrier on Layer Semicon- 
ductors: GaSe," /. Phys. Chem. Solids (in press). 



D. Metal Contacts to Photoconductors, R. J. Sfim 

1. Introduction 

Recent developments in the physics of metal- 
semiconductor contacts indicate that current models of 
photoconductors may have to be re-evaluated. It has 
been found that photoconductive gains greater than 
unity are possible even when the contact is blocking, 
i.e., when the conduction electrons in the metal are 
separated from the photoconductor majority carriers by 
a potential barrier (depletion layer). 

All photoelectric devices, in which the injection of 
carriers is controlled by ohmic or blocking contacts, 
depend on the metal contact properties. The injection of 
carriers may give rise to injection luminescence by visible 
radiative recombination, and, in the case of the illumi- 
nated metal-photoconductor contact, the photovoltaic 
effect shows promising application for large-area solar 
arrays. The perfonnance of a photoconductor also de- 
pends critically on the type of metal contact. Since it now 
appears that the degree of blocking of "blocking con- 
tacts" to a photoconductor is dependent upon the illumina- 
tion, as well as the photoconductor surface history, further 
investigations of metal contacts to photoconductors are 
being carried out at the Laboratory and by Prof. K. W. 
Boer and his group at the University of Delaware. 



In this article, the concept of photoconductive gain 
and the general model of bloclring contacts on lightly- 
doped semiconductors are briefly reviewed, plus methods 
for determining the potential barrier height of a metal- 
semiconductor contact. The results of thfse methods for 
various metals on cadmium sulfide (CdS) crystals will be 
presented in future articles, along with preliminary re- 
sults from an analysis using stationary high-field domains 
in the range of negative differential conductivity. The 
CdS is the photoconductor of greatest interest because 
of its very high light-to-dark ratio of current, and its 
sensitivity in the visible region of the spectrum. 

2. Photoconductive Gain 

For a photoconductor of unit cross-section exposed to 
a uniform area excitation which generates free electrons 
at a total rate of F/s, the total number of electrons of 
charge q in the steady state is 

N^Fr (1) 

where t is the electron lifetime. The photocurrent Ip is 

/p = Nq/Tr (2) 

where T, is the transit time from the cathode to the 
anode.^ For an electrode separation L, applied voltage V, 
and free carriers with themial velocity v and mobility /i, 
the transit time is given by 



Thus 






I, = qFG=^V 



where the photoconductive gain G is defined by 






(3) 



(4) 



(5) 



It can be seen that, for a fixed geometry and voltage, the 
gain is affected by the material parameters t and ya. Up 
to this point, it has been assnmed that both contacts are 
ohmic, i.e., the carriers are free to leave and enter the 
crystp' without encountering any potential barrier (due 



'Since it is CdS that is being considered, a material in which the 
hole mobility is very much lower than the electron mobility, any 
hole contribution to the photocurrent will bo neglected. 



78 



i?l SPACE PROGRAMS SUMMARY 37-51, VOL. »l 



to a high metal work function or to surface states on 
the photoconductor). In order to allow for possible con- 
tact effects, the gain G is now viTitten in an equivalent 
out more operational sense: G defined by the ratio of 
the photocurrent to the total number of photons absorbed 
per second multipUed by the electron charge q. Thus 



G = 1,/aqLW 



(6) 



wLers W is the width of the crystal, and a is the number 
of photons absorbed per unit cross-section per second. 
Measured this way, gains much larger than unity have 
been reported for CdS with gold contacts,' though, as 
shall be seen, gold is considered to be highly blocking 
on CdS. 

3. Blocking Contacts en Lightly-Doped Semiconductsrs 

When a clean metal surface is bi ought closer and 
closer to a clean semiconductor surface while maintain- 
ing an electrical circuit between them, the electric field 
between them, due to the difference in the respective 
work functions of the materials, ind ices an electric 
•charge on, or near, the two surfaces. In the semicon- 
ductor, this charge can be manifested by (1) a space- 
cha-'ge layer caused by ionized impurity (donor) atoms, 
and (2) a surface charge induced in surface states. The 
surface staJes arise from the termination of the crystal 
lattice (Tamn; states), and possibly from an impurity 
interfacial layer. The role of these surface states in 
affecting the barrier energy depends upon their number 
and relative energy wf^h respect to the Fermi level. In 
CdS, or other more ionit crystals, Tamm states appear 
to play a minor role (Ref. 1;. 

If the surface states are negligible, the barrier height 
<t>B, as seen from the metal side, obtained when the metal 
surface is in intimate contact with the i°miconductor, 
should be simply 



4>B — 4'm ~ E\ 



(7) 



where i^m is the metal work function and £j is the elec- 
tron affinity of the semiconductor (Fig. 6). The total 
potential energy change that the carrier must have in 
passing through the depletion layer (formed when E^ 
< <i>„) is <t>„ — <^„ where <^, is the semiconductor work 
function. This rise in potential energy manifests itself in 
the semiconductor by the diffusion potential q Vd — 



'Boer, K. W., and Voss, P., "Light Dependence of an Effective Work 
Function of Gold Contacts on Photoconducting CdS," to be pub- 
lished. 



<^B — <^n, where <^„ is the Fermi energy measured from 
the conduction band edge. Th" solution of Poisson's 
equation for the depletion layer (Ref. 2) yields the 
following relations for A^,, the thickness of the barrier; 
E,, the electric field at the contact; and \li,{x), the poten- 
'al energy in the barrier 



- 2« ~1 ' » 


(8) 


^st = (<^B ~ <^b) 


(9) 


*r{x) = ^{x-KY 


(10) 



and 



In these expressions, c„ is the static-semiconductor dielec- 
tric constant and 2Vo is the ionized donor density 
assumed to be equal to the total impurity density in 
the depletion layer. When bias V is applied to the 
semiconductor, qV is simply added in the parentheses 
in Eqs. (8) and (9). 

An image force correction lowers the barrier height, 
as shown by the dashed line in Fig. 6. The change in 
barrier height, A<f,B, resulting from this correction and 
from the application of external voltages, is given by 
(Ref. 2) 



A<^B 



^ r q'Np {^B - 4>n ^ qV 



kT)T 



(11) 



where e« is the high-frequency semiconductor dielectric 
constant. The term kT arises from the contribution of 
the mobile carriers to the electric field. 

Turmel penetration of the top of the barrier can be 
expressed as an apparent lowering of the barrier given 

by (Ref. 2) 



A<^B — XcE. 



(12) 



where E, is the surface electric field given by Eq. (9) 
and Xc is a critical tunneling length of about 10"' cm. 
This last expression gives only an approximate estimate 
of the actual importance of tunneling. 

A correction factor, which would be important if 
surface states are present, has recendy been suggested 
(Ref. 3) based on a surface-state model by Heine (Ref. 4). 



jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



79 



VACUUM 



METAL 



SEMICONDUCTOR 




Fig. 6. Scholtky model for metal-semiconductor contact with zero applied bias 



The average volume charge density of these states can 
be written 



p - -^-^ exp {- x/d) 



(13) 



where N, is the number of surface states per unit area, 
and d the penetration distance of the charge in Aese 
states (equal to about 5 X lO"' cm). If this charge con- 
tribution is included in Poisson's equation, the lowering 
of the barrier height is calculated to be 



£l4,b = dE, In (qN,/CoE,) 



(14) 



where E, is given by Eq. (9). 



As seen in Eq. (8), the barrier thickness decreases as 
the impurity concentration increases. For No > 10*^ cm-', 



Ao is small enough that the barrier presents a finite trans- 
parency to electrons with energies lower than <^b; i.e., the 
electrons can tunnel through at energies near the Fermi 
level (field emission) and at energies above the Fermi level 
at temperatures above absolute zero (thermionic field 
emission). Theories have been developed which satisfac- 
torily account for the obiserved current-voltage character- 
istics (Ref. 5). However, in the case of more lightly -iloped 
semiconductors, which includes photoconductors, 
Schottky barriers are not as well understood for reasons 
discussed in Subsection 4. 

4. Techniques and Analyzing Barriers 

One important tedmique used in analyzing barriers 
(principally GaAs and Si) is the measurement of the 
/ — V characteristic of the contact. It has been found 



80 



JH SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



empirically that the current density can be given by 
(Ref. 6) 



(15) 



In Eq. (15\ the contact area is S, To is a temperature- 
independent parameter that varies from contact to 
contact, and A* is the Richardson constant for the semi- 
conductor modified to take into account the temperature 
variation of the barrier height. This expression would be 
identical with the theoretical expression obtained from 
the so-called diode theory (Ref. 2) for an ideal Schottky 
barrier if To were identically zero, and if A* were equal 
to the Richardson constant A= A^qm'k'/h" (where k is 
Boltzmann's constant and m* is the effective mass of the 
electron). That the two expressions are not the same 
results from the following observations: 

(1) At any voltage and temperature, the experimentally 
measured forward current (semiconductor positive) 
is higher than that predicted by the diode theory. 

(2) The rate of increase in current with applied bias is 
smaller than the predicted rate. 

(3) The difference between the experimentally meas- 
ured and theoretically predicted currents becomes 
larger as the temperature and bias become lower. 

(4) The experimental / — V characteristics become in- 
dependent of temperature at low temperatures. 

These last points suggest that quantum mechanical tun- 
neling is present to a much larger degree than expected 
for a large barrier thickness [derived from Eq. (8) for 
No < 10" cm-'], xt is now thought that the actual barrier 
length Ao is much smaller than formerly believed. This 
could very well be due to the presence of deeper traps 
lying above the Fermi .level in the vicinity of the contact 
and are thus ionized. These should be included in the 
value of No-* Such traps are indicated by the squares in 
Fig. 6. Since traps are so very important in the II-VI 
compounds, such as CdS, this point may be quite impor- 
tant in future work on CdS. The effects of these traps 
can be included, in principle, by capacitance measure- 
ments made at very low frequencies, i.e., with periods 
longer than the trap relaxation times. 

The mention of this last type of measurement leads to 
a second means of investigating the barrier of a metal- 



*F. A. Padovam, private communication, 196S. 



jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



semiconductor contact. Since the barrier width Ao changes 
with changing applied bias, the space-charge layer can 
be represented by an effective capacitance (per unit area) 
C = €„Ao = €o (dEJdV). From Eqs. (8) or (9), vdth an 
applied bias V, we obtain 



2N/,€« 



2(<t,B - <f>n + qV) 



(16) 



Thus, a plot of l/C" vs reverse bias (—V) will give tlie 
diffusion potential qVo = ^s — <t>« as an intercept and 
the carrier concentration Nj, from the slope. 

A third, and perhaps the most useful, technique for 
measuring barrier heights is the measurement of the 
photoresponse of the barrier. When light is incident upon 
the contact, either entering from the semicondu'Jtor side 
(back-wall configuration) or through a semi-transparent 
metal contact, the following two distinct photoexcitation 
processes can occur: 

(1) Photoemission of electrons in the metal over the 
barrier. 

(2) Excitation of carriers in the semiconductor from 
either band-to-band transitions, or from impurity 
levels within the forbidden gap. 

If one eliminates the possibility of process (2) by choos- 
ing wavelengths greater than that corresponding to the 
band gap, and reducing the amount of light entering the 
crystal by using the front-wall configuration and a fairly 
opaque metal contact (but not so thick that the hot 
electrons cannot reach the interface because of inelastic 
scattering), one can obtain the barrier height from 
process (1). For photon energies greater than a few kT 
above the barrier height, the photocurrent will bo pro- 
portional to the square of the photon energy, and an 
extrapolation to zero response will give the energy of the 
barrier potential. 

Using two of these three techniques will, in principal, 
give self-consistent information about the barrier. The 
use of these mea.surements on photoconducting (and, 
thus, highly insulating) semiconductors entails special 
problems besides those encountered with normal semi- 
conductors and are not mentioned here. Future articles 
vsdll go into more detail on these problems in regard to 
the CdS investigation. 

5. Stationary High-Field Domain Analysis in CdS 

In addition to one or two of the above techniques, it 
is hoped that investigations using stationary high-field 



81 



domains in the range of negative differential conductiv- 
ity will be useful for analyzing the metal-photoconductor 
interface in CdS. This technique, many aspects of which 
are currently being investigated at the University of 
Delaware, will be presented in the next article along 
with data obtained at metal contacts evaporated on air- 
oleaved CdS crystals. A review of the literature, with 
regard to experimental determinations of barrier heights 
on CdS for different metals, will also be presented. 

Work is progressing to devise a system to vacuum- 
cleave crystals of CdS before depositing the metal. 
Metal contacts made in this manner should not have any 
interfacial layer, such as an oxide, and thus allow some 
comparisons to be made between the high-field domain 
analysis and any of the three techniques discussed in 
this article. 

References 

1. Mead, C. A., "Surface States en Semiconductor Crystals; Barriers 
on the CD( Se:S ) System," Appl. Phys. Lett., Vol. 6, p. 103, 1965. 

2. Henisch, H. K., Rectifying Semiconductor Contacts. Clarendon 
Press, Oxford, England, 1957. 

3. Parker, G. H., McGill, T. C, Mead, C. A., and Hoffman, D., 
"Electric Field Dependence of GaAs Schottky Barriers," Solid 
State Electr., Vol. 11, p. 201, 1968. 

4. Heine, V., "Theory of Surface States," Phys. Rev., Vol. 138, 
p. A1689, 1965. 

5. Padovani, F. A., and Stratton, R., "Field and Thermionic-Field 
Emission in Schottky f arriers," Solid State Electr., Vol 9, p. 695, 
1966. 

6. Padovani, F. A., and Sumner, G. G., "Experimental Study of 
Gold-Gallium Arsenide Schottky Barriers," /. Appl. Phys., Vol. 
36, p. 3744, 1965. 



E. Pre-ignition Characteristics of Cesium 
Thermionic Diodes: Part il, K. Shimada 

1. Imroduction 

Pre-ignition volt-ampere curves of thermionic diodes 
can be divided into two regions: (1) the Boltzmann-type 
region, and (2) the apparent saturation region (Ref. 1). 
However, the current through a diode in the apparent 
saturation region usually does not assume a constant 
value; it increases slowly as the applied voltage increases. 
Two separate physical mechanisms are responsible for 
the increase; they are: (1) a siurface effect, and (2) a 
cesium gas effect (SPS 37-50, Vol. Ill, pp. 122-125). 

This article discusses the pre-ignition characteristics 
of a diode having an interelectrode distance of 0.0045 in. 



(coi.:pared with 0.028 in. for the diode previously tested). 
The results are qualitatively consistent with those previ- 
ously discussed in that the functional dependence of 
the rate of current increase on emitter temperature and 
cesium reservoir temperature is similar. However, the 
rate of current increase in the avalanche region of the 
volt-ampere ciuve was noticeably different in the present 
diode from that of the previous diode. Such a difference 
seems reasonable since the increase of current in the 
avalanche region is governed by the volume ionization 
of cesium atoms, and, hence, by the cesium pressure and 
the interelectrode distance. 

2. Test Diode 

The cesium thermionic diode used for this experiment 
was the SN-107. The emitter and collector, fabricated of 
rhenium, were assembled in a manner determined to 
minimize the collection of spurious electrons emitted 
from the heat-choke area (Fig. 7). The area of the planar 
part of the emitter disc was 2.00 cm^; the nominal inter- 
electrode gap was O.0O45 in. 

For subsequent analyses of the data, the actual emis- 
sion area was assumed to be the dimensional area of 
2 cm^ This assumption was justifiable for this diode 
according to the result of a ciurent measurement that 
agreed with one obtained from a test vehicle (with guard 
rings) whose emission area was accurately defined. It 
should be noted, however, that the agreement was ob- 
tained for the ignited mode, and that no data are avail- 
able for the pre-ignition mode where spurious emission 
may contribute to the net current. 

Currently, a theory is being developed that will enable 
a calculation to be made of the electron emission from 
the heat-choke area, which has a temperature gradient 
and corresponding variations of the work functions. The 
theory will be cross-checked against the measurements 
performed in a guard-ring research diode in the near 
future so that uncertainties due to spurious emission in 
the present results on the pre-ignition characteristics can 
be clarified. 

No attempt has been made in this article to correct for 
any spurious emission that may have existed. 

3. Pre-ignition Volt-Ampere Curves 

The diode undo; test was operated at relatively low 
emitter temperatures Tj. (130O''K-1600°K) and cesium 
reservoir temperatures Tc, (453°K-553°K), vidth the 
ratios Tg/To, being such that the emission was basically 



82 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. HI 



HOHLRAUM 

AREA OF 
POWER 
OUTPUT 



HEAT CHOKE 
SECTION OF 
EMITTER 
SUPPORT 
STRUCTURE 




ELECTRON 
BEAM 
WELD 



COLLECTOR 



IMMERSION 
THERMOCOUPLE 
HOLE 




EMITTER LEAD 
STRAPS 



EMITTER 



PRE- FABRICATED 
SEAL 



EB WELD 




■ RADIATOR 



CESIUM 
RESERVOIR 



Fig. 7. Testdiod* 



electron-rich (ion-richness ratio /3 < < 1). Under such 
conditions, the current through the diode was limited 
by the electron-space-charge sheath at the emitter. 
Moreover, the diode was operating in a non-collision- 
dominated regime since the mean-free-path of electrons 
ranged between 10 and 0.3 times the interelectrode gap, 
depending on Tc,. 

The volt-ampere curves were obtained by a sampler 
(SPS 37-49, Vol. Ill, pp. 130-132), and displayed on 
linear and semi-log x-y plotters. Simultaneous acquisition 
of two x-y plots increased the accuracy of current mea- 
surements since the semi-log plot showed the Boltzmann 
region of the volt-ampere curve (where the current is 
small) in full detail. Typical results are shown in Figs. 8 
and 9. In an output-voltage quadrant (negative-voltage 
part), the current increased sharply with voltage in a 
Boltzmann-like manner. The Boltzmann-like region is 
followed by the apparent saturation region in which two 
sub-regions, the Schottky-like and the avalanche regions, 
are observed. The rate of current increase in the Schottky- 
like region is nearly constant for a given emitter temper- 
ature, as shown in Fig. 9 where Tg = 1400°K. The current 
increases more rapidly in the avalanche region until the 
volume ionization in the diode causes ignition. The rate 
of current increase in the avalanche region depends on 
the cesium reservoir temperature. To demonstrate the 
logarithmic dependence of currents on voltages more 
clearly, the normalized currents I/h (measured current/ 
apparent saturation current) have been plotted against 
voltage corrected for the contact potential (measured 
voltage plus emitter work funcHon minus collector work 
function) as shown in Fig. 10. Three noticeable features 
are as follows: 

(1) All curves exhibit two d'stinct regions differen- 
tiated from each other by the rates of current 
increase. 

(2) The rates of current increase in the Schottky-like 
region are the same, independent of the cesium 
reservoir temperature. 

(3) All curves converge at the zero voltage (corrected). 

Attempts were made to express the normalized cur- 
rents as a fimction of voltage by the relation 

//Jo = exp {k, (V - V,)} + exp {h (V - V,)} (1) 

Here / is the measured current at a corrected voltage V, 
lo is the normalization (apparent saturation) current, and 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



83 



2.25 



2.00 



1.75 



ISO 



5 1.25 

UJ 
K 

5 

u 

9 1.00 



0.75 



0.50 



0.25 



1 1 ^ 

EMITTER 
TEMPERATURE T^ • 1400* 

COLLECTOR 


\ 






TEMPERATURE r^ . 536 -547 'K ^ J 

CESIUM RESERVOIR 
TEMPERATURE 


\ 




~ct ' " 


VMniMouc 












r„ • 55 

533 -V 


3.K-.^ 




\ 






■ 513 -^ 
493—. 
473— V 




>/ 








453-^ 
















k|N 




i 






^ 




^ 


^ I 
^ 




y 


$^ 








-2.0 


LO « 


> 1 


2 


s 





4.0 



APPLIED VOLTAGE, V 



Fig. 8. Typical volt-ampere curves— linear plot 



6 



lOO 



a. 

X 

D 



O 

o 
o 





































/ 














I 


L 












Tcs--y^ 


)3»K — ^ 


7 


r 


^ 










5:3 — 


^■s 


vSjl / 


/ 










^ 


f/ 


/ 












// / 


/ ^ 


^ 


>: 






^ 


""^^c^ 


t 


-453 








/ 




\ 


^ — 473 
— 493 








III 






__ _ 






II 


1 

EMITTER TEMPERATURE 
Tgr I400»K 

COLLECTOR TEMPERATURE 
Tc'- 536-547'K 

CESIUM RESERVOIR TEMPERA 
;^,> VARIABLE 






II 






III 


TURE 















-2.0 -10 



10 2.0 

APPLIED VOLTAGE, V 



30 40 



Fig. 9. Typical volt-ampere curve>-seini-log plot 



84 



JPL SPACE PJtOGMMS SUM/MAKY 37-51, VOL. Ill 




O.S 1.0 1.5 2.0 2.S 3.0 

CORRECTED APPLIED VOLTAGE , V 

Fig. 10. Normalized current vs corrected 
applied voltage 



10' 
6 

4 



IQO 



O 4 

a 



10- 

6 

4 









/ / 








CALCULATED 


-X — ■ 


v/ 










\ 


V 








MEASURED -\ 


Kl 








J-^ 


^ 


r 


Yi 


<:. 


){0.33(V-035)} 








/ 












/ 












f - 


exp{2.9 (V-174)} 






^^ 




































___. i __ 








EMITTER 

TEMPERATURE ^^^WOCK 

COLLECTOR 

TEMPERATURE ^.54I«K 

CESIUM RESERVOIR 

TEMPERATURE Tg,'%\^'Y. 








/ 






/ 






POINTS CALCULATED FROM 
exp{0.33(V-0.35)} + 

exp{2.9 (V-1.74)} 



O.S 1.0 I.S 2.0 2.5 3.0 35 

CORRECTED APPLIED VOLTAGE, V 

Fig. 1 1 . Comparison of meat ured and calculated 
normalized current 



I//o is the normalized current. The first term on the right- 
hand side of Eq. (1) is the contribution by tlie Schottky- 
like ciurent, and the second by the avalanche current. 
For example, at Ts = 1400°K and Ti. = 513°K, an 
empirical expression for the normalized current is 



///„ = exp {0.33 (V - 0.35)) + exp {2.9 (V - 1.74)} 



(2) 



The two terms on the right-hand side of Eq. (2), as well 
as l/h, are shown in Fig. 11. The calculated values, indi- 
cated by open circles, agreed excellently with the meas- 
ured values. Matching of the measured I /la with Eq. (1) 
is now being carried out for all temperatures, and the 
results will be reported as they become available. Pre- 
liminary analyses of ^, and k^ yield results consistent 
with those of previous analyses (SPS 37-50, Vol. Ill, 
pp. 122-125). The voltage coe£Bcient k, increases from 
0.1 to 0.6 as 10»/r£ increases from 0.62 to 0.76, and )k, is 



in the range between 3 and 4, but independent of Tg for 
a given Tc- 

4. Conclusions 

Pre-ignition volt-ampere curves for a cesium tlierm- 
ionic diode, operated at relatively low temperatures, 
exhibit Schottky-like and avaianche regions prior to igni- 
tion. The current increases exponentially with the diode 
voltage at difFerent rates in the two regions. The rate in 
the Schottky-like region is determined by the emi*+<n- 
temperature and is nearly independent of both the 
cesium resei-voir temperature and the interelectrode gap. 
On the other hand, the rate in the avalanche region is 
determined by the cesium reservoir temperature and by 
the interelectrode gap, but is independent of the emitter 
temperahare. Therefore, it may be concluded that the 
current through the diode in the Schottky-like region is 
mainly controlled by the emitter surface effect, whereas 
^e current in the avalanche region is controlled by the 
cesium gas effect. 



in SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



85 



These findings should be verified by using a cesium 
thermionic diode equipped with a guard ring to ehmi- 
nate spurious currents. 

Reference 

1. Bullis, R. H., et ed., "The Plasma Physics of Thermionic Con- 
verters," IEEE Report on the Thermionic Specialist Conference, 
pp. 9-29, Oct. 1965. 



F. Thermionic Diode Switches. Luebbers 

1. Introduction 

CertP.in unique characteristics of the thermionic diode 
allow its application to power switching. To investigate 
switching feasibility, a test circuit was designed and 
experiments were performed. The results of these experi- 
ments showed a dc-to-ac conversion efficiency in excess 
of 50% ; however, values as high as 85% may be easily 
reached. 

2. Diode Characteristics 

The thermionic diode is conventionally employed as a 
high-temperature power source. Heat energy supplied to 
the electron-emitting surface (emitter) brings its temper- 
ature to incandescence causing it to serve as an efficient 
electron emitter. The emitted electrons traverse a cesium- 
vapor-filled interelectrode gap (typically 0.002 to 0.030 
in.), and arrive at the collecting electrode (collector) at a 
higher potential energy than they initia'W possessed at 
the emitter. A load connected between the emittei and 
collector electrodes is supplied this potential energy and 
transforms it into useable electrical power. Under certain 
temperature conditions, the power-generating diode ex- 
hibits dual-m.ode properties that could be ppplied to 
switching; however, the efficiency of this switch would 
be extremely poor compared with that of a power- 
consuming diode, as will be discussed in this article. A 
typical volt-ampere characteristic for a power-generating 
diode is shown in Fig. 12 Two distinct modes of oper- 
ation are evident — the ignited and unignited modes. If 
the diode was to be employed as a switch, a resistive 
load would be placed across the diode via transformer 
coupling. The optimum load resistance would be that for 
which the coupled load r sistance matched the diode 
internal impedance. The converter can be switched by 
short-duration pulses between the ignited and unignited 
modes (between points A and B of Fig. 12). This change 
in voltage results in ac power output. The main factor 
limiting efficiency, with this scheme of power switching, 
is the small change in voltage mcurred in going from the 
unignited to the ignited mode. (A change of less than 



so 



80 



I- 

z 

UJ 40 

tr 
tt 

■D 
O 



20 



EMITTER TEMPERATURE - I800°C 
CESIUM RESERVOIR TEMPERATURE 



280°C 




-20 -1,5 -10 -0.5 

OUTPUT VOLTAGE, V 

Fig. 12. Power-generating thermionic diode exhibiting 
dual-mode characteristics 



0.5 V is observed in Fig. 12.) In practice, this method 
of switching would yield efficiencies of the order of a 
few percent and be extremely sensitive to temperature 
conditions. 

'iTie attractiveness of the thermionic diode for switch- 
ing, as described herein, is not its capability to produce 
power, but to act as a passive switching element. When 
both the emitter and cesium-reservoir temperatures are 
lowered by approximately a factor of two from the 
power-producing temperature values, the volt-ampere 
characteristics shown in Fig. 13 result. The theoretical 
implications of these characteristics are discussed in 
SPS 37-44, Vol. IV, p. 59 and Ref. 1, and are shown here 
in contrast to the power-producing characteristics of 
Fig. 12. Once again, two distinct modes of operation are 
observed; however, under the low-temperature condi- 
tions, the voltage variation across the diode between the 
two modes has increased dramatically. This large voltage 
separation is desirable if an efficient switch is to result. 

Figure 14 shows how a thermionir* diode might act as 
a switch. The diode is connected in senf j with a power 



86 



in SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 




-3 2 

OUTPUT VOLTAGE, V 



2 4 

APPLIED VOLTAGE, V 



Fig. 13. Powcr-contuming thermionic diod* •xhiblling 
dual-mod* proportiot 



circuit require careful design. Since the pulsing circuit 
operates into a nonlinear load, its optimum design is 
rather complicated This design was rot considered par- 
ticularly pertinent to the problem of proving switching 
capability, and, therefore, received little consideration. 
The pulsing circuit consisted of a free-running multi- 
vibrator, operating at 1000 Hz, and two channels of 
amplification to provide the necessary sequence oi posi- 
tive and negative on-ofF pulses. These pulses were then 
applied directly across the switching diode. 

Since the ultimate performance of a thermionic diode 
switch will be greatly influenced by the series step-up 
transformer, the transformer design received careful 
attention. Only the high-lights of these considerations 
will be discussed here. 




■^l TO SWITCHING 
^J PULSE TRAIN 



POWER -i- 
SOURCE 



SWITCHING 
DIODE 



L 




STEP-UP TRANSFORMER- 

Fig. 14. Tott circuit used in switching 

source whose voltage is less than the ignition voltage of 
the passive diode switch, and a step-up transformer is 
connected to an appropriate load resistance, R,,. A load 
line for such a circuit arrangement is included in Fig. 13. 
By a sequence of short-duration positive and negative 
pulses, the diode operating point is alternately switched 
between points A and B (shown in Fig. 13), and the 
resulting variation in voltage appears across the load 
resistance. For the volt-ampere characteristics of Fig. 13, 
the maximum switching efficiency (ac power output/dc 
power available) may be calculated to be 85%. This 
relatively high efficiency makes the thermionic diode an 
attractive switching device for use in hostile environ- 
ments where more conventional low-temperatwe devices 
would fail. 

3. Detiign Considerationt 

The circuit used for the experimental portion of these 
tests is shown in Fig. 14. Because of its simplicity, only 
the transformer and the pulsing circuit portions of the 



a. Transformer core selection. To fully utilize the 
high-temperature (900° C) characteristics of a thermionic 
diode switch, it should be used in conjunction with a 
high-temperature transformer. Present technology indi- 
cates that the iron-cobalt alloys offer the desired high- 
temperature capability. Reported Curie temperatures of 
900° C, and saturatioT' inductions as high as 23 kC, nllow 
switching and voltage step-up to occur in the immediate 
vicinity of the power source. 

The availability of a conventional (selection) t.ans- 
former core dictated its use rather than the high- 
temperature core specified above. Since the electrical 
performance of the experimental core was found to be 
comparable witi^ the characteristics specified for iron- 
cobalt cores, this substitution of core materials will not 
significantly detract from the primary objective of prov- 
ing switching feasibility. 

b. Transformer specifications. From the discussion of 
diode characteri.^vics {Subsection 2) and Fig. 13, one 
would expect the input to the transformer primary to be 
a square wave of approximately 2.5 V-peak-to-peak ampli- 
tude (i.e., ^-he change in voltage in going from point A 
to B in Fig. 13). For this input voltage V, the product of 
the primary number turns, Sp, and the core cross-sectional 
area, A, may be easily calculated from Faraday's law 



-Np 



d<i> 
IT 



NpA 



dB 



or 



NpA: 



\B/\t 



(1) 



(2) 



iH SPACE mOGHAMS SUMMAMY 37-51, VOL. Ill 



87 



where * is the core magnetic flux, B is the core flux 
demit)', and Jt = 1/2 (ac output frequency)'. If we 
assume a linearly increasing flux and a h'^quency of 
1000 Hz, the product iV,A is found to l»e 25 tmn-cm=. 
The number of primar>' tumi. was set equa^l to S, thus 
requiring a core cross-sectional area of approximately 
3 cm'. As a final check of these calculations, the flux 
build-up within the core was estimated and found to 
increase too rapidly and cause core saturation. To avoid 
this effect, the priman- inductance wa.' increased by 
enlarging the core cross-si^tional area by a factor of 3. 
The final specifications for the experimental transformer 
were as follows: 

(1) Core cross-sectional area - 9 cm=, 

(2) Core volume = 6? * cm'. 

(3) PHmary turns = 8. 

(4) Secondary turns - 88. 

These specifications were met by a toroidal core wound 
with the appropriate gauge wire and number of turns. 

Figure 15 shows the experimental transformer effi- 
ciency {power out/power in> versus frequency. For the 
design point of 1000 Hz, a transformer efficiency of 85% 
is observed in Fig. 15. This transformer efficiency reduces 
the ideal conversion efficiency from 85% (calculated 
from Fig. 13) to 72^. Considering the physical ood- 
stiuction of the transformer, this efficiency is reasonable. 



103 






__, 






- .n 


u ec 
u 

!Z 

& ( 
eo 










r^ 













«0a 600 803 1000 1200 HOO KOO 

FfrEOuE^cY, hi 
Fig. 1 5. Tfonif Bfintr cfficitnqr vi frequency 

4. Experimental Kesults 

The experimental ciraiit is shown in Fig, 14 and dis- 
cussed in Subsection 3. The experiments performed on 
the circuit consisted of the following: 

(1) Measuring ac power output versus load resistance 
for the 400-, 1000- , .md 1.50O-Hz frequencies at 
fixed emittsr and cesium -reservoir temperatures. 

(2) Measuring dc-to-ac conversion efliciency versus 
load resistance at the 400-, 1000-, and 1500-Hz 



frequencies for fixed emitter and cesium-reservoir 
temperatures. 

{3} Measuring ac power output versus load resistance 
for several different cesium-reservoir and emitter 
temperatiu-es (frequency = 1000 Hz), 

Each of the above measurements is briefly discussed 
below. 

a. Ac power output venus load resutance. To simu- 
late the low internal resistance and low output voltage 
of a thermionic generator, a 1,5-V Ag-Zn battery was 
used as a power source. One would expect this low volt- 
age power source to have poorer performance than the 
S-V power source used in earher calculations since the 
switching diode voltage drop represents a significant por- 
tion of the available 1.5 V. Figure 16 is a typical output- 
voltage waveform observed across a 30-Q load resistance. 
The circuit parameters are also specified in Fig. 16. The 
dc powtT source (battery) was ac- modulated by the 
thermionic diode switch as indicated in Fig. 14, The 
droop in positi^'e voltage curve, seen in Fig. 16, is caused 
by nonlinear effects experienced in the transformer core 
when used in the specified single-ended mode of opera- 
tion (current passes through the transformer only in one 
direction). ''Tie negative voltage droop is the start of a 
resistance-inductance (B-L) decay exponential experi- 
enced when the thermionic diode is turned off. 

A plot of tlie resulting ac power output versus load 
resistance is shown in Fig. 17. The 400-H2 data clearly 
illustrate transformer deficiency, and show that thb is 
not a desirable operating frequency. The upper curves 



EMITTER TEMPERATURE = 1100" C 
CESIUM RESERVOIR TEMPERATURE 
FREQUENCY = ;000 Hi 
LOAD RESISTANCE = 30 SI 



172° C 



< 

o 

> 



3 
0. 



3 

o 




TIME- 



Fig. 16. Typical «utput-voltoge wovifonn 



Si 



JH SMCE PROGRAMS SUMfAARY 37-5), VOL. Ill 



"— i 



2.5 












\ 

EMITTER TEMPERATURE = 900°C 










^- -. 






CESIUM RESERVOIR "^cMPERATURE = IST'C 

1 1 


OUTPUT, W 

IS) 

b 






-^ 














/ 


w 






;n 


^^ 




^ 




q: 

UJ 
0. 


/ 








1/^ 


^""--^ 




" 1 


~~ — 


/ 


FRECUENCV, Hz = 1500 — ^ 


/ 


"■"^ 










/ 

/ 




-^ 


lOOO ^ 

400 — -7 








^ 


-_ 


1.0 







40 50 60 

LOAD RESISTANCE, il 

Fig. 17. Power output vs load resistance 



80 



100 



>• 
o 

z 

UJ 

o 



z 
o 

'& 

UJ 

> 

z 
o 
o 




50 60 

LOAD RESISTANCE, Q 
Fig. 18. Conversion efficiency vs load resistance 



exhibit the anticipated behavior with fallofiE at both high 
and low values of load resistance. The low-resistance 
falloff may be attributed to resistance mismatch between 
the load and efiFective source internal resistance. The 
high resistance falloflF includes both load mismatch and 



some transformer saturation. This fallcfif is common to 
all of the curves obtained in these experiments. 

h. Dc-to-ac conversion efficiency. Figure 18 is a plot 
of the conversion efficiency versus load resistance for 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



89 



three different frequencies. Again, the 400-Hz data are 
low, as anticipated, and the 1000- and 1500-Hz daia are 
comparable up to a load resistance of 50 O. At higher 
values of load resistance, transformer core saturation 
becomes evident in the 1000-Hz data. Conversion effi- 
ciencies of approximately 50% are observed. If a higher 
voltage power source weie used, the efficiency would 
increase. 

c. Ac power output versus load resistance. Ac power 
output versus load resistance, \\ith the cesium-reservoir 
temperature as a parameter, is plotted in Fig. 19. In con- 
trast to those of Fig. 17, these data were taken at a fixed 
frequency of 1000 Hz and an emitter temperature of 
1100°C. The effect of increasing the cesium-reservoir 
temperature is that the diode's internal resistance de- 
creases, and the output power increases. As would be 
expected, the shift to a lower optimum load resistance 
is also accompanied by an increased power output. 

5. Summary 

The thermionic diode has been successfully u'.od to 
ac-modulate the power output from a 1.5-Vdc power 
source with a conversion efficiency in excess of 50%. 
A maximum efficiency of 85% is predicted for a 3-Vdc 
power source. 

These relatively high efficiencies make the thermionic 
diode an attractive switching device for high-tempera':ure 
applications as, for example, in a thermionic nuclear 
reactor. The high-temperature and radiation-resistant 
properties of the thermionic switching diode would per- 
mit its location in the imniediate vicinity of the nuclear 
power source, thereby reducing power lost to the current- 
carrying conductors. Conventional semiconductor power 
components would have to be located at a remote posi- 
tion where temperature and radiation levels would be 
tolerable. 



Reference 

Shimada, K., and Luebbers, S., "\nomalous Electron and Ion 
Currents in Plasma-Mode Operation of a Thermionic Energy 
Converter," in Advances in Energy Conversion Engineering, 
ASME Conference, 1967. 



40 



0. 



o 
a: 

c 

Q. 



55 



30 



2 5 



2 



! ! 

EMITTER TEMPERATURE = IIOO°C 
CESIUM RESERVOIR TEMPERATURE 


•c 


172 


- 165 








/ ' ^\. 


/ ^ 
/ M ^' -- 


-^ — X 1 



I 



N 



10 20 30 

LOAD RESISTANCE,. a 

Fig. 19. Power output vs load resistance 



90 



JPl SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



\ N68-31405 



VIII. Materials 

ENGINEERING MECHANICS DIVISION 



A. Effect of Notch Severity on Cross-Rolled 
Beryllium Sheet, R. Moss 

1 . Introduction 

Brittle materials such as beryllium (Be) are considered 
notch sensitive. Presence of a sharp notch is believed 
to reduce the material strength and ductility so greatiy 
that it is of questionable value in structural applications. 
Unfortunately, almost no work has been done to demon- 
strate quantitatively the effect of machined notches on 
the strength of cross-rolied Be sheet as a function of 
mateiial variables, or notch severity; in particular, the 
effect of sharp notchei has not been examined in any 
detail. Some data does exist on the effects of relatively 
dull notches in hot pressed block. Previous data on hot- 
pressed Be block showed an increase in notched/ 
unnotched strength for a stress concentration factor 
(Kt) =« 3 to 4, and a reduction for K( between 3 and 5 
(Refs. 1-7). Some of the references give conflicting results 
for Kt between 3 and 4. Existing data on cross-rolled 
sheet show no notch strengthening even at K( < 2 
(Ref. 8). The series of tests reported here was intended 
to determine whether material other than hot-pressed 
block would show any notch strengthening at Kt < 5, 



and what the effects of different Kt, process history, and 
composition had on the transition from strengthening to 
weakening of Be. 

2. Test Results and Discussion 

This article presents preliminary data on the effect 
of .sharp notches in cross-.oUed ingot and powder sheet. 
Early results indicate that the expected severe reduction 
of notched/unnotched strength did not occur in the 
materials and sample configuration studied. Vendor 
analyses and properties of these materials are given in 
Table 1. Additional tests on a second grade of ingot 
sheet and two more grades of powder sheet are in 
progress. 

Sample*; tested were double-edge-notched sheet tensile 
specime'.is 0.025 in. thick, V* in. wide, with a %-in. gauge 
length This represents a sheet thickness of interest in 
spacecraft applications. After rough blanking, 0.002 in. 
wa'. etched from each specimen surface to remove micro- 
cracks and surface damage. Notches having a severity of 
Kt = 3.2 to 8.3 then were formed by electrical discharge 
machining. This Kt range spans the region in which 
notch sensitivity should be apparent. Both longitudinal 



JPl SPACE PROGRAMS SUMMARY 37-51, VOL. HI 



91 



Toble 1 . Vendor-reporttd properties of beryllium 



Proptrty 


Powder >h*«l 
Hlt-379 


Ingel ihMl 
l$.318 


Compoiilion, % 






B.O 


1.58 


0.32 


C 


100 


0.064 


F« 


0.092 


0.114 


Al 


0.056 


••.057 


Mg 


0.005 


0.004 


Si 


0.048 


0.064 


Other metali 


0.04 (max) 


0.04 (max) 


Be assay, % 


98.48 


99.41 


Grain size, Mm 


60 


60 


Tensile strength (longitudinal), lb/in.' 


80.500 


49,400 


Tensile strength (transverse), lb/in.' 


79,300 


63,700 


Yield strength (longitudinal), 0.2% 


55,000 


41,200 


Yield strength (transverse), 0.2% 


55,700 


47,500 


Elongation (longitudinal), % 


22.0 


3.0 


Elongotior (transverse), % 


16.0 


3.0 



and transverse samples were tested. Tensile tests were 
run at a constant crosshead rate of 0.05 in./in./niin, 
with results recorded directly on an x-y plotter. 

Results are shown in Fig. 1. It is apparert that the 
expected severe loss of strength at K( 5; 4 did not occur 
in powder sheet, or longitudinal samples of ingot sheet; 
indeed, a slight trend toward strengthening seems to be 
present at Kj < 6 for longitudinal samples. The trend for 
transverse ingot sheet samples was in the direction of 
reduced notched/unnotched strength ratio. Data scatter 
is too great to justify drawing simple curves, so scatter 
bands are shown. This scatter is to be expected for a brit- 
tle material such as Be; it is not unreasonable considering 
the normal scatter of unnotched cross-rolled powder sheet 
is ±4.5% (3-<7 level, 90% probability, 95% confidence) 
(Ref. 9). Actual K, was calculated for each sample, using 
measured dimensions and the standard nomographs 
(Refs. 10 and 11). 

There are several possible explanations for the differ- 
ences between this data and previous notched/unnotched 
tensile test results. Most of the existing data was obtained 
from hot-pressed block. Sheet properties are significantly 
different from hot-pressed block in regard to strength, 
elongation, and anisotropy of mechanical properties. 
Another possible cause of reported notch sensitivity is 
the presence of machining damage on the sample sur- 
face. Some of the early work (Refs. 2 and 3) was done 



b 

X 

& 

z 



UJ OTO 





1.30 



POWDER SHFE;T HR-379, LONGITUDINAL 
--D POWDER SHEET HR-379, TRANSVERSE 

-4 1 1 1 



(b) 



















J 




"^ c 












^^ 


8 






oO 






*^ 




-8^ 

Q 

n 






<^ 








=^=J 


H^ 




- IN ROT 


SHEET ^-. 


D 




— 

— 


IS-3i8, longitudinal"""-^'^! ^ 

- INGOT SHEET ""^^n C 

IS-3ie, TRANSVERSE *^v 

1,1. •* 







40 SO so 
SEVERITY *> 

Fig. 1. Effect of notch severity and testing direction 
on the notched/unnotched tensile strength of Be: 
(a) powder sheet, (b) ingot sheet 

before the need for post-machining etching was estab- 
lished; other reports did not describe sample preparation 
(Refs. 5 and 6). It is likely that these samples contained 
microcracks, giving much higher effective Kt values than 
those measured and reported. Avoidance of surface 
cracks was a major objective of sample preparation in 
this program. Therefore, it is believed these results may 
be more representative of notch effects in samples without 
microcracks or twins. Sample geometry should be con- 
sidered also. It is possible that another sample geometry 
would give different results. 

The reason for the directionality of ingot sheet notch 
sensitivity is somewhat puzzling. One possible explana- 
tion is the well-knov/n anisotropy of Be sheet mechanical 



92 



JH SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



properties. Reported grain size of this sheet is < 60 fim. 
A large amount of rolling is required in order to obtain 
fine grain sizes in ingot sheet. This might have introduced 
severe texturing, making the sheet more susceptible to 
crack propagation in one direction than the other. X-ray 
diffraction studies revealed appreciable texturing in the 
ingot sheet. Comparison with the powder sheet is in 
progress. 

3. Conclusions 

Although the test data obtained are preliminary and 
subject to further verification, results suggest that the 
presence of sharp notches in Be sheet need not cause 
catastrophic failure. Ingot sheet was weakened in the 
transverse direction, but was not weakened significantly 
in the longitudinal direction. It would be misleading 
to suggest that rolled Be sheet is not notch sensitive; 
however, in structures which use thin gauges of Be, 
there seems to be more tolerance for defects than gen- 
erally anticipated. Similar resistance to crack propaga- 
tion in 0.051-in. cross-rolled powder sheet was reported 
by others (Ref. 12). 

References 

1. Fellman, R. B., et al, Final Report, Development of High 
Strength Beryllium Materials for Structural Applications, Vol. 1, 
Report 675D519. General Electric Company, Re-entry Systems 
Division, Los Angeles, Calif., Feb. 1967. 

2. Crawford, R. F., and Bums, A. B., Strength, Efficiency, and De- 
sign Data for Beryllium Structures, ASD-TR-61-692, AD290770. 
Lockheed Aircraft Corporation, Sunnyvale, Calif., Feb. 1962. 



3. Hodge, W., Beryllium for Structural Applicatioris, DMIC Report 
16S. Battelle Memorial Institute, Columbus, Ohio. May 18, 
1962. 

4. Kesterson, R. L., The Cryogenic and Ambient Tensile and 
Compression Properties of Hot-Pressed Block Beryllium, 
WANL-TME-1619, N68-13893. Westinghouse Astronuclear 
Laboratory, Large, Pa., June 1967. 

5. Beryllium Thermal Shock Testing, preliminary report to NASA 
Research Advisory Committee. Westinghouse Astronuclear 
Laboratory, Large, Pa., Jan. 1967. 

6. Beryllium Fracture Mechanics, preliminary report to NASA 
Research Advisory Committee. W'stiughouse Astronuclear 
Laboratory, Large, Pa., Jan. 1967. 

7. Campbell, J. F.., Mechanical Properties of Beryllium at Cryo- 
genic Temperatures, Including Notch-Specimen Data, DMIC 
Technical Note. Battelle Memorial Institute, Columbus, Ohio, 
Nov. 5, 1965. 

8. Finn, J. M., Koch, L, C, and Muehlberger, D. E., Design, 
fabrication, and Test of an Aerospace Plane Beryllium Wing- 
nox, AFFDL TR-67-38. McDonnell Douglas Corporation, St. 
i^ouis. Mo., Mar. 1967. 

9. King, B., Nev> Grades of Beryllium— Their Meaning and Use, 
paper presoiitei' ' ^\E Manufacturing Foriun, Los Angeles, 
Oct. 2-6, 1967 ;.sh Beryllium Company, Cleveland, Ohio. 

10. Peterson, R. E., Stress Concentration Design Factors. John 
Wiley & Sons, New York, N. Y., 1953. 

11. Neuber, H., Theory of Notch Stresses. Translation published 
by J. W. Edwards Co., Ann Arbor, Mich., 1946. 

12. Finn, J. M., Koch, L. C, and Muehlberger, D. E., Design, 
Fabrication, arid Ground Testing of the F-4 Beryllium Rudder, 
AFFDL-TR-6;-68. McDonnell Douglas Corporation, St. Louis, 
Mo., Apr. 1967. 



jn SPACE PROGkAm SUMMARY 37-51, VOL III 



93 



fRKSDlbJG^PAGE BL^N1^ HOT,flU*EB. 



N 68-37406 



IX. Aerodynamic Facilities 

ENVIRONMENTAL SCIENCES DIVISION 



A. Heat Transfer Study of 60-cleg Half-Angle 
Cones, M. F. Blair 

With the objective of determining the applicabihty of 
presently available theories of laminar convective heat 
transfer in planetary gases, a study of 60-deg half-angle 
cones is being carried out in the JPL 43 in. hypersonic 
shock tunnel and the JPL 12-in. free-piston shock tube. 
The bodies under investigation are three 60-deg half- 
angle blunted cones (Fig. 1) with various edge radii. All 
three cones have a bluntness ratio Rn/D of 0.10 while 
the shoulder radius/body diameter ratios R,/D are 0.05, 
0.025, and sharp. Heat transfer distributions are currently 
being measured and will eventually be compared to 
values predicted by using measured pressure distributions 
as input to a convective heat transfer computer program. 

Measurement of the pressure distributions, carried out 
entirely in the JPL 43-in. hypersonic shock tunnel, has 
been completed. This tunnel is driven by a 3-in. inside 



diameter shock tube which is operated in the reflected 
mode (tailored interface). The shock tube driver gas for 
all cases was Ho while the driven gas, and hence the 
tunnel working medium, was Na. The working section 
Mach number was about 12.5 while the total enthalpy 
was approximately 1800 Btu/lbm. The flow Reynolds 
number was about 4.2 X 10\ ft. 

The pressure study consisted of measurements at 45- 
deg increments of roll angle for the following angles of 
attack; a = 0, 5, 10, 15, 20, and 30 deg. Samples of the 
data obtained for the body of Fig. 1 at a = deg are 
shown in Fig. 2, and for the body pitched to o = 15 deg 
in Fig. 3. The symbols represent the numerical average of 
data taken while the error bars show the extremes of data 
taken from all runs. Average points represent results 
from two to four tunnel runs. Also presented (Fig. 4) is 
a diagram of isobars that resulted from radially cross- 
plotling the curves of Fig. 3. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



95 



M 




12 



1^ 

UJ 
ce 

3 



UJ 
UJ 

u 



08 



06 



£ 04 



a. 

D 

to 



02 





] 




/?5/i9 = 005 
Rp/D -- 10 




^ 


^ 


> — a 




^ 




T 

ROLL 
ANGLE, 


V* 


A 


^—4^ 

!< 






Y 


deg 
O 
A 45 
• 90 — 
135 
A 180 


r • ~ 


"• 


I A 


^ 1 


r^ 










Y ' 


o 


7 
































I 





1 2 3 4 S 6 7 

S/R„ 
Fig. 3. Pressure distribution along 60-cleg half-angle 
cone at 15-defj angle of attack 





p/PJ- 




p/Pr 


o 


000 


<} 


825 


a 


975 


0' 


775 


A 


950 


c» 


725 


A 


925 


< 


700 


D 


900 


<- 


675 


D- 


875 


> 


650 


o 


850 


.d 


600 









528 



R„/D ■- 010 
Rg/D = 05 
a = 15 deg 



Fig. 1 . Typical 60'deg half-angle blunted cone 
with 0.3-tn. edge radius 



12 










/?„/£> = 10 
R,/D-- 005 




' 1 

UJ 
K 

!3 06 

gO.4 

<n 

02 




W 


T- 


^-^ 


Hy-^ 


I _ 


Tt 












1 


r ^ 


1 
































I 
















\ 







1 2 3 4 9 6 7 1 

S/R„ 

Fig. 2. Pressure distribution along 60-deg half-angle 
cone at 0-deg angle of attack 




Fig. 4. Isobar diagram prepared from curves of Fig. 3 



96 



iPL SPACE PROGRAMS SUMMARY 37-51, VOL III 



N 68-37407 



X. Environmental and Dynamic Testing 

ENVIRONMENTAL SCII^NCES DIVISION 



A. Low-Frequency Plane- Wave Sound Generator 

and Impedance-Measuring Device, C. 0. Hoyei 

1 . Introduction 

In tiif field of acoustic tfstiri^ of vpucetraft aiul sub- 
systems, tlic production of c^rroct sound power spectra 
is II n important rc'C]iiirt'ini'iit f(tr propei cii'irotimrntal 
qiialiiication testing. At the pn-scnt time, studies are being 
made to cievel()i) high-inteii'iity smiir.I generators with 
broader freqiit'ucy response cliaracteristics. Iiivestij;ations 
of the response cliaracterisliis of acoustic boms wil! eotn- 
plement tliesc sound generator studies (Ref";, 1 and i). 

To empirically determine the response characteristics 
of any acoustic horn, a device providing plane-wave 
icoustit inputs over the freqnency range of interest is 
needed. Tliis device must also he capable of providing 
ai.oustic measurements within the born to determine its 
response characteristics. The design should be such that 
the risponse characteristics of the act. -i^tie source and 
of thi' termination can be malbematii.-ally eliminated, 
thus providing only the response chancteristics of tbf 
horn. With this information, the precise eontribntion of a 
given horn to any acoustic system can l)e determined in 
advance of the actual system assembly. 

A low-frequency plane- wiivc sound generator and 
impedaiiee-measuriiig device (Fig. 1) was r'esigned to 



fulfill these rcquircmcrts (Ref. 3). This device (herein 
called impcdance-mcasurii.j; device) provides undistorted 
sinu.soidal acoustic signals in the range from 10 to 400 liz. 




Ftg. 1. Lew'freqiuoncy plane- wov« found gonorator 

and Impodanca-meaturing davic*, allaehoH 

to vibration shakcir 



JPL 5MCE PROGRAMS SUMMARY 37-51, VOt, HI 



97 



It (.■staMislu's thf aitMLslic pri'ssiiros, particle vclticities. 
and tUv phase rebtionsliip bWwL-vn thi" prtssurcs ;md 
\*cl()cities a* tin- input to the atniistic clcnient iimitT invps- 
tigation. These same paraiiifters are determined, either 
hy measurement or hy analytieal prediction, at the output 
of the acoustic dement. With these data, an accnra'e 
detennination of the element impedanttr prop<'rtics can 
be made it a given frcjueney and the element response 
characteristics determined. Tliis process is rciH'atcd at 
enough frcqwencics within the fiequency hand of interest 
to provide adequate resohition. 

2. Design 

This device is designed to determine the acoustic im- 
pedance in terms of sound pressures and \oinme \eJoci- 
ties as measured at the input of an attached acoustic 
cbnient. Basically, the device rt quires only nndistorted 
shaker output pressure signals, which are compatible with 
the sensitivities of tlie monitoring ac-ccloruineter and mi- 
crophoni'. to provide accurate aconslic impedance infor- 
mation. (Mathematical derivations are drscrilKKl in detail 
in Ref. 3.) 

3. Test Configuration 

A tjpical test confifji'ration (Fig. 2) consists nf the fol- 
lowing c«mpone;its: 

(1) CijUndrical Utbe. This I. liam hibe provides 

the vohime area for the soimd soiirc<' over the 
length of the c\Iindrical tube and the aUgnmcnt 



guide for the piston. The tube al.so provides the 
means for attaching a monitoring microjjhone for 
tile throat pressure and an acoustic element or a 
varia hie- impedance tube. 

(2) Piston. This unit is driven hy a vibration shaker. 
The piston creates plane-wave sound fluctuations 
as it travels in the cylindiical tube under very clcwe 
t<jlcr.iucc. An acc-elerometcr is ntounted, with its 
axis parallel to the piston motion axis, to measure 
the piston's acceleration. The fac-e of the piston 
defines thp "source " of the acoustic pressure flue *m- 
ations. .A mi'jroplione can be installed in the face 
of the piston to measure the sound level and 
amount of distortion. 

(3) Vihratitm shaker. This unit imparts oscillator)' mo- 
tion to the piston and is attached to the cylindrical 
tube with a nuuiiting rmg. 

(4) Acoustic element. The acoustic clement or horn to 
be analyzed is attached to the cylindrical tube so 
that the output of the tube bec-omes the input over 
its length fo; an output of the horn. 

(5) Variable-impedance tube. For calibration purposes, 
a \ ariablc-imi>edance tube (blocked tutx) is in- 
stalled on the cyhndrical tube. The diameter of tlie 
blocked tube is 1.375 in. witli the inside diameter 
flared for a distance of 1.25 in. from the mating 
end to assure a smiH)th transition between the two 
tubes. 



PLANE -NAVE 
TUfiE 




Fig, 7 Typical t«st configuration, using Kyperfaslic horn with fx termination 



JPL SPACE PROGRAMS SUMMARY 37-5', VOL. 1(1 



(6) Pltine-icaci' tuhr. For U'st purposes, a pliinewave 
tnlu' may he .ittiK'licci t(i the iDOntli of the acoustic 
(Iciiu'iit ti> pi'()\'idc a i>f tcnniiiatioii for thi* tiorn. 

In a typifal test setiiji (Kig. 2), tlie iii'iK'dunte- 
measiiring devUf is attiiclietl to llie throat of a (liypcr- 
l>olk') 'itini. Tlif nuHirt of the horn is uttachfd to a 
plane-wave tohe. whieti is paeked with ahsor'ntit mate- 
rial to prtniile a fx* termination for ttie horn. Tliis eon- 
fij^iiration allows a comparison hetween the measured and 
the predicted (the(»retieaU respfinsc characteristics of a 
finite-leiijjth horn, 

4. Calibration 

The impedanee-measiiritij; device was calibrated iisiiif^ 
a \aria!)le-lenj;th hlocked tnhe (Fig. 3) that pn>\ided a 
known loadinjj impedance »\('r a fre<iiieiicy range from 
&) to 42.5 Hi'.. .\ computer program was written to imple- 
ment the equations. The data indicate ver)* gotxl results 
over the fre(iueiKy range of interest (60—125 (Iz). and 
pro\ ide a \erificatton of hotli the device and the ealihra- 
tion technique. 

There are sevend possifjle sourci s of ermr in the cali- 
hration of the tube and the de\ iue. 

^l) Variable- impedance tid)e. 



(a) Tlie tube w 'U will resonutc at particular fre- 
quencies within the frequency baud of interest. 

(b) The microphone head nsi'd to measure the 
tJiroat pressure has a finite area rather than 
being a iKiiut. Tins source of error would in- 
trea,se with an increasii' in frequtiicy. 

('2) Impedance-measuring device, 

(a) The microi)hone, which monitors the sound 
press' rre levels at the output of the device, 
monitors pri'ssures over a finite area (0.5-in.- 
diain circle); therefore, it is not a single point 
nu)nitor as requiied b\ the accompanying 
mathematical theory. 

(b) The phase angle between the acceleromcter 

output voltage signal and the microphone out- 
put voltage signal is very crttieal. 

(cj Small errors may be introduccJ l)y the manual 
reudtHit of the aeci'leronietcr and the micro- 
phone output signal levels. (These ou^puts have 
not. as yet. lieen digitized.) 

This calibration technique, which uses a blocked tulx*, 
provide.s a very wide dynamic range for calibrating the 
unit, since the tulK" will rellect impedances ranging from 
zero to infinit> , depending on the value of a)t KL, where 




Fig. 3. Variabl«-inipeclQn(« tub* attochad to devic* 



JPL SPACE PROC-RAMS SUMMaHY 37-51, VOL. Ill 



99 



wave number K, em' s= 2ir//c. This method provides a 
very acci'rate technique by selecting values of KL — 
and ±(2n + l)ir/2, where n = 0,1,2, ■ • • , since the 
value of cot KL changes very rapidly with KL. For these 
values of KL, any errors, such as described in Para- 
graph (2-a), will greatly affect the value of the output 
impedance. The values of KL for these calibration runs 
were chosen such that cot KL was a fixed val e for each 
run and had, for all the runs, a range of absolute values 
of 0.303^ cot KL^ 1.000. 

5. Conclusions 

The data obtained from the calibration runs indicate 
that accurate acoustic impedance information can be ob- 
tained. Calibration runs have verified the mechanical 



design of the device as well as the accompanying mathe- 
matical analysis. Use of this device requires accurate 
methods of data acquisition, such as digital readout of 
phase, acceleration, and acoustic pressure, and maintain- 
ing undistorted input signals at the output of the device. 

References 

1. Olson, H. R., Acotiitical Engineering, pp. 103-114. D. Van 
Nosfrand Co., Inc., Princeton, N. J., 1957. 

2. Hayes, C. D., Acoustic Spectrum Shaping Utilizing Finite 
Hyperbolic Horn Theory, Technical Report 32-1141. Jet Pro- 
pulsion Laboratory, Pasadena, Calif., Aug. 15, 1967. 

3. Hayes, C. D., and Lamers, M. D., Low-Frequency Plane-Wave 
Sound Generator and Impedance-Measuring Device, Technical 
Memorandum 33-376. Jet Propulsion Laboratory, Pasadena. 
Calif., Mar. 1, 1968. 



100 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL III 



Nfi8-37'108 



XI. Solid Propellant Engineering 

PROPULSION DIVISION 



A. Molecular Momentum Transfer From 
Regressing Solid Propellant Surfaces, 

O. K. Heiney 



(2) There is a directional equiprobability of molecular 
emission in the half hemisphere bounded by the 
propellant surface. 



1 . Introduction 

One of the more enduring suppositions of proptllant 
deflagration is that of impulse propulsion. In essence, the 
hypothesis assumes a significant impulse pressure will 
be generated by an exchange of momentum between 
burning gas molecules and the surface of the propellant, 
from which these molecules were emitted. Tlie following 
analysis briefly outlines the argument ard development, 
largely on a molecular basis, that serves as justification 
for this effect, then considers more comentional gas- 
dynamics and ballistics which predict an effect of much 
lower magnitude. Finally, the experimental procedure 
used to adequately demonstrate that the lower predicted 
value of impulse pressure is the correct expression is 
described. 

Symbols used in this article are defined in Table 1. 

2. Analysis 

a. Impuhe pressure. Reference 1 is the generally 
quoted analysis for warranting the anticipation of this 
impulse effect. The development and assumptions pre- 
sented below are those given in this reference: 

(1) Tliere is a 100% conversion of the heat of com- 
bustion of the propellant into the kinetic energy 
of the gas molecules. 



(3) A mean molecular emission velocity may be defined 
which is a funcMon of the total combustion energy 
potential of this propellant. 



The specifi'; energy potential of the propellant is g'ven 



as 



M, = 



2 n»x "x Vi 



y-1 



2 2 »njf *»A 



The mean velocity Ve is 



v. = 



Then, 



2 wi/. n* Vi 

K= 1 

n 

2 "Ia "k 

A=l 



2F,g\^ 



H^) 



which relates this postulated emission velocity to the 
impetus Fp of the propellant. 



JPL SPACE PROGRAMS SUMMARY 3/-5I, VOL. f/l 



101 



Table 1. Nomenclature 



c* 


characteristic velocity 


F. 


impetus of propellant 


g 


acceleration due to gravity 


rriK 


mass of K-type molecule 


n 


dimensionless burning rate exponent 


riK 


number of K-type molecules 


P. 


chamber pressure 


P. 


impulse pressure 


R 


gas constant 


r 


burning ra*e 


Ss 


burning surface 


Tr 


temperature of flame 


u. 


specific energy potential of propellant 


V, 


gas velocity 


V, 


velocity of K-type molecule 


V, 


mean molecular ejection velocity 


r 


flow factor 


y 


ratio of sptL ific heat 


Ps 


gas density 


pp 


propellant density 



Using geometrical arguments, the development then 
states that the eflFective impulse pressure generated is 
equal to only one-fourth of the mass emitted at this 
velocity, giving 



where 



Then, 



d{mV) 
dt 


4 


dm 
dt 


dm _ Pp Sb r 
dt g 




_PpSBr 1 


'2Fp 


lY 



(1) 



or for an end-burning configuration 

,r/2Fpg\'i 



Ppr/2Fpg\' 



(2) 



which is the predicted impulse pressure with the given 
assumptions. 



b. Conventional mass balance approach. The mass 
balance equation is given as (see Fig. 1) 



,rS, 



PpTSb — Pi^ Vg 



AV„ 



(3) 



For end burner 




Ps 


(4) 


Pc = PsFp 


(5) 


Then, substituting Eq. (5) into Eq. (4) gives 




^' Pc 





for an impulse pressure of 



P, 



PJr'Fp 

gPc 



(6) 



Equations (2) and (6) are fundamentally different in 
both form and effect prediction. It can be seen, however, 
that for either equation this predicted impulse pressure 
is quite low. In fact, it is for most purposes a second 
order effect. Figure 2 gives a plot of the impulse pressure 
predicted by both equations as a function of chamber 
pressure. It dn be seen that the conventional gas dynamic 
approach indicates a pressure 40 times lower than the 
molscular momentum transfer approach at low pressures. 




4g 



Fig. 1. Moss balance approach 



102 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



While at higher pressure (e.g., 10,000 psia), *he diflference 
is well over three orders of magnitude. It must be under- 
stood that these figures are for a given propellant formu- 
lation and burning rate, as both Eqs. (2) and (6) are 
highly sensit \ e to the deflagration rate dependence on 
pressure. An end-burning config;iration was also assumed, 
ff charges are perforated to increase the burning area, 
Eq. (6) can be multiplied by the burning area to chamber 
area ratio. The analysis for Eq. (2) wou''^ completely fail, 
however, as the majority of the mi.-lecuic.^ would be 
"ejected" radially rather than longitudinally. 

To determine wh'ch, if either, of the expressions is 
correct, an experimental program was undertaken to com- 
pare the thrust generated by a 3-in.-diam end-burning 
motor. This motor was fired at atmospheric pressure with 
a constant 7.07-in.- burning surface. 

As can be seen from Fig. 2, the molecular momentum 
exchange equation would predict a thrust of 1.98 lb, while 
conventional ballistics would predict a thrust of 0.044 lb. 



The propellant used was of the aluminized composite 
rubber base type. An impetus F^ for the propellant was 
determined from the C* value by the simple relationships 
(Ref. 2) 



Pr 


- fir,, 




RT„ 

r- 


F„ 


= c*-' r 



The C* of 4890 ft s for the formulation gave an impetus 
of 312,400 ft-lb/lb, which is quite typical of average gun 
propellant impetus values. Other parameters of the pro- 
pellant are: 

y = 1.14 
r, at 1000 psia = 0.37in./s 

Pp =- 0.065 lb /in. ' 
Tf = 2743° K 
n = ~0.5 



UJ 

cc 

o 
<n 

CO 

iii 
o. 

UJ 
CO 

s 



lO' 


























6 

4 

2 

.^0 
















J 








. /^ 










1 










1 


>1 










1 

1 








/ 


r"^ 


-EQ (2) 










1 ^ 


V 


T I 








1 


lo*- 








.&^ 


r 1 
















6 
4 




^ 


r 
























r 


y 














~1 










u 






























,^-\ 
































lU ' 








_ 






















6 

4 

2 
































































^. 












































/— EQ(6) 


lu 






[ i^ 


^ 




















6 ( 
4 

2 


I 


\^\ 


J V 


r 


r^ 


1 <' 


I 


o 




^ 




( 


T 




r 










"v 




v-^ 


? — c 


>< 


|)^ 














1 




1 


















L_,. , — 






1 






. 





6 10 2 4 6 10 2 

CHAMBER PRESSURE, psia 



4 6 10 



Fig- 



2. Impulse pressure predictions as function 
of chamber pressure 



3. Experimental Procedure 

The test configuration initially used is illustrated in 
Fig. 3. The load cell utilized had a maximum thrust 
capability of 2 lb and a resolution accuracy of ±0.002 lb. 
The wheeled suspension system was found to be too 
crude for the delicate thrust measurements. A suitably 
sensitive suspension system that was successfully utilized 
is illustrated in Fig. 4. The system was based on ballistic 
suspension of the motor and proved quite effective. 

Figure 5 illustrates the plume developed from the 
3-in. motor during a firing. The fiducial lines on the 
thrust stand are 1 ft apart. In general, the plume was 
quite impreSL've and one could legitimately suppose a 
sizable thrust was being generated. During firings for 
which data was developed, a ± 1-osid pressure gage with 
a resolution of ±0.01 psi indicated ihat chamber pressure 
and ambient pressure differentials were not measurable. 
Thrust measurements during the first firing showed a 
constant thrust of 0,046 lb for the 60-s firing diir<ttion 
while the second firing had a constant thrust of 0.'>42 lb 
for a like period. Within the limits of the load cell rtro- 
lution, these values are as predicted by the balHstic 
analysis of Eq. (6). A shortened chamber pressure and 
thrust curv^ are shown in Fig. 6. It can be seen that the 
only noticeable pressure increment occurs at ignition 
and then falls to zero. 



iPl SPACE PROGRAMS SUMMARY 37-51, VOL. /K 



103 




Fig. 3. Iniilai solid propellant motor test configuration using wheeled suspension sy;H' n 



104 



JPL SPACE PROGRAMS SUMMA'f 3751. VOL Iff 




Fig. 4. Sensitive suspension system for motor 
thrust measurements 




Fig. 5. Plume developed from firing a 3-in. 
solid propellani motor 



4. Concluiion 

The result'.- of this study indicate that the molecular 
monicntiim cxthiingc impulse pressure development is 
erromous. Thiis is primarily due to a fundamiTital physical 
misconception between tho mean and net g;c- velocity, 
which is contiiincd in the assumptions. 



Fig. 6. Chombcr pressure and total thrust curves 
for motor firing 

.\lso implicit in tJiesc results is thp fact that a significant 
KilUstif effect is not obtaimihl*' from the impulse pressure 
f'uncept. ConsiLk'rable effort nn*! been expended on the 
various "traveling charge" systems of gun ballistics in an 
attempt to uiilize this intpulse prt-ssurc phenomenon. 
Tliese experiinenls usually failed due te propcllant physi- 
cal property consideratioiis. If they had .not, however, it 
would have been seen that fundr. mental physical miscon- 
ceptions wore present in the basic liypotlieses. 

References 

I. I. IV, L., iiiiil LiiiiHiT, K., "Till- liitfricir nallislifs iif the Impiil^ 
l'r()|«iKii)ii tiiiii." Ciitlmiic Univcn-ity of Amfric.i. Wiishinclon, 
t),C., .Vit;. 1951. 

■1. ttiiucit. C, Uiirtlfj-, C. K.. iinil MiU^. M. .\f.. ScUd Propeitant 
ftiiiki'l.\. I'Ttnictiiii .\i'r(ui;iii(iiiil I'Hpcrli.nks, I'riuf'Um. N. J„ 

mm. 



B. T-Burner Studies, £. H. Perry' 

1. Introduction 

One of the primary objectives of the current T-burncr 
studie-f at J PL is to gain a more thorough understanding 
of tlie burner itself, Experiments were conducted to mea- 
sure the ncoustic losses of a l.S-in.-diam T-bumer. 



'Culifurtiia lii>titiilfo[ Tfehniitntity, I'cs^tltni , Calif. 



in SPACE PROGRAMS SUMMARY 37-51, VOL lit 



.05 



Although the measurements were made under "cold" con- 
ditions in the burner, a basis is provided for understand- 
ing the losses observed during test firings. 



2. Th«er«tical Acoustic Losstt 

The acoustic field withir a T-bumer during a firing 
consists of a standing wave of wavelength 2L, where L 
is the length of the burner cavity. This field is maintained 
by the burning propellant at the ends of the cavity and 
accordingly decays after burnout of the propellant. The 
decay is approximately exponential in time with the time 
required for the acoustic pressure to drop by a factor 
of e, defined as the "decay time" of the burner. Usually, 
however, reference is made to the "decay constant," which 
is the reciprocal of the decay time. 

It is well estabUshed that a sound wave traveling 
through a tube is attenuated at a rate proportional to the 
square root of the frequency ard inversely proportional 
to the tube's radius. This decay is due to viscous and 
thermal dissipaHon near the tube wall. Reference 1 gives 
the following expression for the decay constant associ- 
ated with these wall losses: 



«« = W^[(v)'* 



(K)W(y-l)]^ (1) 



where 



, ~ kinematic viscosity coefficient 
K = thermal diffusivity coefficient 

y = specific heat ratio 

/ — frequency 
R = tube radius 

In addition, there are thermal losses associated with 
the reflection of the wave from the ends of the cavity, 
llirough arguments similar to those used to derive 
Eq. (1), one can show that the decay constant for such 
end losses is given by: 



a. = (47r*:)*4(y - 1} 



.m 



(2) 



Since the T-bumer is a vented cavity, the possibility- 
exists for acoi'stic radiation from the exhaust vent. How- 
ever, the center of this vent is located precisely at the 
pressure node of the standing wave in the cavity. There- 
fore, if the diameter of the vent is small compared to the 
ty length, any radiation losses from the vent can be 



expected to be very small. In the present experiments, the 
ratio of vent diameter to cavity length never exceeded 0.06. 

Thus, it appears that the only losses in the "cold" 
T-burner should be those due to dissipation at the walls 
and ends of the cavity. If this is indeed the case, the 
decay constant of the burner should be the sum of the 
wall and end decay constants. That is, if a is the burner 
(Iv.cay constant, then 

o = a„ + «e (3) 

where aw and a« are given above. 



3. Expcrimtntal Proc*dur«t and Rasulli 

An acoustic environment simulating that encountered 
during a firing was provided within the cavity by a sound 
driver unit outside. An audio oscillator was used to drive 
this unit at the standing-wave frequency of the cavity. 
The sound introduced into the cavit>' through a small 
hole at one end was observed by a 0.25-in.-diam con- 
denser microphone at the opposite end. Figure 7 illus- 
trates the arrangement used. 











MICROPHONE 7 










\ 








h^ 


AMPLIFIER 






SOUND \ 
DRIVER / 


T-BURNER 












/ 




1 1 














11 


















AUDIO 
OSCILLATOR 






STORAGE 
OSCILLOSCOPE 



C'-x 



Fig. 7. Block diagram of •xporimontol arrangemoni 



By abruptly turning off the sound driver and observing 
the subsequent decay ci' the standing wave, the decay 
constant of the burner was determined. Burner lengths 
ranging from 7 to 42 in. were used to obtain a range of 
frequency. 

Figure 8 illustrates the behavior of the decay constant 
as a function of frequency at atmospheric pressure. For 
the purpose of comparison, the values of the decay con- 
stant predicted by Eq. (3) are plotted along with the 
experimental values, llie agreement is seen to be fairly 
good over the entire frequency range. The experimental 
values all lie above those given by the theory, which is 
to be expected since there are small losses associated 
with the sound lead-in and detection devices. 



106 



in SMCE noQUAm summary 37-51. vol. hi 



< 
o 



z 



8 



2S 



20 



n " 



10 






/ 

olt 1 I I L 



O EXPERIMENT 
EO (3) 



_L 



200 400 600 800 1000 1200 1400 

FREQUENCY. Hz 

Fig. 8. Dacay constant a% a function of froquoncy 
at atmoiplioric protsuro 



4- 



2|_ O EXPERIMENT 
THEORY 



L 



_L 



J_ 



SO 100 ISC 200 2S0 

PRESSUKF., psia 



300 350 



F.g. 9. Decay constant as a function of prossuro 

Figure 9 presents the experimental and thewetical 
values of the decay constant as a function of mean cham- 
ber pressiu-e. To obtain these measurements, the appa- 
ratus was placed in a chamber pressurized with nitritgen. 
All of these measurements were made at a frequency of 
530 Hz. As can be seen in the figure, the agreement be- 
tween theory and experiment becomes progressively 



worse as the chamber pressure increases. The cause of 
this condition is not completely understood at present, 
although it might be due to losses associated with im- 
proper fitting of some of the burner sections. Small gaps 
between adjoining sections have been found to give rise 
to very large losses; possibly these losses increase with 
pressure, which would explain the above results. 

Tlie final phase of the experiments consisted of an 
attempt to measure the acoustic losses associated with 
the vent. A plug was made to fit into the vent so that 
the latter could be completely closed ofiF, thereby elimi- 
nating the possibility of any acoustic radiation from the 
vent. Decay measurements obtained with the vent thus 
closed were compared with those obtained with it open. 
Any difference Ijetween the two sets of measurements 
was too small to be detected, which indicates the vent 
losses are indeed small as suggested above. 

4. Appiicatioii of Results 

There is evidence that the above "cold" burner analysis 
applies also to the losses observed during actual T-bumer 
firings, Figure 10 presents decay constant data reported 
in R''f. 2. for two similar composite propellanis denoted 
as A l*^ and A-14. The empirical curve through the data 
assumes the square-root dependence suggested by Eq. (3). 
The rather good fit suggests that the acoustic losses of 



I 




2000 4000 6000 

DECMr FPl^QUENCY, Hx 

Fig. 10. Decoy constant as roportod in Rof. 2 
for actual T-burner firings 



■000 



ti. SPACE nOGKAMS SUMMAHY 37-51, VOL. Ill 



107 



^'f 



the T-burner are described rather well for these two pro- R«f«rtncM 

pellants by an equation simUw to Eq (3). It should be j landau. L. D.. and Llf^chltz. E. M.. FluUi Mechanic, p. 303. 

mentioned that other data of this reference exhibit a Addison-Wesley Publishing Co., Inc., Reading, 4ass., 1659. 

similar behavior. Future studies are expected to show, „ „ „ ^ „ . . „ , „ ,,„. ^ c nj » i 

iU iLj iU •. ^l 1 V .1. ._i 2. Horton, M. D. Testing the Dynamic Stability of Solid Propel- 

among other things that the losses have the geometric ^^. Technique, and Data, NAVWEPS Report 8596. NOT? 

dependence indicated in Eq. (3) as well as the frequency xr ?910, pp. 34-35. U.s. Naval Oidnance Test Station. China 

dependence discussed above. Lake, Calif., Aug. 1064, 



101 JPL SfACE M06XAMS SUMMAMY 37-51, VOL. Iff 



N 68-37409 



XII. Polymer Research 

PROPULSION DIVISION 



A. Investigfrion of the Transport Characteristics 
of on lonene Membrane, 

H. y. Tom and J. Moacanin 

1 . Introduction 

The battery separator material is one of tlic key factors 
that determine the lifetime of a silver-zinc battery. 
Ideally, the battery separator membrane should allow 
charge transfer to carriers such as OH", but should pre- 
vent silver and zinc ionic species from leaving their 
respective half-cells and thus avoid internal short circuits. 

The objective of this work was to initiate a systematic 
study of the various transport characteristics of mem- 
branes to ascertain the chemical and morphological 
requirements that lead to desirable permselective prop- 
erties. In free difiFusion, the solvent and solute move 
relative to each other. Hence, only one transport coeffi- 
cient would be required to relate flow and concentration. 
Imposing a membrane would require additional factors 
that must consider the interaction of the solute and 
solvent with the membrane. Another consideration that 
influences transport is the pore size in the membrane. 
Such membranes can then be experimentally tested for 
their permselectivity by the number of coefficients re- 
quired to describe the transport of ions, using the 



formalism of irreversible tfiermodynamics, provided the 
process is just slightly off equilibrium (Refs. 1 and 2). 

This portion of the study was performed on ionene 
membranes (Ref. 3). In ionenes, positive quaternary 
ammonium ionic groups are incorporated along the 
hydrocarbon backbone and their charge density can be 
varied in a systematic fashion to assess their effect on the 
transport coefficients. Although the current polyethylene- 
graft-acrylic-acid separator also has a hydrocarbon back- 
bone, the acrylic acid branches are distributed at random; 
whereas, in ionenes, the charged groups are distributed 
in a uniform manner. This article covers the electrical 
properties of cells prepared with ionene membranes. 
When the concentration data that are presently awaiting 
analysis become available, an article demonstrating the 
presence or absence of preferred ionic transport will be 
presented. 

2. Motorials and Equipmont 

Materials procured for this study consisted of N, N, N', N'- 
tetiamethylhexanediame, 1,6-dibromohexane, tetrachloro- 
o-benzoquinone (TCBQ), reagent grade potassium chlo- 
ride, grade 72-51 polyvinyl alcohol (PVA), and battery 
separator membranes. De-ionized distilled water was used 
throughout the investigation. 



Jn SPACE PROGRAMS $UMMkkV 37-51, VOL. Ill 



109 



Transiwrt cells were fabricated from pyrex glass 
(Fig. 1). Th^ glass joint holding the two half-cells has 
a grooved flange to permit the installation of an O-ring. 
The membrane is mounted on the flange of one cell with 
the O-img pre-iiistalled. The flange from the other cell 
is then brought in contact with the membrane and the 
assembly clamped. 



VOLUME 
MEASUREMENT, 
3-mm diom 



-ST 




UnO. 



CLAMP- 

Fig. 1 . Transport cells for ionene membrane tests 

The horizontal arms with a 3-mm bore diameter are 
used for volume measurement. Since both arms are at 
the same height, flow of liquid across the membrane can 
occur without change in hydrostatic pressure. The volume 
measurement is good to ±14 /xl with a volume of about 
125 ml/cell. Glass joints were also included to permit the 
insertion of platinum electrodes. 

Equipment required for electrical measurements con- 
sisted of a high-impedance dc millivoltmeter, an ac im- 
pedance bridge for resistance, a regulated dc power 
supply, and an electrical timer. Platinum blackened elec- 
trodes were obtained by electroplating O.OlO-in.-diam 
pladnum wire. 

3. Membrane Fabrication 

The membranes made for this study were prepared by 
combining N, N, W, N'-tetramethylhexanediame and 1,6- 
dibromohexane on a 1:1 gram molecular weight basis 
(Ref. 1). The synthesized copolymer designated as a 
6,6-ionene was weighed ai d added to PVA and TCBQ 
in different proportions. TTie PVA and TCBQ weight 
ratio was maintained at 100:1. Water was added as 
needed. PVA was prepared as a solution by heating 
water to 100°C and adding PVA for sr.persaturation. 
Any insoluble PVA was removed by filtration. 



The water mixture with the membrane ingredients was 
shaken, then cast the next day onto glass slides. The 
water was allowed to evaporate, and the films were later 
heat-treated at 100°C for 1 h. These membranes were 
then stored in petri dishes. 

One membrane (50 wt % ionene) was inspected with the 
stereoscan electron microscope. The dry-mounted sample 
was found to be pinhole free; for comparison the same 
sample is shown with a puncture made with a 250-/itm 
pin (Fig. 2). This result indicated that the fabrication 
procedure was satisfactory and that all the membranes 
should be free of pinholes. Further investigation on this 
point is being c-ontinued and will become a routine pro- 
cedure for membrane characterization. 

For chemical analysis (by Gulf General Atomic, San 
Diego, Calif.), the samples were first treated with neutron 
irradiation and then assayed in batches for potassium 
and chlorine as radioactive elements in a scintillation 
counter. A standard and a blank were always included 
with each batch. 

4. Experimental Procedures 

The membranes, water-prewetted or dry, were mounted 
first; the platinum electrodes were inserted next; then, 
the cells were filled with their respective bathing media. 
Once the media came in contact with the membrane, a 
timer was activated. The transport apparatus was then 
placed in an ultrasonic cleaner and vibrated for 2 min 
to remove any entrapped atmospheric gases. The cells 
were then transferred to a bench where an ac impedance 
bridge and a millivoltmeter were connected to each 
electrode. The high-impedance millivoltmeter which con- 
tinuously monitored the potential difference across the 
membrane was assumed not to draw current from the 
system. The ac impedance bridge was activated only 
when the membrane resistance was measured. The bridge 
was energized by a 400-mV, 1000-Hz internal source. 
/Jiquots of 50 /*1 were removed periodically from each 
cell. 

Membrane thicknesses were measured to ±31 /ttm with 
the aid of a calibrated filar eyepiece and a stcreomicro- 
scope. A piece of the membrane was excised from the 
remaining stock material in the immediate area where a 
larger piece had been previously removed for the trans- 
port study. Its thicks :.»s was measured while dry, in 
water, and in salt solution. For the analysis of the trans- 
port experiments, the thidone. was taken to be the 
average dimension in v^ater and ^alt solution. 



110 



JPL SPACE PROGkAMS SUMMARY 37-51, VOL. Ill 





:<.A 



^ 

X 



■*M 


'■-'M 


^n^ 



(o) Hir ' MAGNIFICATION OF MEMBRANE FREE OF 
Pi., holes 



lb) LOW MAGNIFICATION OF SAME MEMBRANE 
WITH 25U-^m PIN HOLES 




(c) HIGH ^f^iGNtFICATION OF A PIN HOLE 



Fig. 2. Electron micrographs of a SO-wt- % ionene membrane and Ihe detecllon of pinholes 



5. ttesulU 



At different Intervals of timt", ffu- volume (liBcrcncc 
of <-ai.'li cell, lie resistance i\l fOfK) Hz, potential difFercnee, 
and aliqiiots from each cell were obtaiiu'd. Tlie [jotential 
observed is that Kt'iieratcd b>- the two cells, which act 
;.y concentration half-cells. f'oncentraHons are not in- 
< hided at this time as the c!ieinic;d analyses are incom- 
pkte, Tlic volume changes are shown in Fig. 3, 

Diirint* the first 3000 .s, the ionene membranes (Fig, 3) 
exhibit fairly rapid volume changes. Tlie initial phase is 



followed hy a decrease in the rate of vohime change with 
some iiulieatioii I hat steady state is approachc<l. This is 
best illustrated in I'ig. 3f where the volume change for 
tlie 70-wt-5E ionene iimtent is essentially a straight line. 
The battery separator (Fig. 3g) also shows an initial 
rapid phase followed by a slower phase. For the battery 
.separator, Iiowevcr, the initial phase takes only 1500 s. 
It is interesting that the initial phase, the incubation 
period, of the ionene membranes appears to be inde- 
pendent of its thickness. For the battery separator, this 
point could not be chi;cked since only one thickness was 
available. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



m 





»» 


-(o) wt % lONENE kCMBRANE, 282 /im THICK. MOUNTED WET 




6 

4 

2 
I0> 


V 
/ / 




4. 


« 


*"/ / 




i 


4 
2 


/ / 


- 




«! 


- 






6 








4 


J 






2 


. t"^" 






IflO 


L 


1 1 1 1 



I0« 



2 ^ 

'0' ^ 5 



I0«- 



1 ^ 

-i =? 2 



I 



3 

TIME, s X lO' 



J 10° 



10' h 

6 
4 



KJO 



■(< 


1 -T- 1 r ' I I 

) 1 wt % lONENE MEMBRANE, 590 fin THICK, MOUNTED DRY 




i<^ 


- 


/ 


- 


y 


- 


^'X 


- 




- 


1 1 1 1 1 1 



I0« 



w" 



UJ 
I 

i 

i 
■? 



3 4 

TIME, s X lo' 



Vfi 




Kit 



-(d) 20 wt % lONENE MEMBRANE, 9:j/im THICK, MOUNTED DRY 



i * 



=! 2 h 



- 6 P 5 



3 4 

TIME, > X 10' 



10' 

6 - 

4 



lOO 




10' 



Ul 



P 

I 

< 



3 4 

TIME, t X 10* 



I0«> 



Fig. 3. Temporal raspensM 



ifl SMCE PROGRAMS SUMMARY 37-51, VOL Iff 



s 

3 




10' 


: i 


I 




I 1 r— ■ 
(g) BATTERY SEPARATOR 
MEMBRANE, 442 /im 


4 
2 


' r 

- \ 




THICK, MOUNTED WET 


nf 


: ) 


( 




. 


6 
4 


■1 


\ 






2 


- /■ 






" 


KJI 


j 








6 


i 
j 




\ 




4 


- i 




\ 

1 

\ 


- 


2 


~ 








KlO 


t 


1 


1 


1 1 1 



1 

< 
z 

1^ 

4 2 

< 

I 



I0-' 



3 4 

TIME, $ X 10' 



I0> 
6 - 

4 - 



I0«- 

6 - 

4 - 



6 - 

4 - 



lo"! 



(f ) 70 wt % lONENE MEMBRANE, 10,104 /im THICK. 
MOUNTED WET 




f"<,- 



^<. 



^^: 



^.. 



-^ig 



02 



2 
< 



9 4 5 

TIME, t X 10* 



wo 



O '/OLUME INCREASE 

IN SALT CELL 

O VOLUME DECREASE 

IN WATER CELL 

A POTENTIAL DIFFERENCE 

D AC RESISTANCE 



Fig. 3 (contd) 



Ifl SPACE PJtOGMMS SUMMAHY 37-51, VOL. Iff 



113 



Once the slow phase begins, the suiii of the volume 
differences may not be zero. However, where those sums 
are not zero, the membranes were mounted dry; those 
membranes whose values are about zero were mounted 
wet. To assess the swelling behavior, the membranes were 
buthed in both water and in potassium chloride solutions. 
Results in Table 1 show that the membrane thickness, 
including the separator material, is essentially unaffected 
by the media until the ionene content is 20 wt X or greater. 
When the ionene content exceeds 20 wt %, the membrane 
prewetted with water contracts markedly when plunged 
into a salt solution. 

One variable sensitive to the volume differences is the 
potential difference (^) across the cell. The voltage in 
absolute units is used to generate the curves in Fig. 3. 
However, whether or not the potential difference is a 
measure of the ionic concentration across the membrane 
is still inconclusive as the aliquot concentrations have yet 
to be completed. Nevertheless, this measurement is 
certainly more sensitive to the volume changes than 



resistance and must necessarily be reflected in the ionic 
concentrations. 

6. Conclusion 

The results suggest that as the membrane absorbs 
water there ensues a decrease in the total volume in the 
cell (liquid plus membrane). One possible explanation 
is that when water is absorbed by the membrane, the 
hydration sphere around the quaternary ammonium ion 
reduces the specific volume of water in the sphere and 
thus leads to a negative volume of mixing. Volume con- 
tractions are well known for mixtures of salt solutions 
and water. 

This volimie decrease is evidently unrelated to the 
incubation period since it exists whether the membrane 
is mounted wet or dry. It may be argued that the initial 
phase is an artifact since the surface tension in the 
capilhry of the water cell may be large enough to pre- 
vent flow. Flow begins at some later time when enough 
ions are transported across to reduce the surface tension. 



Table 1 . Temporal response of membrane to bathing media 



THickntu 

•fdry 

mnnbran*,* 

/rni 


Tim* in 

2-M 

potattium 

dilerid* 


Tliirkn*u of 

ri*Mbran* 

i 1 wlulien, 

^m 


Tim* in 
water, t 


Thickn*(t of 
m*mbran* 
in water. 




Thicknax 
of dry 

nmnbranSf 


Tim* in 

2-M 

petatsium 

chiorida 
•eiuHen, t 


Thitknatt of 

m wfTi orofi w 

in Mlulion, 

Mm 


Tima in 
water, s 


Thicknat' of 

III V HID FQ nw 

in water. 
Mm 


Centroi 




50 wt % ien*n* 


79 
109 


100 

300 

600 

1000 


116 
99 
92 
93 


400 
3500 


101 
93 


796 


1070 
1230 
1320 
1410 
1934 


990 


670 
1030 
1460 
2490 


17797 
17999 
16447 
17688 


Conlral 


273 


170 

250 

42S 

1400 

7000 

2700 


261 
358 
225 
231 
248 
267 


400 

800 

1000 

1350 

6500 


223 

318 
310 
290 
296 


70 wl % ionMM 


45 


600 
1700 
2080 


1226 
1188 
1160 
1069 
1046 


(pra 

00 


waited) 

19162 


1 Wt T» ioiMn# 


45S 


800 
1S00 
1800 
19W 
2000 


599 
714 
574 
672 
559 


100 
500 
800 

1400 
2900 


545 

621 
636 
613 
621 


■aHary laparater 


402 
313 
301 
341 
335 
335 
323 
313 
311 
304 


200 

400 
1400 


396 
448 
438 


200 
500 

900 
3700 


527 
464 
403 
445 


20 wl % Imimi* 


827 


4«0 
S75 
750 
900 
1800 


586 
611 
594 
584 
583 


1500 


1247 





114 



Jn SPACE nOGHAMS SUMMAHY 37-51, VOL. Ill 



If this were so, the duration of the induction period R«f«r»ne«f 

would depend on the thickness; however, no correlation , .^ ^ , ^ „ ^. . , ^ , ,1,1 » 

. , , y , , ., I 1. De Groot, S. R., Thermodunamics of lrrev<!rsihle Proceasea, 

with thickness was observed. Moreover, even when some j^^^j^ Kolland 1951. 

membranes were mounteJ prewetted, the incubation ^ ^^^^ ^ J^^ Katchalsky, A.. Tmns. Faraday Soc, Vol. 59. 
period was not reduced. Thus, the evidence seems to pp igjg ^931 a„j 1941 jgea. 

strongly indicate that the initial phase is eal, although 3 Rembiuai. A., Baumgartner, W., and Eisenbarg, A.. /. Polym. 
the mechanism is unknown. Sci., Part B, Vol. 6, p. 159, 1968. 



V 



Jn SPACE PROGRAMS SUMMARY 37-51, VOL. HI 115 



%• 
t 



^68 



-S 



41^ 



^ * 



XIII. Research and Advanced Concepts 

PROPULSION DIVISION 



A. Laminarizalion in Nozzle Flow, 

L. H. Back, R. F. CuHtl, and P. F. Ma$tiw 

1. Introduction 

Turbulent boundary layers under certain flow accelera- 
tion conditions can undergo reverse transition toward 
laminar boundary layers. This phenomenon offers the 
advantage of a reduction in convective heat transfer and 
is of consideiable interest since it can sometimes be pro- 
moted in rocket nozzles. The reverse transition process, 
referred to as laminarization, has been found to occur 
when values of the parameter K = (vt/ul) {du,/dx) ex- 
jeed about 2 X 1(H. (Symbob used in this article are 
defined in Table 1.) 

To better understand the conditions under which lam- 
inarization occurs and the effect of laminarization on 
the fricticAt coefficient, an investigation of the structure 
of the boundary layer was undertaken in a nozzle. 

2. Tott Conditions and Apporatus 

The nozzle used for die tests (Fig. 1) resembles a 
oonflguratioa used for rocket engines in v«dbich the com- 



bustion chamber is an integral part of the convergent 
portion of the nozzle. The conical half-angle of conver- 
gence was 10 deg, the inlet diameter 5.00 in., and the 
throat diameter 1.59 in. The nozzle was also instrumented 
so that heat transfer meastvements could be made. 

Boundary layer measurements upstream and within 
the nozzle were made at the stations noted in Fig. 1, 
where the free-stream Mach numbers were 0.066 and 
0.19, respectively. Compressed air was used and data 
were obtained over a range of stagnation pressures 
between 15 aud 150 psia and at a stagnation tempera- 
ture approrimately equal to the ambient temperature of 
540° R. Consequently, the flow was essentially adiabatic 
in the boundary layer region where the measurements 
were made. The boundary layer was turbulent at the 
nozzle inlet with a thickness of about % the inlet radius 
of the nozzle. 

Flattened pitot tubes 0.005 in. high were used li 
measure impact pressures; the tubes were moved 
mechanically norma! to die waU by a micnmieter load 
screw. 



116 



jn SPACE PJtOORAiMS SUMMARY 37-51, VOL HI 



£ 



M 

I 



r 



APPROACH 
SECTION 
BOUNDARY 
LfllTER 
STATION 
z «-2.l4 m. 

AIR FLOW 

^ 

I 



r,--\.eOin. 



^g 



TANGENCV 
z = 0.497 in. 
7 - 0.219 in. 



,10 dag 



tT 



£1 



I 

NOZZLE CONTOUR- 



NOZZLE BOUNDARY 
LAYER STATION 
z« 6.67 In. — K-.,^ 
1 I 



TAN6ENCY— >J 
/» 9.839 in. j 



<^« 1.987 in. 

mtm^^r 

THROAT 
.? = 10.184 in. 
/•M-0.795in. 



4C0 
^300 

§ 

UI 200 

1 

« WO 
til 








1 
















/ 


/ 


























> 


/ 


/ 






















^^<r 


^ 


c^ 


^-^ 














( 




r\^~\ . 


rn:><3 


oo-- 





























\J\J ' 








6xlor« 
Ay mr6 






TEST 


P238 
























PARAMETER /T- 

o 


9vm~€, 










'^^ 


^- 


___^ 


10 = 15.0 










/« 














P24I 








149.6 

















-3 



2 3 4 5 6 

AXIAL DISTANCE z. in. 

Rg. 1 . Variation of flew variables 



10 



12 



3. Experimental Results 

The free-stream velocity variation obtained from the 
measured waU static pressures for isentropic core flow 
(y = 1.4) is shown in Fig. 1. This distribution is essen- 
tially independent of stagnatim pressure. The parameter 
K, indicated for two stagnation pressure tests, is highest 
in the inlet regimi of die nozzle. It then diminishes akmg 
the nozzle and is larger for the lower stagnation pressure 
test since Ka (1/pt) for nozzle flow. 

At the approach section station, velocity pro&les 
(Fig. 2) are seen to be typical of a turbulent boundary 
layer. In the representation of tt and y*, the wall shear 
stress r was determined in the approadi sectitm by fitting 



the profiles to the law (rf the wall which was taken in 
the form 



tt* = 5.5 + 2.5 biy*. 



sr>30 



(1) 



The velocity distribution is seen to agree well with the 
law of the wall relation. In the outer part of the boundary 
layer, the wake-like behavior found in many tiirbulent 
boundary layers is evident (Ref. 1). 

The e£Eect of flow acceleration on the velocity prt^es 
is shown in the lower half of Fig. 2 at the nozzle station. 
At the higher stagnation pressure, the profile becomes 
•vlatively flat in the outer part of the layer. The wake- 
like behavior found upstream has disappeared and. 



JPL SMCE NOGMMS SUMMAKY 37-51, VOL. Ill 



U7 



+ 
a 
>- 
l- 

o 
o 

-I 

(/> 

!2 

_i 

z 
o 

CO 



40 
95 


Al 


PROAC 


^s^ 


ATI 


or 
































s.A 










































C 

13 

□ 


D 




^ 






































C 


° c 

Q 


D 
D 


y 


-^ 


y 


^ 




































( 
Qi 


1 


r 
c 


^^ — LAW OF THE WALL Eq.I 


























rt^ 


y 


5r 


r^ 


^ 






























KARM/ 


N SUBl 


.AYE 


R- 




V 


/ 


r 






TES 

.O P23 

Q P2' 


T psia 
S 15. 

n 149. 


7 

■■ 

5 

6 5 


f 

35 
55 


/ 

4 
4 


* 

580 
6100 ( 


9 
2 

.22x1 
3.77x1 


0-5 
















5 






/ 


/ 


/ 


/ 


























D* 2 4 6 10* 

DIMENSIONLESS DISTANCE /'*' 

Fig. 2. Velocity profiiu in the approach tocHon and nozzle 



118 



»l SPACE PROGRAMS SUMMARY 37-51, VOL. Iff 









although there is some curvature of the profile nearer 
the wall associated with the effect of acceleration, a 
fair fit is still found to the law of the wall. The friction 
coefficient of c//2 = 1.83 X 10-' obtained from this fit 
is about 10% higher than the value that might be 
inferred from the Blasius turbulent boundary layer 
relation 



c, _ 0.0128 

2 /peU^SV 



(2) 



The value of K corresponding to this higher pressure 
test is 0.24 X 10-«. 

A drastic change in the structure of the boundary layer 
in the nozzle occurred at the lower stagnation pressure 
where K is an order of magnitude higher (2.4 X lO^*). 
The slope of the velocity profile {du/dy) is considerably 
reduced near the wall. In fact, the measurements near 
the wall can be linearly extrapolated to the wall, and 
the friction coefficient so deduced is c//2 = 1.67 X 10"'. 
This value is consistent with that obtained by fitting 
the Blasius flat plate laminar velocity profile f (ij) 
(Ref. 2) to the measured values near the wall. The fit 
specffies 71 in terms of the experimental value of y/$, 
and the friction coefficient is then determined from the 
slope of the exact solution f^ at the wall: 



2 



PeM = 



^(".fj) 



dr, djy/e) 
d(ym dy 



f" 



d(y/e) 



{^) 



The Blasius profile, however, deviates from the measured 
profile at points away from the wall because the boundary 
layer that has apparently become laminar-like near the 
wall experiences flow acceleration. A better fit is afforded 
by the Hartree wedge flow profile for ;3 = 2 (Ref. 3) 
shown in Fig. 2, and this profile yields a somewhat higher 
friction coefficient of C;/2 = 2.07 X lO"'. Other accel- 
erated laminar flow profiles for convergent channel or 
sink flow (Ref. 2), or perhaps more appropriately for 
conical channel or sink flow (Ref. 4), would fit the 
measured profile about as well as the Hartree profile for 
j9 = 2 and yield friction coefficients no more than 5% 
higher than that deduced from the Hartree profile. 

To illustrate the reduction in the wall friction because 
of the apparent laminarization near die wall for the lower 
stagnation pressure test, the friction coefficient C//2 = 
2.07 X 10-' is about 25% below the value that might be 



inferred from the Blasius turbulent boundary layer rela- 
tion Eq. (2). It is noteworthy that the reduction in wall 
friction occurred in a relatively high Reynolds number 
flow with the throat Reynolds number [(p,UeD)//i«]«» = 
6.5 X 10° for the lower stagnation pressure test. 



TabI* 1. Nomenciatiir* 



Cf friction coefficient, -^ = — ^-r- 
2 p,ul 

D nozzle diameter 

K laminarization parameter, -~ —y^ 

Pt stagnation pressure 

r tube or nozzle radius 

Tt stagnation temperatiure 

u velocity component parallel to wall 

tt dimensionless velocity, 

X distance along the wall 

y distapce normal to wall 

I 

y* dimensionless distance, - 



fe)" 



ifX^ 



z axial distance 

a angle between waU a jd axis 

y specific heat ratio 

8* displacement thickness 



/»« 



momentum thickness 

,/ 8*cosa\ r°° « /- «\, xj 

ei^r ^yj^ -^i--.yr-ycosa)dy 

ft, viscosity 
•> kinematic viscosity 
p density 
r wall shear stress 
Subscripts 
e condition at free-stream edge of boundary layer 



jn SPACE PROGKAMS SUMMARY 37-51, VOL. Ill 



119 



An indication of the region in which laminarization 
occurred near the wall in the nozzle at the lower stag- 
nation pressure is shown in Fig. 2. Inference from the 
agreement with the Hartree profile f or /8 = 2 suggests 
that the boundary layer was laminar-like out to a loca- 
tion where y* is about 30, a value associated with the 
viscous sublayer of a normal turbulent boundary layer 
and at which location laminar transport is small com- 
pared to turbulent transport. However, Launder (Ref. 5) 
still detected tiu-bulent fluctuations close to the wall with 
his hot wire surveys in a similar laminarized boundary 
layer. Farther away from the wall, the velocity profile 
(Fig. 2) indicates that some turbulent transport still exists. 

Thus, in the experiments discussed here, the velocity 
profiles measured upstream and within a conical axisym- 
metric nozzle revealed a strong effect of flow acceleration 
on the structure of an originally turbulent boundary 
layer. When values of the parameter K exceeded about 
2 X 10"*, the boundary layer became laminar-like near 
the wall because of flow acceleration, and the wall fric- 
tion was correspondingly less than that associated with 
a turbulent boundary layer. 

References 

1. Coles, D., "The Law of the Wake in the Turbulent Boundary 

Layer," /. Fluid Mech., Vol. 1, pp. 121-226, 1956. 

2. Schhv^hting, H., Boundary Layer Theory, Sixth Edition. McGraw- 
Hill Bo.* Co., Inc.. New York, 1968. 

3. Hartree, D. R., "On an Equation Occurring in Falkner and 
Skan's Approximate Treatment of the Equations of the Boundary 
Layer," Proc. Cambridge Phil. Soc., Vol. 33. pp. 223-239, 1937. 

4. Crabtree, L. F., Kuchemann, D., and Sowerby, L., "Three- 
Dimensional Boundary Layers," in Laminar Boundary Layers, 
p. 427. Edited by L. Rosenhead. Oxford University Press, New 
York, 1963. 

5. Launder, B. E., Lamirtarization of the TurhuleTa Boundary 
Layer by Acceleration, Report No. 77. Gas Turbine Laboratory, 
Massachusetts Institute of Technology, Cambridge, Mass., 1964. 



B. Liquid-Metal MHD Power Conversion, 

D. G. Ellion, L. G. Hays, and D. J. Cerini 

1. Introduction 

Liquid-metal magnetohydrodynamic (MHD) power 
conversion is being investigated as a power source for 
nuclear-electric propulsion. A liquid-metal MHD system 
has no moving mechanical parts and operates at heat- 
source temperatures between 1600 and 2000^. Thus, 
the system has the potential of high reliability and long 
lifetime using readily available containment materials 
such as Nb-l%Zr. 



In the MHD cycle being investigated, liquid lithium 
is (1) heated at about 150 psia in the reactor or reactor- 
loop heat exchanger; (2) mixed with liquid cesium at the 
inlet of a two-phase nozzle, causing the cesium to 
vaporize; (3) accelerated by the cesium to about 500 
ft/s at 15 psia; (4) separated from the cesium; (5) decel- 
erated in an AC MHD generator; and (6) returned 
through a diffuser to the heat source. The cesium is 
condensed in a radiator or radiator-loop heat exchanger 
and returned to the nozzle by an MHD pump. 

A 50-kW conversion system, which is to be operated 
with room-temperature NaK in place of lithium and 
nitrogen gas in place of cesium vapor, has undergone 
closed-loop tests with water and nitrogen. Cycle improve- 
ments have been studied and efficiencies of 8 to 11% 
were found to be theoretically possible through separator 
improvements or multistaging. 

2. NaK-Nitrogen Conversion System 

The conversion system was assembled without genera- 
tor coils for water-nitrogen testing. Figure 3 shows the 
nozzle, sepajator, generator housing, diffuser, liquid 
return lines, nitrogen lines, and the starting and makup 
systems. The coils wrapped around the liquid return 
lines are heaters which will serve as the electrical load 
for the 'generator in the NaK-nitrogen tests and maintain 
the NaK at room temperature. 

Twenty 5- to 10-min runs were made to determine 
the starting conditions and closed-loop operating limits. 
The system was started by turning on the nitrogen and 
then u);ecting water from the start tank at 140 psia and 
I'JiO 11' i while feeding 5 Ib/s of water from the makeup- 
flow r -gulator which was set to maintain 150-psia nozzle 
inlet pressure. When the nozzle pressure exceeded 140 
psia the start-tank flow stopped and back flow was pre- 
vented by check valves. The makeup regulator then con- 
."^inued o inject liquid until 150 psia was reached, after 
which he regulator continued to supply water to replace 
the i.o Ib/s lost with the nitrogen. The start sequence 
.equired about 5 s. Various settings of the nitrogen flow 
rite and the start tank and makeup regulator pressures 
were tried in the first few runs until the smoothest pres- 
sure buildup was achieved. Pressiure oscillations occurred 
with .some settings and closed-loop operation was not 
sustained after the start-tank flow ceased. 

After several runs the generator channel was mspected 
and it was found that the laminated vanes for eddy- 
cun ;nt suppression at the generator inlet and three of 
the laminated slot plugs were missing. The tests were 



120 



Jn SPACE PROGRAMS SUMMARY 37-51, VOL III 




Fig. 3. Liquid-metal MHD reference system 



continued and operation wa,. obtained at several mixture 
ratios at each of tlif three nozi;le p.cssures selected for 
the NaK tests; 150, 190, and 2.30 psia. The vanes and 
slot plugs were then replaced, and a second scries of 
runs was made. Some vanes were again lost, and Iwtter 
anchoring techniques will be required in the NaK tests. 

3. High Efficiency Cycles 

Thermodynamically, Hquid-metal MHD cycles using 
two components, such as c< sium and h'thium, and employ- 
ing a regenerative heat exchanger between the ccsmm 
vapor and cesium condeii.sate lines are limited only by 
the Camot efficiency 1 — Tn/T„ since heat input and 



output are at es.wntiall) constant temperature. However, 
friction losses in the pre.sent design concept limit the 
efficiency to about 25% of the Camot value at space 
powerplant conditions of Tj/T, s 0.7, or an efficiency 
of about Q% (half the eflSciency of turbine and ther- 
nnonic conversion sj'stems). 

The main friction losses are in the separator and gener- 
ator and can be reduced in three ways: (1) decreasing 
the separator width to decrease the generator surface- 
to-volume ratio, (2) finding a method other than surface 
impingement for coalescing the flow, and (S) reducing 
the velocity of the liquid metal through multistage opera- 
tion at reduced pressure ratio per stage. 



JPL SPACf PROGRAMS SUMMAHY 37-5), VOL. lit 



121 









GENERATOR INLET WIDTH/HEIGHT RATIO c/ZIz 






S2 


16 


8 


3 




12 


NOZZLE EXIT PRESSURE, psio 


«jO 







10 


r 


.- WITHOUT FRICTION 


15 




s» 


8 


- 




JS^--:===^ 


,:=:==== 


>■* 

o 

z 








15 




o 

b. 
Il- 

UJ 


6 


r' 


-"■'■■''with FRICTION 






UJ 

-I 






c 
/ 




^ 


>- 
o 


4 


- 


/■ / ^ 


,y 


GENERATOR 




Tf^ 


1 
1 


-^■^^r^ 




2 


- 


1 ' 

♦ L 


1 ^^^ 


^"^ — SEPARATOR 




^^— NO 


ZZLE 









1 


1 


1 



2 4 6 

NOZZLE EXIT HEIGHT/WIDTH RATIO h^/c 



10 



Fig. 4. Effect of reducing separator width and eliminat- 
ing separator friction on cycle-efficiency 

a. Separator improvements. Figure 4 shows the effi- 
ciency gains possible through separator width reduction 
and frictionless coalescence. The operating conditions, 
using cesium and lithium as the working fluids, are: (1) 
ISOOT nozzle inlet temperature, (2) 300-kW electric 
output, (3) nozzle performance as calculated from Ref . 1, 
(4) turbulent skin friction on the separator, (5) generator 
performance as given in Fig. 4 of Ref. 2 (compensated 
case), (6) 80% diffuser efficiency, and (7) 20% cesium 
piunp efficiency. The lower curves show cycle efficiency 
as a function of height-to-width ratio hi/c at the separator 
inlet for condensing pressures of 10 and 15 psia, the 
latter giving minimum radiator area. The cycle efficiency 
increases from 6.3% with a square inlet to 8.0% with a 
height-to-width ratio of 10. About 20% of the increase 
is due to the increased Reynolds number of the thicker 
liquid film on the separator surface, and the remainder 
is due to the increased width-to-height ratio of the gen- 
erator channel c/hi (Fig. 4) which reduces the generator 
siuface-to-volume ratio. 

A separator with a square inlet, matching a circular 
nozzle exit, has been assumed in past cycle studies and 



has been the only type tested. Liquid impingement on 
the sidewalls of such a separator has been small and it 
may be possible to increase the height-to-width ratio to 
4, for a 1-percentage-poinf efficiency gain, and even 10, 
for a 2-percentage-point gain, without excessive sidewall 
impingement. A nozzle with a ratio of 3.8 is being fabri- 
cated to investigate narrow separators. 

The upper two curves in Fig. 4 show the efficiency 
attainable without separator friction. Even with a square 
nozzle exit, the efficiency is 9.3% at 15 psia condensing 
pressure and 10.2% at 10 psia. If, in addition, the fric- 
ticnlessly coalesced liquid can be delivered to the gen- 
erator •'t the more favorable aspect ratios, then cycle 
effici). .es could reach 11 to 12%. To determine to 
what extent such gains can be realized in practice, a pair 
of nozzles are being fabricated for impingement of the 
flows on each other instead of on a solid surface. 

b. MuUiatage cycle. An efficiency improvement to the 
9-11% range can be achieved without separator changes 
if power is extracted at intermediate stages of the expan- 
sion process. The improvement results from lower separ- 
ator losses in stages at higher pressure and improved 
generator efficiency resulting from the lower liquid veloc- 
ities. 

A liquid-metal MHD cycle in which power is extracted 
in five stages is shown in Fig. 5. Lithium and cesium are 
expanded (at 1800° F) from 137 psia to a pressure of 88 
psia in the first stage, producing a velocity of 245 ft/s. 
The two-phase mixtiure impinges on a separator where 
the lithiimi liquid is separated from the cesium vapor. 
The separated lithium enters an MHD generator at 
about 233 ft/s, and power is extracted at constant pres- 
sure, reducing the velocity to 50 ft/s. The lithium stream 
is then re-mixed with the separated cesium vapor in the 
second nozzle and the mixture is further expanded from 
88 psia to 57 psia, giving a velocity of 240 ft/s. The pro- 
cess is continued through succeeding stages until the last 
stage is reached, where sufficient dynamic head is 
retained in the lithium at the generator exit to return 
the lithium through a diffuser to the heat source and 
first-stage nozzle. The cesium vapor from the last stage 
passes through a regenerative heat exchanger to the radi- 
ator (or other heat sink) where it is ccmdensed. It is then 
returned by a pump through a regenerative heat 
exchanger to the first-stage nozzle. 

An analysis was made of the performance of this cycle 
with 3, 5, and 7 stages for a few specific values of nozzle 
exit pressure and lithiiun-to-cesium mass ratio re. The 



122 



Jn SPACE mOGkAMS SUMMARY 37-51, VOL. Ill 




STAGE 3 



XI n_ 









n n 


SEPARATOR 




GENERATOR 













PUMP 



RADIATOR 



7f 




Fig. 5. Five-stag* liquid-metal MHD cycle 



assumptions were the same as those used in the analysis 
of the single-stage cycle, and the generator efficiency 
was obtained as a function of velocity from Fig. 7 of 
Ref. 2. 

The result, expressed in Fig. 6a, shows cycle efficiency 
to increase with the number of stages at the chosen con- 
ditions of a mass ratio of 9 at a condensing pressure of 
14.5 psia. Tlie cycle efficiency increases from 6.3% for 
the single-stage reference design system to 9.3% for 
seven stages. For a condensing pressure of 9.5 psia, the 
efficiency rises from the single-stage value of 6.5% to 
9.9% for seven stages. Further increases are attainable 
through variation of the condensing pressure and mass 
flow ratio, since the use of multiple stages lowers the 
frictional losses of the system. For five stages, Fig. 6b 
shows the efficiency to increase from 8.5% at a back 



pressure of 20 psia to about 9.8% at 8 psia. The use of 
seven stages at this condensing pressure should result in 
an efficiency in excess of 10% . Further increases are also 
possible by increasing the mass ratio. For example. Fig. 6b 
shows that increasing te from 9 to 15 at 14.5 psia increases 
the cycle efficiency from 8.7 to 9.5% . 

Increasing the number of stages at constant condensing 
pressure reduces the specffic radiator area in proportion 
to the increa'se in efficiency. For example, increasing 
the mmiber of stages from one to seven decreases the 
isothermal, c = 0.9, radiator area from 3.8 to 2.4 ft*AWe. 
For a 300-kWe space power system, this would corres- 
pond to decreasing die isothennal radiator area teom 
1130 to 720 ft*. 

Multistage systems thus appear to offer ctmsideiable 
performance advantages over a single-stage system when 



Jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



123 



10 






UJ 



u 

u 

-I 
u 

> 



(a) 




1 
f, -9 












J^ 




^ 




/^ 




^ 






/^ 


r 


^ T 

i-r, = 9 
P, = 14.5 






^ 













S 4 B 

NUMBER OF STAGES 




10 IS to 

CONDENSING PRESSURE p,, ptio, AND MASS RATIO r^ 



u 



Fig. 6. Multittag* cycle •fficiency as a function of: 'a) number of stages, and (b) condensing 

pressure and mass ratio 



the major losses are taken into consideration. In addition 
to performance gains, increased reliability and operating 
life should be possible because of the lower liquid-metal 
velocities. For example, the reference single-stage system 
has a nozzle exit velocity of 513 ft/s while a five-stage 
system has an exit velocity of 245 ft/s. Furthermore, the 
nozzles in a five-stage system are subsonic so that further 
reduction in friction may be possible through reduction 
in separator area with the convergent flow. 

References 

1. Elliott, D. C, and Weinberg, E., Acceleration of Liquids in 
Two-Phase NozxJes, Technical Report 32-987. Jet Propukion 
Laboratory, Pasadena, Calif, (in press). 

2. Elliott, D. G., "Performance Capabilities of Liquid-Metal MHD 
Induction Generators," paper to be presented at the Symposium 
on Magnetchydrodynamic Electricd Fower Generation, Warsaw, 
Poland, July 24-30, 1968. 



C. Evaluation of the SE-20C Thruster Design, 

r. D MasBk 

1. Introduction 

Improvements in thruster efiBciency due to configura- 
tion changes have been reported in Refs. 1 and 2. These 
changes resulted in the SE-20B thruster design (solar 
electric 20-cm-diam thruster, modification B). Since many 
of the changes were made without cmnplete thruster 
redesign, only minor consideration was given to weight, 
fabrication, and packaging. A new design (SE-20C) dis- 



cussed in this article includes previous modifications but 
with variations required to reduce weight, provide 
strength, and ease assembly and mounting. Since these 
small variations in design might change thruster perfor- 
mance, the SE-20C thruster must be evaluated in detail. 
Thruste/ construction, weight, and performance are con- 
sidered in this work. 

2. Thruster Construction 

The basic elements of the present 20-cm-diam thruster 
are shown in Fig. 7. The general size and shape of the 
ferromagnetic elements were established in Refs. 3 and 
4. As in the initial design (Ref. 1), assembly ease and grid 
alignment were basic considerations. Use of previous 
grid designs was also required to allow interchangeability 
and to avoid the expense and time of new grid fabrica- 
tion. Thus, the specific dimensions of the housing, anode, 
support rings and brackets were determined by the 
existing grid design. 

Front and rear support rings niount the bar electro- 
magnets and provide a magnetic flux path. Bar elec- 
tromagnets were chosen (1) to provide for the possibility 
of using permanent magnets, (2) t^ tulow the magnetic 
field to be adjusted in performance mapping, and (3) for 
low power since the magnetic flux is used more e£Bciently 
than with conventional solenoidal designs. 

The mount assembly was designed to mate with Ae 
gimbal elements of the thrust vector alignment system 
(Ref. 5). High-voltage isolation is included in the mount 



124 



JPL SMCE nOOMm SUMMAKY 37-51, VOL. Ill 



assembly by four Alite insi'la«-ori. Propellati* is intro- 
duced, as in previous designs, through the side of the 
thruster at the center of the anode. 



3. Wtight Summary 

A weight breakdown for the SE-20C thruster is pre- 
sented in Table 2. The total weight of 4.06 kg (8.96 lb) 
includes a ground screen, connector halves, 10,000-h 
(estimated life) grids, and feed system up to the 
vaporizer. 

Table 2. SE-20C thruster waight summary 



Component 


Weight, 9 


Housing 


378 


Screen grid pole piece 


135 


Support ring, forward 


114 


Support ring, oft 


128 


Anode 


300 


Rear plate 


240 


Cathode pole piece 


70 


Cathode (Hughei oxide) 


135 


Screen grid 


106 


Accelerator grid 


675 


Magnet (8 each) 


640 


Accelerator mount aisembty (8 each) 


168 


Ground screen, forward otiembly 


128 


Ground screen, oft assembly 


246 


Anode and ground screen insulators 


40 


Mount assembly (pod, insulators, and cover) 


206 


Connector halves 


255 


Feed system (vaporizer, isolator, and manifold) 
Total 


100 


4064 



The need for ferromaguntic parts places certain re- 
strictions on thruster weight. Aluminum has been used 
in certain parts as indicated in Fig. 7 but cannot be 
used extensively. Additional weight reductions (approxi- 
mately 10%) appear possible by reducing thicknesses. 
However, the effect of these reductions on the magnetic 
field shape and strength (or power) and on structural 
strength must be evaluated. 

4. TcttitMuits 

a. Grid ttabiUty. Initial testing of the SE-20C ti ruster 
resulted in relative higjti discharge efficiency but 8hi>wed 



high accelerator impingement for flow rates above 6 g/h. 
The impingement could be reduced bv increasing the 
total ion beam accelerating voltage (from 4.0 to 5.5 kV 
at 6 g/h). This indicated that the grid spacing, nominally 
0.178 cm, had increased substantially. 

Bench tests were conducted using dial indicators to 
measure screen and accelerator deflections. The grids 
were heated to simulate cathode and plasma radiation 
heating using lamps and a heat gun. The results of these 
tests are as follows: 

(1) The accelerator deflected up to about 0.025 cm 
toward the screen when heated in the center 
region. As the housing and outer portion of the 
accelerator were heated, tho deflection decreased. 

(2) The direction of screen grid deflection depended 
upon its initial setting. When heated centrally, 
deflections up to 0.125 cm occiured in the direction 
of the initial bow. As with the accelerator, heating 
the housing reduced the screen deflection. Since 
fabrication always produces a slight bow, the initial 
assembly must force the screen to deflect toward 
the accelerator. This reduces the grid spacing with 
heating and is much more desirable than increased 
spacing. 

As a result of these bench tests, a method for providing 
an initial positive deflection (towaid the accelerator) was 
devised. The outer 0.4'"5 cm of the housing side of the 
screen grid was chamfered at an angle of 1.5 deg. This 
slight chamfer produced an initial bow of about 0.05 cm 
at the center. Thruster operation with this configuration 
(with a 0.178-cm spacing at the outer edge) showed low 
impingement rates at all flow rates. However, the close 
spacing, probably as low as 0.05 cm during start-up or 
fast power level changes, caused sparking between the 
grids. 

In addition to low impingement with the pre-bowed 
configuration, the thruster could be operated with lower 
accelerating voltages. A beam current of 1.0 A was 
obtained with a total voltage of 3.5 kV. This result 
verified the conclusion that the initial impingement 
difficulties and observations were caused by a large 
grid spacing. 

Previous difficulties with high impingement rates have 
been attributed to magnetic field or plasma density dis- 
tributions. Many of these problems may be resolved with 
the more controlled grid configuration. 



JH SPACE PROGRAMS SUMMARY 37-51, VOL. If I 



125 




u 

o 

M 

lit 

« 



e 

.2* 

*5 

at 
•0 
w 
'S 

o 
a 

n 



126 



Jn SMCE nOGRAMS SUMMARY 37-51, VOL 11/ 



b. Performance. Thruster performance can be easily 
evaluated by considering only the discharge loss per 
beam ion. All other losses, although significant in deter- 
mining thruster e£Bciency, are not important in comparing 
die SE-20B and SE-20C designs. 

Discharge loss as a function of propellant utilization, 
propellant flow rate, and magnet current is presented in 
Fig. 8. A comparison of this data with that obtained in 
the SE-20B thruster (Ref. 2) is shown in Fig. 9 for a 
magnet current of 2.0 A. Higher losses (about 15 eV/ion 
at 80% utilization) and higher slopes are indicated for 
the SE-20C thruster. Since both magnet designs are 
nearly identical, the difference in performance is attrib- 



uted to the minor differences in the ferromagnetic parts 
(thicknesses and construction). 

The discharge losses of the SF,-20C at slightly higher 
field are equivalent to the SE-206 thruster as shown in 
Fig. 8. The higher loss is attributed to a somewhat higher 
magnetic flux resistance in the new design due to thinner 
ferromagnetic elements. With the small differences 
noted, performance of the SE-20C is quite similar to the 
SE-20B design. 

Rcf«r«nc*t 

1. Ma«ek, T. D., Expertmentd Studie* With a Mercurv Bombard- 
ment Thnuter System, Techm-al Report 32-1280. Jet Fropulsion 
Laboratory, Pasadena, Calif, (in press). 



300 



250 



V) 
tu 
CO 

I/) 
o 



tu 

cc 

< 

X 

u 



200 




70 so 

UTILIZATION EFFICIENCY, 7. 

Fig. 8. Si-20C thrutttr parfemanc* dcrta 



JH SMCC nOGKAMS SUMMAItY 37-51, VOL. 



t27 



2. Masek, T. D., and Pawllk, E. V., "Thrurt System Technology 
for Solar Electric Propulsion," AIAA Paper 68-541, AIAA Fourth 
Fropubion Joint Specialists Conference, Cleveland, Ohio, June 
10, 1968. 

3. Bechtel, R. T., "Discharge Chamber Optimization of the Sert II 
Thruster," AIAA Paper 67-668, AIAA Electric Propulsion and 
Plasnittdynamics Conference, Colorado Springs, Colo., Sept. 
1967. 

4. Pawlik, E. V., Scaling of a High-Performance Ion Thrutter, 
Technical Memorandum 33-387. Jet Propulsion Laboratory, 
Pasadena, Calif., Apr. 1968. 

5. Reada, P. D., and Mankovitz, R. J., "Attitude Control of an 
Electrically Propelled Spacecraft Using the Prime Thrust Sys- 
tem," paper to be presented at the ASME 1968 Aviation and 
Space Conference, Los Angeles, Calif., June 1968. 



D. Radial Distribution of Enthalpy in a 

iHigh-Temporaturo Swirling Flow, P. F. Maulv 

1. Introduclien 

In arc heaters and plasma electrical propulsion devices, 
gas is sometimes injected tangentially upstream of the 
electrodes in order to introduce swirl into the flow. 
Although certain advantages may be gained from the 
swirl, one of the disadvantages may be an increase in the 
coiivective heat transfer rates as shown in SPS 37-24, 
Vol. IV, pp. 105-108, for flows through nozzles. Con- 
sequently, swirling flows are being investigated to acquire 
a better understanding of heat transfer to electrode and 
other surfaces :>o that improvements can be made in 



300 



250 



ill 

<n 
■J) 
o 



UJ 

o 

< 
I 
o 
« 
o 



200 



OPEN SYMBOLS: SE-20C 
SOLID SYMBOLS- SE -208 



FLOW RATE, g/h 
9.40 




70 80 

UTILIZATION EFFICIENCY, % 



ng. 9. Comparison of Si-20C ana SE-20B tliruttor porformonco data 



1 



128 



jn SMCf nOOHAMi SUMMAtY iT-SI, VOL iff 



predicting the cooling requirements of piasma devices. 
Other effects ako being investigated but not discussed 
here include severe wall cooling, acceleration, ionization, 
and applied magnetic and electric fields. 

Frtmi a heat transfer viewpoint the important flow 
variable is the radial distributimt of the enthalpy. When 
evaluated at the wall, the slope of the enthalpy is related 
to the wall he^tt flux. This distribution is generally 
dependent on many factors; in particular, for a swirling 
flow it depends upon the amount of swirl, Le., the ratio 
of tangential to axial velocity. Consequently, a knowledge 
of the enthalpy distribution is essential for evaluating 
the theoretical methods now being advanced for pre- 
dicting the convective heat transfer. The discussion in 
this article pertains primarily to the feasibility of using 
a calorimetric probe to determine the radial distribution 
of the enthalpy in a confined swirling flow of a high- 
toraperature gas. 

2. ExparimMital Appf.<atws 

The experimental apparatus (Fig. 10) was fabricated 
to evaluate radial distributions of enthalpy and tangential 
velocity, and longitudinal distributions of wall heat flux 
in a constant-diameter duct. Arc-heated argon enters die 
duct throu^ one port near the endwall. The gas then 
flows through the duct and discharges through a 
convergent-divergent lUKzzle attached to the other end. 



ARC HEATER 




0.3 diom 
THROAT 



DIMENSIONS ARE INTEhKAL IN INCHES 

Fig. 10. TMt apparatus 



After leaving the nozzle the gas flows into a vacuum 
system. Several tests have been conducted in which the 
enthalpy distribution was obtained by radially traversing 
a calorimetric probe at the location shown in Fig. 10. 
Details of the probe and the associated data analysis 
procedure appear in SPS 37-46, Vol. IV, pp. 153-161. The 
probe used in the swirling flow investigation was straight 
with the tip pointing in the radial direction; hence, the 
local impact and static pressures were not measured. 

The walls ?*^ the apparatus consisted of many indi- 
vidual circumferential coolant passages for determination 
of the wall heat flux distribution an'l the endwall con- 
tained numerous pressure taps for the purpose of evaluat- 
ing the radial distribution of the tangential velocity. 
The velocity and the heat transfer results are not dis- 
cussed here, however. 

3. Results 

The distribution of the enthalpy as determined by the 
calorimetric probe is shown in Fig. 11 for one test in 
which the pressure in the duct was 3.9 psia and the stag- 
nation temperature was approximately 3000° R. Trends of 
the other tests are similar. At the probe location the 
Reynolds number of the main gas stream was 490 based 
on the average mass flux and duct diameter with vis- 
cosity evaluated at the average free-stream temperature. 
The Mach number based on the average axial velocity 
was 0.01 and the ratio of the maximum tangential to axial 
velocity was approximately 5. 

Figure 11 indicates a symmetrical enthalpy distribution 
and shows that the edge of the thermal boundary layer 
was approximately 0.1 in. from the wall. The maximum 
values on either side of the centerline resemble the trends 
in stagnation temperature distributions observed in vor- 
tex flows near room temperature that are discussed in 
SPS 37-33, Vol. IV, pp. 133-141. It has been verified, 
however, that probes introduced into a swirling flow 
can have a significant effect on the flow field (Refs. 1 and 
2). An influence of the probe (m the enthalpy distribution 
shown in Fig. 11 appears to be evident when companng 
the integrated average enthalpy based on the probe data 
with the average obtained by an energy balance which 
takes into account the applied electric power and heat 
transferred to the coolant. The probe average is about 
17% lower than the enthalpy determined from energy 
balance. A comparism cf the average enthalpies obtained 
by these two methods in nonswirling flows shows better 
agreement (SPS 37-47, Vol. Ill, pp. 103-116, and SPS 
37-46, Vol. IV, pp. 153-161). The low probe readings 



Jn SPACE PROGRAMS SUMMARr 37-57, VOL. Iff 



IW 



UJ* 



a: 

UJ 



111 

(9 
U. 

o 

>- 



< 

X 



700 
600 


1 
TEST 57-20H 

Pt • 3.9 psia 


















^.^DUCT C 






















,^ 


^, 






,^ 


s. 








4C0 














1 


Y 


\ 


k 


J, 


V 


N 


\ 






y 






I 


L , 




/ 


\ 


II.' 1 
-AVERAGE BASED ON ENERGY 
BALANCE ='.405 Btu/lb 


s 


^^ 




300 

200 
100 


f ~l 


r 




V 


p 


I 




C 


> 








^ AVERAGE BASED ON PROBE 
DATA' 335 Btu/lb 











































0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 

DISTANCE FROM WALL, in. 
Fig. 1 1 . Radial distribution of enthalpy 



1.0 



I.I 



1.2 



19 



14 



I.S 



may have resulted from the integrated average enthalpy 
being based on f HtdA instead of J puHt dA. Avaflable 
information was insufficient to determine the mass flux 
(pu) distribution. It is also possible that the low probe 
average was caused by some of ^he cool gas in the 
boundary layer near the duct wall flowing radially in- 
ward along the outer wall of the probe tube and then 
entering the probe tube during sampling. Thus, the heat 
transfer measurements that are made on the sampled 
gas would indicate a lower enthalpy of the main gas 
stream at a particular radial position than would exist 
if the probe were not in the dt'ct. The gas at the outer 
radii of the duct has a tendency to flow radially inward 
along the probe wall because of a reduction in tangential 
velocity caused by the boundary layer formed on the 
probe. Thus, locally, the radial pressure gradient dp/dr 
is not balanced by the centripetal acceleration pv'/r 
maintained by the tangential velocity and, hence, radial 
flow occurs. Such radial flow can also occur in the wake 
of the probe. 

4. Conclusions 

Despite the apparent low average enthalpy determined 
from the probe data, the location of the edge of the 
thermal layer and the general distribution of the enthalpy 
are significant results. Near the duct wall the radial 
pressure gradient is comparatively small; hence, the 
radial flow there would not be large and the value of 
the enthalpy at the edge of the boundary layer is prob- 
ably realistic. 



References 

1. Roschke, E. J., "Flow-Visualization Studies of a Confined, Jet- 
£>riven Water Vortex," Tedinical Report 32-1004. Jet Propulsion 
Laboratory, Pasadena, Calif., S^t. 15, 1966. 

2. Pivirotto, T. J., "An Experimental and Analytical Investigaticm 
of Concentration Ratio Distributions in a Binary Compressible 
Vortex Flow," Technical Report 32-808. Jet Propulsion Labora- 
tory, Pasadena, Calif., Mar. IS, 1966. 

E. Some Effects of an Applied, Transverse 
Magnetic Field on Heat Transfer With 
Swirling and Nonswirling Gas Flow, E. J. Roschke 

1. Introduction 

An apparattis for studying convective heat transfer from 
partially ionized gases in a transverse magnetic field was 
described in SPS 37-47, Vol. Ill, pp. 120-128. Modifica- 
tions of this apparatus and some preliminary he&t transfer 
results were discussed in SPS 37-49, Vol. Ill, pp. 199-201. 
This work is an initial step towards increasing the under- 
standing of energy transfer processes that occur when a 
flow of ionized gas interacts with electric and magnetic 
fields. Such iuformation is important in the prediction of 
electrode heat transfer and is also necessary for the design 
of magnetogasdynamic generators and propulsion devices 
such as magnetoplasmadynamic arcs. The purpose of this 
article is to present the effects of lioth magnitude and 
direction of an applied, transverse magnetic field on beat 
transfer from partially ionized argon that have been 
investigated in two tests, with and without swirl in the 
flow. 



130 



jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



Symbols used in this article are defined in Table 3. 

2. DMcription of Apparalut 

The in-line arc configuration used for the present heat 
transfer experiments is shown in Fig. 12. The portion of 
the apparatus of immediate interest is the 2- X 2-iii. 
square channel which is approximately 13 in. long. An 
irJ . .. "ction is provided to promote adequate mixing and 
flf / de- elopment of the high-temperature gas stream 
s iplied by the electric arc heater. The test section (total 
leagth 4 in.) is the downstream portion of the channel; 
the four walls of each 1.0-in.-long segment are individ- 
ually cooled so that heat transfer may be determined 
by calorimetry. [The walls are designated A, B, C, and 
D, clockwise, looking downstream (Fig. 12).] Flow is 
exhausted from the system by means of a 2.88-in.-diam 
circular duct approximately 19.3 in. long. All experiments 
are conducted at the short-circuit condition with zero 
load factor. 



Tabia 3. Nem«nclatur* 



b 


channel height, 2 iiL 


k 


thermal conductivity of gas (Ref . 1) 


m 


mass flow rate of gas 


P 


static pressure, absolute 


q 


heat flux 


Q* 


non-dimensional heat flux 


Qt 


non-dimensional heat flux at zero magnetic field 


Re 


Re>-nolds number based on mass flow rate, for 




square channel Re = m/iib 


r, 


inlet gas temperature, at c«iter of first test- 




section segment 


r« 


gas-side wall temperature 


M 


gas viscosity (Ref. 1) 



CATHODE 




TO 
VACUUM 
PUMP 



INSULATOR - 



■ INSULATOR 
■TANGENTIAL GAS INJECTOR 



SECTION X-X 
FLOW ® ^^^^ ' ' 



e 



^^ COPPER 



DIMENSIONS IN INCHES 

PRESSURE TAPS AND COOLANT 
PASSAGES OMITTED FOR CLARITY 

EACH SffiMENT OF TEST SECTION HAS 
AXIAL LENGTH OF lln. 



STAINLESS STEEL 



Fig. 12. In-1in« arc c«nfigurotlon for hocrt mnsfor oxporimontt (tido viow) 



Jn SPACE PROGRAMS SUMMAKY 37-51, VOL. Iff 



131 



The axial location of the test section with respect to 
the magnet pole pieces is also shown in Fif. 12. Heavy 
arrows indicate the normal direction of the applied mag- 
netic field, termed "forward field." In the case of "reverse 
field," the arrows point in the opposite direction to that 
shown. 

Experience has shown that tang.ntial gas injection 
upstream of the anode (Fig. 12) is generally superior to 
radial iujectM.m because higher arc efficiencies are 
obtained for the same applied elect ic powjr. Tlius, more 
heat may be added to the gas resJliiig ia a higher tem- 
perature stream. In addition, it his oeen found that 
improved stability may be obtained at higher power 
levels. For this reason, most of the fxperiments have 
been obtained using tangential injcctio:>. (The direction 
of the gas injection in Fig. 12 would je clockwise, looking 
downstream.) However, the presenf» of swirl in the flow 
complicates the interpretation of data as well as any 
theoretical analysis that might be attr>mpted; therefore, 
comparisons of data with and withon swirl are desirable. 
Currently, a series of tests employing radial injection is 
being conducted. The first results are presented here. 

3. Method of Presenting Data 

To study the effect of the magnitude of the applied 
magnetic field on heat transfer, it is necessary to find 
some basis of c<Mnparison for a series of tests in which 
the only parameter varied deliberately is the magnitude 
of the applied field. The reason for this is that changes 
in the applied field produce internal changes in the gas 
which are often accompanied by changes in the voltage 
in the electric arc-heater. Thus, the initial energy content 
and temperatiu^ of the gas usually varies considerably 
with varying magnetic field. An approximate correction 
for this is obtained by using the non-dimensional heat 
flux defined by 



Q* = 



qb 



HT, - r„) 



(Also see the theoretical analysis of Back, Bef. 2.) In the 
present application, Tt is obtained as a bulk or average 
value of temperature at the center of the first test-section 
segment by means of an energy balance applied to the 
system up to that axial location and by ase of a moUrer 
chart for argon. At that axial location, the nugnitude 
of the magnetic field is 94% of the peak value. The 
thermal conductivity of the gas is obtaiiied at Ti and 
the local wall pressure using the results given in Bef. 1 
Gas-side wall temperatures are of the order of IWF in 
these experiments. Heat transfer results are presented 



for the second segment of the test section, however, using 
Ti as discussed. 

To isolate and clarify the effect of magnetic field still 
further, values of Q* are normalized with respect to their 
values for each wall when the magnetic field is zero. 
Thus, the parameter used is Q*/Q*. which, in effect, 
reduces the results for the four, walls to a comparablt 
base value so that trends with varying magnetic field 
are more easily evaluated. 

Changes in heat transfer brought about by the mag- 
netic field through joule heating are independent of the 
direction of the induced ciurent in the gas (only its 
magnitude, Bef. 2). Thus, the vertical orientation of 
the applied, transverse field is theoretically unimportant 
when the gas has axial motion alone and there are no 
Hall effects to cause transverse Lorentz forces. With swirl 
present, this would not be necessarily true. Tv,'o - ^s 
have been selected for presentation, one utilizing tangen- 
tial injection and the other utilizing radial injection. 
Conditions iu these tests are given in Table 4. 

Table 4. Nominal test conditions at zero magnetic field 



ParaaMtor 


Test 1 07-1 SH 
iniector 


Ten 
107-2SH 
Aadion 
injector 


Applied power le ciccl.k 
ore, kW 


49.5 


34.8 


Actual heal input to 
got, Btu/t 


31.2 


18.1 


Matt flow rote m, Ib/t 


0.007 


0.007 


Inlet ttatic pretture pi, ptio 


0.85 


0.83 


Inlet static temperature Ti, *t 


16,700 


11,750 


Inlet Reynoldi number. Re 


260 


280 



4. Experimental Results and Discussion 

With the tangential injector, it has been generally 
found that the largest absolute changes in wall heat 
flux due to the applied magnetic field occur for reverse 
field. Also, the sidewalk generally experience relatively 
greater changes than the upper and lower walls. The 
non-dimen.sional heat-flux ratio Q*/C for test 107-18H 
is shown in Fig. 13a. Although there is some scatter, the 
trends of the data are relatively clear. Heat transfer to 
the upper and lower walls tends to increase with increas- 
ing magnetic field regardless of the direction of the field. 



132 



JH SPACE PKOGItAm SUMMARY 37-51, VOL. Iff 




22 


(b) 




1 
















\ 


\ 


















^ 


^-A 


^ 




















\ 


^ 




^FORWftRD 
FIELD 




f^ 


^ 








s 


^ 


\ 


y 




y 










N 


\ 




y 


,^ 


^ 


,/" 








/ 


p. ^ 




^N, 




y 


Y 




'^, 


-^ 


/ 


^FIELD 




X 


K. 


^ 


--- 


OR 


1 







^0 -8 -6 -4 -2 



2 4 6 8 10 -10-8 -6 

APPLIED MAGNETIC FIELD. KG 



10 



Fig. 13. Heat transfer test results using: (a) tangential injector, test 107-18H; (b) radial injector, test 107-28H 



This trena agrees with the predicted tr-jnd, (Ref. 2). 
The maximum effect measured was a 60% increase in 
heat transfer on lower wall C. Trends for sidewalk B 
and D are different; wall B experiences a marked decrease 
in heat transfer with increasing forward field but wall 
D experiences a marked decrease with increasing reverse 
field, and conversely. This behavior for the sidev.alls is 
thought to be associated with a Hall effect and tends to 
agree with the lateral (side-to-3ide) deflection of the 
exhaust plume observed visually; i.e., an observed motion 
of the plume toward one wall coincides with an observed 
increase in heat transfer at that wall but a more signi- 
ficant decrease in the heat transfer at the opposite wall 
from which the plimfie was deflected. A prediction for 
the deflection of the gas stream due to a Hall effect is 
difiBcuIt to make in this case because of the consequences 
of swirl. 

Comparable data using a radial injector at considerably 
lower power levels and gas temperatures are shown in 
Fig. 13b for test 107-28H. The trends of the curves 'or 
upper and lower walls agree with that obtained for the 
tangential injector, i.e., increasing applied magnetic field 
tends to increase the heat transfer at those surfaces. 
Results for the sidewalls are somewhat different; heat 
transfer to wall D was decreased regardless of direction 
of field, whereas wall B experienced an increased heat 
transfer for reversed field but a decrease for forward 
field. Visually observed deflections of ihe gas were not 
pronounced in this test although deflection towards walls 
B and C were noted for forward field. A noticeably 



stronger gas deflection towards wall B was detected with 
reverse field. 



5. Conclusions 

Based on the limited results obtained in this study, the 
following conclusions are made: 

(1) Surfaces transverse to the applied magnetic field 
experience an increase in heat transfer with increas- 
ing field either with or without swirl in the flow 
regardless of the orientation of the field. 

(2) The largest increases observed are 60% in the case 
of flow with swirl and 60 to 100% without swirl 
compared to results with zero magnetic field. 

(3) Significant changes in heat transfer for walls par- 
allel to the field occur and may be positive or 
negative depending on field orientation and the 
presence or absence of swirl. These observations 
are thought to be associated with Hall effects. 

References 

1. deVoto, R. S., Argon flasma Transport Properties, Technical 
Report 217, Department of Aeronautics and Astronautics, Stan- 
foid University, Stanford, Calif., Feb. 1965. Also available in 
?hys. Fluids. Vol. 10, pp. 354-364. Feb. 1967. 

2. Back, L. H., "Laminar Heat Transfer in Electrically Conducting 
Fluids Flowing Between Parallel Hates," paper accepted for 
publication in Int. J. Heat Mass Transfer. 



jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



133 



F. Some Effects of on Applied, Transverse 
Magnetic Field on Wall Pressure in a 
Square Channel, E. J. Roichlr* 

1. Introduction 

Some heat transfer measurements for partially ionized 
argon flowing in a square channel with a transverse 
magnetic field were presented in Section E. The purpose 
of this article is to present wall pressure measurements 
indicating some of the effects produced by varying both 
magnitude and direction of an applied, transverse mag- 
netic field on pressure within the channel. These are 
companion results to those presented in the previous 
article for test 107- 18H and, therefore, apply for the case 
of swirl present in the flow. 

2. Experimental Apparatus and Measurements 

The arrangement of apparatus was identical to that 
shown in Fig. 12. (Section £); the designations of the 




four walls of the channel are retained. Static pressure 
taps were located at three axial positions of all four walls 
in the inlet section. Each l-in.-long segment of the test 
section was provided with pressure taps at a mean axial 
position, but only on sidawalls B and D. Pressure was 
measured by means of oil manometers which could be 
read to a precision of better than 0.002 psia. The con- 
vention used for orientation of the magnetic field is the 
same as that of the previous article. Static pressure results 
given here were taken concurrently with the heat transfer 
data of test 107-18H; Table 4 of Section E listed the 
appropriate test conditions. 



3. Experimental Results 

Axial distributions of static pressure were generally 
similar to those presented in SPS 37-49, Vol. Ill, pp. 
199-201. The effect of the magnetic field was to increase 
the pressure throughout the channel for a constant mass 
flow rate and to cause a peak pressure to be reached near 
the downstream end of the inlet section (Fig. 14). Values 
of magnetic field listed in the figure are for the down- 
stream end of the test section. Results are shown for 
forward field and for two walls, upper wall A and side- 
wall B. The relative differences observed between the 
two walls at zero field are not only preserved but 
increased with increasing field. 



4 6 8 10 <2 

DISTANCE FROM ANODE EXIT, In. 



1 00 




' T r 'H 

o UPPER WALL A 
'^ SIDE WALL 8 














LOWER WALL C 
n SIDE WALL D 




i 












i 


[ 








lO-mm OIL 


/ 






a 94 

J" 


\ 


A 






V 








T 






i 


\ 


5 

K 90 


I 


k^ 


^. 












J 






^J 


[\ 


RE 
F 


VERSE 
lELO 


FORWA 

FIEL 


RO ~i 

3 // 


/ 
















^ 


^ 


V^ 








r 






84 
82 
80 






^ 


s/^ 


V 


J 


















\ 


\J0 































Fig. 14. Axial distribution of pressure along walls 
of square channel 



-5-4-3-2-1 I 2 3 4 

APPLIED MAGNETIC FIELD, kG 

Fig. 1 5. Static pressure in inlet section at axial 
location 2 in. upstream of test section 



134 



JPL SPACE PROGRAMS ^{itAtAkn 37-51, VOL. Ill 



The eflFect of magnetic field is examined in more detail 
in Fig. 15, where the static pressure for all four walls 
at one axial station has been plotted as a function of 
applied magnetic field at that axial location. The axial 
position selected corresponds to the third pressure tap 
lovation of the inlet section, i.e., at an axial distance of 
7 in. from the anode exit (Fig. 12), which is the region 
of peak pressure (Fig. 14). Two results are apparent from 
Fig. 15: (1) the pressure increases with increasing field 
regardless of the direction of the field, and (2) the effect 
is much more pronounced at the upper and lower walls 
of the channel than at the sidewalls. It is also evident 
in Fig. 15 that, where wall C exhibits a higher pressure 
than wall A with forward field, the converse o<curs with 
reverse field. Walls B and D exhibit similar trends. It 
is believed that this observation could result because of 
Hall effects; however, it could also be a consequence 
of swirl present in the flow. 

In the regime of operation of the present experiment, 
theoretically high values of the Hall parameter are pre- 
dicted (SPS 37-47, Vol. Ill, pp. 120-128). Since an induced 
electric field in the axial direction is unlikely because 
the four walls of the channel form a continuous electric 
conductor, a large axial current flow is possible when 
the Hall parameter <ut > 1. Three experimental observa- 
tions tend to indicate that Hall effects were present in 



this experimen,"^: all three indicate the presence of a 
significant lateral (side) force, as well as transverse 
(vertical) component of force acting on the field. Firstly, 
an applied magnetic field had the effect of increasing 
heat transfer on one sidewall of the channel but decreas- 
ing the heat transfer on the opposite sidewall; when the 
direction of the magnetic field was reversed, the heat 
transfer effect also became reversed (see Section E). 
Secondly, the static wall pressure was sUghtly different 
comparing the two sidewalls, or comparing the upper 
and lower walls, and this effect also reversed when the 
field was reversed (Fig. 15). Thirdly, a visible effect was 
produced when the luminous core of the exhaust plume 
(in vacuum tank) was observed during a change in mag- 
nitude of the applied field. A significant lateral motion 
of the luminous core was observed with increasing mag- 
netic field; when the field was reversed, the luminous 
core nioved to the opposite side. 

Thus, the effect of the magnetic field was to increase 
the static pressure throughout the channel regardless of 
the direction of the applied field. Walls transverse to 
the magnetic field experienced a greater increase in 
pressure than did the sidewalls which were parallel to 
the direction of the field. The presence of Hall effects 
during this experiment is considered likely although the 
magnitude of these effects has not yet been established. 



Jn SPACE PROGRAMS SUMMAkY 37-51, VOL. Ill 



135 



*»CSD1HG.PAGE BUNK 



WOT HLMED. 



■', N 68- 3 7411 



XIV. Liquid Propulsion 

PROPULSION DIVISION 



A. Heat-Sterilization Compatibility of Ethylene- 
Propylene Rubber in N^H^, O. F. Keller 

1. Introduction 

This article presents the data covering the last part of 
a series of patch-type tests of an expulsion bladder 
material for the thermal sterilization compatibility study. 
The bladder material is ethylene-propylene rubber (EPR), 
Stillman Rubber compound SR 722-70. The results of the 
first three cycles at 275 and 300° F were previously 
reported in SPS 37-46, Vol. IV, pp. 167-173. The results 
of the last three cycles are reported herein and complete 
this phase of the program. 



2. Tott Proceduro 

Throughout the study, two test sample configurations 
have been used: (1) the circular-type, about 1.5-in. diam 
and 0.037 in. thick, and (2) the rectangular-type about 
1.5 by 2.0 by 0.037 in. tliick. These samples were cut from 
an existing diaphragm-type bladder as shown in Fig. 5 
of Ref. 1. The circular samples have been designated 
Oi and (h; the rectangular samples, bi and bj (Table 1). 
Two circular samples and two rectangular samples have 
been tested in each of three stainless steel containers. 



The average total surface area of the samples exposed to 
hydrazine in each container was 21 in^. The four samples 
in each container were separated from one another by a 
special stainless steel wire rack. 

The propellant containers and the wire racks were 
made of AlSI-type 347 stainless steel. The average 
volume of the containers with test samples removed was 
502 ml. Each container was equipped with an inlet port 
near the bottom and a vent port near the top. A pressuriz- 
ing port was included on the top of each container for 
adjusting the initial test pressure. The average container 
volume to the vent port was 305 ml. 

Each container was filled with hydrazine up to the vent 
port. A fourth container, without patch-test samples, was 
used as a reference, or control, container. The average 
pretest ullage volume for each container, including lines 
and transducers, was 205 ml. 

The foiur containers were mounted in a temperature 
control chamber, and heat-sterilization temperatures of 
275 and 300° F were maintained. The length of time at 
heat-sterilization temperature was 60 h for each test 
cycle, and the maximum number of test cycles at each 



JH SPACE PROGRAMS SUMMARY 37-51, VOL. /// 



137 



test temperature was six. Prior to testing, the containers 
were passivated, using dilute hydrazine at ambient 
temperature and pressure fov a period of 20 h. Initial 
container pressure for the tests at iSTST was 40 psig and 
for the tests at 300°F was 50 psig. 

3. T«stR«sult( 

After the first heat-sterilization cycle, the liquid hydra- 
zine was light brown in color and contained fine black 
material in suspension. This color did not change appre- 
ciably as a result of additional heat-sterilization cycles. 
The liquid hydrazine in the reference, or control, con- 
tainers remained colorless after heat-sterilization cycling. 

Following heat-sterilization testing with containers 2 
and 4, a quantitative chemical analysis of the remaining 



z 
o 

I- 

§ 

< 

(C 

o 



UJ 

z 

Nl 

<t 
a: 
a 

y 

X 




hydrazine was made. The results of this analysis are 
shown in Table 2. The quantities of water, ammonia, 
aniline, and hydrazine were -determined by gas chro- 
matography. The ammrnia content was also confirmed 
by a titration technique. The volumes of residual hydra- 
zine following heat-sterilization testing ranged from 140 
to 270 ml. The quantity of residual hydrazine varied with 
the number of cycles and the quantities of hydrazine 
vapor lost during the venting operation between heat- 
sterilization cycles. 

This series of tests included determination of param- 
eters for Shore A hardness of the patch-test samples, the 
permeation rate of Ae samples to hydrazine, and 
the degradation of the hydrazine resulting from heat- 
sterilization cycling. These parameters have been plotted 
as a function of the number of heat-sterilization cycles as 
shown in Fig. 1. The data indicate a slight increase in 
permeation rates with an increasing number of heat- 
sterilization cycles. Also an appreciable degradation of the 
hydrazine occurs during the first heat-sterilization cycle. 

Previous test results, as reported in SPS 37-46, Vol. IV, 
p. 173, indicated that the average pressure rise in the 
reference containers (with hydrazine on?y) was greater 
than the average rise in the co.itainers with both EPR 
patch-test samples and hydrazine (Fig. 2). To isolate the 
effects of the EPR/hydrazine reaction from the hydrazine 
reaction with the stainless steel containers, four type-347 



07 



05 



03 



0.01 



(b) 




275'F^ 


i^ 


k-522!L. 


p^ 




/\ 


i ^^ 


^ 


rZTST 




^ 




H 


M 


W.SOO'F 


\ 







74 


..^ 


k 












y 


S 


i.275'F 






iV 


1 
70 


1 * 


\ \ 


^ 


':^ 


300*F 








\ 


^ 


r^75*F 










300*f\| 


'v 






66 






^^ 







2 3 4 

NUMBER OF 60-h CYCLES 



Fig. 1. Effvct of hydrazin* on EPR (SR 722-70) afttr 
h*at tt«rilization cycling 



HYDRAZINE ONLY IN 347 -TYPE 
STAINLESS STEEL CONTAINERS 
(REFERENCE) 



ErR PATCH -TEST SAMPLES 
AND HYDRAZINE IN 347- TYPE 
STAINLESS STEEL CC TAINERS 




90 100 190 200 290 

STERILIZATION TEMPERATURE, 'F 



300 



390 



Fig. 2. Tamparatur* •msitivity of EFR patch-t«tt 
samplM immanad in hydrazin* 



138 



if\. SPACE PROGRAMS SUMMARY 11 -Sy, VOL. \\\ 



ISQxiit^ij'x. li 



Itsm* 


SampU No. 


Initial 

Ihltknoit, 

in. 


Final 

thickntii, 

in. 


InlNal 

wtlght, 

fl 


Final 

wtlght, 

9 


Ntt 

incrtatt 

Idtcrtait), 

9 


Incrtatt 
(dtcrtoitl, 

% 


Initial 
rhort A 
hardntii 

±2.0 


Final 

Short A 

hardntii 

±2.0 


Short A 

hordntM 

afttr 

ptrmtatlon 

Itit ±2.0 


1 


66 >' 0920M-a, 


0.034-0.036 


— 


1.3052 


1.3163 


O.OIII 


0.85 


72 


69 


66 


2 


66 X 09201-1-a, 


0.035-0.037 


— 


1.3153 


1.3248 


0.0095 


0.72 


73 


70 


66 


3 


66 X 09201-1-b. 


0.038-0.041 


— 


3.3868 


3.4076 


0.0208 


0.61 


71 


69 


— 


4 


66 X 09201 -Iba 


0.039-0.041 


— 


3.3690 


3.3930 


0.0240 


0.71 


71 


70 


— 


5 


66 X 09201-2-a, 


0.034-0.036 


— 


1.2619 


1.2597 


(0.0022) 


(0 17) 


71 


73 


67 


6 


66 X 09201 -2-Qa 


0.035-0.037 


— 


1.3135 


1.3146 


0.001 1 


0.08 


72 


70 


67 


7 


66 X 09201-2-b, 


0.037-0.041 


— 


3.3094 


3.3211 


0.0117 


0.35 


72 


72 


— 


8 


66 X 09201 -2b: 


0.038-0.042 


— 


3.565V 


3.5777 


0.0118 


0.33 


72 


70 


— 


9 


66 X 09201 -3-a, 


0.036-0.038 


— 


1.3235 


1 313? 


(0.0098) 


(0.74) 


72 


73 


— 


10 


66 X 09201-3-0: 


0.034-0.037 


— 


1.J975 


1.2991 


0.0016 


0.12 


73 


74 


— 


11 


66 X 09201 -3-b, 


0.039-0.042 


— 


3.4381 


3.4360 


(0.0021) 


(0.06) 


72 


73 


— 


12 


66 X 09201 -S-b: 


0.036-0.040 


— 


3.2606 


3.2566 


(0.0040) 


(0.12) 


71 


71 


— 


13 


66 X 09201 -4-a, 


0.034-0.036 


0.033-0.035 


1.2519 


1.7468 


(0.0051) 


(0.41) 


73 


73 


— 


14 


66 X 09201-4-a> 


0.033-0.037 


0.033-0.037 


1.2742 


1.2757 


0.0015 


0.12 


72 


72 


— 


15 


66 X 09201 -4-b, 


0.038-0.041 


0.037-0.041 


3.3060 


3.3020 


(0.0040) 


(0.12) 


71 


71 


— 


16 


66 X 09201 -4-b: 


0.038-0.041 


0.038-0.041 


3.3979 


3.3914 


(0.0065) 


(0.1 ») 


71 


71 


— 


17 


66 X 09201 -S-a, 


0.035-0.038 


0.035-0.037 


1.3380 


1.3368 


(0.0012) 


(O.OV) 


74 


71 


— 


18 


66 X 09201 -S-a: 


0.033-0.034 


0.032-0.034 


1.2221 


1.2196 


(0.0025) 


(0.20) 


74 


73 


— 


19 


66 X 09201 -5-b, 


0.035-0.039 


0.035-0.039 


3.1618 


3.1499 


(0.0119) 


(0.38) 


70 


70 


— 


20 


66 X 09201 -5-b: 


0.036-0.040 


0.036-0.040 


3.2941 


3.2845 


(0.0096) 


(0.29) 


72 


71 


— 


21 


66 X 09201 -6-a, 


0.035-0.037 


0.035-0.037 


1.3142 


1.3176 


0.0034 


26 


72 


72 


— 


22 


66 X 09201 -6-0, 


0.035-0.038 


0.035-0.037 


1.3131 


1.3188 


0.0057 


0.43 


71 


71 


— 


23 


66 X 09201 -6-b, 


0.035-0.039 


0.035-0.038 


3.0955 


3.0970 


0.0015 


0.05 


71 


71 


— 


24 


66 X 09201 -6-br 


0.037-0.041 


0.037-0.041 


3.4006 


3.4011 


0.0005 


0.01 


71 


70 


— 








PrtttM 


rt gtntralion ri 


f crtnct lor ce 


ntrel) lampltt 








25 


66 X 09201 -7-0, 


0.036-0.037 


— 


1.3228 


— 


— 


— 


72 





71 


26 


66 X 09201-7-0, 


0.033-0.034 


— 


1.1773 


— 


— 


— 


73 


-- 


73 


'Compound St 722-70. 






'Initial conto 


nor protsuro 50 p 


•Is. 










»PliiId t«f Itmptnttirm 37S'f (10 )> a 


V3'n 




'Thtu dclo a 


iivmod low bv a 


factor of 10. 










'Fluid toil tomporaturo )00*F. 






01 and « or 


t circular wmplo* 












'^Inltlol contalnor proiiuro 40 piig. 






bi and bi or 


• roctongular fam 


plot. 











JFL SPACE PKOGkAMS SUMMARY 37-51, VOi. Ill 



IOlt; 



L'.'juia 



X 



TabU 1 . Elhylsn* propylant patch-t*»t lampUt h*at->t»rilization tost data 



Shora A 
hordnoM 

aftor 
lormoatlen 
toft±J.O 


Shora A 

dry 
hardnoft 

oftor 
vacuum 

±2.0 


Pormoalion 

mg N,H,/ 

h/ln.' 


Tin* of 

ptrmoation 

toit, 

h 


No. 

of 

cyclot 


Total 

llmo 

at lost 

Ion poralura, 

h 


N>H. 

copcontrolt, 
% 


Containok 
No. 


Proituro 
dotignator 


Praiiuro 

oftor 

two 

cycUi, 

p.l« 


Praiiura 

oftor 

four 

cyclot, 

ptig 


Praitun 
of'ir 

tix 

cyclot, 

Ptig 


66 
66 


69 
70 
73 
72 


0.04S 
0.041 


108 
108 


4' 

4" 
4" 
4' 


250.5 
250.5 
250.5 
250.5 


96.4 
96.4 
96.4 
96.4 






310 
310 
310 

310 


190 
190 
190 
190 


— 


67 
67 


71 
70 
73 
71 


0.057 
0.030 


120 
48 


2" 
2' 
2' 
2' 


130.5 
130.5 
130.5 
130.3 


94.7 
96.7 
96.7 
96.7 






276 
276 
276 
276 




— 


— 


71 
72 
74 
72 


0.049 
0.059 


92.5 
92.5 


6" 
6" 
6' 
6" 


369.5 
369.5 
369.5 
369.5 


95.5 
95.5 
95.5 
95.5 


5 

5 




226 
226 
226 
226 


210 
210 
210 
210 


174 
174 
174 
174 


— 


70 
70 
71 
71 


0.005.!' 
0.0C56' 


92.5 
92.5 


4'' 
4' 
4' 
4"^ 


240 
240 
240 
240 


— 


1 




588 
588 
588 
588 


529 
329 
529 
529 


— 


— 


69 
70 
71 
71 


0.0048' 
0.0054' 


92.5 
92.5 


6' 
6' 
6' 
6' 


360 
360 
360 
360 


— 






694 
694 
694 
694 


625 
625 
625 
625 


530 
536 
536 
536 


— 


69 
69 
71 
70 


0.00S4' 
0.0037' 


70.5 
70.5 


2' 
2' 
2' 
2' 


120 
120 
120 
120 








684 
684 
684 

684 


— 


— 




Xoforanco (or control) centalnon 


71 
73 


72 
73 


0.026 
0.02S 


40 
40 


3' 
3' 


179 

180 


96.1 


6 
10 


P." 


270 
572 


160 
595 


136 
567 





139 



PRECtDlSG-PAGE BLANK HOT FIL^^D. 



TabI* 2. Chtmical onolytis of remaining hydraiin* 
oftw hMrt-starilization 





CMMahMT 2 


CohMmt 4 
% 


HydraziiM (NiH4 
Water (HiO) 
A«iliM(CMNH4 


96.4 
\.9 
0.3 
1.2 


96.7 
1.4 
0.3 
1.4 


■nMiM 
Tetab 


0.06 
99.86 


0.09 
99.89 



stainless steel c<mtaiiKrs were filled with 305 ml of hydra- 
zine and heat-cycled at 300°F for GO h. Initial container 
pressure was approximately SOpsig. Passivation of the om- 
tainers prior to testing was again accomplished using the 
same procedures. During die first 60-h heat-sterilization 
cyde at 300'F, one of the containers was vented because 
the pressure buildup exceeded 1500 psig — a t«itative 
maximum safe test pressure based cm previous test data, 
as Aown ia SPS 37-44, Vol. IV, p. 180, Table 7. The 
second 00-h heat-sterilization cycle at 300°F was termi- 
nated after 50.7 h when another test container pressure 
exceeded the 1500-psig limit. 



The pressure rise per square-inch of .est sample surface 
area was also determined. The pressure rise data tor all 
tost containers after heat-steiilization cycling at 300° F in 
this series of tests were averaged and divided by the total 
patch-test surface area for a typical ccmtainer. This cal- 
culation produced a value of 0.4-psi pressure rise per hour 
per square inch of test sample surface area. The test 
sample prc^ierties were determined before and after heat- 
sterilization cycling only. No attempt was made to take 
into account any changes occurring during the heat- 
sterilization cycle. 

This ieiies of te.^ concludes the patch-type testing. 
Further details concerning the AlfS gcnerant tank 
development program are described in Ref. 1. 

4. Conclusion 

Based on die results of this series of tests, it must be 
concluded that the ethylene propylene material is very 
marginal for expulsion use with hydrazine at a tempera- 
ture level of 275°F. 

ilcfcrwice 

1. KcDer, O. F., and ToA, L. R., ALPS Generant Tank and CM 
MtetMy, Technical Rqport 32-865, Jet Propulsion Labotatoiy, 
Pasadena, Calif., Ff b. 28, 1966. 



I 



in SPACE PrOGMMS SUIAIAAR^ 37-51, VOL. Iff 



141 



PRBCEDING^PAGE BLANK NOT RLMED, 



'i 



N 68-37412 



XV. Lunar and Planetary Instruments 



SPACE SCIENCES DIVISION 



A. Atmospheric Entry Sampling System, s. Rich 

1. Introduction 

In order to analyze the composition of the Mars 
atmosphere with the JPL entry mass spectrometer (see 
Section B), uncontaminated samples of the atmosphere 
must be introduced into the ion source of the instrument 
under molecular flow conditions. To perform this type of 
analysis during the terminal descent phase of a Mars 
ent^,' mission, the capability to continuously sample the 
atmosphere over the Mach No. range < Af < 9 is required. 

'lTic method currently under consideration for obtaining 
un:;ontaminated atmospheric samples during terminal 
des.'ent consists of inserting a sample tube through the 
entry ':apsule nose cap to sample the atmosphere behind 
the bow shock wave. To prevent sample contamination 
by the entry capsule, the sample tube inlet port must be 
located forward of the capsule boundary layer. For the 
VM-8 Mars model atmosphere and a 6.5-ft-diam 60-deg 
capsule with a ballistic coefiBcient of 0.12, the sample 
tube inlet port would have to be located approximately 
0.5 in. in front of the nose cap. 

In order to provide the required molecular flow into 
the mass spectrometer ion source, part of the atmosphere 
flowing into the sample tube must be converted to 
molecular flow and subsequently piped to the ion source. 



To accomplish this conversion, a variable conductance 
molecular leak is being developed. The rate of flow 
through a molecular leak is a function of the sample gas 
molecular weight, differential pressure across the leak, 
and the sample gas absolute temperature. Feedback 
control will be utilized to vary the conductance of the 
molecular leak. This provides a measiu-e of adaptive flow 
control to compensate for atmospheric uncertainties 
which may affect sample inlet pressure and sample inlet 
temperature. By utilizing the mass spectrometer total ion 
current measurement as the feedback control signal, a 
uniform sample flow rate into the mass spectrometer can 
be maintained during the entire atmospheric sampling 
period. Maintaining an appropriate uniform flow rate 
permits mass spectrometer operation at maximum ion 
souice pressure, which provides maximum measurement 
sensitivity during the entire atmospheric sampling period. 

2. Sample Tube Configuration 

Two alternate sample tube conflgurations under con- 
sideration are shoivn in Figs. 1 and 2. Both configurations 
utilize explosive actuators to deploy the sample tube in 
front of the entry capsule nose cap. The nose cap plug 
shown i, 1 Fig. 1 has a higher ballistic coe£BcJent than 
the entry capsule; consequently, the plug falls free of the 
entry capsule after it is forced out of its hole by 
the sample tube. 



JP. SPACE PROGRAMS SUMMARY 37-51, VOL. HI 



143 



NOSE CAP PLUG 



PARALLEL 
CURVES 




BELLOWS 



MASS 

SPECTROMETER 

ION SOURCE 



wm/mm/w/ 

Fig. 1 . Molecular leak deploymenf configuration 




TYPICAL SAMPLE 
EXHAUST PORT 



RIGID 
TUBULATION 



MOLECULAR 
LEAK 



MASS 

SPECTROMETER 

ION SOURCE 



777777777777777- 

Fig. 2. Sample tube deployment configuration 



Advantages and disadvantages of the two sample tube 
configurations under consideration are as follows: 

(1) The configuration shown in Fig. 2 permits the use 
of a smaller diameter sample tube and will require 
a smaller diameter nose cap clearance hole and 
plug. Consequently, less force is required to eject 
the nose cap plug, and a smaller explosive actuator 
can be used. 

(2) In the Fig. 1 configuration, the deployed molecular 
leak aperture is located in front of the nose cap, 
and the atmospheric sample flow does not enter 
the capsule. In the Fig. 2 configuration, the molec- 
ular leak aperture is located inside of the entry 
capsule, and the atmospheric sample flow enters 
the capsule. Entry of the sample flow into the 
capsule may cause a thermal control problem, and 
an additional sample exhaust duct may be required. 

(3) The adaptive flow control problem is more compli- 
cated in the Fig. 1 configuration, since the feed- 
back flow control system must compensate for 
atmospheric heating of the molecular leak. 

(4) In the Fig. 1 configuration, a bellows is required to 
permit extension of the tubulation between the 
molecular leak and the ion source during sample 
tube deployment. Extension of the bellows may dis- 
lodge contaminants entrapped in the bellows wall. 

For sample system simplicity, the configuration in 
Fig. 2 appears preferable; however, further study is 
required to determine the effect on capsule thermal 
control or the effect on capsule configuration if a sample 
exhaust duct should be requL-ed. 

3. Variable Conductance Molecular Leak 

A schematic diagram of a variable conductance molec- 
ular leak currently under development is shown in Fig. 3. 
The conductance of the leak is varied by applying 
current to the heating elements on the inner and outer 
.shells. Heating the outer shell causes it to expand in 
length, opening the leak aperture to increase conduct- 
ance. Similarly, heating the inner shell closes the leak 
aperture to decrease conductance. 

The theoretical conductance of the molecular leak is 
given by the equation 



F = 60.96 



ft' (T/M)^ 
ln(d„M) 



(1) 



144 



Sn SPACE PROGRAMS SUMMARY 37-5?, VOL. Ill 



TYPICAL SAMPLE 
INLET PORT 



TYPICAL SAMPLE 
EXHAUST PORT 



OUTER SHELL 




METAL SEALING 
GASKET 



-SAPPHIRE 

ALIGNMENT AND 
SEALING 
DIAPHRAGM 



HEATING 
ELEMENTS 



INSULATED 
DIFFERENTIAL 
ADJUSTING NUT 



INNER SHELL - 

Fig. 3. Variable conductance molecular leak 

where 

F = conductance, 1/sec 

h = the effective cylindrical apertuie height bebveen 
the optically flat sapphire and the circular metal- 
sealing surface, cm 

do = the outside diameter of the circular metal-sealing 
surface, cm 

dj = the inside diameter of the circular metal-sealing 
surface, cm 

T = the absolute temperature of the flowing gas, "K 

M = the molecular weight of the gas 

Rate of flow through the molecular leak is given by the 
equation 

Q = F{P. - Pi) (2) 

where 

P, — the pressure outside the leak (essentially the stag- 
nation pressure behind the bow shock wave) 

Pi = the pressure on the ion source side of the leak 

Using Eqs. (1) and (2), the range of h required for a 
uniform flow rate of 10"^ torr-l/s was computed for the 
terminal descent phase of a Mars entry mission. A plot 
of the variation in ft as a function of time to impact, 
altitude, and Mach No. is shown in Fig. 4. 



E 
u 

O 









ALTITUDE, ft 








3 2000 5000 12.000 16.000 25,000 40,000 


0.07 
0.06 


\ 


1 




1 1 


1 


1 




\ 


_ 478X10-5 ._ 
Ft- i/2 pV^^ 




0.08 




\ 


1 T CTArMATinM TCUD Ol/ 




\ 


1 r, 5 




004 
0.03 






P, STAGNATION PRESSURE, torr 


0.02 
001 








\ 


k 

















\ 



10 IS 20 

TIME TO IMPACT, s 



25 



30 



2 3 

MACH NUMBER 



6 9 15 



Fig. 4. Theoretical aperture height during terminal 
descent for 6.5-ft-diam 60-deg sphere/cone 

The thermal energy required to produce an aperture 
height h by expansion of the outer shell (assuming no 
heat loss) is given by the equation 



H 



wc Ah 



(3) 



where 



H = the required thermal energy 

w = the specific weight of the outer shell material 

c = the specific heat of the outer shell material 

A = the cross-sectional area of the outer shell 

e = the coeflicient of expansion of the outer shell 
material 

For an outer shell constructed of 304 stainless steel, with 
a cross-sectional area of 0.15 in.', 0.02 Btu of thermal 
energy or an average power of approximately 0.84 W 
during the last 25 s prior to impact, is required to pro- 
duce the maximum h (0.07 X 10"' cm) shown in Fig. 4. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



145 



An estimate of the variable conductance molecular 
leak thermal actuation time constant for expansion of the 
outer shell (assuming no heat losses) is given by the 
equation 



B = 



cwst 
2k 



(4) 



where 

B = the thermal time constant 

Jk = the thermal conductivity of the bonding material 
between the heating elements and the outer shell 

c = the specific heat of the outer shell material 

«> = the specific weight of the outer shell material 

s = the radial thickness of the outer shell wall 

{ = the bonding material thickness between the heat- 
ing elements and the outer shell 

For the outer shell constructed of 304 stainless steel, with 
a wall thickness of 0.05 in. and a 0.003-in. thickness of 
Delta Bond 152 cementing the heating elements to the 
outer shell, the computed thermal time constant is 0.174 s. 

B. Prototype Moss Spectrometer for Planetary 
Atmospheric Analysis, H. R. Meriz 

1. Introduction 

One of the important tasks in planetary exploration is 
to determine the composition and density of the atmos- 
phere of the planet. One way to obtain such information 
is with a flight-type atmospheric mass spectrometer which 
covers the desired mass range with the proper sensitivity. 
A first step in developing such an instrument is to con- 
struct, test, and make a flight evaluation of an engineering 
model. A contract was let in July of 1967 to design and 
construct an engineering model based upon the results 
of the science breadboard mass spectrometer design. 
The instrument was to be incorporated into the Capsule 
Systems Advanced Development (CSAD) program in the 
early Spring of 1968. 

2. Inttrumont Oporation 

A mass spectrometer performs the compositional 
analysis of a gaseous sample by ionizing a portion of the 
gas being analyzed. The ions generated are separated 
according to their individual mass to charge (tn/e) ratios. 



Once separated, the resulting ion currents are detected 
and amplified by an electron-multiplier-electrometer de- 
tection system, the output appearing in the form of 
discrete voltage peaks of different values of m/e. Relative 
abundance measurements are made by an intercompari- 
son of the voltage levels of these peaks 

Mass spectrometers differ only in the method used to 
achieve m/e separation. The double-focusing magnetic 
sector instrument (Fig. 5) first accelerates ions through a 
radial electrostatic analyzer. The radius of curvature of 
the ion trajectories in this portion of the instrument is 
proportional to the energy of the ions, and the ions are 
focused accordingly. The ions are then directed through 
a magnetic field where the radius of curvature of the 
ion trajectories is proportional to the individual m/e 
value of each ion. With a constant magnetic field each 
variety of ion requires a different acceleration voltage 
(and, hence, electrostatic analyzer voltage) to traverse the 
two fixed curvatures of the instrument to be collected by 
the electron multiplier detector. By scanning the accel- 
eration and electrostatic analyzer voltages cyclically be- 
tween the proper limits, a mass spectrum is produced. 
Simultaneous correction of direction focusing and velocity 
focusing inhomcgeneities in this instrument are achieved 
through the proi)er choice of the electrostatic and mag- 
netic analyzer ion optical properties. Hence, high mass 
resolution and sensitivity are simultaneously obtained, a 
result not readily attainable in other types of mass 
spectrometers. 

3. Instrument Description 

The instrument described here is a double-focusing 
magnetic sector mass spectrometer. The critical speci- 
fications of this instrument are shown in Table 1. 

Table 1 . Specifications for double-focusing magnetic 
sector mass spectrometer 



Gladrottotic utctor angle 


90deg 


Electrostatic sector radius 


2.480 in. 


Magnetic sector angle 


60deg 


AAognetic sector radius 


2.003 in. 


Ion source exit slit width 


0.004 in. 


Collector entrance slit width 


0.008 in. 


Source divergence angle a 


Ideg 


Moss resolution VA/M*) ,9^ 


90 


Dynamic range 


1.5 X 10* 


Moss range 


M= 10 to 90 


Scon time (for one spectrum) 


2.8 s 



146 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



ELECTROSTATIC 
ANALYZER ■ 



OUTER 
SECTOR 



ALIGNMENT LENS 
(TEST ONLY) 



MAGNETIC 
ANALYZER 
SECTION 



COLLECTOR SLIT 
SUPPRESSOR 




■SAMPLE INLET LINE 
AND MOLECULAR LEAK 



Fig. 5. Analyzar, ion trajectoryi and optical system of mass spectrometer 



The analyzer vacuum envelope consists of five indi- 
vidual components which enclose the ion source, the 
electric sector, the magnetic sector, and the electron 
multiplier. It is a tfain-walled 304 stainless steel housing 
which is electrically accessible through several feed- 
throughs. Multiple pin feedthrough header's introduce 
voltages from the ion soiu-ce electronics and feed them 
to the filament and to the various focusing electrodes; a 
single pin feedthrough transmits the ion current from the 
electron multiplier to the range-switching electrometer. 

The internal vacuum necessary for operation of the 
analyzer is maintained by an ion pump which is made as 
an integral part of the magnetic sector. 

In addition to the electrometer amplifier and ranging 
circuit, the support electronics consist of the following 
modules: 

(1) The filament supply and emission regulator which 
maintains a constant ionizing electron current. 

(2) The scanning electrode bias supply which provides 
the ion source accelerating potential and propor- 
tional electrostatic analyzer potentials. 

(3) The low-voltage power supply for the various 
modules. 



(4) High-voltage supplies for the ion pump and elec- 
tron multiplier. 

A more detailed description of the instrument com- 
ponents and design considerations are covered in the 
following sections. 

4. Ion Source 

A cross-sectional view of the ion source is shown in 
Fig. 6. A closed ion source design was used to obtain a 
minimum gas flow out of the source, allowing the source 
to be operated at a pressure higher than the rest of the 
analyzer. Electrons are admitted into the ionization 
region through a small aperture, and the ions are with- 
drawn througli another small aperture. The structm'e of 
the electrodes is circular in shape so that they can be 
sealed and insulated from each other by ceramic rings. 
The close fit between the metallic lenses and the ceramic 
rings produces a very small gas conductance, which 
eflFectively seals the ion source. 

The operation of the source at a pressture higher than 
that of the rest of the system has the following advantages 
toward reducing sample distortion: 

(1) Outgassing from the hot filament and the surfaces 
of the system is pumped away, thereby minimizing 
entry of these species into the ionization chamber. 



Jn SPACE PROGKAMS SUMMARY 37-5J, VOL. Ill 



147 



MAGNET YOKE 



MAGNETIC POLE PIECE 



ANODE 



IONIZATION 
CHAMBER - 



REPELLER - 



SAMPLE 






INLET ^^^^^S 



ELECTRON 
ENTRANCE 
APERTURE- 



SPLIT ION 
FOCUS LENS 

FIELD 
TERMINATION 
PLATE 




-RUBY INSULATORS 

r—VACUUM ENVELOPE 



-BAFFLE 



FILAMENT 
FILAMENT SHIELD 

MAGNET YOKE 



OBJECT SLIT 
ION ACCELERATOR 

ELECTRON FOCUSING 
SPLIT LENS, MAGNETIC 
POLE PIECE, AND 
MAGNETIC SHIELD 

ION SOURCE 

Fig. 6. Ion source cross section 

(2) Variations in the ion pump speed have less influ- 
ence on the source pressure. 

(3) A source pressure that is too high for operation of 
either the filament or the electron multiplier is 
permitted, thereby improving th" efiFective ion 
source sensitivity. 

To obtain a high differential pressure between the ion 
source and analyzer, it was necessary to minimize the 
area of the electron aperture but still maintain a good 
electron transmission efficiency. Maximum transmission 
through a small aperture can be obtained by a well- 
focused beam. To obtain such a beam, a shield located 
on the sides of the filament plus an apertiu'e lens located 
between the filament and the electron entrance slit were 
used. The aperture lens was split into two electrodes so 
that misalignment within the electron gun could be cor- 
rected by a differential voltage across these two elec- 
trodes. In addition to the electron gun, a magnetic field 
is employed in the ion source to align the beam for 
maximum stability. This gun configuration should pro- 
vide a 50% transmission efficiency. 

5. Electric Sector 

The electric sector is used to compensate for the effects 
of velocity dispersion in the magnetic sector. It consists of 
the two cylindrical coaxial plates shown in Fig. 7. An 



SECTOR PLATES 



GROUND 
PLANE 




INPUT 
TERMINALS 



Fig. 7. Cross-sectional view of electric sector 

electric potential is applied to each plate, establishing 
a force on the ions that balances their mean centrifugal 
force. The lip on the edge of the plates is used to com- 
pensate for the curved equipotential surfaces which the 
edge produces. These edge corrections are designed for 
a sharp 90-deg comer. The figure also shows a ground 
plane on the sides of the plate. 

Ruby washers are used to insulate the plate from the 
mounting points. Screws are used to hold the plates to 
the washers. The washers are contained in counterbores 
so that they will stay in place even if they are shocked 
to the point of fracture. The ruby washers can be 
shimmed to produce the required parallelism between 
the two plates. Because of the curvature of the plates, the 
assembled clearances can only be measured at the ends. 
Variations in the plate spacing results in either a beam 
spread or a beam displacement at the collector slit. A 
tolerance analysis was performed by the contractor to 
determine the alignment tolerances for the electric sector 
plates. These calculations indicate that the design will 
meet the instrument requirements. 

6. Analyzer Tube and Ion Pump 

The analyzer and ion pump sections were machined 
as a single part. An entrance and an exit tube were 
welded to the analyzer section. The ion pump housing 
forms an integral unit with the analyzer section. This 
unit is illustrated in Fig. 8. 

The ion pump (Fig. 9) consists of two titanium plates 
with a titanium grid mounted between them. The 
plates are operated as a cathode, and the grid is operated 
as an anode. A basic design problem with an ion pump 
is the insulated mounting required for the anode. Since 



148 



JPL SPACE PROGRAMS SUMMARY 37-51, ^01. Ill 




-ION PUMP HOUSING 

Fig. 8. Ion pump housing and analyzer taction 




ANODE 



Fig. 9. internal construction ion pump 

the pvunping action kivolves considerable sputtering, the 
insulating material used in the mounting can become 
coated with a film of sputtered metal. This metallic film 
would, in time, shortcircuit the ion pump power supply. 
To overcome this problem, the insulating material is 
surrounded by a shield held at the anode potential. The 
insulators used to support the grid structure are ruby 



washers. One side of the washer is in contact with a short 
boss which extends from the cathode. The other side of 
the washer is in contact with the anode. A cylindrical 
shield extends from the anode to surround the washer. 

To understand the function of this shield, it is advan- 
tageous to review the nature of the gaseous discharge 
that occurs within the pump. Background radiation pro- 
duces a small amount of electrons in any region where a 
gas is present. These electrons are accelerated by an 
electric field so that they will collide with neutral gas 
particles to produce ions and additional free electrons. 
One accelerated electron can produce many ions and 
additional free elef^trons if it is allowed to travel a long 
distance before it is collected by an electrode. The 
long electron path lengths are provided in a small con- 
tainer by causing the electrons to oscillate. They are 
accelerated toward the anode, which is a grid, but are 
not likely to be collected because of the grid geometry. 
After they pass the anode, they are decelerated by the 
cathode potential. The result is that they oscillate about 
the anode until they strike the grid. The number of 
cycles of oscillation can be increased if a longitudinal 
magnetic field is present. This field coUimates the elec- 
trons as they oscillate about the anode. 

The function of the shields around the ruby washers 
is to invert the electric potentials so that oscillations do 
not occur in the region around the washer. When elec- 
trons are produced by radiation and ionization in the 
region around the washers, they are accelerated and 
collected immediately by the shield. This arrangement 
establishes a short electron path, which greatly reduces 
the ion production and the amount of sputtering around 
the washer. 

7. Electron Multiplier and Housing 

The ion optical path in the mass spectrc!T>eter is termi- 
nated by an electron multiplier. A collector slit is loctited 
at the focal plane which blocks the entry of ion beams 
of other than the correct mass. The ion beam passes 
through the collector slit and strikes the first dynode of 
an electron multiplier and amplifies the ion current by 
secondary electron emission. The multiplier housing sup- 
ports the electron multiplier and also provides a vacuum 
envelope. 

8. Magnet Assembly 

The magnetic assembly provides both the magnetic 
field necessary for mass separation and the field used by 
the ion pump. It consists of a yoke, a C-shaped structure 
of Armco Iron, permanent magnets of Alnico 5-7, and 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. //( 



149 



MOLECULAR LEAK 



MULTIPLIER 
HOUSING 




(b) 



'•d^S&l. 




^' 



Fig. 10. Engineering model most «pectromi>ler: (al vyithout top mounting plate, (b) wtlh top mounting plate 



150 



Jf>t SPACE PROGRAMS iUMMARY 37-51, VOL. Ill 



pole pieces of Armco Iron. The design was to give mag- 
netic intensities of 4000 and 2000 G at the analyzer and 
ion pump sections, respectively. 

9. Electronics Packaging 

The electronics consist of 21 welded wire modules 
incorporated into 6 module assemblies. The modules 
were potted solid with Stycast 1090/11. Metal inserts 
were cast in the modules ^o provide for assembly to the 
mass spectrometer structure. 

10. Structurai Design 

The structural support for the instrument consists of 
two semi-circular 321 stainless steel plates between 
which the analyzer and electronics modules are sand- 
wiched. In addition, a stainless steel stiffener is also used. 
The bottom plate also provides the means for mounting 
the mass spectrometer to the CSAD nose cone. Figures 
10a and b show the instrument with and without the top 
mounting plate. 

11. Auxiliary Equipment 

To facilitate the testing of the instrument a variable 
leak assembly was made an integral part of the instru- 
ment. The leak assembly is supported by the vertical 
stiffener. A valve was included so that the ir.ctrument 
could be connected to a commercial vacuum system to 
bake out the analyzer and could also be used for pre- 
liminary testing. The valve was subsequently removed 
from the instrument by pinching off at the interconnect- 
ing copper tubulation. 

12. Preliminary Results 

The analyzer assembly was completed during the 
third quarter of FY 1968. Preliminary tests were per- 
formed; electronic component selection was performed 



on the electronic modules; the modules were potted; 
final assembly was completed; and pinch-off performed. 

There was not sufficient time to obtain quantitative 
measurements of the mstrument performance; qualitative 
measurements showed, however, that the 'closed ion 
source provided an order of magnitude improvement in 
sensitivity over that exhibited by the science breadboard. 
The closed ion source design also allowed measurement 
of the oxygen peak. One area of ion source performance 
that was not up to expectation was electron beam effi- 
ciency. Rather than the predicted electron transmission 
of 50%, a value of about 10% was obtained. The elec- 
tron gun design called for the filament shield to be at a 
slight negative voltage with respect to the filament. The 
design of the emission regulator circuit prevented the 
application of such a potential, so the shield was con- 
nected to the filament. The functional performance of 
the instrument as observed during the qualitative testing 
showed that the resolution was equal to that of the 
breadboard unit. 

The testing of the unit revealed one major problem: 
The design of the shield, described in Subsection 6, 
proved inadequate, and a metallic film was deposited 
on the ruby washers that were used as the insulating 
material in the construction of the ion pump. This created 
a short across the ion pump supply, thereby shutting off 
the pump. In order to deliver a functioning unit to the 
CSAD program, a temporary adjustment wis made, and 
the pump was able to maintain the system pressure at the 
proper level. New ruggedized supports have been de- 
signed and will be installed as soon as the unit is returned. 

A sterilization cycle was performed on the instrument. 
No degradation in perfonnance was noted. Tlie instru- 
ment was delivered to the CSAB program for inclusion 
in the capsule system. The unit functioned properly 
during the subsequent subsystems and system tests and 
sterilization performed on the capsule system. 



JPl SPACE PROGRAMS SUMMARY 37-51, VOL III 



151 



N68 3 »^1^ 



XVI. Space Instruments 

SPACE SCIENCES DIVISION 



A. A Pulse-Height Analyzer for Space 
Application/ W. J. Schneider' 

1. introduction 

A number of scientific experiments performed from 
space vehicles make use of nuclear pulse spectrometry. 
The Jl'L program described here was designed to pro- 
vide a pulse-height analyzer of sufficient precision and 
versatility to be suitable for any of a num'oer of such 
spacebome experiments. The analyzer may be com- 
manded in flight to perform pulse-height analysis, time 
analysis, or multiparameter analysis. Instruction storage, 
data storage, and data readout, including a data com- 
pression option, are provided internally. (See Table 1 
for the nomenclature used in this article.) 

The analyzer incorporates the basic functional capa- 
biUties found in laboratory analyzers, with the exception 
of the linear amplifier and display sections (Table 2). 
The analog/digital converter (A/DC) has an input-pulse 
voltage range of 0.0 to 10.0 V with 19.5 mV resolution. 
Seven- and eight-bit resolutions are also available with 



'Work on the analyzer performed at Fabri-Tek Instrument Corp., 
Madison, Wis., for JPL (under Contract No. 951302) by R. 
Schumann, under the technical direction of the author. 
'Member of JPL Technical Section 314. 



corresponding memory subdivision. Coincidence and 
anti-coincidence modes are also provided, together with 
a live timer. Full-scale conversions are made in 128 /ts, 
regardless of resolution. 

The memory section of the analyzer stores 512 eighteen- 
bit words. The cycle time for "read-add/one-write" is 
slightly more than 5 fis. In pulse-height analysis, the first 
address contains live-time data, the last contains overflow 
or off-scale pulse count, and the remaining 125, 253, or 
509 addresses contain spectral-density data. 

The logic section of the analyzer accepts and stores 
externally generated instructions in its insti 'ction regis- 
ter. Available instructions include: the analyzer modes; 
pulse-height analysis, time analysis, combined pulse- 
height and time analysis; two multiparameter modes; 
and a multiscaler mode. (A full description of the avail- 
able instructions is given in Table 2.) Instructions stored 
in the register reorganize the analyzer's functional ele- 
ments and control logic to fill the requirements of the 
commanded mode (Fig. 1). For example, in the pulse- 
height analysis mode, the receipt of a pulse for analysis 
causes an initiate-storage signal from the A/DC that, 
in turn, initiates a pulse sequence in the programmer. 
These pulses are routed by the programmer, under con- 
trol of the instruction register, throughout the analyzer 



152 



JPL SPACE PROGRAMS SUMMARY 37-51. VOL. Ill 



as follows: 

(1) Clear address register. 

(2) Transfer contents of the pulse-height scaler to 
address register. 

(3) Clear the data register 

(4) Read the memory. 

(5) Advance data register. 

(6) Write the memory. 

(7) Clear the pulse-height counter. 

(8) Enable the A/DC. 

Table 1. Nomenclature 



A/DC 


analog/digital converter 


A/DCA 


analog/digital converter advance 


ADS 


advanced data scaler 


ANTI 


anti-coincidence mode signal 


APHS 


advanced pulse-height scaler 


CLOKF 


1-MHz clock 


COIN 


coincidence mode signal 


COINP 


coincidence mode command 


DISCH 


discharge flip-flip 


FETCH 


externally produced pulse calling for new 




data during readout 


FF 


flip-flop 


HISEN 


high sensitivity threshold command 


INDRN 


initiate rundown signal 


INITS 


initiate-storage-cycle command 


LGO 


linear gate open 


LIVEF 


live-time flip-flop 


MEASF 


measurement mode flip-flop 


MP 


mulHparameter 


MSC 


multiscaler clock 


PCHO 


pulse-height scaler zero 


PHFF 


pulse-height analysis mode flip-flop 


PHSIG 


pulse signal to be analyzed 


PO 


memory-busy flag 


RDS 


reset data scaler 


REJF 


reject flip-flop 


RNDWN 


rundown signal 


RTFF 


reset command for the T flip-flop 


STA 


start analysis 


STOP 


stop analysis 


STR 


start readout 


T 


synchronizing flip-flop set when an input 




signal is detected and cleared after the 




memory cycle 


TMRS 


telemetry bit sync 



Durmg readout, the instruction register is used to 
assemble the output data and shift it to telemetry. The 
first 18 bits shifted out contain all of the program instruc- 
tions. Simplified instructions for the readout process 
are generated from the shift counter. These i istructions 
cause reading of the addresses sequentially a;id the 
transfer of addresses and data into the instruction rr „ Jter. 

Table 2. Pulse-height analyzer specifications 



Item 


Capability 


Analog/digital cenvarter accuracy 


Input rang* 


Oto lOV 


Quantization 


9 bits, 51 1 levels of 19.5 mV each 


Zero itability 


±0.04% from -5 to -|-45*C 


Gain itability 


±1% from -5 10 -|-45'C 


Linearity 


±0.2% from best fit over upper 96% 




of scale 


Slope 


±4% from average over upper 96% 




of scale 


Count-rate effect 


±0.05% from 1 to 10' pulses/s 


Analysis time 


128 lit 


Noise (uncertainly) 


1 mV 


Memory capacity 


V 'ords 


812 


Bits/word 


18 


Access time 


1.2 /US 


Read-modify-write time 


25 ms 


Functional capability 


Pulse-height analysis 


1 28, 256, or 51 2 with automatic gain 


Channels 


change 


Coincidence 


Non-coinciaence, onticoincidence. 




coincidence 


Threshold 


High, low 


Time analysis 




Channel width 


1,2, •••,64, 128 fis 


Multiscaler analysis 


External clock is required 


Multiparameter 


MP9, 9 external bits, plus internal 




A/DC bits 




MP18, 18 external bits 


Memory subdivision 


Quadrant and half routing 




Overflow count in last channel of 




sector 


Readout 




Normal: Full address 


27 bits/address 


and data 




Condensed! pulse- 


14 bits/address 


height analysis 


3 bits of address 


only 


8 most-significant data bits 




3 bits of data multiplied 


Power 




Standby 


7.15 W 


20,000 evenls/s 


12.2 W 


Complexity (upproximottly) 


500 IC flatpocks 




400 discrete semiconductors 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



153 



m I 




A/DC 



I COINP > 



CLOCK 



DIGITAL 
INTERFACE 



-] 

INITS 



LA/r^rA_J CLOCK 

A/DC '^"^^" nOMDER 



-ADS — I 



PROGRAM 
PULSER 



MSC 



STA 
STOP 

SIR ; 

FETCH. 



V. MULTISCALER 
/" CIOCK 



CLOCK 

START ANALYSIS 
STOP ANALYSIS 

START READOUT 
READOUT SYNC 



TIME ANALYSIS 
DIVIDER 





READOUT 
LOGIC 







■APHS- 



PULSE-HEIGHT SCALER 



1 



TRANSFER | 



READ 

READ PULSE -► 

WRITE 

WRITE PULSE-W 

RDS 
ADS 



ADDRESS REGISTER 



512 words 
18 bits/word 



DATA REGISTER 



INSTRUCTION DECODER/LOGIC 



SHIFT 



PULSE-HEIGHT ANALYSIS ^ 

CONVERSION GAIN (2 bits) 

THRESHOLD SENSITIVITY 

COINCIDENCE 

ANTI-COINCIDENCE 
TIME ANALYSIS 

TIME BASE (3 bits) 
MULTIPARAMETER ANALYSIS 

9 bits EXTERNAL 

18 bits EXTERNAL 
MULflSCALER ANALYSIS 
READOUT MODE 

SYNCHRONOUS 

ASYNCHRONOUS 

DESTRUCTIVE 

COMPRESSED 

I MP DATA BITS 1-9 > 




ac 



TRANSFER 



INSTRUCTION REGISTER 



TRANSFER 



,1 u 



MP DATA BITS I0-I8> 



■ |tmbs> 



Fig. 1. Functional diagram of pulso-hoigh^ analyzer 



154 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. (H 



I I I 



2. Pulse-Height Analysis Mode 

a. Analog/digital converter. The A/DC, shown sche- 
matically in Fig. 2, is a conventional Wilkenson type. 
The input circuit is a capacitively coupled emitter 
follower (Ql). Threshold control is provided (through 
Q2) by holding the emitter of Ql above its base line 
value by the amount on the desired threshold. Thus, 
the peak value of the puise is not altered by the thres- 
hold setting. Emitter follower Ql drives the linear-gate 
emitter follower (Q4). The linear-gate shunt switches 
(Q5 and Q6) are required only to sink the current sup- 
plied from the load resistor (R8) of the linear-gate emitter 
follower. 

The linear gate is normal., pen ..nd is closed after 
a pulse peak has passed. The reference voltage for end 
of rundown is the output of the closed gate and, thus, 



is near ground potential and is independent of the thres- 
hold settings. Since neither the pulse peak nor the 
rundown reference is altered by the threshold setting, 
the position on pulses above the threshold are unaltered, 
while those pulses below the threshold are eliminated. 
The threshold may be altered during data accumulation 
without smearing the spectrum. 

The stretcher amphfler (Q7 through QIO) acts to keep 
the voltage on the stretcher capacitor equal to the linear- 
gate output. Due to the unilateral nature of the charging 
diode (D5), this action is only possible when the linear- 
gate output is greater than the capacitor voltage. Accord- 
ingly, the amplifier is effective in charging the stretcher 
capacitor, while discharge is accomplished through the 
current-sink transistors (Qll and Q12). The constant- 
current-sink transistor (Qll) is necessary to provide 



+ 50 V 9 9 + 12 V 



PHSIG I o— j^ 




MI 




+ 50 V 



R8 



Q5 



04 



06 



rr 
T 



;: 






6 t> ■±- 

-50 V -6V 



LGO 



9+50 V 9+I2V 

m I 



|'+I2 V 



f I 



Q9 



+ 12 V 



D5 



-- -50 V 



07 08 



-ft-M- 



013 



RNDWN 



—^ II 1 

i +12 V 



-50 V 




6-50 V 



Fig. 2. Schematic diagram of analog/digital converter 



JPL SPACE PROGRAMS SUMMARY 37-51, VOt. /// 



155 



negative corrections to the stretcher-capacitor voltage 
during base line keeping. The linear discharge to the 
reference is provided by the switcl'ed sink transistor 
(Q12). 

The current-sink transistor is svifitched by diverting its 
emitter current through D9. The output of the stretcher 
amplifier (Q9) is a convenient source of a rundown signal 
(RNDWN), since it goes negative as soon as the pulse 
peak has passed and remains so until the stretcher - 
capacitor voltage is again equal to the linear-gate output 
when the amplifier regains control. 

The problems associated with pulse spectrometry are, 
in part, those of measurement precision and, in part, 
those associated with pulse-to-pulse interference brought 
on by the random nature and dynamic range of the 
nuclear phenomenon. In this pulse-height analyzer, 
the latter problems are handled by the digital interface 
between the A/DC and the balance of the analyzer 
functions. No means are incorporated to discriminate 
against nearly coincident pulses where their sum results 
in a monotonic increasing pulse. The decision not to 
incorporate pulse-shape discrimination was made pri- 
marily on the basis of the low event rates expected from 
spacebome experiments. 



b. Rundown control. The logic diagram of the analog/ 
digital converter and its interface are shown in Fig. 3. 
The first indication that a pulse has been received is the 
occurrence of the rundown signal (RNDWN). The state 
of the discharge flip-flop (DISCH) follows RNDW N on 
the succeeding negative clock transitions. DISCH gates 
on the constant-current discharge of the stretcher capac- 
itor and, together with the memory-busy flag (PO) and 
the pulse-height analyzer mode control signal, controls 
the analog/digital converter advance (A/DCA) pulses, 
which eventually advance the pulse-height scaler. DISCH 
also sets the T flip-flop that, together with RNDWN, 
causes the initiate-storage-cycle command (INITS). 

c. Program pulser. The storage cycle for all modes of 
the analyzer is controlled by the progra-n pulser. The 
pulser consists of a four-stage serial carry counter opera- 
ting at a clock-derived 1-MHz frequency. When the 
counter is initiated, it generates 15 sequential l-/is inter- 
vals (IPl through 1P15) and locks up in the 16th state 
(PO is true). Since the memory cycle for pulse-height 
analysis is completed during the 15 /is of PO, it is con- 
venient to use PO as a memory busy flag. The program 
pulser is initiated by INITS and sustained by PO through 
the completion of its cycle. INITS = T • RNDWN is 
an indication that a pulse has occurred, that a corres- 
ponding count has been accumulated in the pulse-height 



ANALOG/DIGITAL CONVERTER |CLOKF> 



PHSIG) 



|HISEN> 



.iNEAR 
GATE 



THRESH- ' 
OLD 




LGO 



AMPLIFIER 



i_,H_ _VWV^|_j_ 



[rtffV 



T FF 



-[> {EME> 



i^Si^ 



- ICgITiP> 




4iNrfs> 



{ADS> 



FF = FLIP-FLOP 



Fig. 3. Digital interfac* 



156 



jn SPACE PXOGJtAMS SUMMARY 37-51, VOL. Ill 



scaler, and that the storage cycle should proceed. All 
storage-cycle pulses are decoded from the program 
pulser. 

d. Linear-gate control. The linear gate is controlled 
in two ways; first, by the coincidence pulse in conjunc- 
tion with the "coincidence mode" instruction, and second, 
by RNDWN or T, once a pulse has been detected. Thus, 
the gate will close during every coincidence pulse in 
the anti-coincidence mode. Even if there is no pulse- 
height signal received, an analysis will occur, since the 
threshold circuit output is greater than the output volt- 
age of the closed gate. This is an undesirable mode of 
operation, since it has the effeft of decreasing live time 
and of storing unwanted data in the address corres- 
ponding to the threshold voltage. 

e. Reject circuits. At the end of the normal storage 
cycle, RTFF (at 1P14 time) causes reset of the T flip-flop, 
and the linear gate opens. Should this occur during the 
tail of a pre-existing pulse, an erroneous analysis would 
result. This condition is avoided by rejecting any anal- 
ysis data that occurs within 5 jiis of the linear-gate open- 
ing. This is accomplished by REJF, which prevents the 
advance of the data scaler during the storage cycle. Thus, 
rejection does not perturb the linear portions of the 
analyzer. 

/. Metrory subdivision and analog/ digital converter 
conversion gain. Change in resolution or in conversion 
gain refers to the number of quantization levels used 
to measure pulse height. Conceptually, the discharge 
rate of the stretcher capacitor could be changed by 
altering the magnitude of the discharge generator cur- 
rent. It is preferable to leave the analog circuits of the 
A/DC unchanged and merely to alter the clock fre- 
quency. A change of the clock frequency prior to the 
A/DC gating circuits would result in an increase in 
the uncertainty of the stretcher capacitor discharge prior 
to INDRN. In the analyzer, a counter that can divide 
the A/DCA pulses by 1, 2, or 4 is provided between the 
A/DC and the pulse-height scaler. 

g. Pulse-height scaler. The pulse-height scaler is a 
conventional ripple-carry counter, with provision for 
reset and for parallel output. It also has provisions for 
indicating pulse-height scaler full at a count of 511 and 
for indicating pulse-height scaler zero (PCHO). The 
former is used to prevent overflew and has the effect of 
indicating all overflow or off-scale pulses in address 
511. The zero indicator has a special use during live-time 
determinations, as will be clear later. References made 



here to address 511 imply that the measurements were 
made at maximum resolution. The logic generating the 
full and zero signals is altered — as is the A/DCA 
divider — by the memory subdivision signals indicating 
quarters or halves, as is required. In such a case, the 
most significant stages of the pulse-height scaler are 
conditioned by externally generated quadrant-selection 
pulses. 

h. Memory. The analyzer uses a conventional mag- 
netic core memory containing storage for 512 words, 
each 18 bits in length. The cores are arranged in bit 
planes of 16 X 32 cores. Each particular core in the 
Nth plane then represents the Nth bit in one of the 
512 words. When the memory is to be read, the address 
register is cleared, and the pulse-height scaler states are 
transferred in. This occurs in 4P4 time, 4 fis after the 
end of INDRN. The read signal, generated at 2P7 time, 
energizes the memory address decoding gates. These gates 
consisl of both current sources and sinks which, in com- 
bination with routing diodes, route the read pulse half- 
select currents to one of the 16 Y wires and to one of 
the 32 X wires, simultaneously. The core at the inter- 
section of the energized X and Y wires receives the full 
select current. Such an intersection exists once, and only 
once, on each of the 18-bit planes. The read pulse is 
generated at 1P9 time, starting 1 /is after the decoding 
gates are energized by the read signal. Both read and 
read pulse coexist for 1 /is, and during that time, the 
combined action of the X and Y half-select cxutents 
drive the selected cores to the reset state. The 511 cores 
in each bit plane that receive only one half-select current 
remain in their original state. 

Each core of each bit plane is threaded by a single 
sense wire. If the selected core in the Nth bit plane is 
originally in a 1 state, a voltage will be generated in 
the sense wire as the core is reset. This sense voltage 
is ampUfied and is used to set the Nth flip-flop in the 
data register. In practice, the sense line contains a con- 
siderable amount of noise voltage induced by the leading 
and trailing edges of the read pulse. Time domain 
filtering is used to enhance the sense voltage signal-to- 
noise ratio. 

The data register flip-flops are connected both for 
parallel entry from the sense amplifiers and for counter 
operation. During the pulse-height analysis read-add/ 
one-write cycle, the data register is advanced at 4P9 
time. Immediately, the write decoding gates are energized 
by the write signal at 2P13 time. The write pulse occurs 
at 1P14 time and causes half-select currents in the 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



157 



opposite direction to the read half-select currents to 
be generated. 

Each core of each bit plane is threaded by a single 
fourth wire. This wire is energized with a half-select 
current in such a direction as to oppose the write half- 
select currents. The selected core in a bit plane where 
the fourth, or inhibit, wire is energized remains in the 
reset state. The selected core in an uninhibited bit plane 
is driven to the set state. The inhibit winding is energized 
with signals derived from the data scaler, itself, thus 
permitting rewriting of the modified register contents. 

The time required for a read-add/one-write cycle is 
5.6 fis. As a power conservation measure, both decoding 
gates and sense amplifiers are energized only diuing read 
and write pulses. 

t. Live timer. The live timer provides a measure of 
the time during which the analyzer is available for the 
measuring of pulse heights. This measurement is accom- 
pHshed by sampling the combined REJF and T functions 
with a 100 pulse/s clock. A coincidence of these signals 
sets the live-time flip-flop, initiates a storage cycle, and 
closes the linear gate. Since live-time data are to be 
accumulated in address 1, the address register is 
advanced from to 1 at the IPl time of each storage 
c/cle. The cycle proceeds to 4P9 time when the data 
scaler is to be advanced, indicating that the analyzer was 
interrogated and was found to be live. The advance is 
made conditionally on the state of the pulse-height scaler. 
A pulse-height scaler state other than 0, with the live- 
time flip-flop set, indicates a coincidence between a 
pulse-height signal and a Hve-time clock pulse. In such 
a case, the pulse-height signal must have been at least 
partially stored in the stretcher before the gate closed. 
When this condition is observed, both the live time and 
the signal pulse are lost. 

;. Readout. The instruction register also serves as the 
output register for the analyzer during data readout. 
The readout function, itself, is controlled by the shift 
counter, the address register, and the control logic. 

The first data shifted out of the analyzer is that stored 
in the instruction register, and tells the user exactly what 
the conditions of the analysis were. After the first 9 
bits have been shifted out, the contents of the memory 
address register are transferred into the cleared positions. 
As shifting continues, the address register is advanced 
by 1, and the new address is read from the memory into 
the data register. After 18 additional shifts have occurred. 



the contents of the data register are transferred to the 
shift register. As shifting continues, the address register 
is advanced, transferred to the shift register, and the 
memory is read again. The data sequence is thus 18 bits 
of instruction data, followed by address 0, next data from 
address 1, followed by address 1, and so on, until all 
addresses have been read. Note that address is never 
read out. Address should have no data, since the 
address register has been advanced by 1 as a routine 
part of every storage operation. 

3. Time-Analysis Mode 

The analyzer has been designed to allow measurement 
of energy and die-away spectra of capture gamma rays, 
as might be obtained using a pulsed neutron source. 
When so instructed, the analyzer will, on command, 
begin a time measurement. The measurement consists 
c • l26 intervals of 1 to 128 /xs in duration, as instructed. 
V hen a p-i1se-height signal is received, a storage cycle 
is initiated, and the content of the address corresponding 
to the appropri)' if.lerval is advanced by one. Interval 
timing continues without interruption, but additional 
pulse-height signals do not initiate storage. When timing 
is complete, a second storage cycle is initiated, and the 
content of address 127 is advanced by 1 to indicate 
the total number of timing cycles that have been com- 
pleted. 

4. Combined Time and Pulse-Height Analysis 

Pulse-height analysis on the first pulse-height signal 
proceeds concurrently with tlie time analysis described 
above. The result of the analysis is retained in the pulse- 
height scaler. When the time analysis and both storage 
cycles are complete, the pulse-height data are stored 
in the selected quadrant. 

5. Multiparameter Modes 

In the multiparameter mode the analyzer merely 
provides a means of recording 512 words of 18 bits each. 
On command, such data are transferred into the data 
register of the memory, written into the core, and the 
address register is advanced by 1. When the last address 
has been used, a memory full gate is set, and no further 
inputs are accepted. 

Provision has been made to record 9 bits of multi- 
parameter data, together wdth 9 bits of pulse-height 
analysis data in each address. In this case, a coincidence 
pulse is required to indicate the pulse to be analyzed 
and the dala to be entered. 



158 



JPL fiPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



6. MultiscaUr Mode 

A conventional multiscaler mode has been provided. 
In this niode, interval pulses are provided externally. 
Pulses to be scaled are used to advance the data register 
of the memory. When the interval pulse arrives, a 
storage cycle is initiated, and the accumulated count in 
the data register is stored in the appropriate address. 
During the storage cycle of 16 /*s, the scaler is inactive. 
When the last address has been used, a memory-full 
gate is set and no further inputs are accepted. 

7. Conclusions 

The versatility provided by this analyzer is greater 
than that provided in many laboratory instruments and, 
hence, it should be suitable for most space science 
experiments. Admittedly, this instrument may have an 
excess functional capability and be somewhat less efiS- 
cient (in terms of power and weight) for specific appli- 
cations. However, the excessive costs for developing a 
special instrument for a specific application favor the 
use of this analyzer design. 



B. Quantitative Use of Imaging Systems: 
An Electronic Camera System, 

A. T. Young and F. P. Landauer 

1. Introduction 

The purpose of this work is to develop an imaging 
astronomical photometer with both high photometric 
accuracy and high spatial resolution. Accurate, high- 
resolution photometric data are needed in a wide range 
of planetary and stellar investigations. For example, the 
problems of the clouds of Venus, the nature of seasonal 
changes on Mars, the dynamics and structure of Saturn's 
rings, and fundamental studies of stellar masses and 
evolution, all require such observations. 

At the present time, low-resolution data of high photo- 
metric accuracy are obtained photoelectrically, moderate- 
resolution data of moderate accuracy are obtained photo- 
graphically, and high-resolution data of low accuracy are 
obtained visually. In these conventional techniques, the 
"seeing" (image blmring produced by the Earth's atmos- 
phere) is a major limitation, and has been regarded as 
an insuperable limitation. However, recent advances in 
understanding the "seeing" problem (Ref. 1) have shown 
that (1) the resolution advantage of visual or short- 
exposure photographic observations can be realized in 
longer exposures if the image motion is cancelled by, an 
automatic guider, and (2) the remaining image blurring 



has the eflFect of attenuating high spatial frequencies in 
the image, which can then be "restored" by suitable 
image processing. Such image-restoration techniques have 
been developed by the image processing laboratory at 
JPL, and successfully applied to Mariner IV and Surveyor 
data. 

The "restoration" of high spatial frequencies requires 
that the image be recorded by a linear process, and that 
the signal-to-noise ratio be so high that even the atten- 
uated spatial frequencies are larger than the correspond- 
ing components of the noise. Conventional photographic 
recording is strongly nonlinear, and gives signal-to-noise 
ratios of about 30 to 50 at best. Furthermore, the detective 
quantum efficiency of photography is low, typically a 
few tenths of a per cent, so that telescope time is not 
used effectively. 

These diflBculties can be reduced by detecting and 
recording the image electronically. At the present time, 
the image isocon appears to be the most suitable detector, 
with good linearity, wide dynamic range, and excellent 
signal-to-noise ratio. With slow-scan readout and FM 
recording on magnetic tape, a detective quantum ti- 
ciency of a few per cent and a signal-to-noise ratio of 
at least 100 can be expected. An electronic camera has a 
considerable advantage over photography, since each 
picture element can be individually calibrated by ex- 
posure to a series of knovra light levels. The tape 
recording has the additional advantage of much faster 
conversion of data to digital form than can be achieved 
by scanning a photograph mechanically on a micro- 
densitometer. 

Figure 4 is a block diagram of the entire electronic 
camera system. From a systems point of view, the earth's 
atmosphere must be included with the telescope in deter- 
mining the optical modulation transfer function. At the 
present time, we are concerned with the design of 
the portion of the system above the dashed line. 

2. Electronic Camera System Design 

a. Telescope and Earth's atmosphere. The telescope 
used will be primarily the 24-in.-diam telescope at Table 
Mountain, California. However, it may be desirable to 
use other telescopes (e.g., at McDonald or Kitt Peak), 
and provision should be made for mounting the equip- 
ment on other telescopes. 

The work of Fried (Ref. 1) has shown that there is an 
optimum aperture for a given wavelength and "seeing" 



JPL SMCE PROGRAMS %UMMAkY 37-51, VOL. Ill 



159 



1 



TELESCOPE 



TELESCOPE 



GUIDER 
MECHANISM 



OPTICAL 
CALIBRATION 

SIGNALS 
(PHOTOMETRIC 
AND GEOMETRIC) 



JPL 



ENLARGING 
OPTICS 



FILTERS FOR 

WAVELENGTH, 

APERTURE, AND 

POLARIZATION 



IMAGE 



SERVO 



PHOTOMETRIC 
IMAGE SENSOR 
(CAMERA UNIT) 



POSITION 
SENSOR 



DIGITAL 
TAPE 



ANALOG-TO 

DIGITAL 

TAPE 

CONVERTER 



ANALOG 
TAPE 



ANALOG 

TAPE 
RECORDER 



VISUAL AND 

PHOTOGRAPHIC 

MONITORS 



IBM 360/44 
COMPUTER 
AND DATA 

PROCESSING 



SCIENTIFIC 

PHOTOMETRIC 

AND POSITIONAL 

DATA 



HIGH QUALITY 
PHOTOGRAPHS 



ASTRONOMERS 



INFORMATION 

ABOUT PHYSICAL 

CONDITIONS AND 

PROCESSES 

ON PLANETS 

AND ELSEWHERE 



Fig. 4. Planetary photometry (electronic camera system) 



quality. A review of current knowledge of the seeing 
problem will appear shortly in Sky and Telescope. Be- 
cause the modulation transfer function of the telescope- 
atmosphere combination depends on the telescope aper- 
ture, Fried's work is being extended to annular apertures. 
Aperture filtering will be included in the optical head. 

Because of the importance of atmospheric dispersion 
in high-resolution observations, relatively narrowband 
filters must be used. Partial compensation for dispersion 
is possible, but at the expense of more optical elements 
and two additional continuously varying degrees of 
freedom in an already complex system. Narrowband 
filters are desirable in planetary work, regardless of the 
dispersion problem. 

b. Guider. Several alternative methods of sensing and 
controlling the position of the image are being investi- 
gated. Calculations indicate that adequate bandwidth 
will be available to guide on any naked-eye object in the 
blue and visible; for fainter objects, or in the infrared, 
the accuracy of motion compensation will be limited by 
photon noise. 



c. Camera head and control system. The detailed 
electronic design of the camera system is essentially 
complete. A description of the electronic system follows. 

Camera head. 

Image detection. Recent data on the RCA type-C21093 
image isocon indicate substantial superiority over image 
orthicons. With a close-spaced mesh and 10" fi-cm 
target, linearity and readout efficiency are high, and 
storage (integration) will be possible for several minutes. 
The bialkali photocathode is reported to be stable, 
uniform, and highly sensitive. With an expected detec- 
tive quantum efiiciency of a few per cent, the required 
exposure times to achieve 1% precision per picture 
element will be on the order of 0.2 s for Venus, 4 or 5 s 
for Mars and Jupiter, and 1 min for Saturn. 

PreampHfication. The output signal from the isocon 
will range from a few nanoamperes to several micro- 
amperes. This requires a gain adjustment of from 20 to 
20,000 in the video preamplifier. It is anticipated that the 
existing nuvistor preamplifier will be augmented by 
programmed operational amplifiers to control gain and 
bandwidth. 



160 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



Deflection. An operational amplifier-type deflection 
amplifier will be used but will not be an integral part of 
tho camera head. Deflection waveforms and regulated 
focus and alignment currents will be remotely generated 
in the control console. 

Control section. 

Sweep and format generation. Horizontal line genera- 
tion will be by means of binary division of a crystal- 
controlled clock. The sweep drive signals will be coupled 
to operational amplifier integrators operating through an 
integrate-reset bridge. The vertical drive will be derived 
by binary division of horizontal drive pulses. Thus, both 
the horizontal and vertical line numbers can be controlled 
by digital selection. 

Because image orthicons and isocons are complex in 
operation, two sweep modes will be used: (1) a slow-scan 
mode limited by tape recorder bandwidth versus signal- 
to-noise ratio constraints, and (2) a fast-scan mode 
limited by the reset capabilities of the yoke drivers and 
yokes. The fast-scan mode is for visually monitoring the 
preliminary adjustment of the camera operating param- 
eters; the slow-scan mode is for data recording. It will 
have constant sweep rate and video bandwidth, i.e., the 
raster size will change as the line number changes. This 
is consistent with the requirement for constant optical 
magnification and Nyquist sampling at the optical reso- 
lution limit. The reason for changing the line number at 
all is that if only a small field is required, then consider- 
able time is wasted if an unnecessarily large target area is 
scanned. The formats to be accommodated are 256 X 256, 
512 X 512, and 1024 X 1024 Nyquist samples. The fast 
system differs in that the act've line time is constant, the 
video bandwidth varies, the line number changes, and 
the clock frequency and integration rate is eight times 
faster. 

Exposure and Erasure. Exposure time" will be gen- 
erated in increments of powers of the square root of two 
by binary division of crystal clock. Automatic exposures 
from 1 ms to 45 s (2" ° ms) will be selectable. 

Erasure will be available only during slow-scan opera- 
tion. The sequence will consist of switching the system 
to fast scan, gating the target to a higher potential and 
increasing the beam current for one frame, then returning 
to slow-scan mode. Erasure scans will always be done in 
the 1024-line format. In slow-scan operation, the sweep 
will be disabled doring exposure to eliminate the effects 
of a changing magnetic field on the image section of the 
camera tube. In fast scan, however, no hesitation will 



occur unless the selected exposure time exceeds the 
frame time; thus providing the fastest possible frame rate 
for setup. 

Control of optical functions. Besides controlling the 
camera unit, the control unit must step the optical filters 
and calibration targets through an appropriate calibration 
sequence. The calibration procedure is complex; it pro- 
vides adequate photometric and geometric calibration 
data for each combination of wavelength, polarization, 
and aperture filters selected. The control unit also 
sequences the filters and other optical and electronic 
adjustments during the observational cycle. 

Visual monitor. The electrical characteristics of the 
display monitor will be similar to those of the film 
recorder. The format will be a minimum of 6 in.^ on a 
spherical faceplate. The resolution will be a minimum of 
1000 lines using a dual-mode ph ;phor, i.e., different 
colors for phosphorescence and fluorescence so that when 
proper filters are used, the monitor will be suitable for 
both fast- and slow-scan rates. 

A Tektronix RM 561 oscilloscope will be used for A- 
scope monitor. A type 3B3 time-base unit will be modified 
to accept external sweep from the control section, and to 
enable the delayed sweep gate to function as a line 
brightenei on the monitor when the A-scope is used as a 
line selector. 

Film recorder. A Rongcr Block III film recorder will 
be used to provide 35-mm film output. Reliability and 
stability of the unit is being improved by the replacement 
of the high-voltage and focus power supplies with ultra- 
stable solid-state un'ts. As in the camera head, the yoke 
driver will be a wideband, ultralinear, solid-sta'e opera- 
tional amplifier. It will receive its sweep waveforms 
from the control section. Resolution will be better than 
1000 lines on 2.5 in. of the cathode-ray tube faceplate. 

Tape recorder. Data will be recorded on magnetic 
tape using an Ampex FR1400 or equivalent operating at 
120 in./s. The tape will be similar to Ranger and Surveyor 
analog tapes, i.e., frequency modulation by video data. 
Separate tracks will be used for vertical and horizontal 
sync signals because there is no reason to generate com- 
posite video. Sufiicient telemetry data will be recorded 
to provide a record of the system operating parameters. 
A file number will also be recorded to aid in data 
extr ction. 

R«f«renc« 

1. Fried, D. L., /. Opt- Soc. Am., Vol. 56, p. 1372, 1066. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



161 



HMMlill 



iwiui 



«1 



m 



C. On the Slow-Scan Characteristics of the 
WX30691 SEC Vidicon, K. J. Ando 

1. Introduction 

A continuing task of the JPL image detector laboratory 
is the evaluation of new imaging devices for possible 
application in future interplanetary missions. One type 
is the secondary electron conduction (SEC) vidicon. The 
SEC vidicon is particularly suited for space applications 
due to its inherent simplicity and high sensitivity. SEC 
tubes have been selected for various future space systems, 
the most important ones being the Apollo mission and the 
Apollo Telescope Mount program. 

For any space application, the imaging device must be 
sufficiently rugged to withstand the severe environment 
of launch and a long-term flight. Further study and 
developmental work will be necessary to determine 
whether ruggedization of a SEC vidicon is feasible. 
Present results indicate that it may be. The SEC vidicon 
has already met many MIL specification shock and vibra- 
tion requirements which specify typical levels encountered 
on airborne flights. 

The present article discusses some results of the eval- 
uation of the Westinghouse WX30e91 SEC vidicon. The 
main purpose of this work was to determins ^he slow-scan 
capabilities of the WX3U691 and provide information on 
the gentral characteristics of the SEC vidicon. 

2. Brisf Description of the SEC Vidicon 

The SEC process and the SEC vidicon are described 
extensively in a series of papers (Ref. 1) by the Westing- 
house group which dev»^loped it. Thus, only a brief 
description will be given here. 

Thf SEC vidicon has many features and characteristics 
which are identical to those found in an image orthicon 
and a conventional vidicon. Figure 5 shows a simple 
schematic diagram of an SEC tube. Basically, the SEC 
vidicon consists of an image intensifier section coupled 
to a vidicon readout section. The tube has a fiber optic 
input window which couples light from the image plane 
to a hemispj-.encal photocathode layer. The secondary 
electrons from the photocathode are accelerated and 
focused onto the SEC target. The resultant charge is 
stored on the surface of the target. Readout is accom- 
plished by a reading beam in the conventional manner. 

The unique feature of the SEC vidicon is the target, 
which is depicted schematically in Fig. 6. It consists of 



PHOTOCATHODE 



SUPPRESSOR 
MESHy 




DEFLECTION AND 
FOCUS COIL- 



r///////////^ 



c_ 



SEC VIDICON 
TARGET 
TO 

PREAMPLIFIER 



I ELECTRON 
GUN 



Fig. 5. Schematic diagram of the SEC vidicon 



7 keV ELECTRONS 
FROM 
PHOTOCATHODE 



AI2O3- 



SUPPRESSOR 
MESH 



SIGNAL PLATE 




KCI LAYER 



TO 
PREAMPLIFIER 



Fig. 6. SEC vidicon target 

three layers. A thin layer of aluminum and a porous KCI 
layer are evaporated onto an alumina substrate. The 
KCI layer is evaporated in such a manner that it has an 
e;:tremely low mass thickness of between 10 and 100 
jiig/cm^. (For comparison, a solid KCI layer 20 am 
thick would have a mass thickness of 1.984 g/cm' 
X 20 X 10" = 4000 /ig/cm^) In operation, the electrons 
from the photocathode bombard the target with an energy 
determined by the photocathode voltage. Typically, the 
electron energy is about 7 keV. The incident electrons 
pass through the alumina and the aluminum signal elec- 
trode layer and generate secondary electrons in the 
KCI layer. 

The alumina and the aluminum layer are su£Bciently 
thin such that the transmittance for 7 keV electrons is 
high. The secondary electrons generated in the KCI 
layer are swept to the signal electrode by the reverse 



162 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. 11/ 



bios applied between the signal electrode and the XCl 
layer's surface which has been charged to cathode 
potential by the electron beat). The signal elect, ode is 
t\'picaUy biased at between 10 and 20 V. The collection 
efficieucv for the secondaries is larRelv due to ihc many 
voids in the tftrget, Surre of the secondary electrons wilt 
reeombine witli posili%e charge recombination centers, 
but most will reach the signal electrode. T>pical electron 
gains between 100 to 200 have been achieved in SEC 
targets. The resulting positive image pattern is neutral- 
ized by the electrons from the scanning beam. The target 
current which discharg"s the KCl layer through the signal 
electrode during readout constitutes the video signal. 

The most outstanding; feature of the SEC t.^gct is 
probably its high resistivity, which permits charge inte- 
gration and storage for extended periods of lime. For 
lovv-light-level detection, time exposures up to several 
hours cuii be utili/cd with no reciprocity failure, in 
addition, the SEC target is capable of storing -ignals for 
several days without any significant image degradation. 

3. Tett Proctdurei 

Most of tlie data were taken on two WX.'SOegi tubes 
purchased from Westinghousc. A photograph of the 



WX30691 SEC vidicon is shown in Fig. 7 Additional 
data and experience were acquired from the evaluation 
of two Ai}oUo SEC tuhes during the previous year, and 
disc'is.sions with \Vestinghou.':e engineers at Elmira, 
New York, in connection with the SEC vidicon niggedi- 
zation proposal. 

The W.\30691 has a 25-mm photocatliode, The Input 
raster size is 0.6 X 0.8 in. although other ra.'iter sizes 
bounded by a 1-in, diameter can be used. The input 
format si/e is thus K)% larger in linear dimensions than 
a standard vidicon (4 X ^i in.) The read section of the 
WX.30691 is identical to that of a conventional 1-in. ;dl 
magnetic vidicon so that standard deflection yokes and 
focus coils can he uti'-zed. 

The \V.\G0691s were evaluated in a camera head 
designed specifically for SEC vidicons (.'JPS 37-48, Vol. Ill, 
pp. H3-I46) and the vidicon test facilities. The WX30e91s 
were initially operated at EI.A rates to optimize alignment 
currents, set size and centering, and optical-cleetrical 
focus. For slow-scan tests, line and frame rates were set 
to yield 600 noninterlaced television lines per picture 
height. A solenoiJ-drivcn Wollensak leaf shutter was 
used to shutter the WX.'30691 during slow-scan operation. 







Fig. 7 WX30691 SEC virilcen 



JFL SMCE PROGRAMS SUMMAHY 37 -5 J, VOL. Hi 



1f3 



Since the sensitivity depends critically on the image 
area scanned, the fiber optic input faceplate was masked 
with a C.6 X 0.8-in. template. The proper raster size was 
set at all scan rates by adjusting size and centering 
controls, using the template as a reference. Static transfer 
and sensitivity measurements were made with a tungsten 
source (2875° K), a series of calibrated Iconal neucral 
density filters, and a KietLley electrometer. 

4. Results 

The static transfer characteristics at EIA rates of the 
WX30691 for various target voltages Vr are shown in 
Fig. 8. The djTiamic range extends typically over two 
orders of magnitude in illuminance. Since flat field 
illuminance was used, signal current as measured by the 
Kiethley can be expressed as a peak c- rrent if the 
blanking time is taken into consideration. The gamma 
typically varies from approximately unity at lower illumi- 
nance levels to about 0.5 at the saturation pomt. Beyond 
the maximum point indicated on each of the curves, the 
iarget is saturated at the suppressor mesh voltage. This 
provides a Tcnee" in the transfer curve. The "knee" region 
is not as extended as in an image orthicon, and operation 
in this region is not recommended due to unage "bum in" 
and an abrupt change into a "crossed over" mode which 
produces "blacker than black" areas on the monitor. 



Ul 
O 



O 
IE 




The operating range can be extended to higher illumi- 
nance levels by decreasing the target gain via a decrease 
in the photccathode voltage Vpc. This can be done in a 
dynamic fashion by an automatic gain control loop which 
samples the video level and varies the photocathode 
voltage accordingly. Such a system is incorporated in the 
Apol > lunar TV system. There Ss no perceptible degra- 
dation in resolution with photocathode voltages from 
3 to 8 kV. The target gain dependence on photocathode 
voltage was determined at EIA rates by measuring the 
signal current as the photocathode voltage was varied 
with a constant target voltage. Figure 9 shows the 
resultant relative target gain versus photocathode voltage 
curve for the WX30691. Maximum target gain occurs at 
7 kV. At higher photocathode voltages, the cross section 
of the target for primary electrons decreases and the gain 
accordingly drops ofiF. At lower photocathode voltages, a 
substantial portion of the primary electrons are stopped 
by the Al and AKOj substrates with a resultant sharp 
decrease in gain. 

Of particular importance for space applications are the 
slow-scan characteristics of the SEC vidicon. EIA rate 
operation requires a disproportionately large bandwidth 
and is not compatible nor feasible for Mariner class 
missions. It is anticipated that future niissions will still 



300 


1 1 ! 1 
ILLUMINATION = 7.8 x 10"* ft-cd 

Wj- ' 12 V „t(ffi^^ 


^■^ 


«%- 


250 
? 200 






**oo<J 






-& 


f 












/ 










TARGET 






/ 














6 










RELATIVE 

5 
o 




J 


/ 
























50 




6 














/ 















rrtf^ 


¥^ 













TARGET ILLUMIN.;T!0N, ft-cd 

Fig. e. Transfer characteristics of the WX30691 SEC 
vidicon at EIA rales 



4 5 6 7 

PHOTOCATHODE VOLTAGE, kV 



Fig. 9. Relative target gain vs photocathode voltage 
for the WX30691 SEC vidicon at EIA rates 



164 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Iff 



utilize slow scanning as a means for bandwidth and data 
reduction. To my knowledge, there has not been any 
extensive evaluation or data on the slow-scan capabilities 
of the SEC vidicon due to its limited applicability. The 
Apollo lunar SEC camera utilizes frame times of 0.1 and 
1.6 s, whereas the Apollo telescope mount SEC camera is 
an EIA system. Future Mariner-class missions will require 
operation of the SEC vidicon at longer frame times unless 
another means is developed to buffer down the data rate 
from the vidicon. 

To determine the slow-scan capabilities of the SEC 
vidicon, the WX30691 was evaluated at several slow-scan 
rates. Figure 10 shows a typical transfer characteristic 
at a frame time of 1 s. Slow-scan operation requires some 
means of shuttering to stop motion since the long frame 
times do not permit an open shutter mode without image 



OC 
(£ 

o 

_J 
< 
z 
o 
</> 



2 

10-8 


r, = is 


._ = <»o 


ms . 








































A 


—Li 








1 

1 


















/y^ 


y^ 




























X 










2 
10-9 














// 


/ 


Y 
















i 
[ 






y^ 












- 


_ 


































' 


/ 






















' 


-- 






/ 




















10-10 




^ 


/ 


r 













































10"' 2 4 6 I0-* 2 4 6 10"* Z 4 6 10 

ILLUMINANCE, ft-cd 

Fig. 10. Light transfer curve for a frame time of 1 s 



6 r 

4 



(- 
Z 
UJ 
i 10-9 

CJ 

< 

2 4 



l/x 



7^^ 



t 



-Q^-^ 



«4T 



X I s 

A 100 ms 

D 40 ms 

O 20 ms 



I0-* a 4 6 IO-< 2 4 6 iO-' 2 4 6 

ILLUMINATION, ft-cd-s 

Fig. 1 1 . Light transfer curves for different shutter 
speeds at a frame time of 4 s 



smear. In order to check reciprocity between shutter 
speeds and total light exposures, a series of transfer 
curves were obtained at various shutter speeds between 
1 s and 20 ms. Figure 11 shows the superposition of the 
transfer curves taken with the different shutter speeds. 
The scatter in the data points is well withhi the measure- 
ment uncertainties. Therefore, within the limited range 
of shutter speeds utilized, it can be said that shutter 
speed-light reciprocity holds. 

Figure 12 shows the variation in signal current as a 
function of frame time for a number of constant light 
energy values within the normal operating region of the 
WX30691 at a target voltage of 16 V. As anticipated, 
the signal current drops off proportionately with scan 
rate. Although the signal current can be increased to 
some extent by higher target voltage operation with some 
loss in dynamic range, the WX30691 is limited to frame 
times that do not exceed 10 s if scanned in the conven- 
tional manner due to the degradation in signal-to-noise 
ratio arising from the decreased signal output. Mariner 
Mars 1969 type slow-scan vidicon, for example, has a 
typical signal current of 3 nA at 0.1 ft-cd-s at a 42-s 
frame time compared to an extrapolated maximum cur- 
rent of approximately 0.2 nA at the same frame time for 
the WX30691. The WX30691 can be operated at longer 
frame times, however, if special scanning techniques are 
utilized to achieve short beam dwell times under slow- 
scan conditions. Such techniques were used in the 
Uvicon SEC camera (Orbiting Astronomical Observatory 




Fig. 1 2. Signal output vs frame time 



■in SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



165 



satellite) and the Mariner Mars 1964 vidicon camera to 
maintain signal current output at slow-scan rates. 

The higher signal current for the slow-scan vidicon is 
due to the higher electron charge density stored on the 
target for a given exposure time. The higher electron 
charge density for a Mariner Mars 1969 slow-scan ^-'dicon 
is the result of its higher target capacitance (=« 10,000 
pF/cm= vs « 200 pF/cm= for the WX30691). The lower 
capacitance of the SEC cannot be compensated for by a 
larger voltage excursion on the target since the maximum 
practical voltage excursion is limited to about 6 V. 
Larger voltage excursions result in a "beam pulling" 
phenomenon which lowers resolution. In addition, the 
operating region of the WX30691 is typically two orders 
of magnitude lower in illuminance. The combined photo- 
cathode and target gain of the WX30691 is not large 
enough to offset this in terms of the total charge stored 
on the target. 

Let us now consider the signal-to-noise ratio of the 
WX30691 at slow-scan rates in the light of its signal 
current output. As in all photocathode devices, the signal- 
to-noise ratio is shot noise limited by the less than unity 
quantum efficiency of the photocathode material. A 
typical figure quoted for the sensitivity of an S 20 photo- 
cathode is 150 A/lm. If we take the photo flux in 1 ft-c of 
illuminance as 1.1 X lO'Vs for a 2850° K source, 150 A/lm 
conesponds to a quantum efficiency of 9%. The shot 
noise in the photoelectron current is given by 



/,. - (2et, Af)^ 



(1) 



Utilizing Eq. (1), the shot noise signal-limited signal- 
to-noise ratio can be written as 



S/N 



m 



(2) 



sensi- 



where A = area of the photocathode in ft*, S 

tivity of the photocathode in j*A/lm, 7 = illumination in 

ft-cd-s, n = number of pixels, and e = 1.6 X 10"" A-s. 

Figure 13 shows the limiting signal-to-noise ratio for 
the VVX30691 for some typical resolution requirements. 
This signal-to-noise ratio is never achieved in practice 
because the preemplifier further degrades the signal-to- 
noise ratio, and the curves in Fig. 13 should be con- 
sidered as upper limits. 

The preamplifier noise level is not eas.uy determinable 
since it depends on the noise parameters 5or the specific 




TARGET ILLUMINATION, ft-cd-s 

Fig. 13. Limiting signal-to-noise ratio for the WX30691 
SEC vidicon for some typical resolution requirements 



input device and bias conditions. However, the pream- 
plifier noise levels can be characterized by utilization of 
the concept of noise figure. Preamplifier noise is assumed 
to contribute additive "white" noise. This is a reasonable 
assumption since slow-scan operation requires only 
limited video basebands which can easily be shifted 
beyond the 1/f noise region of transistors by carrier or 
chopping techniques. Adding the vidicon shot-noise and 
load-resistor thermal noise in quadrature, the total 
equivalent input noise current can be expressed as 



= (4fcrA/Fr*^20ai. + -^y 



(3) 



where a is a parameter which characterizes the noise 
buildup in the stored charge, F = noise figure for the 
preamplifier, Rt = load resistor, », = signal current from 
vidicon, and A/ = video bandwidth. 



In practice, 1/Ri»20ai,. Therefore, 
The signal-to-noise ratio is thus 



(4) 



(5) 



For a given resolution requirement, the minimum 
bandwidth is given by 



A/ 



0.5 N^N^ 



(6) 



where Ni and Ny are the horizontal and vertical resolu- 
tions in TVL/picture height, h = blanking fact^i = 



166 



i?L SPACE PROGRAMS SUMMARY 37-51, VOL. Iff 



T,/T, (unblanked), X/Y = aspect ratio, and Tf = frame 
time. 

Since the signal current output is approximately pro- 
portional to 1/Tj up to frame times of 10 s, from Eqs. 
(5) and (6) it follows that 



S/N ^ (Ri A/)^ 



(7) 



Thus, the signal-to-noise ratio can be maintained at 
slow-scan rates up to 10 s by increasing Rt proportion- 
ately as the bandwidth is decreased. For typical values 
of 3 dB for the noise figure (Rt = 500 O, 1 , -Is, and 
N.r = Nv = 600 TVL), the maximum highlight signal- 
to-noise ratio calculated for the WX30691 is 34 dB. A 
typical highlight signal-to-noise ratio for a Mariner Mars 
1969 vidicon-preamplifier combination is approximately 
50 dB. The higher signal-to-noise ratio is a result of the 
higher signal output of the slow-scan vidicon within its 



operating region, lower bandwidth due to the longer 
frame time, and a larger Rl- 

5. Conclusions 

The feasibility of operating the WX30691 at slow-scan 
rates up to 10 s was demonstrated. Longer frame time 
operation is not practical using conventional scanning 
techniques. Although there are other versions of the 
25-mm SEC vidicon modified to favor specific system 
requirements, the performance characteristics of the 
WX30691 are typical of what can be expected from a 
25-mm SEC vidicon. Further evaluation of the resolution, 
image storage, and spectral response of the WX30691 
is in progress. 

Reference 

1. Goetze, G. W., et al.. Advances in Electronics and Electron 
Physics, Report 22A, pp. 219-262. Westinghouse Electric Cor- 
poration, Elmira, N. Y , 1966. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



167 



PRECEDING- PAGE BLANK NOT FILMED. 



N68-37414 



XVII. Science Data Systems 

SPACE SCIENCES DIVISION 



A. Digital Techniques for Generating a Time- 
Dependent Acceleration Voltage for a 
Mass Spectrometer, M. Perlman 

1. Introduction 

A mass spectrometer can be used to determine the 
composition and relati.e abundance of the constituents 
cf a planetary atmosphere. This article discusses a tech- 
n_qr.e for the digital generation of the acceleration volt- 
age of such an instrument. 

The instrument first considered was a single-focusing 
mass spectrometer (Ref. 1), the essential components of 
which appear in Fig. 1. The instrument portion is shown 
in its mechanical configuration, whereas the suppo/t elec- 
tronics are represented by functional blocks. 

2. Instrument Operation 

The gas to be analyzed is introduced into the ioniza- 
tion chamber, where a portion of it is ionized when 
bombarded by an electron beam that is parallel to the 
source exit slit. The high-voltage sweep produces an 



electrostatic field that accelerates the ions through the 
source exit slit with approximately uniform energy The 
resulting ion beam is deflected by the electromagnetic 
field of the analyzer (permanent) magnet such that, at a 
given value of v (high-voltage sweep), all ions with a 
particular mass-per-unit-charge are focused on the col- 
lector defining slit. The ion current is collerted and fed 
into a sensitive G,jerational amplifier called an electrom- 
eter. Automatic scale switching provides a large dynamic 
range. 

.\ monotonically varying v is used to separate ions 
with different masses-pcr-unit-charge. A plot of the ion 
current versus time (resulting from a monotonically vary- 
ing i>) yields a spectrogram. The location of a peak in 
time identifies the associated mass-per-unit-charge, and 
the amplitude of the peak gives the relative abundance. 

The instrument's resolution is an important parameter. 
The mass-per-unit-charge, M/q, is in atomic mass units 
(amu) where the isotope "^ O is taken to be 16. It differs 
slightly from the chemical scale of atomic weights (Ref . 2). 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



169 



ANALYZER 
MAGNET 



FOCUSING 

SLITS 
TRAP- 



ELECTRON 
SUPPRESSOR - 




COLLECTOR 



SHORT 
LEAD 



lON- 

PUMP 

SUPPLY 



ION PUMP 
TOTAL ION CURRENT 



FILAMENT 



REPELLER 



EMISSION 
REGULATOR 

AND 

ION-SOURCE 

SUPPLY 



AUTOMATIC 

SCALE 

SWITCHING 

ELECTROMETER 



DATA- 
CONDITIONING 
SYSTEM ' 



CLOCK 

mND 

PROGRAMMER 



■DATA 

READ DATA 
COMMAND 



RESET BUFFER 
• AND START 
SEQUENCE 



HIGH- 
VOLTAGE 
SWEEP 



LOW-VOLTAGE 
POWER 
SUPPLY 



CAPSULE 
POWER 



Fig. 1. Single-focusing mass spectrometer 



Hereafter, the amu will be referred co as mass (m). The 
resolution of the instrument is defined at a particular m as 



where 



m 



_ m + {m + i) 




^wt / r,% {m + i) — m (7) X 100% 



and X and 1 are time measurements. The resolution of 
the instrument described in this report is 



J!L\ =26 



(1) 



That is, at mass 25, the instrument can distinguish peak.' 
diflFering by one unit. 

3. Parameters for Determining the Acceleration Voltage 
Curve 

a. Ion balliatics. The ion ballistics of the instrument 
in Fig. 1 are expressed 



-^[(f>I 



(2) 



ro 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. HI 



vhere 



I. 
f 





R = 3.81 cm 




B = 3,780 G 




M . . 

— = m IS in amu 


and « is in volts. 




Thus 






m{t)vit) = 10,000 



(3) 



At time t, the velocity (which is proportional to v) and 
the mass, M/q, of the ions determine its radius of deflec- 
tion, which must be 3.81 cm, to be focused on the 
collector-defining stit. An accelerating voltage that decays 
exponentially can be approximated by the discharge of 
a capacitor through a resistor. The ^ width of the ion 
peaks over the entire mass range are nearly the same for 
the exponential accelerating voltage, 



v{t) = u(0) exp 



(-7) 



(4) 



Unfortunately, ion peaks will not appear linearly sep- 
arated in time as indicated by 



,^, 10,000 



(t) 



(5) 



A linear separation of ion peaks, vdth respect to time, 
is desirable when interpreting a spectogram. The form 
required for m{t) is 



m{t) = at + m(0) 



Thus 



v{t} = 



10,000 
at + m{0) 



(6) 



(7; 



The hyperbolic (i.j., inverse) acceleration voltage ex- 
pressed in Eq. 7) caimot be as readily generated by 
analog method'^ as the exponential. 

Unlike the exponential case, the base width of the ion 
peaks varies directly with the amu interval. 

b. Mass range. The mass range is 10 to 45 for the 
instrument in question. Thus, v{t) must vary from 1000 
to 222.22 V (a lower iLiiit of 220 volts is actually used). 

JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



This places the ion peak associated with mass 45 within 
the spectrum. 

4. Hyperbolic Curve Generation Utinc; Digital 
Techniques 

a. The derivation of successive decremented dc volt- 
age leveh of fixed duration. The calculus of finite 
differences (Bef. 3) yields the discrete relationships 



rf^'^) = at(k) + mt(0) = ak + 10 
^ ntik) 

., . _ 1000 1000 _ 

"^^^^ ~ aft(k) + 1-0-^ + 1 

t{k) = fcforfc = 0,l, •••,2'-l 



v{k) 



(8) 



and r is an integer. From Eq. (8), where v{2' — 1) = 
220 V. 

39 a 



11(2'- - 1) 2'- - 1 
The quantization required for v in quanta is 



R = 



V{\}) 



[2^(a + l)-(2g + l)][o-H] 



T 



At) (2' - 2) 

Where Av{k) is the forward difference, 

Av{2'- - 2) = v{2' - 1) - v{2^ - 2) 

Note that Av{2'' — 2) is smallest change v undergoes. 

voltage quantization R_ 

time quantization ~ 2' 

^ 

(9) 



a 



5.8forr>5 



Thus, if time is quantized with r bits (r > 5), voltage 
must be quantized to r + 3 bits to recognize Av{2' — 2). 
(See Fig. 2 for an illustration of this method.) 

Time is quantized by means of a feedback shift reg- 
ister (FSR) operating synchronously with a constant 
clock frequency. The 9-stage FSR is cycled through 512 
internal states. The assertion outputs of the 9 stages 
represent a 9-bit non-weighted code. A 2-level diode 



171 




9-bit NON- WEIGHTED 



AMPLIFIER 



I J I TRANSLATORj [_ 



Fig. 2. H,perbolic curve generator with 
time quantization 

and-or matrix with 12 outputs translates the 9-bit non- 
weighted to a 12-bit weighted (positional) code. The 
12-bit representation is converted to a dc voltage level 
proportional to the magnitude of a 12-bit binary number. 
This is the function of the digital-to-analog converter. 
The 1000- to 220-V hyperbolic sweep appears at the 
output of the higli-voltage operational amplifier. Suc- 
cessive decremented levels of a fixed duration appear at 
the output of the digital-to-analog converter. 

The number of diodes in the and-or matrix, which 
represents the 9-input/12-output truth table in disjunc- 
tive canonical form, is 4608 for anding and 3054 for oring, 
or a total of 7662 diodes. A silicon-on-sapphire micro- 
electronic implementation of the diode and-or matrix is 
currently under test (see SPS 37-47, Vol. Ill, pp. 169-174). 

A minimization program based on J. P. Roth's extrac- 
tion algorithm (Refs. 4, 5, and 6), which is applicable to 
single-output Boolean functions, has been written for the 
IBM 7094 general-purpose computer. This program in- 
corporates a transformation for handling multiple-output 
combination logic. An n-input/m-output problem is 
transformed in*o an imaginary (m -f- n) input/single- 
output problem. The minimization of the single- output 
function yields the minimization of the simultaneous 
Boolean functions representing the original multiple- 
output problem in 2-level and-or form (see Ref. 7). 

The minimization program was used to find an ap- 
proximate minimum cover. A reduction of 738 diodes, or 
9.6%, was realized in 4 h 12 m of computer running 



time. This program '/as the only one found that could 
handle the 12 Boolean functions of 9 variables. It has 
since been improved, particularly for the approximate 
minimum cover options. Further runs will be made with 
the improved program. 

b. The derivation of successive and equally decre- 
mented dc voltage levels of varying duration. In this 
method, v{kj is the independent variable. Thus 



v{k) 



1000 



ai{k) 



+ 1 



'^(it) - *(^) - i / iooo-«(fc) 



t{k) 1 / 1000-u(fc) \ 

^-l-^\ v(k) ) 



(10) 



where 



780fc 



v{k) = 1000 + ' _ ^, for k = 0,1, ■■■,2'- 1, 



and r is an integer. Therefore 
/ix 780 



A,/n 1 /_1000Au(fc) \ 



2A?(0) = ^ ' 



100 (2'- - 1) 

Note that the quantum at M(0) must be halved to ensure 
that two successive one levels are separated by a zero 
level. Thus 



At{0) 



1 



11 



100 (2^ - 1) 



The required time quantization is 



Ri 



t(2' - 1) 100(2'- - 1) 



Af(0) 



11 



Rt 



8<-^<9.1forr>4 

Thus, if voltage is quantized with r bits (for r > 4), then 
time must be quantized with r + 4 bits to recognize 
Af(0). For 512 equal changes in voltage results, from 



172 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Iff 



deriving 512 unequally-spaced clock pulses to decrement 
a binary counter, 

^ = 4638 quanta 

The 14-stage FSR in Fig. 3 cycles through 4639 states 
out of a possible 8192 states. The remaining states are 
treated as "don't cares." The 2-Ievel diode and-or selec- 
tion matrix converts 512 of the 4639 thirteen-bit repre- 
sentations of internal states to timing pulses. The timing 
pulses are properly spaced in time such that the binary 
counter, which thoy decrement, sequences through bi- 
nary representations of a hyperbolic curve. 



are required to recognize At)(30). Thus 
255 



«(fc)- 



11 31 



forfc-0,1, • -,31 



The largest 8-bit binary number, 255, is used to repre- 
sent 1000 V. The feedback function for the 5-stage 
FSR is 

at = ai-3 0a*-5@ai., aj., oj., a'^_, a^., 

where (+) denotes the exclusive-or, prime (') denotes 
complementation and and is denoted by juxtaposition. 



MEMORY 
ELEMENTS 



.x„=l3-blt NON-WEIGHTEO 
CODE 
9 -bit WEIGHTED CODE 




[ (_MATRIX _j I 



DIOITAL-TO -, 
I ANALOG 
LCONVERTERj 



Fig. 3. Hyperbolic curve generator with 
amplitude quantization 

The number of diodes in the and-or matrix, which 
represents a 13-input/single-output truth table in can- 
onical form, is 6656 for ending and 512 for oring, or a 
total of 7168 diodes. This is 494 fewer diodes than needed 
in the (canonical) 9 X 12 matrix discussed previously. 

The single Boolean function of 13 variables has not 
yet been subjected to minimization. A higher percentage 
of diode reduction than that for the 9 X 12 matrix is 
anticipated where Muller coding (i.e., multi-output to 
single-output transformation) introduces new prime im- 
plicants in addition to expanding the number of inputs. 

5. Examplei of Hyperbolic Curve Generation with 
2^^ Quanta 

a, Sucf .dve decremented dc voltage levels of fixed 
duratioit. Since time is quantized with r = 5 bits, 8 bits 



The Boolean variable aic-i represents the state of the 
»th stage at clock-pulse interval (CPI) k. Successive in- 
puts and outputs of a 5 X 8 matrix appears in Table 1, 
Note that c*-, has been replaced by Xj. A plot of 
Z = Zi Zo • • • Z, in decimal versus k appears in Fig. 4. 

The 8 Boolean functions of 5 variables were minimized 
simultaneously under an approximate minimum cover 



275 



250 



> 

o 



225 



200 



5 ITS 

< 



O ISO 






»- 

8 





























l-l 














L-i 
















-n 














L 
1 
















K 














'^ 


-^ 














'-K- 




-. 



125 



100 



75 



50 

5 10 IS 20 25 30 35 

Fig. 4. Output Z (prior to amplification) in volts 
Yd r (k) = fc 



JPL SPACE PROGRAMS SUMMARY 37-5), VOL. /(/ 



173 



TabI* 1 . Nonw«ighttd-ta-w«ight*d cod* trantlalor 





—\ 


























Owlpul Z 


k 


XI 


XI 


xs 


m 


X6 


Zi 


z. 


z. 


Zt 


Zs 


z< 


27 


z» 


(prior to 

amplifl- 

callen), V 





















1 


1 


1 










255 


1 





1 













1 


1 














229 


2 








1 










1 





1 













208 


3 










1 










1 


1 











190 


4 





1 








1 







1 









1 




175 


5 







1 













1 











1 





162 


6 




1 





1 













1 











',51 


7 





1 


1 





1 



















c 


142 


8 








1 


1 

























133 


9 










1 


1 









1 











126 


10 




1 








1 









1 











119 


11 




I 


1 















1 












113 


12 




1 


1 


1 
























107 


13 




1 


1 


1 


1 















1 






103 


14 





1 


1 


1 


1 























98 


15 








1 


1 


1 










1 











94 


16 











1 


1 










1 












90 


17 













1 










1 











87 


18 




1 



















1 












83 


19 





1 


1 
















1 













80 


20 







1 


1 

























78 


21 




1 





1 


1 






















75 


22 




1 


1 





1 






















73 


23 





1 


1 


1 


























70 


24 







1 


1 


i 























68 


25 





1 





1 


1 
























66 


26 







1 





1 
























64 


27 





1 





1 
























62 


28 








1 





1 


















1 


61 


29 











1 






















1 


59 


30 














1 






















58 


31 






































56 



option In Table 1, 10000 is the initial state and the 
singul'tr state 00000 is the tenninal state, which remains 
unt.i' tne first stage is set (i.e., Xi is made a one). This 
initi.' state yielded the best minimum :»ver of all die 
possible 32 initial states. Tlie efiect of using a different 
initial state is to cyclically permute the input states rela- 
tive to the fixed output states. A total of 293 diodes is 
associated with each of 32 canonical truth tables. A re- 
duction of 119 diodes or 40.6%, was realized with 1000 
as an initial state. The initial state of 10101 yielded 
the smallest reduction (67 diodes, or 22.8%). Each of the 
mi., nization runs required less than 2 min of IBM 7094 



computing time, including pre-processing, extraction, and 
post-processing time. 

b. Successive and equaUy decremented dc voltage 
levels of varying duration. Since voltage is quantized 
with f = 5 bits, 9 bits are required to generate Af(0). 
Thus 



'^''^ 39^ v{k) ) 



where 



v(k) = 1000 - 



780k 
31 



fork = 0,1, ■■■,31 



2At(0) 



22 
3022 



*)y = 274 72 

Thus, thLty-two 9-bit combinations are to be selected 
from a total of 276 successive states (or 275 time inter- 
vals) of an FSR. 

7{k) = [274.72*(fc)+0.5] 

represents time in quanta. The brackets denote the in- 
teger portion of t{k). 

The feedback function for the 9-stage FSR is 

The FSR will cycle through 276 of a possible 512 states. 
The remaining states are treated as "don't cares." The 
word detector \V may also he used to inhibit the dock, 
thereby holding the FSR in ..tate ?^6, corresponding to 
220 V, or the end of the high-volti.je sweep. 

The thirty-two 9-bit combinations and the correspond- 
mg t{k), for which a 9 X 1 matrix will furnish a time 
pulse to the binary down counter (Fig. 5), appear in 
Table 2. The output of the counter Z = Zi Zj ••• Z, is 
repnjsented decimally where 31 corresponds to 1000 V 
and i 1 corresponds to 220 V. 

Ihe number of diodes in the andr-or matrix, which 
represent the 9-input, single-output truth table of 
Table 2, is 288 for anting and 32 for oring, or a total 
of 320 diodes. Of the possible 276 states, 110101011 was 
used as an initial state in forming Table 2. A reduction 



174 



jn SPACE PROGRAMS SUMMARY 37-51. VOL. Ill 



of 144 diodes, or 45%, was obtained when minimized 
under an approximate minimum cover option. The total 
running time was 0.67 min. 

Since this solution (176 diodes required) was compar- 
able to the best solution found for the method in 
Subsection 5-a (174 diodes required), no other initial 
state was tested. 

References 

1. Duckworth, H. E., Masd Spectroscopy, Cambridge University 
Press, New York, 1958. 

2. Leigliton, R. B., Principles of Modern Physics, McGraw-Hill 
Book Company, Inc., New York, 1959. 

3. Hamminjt, R. W., Numerical Methods for Scientists and Engi- 
neers, McGraw-Hill Book Company., Inc., New York, 1962. 

4. Roth, J. P., "Algebraic Topological Methods in Synthesis," Prj- 
ceedings of an International Symposium on the Theory of Sv>itch- 
ing, April 1957, in Annals of Computation Laboratory of Harvard 
University, Vol. XXIX, pp. 57-73, 1959. 

5. Roth, J. P., Algebraic Topological Methods for the Synthesis of 
SuAtching Systems in n-variables, ECP56-02, The Institute for 
Advanced Study, Princeton, New Jersey, April 1956. 

6. Miller, R. E., SuHtching Theory, Volume I: Combinational Cir- 
cuits, John Wiley & Sons, Inc., New York, 1965. 

7. Muller, D. E., "Application of Boolean Algebra to Switching Cir- 
cuit Design and to Error Detection," iR£ Trorw. — Electronic 
Computers Vol. EC-;}, September 1954. 



35 



30 



Table 2. An array for a 9 X 1 diode selection matrix 



z 
o 



25 



-I 

o. 

Z 20 
< 



B 15 
I 



:3 
p. 



10 

















\ 














\, 














\ 


K 














\ 


1 














^ 


"S 

















L_ 


■^ 






1 



40 



80 



120 160 

/?*), QUANTA 



200 



240 260 























Output Z 


«1 


«2 


*s 


'• 


~t 


«» 


«7 


»8 


»» 


m In 


omplin- 
(otlen), V 


1 







1 





1 





1 


1 





31 







I 


1 





1 





1 





s 


30 










1 


1 


1 





1 




4 


29 
















1 


1 


1 





6 


28 


1 










1 











1 


9 


27 







1 


1 








( 








11 


26 
















1 


1 


c 





14 


is 


1 










1 











1 


17 


24 










1 











1 





20 


23 




















1 








23 


22 


1 


























26 


21 














1 














30 


20 











1 














1 


34 


19 








1 


1 











1 





38 


18 





' 














1 


1 





42 


17 





1 


1 


1 








1 








47 


16 





1 





1 








1 


1 


1 


i? 


15 








I 


1 





1 





1 





58 


14 


1 





1 


1 














1 


64 


13 








1 





1 




1 


1 





71 


12 





1 


1 


1 







1 








78 


11 





1 





1 


r- 





c 


1 





87 


10 




1 





1 
















96 


9 




1 


1 


1 


1 










1 


106 


G 










1 








1 








118 


7 










1 


1 




I 


1 


1 


131 


« 




I 








1 










1 


147 


5 










1 


1 











1 


164 


4 




1 


1 


1 


1 




1 


1 





185 


3 







1 


c 







1 





1 


209 


2 







1 


1 

















238 


1 







1 




1 


1 




L 


1 


1 


1 


275 






Fig. 5. Output Z (prior to amplification) In voitt 
vs f (k) in quanta 



B. Capsuie System Advancer' Development 
Woven Piated-Wire Wmory, p. 8. White/iead 

1. Inlnii^uction 

An 8,192-bit woven plated-wire memory has t-^en de- 
veloped for JPL by the Librascope Group of General 
Precision Systems, Inc., for use in the entry data system 
of the Capsule Systeni Advar.c-c.d Development (CSAD) 
program. The plated-wire 'jacV was built in a fii^t i,oa- 
figuration by Libra$cc<pe, but tht electronics for the 
memor>' v/ae built in breadboard form by JPL using 



JPL SPACE PftOGRAMS SUMMAHY 37-51, VOL. Ill 



175 



designs submitted by Lihrascope. Two stacks were man- 
ufacturt'd: one for use with the breadboard electronics, 
and the other for testing the effects of sterilization and 
shock environments. 

2. Background 

Prior to the CSAD program, a pmstram had been 
undervvay at JPL for the development of a low-ixiwer, 
non-destructive readout, plated-wire memor>'. A contiact 
had been let to Lihrascope. a breadboard model pro- 
duced, and a flight -qualified engineering model was to 
be delivered by the e.id of F\*6S. A description of this 
niemor>' is given in SPS 37-45. Vol. IV. pp. 22S-!«4. This 
memory was to have had a capacit>* of 20,4S0 bits. l,0iJ4 
words of 20 bits each. Data transfer into and out of the 
memorj' was serial, i.e., one bit at a time. Addressing, 
however, was random access by 20-bit word. .\n internal 
bit counter controlled which of thf» 20 bits was bcir J, 
selected. 



When the requirements for the CS.\D memor>' Ix'- 
came known, it was decided to modify the existing c-on- 
traet and produce a memon- compatible with this new 
project. TIk" memory size was changed to S,192 bits, with 
each hit random accessible, and the electronics were 
redesigned in order to more fully utilize integrated cir- 
cuits. Power requirements wen; relaxed in order to make 
this utilization more feasible. 

The functional characteristics of the memory are given 
in Table 'I. The memory operates in the non-destructive 
readout (NDRO) mode, and can transfer data at 100,000 
bits/s. The memorv' requires 200 m\V of power on 
stand by. 1 \V when v\Titing at 100.000 bits/s, and 
800 m\V when reading at 100,000 bits/s. 

3. Woven Pfafed-Wire Stack 

a. General description. The woven plated-wire stack 
is shown in Figs. 6 and 7. The stack consists of a single 




y i 



Fig. 6, Top of plat«tl-wir* itack 



m 



Jn SPACE PROGRAMS SUMMAKY 37-51, VOi. Ill 



Table 3. Memory choracteriilics 



Copadty 


B,l93biti 


Storoga *)afl>«nl 


Plata d wire 


AddrMiing 


Random occeu by bit 


E>ato tr«nsf*f mod* 


Bit serial, reod ond write 


Dafo tronifer rot* 


to 100,000 biti/i 


R«odovt mode 


NoA'dettTucKve 


Vdalility 


Non-TOlotlle 


Inpvl lipnolt 


Clo4, reod/write, addreti linet (1-131, 




dotti inpul 


Output iigr.alt 


Do Id output 


Supply Yohogct 


+ 13 V ± 10%, +S V t 7%, 




- 3 V ± 7% 


Power QHuimpiion 


200 mW during standby 




1 000 mW during write at 1 00 K bitt/t 




BOO mW during read at 1 00 K b!l>/i 



8,192-bit plane. On either side of a 2-layer printed circuit 
board arc two 4,096-bit mats. The printed circuit board 
measures 5^ X 6^ X 14 in., and the mats are 3V1 X 21^ 
in. The top of the plane is sho%vn in Fig. 6. The 4,096-bit 



mat consists of 64 plated digit wires (shown running from 
front to rear in Fig. 6), and 64 word coils (running hori- 
zontally). The intersection of a digit wire and a word coil 
form a bit location. Woven along with the 64 plated 
wires are 16 unplatcd wires used as return lines — one 
return line for every 4 plated wires. The 4 rows of diodes 
on either side of the mat [jrovide decoding for the word 
coils. The 2 rows closest to the mat provide decoding for 
til. word coils in the mat on the top of the printed circuit 
boiird, and the 2 outer rows pro\iile decoding for the 
word coils in the mat on the underside. 

The stack is coated with a polyurethane resin. Soli- 
thane 113. This encapsulant provides the necessary ad- 
hesion to ensure that the mats are securely bonded to 
the printed circuit hoard, and yet provides sufficient 
llexibiht\' to allow for contraction and expansion of the 
plated wires. Strain relief for the plated wires is provided 
by small coils of magnet wire located at the front of this 
mat. At the rear of the mat are plated- thro ugh holes 
that connect each plated vi ire to the corresponding plated 
wire in the mat on the underside. Figure 7 shows the 




/ - - 



'!"''i-"f„u,^. 



m'^ 




^^«^Xl 



^mi/f 




Fig. 7. Undertid* of ploted-wir* stack 
JH SMC£ PftOGMMS SUMMARY 37-51, VOL. Ill 



177 



underside of the plane. At the rear of the stack are 
the plated-through holes coming from the other side. 
At the front of the mat, the plated wires and unplated 
wires are bused together, and strain relief is provided. 
Along the et'ges of the prii ted circuit board, the Soli- 
thane 113 has been removed. In a flight configuration 
the stack would be bonded along these exposed surfaces 
to the web of a blivet. A recess would be provided in 
the center of the web to accept the mat. 

b. Weave. The 4,096-bit mats are woven on a textile 
loom with 9-mil unplated wire as the warp and No. 41 
AWG magnet wire as the woof. In the loom, the 9-mil 
unplated wires are held parallel (40 mils center-to-center). 
Alternate wires are raised, and the magnet wire is 
threaded through. One continuous strand of magnet wire 
forms a word coil. As shown in Fig. 8, a word coil con- 
sists of two loops of wire at each bit location. Word 
coils are separated by strands of magnet wire that form 
spacers. Alternate word coils are terminated on opposite 
sides of the m ' 

Once the weaving process is completed, the mats are 
taken from the loom and hand-soldered to the printed- 
circuit board, at which time 64 of the 80 unplated wires 
are removed and replaced with 8 mil magnesium-copper 
wire plated with permalloy. The 64 plated wires become 
the digit lines, and the 16 remaining unplated wires 
become the return lines. 

c. Operation. The magnetic action of the plated wire 
is described in SPS 37-45, Vol. IV, pp. 230-231 and will 
only be summarized here. 



The plated wires are formed by electroplating a thin- 
film of permalloy onto 8 mil magnesi'ir. ^opper vv.». 
the presence of a circumferential magnetic field. Thus, 
under quiescent conditions, the magnetization vectors 
along the wire lie in one of the two "easy" circumferen- 
tial directions. 

In order to write into the memory, a current is directed 
through one of the word coils. In that portion of the 
plated wire enclosed by the word coil, the magnetization 
vector is rotated until it is just short of being in the axial 
direction. A current through the plated wire then "tilts" 
the magnetization vector so that when the word current 
is removed, the vector will rotate back to the circumfer- 
ential direction desired. Circumferential magnetization 
represents a one in one direction, and a zero in the oppo- 
direction. 

Data is read out of the memory by applying a word 
-unent only. This current causes the magnetization 

ctor to agiin rotate just short of the axial direction. As 
the vector rotates, it causes a small voltage of a given 
polarity to appear at the ends of the digit line. Detection 
of the polarity of this signal determines whether a one 
or zero was stored in that bit location. When the current 
pulse is removed, the vector rotates back to its original 
position and the data is retained for future access. 

4. System Design 

a. Block diagram. The block diagram of the complete 
system, plated-wire stack plus associated electronics, is 
shown in Fig 9. The external signals are indicated by 
small circles in the figure and include the input and 
output signals, and the supply voltages, given in Table 3. 



SPACER - 



DECODE 
DIODE 




PLATED ^ 
DIGIT (T 
WIRE 







DECODE 
DIODE 



IJ 







-SPACER 



WORD COIL A 

Fig. 8. Word coils 



b. Initial operation. The operation of the memory 
begins with a 2 /xs clock pulse sent to the memory from 
the data system. This clock pulse is received by the tim- 
ing generator, initiating an 800 ns countdown. During 
the countdown, switched voltages (SVs) 1, 2, and 3 are 
turned on to provide power to various portions of the 
memory electronics. When the countdown is completed, 
the tinung generator checks to see that the clock pulse 
is still being received. If so, the timing generator then 
initiates either a write cycle or a read cycle, depending 
on the condition of the read/write line into the memory. 
In either case the data output flip-flop in the read ampli- 
fier is reset to zero. 

c. Read cycle. If the read/write line is high, then a 
read cycle is initiated after the 800 ns countdown. The 



178 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. /(( 



&; 



SV 3- 



VOLTAGE 
SWITCHES 



• SV I 
>SV 2 
•SV 3 



ADDRESS LINES (1-3) 

XII 

- iRTER! 



1 3 INVERTERS 



SV2 
CLOCK c 



SV2 
DATA 

INPUT 
READ/ 

WRITE 



+ I5VC 
+ 5VC 
-3VC 



TIMING 
GENERATOR 



ADDRESS LINES (10-13) 

JJJl 

SV 3— Ht inverters! 



SV I 



8A 

SWITCHES 



timing 

LOGIC 



_ CLEAR DATA 
FLIP-FLOP 

SV I' 



•DT I 

•DT 

►WORD PULSE TIMING 

■ STROBE 




4X16 

TRANSFORMER 

SELECTION 

MATRIX 



m 



DIGIT 
LINES 



[ji] 



RETURNS 



SV 1- 



READ 
AMPLIFIER 



T 



DATA OUTPUT 

STROBE 



lE 



MEMORY STACK :8K 
128 DIODE MATRIX | 



1 




1 

PLATED 
WIRE 
MAT 

4K 


4K 





H 



40 
SWITCHES 



•SV I SV I- 



DIGIT 

CURRENT 

SINK 



TT 



~ri3 

SV 3-H2 inverters! 

address LINES(8,9) 



,, (m 



16 B 
SWITCHES 



TW 



SV 3—^4 inverters] 

address lines (4-7) 



word pulse 
generator 



CLEAR DATA DT I DT 
FLIP-FLOP 



WORD PULSE 
TIMING 



NUMBERS WITHIN BOXES REFER TO 
THE NUMBER OF SIGNAL LINES 



Fig. 9. Memory system block diagram 



word-pulse generator causes a 160 ns, 400 mA pulse to 
pass into one of the A switches, through a word coil in 
the stack, and out one of the B switches. Address lines 
1 through 3 select which of the eight A switches is turned 
on, and address lines 4 through 7 select which of 'lie 
sixteen B switches is turned on. The word pulse causes a 
"readback" voltage co appear on all 64 plated-wire digit 
lines. The transformer selection matrix determines which 
of these 64 signals reaches the read amplifier. This ma- 
trix consists of 64 transformers, one for each of the 
plated-wire digit lines. Address lines 10 through 13 select 
one of the C switches, and address lines 8 and 9 select one 
of the D switches. The combination of C and D switches 
select one of the transformers in the matrix. The trans- 
former that is so selected allows the signal from its cor- 
responding digit line to be passed through to the read 
amplifier. The read amplifier then takes this signal 
(typically 6 mV) and amplifies it. The timing generator 
and timing logic generate a strobe for the amplified 
readback signal. If the readback signal has a positive 

JPL SPACE PROGRAMS SUMMARY 37-51, VOt. Ill 



polarity at the time of the strobe, the data output flip- 
flop in the read amplifier is set to a one. If the readback 
signal has the opposite polarity, the data output flip-flop 
remains reset to zero. The output of this flip-flop be- 
comes the data-output signal from the memory. 

d. Write cycle. If the read/write signal is low, the 
memory initiates a write cycle, employing a bipolar 
write scheme, after the 800 ns countdown. For example, 
if a one is to be written into the memory, a zero is first 
written followed immediately by a one. This method 
ensures that an equal number of ones and zeros are 
written into every bit location in the memory, thus re- 
ducing the possibility of "creep" in the plated wire. 
(Creep is the enlargement of an area of magnetization 
caused by repeated writing of data of the same polarity 
into a given bit location.) 

In order to write data into the memory, a 94-mA cur- 
rent pulse is drawn from one of the C switches, through 



179 



the primary of one of the transformers in the selection 
matrix, through one of the D switches, and down to the 
digit current .>;ink, A 94-inA current is then induced in 
the secondary of the selected transformer. The polarity 
of this current is determined by the data to be written 
into the memory. This current flows through the cor- 
responding digit Une, and its associated return wire, ai.d 
lasts for about 320 ns. During this time, the word pulse 
generator is activated for 1 60 ns. The toiiicidence of the 
word pulse and the digit current causes data to be 
written into the corresponding bit of the memory. Be- 
cause of the bipolar \vrite sclieme used, however, the 
data that has lieen written is the complement of that 
desired. A digit current pulse of the opposite polarity is 
immediately initiated and the word pulse is repeated. 
writing the correct data into the memory. 

e. Return to standby. After either the read or the 
write cycle i.s completed, the memory turns off SVs 1. % 
and 3. re>-Iur:ng power to standby mode. The timing 
generator then waits for the next clock pulse in order to 
re-initiate action. 

5. Test Results 

a. Stack tests. Before delivery from the contractor, the 
t\vo stacks were tested using the following procedure for 
each bit in the plane: 

(1) A zero was written into a bit location 1600 times, 
using a word current S% higher than nominal and 
a digit current 10% higher than nominal, 

(2} A one was written into the same bit location once, 
using n word current 5% less than nominal and 
a digit current 10% less than nominal. 

(3) The digit currents used to write the original zero 
were repeated 1600 times, then the word currents 
were repeated 1600 times. Since the digit cur- 
rent and word current.*; never coincided, the data 
should not have changed. 

(4) The data was then read out of the bit location on 
an oscilloscope. If the polarity of the signal did not 
correspond to a one, or if the output signal was 
less than 2 mV, the bit was recorded as question- 
able, 

(5) The procedure was repeated for data of the oppo- 
site polarity ui the same bit location. 

The first stack was tested only at 25°C. About 80 bits 
on the top and 60 bits on the underside of the plane 
were questionable — either low voltage or incorrect po- 



larity. With the exception of two wires on the underside, 
the vast majority of the errors were in those wires near 
the edges of the mat. The nature and location of the 
errors indicated that the word currents flowing in 
the printed circuit board were interfering with the read- 
out signal. The etched lines that carry the word currents 
are parallel to the digit lines, and only about ^4 in. from 
tiiose lines on the edges. 

The second stack was tested at -20, 25, and 90° C. 
The nominal currents used at the various temperatures 
are as follows: 





Currtnf, mA 


nisir 


Wdfd 


-20 
25 

90 


1)6 
94 
83 


460 
400 
329 



The number of questionable bits remained much the 
same over the three temperatures — about 90 on the front 
and 30 ou the back. As with the first stack, most of the 








111! II mw . 




Fig. 10. Syitem breadboard 



ilo 



JPl SPACE PROGRAMS SUMMARY 37-51, VOL. Id 



errors occurred near the edges of the mats. The output 
voltage, however, did vary with temperature. At — 20°C 
the average ouiput was about 4.0 mV, at 25° C it was 
about 5.5 mV, and at 90° C it was up to about 7.0 mV. 

The second stack will be subjected to an environ- 
mental test program that will include sterilization, shock, 
and vibration. After each test the electrical performance 
will be monitored to detect any degradation. 

The errors in both stacks are well understood and 
could be eliminated by redesign of the printed-circuit 
boards. However, because of the limitations in funds 
and the pressures of the CSAD schedule, it was decided 
to accept the stacks without further modification. When 
the memory breadboard was operated as part of the 



Entry Data System of CSAD, those plated wires with 
bit error; were not used for storage. The remriining 
capacity of the memory was sufficient for CSAD require- 
ments. 

b. Breadboard teats. The plated-wire memory bread- 
board shown in Fig. 10 consists of the first stack (the 
lower left-hand comer) plus the associated electronics. 
The total parts count, including the diodes on the stack, 
is 717. The system was tested only at 25°C and, in gen- 
eral, operated properly. As expected, there were bit errors 
in the plated wires along the edge'j of the mats — about 30 
on the front and 40 on the back. 

The breadboard memory will be used for temperature- 
margin and extended-life tests. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



181 



N68-37415 



XVIII. Lunar and Planetary Sciences 



SPACE i-CIENCES DIVISION 



A. Scattering in the Twilight Atmosphere 

of Venus, K. O. Abhyankar 

1 . Introduction 

Earlier computations of the scattering of light in the 
atmosphere of Venus madt by Horak (Ref. 1) and Harris 
(Ref. 2) had shown that the observed visual brightness of 
Venus at phase angles greater than 120 deg exceeds the 
predicted theoretical brightness for isotropic and Rayleigh 
scattering phase functions (Fig. 1). The objective of this 
work was to test whether all or a part of this discrepancy 
could be caused by the neglected efiFect of sphtricity of 
the Venus atmosphere as suggested by Harris. 

2. Computational Factors and Results 

By an appropriate geometrical consideration, which 
obviates the usual necessity of approximating each ele- 
ment of the spherical atmospheric shell by a plane par- 
allel slab, it was possible to resolve the problem of 
scattering by the spherical twilight atmosphere into a 
series of separate problems that can be treated by the 



ordinary plane parallel technique. In this procedure, de- 
scribed in detail elsewhere, the disk of Venus is divided 
into four partly overlapping regions, each of which is 
illuminated in a difiEerent manner. The excess flux con- 
tributed by the twilight atmosphere is then easily com- 
puted by using the available tables of scattering functions 
for plane Rayleigh atmospheres of diflFerent optical thick- 
nesses due to Coulson, Dave, and Sekera (Ref. 3) and 
Sekera and Kahle (Ref. 4). The Rayleigh scattering op- 
tical depths required for this purpose were derived from 
two models of the Venus atmosphere; one was the stan- 
dard model of Kaplan (Ref. 5), and the other was a new 
extreme model quite similar to Kaplan's but more con- 
sistent with the recent data obtained from Venera 4 and 
Mariner V measurements. The latter model (Table 1) is 
cooler, denser, and more compact than Kaplan's model. 

The total fluxes at various phase angles were computed 
for three wavelengths: V (5550 A), B (4550 A), and 
U (3700 A). The computed phase curves for the V wave- 
length (Fig. 1) show that Rayleigh scattering alone is not 
sufficient to account for the excess observed brightness 



182 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



0.20 



0,15 



S 10 



0.05 




TabI* 1 . New •ytrcme model of Vvnus atmotphero* 



120 140 160 

PHASE ANGLE a. deg 

Fig. 1 . Visual phase curves of Venus 



Htlghl, km 


Tamparatwra, ^K 


Pr*(tur*, dyn/tm' 


D«n(lty, g/cm'' 





550.0 


2.20X10' 


2.00X10' 


3 


503.0 


1.53X10' 


1.52X10' 


10 


456.0 


1.01X10' 


1.11X10' 


15 


409.0 


6.44 X 10" 


7.89X10" 


20 


362.0 


3.87X10* 


5.36X10' 


23 


315.0 


2.16X10' 


3.44X10' 


30 


268.0 


1.08X10' 


2.03X10' 


35 


221.0 


4.92X10" 


1.12X10" 


40 


219.0 


1.85X10" 


4.25X10' 


45 


217.1 


6.79X10* 


1.57X10' 


50 


215.2 


2.42X10' 


5.64X10^ 


55 


213.3 


8.40X10" 


1.97X10' 


60 


211.4 


2.91 X lO' 


6.90X10' 


65 


209.5 


1.01X10" 


2.42X10* 


70 


207.6 


3.50X10' 


8.45X10' 


75 


205.7 


1.18X10' 


2.18X10' 


80 


20:>.8 


3.89X10' 


9.57X10" 


85 


201.9 


1.28X10' 


3.18X10' 


90 


2M.0 


4.21 


1.05X10' 


95 


210.0 


1.285 


3.07X10' 


100 


220.0 


0.421 


9.59X10'" 


110 


— 


— 





•Compowtion 


= «5%C02. M%N 


M«on mol*cular wfllcht = 41 .6. 



at large phase angles. The main contribution to the ob- 
served brightness of Venus at inferior conjunction must 
be concluded to have come from particulate or condensate 
matter that scatters about one order of magnitude more 
efficiently in the forward direction than a Rayleigh 
scatterer. 

To determine the possible nature of the scattering 
particles, the efficiency factors for V, B, and U wave- 
lengths were obtained by combining the visual observa- 
tions of Danjon (Ref. 6) and B-V, U-B colors measured 
by Knuckles, Sinton, and Sinton (Ref. 7), and comparing 
them with the computed brightness in the three colors 
at inferior conjunction. They were found to be 6, 10, and 
17 for V, B, and U, respectively, in the case of the new 
model, and 2, 4, and 7, respectively, for Kaplan's model. 
The variation of the efficiency factor with color is caused 
partly by the variation of the phase function of the par- 
ticles with wavelength and partly by the variation of their 
extinction coefficient with wavelength. From the scatter- 
ing functions of water drops given by Deirmendjian 
(Ref. 8) for his haze model M, and from the extinction 
coefficients for dielectric particles tabulated by Penndorf 
(Ref. 9), it was found that the above efficiency factors 
were consistent with a haze model consisting of water 
drops of 0.1- to 1.0-/im radius, assuming a haze thickness 
. ' ''O km. For both models of the Venus atmosphere con- 
sicc \ here, the total amount of water in a column of 



1-cm- cross section above the lowest layers visible at 
inferior cor junction (above the height of 30-35 km) comes 
out to be close to 10" g/cm-; i.e., about 0.01 nm of pre- 
cipitable water above that level. This amount of water 
is too small to be detected by spectroscopic means. The 
total amount of water in the line of sight at inferior con- 
junction would be about 0.5 iixn. 

3. Atmospheric Contribution 

The curves in Fig. 2 indicate the relative contributions 
/( of the various layers of the atmosphere to the bright- 
ness of Venus at inferior conjunction. It is seen that the 
contribution to visible radiation (V, B, and U) comes 
mainly .'icm the layers between 3(' and 55 km in the new 
model and f' -;n 35 to 90 km in Kaplan's model. In both 
cases the densities in the effective layers range from 
2 X 10"' to 2 X 10" g/cm'; the larger contribution in 
Kaplan's model is due mainly to the larger geometrical 
depth of the effective layers. It is also seen that in both 
models the V, B, and U radiations come from successively 
higher layers due to the increase of the scattering coeffi- 
cient f'om V to U. The range in height between V and U 
is about 10 km in the new model and about 20 ^-m in 
Kaplan's model. However, the geometncal thickness of 
the contributing layers is approximately the same for 
all the three colors, about 15 km in the r^w model and 
about 30 km for Kaplan's model. These values are in 



Jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



183 



good agreement with the minimum thickness of 15 km 
derived by SchilHng and Moore (Ref. 10) from the ob- 
served cusp extensions of Venus. 

Rsferences 

1. Horak, H. C, Aitropluji. J.. \'ol. 112, p. 445, 1950. 

2. Hams, IX, Plar>"t.s and Satellites, \'ol. Ill, Chap. 8 p. 311. 
Edited by G. P. Kiiipcr. I'nivtTsity of ChicaKo Press, diitimn. 
111., 1961. 

3 Coulson, K. L., Dave, J. W. and Sekt-ra, Z., Tables Rehted 
to Radiation Emerfiinfi From a Planetary Atmosphere With 
Rayleifih Scatterina. Univcrsits of California Press, Qerkcle) 
,ind '-OS AnKele.s, Calif., 1960. 

4. Sckera, Z., and Kahle, A. B., Raud Coiixiration Report 
R-452-PR, Santa Moniea, Calif., 1966. 

5. Kaplan, L. D., A Preliminary Model of the Venus Atmosphere. 
Technical Report 32-379. Jet Propulsion Lahorator>-, Pasadena, 
Calif., Dec. 12, 1962. 

6. Uanjon, A., Bull, .\iiron . Vol. 14. p. 315, 1949. 

7. Knuckles, C. F., Sinton, M. K., and Sinton, \V. M., Lowell 
Ohser\ator>- Bulletin 115, Vol. 5, No. 10, p. 153, 1961. 

8. Deirmendjian, D., Appl. Opt., Vol. 3, No. 2, p. 187, 1984. 

9. Penndorf, R. B., /. Opt. Soc. Am., Vol. 47, No. 1 1, p. 1010, 1957. 
10. Sthillirg, G. F., and M(X)re, R. C, Ra: 1 Corporation Memo- 
randum R\I-.5.386-PR. Santa Monica, Calif.. 1967. 




04 06 

Fig. 2. Contribution of various otmotphoric fcyori 

to brightntts of Vtnus at inforior 

conjunction (a = 1 80 dog) 



B. Water Vapor Variations on Venus, R. A. .5i:horn, 
L. D. Gray, E. S. Barker.' arid R. C. Moore= 

An extensive series of spectroscopic observations of 
\'enus in the 8300-A H..O band was carried out during 
1967. The purpose of this study was to try and reconcile 
the conflicting estimates of the water vapor abundance 
"above the cloudj" of Venus by a homogeneous set of 
observations covering a large range oi phase angles, a 
long period of time, and a variety of regions on the disk 
of the planet. 

Early results in the near infrared (Refs, 1- 4) gave values 
of IV* (the amount of precipitable HnO in a vertical col- 
umn through the atmosphere of Venus "above the 
clouds") ranging from 52 to 222 urn of precipitable H.O. 
More recently Belton and Hunten (Ref. 5) and Spinrad 
and Shawl (P ',6) detected doppler-shifted Venus com- 
ponents to the 8189.272-A H.O line. Belton and Hunten 
observed a small region near the center of the disk and 
•estimated the equivalent width of this weak feature as 
20 m \, which corresponded to 317 jum of precipitable H^O 
in the total path (rjtv' = 317 ^m, where ?; is the effective air 
mass). Spinrad and Shawl, using a spectrograph slit set 
parallel to the terminator, found the 8189-A feature to 
have an equivalent width of about 15 mA at the center 
of the disk and less at the poles. They estimated 
?;tt° = 250 ;tm and w* = 60 /tm at the cente. of the disk. 
.\ later discussion of the Kitt peak data (Ref. 7) gave an 
equivalent width of 15 mA, identical with the result of 
Spinrad and Shawl. 

While Venus observations were being made at JPL, 
Owen (Ref. 8) observed the 8200-A H,0 band on Venus 
and found no evidence of Cytherean H^O. He set an 
upper limit of w* < 16 nm and suggested that the faint 
8189-A "Venus" feature was a solar line. In addition, 
Connes, et al. (Ref. 9), set an upper limit of w' < 20 iitn 
from H..0 bands in the region 1 < A. < 2 jim, while Kuiper 
(Ref. 10) set an upper limit of a few microns of HjO from 
observations of the 1.4-;am HoO band. The low H^O limits 
at longer wavelengths do not necessarily contradict the 
larger H-O abundances derived from the 8300-A band 
(Ref. 11) but those of Owen clearly do. The results ot 
observations at jPL in the 8200-A region were negative 
from April 5 through June 23, 1967; however, observations 
in November and December 1967 gave positive results. 

The methods of observation and reduction used in this 
study are the same as those used previously in a stud/ 



'University of Texas, Austin, Te\,is. 

■Rand Corporation, Santa Monica, California. 



184 



JPL SPACE PROGItAMS SUMMAHY 37-51, VOL. Ill 



of H2O abundance and variabilit>- on Mars (Ref. 12). The 
spectra were taken with the 160-cm focal length camera 
of the 82-in. Struve Reflector Coude spectrograph. All 
plates were ammonia-hypersensitized IV-N emulsions and 
utilized a projected slit width of 20 ^m. The spectra used 
in this study are listed in Table 2. 

About 30 uncontaminated H2O lines (of varying /-value) 
were inspected in the 8200-A band on each plate. Com- 
parison with earlier work at J PL on Mars and examina- 
tion of the visibility of weak solar lines of known 
equivalent width in the vicinity of strong terrestrial H..O 
lines show that Cytherean lines with equivalent widths 
of >8 mA for the 4-.\/mm spectra and >4 mA for the 
2-.\ mm spectra should be detected. 

None of the blue-shifted spectra showed any trace of 
\'enus H..0 lines. Negative results on the 8176.975-.\ 
H;0 line and the particular case of ij = 4 will be used 
tc compare these results with those of Spinrad and Shawl 
and Owen. According to Rank, et al. (Ref. 14), the in- 
tensity of the 8176.975-A line is S„ - 0.077 (cm-m-atm) ' 
(almost exactly the same as the intensity of 8189.272). 
This intensity leads to an Ufyper limit of a* = 16 /im for 
the 2- A mm plates and w* = 32 jum for the 4-A/mm 
spectra of this study. These upper limits are consistent 
with Owen's limited simultaneous observations and 
Kuiper's 1.4- and 1.9-jnm results during the same period. 

In contrast, all of the red-shifted spectra from Novem- 
ber and December 1967 show positive evidence of 
Cytherean H..O features, which appear weaker at the 
poles than at the equator. The Venus features appear 
only near the sirorgest lines of the 8200-A band; i.e., 
8164.54, 8169.995, 8189.272, 8197.704, 8226.962, and 
8282.024 A (all of which are low /-value lines). In faot, 
the visibility of the Venus lines is strictly proportional 
to the strength of the corresponding terrestrial lines 
(Ref. 15). 

The 8189-, 8164-, and 8226-A Venuj. K,0 lines have an 
estimated equivalent width of 8-10 mA on the plates 
used for this study (evidently the solar line near 8189 A, 
suggested by Owen, did not affect these measurements). 
This compares with 15 mA for the 8189-A feature esti- 
mated by Belton and Hunten in 1965-1966 and Spinrad 
and Shawl in 1964 and the upper limit of 4 m.\ set by 
Owen and J PL observers earlier in 1967. Evidently the 
water vapor "above the clouds" of Venus varies with 
time. If 1; = 4 is adopted for comparison purposes, it can 
be found that w* = 30-40 /ju of precipitable H2O. 



Table 2. Venus H^O observations 



Dot*. 1967 


CraKnf;' 


i.dn' 


A\,,A' 


Petition of 
,li.- 


6lu«-shiftMl tpKlra 


AprS 




52 


-0.284 




Apr 26 




61 


-0.324 




Apr 27 




61 


-0.326 




Apr 28 




62 


-0.328 




Apr 30 




63 


-0.331 




Apr 30 




63 


-0.331 




May) 




63 


-0.333 




Moyl 




63 


-0.333 




May 1 




63 


-0.333 




Moyl 




63 


0.333 




Mayl 




63 


-0.333 




May 2 




64 


-0.334 




AAay23 




74 


-0.364 




May 24 




75 


-0.365 




May 24 




75 


-0.365 




May 29 




77 


-0.372 




Junll 


1 


89 


-0.382 




Jun19 




89 


-0.382 


1 


Junl9 




89 


-0.382 




Jun19 




89 


-0.382 




Jun22 


III 


91 


-0.382 




Jun23 




92 


-0,382 




■•d-thift*d spKtro 


No* •• 




87 


+ 0.352 




N 16 




86 


+ 0.352 




Decll 




73 


+ 0.339 




3ecl2 




72 


+ 0.338 




Dec12 




72 


+ 0.338 




Dec12 




72 


+ 0.338 




Dec 17 




70 


+ 0.318 




D«cl7 




70 


+ 0.318 




Dec 18 




69 


+ 0.317 




Dec 18 




69 


+ 0.317 




Dec 18 




69 


tO.317 




Dec 19 




69 


+ 0.316 




Dec 19 




69 


+ 0.316 




Dec 20 




68 


+ 0.315 




Dec 20 




68 


+ 0.315 




*Oiip«rsion ot 8300 A. 4.1 A/mm for groting 1; 2.1 A.*'ffim for gratif 


g 111. 


opionetocat.tnc arjim b*twee:i swn or-d eorth. 




'Doppler shift occsrdini le N!*hau> and Petri* (Ref. 13). 






« to pole noor 


termlnotor; 2 = pcraliel te 1, but near limb; 3 — porollal te «quc 


tor n«ar South 


Pol9; 4 = porollel to eqiiotor through iub-«orth point; £ — poro 


lei to equator 


near North Pole. 





The modem observations of H2O on Venus are compiled 
in Table 3, including the recent positive result of Kuiper. 
The evidence presented in Table 3 seeirs to argue strongly 
for a real variation of the observable Cytherean water 
vapor, although there is no recognizable pattern to the 
variations in the available data. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. HI 



185 



The confirmation of this variation, a study of the vari- 
ation with time and phase angle if it is confirmed, and 
the confirmation of possible variations over the disk of the 
planet are obvious questions to be solved by further 
observations. 



References 

1. Dollfus, A., "Contribution au CoUoque Caltech-JPL sur la Lune 
et les Planetes: Venus," in Proceedings of the Caltech-JPL 
Lunar and Planetary Conference, Sept. 13-18, 1965, p. 187. 
California Institute of Technology and Jet Propulsion Labora- 
tory, Pasadena, Calif., June 15, 1966. 

2. Bottema, M., Plummer, W., and Strong, J., "Water Vapor in 
the Atmosphere of Venus," Astrophys. J., Vol. 139, p. 1021, 
1984. 

3. Bottema, M., Plummer, W., and Strong, J., "A Quantitative 
Measurement of Water Vapor in the Atmosphere of Venus," 
Ann. Astrophys., Vol. 28, p. 225, 1965. 

4. Strong, J., "Balloon Telescope Studies of Venus," in Proceed- 
ings of the Caltech-JPL Lunar and Planetary Conference, 
Sept. 13-18,1965, p. 147. California Institute of Technology 
and Jet Propulsion Laboratory, Pasadena, Calif., June 15, 1966. 



5. Helton, M., and Hunten, D., "Water Vapor in the Atmosphere 
of Venus," Astrophys. J.. Vol. 146, p. 307, 1966. 

6. Spinrad, H., and Shawl, S., "A Search for Water Vapor on 
Venus-A Confirmation," Astrophys. J., Vol. 146, p. 328, 1966. 

7. Belton, M., Hunten, D., and Goody, R., The Atmospheres of 
Venus and Mars. Edited by J. Brindt and M. McElroy. Gordon 
and Breach, Science Publishers, Inc., New York (in press). 

8. Owen, T., "Water Vapor on Venus— A Dissent and a Clarifica- 
cation," Astrophys. J., Vol. 150, L121, 1967. 

9. Connes, P., et al., "Traces of HCl and HF in the Atmosphere of 
Venus," Astrophys. J., Vol. 147, p. 1230, 1967. 

10. Kuiper, G., Pub. Lunar Planet. Lab. (in press). 
li. Hunten, D., Belton, M., and Spinrad, H., Astrophyt. J., Vol. 150, 
L125, 196'7. 

12. Schom, R., et al., "High-Dispersion Spectroscopic Observa- 
tions of Mars II: The Water Vapor Variations," Astrophys. J., 
Vol. 147, p. 743, 1967. 

13. Niehaus, W., and Petrie, T., Tables of Stellar and Planetary 
Doppler Shifts from 1962 to 1982, Standard Oil Co. of Ohio, 
1961. 

14. Rank, P., et al., Astrophys. J., Vol. 140, p. 366, 1964. 

15. Moore, C, Minnaert, M., and Houtgast, J., The Solar Spectrum 
2935 A to 8770 A, National Bureaj of Standards Monograph 61, 
United States Government Printing Office, Washington, Dec. 
1966. 



Table 3. Estimates of H.O abundance in a vertical column "above the clouds" of Venus 



Dot* 


i, d*g* 


DirocKon of shift 


Wavelength of H,0 
banddl, /im 


* b 

w , /un 


Observers 


Jun 22-23, 1959 


90 


Red 


1.38 


70 


Dollfus (Ref. 1) 


Feb 21, 1964 


65 


Blue 


1.13 


52-222 


Strong (Re*. 4) 


Apr 28, 1964 


101 


Blue 


0.82 


60 


Spinrad end Shawl (Ref. 6) 


Apr 29, 1964 


102 


Blue 


0.82 


60 


Spinrad and Shawl (Ref. 6) 


Nov 17, 1964 


51 


Red 


0.82 


60 


Spinrad and Shawl (Ref. 6) 


Nov 1965 


~90 


Blue 


0.82 


^125 


Belton and Hunten (Ref. 5) 


May 1966 


—70 


Red 


0.82 


=^125 


Belton and Hunten (Ref. 5) 


Jun-Jul 1966 


60-40 


Red 


1<X<2 


<20 


Connes, et ol. (Ref. 9) 


Apr 1967 


~55 


Blue 


0.82 


<16 


Owen (Ref. 8) 


May 24, 1967 


75 


Blue 


1.4, 1.9 


-0 


Kuiper (Ref. 10) 


Jun 11,1967 


85 


Blue 


1.4,1.9 


-0 


Kuiper (Ref. 10) 


Apr-Jtn 1967 


52-92 


Blue 


0.82 


<32,<16 


This study 


Nov-Dec 1967 


87-68 


Red 


0.82 


30-40 


This study 


Nov 1967 


-80 


Red 


1.9,2.7 


— 1 


Kuiper (Ref. 10) 


"Planetocentric ongl* b*tw«*n 


lun and earth. 










''Amount of procipitoblo H2O. 













186 



jn SPACE PROGRAMS SUMMARY 37-5 T, VOL. Ill 



N 68-37416 



XIX. Physics 

SPACE SCIENCES DIVISION 



A. Auroral Arcs: Result of the Interaction of a 
Dynamic Magnetosphere With the Ionosphere, 

G. Afkinson 

1 . Introduction 

This article presents a theory to explain the occurrence 
of aurora in the form of arcs. The high-latitude auroral 
arcs are caused by electrons with energies of several 
thousand electron volts. These electrons travel down mag- 
netic field lines from the outer magnetosphere until they 
collide with, and excite, particles in the atmosphere. The 
excited particles then emit light, thereby giving rise to 
what is called an aurora. The occurrence of aurora at 
high latitudes is believed to be the result of the structure 
and large scale properties of the magnetosphere. The most 
baffling feature has been their tendency to adopt the arc 
structure; i.e., thin parallel sheets of precipitating elec- 
trons, greater than 1000 km in east-west extent, a few 
hundred kilometers high, and yet less than 1 km thick 
in the north-south direction. The average separation be- 
tween sheets is 30-40 km; the sheets lie along the mag- 
netic field lines, which are nearly vertical at these high 
latitudes. The present theory explains this structure. 



Two basic assumptions are made about the magneto- 
sphere: 

(1) There is a region in the outer magnetosphere capa- 
ble of SU11 plying electrons with the required 
energies. 

(2) There ar ; large scale electric fields in the mag- 
netosphere causing plasma flow. 

Both of the assumptions are consistent with most of the 
current models of the magnetosphere and are supported 
by strong experimental evidence. 

2. Structui-al Thee y 

Because of tlic high electrical conductivity parallel to 
magnetic field lines, the field Unes approximate lines of 
equipotential. Tliis may sometimes require that large 
electric currents (electron flows) occur parallel to the 
magnetic field lines. An auroral arc is such a current. 

The aurorul arc system is a regenerative or self- 
maintaining system. The current of precipitating electrons 
produces a reipon of intense ionization in the ionosphere 
as shown in 1 'ig. la. Such a high conductivity region in 



JPL SPACE PROGRAMS SUMMARY 37-5 T, VOL. Ill 



187 



(0) 



AURORAL 
ELECTRONS 



B, 



• V, ( FLUX TUBE 
FLOW) 



MAGNETOSPHERE 



IONOSPHERE 



(b1 



ELECTRON 
FLOW 



I 2 

+ 

,+ .+ 



V^ l-:-i 



-♦>£■; - MAGNETOSPHERE 



^^^m 






lONOSPHERE 



Fig. 1. Vertical section through the ionosphere and 

lower magnetosphere: (a) precipitating electrons 

producing polarization electric field Ex, (b) E^ 

mapping to the magnetosphere as E^ 



the ionosphere produces a polarization electric field Ej- 
(Ref. 1). 

If the magnetic field lines are to be lines of nearly con- 
stant voltage, then an electric field E^ ~ Ex must exist in 
the magnetosphere (Fig. lb). This requires that there be 
regions of positive and negative space charges shown. 
All of the plasma in the magnetosphere is flowing in the 
X direction; the only way the region of space charge in 
the magnetosphere can remain stationary is for vertical 
electron flows (negative currents) to occur as shown. 
Arrow 1 is the auroral arc. Thus, the precipitating elec- 
trons cause the region of high electrical conductivity in 
the ionosphere, which in turn causes the precipitation of 
electrons. 

The final downward flow of electrons (arrow 4) trig- 
gers the next arc, so that Fig. lb is only one cell in a 
series of parallel arcs of great extent in the «/ direction. 

It is possible, using a few simple assumptions, to pre- 
dict the following: electron precipitation rates, ionosphere 



electron and ion densities, arc thicknesses, and distances 
between arcs. These predictions are in reasonable agree- 
ment with the observed values. In addition, the theo- 
agrees with recent ionosphere measurements of electr 
fields and magnetic distortions. 

3. Solution 

A set of equations has been developed that describe 
the system, and a steady-state solution has been obtained 
for the special case Ej. = E^; i.e., infinite conductivity 
along magnetic field lines. The solutions are shown in 
Fig. 2. The top curve shov.'s electric field variation with 
distance; the second, height-integrated ionosphere cur- 
rent; the third, vertical current density; and the fourth, the 
height-integrated (in the ionosphere) number density of 
electrons or positive ions. Some of the quantities become 
infinite at the arcs because the conductivity has been 
assumed infinite. 

The solution has three main uses: 

(1) It shows the existence of an oscillatory solution. 

(2) It predicts spacing between arcs. 

(3) It allows a more detailed study of cause and effect. 

One unexpected result was the requirement of a minimum 
average particle energy for auroral arcs to form (600 eV 
for the values used in this solution). 




-1 


1 


1 1 1 1 1 1 


-3 


^ 


' 1 


1 


1 


1 1 






+(0 
1 




1 




1 


1 1 


■^ 


^ 


1 


1 


1 


1 1 






*" 


- ;a:. 


\ 1 

V — 


^ 


K - 



-20 -10 10 20 30 40 50 60 70 

X, km 

Fig. 2. A solution to the infinito parallel conductivity case 



188 



Wl SPACE PROGRAMS ^{itAN^MCi 37-51, VOL. Ill 



The plots are only quantitative in the region 
< cc < 50 km. Outside of this, the curves are intended 
to be schematic. 

Reference 

1. Bostrom, R., "A Model of the Auroral Electrojets," /. Giophyn. 
Res., Vol. 69, pp. 4983-4999, 1984. 



B. Rates and Mechanisms of the Gas Phase 
Ozonation of Ethylene and Acetylene, 

W. B. DeAlore 

1 . Introduction 

Reactions of ozone with unsaturated hydrocarbons are 
key processes in air pollution, and also constitute an inter- 
esting class of molecule-moltcule reactions that have not 
been studied in detail. This study describes gas phase 
rate measurements on the ozonation of CjHi and C^.H.. 
The results show that these two reactions, although for- 
mally similar, are fundamentally different with respect 
to detailed reaction mechanisms. Evidence has been 
found (1) that acetylene is inert to the ozonide-type reac- 
tions, which are characteristic of olefins; and (2) that 
acetylene reacts instead by a separate path, which has a 
higher collision efficiency and higher activation energy. 



2. Experimental Methods 

The reactions were carried out in a cylindrical metal 
cell coated on the inside with Kel-F grease. The cell tem- 
perature could be lowered to any desired point by flow- 
ing chilled N; gas through copper tubing wrapped 
around the cell. In this manner the temperature could 
be controlled to within d=0.2°C. To avoid temperature 
gradients due to self-heating, the gas mixtures were 
stirred vigorously with a small magnetically driven stirrer 
mounted in the cell. The O3 concentrations were about 
10 * M, and the hydrocarbons were present in 2- to 25-fold 
excess. In most cases the mixtures were pressurized with 
argon to approximately 1 atm. For C2H4, the temperature 
range was -48 to — 95°C, and for C2H2 the range was 
+ 10 to — 30°C. The reaction rates were measured by 
following the decay of O3 absorbance at 2537 A, following 
rapid mixing of the reactants. In some of the experiments 
with C-iHi, aerosol formation caused a transient baseUne 
shift and this interfered v/ith the spectrophotometric mea- 
surements. Fortunately, this effect could be minimized by 
effective stirring of the reaction mixture. Also, elimination 
of the argon pressurization reduced the aerosol interfer- 



ence. Little or no aerosol formation was observed sviih 
C,H,. 

Most of the rates were measured under conditions 
v/here hydrocarbon excess was moderate and were plotted 
according to the equation 



kt 



1 Inl^'J'tS]' 



[S]''-[03]'"'"[Sr[03]' 



(1^ 



where 



[S] ^ concentration of C2H4 or CjHa 



Since only the Oj concentration was monitored, the hydro- 
carbon concentration at any time t was calculated on the 
assumption of a 1:1 reaction stoichiometry. The validity 
of this assump was borne out by the experimental 
results. In a few cases where the hydrocarbon excess was 
large, the following pseudo-first-order equation was used: 



where 



3. Results 



ln[03]'=ln[03]-]t't 



(2) 



a. Rate measurements. Figure 3 shows C2H4 data 
plotted according to Eq. (1), for those experiments in 
which the cell was pressurized to 1 atm with inert gas. 
In general, good straight lines were obtained, although 
in a few cases aerosol formation caused some error in 
determination of the initial O3 concentration, which re- 
sulted in high intercepts. Figure 3 also shows the C2H4 
data for experiments with no pressurization. The plots 
are excellent straight lines and show adherence to Eq. (1) 
for up to at least 902 completion of reaction. The rate 
data from Fig. 3 are summarized in Table 1 and are 
plotted in Arrhenius form in Fig. 4. 

Data points from experiments with and without pres- 
surization, and for various concentrations of O3, all fall 
very nearly on a straight line (Fig. 4). The extrapolated 
Arrhenius line passes through the room temperature 
point of Hanst, et al. (Ref. 1). The rates of Bufalini and 
Altshuller (Ref. 2) are somewhat higher than those of 
this study. The following rate expression was derived 
from the slope and intercept of Fig. 4. 



log fec,H4- 6.3- 4.7/2.3 RT 
where k is expressed in M"' s"'. 



(3) 



X- 



iPl SPACE PROGRAMS %UMMARY 37-51, VOL. Ill 



189 



'hi 

o. 







Fig. 3. Second-order plots for O3-C2H4 reaction 
at various temperatures 



Table 1 . Summary of rate data for the 0.,-C:iH4 
and O3-C2H2 reactions 



Initial concentrations,* 
M X 10' 


Proituriiing 
gai" 


Temperotuie, 


k,M '.' 


0, 


C=H, 


o= 


Or 


-CjHt reaction 




0.716 


1.963 


14 


None 


-40 


83.0 


0.626 


1.455 





Argon 


-48 


44.0 


0.892 


1.980 


5 


Helium 


-57 


27.0 


0.544 


1.912 


9 


None 


-65 


26.0 


1.057 


3.420 





Argon 


-75 


11.0 


0.728 


4.330 


14 


None 


-80 


9.0 


0.564 


2.909 


9 


None 


-85 


7.5 


0.544 


6.450 





Argon 


-85 


5.3 


0.434 


4.493 


5 


Argon 


-90 


4.2 


0.462 


11.040 





Argon 


-95 


3.0 


0, 


C,H, 


o.. 


0.r-CiHj reaction 


0.308 


1.902 


32 


Argon 


10 


11.8 


0.301 


2.618 


32 


Argon 





6.3 


0.335 


4.980 


32 


Argon 





5.0 


0.510 


6.530 


32 


Argon 


-15 


1.4 


0.355 


2.141 


32 


Argon 


-25 


1.0 


0.463 


7.000 


32 


Argon 


-25 


0.8 


0.435 


26.800 


32 


Argon 


-30 


0.3 


"Concvntratloni of Oa are approxitr 


Ota. 






''PrMiur* opproxlmotvly 1 otm. 









■>e" 



S 2 





1 1 
REF 2 
D REF 1 

O THIS ARTICLE, 

CELL PRESSURIZED 


\\ 


• THIS AR 
CELL ^ 


TICLE, 

lOT PRESSURIZED 




K/ 


— ARRHENIUS 
(REF ?1 


LINE 




\ 


^. 




THIS 
ART 


CLE / 


^ 


s. X 








x 











(I/7-) X 10 3 
Fig. 4. Arrhenius plot of ethylene data 

The rate data for C2H2 are shown in Fig. 5. In this 
case aerosol formation was not noted, and the rates gave 
good straight Unes in every case. The rate data are also 
summarized in Table 1, and the rate constants are plotted 
in Arrhenius form in Fig. 6. The Arrhenius line from this 
study passes through the room temperature point of Cadle 
and Schadt (Ref. 3), but otherwise agrees very poorly 
with the rate parameters reported by them. From this 
study, the rate parameter.<; are 



log itc,nj = 9.5- 10.8/2.3 RT 



(4) 



190 



JPL SPACE PROGRAMS SUMMARY 37-51. VOL. Ill 



20 



i.e 



o 1.6 
3 



X 
.O, 



14 



1.2 



1.0 
S 



1 










s 


k 










\ 


-30 "C 










N. ^ 


) 










\ 








X 
M 



ICC 


,0 










. 










V 






-25 
-25 




Uerf 


r^ 


'""^^ 



20 30 

', min 



40 



50 



Fig. 5. Second-order and pseudo-first-order 
plots of acetylene data 

b. Reaction stoichtometry. The straight hne relation- 
ships obtained in the rate plots of Figs. 3 and 5 provide 
confirmatory evidence that the reaction stoichiometry 
was very nearly 1:1 because the latter assumption was 
used in the calculations. At high onversions the observed 
rates would have been fairly sensitive to any deviation 
from the assumed stoichiometry, particularly in cases 
where the hydrocarbon excess was not great. In addition, 
in several experiments the hydiocarbon loss was deter- 
mined analytically after the reaction was complete. 
Within an experimental error of about 30%, the results 
agreed with the postulated 1:1 stoichiometry for both 
C2H4 and C.H2. 

4. Discussion and Conclusions 

The most surprising result of this work is the finding 
that the C2H2 ozonation reaction has a much hi<^her acti- 
vation energy and pre-exponential factor than the C2H4 
reaction. As shown in the following paragraphs, this sug- 
gests very strongly that the two reactions are funda- 
mentally dissimilar and do not both involve a 1,3 dipolar 
cycloaddition of O3 to a x-bond of the hydrocarbons. 




(i/r) X 10"' 
Fig. 6. Arrhenius plot for acetylene data 

Rate measurements over a sufficiently wide range of 
temperatures provide an important clue to the nature 
of initial reaction structures because the pre-exponential 
factors derived from such measurements are related to 
activation entropies by the following equation from tran- 
sition state theory (Ref. 4, p. 199): 



A(M-'s-') 



■^(RT)expf 



(f) 



(5) 



for a reaction of molecularity 2. The activation entropy 
AS^ is in turn related to the structure of the transition 
state, so that in some cases a distinction can be made 
between possible structures which are widely different in 
entropy. 

Table 2 shows some possible transition state structures 
for the reactions of O3 with C2H4 and C2H2. The en- 
tropies of each were estimated by assuming that they are 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. ttl 



191 



Table 2. PetsibU transition state structures and 

corresponding estimated A-factors for 

ozonation of C^H^ and C^H.. 



Traniilion itat* •quilibrlum 


Eitimattd 

•nirepy of 

tronilllon ttat» 

atlS'-C, 

glbbi/moU 


A.M't' 


O. + CH, ♦=s 


" o 

o 

_H,C— C— H,_ 




/O" 


\0" 


r o— o— oi: 

O, + C,H, J=i 1 1 

Lh.C — CHi J 


75" 


10"' 


Oj + CHi f^ 


[A] 

.HC = CH. 




69.2' 


10' = 










r o— O— O"]: 

O, + C=H, ^ 1 1 

L HC - CH J 


70.5" 


10'' 


Oj + CjH. 5-! 


/ 

_H 


J 


63.4' 


10'" ■' 










■From S" of tho hydrocorben onolog cyctopftnton*. 




'From $" of Ih* hydrocarbon analog molhylcyclobotono, calculatod by 


group oddi- 


tivify rules. 




''From S" of tho hydrocarbon analog cyclopentono. 




■•From S° of Iho hydrocarbon onolog molhylcyclobolono, colculotod by 


group add!- 


tivity ruloi. 




''From S" of tho hydrocorbon analog n-p«nfan«. 





equal to S" for the hydrocarbons of analogous structure, 
and the corresponding A-factors were calculated from 
Eq. (5). 

Two results from Table 2 should be emphasized. First, 
the experimental A-factor of 10° '' M' s ' for C2H4 is in 
remarkably close agreement ■with the predicted value of 
106.4 j^-i j-i for a cyclopentane-like transition state. This 
provides strong evidence that the initial adduct is indeed 
a five-membered ring, rather than a four-membered ring 
as has sometimes been suggested. The low collision effi- 
ciency of Oa-olefin reactions can be explained on a col- 
lision theory basis in terms of a stiict steric requirement 
for ring formation. 

Secondly, Table 2 shows that the transition state for 
the Os-CaHz reaction cannot have a five-membered ring 
structure because the predicted A-factor of 10'-' M"' s' 



is much lower than the observed 10°'^ M~' s'. Instead, a 
loose, open chain structure similar to n-pentane is re- 
quired to explain an A-factor of the observed magnitude. 

F'rom a steric point of view, the 0.,-C2H4 reaction has 
d collisional efficiency more than 1000 times higher than 
the O3-C2H2 reaction. Nevertheless, the acetylene reac- 
tion is still much slower at ordinary temperatures because 
the activation energy is more than twice as high. The 
high activation energy of the acetylene reaction is con- 
sistent with the postulate of a different reaction mecha- 
nism for this reaction, and the magnitude of the activation 
energy suggests that both 7r-bonds are attacked. 

The question remains as to why the low-energy reaction 
path of the Oi-CaH, reaction is unavailable to the 
O.i-CjH; reaction. The answer does not lie in ring strain 
because the strain energies of cyclopentane and cyclo- 
pentene are only slightly different. Neither can the answer 
be found in terms of a low-activation entropy for forma- 
tion of the cyclopentene-like transition state because, as 
shown in Table 2, the A-factor fer such a process should 
actually be higher than that of the O3-C2H4 reaction. 

Since the energy of a 7r-bond in CoH^ is almost identical 
to the 7r-bond energy in C2H4, there seems to be no way 
of escaping the fact that, from both an energy and an 
entropy point of view, the five-membered ring transition 
state should be as accessible in the C-CjHa reaction as 
it is in the Oa-CjHi reaction. Failure of the reaction to 
proceed in this manner can then only be attributed to 
insufficient energy release from the new bonds that are 
formed. The situation is illustrated schematically in 
Fig. 7. At point A in the C-CaH, reaction, formation -of 
the two new C-O bonds has provided more energy than 
the amount that was required to disrupt the original bond- 
ing in the reactants O,, and C2H4, thus resulting in a 
change to a negative slope of the energy curve along the 
reaction coordinate. On the other hand, point A in the 
O3-C2H2 reaction represents only a point of inflection, 
presumably involving rupture of both w-bonds in CjHj. 
It must be emphasized, of course, that Fig. 7 represents 
only relative energy requirements, and that the reaction 
does not have to pass through point A in order to reach 
point B. 

The presently proposed mechanism for C2H2 ozonation 
is in disagreement with the commonly accepted assump- 
tion that ozonation of acetylenic compounds is analogous 
to the corresponding olefin reactions (Ref. 5). However, 



192 



JPL SPACE PROGRAMS SUMMAkY 37-51, VOL. Ill 



Oj+HC=CH 




03+H2C = CH2 




REACTION ► 

Fig. 7. Schematic representation of reactions of 
O, with acetylene and ethylene 

relatively little work has been done on the acetylene reac- 
tions, and much of that was in the liquid phase. Also, the 
conclusions were based mainly on product analysis, which 
often is insensitive to detailed reaction mechanisms. 

In an earlier report, the rate constant of the O.-C^H, 
reaction was measured in liquid argon at 87.5°K (SPS 
37-49, Vol. IV, pp 273-278). The result was log it = -3.8, 
and it was suggested on the basis of a semi-empirical treat- 
ment of the effect of solvent on reaction rates (Ref. 4, 
p. 409) that the gas phase rate at 87.5°K should be lower 
by a factor of 10'"; i.e., log fc (gas phase) — —5.4. From 
Eq. (3), the extrapolated experimental gas phase value 
at 87.5°K would be \ogk = —5.5, which is in very good 
agreement with the predicted value. 

References 

i. I^Ianst, P. L., ct al., Atmospheric Ozonc-Okfin Reactions. The 
Franklin Inslitute, Philadelphia, Pa., 1955. 

2. Bufalini, J. J., .ind Aii^h-iller, A. P Can. ). Chem., Vol. 43, 
p. 2243. 1965. 

3. Cadle, R. D., and Schadt, C. /. Chem. Phys., Vol. 21, p. 163, 
1953. 

4. Glasstone, S., La'dler, K. J., and Eyring, H., The Theory of Rate 
Processes, pp. 199 and 409. McGraw-Hill Book Co., Inc., 
New York, 1941. 

5. Bailey, P. S., Chem. Rev.. Vol. 58, p 956, 1958. 



C. Prediction of OH Radical Microwave Lambda 
Doubling Transitions Below 1 20 GHz, 

R. L. Poynter and R. A. Beaudef 

1. Introduction 

A number of anomalies has been observed in the 18-cm 
OH interstellar raJio lines. These radio sources appear 
to vary widely in observed properties (Ref. 1). Of 50 or 
more radio sources that have been documented at this 
time, only two appear to be anywhere near "nonnal," as 
defined by the thermally expected absorption line inten- 
sities that would occur at the presumed temperatures in 
interstellar space. The remainder of the OH radio sources 
shows either or both emission and ab^'jrption features, 
frequently in all possible combinations, i his obser\'ation 
indicates tha* the OH radio sources are generally not in a 
state of thermal equilibrium. Several mechanisms have 
been proposed (Refs. 2 and 3) to explain the observations. 
Each mechanism involves, in some way, an excitation 
process coupled with a cascade decay of the molecules 
into the ground rotational state. It has been proposed that 
if such a nonequilibrium distribution of OH molecules 
exists, there should be a finite population of OH in the 
higher rotational states, and that the lambda doubling 
transitions associated with thess rotational states should 
be observable. Zuckerman, Palmer, and PenHeld (Ref. 4) 
searched for the lambda doublets belonging to the lowest 
rotational state, / = 1/2, of the excited -tt,/, electronic 
state, which is 140 cm' higher in energy than the ground 
V,/.. electronic state. Although these transitions had not 
been observed in the laboratory, their location in the 
frequency spectrum had been predicted from a set of 
molecular constants derived from the microwave spec- 
troscopic studies of Dousmanis, Sanders, and Townes 
(Ref. 5). Unfortunately, these constants, based on rela- 
tively few observed lines in the spectrum, predicted a 
position that turns out to be 50 MHz removed from the 
correct value (Refs. 4 and 6). Zuckerman, et al., failed to 
observe these lines for this reason. 

The present research does not resolve the anomalies 
that have been observed by the radio astronomers. It does 
define precisely the higher OH lambda doubling fre- 
quencies where further astronomical searches could be 
made for the purpose of studying the cascade decay 
processes. 

2. Experimental Data 

New measurements have been made of the OH micro- 
wave transitions in the range of 8.2 to 40 GHz. An accu- 
rate fit of these transitions has been achieved with a newly 



Jn SPACE PROGRAMS SUMMARY 37-51, VOL. /(/ 



193 



determined set of molecular constants. The analysis shows 
that there is a second complete set of detectable transi- 
tions belonging to the 'it,/^ state. Two of these predicted 
transitions have been observed. 

Because the low-frequency limit of the spectrometer 
at JPL is 8.2 GHz, the lambda doubling transitions below 
this frequency limit could not be measured directly. 
However, enough higher frequency transitions have been 
observed that, if the / = 3/2, ^Tr^s transition as observed 
by Radford (Ref. 7) is included, a fairly complete analysis 
of the microwave spectrum can be obtained. 

The calculated transition frequencies were obtained by 
exact diagonalization of the molecular Hamiltonian that 
was given by Dousmanis, Sanders, and Townes (Ref. 5); 
the molecular constants that were used in this study are 
essentially those defined by them. However, two centrifu- 
gal distortion constants are nece; ,ary to give a satisfactory 
fit of the experimental data. These are defined by the 
following two equations: 

(S|BL„|n)-<i|B„L„|lI>[l-/(/+l)D/Bs] 

(2:|AL„|n> = (i:|A„L„|ii> [1-7(7+ l)8/Bi] 

Here D represents the e£Fect of centrifugal stretching on 
the internuclear distance and 8 represents the effect of 
rotation on the electronic distribution. Of the eight mo- 
lecular constants required for the lambda transitions, 
three were obtained from the optical OH studies of Dieke 
and Crosswhite (Ref. 8). The lambda doubling transitions 
were inserritive to these three parameters. The remainder 
of these constants were evaluated from the microwave 
spectra by the application of least squares methods. Four 
additional constants A, B, C, and D are required to de- 
:cribe the nuclear hyperfine splittings. Of the four con- 
stants, only one, D, is sensitive to the aF = transitions. 
The aF = ±1 hyperfine transitions depend primarily on 
the other three constants. 



3. Rtsults and Discuttion 

A computer program has been written to perform the 
diagonalization and frequency calculations. This pro- 
gram has been modified to work with a least squares 
program for evaluating the molecular parameters. The 
program includes computation of Einstein A coefiBcients 
and intensities. The accuracy of the present analysis gives 
considerable confidence in predicting other low lying 
lambda doubling transition frequencies. The transitions 
that result from this analysis are given in Table 3, along 



with the Einstein A coefficients for the hyperfine compo- 
nents, and the intensities for an assumed temperature of 
SOCK, which represents normal laboratory conditions. 

These frequencies differ by a significant amount from 
other values that have been reported. The differences 
result (1) from more accurate frequency measuremeiUs, 

(2) from least square fitting the new microwave constants 
that have been obtained using these frequencies, and 

(3) from the use of two centrifugal distortion constants. 

The nuclear hyperfine constants obtained here are in 
excellent agreement with those that Radford (Ref 9) de- 
termined by electron spin resonance methods. In spite 
of this agreement, however, there remain some minor 
deviations between the calculated and observed hyper- 
fine splittings. These deviations do not affect the general 
line predictions to any significant extent, because the 
absorption lines that are well-measured are fitted to 
high accuracy. The AF = transitions that deviate by 
±1.0 MHz have not been measured in this work; some 
doubt exists about the accuracies of these frequencies. 
One suspects that the measurements of these lines may 
be off by as much as ±1.0 MHz, which would be con- 
sistent with the errors observed in the '^3/2, J = 9/2 tran- 
sition frequencies. The minor deviations, ±0.4 MHz, that 
are observed in the aF = ±1 components of the 
"f ./-•, 7 = 3/2 and 7 = 9/2 transitions are caused by very 
small residual errors in the hyperfine coupling constants. 
This point (within the experimental error) has been veri- 
fied at JPL by using the measured frequencies of the 
"TTi/.J = 1/2 and ^7r3/a,7 = 5/2 transitions by Radford.' 
No changes are obtained in the lambda doubling molecu- 
lar parameters. 

The new lambda doubling constants are listed in 
Table 4. The nuclear hyperfine coupling constants are 
those given by Radford (Ref. 5). The predicted and ob- 
served line frequencies for all observed OH lambda doLi- 
bling transitions in the microwave spectrum up to 40 GHi 
are given in Table 5. 

Values of the Einstein spontaneous emission coeif - < 
were calculated for a dipole moment of 1.66 ±0.01 L). 
The A coefficients for the Vs/j, J = 3/2 transitions agree 
with the values reported by Turner (Ref. 10), Carrington 
and Miller (Ref. 11), and Lide (Ref. 12). 

Several additional comments may be made about 
Table 3. Laboratory measurements appear to be feasible 



'H. E. Radford, private communication, Apr. 1968. 



194 



JPL SPACE PROGRAMS SUMMARY 37-51, VOi. Ill 



Tabit 3. Lambda doubling and hypcrfint Iraniitiont' 



J 


F(n 


nn 


rra^iMncYf 
MHi 


A IF, m 


InUnnltv' 




J 


Hf\ 


Fill 


Fraqucncv, 
MHi 


A IF, ff) 


Inltniily'' 


'ir,„ ilata 








'"■,/, (tat* 




1.5 


1.0 


2.0 


1611.844 


1.29 X 10 " 


2.01 X 10 ' 


0.5 


0.0 


1.0 


4660.457 


1.08 X 10 • 


9.23 X 10 • 


1.5 


1.0 


1.0 


1665.403 


7.11 X 10 '■ 


1.11 X 10* i 


0.5 


1.0 


1.0 


4750.390 


7.64 X 10" 


6.52 X 10 * 


1.5 


2.0 


2.0 


1667.349 


7.71 X 10" 


1.20 X 10 • 




0.5 


i.c. 


0.0 


4764.990 


3.86 X 10" 


3.29 X 10 ' 


1.5 


2.0 


1.0 


1720.908 


9.42 X 10 " 


1.47 X 10' 




1.5 


1.C 


2.0 


7749.235 


1.87 X 10 " 


1.19 X 10' 


2.5 


2.0 


3.0 


6016.520 


1.09 X 10" 


1.14 X 10" 




1.5 


1.0 


1.0 


7761.329 


9.37 X 10" 


5.97 X 10 ' 


2.5 


2.0 


2.0 


6030.731 


1.53 X 10 • 


1.60 X 10 ' 




1.5 


2.0 


2.0 


7819.650 


1.04 X 10' 


6.59 X 10-* 


2.5 


3.0 


3.0 


6035.059 


1.57 X 10 • 


1.64 X 10 ' 




1.5 


2.0 


1.0 


7831.744 


1.16 X 10 " 


7.36 X 10' 


2.5 


3.0 


2.0 


6049.270 


7.90 X 10 " 


8.26 X 10 ' 




2.5 


2.0 


3.0 


8116.852 


4.25 X 10 " 


1.67 X 10' 


3.5 


3.0 


4.0 


13441.927 


3.40 X 10 " 


2.02 X 10 • 




2.5 
2.5 


2.0 
3.0 


2.0 
3.0 


8135.160 
8188.947 


6.00 X 10" 
6.24 X 10 " 


2.35 X 10' 
2.45 X 10' 


3.5 


3.0 


3.0 


13434.605 


9.17 X 10 • 


5.45 X 10' 




2.5 


3.0 


2.0 


8207.255 


3.14 X 10" 


1.23 X 10' 


3.5 


4.0 


4.0 


13441.374 


9.26 X 10 • 


5.51 X 10 • 




3.5 


3.0 


4.0 


5447.828 


4.41 X 10 " 


8.84 X 10' 


3.5 


4.0 


3.0 


13434.051 


2.64 X 10" 


1.57 X 10 • 




3.5 


3.0 


3.0 


5472.064 


1.21 X 10" 


2.42 X 10' 


4.5 


4.0 


5.0 


23838.799 


7.09 X 10 " 


2.03 X 10' 




3.5 


4.0 


4.0 


5522.693 


1.25 X 10" 


2.51 X 10' 


4.5 


4.0 


4.0 


23817.616 


3.11 X 10' 


8.92 X 10 • 




3.5 


4.0 


3.0 


5546.929 


3.62 X 10 " 


7.26 X 10 • 


4.5 


5.0 


5.0 


23826.634 


3.13 X 10' 


8.96 X 10 • 




4.5 


5.0 


4.0 


194.888 


9.06 X 10" 


7.73 X 10-" 


4.5 


5.0 


4.0 


23805.451 


5.78 X 10 " 


1.66 X 10' 




4.5 


4.0 


4.0 


165.958 


2.46 X 10 " 


2.10 X 10" 


5.5 


5.0 


6.0 


37014.272 


1.19 X 10 • 


1.39 X 10" 




4.5 


5.0 


5.0 


117.905 


8.86 X 10" 


7.56 X 10-" 


5.5 


5.0 


5.0 


36983.501 


7.71 X 10 " 


9.01 X 10 " 




4.5 


4.0 


5.0 


88.975 


7.05 X 10" 


4.07 X 10 " 


5.5 


6.0 


6.0 


36994.485 


7.74 X 10 • 


9.04 X 10 • 




5.5 
5.5 


6.0 
5.0 


5.0 
5.0 


8613.650 
8581.184 


4.09 X 10" 
2.63 X 10" 


1.24 X 10' 
8.00 X 10' 


5.5 


6.0 


-.0 


36963.714 


1.00 X 10 • 


1.17 X 10"' 




5.5 


6.0 


6.0 


8535.274 


2.59 X 10 " 


7,89 X 10 ' 


6.5 


6.0 


7.0 


52759.426 


1.75 X 10 ' 


7.06 X 10 ' 




5.5 


5.C 


6.0 


8502.808 


3.33 X 10" 


1.01 X 10 • 


6.5 


4.') 


6.0 


52721.719 


1.57 X 10 ' 


6.34 X 10 ' 




6.5 


7.0 


6.0 


19597.064 


2.79 X 10" 


2.54 X 10' 


6.5 


7.0 


7.0 


52734.387 


1.57 X 10 ' 


6.36 X 10 ' 




6.5 


6.(1 


6.0 


19561.963 


2.50 X 10* 


2.28 X 10 • 


6.5 


7.0 


6.0 


52696.680 


1.51 X 10 • 


6.10 X 10' 




6.5 


7.V 


7.0 


19517.842 


2.49 X 10 • 


2.26 X 10' 


7.5 


7.0 


8.0 


70886.167 


2.16 X 10 • 


2.a0 X 10 ' 




6.5 


6.0 


7.0 


19482.740 


2.38 X 10" 


2.17 X 10 • 


7.5 


7.0 


7.0 


70843.272 


2.81 X 10 ' 


3.32 X 10 ' 




7.5 


8.0 


7.0 


32955.468 


8.26 X 10" 


1.90 X 10' 


7,5 


8.0 


8.0 


70857.368 


2.81 X 10 ' 


3.33 X 10' 




7.5 


7.0 


7.0 


32918.396 


9.80 X 10" 


2.25 X 10' 


7.5 


8.0 


7.0 


70814.457 


2.08 X 10 • 


2.46 X 10 ' 




7.5 


8.0 


1.0 


32875.772 


9.77 X 10 • 


i.25 X 10' 


8.5 


8.0 


9.0 


91229.499 


3.01 X 10' 


8.88 X 10 • 




7.5 
8.5 


7.0 
9.0 


8.0 
8.0 


32838.700 
48522.791 


7.21 X 10 " 
1.73 X 10" 


1.66 X 10' 
8.52 X 10 " 


8.5 


8.0 


8.0 


91182.618 


4.57 X 10 ' 


1.35 X 10' 




P5 


8.0 


8.0 


48^84.229 


2.62 X 10 ' 


1.29 X 10' 


8.5 


9.0 


9.0 


91197.927 


4.58 X 10 ' 


1.35 X 10 " 




8.5 


9.0 


9.0 


48442.865 


2.62 X 10' 


1.29 X 10 ' 


8.5 


9.0 


8.0 


91151.046 


2.69 X 10' 


7.92 X 10' 




8.5 


8.0 


9.0 


48404.304 


1.54 X 10" 


7.57 X 10 " 


9.5 


9.0 


10.0 


113640.690 


3.68 X 10 • 


2.30 X 10 • 




9.5 


10.0 


9.0 


66149.563 


3 'JO X 10" 


2.69 X 10 " 


9.5 


9.0 


9.0 


113590.676 


6.95 X 10 ' 


4.35 X 10 ' 




9.5 


"^.0 


9.0 


66109.863 


5.66 X 10 • 


5.07 X 10' 


9.5 


10.0 


10.0 


113607.018 


6.96 X 10 ' 


4.35 X 10 • 




9.5 


10.0 


10.0 


66069.56& 


5.65 X 10 ' 


5.06 X 10-' 


9.5 


10.0 


9.0 


113557.003 


3.32 X lO* 


2.08 X 10 ' 




9.5 


9.0 


10.0 


66029 866 


2.70 X 10" 


2.42 X 10 


■EInit 


■In A co«f 


flci«nH, A 1 


f. ff\=Arr. on 


9lv«n In f*. Valwi 


n alon lor th« Vi/>, 1 = 3/3 its 


• or* for 


.emporiioi 


*-j troniitiens obso 


rvod by radio ottror 


omir. 


"Inf. 


:!;iM or* 


or t*mp«rat 


>r« e» JOO'K. 

















for OH lines with intensities larger than 10"' cm *. How- 
ever, Unes this weak are at present marginally detectable 
and will require considerable care if they are to be ob- 
served. This results from the relatively low covicentration 
{3% or less) of OH that can be genciated by present 
methods, and from spectrometer- ^sensitivity, which in this 
case bas been measured as lO"* cm*'. The method of gen- 
eration made use of the well-known H -I- NO, reaction. 

In the 'tti/j state, the transitions are observed to rise 
to a maximum frequency, recede toward zero, and rise 



again. This e£Fect is produced by an inversion in the 
lambda doubUn^ energy levels that occurs between 7=3.5 
and / = 4.5. As / p proaches this inversion point, the 
lambda doubling energy splittings decrease. There are 
no restrictions on level symmetry, so that the traasitions 
for 7 larger than 4.5 are allowed, although they are gen- 
erally weaker than those ot 7 = 3.5 and below. The net 
effect is to produce a sequence of transitions that have 
the appearance of Q, P, and R branches although they 
are not. The low-frequency lines exhibiting the analogous 
effect have been observed in the isotropic molecula).' 



JPl SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



195 



species OD but apparently the effect was overlooked for 
neither comments nor explanation of this feature were 
reported (Ref. 5). The complete spectrum is plotted in 
Fig. 8, with appropriate identi*iration of the 'wo 
'branches" of the -Vi/j state, according to the dir>..lJon 
in which absorption transitions would occur. 

As can be seen from Table 3, the / = 1/2, -iri/-., AF = 0, 
and AF = :tl hyperfine transition frequencies are about 
33 and 42 MHz, respectively, from where Zuckerman, 
et al., attempted to search. Thus, it would seem that their 

Table 4. Molecular constants for OH, assuming 
c = 2.997929 X lO'" cm/$ 



Conilont* 


Vohi*. MHi 


Sigma itale energy, Et — E> 


97979»,I00.0'' 


Rolaftonat constant for signo stotc. Si 


!iO8,478.0^ 


Rotational constant for ^ state, k 


555.066.0'' 


Spin orbit coupling constant. A... 


-4,163,508.0 ± 360 


(i;|«i.|n> 


377,3«2.2 ± 16 


a\{tt + A) l,\l\) 


-1,531,211.0x60 


Centrifugal distortion constr ', 


107.599 ± 0.27 


S 


-44.539 ± 0.12 


X = (A,./«.) 


-7.5009 ± 0.0001 


•As dwi«<d fnm Hiii woA. X oer«« foirir well . 


ritk belli Hie opticel. I»e<. «| end 


electron paramoflnetic resomince (lef. 9) retulti. 




'Kef. 8. 





conclusions about the upper limits of the intensities of 
these transitions must be invalid. Another search would 
be worthwhile for these and other low-frequency OH 
transitions in the interstellar medium. 



References 

1. Robinsun, B. J., and McGee, R. X., Aniiu. Rev. Astron. Astro- 
phys.. Vol. 5, pp. 183-212. 1967. 

2. Cook. A. H.. Nature, Vol. 210. p. 611. 1966. 

3. Litvak. M. M.. et al., Phy-i. Rev. Lett., Vol. 17. p. 821. 1966 

4. Zuckeman. B., Palmer, P.. and Penfield. H.. Nature, Vol. 213. 
p. 1217. 1967. 

5. Dousmanis. G. C. Sanders. T. M., Jr.. and Townes, C. H., 
Phys. Rev., Vol. 100. p. 1735. 1955. 

6. Ban^tt. A. H.. IEEE Trans. Mil. Electron., MIL-8, p. 156. 
1964. 

7. Radford. H. E.. Phys. Rev. Let!., Vol. 13, p. 534. 1964. 

8. Dieke, G. H., and Crosswhite. H. M.. /. Quant. Spec. Rad. 
Transfer, \o\. 2, p. 97. 1962. 

9. Radford. H. E., Phys Rev., Vol. 126, p. 1035. 1962. 

10. Turner. B.. Nature. Vol. 212. p. 184. 1966. 

11. CarrinRton. A., and Miller, T. A.. Nature, Vol 214, p. 998, 
1967. 

12. Lide. D. R., Jr., Naftirc, Vol. 213, p- 694, 1967. 



Table 5. Comparison of observed and calculated frequencies in OH 



Electfonic state 


J 


f;— ff 


Frequency, MHx 


Frequency difference 

(calculated — ebterved), 

MHz 


Experimental 
error limits 


Calculated 


Observed 


"-^.= 


3/2 


J-> 1 


1611.844 


1612.231' 


-0.387 


0.002* 






1 -» 1 


1665.403 


1665.401* 


+ 0.002 


0.002* 






2-»2 


1667.349 


1667.358' 


-0.009 


0.002* 






1 — 2 


1720.908 


1720.533* 


+ 0.375 


0.002* 


'cr, . 


3/2 


1 — ♦ 1 


7761.329 


7760.36* 


+ 0.97 


1.0" 






2->2 


7819.650 


7819.92* 


-0.27 


lO' 


I,- , 


5/2 


2 — 2 


8135.160 


8135.51" 


-0.35 


I.O" 






3 — 3 


8188.947 


8188.94'' 


+ C.007 


1.0' 


'^•.r. 


7/2 


3 — 3 


13434.605 


13434.62 


-0.015 


0.01 






4—4 


13441.374 


:3441.36 


+0.014 


0.01 


'^,n 


13/2 


7 —*7 


19517.868 


19517.55 


+ 0.32 


0.3 






6 — 6 


19561.932 


19562.08 


-0.15 


0.3 


V„ 


9/2 


4—5 


23805.451 


23805.13 


+ 0.32 


0.01 






4— » 4 


23817.616 


23817.64 


-0.024 


0.01 






5-»5 


23826.634 


23826.62 


+ 0.014 


0.01 






5— » 4 


23838.799 


23838.46 


+ 0.34 


0.01 


=^« 


11/2 


*-»5 


36983.501 


36983.47 


+ 0.031 


0.03 






6— »6 


36994.485 


36994.43 


+ 0.055 


0.05 


•Otnerved by Hodferd (Ref. 7). 














■Its estimeted to be levcti ler«er Hion they reported. 







196 



iPl SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



■4- 


1 


+ 


^ 


t= 


fc 


f 


t 

+ 


t 

1 


»: 


): 


>> 




^ 


>- 


tr 




a: 


K 


h- 




Ul 


UJ 


UJ 


7 


2 


5 


f 




2 






>- 


in 


<n 


<n 


7 


z 


z 


O 


o 


o 


H 


l- 


1- 








(/) 


eo 


<o 


7 


z 


z 


< 


< 


< 


ir 


or 


rr 


H 


1- 


H 


UJ 


UJ 




K 


H 




■a 


<. 


<i 


t- 


\- 


1-1 


cn 


Ui 


(/) 


^ 


^ 


Si 


1: 


*> 


t 


rj 


eg 


N 


< 


m 




H 


H 


UJ 


UJ 




en 


U) 


1- 







c 






o 








n 




."t 


lO 




w 
C 
O 






*• 


C\J 




• 




N 


> 




X 





IP 


o 


« 




>- 
u 


S 




z 






UJ 




* 


s 


E 




UJ 






ac 









8 



z 
o 

CO 

d> 



— N 

I I 



A-ilSN3iNI 90T 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



197 



D. An Ion CycSotron Resonance Study of the 

Escape of Helium From the Earth's Atmosphere, 

J. King, Jr., and D. D. E//eman 

1 . Introduction 

Helium atoms are systematically being lost from the 
earth's atmospheie. This conclusion is based on many ion 
pr ' " experiments, primarily by Hale (Ref. 1). For ^he 
I : ate ns to exist in steady-state concentrations in the 

mosf nere, they must be lost at a rate comparable to 
I icir rate of production by radioactive decay (^10" atoms- 
cm"--s''). The fact that this rate is approximately the 
same as for He photo-ionization (Ref. 2) in the upper 
atmosphere suggests that its escape could be explained 
by an ion-molecule reaction mechanism which yielded 
He atoms with adequate kinetic energy. Generally, it has 
been assumed that a dissociative charge transfer reaction 
with N, predominates (Ref. 3). The reaction with N. is 
more probable than that with O2 because of the greater 
abundance of the former in the upper atmosphere. Labo- 
ratory studies of the He*-N2 reaction have been made 
using a crossed-beam technique (Ref. 4). In those colli- 
sions that lead to N'^ production, the process is observed 
to have quasiresonant form. Because of these findings, the 
generally accepted mechanism is the accidental near- 
resonant charge transfer reaction 

He-(^S) + N^'S;)-* He(>S) + N*(C^S;,u = 3,4) 
followed by predissociation 

-»He('S)-^N(*S)-^N*(^P) (1) 

in accordance with the Franck-Condon principle. 

The primary objection to this mechanism is that it 
does not produce He atoms with sufficient energ)' (2.4 eV) 
to escape the earth's gravitational field. This fact has led 
to an alternative mechanism in which the He*, produced 
by solar photo-ionization, charge exchanges with O2 
instead of Nj (Ref. 5). The He^-Oz reaction is exothermic 
by 5.8 eV which, if completely localized in the He frag- 
ment, gives it more than enough energy to escape. 

The basic problem with this latter mechanism has been 
mentioned previously; i.e., Nj is much more abundant 
in the upper atmosphere and asy charge exchange is more 
likely to occur with N2 than with O2. 

A more attractive mechanism is for He'^ to charge 
exchange with Nj in a non-near-resonant process in which 



N J is produced in the ground state (X "DJ ) and the reac- 
tion is exothermic by 9 eV. A test for this mechanism 
is to look for N^ as a stable product since the C ^2; state 
of Reaction (1) is known to predissociate in 10' s (Ref. 6). 

It can be inferred from the spectroscopic studies of 
Inn (Ref. 6) that N!, is produced as a stable ion in He^-Na 
systems. A more direct study has recently been performed 
by Wameck (Ref. 7) using tandem mass spectrometers. 
He concluded that N* and N* are produced with about 
equal efficiency in the system. 

The difficulty in unequivocally determining the N* 
and Nt products in most mass spectroscopic . .periments 
is that these ions are also produced initially by the same 
souice used to ionize the He. Thus, the initial ions must, 
in some way, be differentiated from the product ions. 

2. Experimental Procedure 

The technique of ion cyclotron double resonance 
(ICDR) is ideally suited for selectively studying a par- 
ticular ion-molecule reaction. This method, which has 
been described previously (SPS 37-46, Vol. IV, pp. 205- 
208), involves the simultaneous RF heating of one type 
of ion while a second type is being observed under cyclo- 
tron resonance conditions. When the first type of ion is 
heated with a strong RF electric field, E-^ (t) at mj, large 
changes should occur in the concentrations of the other 
types of ions, provided they are coupled vdth the first 
type through charge transfer. These changes are detected 
with a weak RF electric field, Ei{t) at frequency mi, 
through changes in the intensity of the observed ion 
spectra. The amplitude of the field E2 is modulated and 
the signal at u>i is detected with a phase detector refer- 
enced to the modulating frequency. With this setup only 
those additional ions produced by the RF heating are 
observed. 

3. Results 

The ICDR technique was used to study the production 
of N* and N* in the He*-N2 system when He* is sub- 
jected to RF heating. The ion production was studied 
as a function of He* energy. The He ion energy can be 
varied by varying the amplitude E of the irradiating 
field, E, (t) = E sin wt. The results in Fig. 9 show that 
both N* and N* are produced in the He*-N2 system. 
The ordinate denotes the amplitude of the double reso- 
nance signal that is proportional to the number density 
of N* or N* ions produced by RF heating of He* (SPS 
37-50, Vol. Ill, pp. 231-236). The abscissa is the ampli- 
tude of the irradiating RF field and is proportional to 



198 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. (If 



>- 
I- 



UJ 

H 
Z 



7.0 








/• 








•/ 


/ 


4.0 






/ 






A 


i 


. 9 


3.0 
2.0 


^J&— C^ 






/ 






h 


r 


















30 



40 



10 20 

ION ENERGY, V/m 

Fig. 9. Variation of N^ and N^ production with energy 
of HE* ions (P = 7.5 X 10-^) 



tlie energy of the He* ions. The surprising result in Fig. 9 
is the levehng ofiE of N* production at high He* energies. 
An explanation of this phenomenon is advanced in the 
following paragraphs. 

4. Discussion 

The results conclusively show that both N* and N* are 
produced when He* ions bombard neutral Nj. In order 
for the neutral He atom, produced in the charge exchange 
reaction, to have sufficient kinetic energy to escape the 
earth's gravitational pull, the N* can either be .n the 
B^2; state or the ground state (X=2J). The former state 
is 3.14 eV above the ground state (Ref. 8) and based on 
the difference in ionization potentials between He and 
N^; the reaction leading to the B ''2* state is exothermic 



by approximately 6 eV. As noted earlier, this is more 
than enough to allow He to escape from the upper 
atmosphere. 

The N* production can be explained by Reaction (1). 
The N;,, initially formed in the C^S* states, predissoci- 
ates in 10" s to form N* and N. Because of the limitations 
of the ion cyclotron resonance spectrometer, it is impos- 
sible to observe such a short-lived species. The plateau 
in the N* curve in Fig. 9 shows that the formation of 
the C ''2* state does not continue to increase with increas- 
ing He* ion energy. This could be caused by the fact 
that as the He* velocity increases there is not sufficient 
time for it to form a complex with N, to produce the 
C ''S* state. Since the state is formed by the simultaneous 
ionization of one electron and excitation of another, the 
two species must be in close contact for a reasonable 
length of time. However, the production of N* in the 
ground state is not limited by this requirement since the 
electron can jump from the Na to the He* over relati/o./ 
large distances, similar to the modified stripping mecha- 
nism proposed by Herman, et al. (Ref. 9). 

To understand why N* production becomes constant 
rather than decreases, one must consider the details of 
the RF heating of the He* ion. The ions are heated 
through power absorption from the irradiating RF field. 
The power absorption equation is (SPS 37-50, Vol. HI) 



AW = 



n* e- E- 



Vo 



(2) 



4m (w — o)o)^ + vl 
where 

n* = ion density 
e = charge on the electron 
E = electric field strength 
0) = oscillator frequency 
0)0 — cyclotron frequency of the ion 
vp = collision frequency for momentum transfer 

As can be seen, maximum power absorption occurs at 
resonance when u = wo. However, the experiments are 
performed by sweeping 4he frequency «> from off reso- 
nance, through resonance, and past resonance. When the 
frequency is off resonance, the He* ions are absorbing 
less energy and their velocity is less. Tl^ v can thus form 
complexes with Nj and produce the C^2; state. The 
number of these less energetic ions remains rather con- 
stant as E in Eq. (2) increases. 



jn SPACE PROGRAMS SUMMARy 37-51, VOL III 



199 



To test this hypothesis, experiments with argon (Ar*) 
and neon (Ne*) are being initiated. 

References 

1. Hale, L. C, "Ionospheric Measurements with a Multigrid Re- 
tarding Potential Analysis," Abstract, /. Geophys. Res., Vol. 66, 
p. 1554, 1961. 

2. Nicolet, M., "Helium, An Important Constituent in the Lower 
Exospliere," /. Geophys. Res., Vol. 66, p. 2263, 1961. 

3. Stebbings, R. F., Rutherford, J. A,, and Turner, B. R., "Loss of 
He* Ions in the Upper Atmosphere," Planet. Space Sci., Vol. 13. 
p. 1125, 1965. 

4. St Sbi-es P. F., Smith, A. C. A., and Ehrhart, H., "Dissociati-'e 
Charge Transfer in He*-0: and He*-N: Collisions," /. Chetn. 
Phys., Vol. 39, p. 968, 1963. 

5. Bates, D. R., and Patterson, T. H. L., "Helium Ions in the Upper 
Atmosphere " P/anet. Space Sci., Vol. 9, p. 599, 1962. 

6. Inn, E. C. V., "Charge Transfer Between He* and N.," Planet. 
Space Sci., Vol. 15, p. 19, 1967. 

7. Wamec;., P., "Studies of Ion-Neutral Reactions by a Photoioniza- 
tion Mass-Spectrometer Technique. IV. Reactions of He* and N: 
and Oj," /. Chem. Phys., Vol. 47, p. 4279, 1967. 

8. Herzberg, G., Molecular Spectra and Molecular Structure: 
Volume I, Spectra of Diatomic Molecules, p. 554. D. Van 
Nostrand Co., Inc., Princeton, N. J., Feb. 1963. 

9. Herman, Z., et al., "Crossed-Beam Studies of Ion-Molecule Re- 
action Mechanisms," Discuss. Faraday Soc. (to he published). 



E. Shape of the Magnetosphere, 

G. Atkinson and T. Unti 

As the solar wind passes the earth, it confines the 
earth's magnetic field within a cavity called the magneto- 
sphere. In recent years satellite data have shown the 
cavity to have a shape more complicated than had been 
anticipated. A neutral sheet has been observed, indicat- 
ing that some of the field lines are dragged great distances 
downstream by the solar wind, strongly distorting the 
shape of the cavity and the magnetic field within it. While 
a number of attempts have been made to calculate the 
shape of the magnetosphere, the calculations have failed 
to include the eflFect of the neutral sheet satisfactorily. 
This article reports on the calculations that have been 
performed taking the neutral sheet into account. The cal- 
culations yield possible shapes for the magnetosphere 
that are illustrated in Fig. 10. Each of the shapes is deter- 
mined by the amount of magnetic flux contained within 
the tail portion of the magnetosphere. 

The calculation parallels a previous calculation made 
by Dungey (Ref . 1). The problem of calculating the shape 
of the magnetosphere regarding the solar wind as a par- 
ticle gas is known as the Chapman-Ferraro problem, and 



=5 3 



< 

z 
o 

CO 

z 

UJ 

s 
o 
z 
o 





1 1 


1 
1 


1 1 


1 


- 


/ / 


^ 




CASE C _ 

1 4 55 

2 2.64 

3 2.00 

4 1.35" 

5 0.49 


- 


i> 


'f 




o UPSTREAM LIMIT OF 
RETURN CURRENT 
FROM NEUTRAL SHEET 




IT 




• NEUTRAL 


POINTS 




ORIGIN-^ 

1 1 


Mil 
1 23 4 


1 
5 

1 i 


1 



2- 



-2-10 I 2 3 4 5 

X, NONDIMENSIONAL UNIT 

Fig. 10. Boundary of the two-dimensional 
magnetosphere 

was shown by Dungey to have an exact solution if certain 
simplifying assumptions were made. The most drastic 
simplification was to treat the problem in two dimensions 
only. The other assumptions are: 

(1) The surface of ttie cavity is thin. 

(2) The field is completely screened; i.e., plasma pres- 
sure and momentum in the interior are unimportant. 

(3) Thermal velocities of the streaming particles are 
neglected. 

(4) The particles are specularly reflected at the surface 
of the cavity. 

Using Dungey 's assumptions, a new exact solution to tlie 
Chapman-Ferraro problem was found in which the pa- 
rameters of a neutral sheet are determined along with 
the shape of the field lines. 

Since the problem is two-dimensional, the method of 
complex potentials can be applied. A scalar potential ^ 
and vector potential ^ are introduced, such tli,at the mag- 
netic field H = V^, and "i is the stream function, constant 
on a magnetic line of force. The two-dimensional repre- 
sentation of the magnetosphere will be determined when 
^ is known as a function of x and y. 

The free boundary, formed by magnetic fieH lines 
along which * = 0, is not known as a function of 
z = X + ft/. All that is known is that * = ^ + i* must be 



200 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. /// 



an analytic function of z; but, then, z must also be an cation of Fourier transforms then yields the function z (*). 

analytic function of *. Now, the free boundary in z space Integrations were calculated on the IBM 7094 computer, 

becomes a very simple known boundary in potential The results, reduced to unit dipole, are shown in Fig. 10, 

space, * = <^ + i*. Therefore, it is only necessary to find in which the boundary of the two-dimensional magneto- 

that function z(*) which satisfies the Laplace equation sphere is given for graded values of tail flux C. 
and reduces to the proper boundary conditions in the 
potential plane. To find this function, a conformal trans- 
formation is made that maps the given boundary in * •fer«nc« 

space onto the abscissa in w = u + iv space. An appli- i, DunKey, J. W., }. Gvuphys. Res., Vol. 66, p. 1043, 1961. 



i 

, JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 201 

i 
i. 



PRECEDING PAGE BLANK NOT,FlU*tO, 



N68-37417 



XX. Communications Systems Research 

TELECOMMUNICATIONS DIVISION 



A. Coding and Synchronization Studies: A General 
Formulation of Linear Feedback 
Communications Systems With Solutions, 

5. Butman 

1 . Introduction 

A feedback communication system is a two-way system 
in which the state of a message at the receiver is made 
available to the transmitter. Although the benefits of 
feedback are greatest when the feedback hnk is noise- 
less, Shannon was able to prove (Ref. 1) that it cannot 
be used to exceed the capacity of a memoryless channel. 
It is possible, however, to exceed the capacity of a chan- 
nel uAth memory (Ref. 2). Furthermore, feedback simpli- 
fies the coding and decoding effort and provides a lower 
error than could otherwise be achieved. These consider- 
able advantages are obtained at the expense of the feed- 
back link, which could be put to better use. Oftentimes, 
however, the return path is idle and should be used to 
benefit the forward link. In space applications, a rela- 
tively inexpensive high capacity tip-link could be sacri- 
ficed for a more efficient exploitation of the doum-link 
whose capacity is small due to weight restrictions required 
for take-o£F. 

This article is concerned with linear feedback com- 
munication systems as originally studied by Ellas (Refs. 3 



and 4), later by Green (Ref. 5), and more recently by 
Schalkwijk and Kailath (Ref. 6), Schalkwijk (Refs. 7 
and 8), Schalkwijk and Bluestein (Ref. 9), Omura (Ref. 10), 
and Butman (Ref. 11). The techniques used include the 
Robbins-Monro method of stochastic approximation 
(Ref. 12) used in Refs. 6 and 7, center-of-gravity (Ref. 8), 
Bellman's dynamic programming (Ref. 13) used in Ref. 10, 
directed graphs introduced by Elias (Ref. 4), and Kalman 
filtering (Ref. 14) in Refs. 10 and 11. 

However, none of these techniques are adequate to 
handle the general linear feedback communication prob- 
lem to be considered here. With the exception of Elias' 
work,' they fail to provide the correct approach to the 
noisy feedback problem even in the case of only one feed- 
back iteration. Furthermore, in the case of a white gaus- 
sian noise channel with a noiseless feedback link, where 
all of these techniques have been successfully used, the 
results do not agree completely and the discrepancies are 
not adequately explained. In addition, the techniques 
are applied only after specific linear relationships are 
assumed to hold between the forward and feedback sig- 
nals and between the feedback signals and the receiver's 
estimates of the message. These assumptions represent 
unnecessary constraints which confine the search for the 

'The principle of optimality of dynamic programming used in 
Ref. 10 is not generally applicable to feedback systems. For 
counter examples see Chap. 10 of Ref. 15. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



203 



optimum to a subset of the class of all possible linear feed- 
back codes and allow the possibility of the existence of 
better schemes. 

A complete linear formulation in terms of arbitrary 
linear operations at the transmitting and receiving points 
is presented in Subsection 2 for systems with additive 
noise in both the forward and feedback channels, includ- 
ing noise which is colored and correlated between chan- 
nels. The optimum decision rule is derived in the case 
of gaussi&n noise, and the signal sv lection problem is 
stated for both the forward and feedback signal sets sub- 
ject to an average power constraint on each. The gaussian 
assumption is a convenience since the problem is identical 
for any additive noise and a minimum mean-square error 
receiver. 

Noiseless feedback is considered in Subsection 3, where 
the optimum sequential forms for the forward signals 
and the estimates at the receiver are derived. Also, a 
theorem is stated giving sufficient conditions for achiev- 
ing channel capacity with a double-exponential decreas- 
ing error rate using partially optimum codes. There are 
more than a countable variety of such codes. The effect 
of noiseless feedback on a channel with memory is exam- 
ined in the example of first-order Markov noise. The code 
used, although not optimum, achieves the theoretical 
capacity of the forv/ard channel when the bandwidth is 
infinite and exceeds the theoretical capacity when the 
bandwidth is finite. 

The noisy feedback problem for a system with inde- 
pendent white noise in each channel is treated in Sub- 
section 4, where the optimum code for one feedback 
iteration is determined. Further penetration is algebraically 
unmanageable. However, successive iteration of the avail- 
able result yields a better scheme than the iterative 
scheme suggested by Elias in Ref. 4. In addition, its 
asymptotic behavior is easily found in closed form, 
thereby determining a useful lower bound. This lower 
bound approaches the upper bound for noisy feedback 
for large signal-to-noise ratios in the forward hnk. 

2. Formulation of the Problem 

A linear feedback communication system using a se- 
quence of N signals to transmit a message 6 is illustrated 
in Fig. 1. Each signal is formed by amplitude modulating 
a basic pulse of unit energy and duration \<i W, where W 
is the bandwidth. The pulse is detected by a matched 
filter whose output is the amplitude corrupted by the 
additive noise in the channel. The sequence of amph- 



/-i 



r, ' s, ■¥ n, 



'I -/ ' '"( 



<z> 



"' ^;?,*'/'/ 



Fig. 1 . A linear feedback communication system 

tudes Si,Sj, • • • ,Sx is the code in the forward channel, 
and the sequence r„ra, ■■ ,rif is the set of noisy obser- 
vations. Similarly, the feedback code is the sequence of 
feedback amplitudes Ui.Uj, ■ • • , «w-i which are observed 
by the tra. smitter as u,, Ua, " • • , Vn-i. The process begins 
with s, = gi6 being sent and fj being received. The first 
feedback signal is Mi = bnfi, and it is observed at the 
transmitter as v,. The second signal is now assumed to be 
a linear function of 6 and u,, thus, Sj = g^O + flnUi. In 
general, the ith signal and observation at each point is 
given by 



r,^s, + ni. 



«i ■='2biiri 



i=l,2, ■■ ■ ,N 



V, — Ui +mi , i = 1,2, ■ • ■ ,N — 1 



(1) 



(2) 



(3) 



(4) 



where n^n-^, ■ ■ ■ ,nx and mi,m2, • ■ ■ ,mjf.-i are zero 
mean gaussian random variables representing the addi- 
tive noise in the forward and feedback channels, respec- 
tively. The last feedback signal u^ is not used and is 
therefore not considered. Let A and B he N XN lower 
triangular matrices with the main diagonal of A and the 
last row of B identically zero, and let g, m, n, r, s, u, and v 
be N-dimensional column vectors oi N XI matrices. 



Then 



s = ge + A\ 

r = s + n 
u = Br 
V = u -f m 



(5) 
(6) 
(7) 
(8) 



204 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



9 — K+H 

A 

¥ 



MAXIMUM 
LIKELIHOOD 



©■ 



Fig. 2. Matrix formulation of the feedback 
communication process 



and the system is equivalent to the 
system of Fig. 2. 



jctor feedback 



Equations (5) to (8) mUy be solved for r, s, and u as 
linear functions of the random noise vectors m and n and 
the random variable 6. Thu-, substituting Eq. (6) into 
Eq. (7) into Eq. (8) and the result into Eq. (5) gives 

s = (I - AB)-' (gfl + Am + ABn) (9) 

r = (I - AB)-' {g9 + Am + n) (10) 

and 



u = B(I - AB)-' (ge + Am + n) 



(11) 



where I is the N XN identity matrix. Note that the 
inverse of I — AB exists because the product AB is a 
lower triangular matrix with zeros along the main diag- 
onal, whereupon I — AB must be a lower triangular 
matrix with ones down the main diagonal and det 
(I — AB) = 1. The average energy transmitted in the 
forward and feedback directions is E — E [s''s] and 
E' = E [u''u], respectively, where E [ • ] is the expecta- 
tion operator. The term s''s = tr [s''s] =tr[ss''], where 
tr [ • ] is the trace operator which is invariant under cj clic 
permutabon of the argument, and the superscript T de- 
notes transpose. Since the expectation and trace operators 
commute, it follows that 



where is statistically independent of m and n, a| =E [6'], 
K„ = E [mm''], and K, = E [nn''] are covariance ma- 
trices of the noise, K„„ = E [mn''] is the cross-covariance 
matrix, and 



IV — Kn "r AJVfnn ' '*mn^ "■" AlVmA 



(14) 



The average power used in the forward and feedback 
channels is then 

P = E/T = 2WE/N and P' = 2WE'/{N - 1) , 

respectively, where T = N/2W is the duration of the N 
forward signals and (N — 1)/2W is the duration of the 
N — 1 feedback signals. 

' he optimum decision rule for an equiprobable source. 
Given A, B, and g, the decision rule for minimum error 
is for the receiver to select the message foi /^hich the 
a posteriori probability p{d\r}is a maximum over all the 
possible messages in the message set 0. In general, this 
rule depends on the a priori probability p (9) because by 
Bayes' nile 



p(e\r) = p(t\e) 



P(0) 
P(r) 



(15) 



However, p (6) = i/M is independent of 6 when is a 
set of Af equiprobable points, such as the M uniformly 
spaced points in the interval [ — L,L]. m this case, it is 
equivalent for the receiver to select the message that 
maximizes p{r\0). The vector r is a sufficient statistic for 
estimating 6, another sufficient statistic is 



y = (I - AB) r 
= gfl -1- Am -I- n 



(16) 
(17) 



Since y is a linear function of the gaussian vectors m 
and n, it must be conditionally normal with conditional 
mean E [y|d] = gO and covariance matrix 

E[y-g9)(y-g0r\9]^K 



and 



E = tr [(I - AB)-' (al gg"- + AK„A' + AIU.B'A' -I- ABK^ A' + ABK.B'A'') (I - B''A'-)-'] 



E' = tr [B (I - AB)-' (crjgg' + K) (I - B'A'')-' B'] 



(12) 



(13) 



Sn SPACE PROGRAMS SUMMARY 37-51. VOL. Ill 



205 



Thus, 



p{y\e) = [(2:r)''detK]-^4exp[ - |-(y - gOrK-My - gO)] 



Now, it is obvious that selecting to maximize p(,y\d) is the same as minimi' ing the quadratic form 

(y - gey K-' (y - gd) = (y - g9^)^K-' (y - g^^) + (6- ^,y g'K- g 



(18) 



(19) 



where 



9,= 



g^K-'y 
g'K-'g 



(20) 



can be any point on the real line. Thus, the optimun: 
decision procedure for the receiver is to select the 
Tnaximum-likelihood estimate of 6 as the poiin 6*e& 
which is closest to 5^. 

It can be verified easily that 6s is the minimum-variance 
unbiased linear estimate of 9 given r. Note that Os is 
also a sufficient statistic for estimating 9 at the receiver 
and that 9s is conditionally normal with conditional mean 
E[^s\e]^0 and variance E [^s - Oy\e] = l/g'^K' g. 
Therefore, 

PiOsie) = (^)^xp [- il - ey g] (2i) 



where 



p<,;, = <7Sg^K-'g 



(22) 



is the signal-to-noise ratio E [e']/E [Oy - oy] at the re- 
ceiver after N observations. A quantity closely related 
to 9s is 



Os 



6n poN 



1 + 



PoU 



(23) 



which is the minimum-variance (biased) linear estimate 
of 9 given the vector of observations r. Note that 



E[{es-9y] 



1 + 



PoN 



<EU9s-9y] 



and that 9s is the value of 9 that maximizes p (0 1 r) when 
p (9) is gaussian with zero mean and variance al, whereas 
6s maximizes p(r|d) regardless of the distribution on 0. 



The probability of error given tliat 6 was sent is the 
probability that \6* - 6's\<\6 -6s\ for some 9* =^9. 
Since the nearest neighbor distance is 2L/(Af — 1), the 
condition for an error when is one of the Af — 2 interior 
points of [ — L, L] is 



\9s-6\- 



r 3a; 1 "' 
1 Lw'-iJ 



M 



where ai = U (M + l)/3 (M - 1). Thus, the conditional 
probability of error is 



^=[ p{6s\9) 

yi»v-«li!t/(m-i) 



d6 



— erfc 



(24) 



where 



erfc (x) 



2 C' 



exp( — x')dx 



When 6 is one of the end points ±L, the condition for 
an error becomes ±6^L{M — 2)/(Af — 1), respectively. 
As in this case the conditional error probability is neg- 
ligibly lower; the average and conditional error proba- 
bilities are nearly equal. 

From Eq. (24), it is clear that P, decreases monoton- 
ically with p„.v. Consequently, A, B, and g .<ihould be 
chosen to maximize pos or In (1 + pos) and to satisfy the 
average energy constraints as given by Eqs. (10) and (13). 
Other constraints are not considered here. Conceptually, 
we can extremize the Hamiltonian 



F = ln(l-rp„,v)-AE-,tE' 



(25) 



where X and /i are Lagrange multipliers, by setting the 
derivative of F with respect to each of the total of N' 
unknown elements in A, B, and g equal to zero and solv- 
ing the resulting set of N^ nonlinear equations. Prac- 
tically, this is an extremely difiBcult, if not impossible, task 



206 



JPL SPACE mOGRAMS SUMMAHY 37-51, VOL. Ill 



for N ' -2 unless the feedback channel is noiseless, that is, 
unless the constraint on the feedback energy i. removed 

(m - 0). 

3. Noitclcfs Feedback 

The absence of feedback noise is indicated in the gen- 
eral formulation by the vanishing of m, K,», and Km„. 
llicrefore, K = K„ is independent of A and B, and the 
signal-to-noise ratio p„.v = trl g''K-' g depends only on g. 
The feedback energy £' is no longer a constraint, be- 
cause it can be scaled down fo t^E' for |c| arbitrarily 
small simply by scaling B to cB and A to Ac'. This leaves 
E, which now depends only on the product AB, iniaf- 
fected. 

Define the lower triangular zero-main-diagonal matrix 
C = (I - AB)-' AB. Then 

(I-AB)'=I + C, AB = (I + C)'C 

s = (I + C) g0 + Cn (26) 



and 



E-ari|(I-l-C)glp + tr[CKCn 



(27) 



where || • || is the Euchdian norm, || x ||^ = x'x = tr [xx'']. 
The N {N - l)/2 arbitrary elements of C can be chosen 
to minimize E independently of g and thus independently 
of pojr. Let Q be the lower triangular nonsingular "whiten- 
ing" matrix defined by the factorization Q''Q = K', aid 
letf = (T.Qg, then p„v - !!/!!=. 

Now, the result of the minimization of £ with respect 
to C (Ref. 11) is the functional form of the optimum 
linear coding and decoding operations. Thus, 



Si---gi{6-ei.,) 



(28) 



where 






..)] 



(29) 



is the minimum variance (biased) estimate of 9 given 
r„Ti, ■ ■ ■ ,ri and 



is the signal-to- noise ratio associated with 6,, 

1 + P„i = <Tl/E [{0 - OiY] 

Equation (29) provides a recursive decoding procedure 
for the receiver, since 0* is determined from 

A ^ 

Os "= 6s(l + Pok)/pos 

In addition, it gives a recursive procedure for generating 
the forward signals, because Eq. (28) provides 



Si 



S, 'v 



gi g; 



Gi , 



(31) 



The order of the linear difference equation (Eq. 29) is 
determined by the order of the noise in the forward chan- 
nel, which determines the elements qiiy of the matrix Q. 
Thus, mth-order autoregressive noise is characterized by 
the vanishing of q.i for j <i — m, which reduces Eq. (29) 
to a linear difference equation of order m + 1 . 

Now, the expected forward energy 
where from Eq. (28) and the fact that 

A •" f><i(«-I) 

then 



ei 



g' 



•\ + 



Po(i-l) 



(32) 



Pci=n+n+ ■ ■ ■ + n 



(30) 



is the energy in the tth signal, must be minimized over g 
subject to the constraint al g''K'' g ~ p.* = |i f || ^ This 
leads to the algebraic problem of solving the N nonlinear 
coupled equations 

T^[E-Xln(l + p.v)]=0 
pg< 

where X is a Lagrange multiplier. The transformation 
f = o, Qg does not simplify the problem, and the equa- 
tions are too difficult to solve in closed form except when 
the forward noise is white, or more generally, when 
K = Kn is diagonal A good choice of g, which reduces 
to the optimum choice if K is diagonal, is obtained as 
follows. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



207 



Note that po% as given by Eq. (30) satisfies the identity where 
i 

'"— n('"TT^) ™ 

which from Eq. (32) is 



(34) 



where 



(^y-K-Zfif)' <"• 



Next, let <rj = l/qu, pi = ei/<r! and note from Eq. (32) 
that specifying the energies ei.e^, ■ ■ ■ ,eti determines 
only the magnitudes |g.|,|g2|, • • • ,\gx\- Therefore, 
the sign of gi can be chosen to give 



i^^'-HUm' <-' 



independently of the signal energies. Consequently, 
Eq. (34) becomes 






n (1 + Pi) 



witti equality if and only if K is diagcMial. 



(37) 
(38) 



Now, the choice of signal energies that maximizes the 
lower boimd (Eq. 38), and which is the optimum when 
K is diagonal, is found from 



iZ[--{-^)]- 



/=1 



Thus. 






and e = E/N > a? — w' for all «. Otherwise, it is neces- 
sary to omit the signal corresponding to the largest oi 
and to reduce N until the condition e > a? — a" holds. 
This will not Le necessa:y if £ is sufficiently large or if 
<r? — <r' for all i, in which case e, = e and pi = p = e/a" 
for all t. 



Optimum code for the additive white gaussian noise 
channel. From Eq. (39) and the associated discussion and 
the fact that K = <r* I is diagonal, where tr" = No/2 is the 
two-sided spectral power density of the white nmse, it is 
clear that the optimum choice of signal energies is 
ei = e = P/2W so that Pi = p = P/NoW. Therefore, 
Eq. (34) becomes 



Eq. (32) gives 



l+p,i = (l + pY 



g,=-^[p(l-hp)*-]^ 



(40) 



(41) 



(42) 



Eq. (29) leduces to 

and Eq. (31) gives 

«* = (H-p)^(si-,-Y^n.,) (43) 

where the initial conditions aie Si — giO and Oo = 0. 
The probability of error from Eq. (29) is exactly 
3 (exp2Cr)-l'|^ 



r£ ( exp2Cr)- l'| 
'[2 (exp2RT)-lJ 



^'='^'^if^^m---^\ (^) 



where 



^_ [ln(l + poy)] 
^~ 2T 



Ci = e + ff* — <rf 



(39) 



= Win 



('+w) 



nats/s 



(45) 



208 



JPL SPACE PROGRAMS SUMMARY 37-51. VOL. Ill 



is the theoretical capacity of the channel and R = (ln M)/T 
is the rate of the message source in nats/s. It then follows 
from the asymptotic expansion of the error function inte- 
gral that P, decreases to zero with increasing T for all 
R < C as the doubly exponential function 



-(10 



exp { -(C - R) r - 1.5exp2(C - R) T) 

(46) 



From Eq. (44), it is also evident that P, = erfc(3/2)Vi for 
R = C and P, -* 1 as r-^ 00 for R>C. 

Non-optimum codes. The choice of signal energies, and 
therefore the choice of g, is not critical for achieving the 
doubly exponential decrease of error or even channel 
capacity. Note that P„ as given by Eq. (24), decreases 
to zero if and only if pos increases to infinity, which 
in turn requires that E increase to infinity, because 
1 + £/<r^ ^1 + p„yi^ exp E/<T*. Consequently, we define 
the critical rate cf a code, Re, by the two conditions: 



— = !im \ Pi 



R(.= lim 7r=;ln(l + pox) 



(47a) 



2T 



= lim -rp 



Sln(l + Pi) 
H pi 

i = l 



(47b) 



with equality for additive white gaussian noise, and prove 
the following theorem. 

Theorem 1. If the sequence {pf}"i converges to a 
limit p, then the critical rate Re is given by 



Rc = 



Win 
P 

Wo' 


I'" 


NoW)' 


ifp- 
ifp = 


NoW 



0, 






ifp = 


00 



(48) 



with equality for additive white gaussian noise. The proof 
is in Ref. 11. 

As an example of codes satisfying the conditions of 
Theorem 1, consider the class of codes in which pi = (l/ip, 
where ^ y ^ 1. The optimum code is given by y = 0, in 



which case R^ = Win 2. Otherwise, Re = P/No = 1 and 
the bandwidth is infinite. Although all these codes achieve 
the infinite bandwidth capacity limit, the growth of W 
with T is determined by y. This is illustrated in Fig. 3. 

First-order Markov channel. First-order Markov noise, 
or first-order autoregressive noise, is characterized by the 
first-order linear difference equation. 



fli = an,,^ + Wi , 



l«l<l 



(49) 



where w is white noise with variance cl- = No/2. Station- 
arity implies that £ [ni] = a^ for all i. Therefore, 

a- " a^vr + a% = <tJ,/(1 — a^). 

The elements of K are fc,, = <T^ai'"", and the elements 
of Q are 



2\W 



9o 



jSi, — a8j(,-i) 



for » = / = I 
otherwise 



ffw 



where 8,, is the Kroenecker delta. 



Now, consider the following not necessarily optimum 
code: Take ei=aVi, ei=<jlcp for t^2 so that P/NoW=p 
as N-* 00, and take gi/gi-i = —XiSgna, where 

*. =|gi/gi-.| 




CODING DELAY 
Fig. 3. Bandwidth vs coding delay for a class of codes 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



209 



for i ^ 2. With this choice of g, Eq. (35) gives 



Eq. (32) gives 



3C?+i = 



1 + poi 

1 + Po(i-i) 



which from Eqs. (37) and (50) becomes 



X?., = 1 + p 



i«IV 



(-^) 



(51) 



where the ':tarting value as given by Eq. (32) is 

P(l + P.) 



Xi = (1 - a') ■ 



Pi 



(52) 



We can select pi in Eq. (52) such that x™ = x, where x is 
the only positive stationary point of Eq. (51), in order 
to obtain Xj = x for all i ^ 2. Thus, Eq. (37) becomes 



giving 



1 + P<,x = (l + P.)x""'-" 



Rc = Wlnx' 



where, from Eq. (51) x is related to p by 

_ x^ (x^ - 1) 



and from Eq. (52) 



1 +Pi 



ix+\a\r 



(|«|x + l)^ 



(53) 



(54) 



(55) 



and 1 + p ^ X' ^ 1 + p (1 + I a I )= with the lower bound 
holding for large values of p and the upper bound as p 
tends to zero (W-» oo). Thus, 

IW\nx\ forW<oo(p>0) 

(1 + I«I)V' forW^oo(p^O) 

(56) 



For comparison, the one-way theoretical capacity of 
the first-order Markov channel is 



W 



'"[(rT^^4 '""-(TTm) 



(57) 



(1+|„|)^' 



K,' 



forW 



which shows that Re exceeds the theoretical capacity of 
the forward link for p^ 1/(1 — jaD" since we obtain 



Rc-C=W 

2|a|W 



'"L(l-«-')x^^(x + |a|)^J 



^ for sufficiently large x 

This does not violate Shannon's theorem (Ref . 1) because 
the channel has memory. In fact, knowledge of a is 
equivalent to having additional or side information at 
the transmitter. It is shown by Shannon (Ref. 2) that 
feedback can, in such cases, increase the capacity. 

4. Noisy Feedback 

The case of most practical interest is when all channels 
are corrupted, independently, by additive white noise. 
The optimum output signal-to-noise ratio for N = 2 in 
this case is 



Pf>2 — Pi + PS + 



P1P2P1 



(l + p,)(l+P2) + p'. 



(58) 



where pi and p2 are signal-to-noise ratios of the two for- 
ward signals s, and Sj, and pi is the ratio of the feedback 
signal u,. The optimum allocation of pi and p2 subject to 
pi + P2 = 2p is pi = P2 = p, hence 



P02 — 2p 



pV 



(1 + pr + p' 



(59) 



Unfortunately, a closed form expression for the optimum 
pov is unavailable for N > 2 because of algebraic diffi- 
culties. E wever, the upper bound 



PoN<Np + (N-l)p' 



(60) 



210 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



where 



A A - 1 



was recently proved by Elias (Ref. 4) by means of a rather complicated circuit theoretical argument. Equation (58) is 
also due originally to Elias (Refs. 3-4). A considerably simplified proof of Eqs. (58) and (59), using the matrices A, B 
and the vector g, follows: 

Derivation of Eq. (58). There is no loss of generality in letting Km = K„ = I and <ri = 1 since this converts signal 
energies to signal-to-noise ratios. In particular, for A/ = 2, p, = gi, p' = b^ (1 -|- g;), p^ = (gj + abgy)' + a" (b^ + 1) and 
Po2 = gi' + gj/(l + CI')- Next, let abgi = kg-., where k will be determined shortly. Then 

p.. = gi\{i + ky + (i + p, + p[)~] 

and 



P02 — pi + 



p,p', (1 + kr + [(1 + p,) (1 + p.) + p'l] k' 

The optimum choice of k is that which minimizes the quadratic denominator, thus 



, „ -piPi 



(l + p:)(l + p, + p,) 
and 



Po2 — Pi ^ Pi ^ 



(l+p,)(l + p,)-fp', 

Proof of Elias upper bound p„s — Np + (N — 1) p' . Since K„„ == when the noise is uncorrelated between channels, 
Eq. (14) becomes K = (I + AA'') and hence 

P„iv = g^(I + AAVg 

= iifr 

where f = (I -t- AA'') '-^g. From Eq. (12) and (I - AB)"' = (I -F C), Np = £ is 

2Vp - II (I -f C) g II ' -I- tr [(I -h C)AA'' (I -^ C)"- -H CC] 
= II (I + C) g II = -I- tr [(I + C) (I + AA^) (I + C)] - H 

where the last line is obtained from the fact that tr [C] s=0 and hence 

tr [(I + C) (I -I- C)] = tr [CC + C + C + I] - tr [CC] + N 

JPL SPACE PROGRAMS SUMMARY 37-51, VOL. IN 211 



From Eq. (13) 

(N-l)p'= ||B(I + C)g||^ + tr[B(I + C)(I + AA'')(I f C'')Bn 

Now, define 

M = (I + AAO^MI - B''A'')-' (I + B''B) (I - AB)~' (I + AA^"^ 

then it follows, after cyclic permutation of matrices under the trace operator when necessary, that 

Np + {N - 1) p' = fMf + trM - N 

which is minimal with respect to f when f is the eigenvector corresponding to the smallest eigenvalue of M. Thus, 
without involving the constraint p<,jr = || f || ^, we have 

Np + {N- 1) p' = K,poy + I (X. - 1) 

where Ki—Xz— • ■ ■ Xjv are the eigenvalues of M arranged in increasing order. Elias' result will follow if it can be 
shown that Xi ^ 1 and Ai = 1. This, in fact, has been proven by S. Father of the California Institute of Technology. His 
proof is as follows: 

(I + AA'')^M-' (I + AA'')^4 = (I - AB) (I + B''B)-' (I - B'^A''') 

= (I + B'^B) ' - AB (I + B'^B)^ - (I + B' B) ' B'^A'^ + AB (I + B'^B)^ B'^A'^ 

Next, apply the identities 

B (I + B'^B) ' = (I + BB^)^ B , (I + B'^B) ' = I - B' (I ; BB'^)^ B 

and 

B (I + B'^B) > B*^ = I - (I + BB'^) • 

to the appropriate terms on the right-hand side in order to obtain 
(I + AA'')HM-' (I + AA'')''^ = I - B'' (I + BB'') ' B - A (I + BB'')-> B - B'' (I + BB'') > A"" + A [I - (I + BB'')-i] A'' 

= (I + AA"") - (A + B') (I + BW)' (A'' + B) 
Therefore, 

M • = I - H 
where 

H = (I + AA')-** (A + B') (I + BB')-' (A' + B) (I + AA')-vi 

is obviously non-negative definite. This is sufBcient to prove that Xj (M) ^ 1. However, the rank of H is equal to the rank 
of (A"" + B) which is at most N — 1 because the last row is identically zero. Thus, at least one of the eigenvalues of H 
must be zero and, therefore, Xi (M) = 1. 

212 JPL SPACE PROCRAMS SUMMARY 37-51, VOL. Ill 



A lower bound. A useful lower bound on the output 
signal-to-noise ratio can be obtained by applying Eq. (58) 
iteratively. Since the result of Eq. (58) is indistinguish- 
able at the receiver from that of an equivalent single for- 
ward signal with ratio p<,2, we can apply Eq. (58) to po2, 
P3, and pj to give 



— Po2 + P3 + 



PoiPiPi 



P03-P02^P3 . (l + p^^)(l + p^) + pj 

continuing in this manner yields 

Po (n-l) PnPn-l 



P«n — Po(n-i) + Pn i 



(1 + po(»-i)) (1 + pn) + p'n-i 



(61) 



Next, consider the asymptotic form of pon when p^ and 
p'n are constants p and p', respectively, {pn — pis the opti- 
mum allocation of forward ratios when the feedback link 
is noiseless, p' = oo.) Equation (61) simplifies to 

Pon Po(n-i) "Pi-"-"!"!! I 



hence 

p<Pon — Po(n-i) <pf 1 + Y Jr I 

np < Pon < np ^1 + j-£— ^ (63) 

Substituting tip for p„„ in the right-hand side of Eq. (62) 
gives the inequality 

Pon ~ Po(n-l) > p( 1 + , , 1 

^V-{l + p)[l + [n-l)p]+p'} (^) 

Summing both sides from n = 2 to 2V gives 

Po»-p>p(l + j-^) 

""r-^- ^(iTT) ^" {- p{i+p)+i+p+p''')\ 



Consequently, 

Pox>Np+ ^^,~/^P'P -0(lnN) (65) 

1 + p 

which equals 902 of the upper bound when p = 10, and 
only 102 when p = 0.1. 

The iterative coding procedure represented by Eq. (61) 
gives a better result than the iterative scheme proposed 
by Elias (Ref. 4), in which iV = 2* signals are coded in 
K stages (concatenated in a sense) via Eq. (59) to obtain 



Pon — 2po()4n) + 



(1 + Po(}4n))' + p'h»)-1 



(66) 



The reason why Eq. (61) is better than Eq. (66) is because 
the feedback signals in Eq. (61) convey more information 
than they do in Eq. (66). This can also be verified nu- 
merically; for example, if p = p' = 1, then after N — 2" 
iterations Eq. (61) yields po«/N = 1.496 (max = 1.5) 
while Eq. (66) gives p„j,/N = 1.400. 

There is cause to suspect that the upper bound Eq. (60) 
is too large for small values of p. For instance, if p = 0, 
then Pon =H independently of p'. This suggests that there 
should be a term Hke p/(l + p) multiplying p' in Eq. (60). 
It is, therefore, not unreasonable to conjecture that the 
results of Eqs. (61) to (65) differ only by a negligible 
amount from the truly optimum linear feedback code. 

5. Conclusions 

The utility of the complete formulation of linear feed- 
back systems introduced in Subsection 2 has been dem- 
onstrated in Subsections 3 and 4, where new results and 
results previously obtained by others were derived from 
a unified approach. The derivations of Subsection 3 com- 
prise a proof of the optimum linear noiseless feedback 
coding procedure not previously published. The formula 

Si = gi (9 - fll-i) 

was previously obtained by Omura (Ref. 10) for a channel 
with white noise under the special assumption that 6i 
satisfies a first-order linear difference equation and the 
receiver selects the message 6* closes to 6.,. This decision 
rule is not optimum when $ is uniformly distributed, 
although it is correct when 9 is a gaussian random vari- 
able. The assumption that di satisfies a difference equa- 
tion is not necessary. It serves only to complicate the 
problem, and represents an additional a priori constraint 
on the signal set. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



213 



The feedback scheme described by Schalkwijk in 
Refs. 6-9 which uses signals that in our notation are 
given by 

Si ^ gi(e - e\.,) 

is also linear because 6i is linear in Fj = col(r,, • ■ • ,r,). 
It can be easily derived from the general formulation, 
however, by including the additional linear constraint 
relationship (I — AB)' g = col (gi, 0, • ■ , 0) that must 
hold when s, = gi (0 — di-^) for then 

E[s\9] =col(g, 6,0, • • ,0) 

(Ref. 11). Because of this constraint, the signal-to-noise 
ratio for the additive white gaussian noise channel is at 
most (1 + p — l/N)", which is somewhat less than opti- 
mum linear result (1 + py and, as N-* oo, the ratio 



(1 + p)" 



exp 



(r^) 



Recently Schalkwijk and Bluestein (Ref. 9) pointed out 
that the rate distortion bound can be achieved in the 
case of a gaussianly distributed source by means of the 
noiseless feedback scheme Sj = g. (^ ~ ^i-i). For a uni- 
formly distributed source, one would expect to achieve 
at least as good a signal-to-noise ratio as that of a gaus- 
sian source of equal variance, since the uncertainty 
(entropy) must be less for the uniformly distributed 
source. Schalkwijk and Bluestein suggest the inferior 
scheme Sj = g, (6 — ^, ,) as "appropriate" for the reason 
that 9i is the maximum a posteriori probability (MAP) 
estimate of 6 when 6 is uniformly distributed (perhaps 
in analogy with the fact that $i is also the MAP estimate 
when 6 is gaussianly distributed). 

However, the MAP estimate is not the minimum vari- 
ance (linear or nonlinear) estimate when 6 is not gaussian. 
Moreover, the MAP estimate when 6 is uniformly dis- 
tributed on [ — L,L] is not 9i but the restriction of 5j to 
[ — L,L]; that is, the MAP estimate is 4>i=Bi, for 
|^i|< L, and ^i = Lsgn^, for \^i\> L. The use of $i 
as a feedback signal in Sj — gi(6 — <j>i) takes us into the 
realm of nonlinear feedback, because ^i is clearly a non- 
linear function of tj. The best linear or nonlinear feed- 
back signal with which to minimize the variance and 
hence the transmitted energy is well known (Ref. 16) to 
be the conditional mean E [^jfi]. Indeed, E [9\r] is the 
center-of-gravity proposed earlier by Schalkwijk in Ref. 8, 
but not used for p (9) uniform. 



Unfortunately, E [0|rj] is linear in fi if and only if 8 
is gaussianly distributed. Thus, although it is possible to 
find E[9|ri] in closed form for 9 uniform on [ — L,L], 
it is impossible to express E [(S — E [^Iri])"] in a work- 
able manner. Nevertheless, since E[S|ri] must be in 
[ — L,L], it is reasonable to use ^i as an approximati .i. 
By the same token, the truncated version of ^i, </> = <»< 
for I Sj I < L and Ji = L sgn 9i for | S* | > L can be used. 
It is then easy to show that 



E [{9 - yol <E[{e- 9ir] - E [{9i -%r] < 



"i 



1 + 



Poi 



Similarly, E [{9 - $iY] <E[{9- ^0' - E [(?* - 9'iY]. 
Although this author has not been able to establish an 
inequality between E [{9 - $'i)=] and E [(9 - $,)'], it is 
evident that the nonlinear feedback signals are better 
than the linear signals. 

With colored noise in the forward channel, the intui- 
tive suggestion of whitening the channel and using the 
white-noise code has been made (Ref. 10). This scheme 
would achieve capacity for the whitened (and hence also 
for the colored) channel, but it would not exceed the 
capacity as predicted by Shannon (Ref. 2) and explicitly 
verified in Subsection 3. Furthermore, pre-whitening is 
a "time consuming" operation which, theoretically, re- 
quires infinite delay and therefore gives no opportunity 
for feedback. Actually, the impossibility of a simultane- 
ously time-limited and bandlimited signal (Ref. 17) im- 
plies the nonexistence of even a white-noise channel. This 
gives added importance to the colored-noise problem. 

Round-trip signal delays, measured in units of pulse 
duration, are easily included. If there are k units of delay, 
the first k rows of the lower triangular matrix A vanish. 
The minimum delay, however, is 1 pulse. With k units of 
delay, time division multiplexing will give 1 + p„,f = 
k{l + p/k)" If the pulse duration is increased fc-fold, 
there will be only N/k feedback iterations and hence 
l + Poy = (1 4-?)"^* < it(H- p/k)" for all k>l. How- 
ever, with k separate multiplex channels it is possible to 
send k independent messages each having I + poy — 
{1 + p/ky for a total capacity of kN\n{l + p/k)^ 
Wln(l-l-P/iV„W). 

References 

1. Shannon, C. E., "The Zero-Error Capacity of a Noisy Chan- 
nel," IRE Trar^. on Inform. Theory, Vol. II-2, pp. 8-19, Sep. 
1956. 

2. Shannon, C. E., "Channels v.ith Side Information at the Trans- 
mitter," IBM Journal, Vol. 2, pp. 289-293, Oct. 1958. 



214 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



3. Elias, P., "Channel Capacity Without Coding," Quarterly 
Progress Report, Research Laboratory of Electronics, pp. 90-93. 
Massachusetts Insti'iite of Technology, Cambridge, Mass., Oct. 
15, 1956. 

4. Ellas, P., "Networks of Gaussian Channels with Applications 
to Feedback Systems," IEEE Trans, on Inform. Theory, 
Vol. IT-13, pp. 493-501, July 1967. 

5. Green, P. E., "Feedback Communication Systems," in Lectures 
on Communication System Theory, pp. 345-366. Edited by 
Baghdady. McGraw-Hill Book Co., Inc., New York, 1961. 

6. Schalkwijk, J. P. M., and Kailath, T., "A Coding Scheme for 
Additive Noise Channels with Feedback — Part 1: No- 
Bandwidth Constraint," IEEE Trans, on Inform. Theory, 
Vol. IT-12, pp. 172-182, Apr. 1966. 

7. Schalkwijk, J. P. M., "A Coding Scheme for Additive Noise 
Channels with Feedback— Part II; Band-Limited Signals," 
IEEE Trans, on Inform. Theory, Vol. IT-12, pp. 183-189, 
Apr. 1966. 

8. Schalkwijk, J. P. M., Center-of -Gravity Information Feedback, 
Research Dept. 501. Applied Research Laboratory, Sylvania 
Electronic Systems, Waltham, Mass., May 1966. 

9. Schalkwijk, J. P. M., and Bluestein, L. L., "Transmission of 
Analog Waveforms Through Channels with Feedback," /£££ 
Trans, on Inform. Theory, Vol. lT-13, pp. 617-618, Oct. 1967. 

10. Omura, J. K , "Signal Optimization for Channels with Feed- 
back," Report SEL-66-068. Stanford Electronics Laboratories, 
Stanford, Calif., Aug. 1966. 

11. Butman, S., Optimum Linear Coding for Additive Noise Sys- 
tems Using Feedback, Ph.D. Thesis. California Institute of 
Technology, Pasadena, Calif., May 1967. 

12. Robbins, H., and Monro, S., "A Stochastic Approximation 
Method," Ann. Math. Statist., Vol. 22, pp. 400-^07, 1951. 

13. Bellman, R., Dynamic Pro:;Tamminp. Princeton U' iversity 
Press, Princeton, N. J., 1957 

14. Kalman, R. E., and Bucy, R. S., "New Results in Linear Fil- 
tering and Prediction Theory," Trans. ASME, Ser. D: J. Basic 
Eng., pp. 95-108, Mar. 1961. 

15. Aris, R., Discrete Dynamic Programmivg, Blaisdell Publishing 
Company, New York, 1964. 

16. Blake, I, F., and Thomas, J. B., "On a Class of Processes Aris- 
ing in Linear Estimation Theory," 'FEE Trarxs. Inform. Theory, 
Vol. IT-14, pp. 12-16, Jan. 1968. 

17. Cabor, D., "Theory of Communication," Proc. Inst. Elec. Eng., 
Vol. 93, pp. 429-441, 1946. 



B. Combinatorial Communication: The Maximum 
Indices of Comma Freedom for the High-Data- 
Rate Telemetry Codes, 

L. D. Boumerf and H. C. Romsey, Jr. 

1. Introduction 

The high-data-rate telemetry project (SPS 37-48, Vol. II, 
pp. 83-130) uses the three biorthogonal Reed-MuUer 



codes with parameters (16,5), (32,6), and (64,7). [Param- 
eters (n, k) indicate that the code consists of 2* binary 
n-tuples.] Word synchronization for these codes is pro- 
vided by modulo £ adding a suitable fixed binary n-tuple 
to each code word before it is transmitted. This n-tuple 
is called the comma free vector and the set of transmitted 
words is a coset of the original Reed-Muller code. If this 
coset is such that all possible n-tuples, which could arise 
from erroneous synchronization of the data stream, diflFer 
in at least r symbols from every word of the coset, then 
the coset is said to be comma free of index r. The maxi- 
mum values of r occurring for the high-data-rate telem- 
etry codes are discussed below. 

2. Previous Results 

The maximum index of comma freedom for the Reed- 
Muller (16,5) code is 2. This fact has been known for 
some time and is due to Stiffler. Because of its importance 
for tha high-data-rate telemetry project, the references are 
cited. In his thesis (Ref. 1, pp. 139-143), Stiffler shows 
that the maximum index for the Reed-Muller (16,4) code 
is 2; this implies that the Reed-Muller (16,5) code has 
maximum index ^2. On the other hand, Stiffler (Ref. 2, 
p. 147) provides a comma free vector of index 2 for the 
(16,5) code. 

The maximum index of comma freedom for the Reed- 
Muller (32,6) code is 7. In fact, all comma free vectors 
of index 7 are explicitly determined in SPS 37-46, Vol. IV, 
pp. 221-226. 

3. The Reed-Muller (64,7) Code 

The maximum index of comn'.a freedom for the Reed- 
Muller (64,7) code (call it ht) is unknown Stiffler (Ref. 2, 
pp. 147-156) has established that 14 ^ h, ^ 26 and fur- 
nishes there a comma free vector of index 14. It is shown 
below that 16:^/64^22 for this code. 

Since there are 2" ( = 2"^"' ~ 140,000,000,000,000,000) 
cosets for this code, in contrast with the 2^" ( = 2'^"° — - 
67,000,000) cosets possessed by the (32,6) code, it should 
be no surprise that the basically enumerative techniques 
used (SPS 37-46, Vol. II, pp. 221-226) for that code are 
of no value here. Instead, using Stiffler s comma free vec- 
tor of index 14 as a starting point, a gradient-type com- 
puter search was made on an SDS 930 in the hope of 
finding comma free vectors of higher index. This search 
resulted in the determination of several hundred comma 
free vectors with index 16, but none of index 17 or higher. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



215 



(Of course, the search was far from exhaustive.) One such 
comma free vector of index 16 is 

00001100 00000110 in 11100 looioioo 

11011100 00010000 11000011 00111001. 

4. The Upper Bound 

Let V = V* be the h-dimensional vector space over 
GF(2). Represent the vectors weV as h-hit "words" 
w= {w„i= 1, ■ ■ ■ , /i}. If S is a subset of V, write D (S) 
for the minimum weight ( = number of I's in the vector) 
of the vectors in S; geometrically, D (S) is the distance 
from S to the origin. Also write D (w) for the weight of 
the vector w. 



Let / be the linear operator on V defined by 

(Jtv): = 

(}w), = u>, , , i=l, ■ ■ ■ ,h 

That is, / shifts w one bit to the right and inserts a zero 
in the first bit position. The operator /* shifts w k bits to 
the right and inserts zeros in the first fc-bit locations. / is 
clearly a singular operator (e.g., /* ^0), but by an abuse 
of the notation write / * for the operator which shifts k 
places to the left and inserts zeros in the last k places. 
Finally, let G,. represent the (2", n + 1) bi-orthogonal 
code. The index of comma freedom L,i of the (64,7) code 
can be defined by 



h, = max min D[w + G^, + /* (u; + Ge*) + /*" {w + Ge*)] 



(1) 



J'-cV«4 A;^ 1. ■ • . ,fi;i 



We shall prove that I^ — 22 by considering the case 
fc = 33. The proof proceeds by means of three simple 
lemmas. 

Lemma 1. Let w e V,6, then 

D(i/; + G,e)^6 

and equality holds if and only if every element of u; + do, 
has weight either 6 or 10. 

Proof. This is an elementary consequence of the stan- 
dard Chebyshev argument (Ref. 2, p. 154) which shows 
that 

and equality can occur only if all the vectors in u> + Ci 
have weight 6 or its compliment 16 — 6. 

Lemma 2. Let weVsi, then 

D(w + G„)^ 12 

Proof. The Chebyshev argument shows that 

.32-(32)''4 



D{W + G,,}: 



13.1 • • 



Hence, it is only necessary to show that D(w + Gas) =" 13 
is impossible. Assume that D(w + G32) is odd, then D (w) 



is odd since the vectors in G32 have even weight. Write 
u) — tViW^ where tr, and w^ are 16-bit words and assume 
(by symmetry) that tu, has odd weight and that Wi has 
even weight. There are two cases to consider. First, let 
D (1^2 + Gic) -— 6. It follows from lemma 1 that 
D (W'i + g) = 6 or 10 for any g€ G,6. It also follows from 
lemma 1 and the fact that D (wi + Gie) is odd that 

D(«;, + G,e)^5 

Hence, let g€G,6 be such that D(iVi + g)— 5. Both gg 
and gg are elements of G32 (where g is the compliment 
of g). Thus 

D(u; + G,,)^D(u,' + {gg,gg}) 

^D(u;, +g) + D(u;,+ (g,g}) 

^5 + 6 = 11 

Similarly, if D(w2 + Gie)^4 (the other case) let 
gcG,„ be such that D{w2 + g)^4. Then 

D(u; + G3.)^D(u;+{gg,gg)) 

^D{w, + {g,i}) + D(w, + g) 
^7 + 4 = 11 
This completes the proof of the lemma. 



216 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



Lemma 3. Let G.,, be any 31-bit code obtained from 
G32 by deleting one of its bit locations. Then for any 
u>eV3, 

D(w f GaO^ll 

Proof. The proof is a simple parity argument. Let 
u>' € V:,2 be the vector of odd weight obtained by filling 
in the "missing" bit of w. It follows that 

D{w + G3.) ^D{w' + G,^) ^ 12 

by lemma 2. But since D (tv' + G.,-,) is odd, the lemma 
is proved. 



Theorem, The index o^ comma freedom l^ of the (64,7) 
bi-orthogonal code is at most 22. 

Proof. Let A: = 33 in Eq. (1); then 
/o4 ^ max D [iv + G,, + P' {w + G^,) + /-" {w + G^,)] 



(2) 



^ max D{w + G,, + /'•' Ge, f /" G,,) 



Let Gj be the group generated by the vectors gi = 
(100, • • • ,00)< Vc4 and g,, = P%. Let G be the group 
obtained from } "G^^ by setting the first and thirty-third 
bits of each vector in /"^'G64 equal to zero. Finally, let 
G' = /^^Gr,4. Then 

G, + G + G' C G„4 + P' G„4 + /-^' G„. 

since the vectors x, y are in /" and z is in Gn4, where 

1 2 3 • • • 32 33 34 • ■■ 63 64 

x=100-- 0-- 

i/ = II- 1 1 O- 

z = l 1 1 • • • 1 • • 

Thus, it follows from Inequality (2) that 

l64^ max D(w + G^ + G + G') 

^ max D(uj + {00,01,10,11}) 

+ max D («; + G'„) + max D{w + G3,) (3) 

KtVn v>f Vj, 



where G31 is obtained from G by suppressing the 
1,33,34, • • • ,64 bit positions of G, and Gf,, is obtained 
by suppressing the 1,2, • ■ • , 33 hi' positions of G'. Since 
both G.n and GJ, are groups of the type defined in 
lemma 3, we have by that lenima and Inequality (3), 

h,^0 + 11 + 11 = 22 

This completes the proof of the theorem. 

It seems likely that 22 is the best upper bound for I^ 
that can be obtained by considering a single shift k. For 
example, the distance 22 is attained for k = 17, 47, 31, 
33, 32 and other values of k. To obtain a smaller upper 
bound, it is presumably necessary to consider several 
shifts simultaneously. 

References 

1. Stiffler, J. J., Self-Synchronizing Binary Telemetry Codes, Ph.D. 
thesis. Cahfomia institute of Technology, Pasadena, Cahf., 1962. 

2. Golomb, S. W., et al.. Digital Communications with Space Ap- 
plications. Prentice-Hall, Inc., New York, 1964. 



C. Propagation Studies: A Map of the Venus 
Feature P, 5. Zohar and R. Goldstein 

Radar studies of Venus have shown that there exist on 
its surface relatively permanent topographic prominences. 
These features rotate with the planet and return to radar 
view year after year. Because of the peculiar rotation 
period of Venus, the same features return very nearly to 
the same apparent position at the time of closest ap- 
proach. The feature known as ;3 is the "brightest" and 
hence most favorable to observe at these times. Several 
other features are brighter, but are on the other side of 
the disk and are not presented to view until the radar 
range is much larger. 

The feature /3, as well as the other prominent features, 
was first located by a technique which is sensitive to only 
one-dimension, radar doppler shift (Refs. 1, 2, and 3). It 
has been established that the reflectivity of these features 
at 12.6 cm is significantly stronger than that of the average 
regions of Venus. They also have the ability to depolarize 
microwaves; that is, if right circularly polarized waves 
are beamed toward Venus, the reflections from the fea- 
tures contain a much larger percentage of right circularly 
polarized energy than the surrounding areas. This indi- 
cates that the features are relatively rough to the scale 
of one v.'avelength (12.5 cm). However, it is not known 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



217 



whether the features are mountains or craters or fields of 
boulders or some other such rough formations. 

In order to gain information about the actual size and 
nature of the region j8, it has been studied with a two- 
dimensional technique utilizing both range and doppler 
shift (Ref. 4). The result is a two-dimensional radar map 



of the area. It is a unique map except for a north-south 
ambiguity, i.e., there are two points, symmetric about 
the doppler equator, which have the same values of dop- 
pler shift and range. The results of our earlier studies, 
taken over several conjunctions of Venus, demonstrate 
that the highly reflective areas are actually in the northern 
hemisphere. 




Fig. 4. Area of radar map 



218 



jn SPACE PROGRAMS SUMMARY 37-51, VOL III 



Tilt' ItKiitioii of thf ruiippi'd rt-gion, in relation U) 'he 
overall siirfaco nf Venus, is indicatec) in Fig. 4 by the 
rectiinKl*' in its upper left piirt. The grid of latitude and 
longitude circles show n here represents a frame of refer- 
ence characterized as follows: T)ie Venus rotutinn vector 
pierces the surface at latitnde —90". The zero meridian 
is the location of the s-ib-eartli point at the 19fi7 inferior 
conjunction. 

The map is shown in Fig. 5, where the darker rogiont 
represent areas of significantly higher than average refi'*e- 
ti\ity. This map was obtained as a w^'ighted average of 
17 probings of Venns. utilising the radar capability of 
the Mars deep space station 210-ft antenna. These experi- 
ments Mere conducted between August 12 and Septem- 
ber 11, 1967. Of the three distinct regions shown here, 
the one located at latitude 26 (ft) was covered by 10 of 
17 obsen'ations. The one at latitude 35~ , previously iden- 
tificil as 8, was covered by si.x observations. The third 
region at longitude 40 was covered by three probings 
only and .shot) Id thus he treatt'd with .some reserve, [x ud- 
ing further exiieri mental verification. 

It is the nature of extended rough radar targe's to show 
statistical variation. This is so because very small change-, 
in aspect angle can cause large changes in reflected 




J,^l\^.:l ■■.... 



». 



M 



-40 



-3» -30 

LONGITUDE, tJeg 



i 

-zs 



power, llcnce, avera^; -s over many ho'srs oi observation 
are needed to produte reliable radar maps. 

Some of the observations have shown a detailed stnic- 
titre for region ft. However, the relatively high noise asso- 
ciated with these observations precludes their use in a 
single observation map. 



R»f*r«nt«s 

i. OoIikU'Tn, R M., "IVIimitmri' VVmi.ii Railar Results," Hadlo 

Srii-nri-, p. 182.1, 1S6.5. 

2 Cnrpi-ntfr. B. L., Astrim. /.. Vol. 71. p 142. IMfl. 

.1. CoWstfin, R. M, Moim and PUmrt.!, pp. 12e-131. Etiited by 
J'nifM^ir \. Diillfii!., Ni)rlli-Holl;inil I'liblisliinR Co., Amsterdam, 
Jhr Netlifrlnmls. lf*67. 

4. Miilili'iiian, 1). 0-. IJolilstHn. R.. and Carpfntcr, R., "A Bcview 
of RaHiir Astmnnmy." IEEE Sprrtnim, Oct.. Nov. 1995. 



D. Propagation Stud:es: Thn Variance of 
Scattering-Law Estimates, o G. Kellf 

1 . Introduction 

If {x,} is a random process representing radar echoes 
from tile surface of a planet, it is known (Ref. 1) that tht 
power spectral density P{f) of the process can be ex- 
pressed in terms of the backseat ter function F{0) (ihc 
ability of the surface to reflect back to the observer a 
signal striking it at angle ^). The relation is 



/•it/s 
P{f}= j F(0) sin (a- sin- - /-) '* dB 



(1) 



where fl(0<fl< 1} is tin rotatii;n constant, which can 
be defined here as the bandv\idth of the spectrum, divided 
by the niimber of samples per second. Fqtiation (1) ha.s 
been inverted to yield 



rw- 



'f, 



P'{f)if -a^ sin'' ey^df 



(2) 



Equation (2), in turn, makes it [xjssibk- Ui estimate the 
hackscatttr function b,' eKpre.-Jsing /*'(/) in terms of the 
covariunces of theproco.ss and by using familiar est-mates 
for the co\ariances. 



Fij. 5. Radar map of a Vcnui region 



'Rfsiilftit Ri-y arih Assoiiulf. 



jn SPACE PROGRAMS St/MMAItr 37-5 T, VOL. ttt 



219 



In this article, we derive asymptotic expressions and 
upper bounds for the variance of such an estimate. The 
estimate is 

F(e) = N''4a'cose'2tv,Ai2xnXn., (3) 

i - 1 n - 1 

where u;, and A, are constants described in Subsection 2. 
The result is that as first N and then K tend to infinity, 



Inserting the estimates 



T] — iV ^ XnXnt.) 



(11) 



in Eq. (9) and truncating the sum in Eq. (9) leads to the 
estimate 



F{0) = 4a^ case's, A,r, 



(12) 



var F ((9) - CK' N' • ICa" cos= (sin (9)-' P (a sin 8) 



(4) 



f or < e ^ -,7/2, and uniformly for bounded away from 
zero; 



var F (0) ~ DK' N- ' • I6a* tt^ F' (0) 



(5) 



and finally. 



varF(e)^DK'N '-lea^T'-cos^ff' max P-(x) (6) 

^ .• ^ 1 

uniformly for ^ 6 ^ 3r/2. [Here C and D are constants 
describing the asymptotic behavior of Wf, see Eqs. (13) 
and (14) below.] 

2. Definitions and Assumptions 

Let {^j} (/ = 1.2, • ) be a real-valued stationary 
gaussian process, with E{x,) — 0, var (x,) — 1. Let 



r, = r^j = cov (Xn, 3Cn+,) 



(7) 



We further assume that the spectral density F (f), which 
is an even function of /, is continuous on the closed interval 

(-1,1)- 



In terms of the covariances, 



P(/) = l+2 2r;Cos(7r;/) 



(8) 



i = i 



Differentiating and inserting in Eq. (2) yields 



F{e) = 4a^cose 2 A,r, 



where 



A, = /• r sin (tt//) • (f - a'' sin'' e)-% df 

J asin» 



(9) 



(10) 



Equation (3) is more genpral than Eq. (12) because of 
th^ introduction of weight factors Wj. We shall regard 
[w,} as an infinite sequence of real numbers in which the 
terms may depend on K and are zero after the Kth term. 
We make four assumptions about the weight factors: 

(1) w^ = 1 for all K 

(2) (t«i, • • • ,«;«} is a nonincreasing sequence of non- 
negative real numbers for each K 



(3) lim K- 2 ;w>5 = C 

K->oo j-l 



(4) lim K' 2 r' w) = D 

K-*ao J c: 1 



(13) 
(14) 



Here C and D are positive constants. 

Note, for example, that w, = • • — Wk = I, and also 
w,^{K-i+ 1)/K, (/ = 1, • • • , K), satisfy these as- 
sumptions; in the first case, Eq. (3) becomes Eq. (12), and 
in the second case, Eq. (3) is the arithmetic mean of the 
r, and thus tends to the Cesiro sum of the r,. 



3. Estimate of the Variance 

In the language of Toeplitz matrices (Ref. 2), 

F{e) = (N + Ky'XWx'- (15) 

where X represents the vector (xi, • • ■ ,x>+K)and Wisthe 
{N + K.)XiN -i- K) Toeplitz matrix given by 

W», ^{N + K) N-' '4a' cos 9- w,.* A,.* (16) 

(Here we take w., = w, and A-, = A,.) 

We can express W m terms of its 'Toeplitz kernel" w (A) 
(see Ref. 2, pp. 16-19) as follows: If 



w{\)=-(N + K)N-'- 4m' cose 2 w^A*e'*^ (17) 



220 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



then Combining Eqs. (19), (20), and (21) gives 

W.=:(2.)>j\--a-(A)./A (18) ,,rF(^)-.[.(N + K)] ^f'[r{^l)wiy)Jdy (22) 

Applying the results in Ret. 2, pp. 217-218, we obtain If we define 

var F(e) = 2 V A^ (19) A (x) - 2 w,A, cos (tt/x) = 2 tv,A, cos (tt/x) (23) 

where A,, • • , Av.k are the eigenvalues of (N + K)' RW, and replace y by ttx in Eq. (22), we obtain 
R being the covariance matrix given by Rt, = r, ^-. 

A ri 

Now using the results in Ref. 2, pp. 219-220, we find var F (S) - 64^' • a* cos^ ej P- (x) A^ (x) dx (24) 

„> V^ ,, > ,., „> fr,. , .1 , or, since P and A are even functions, 

(N + K)-'\ Aj ~ (27r)-' (N + K) - / [r(x)u;(x)]-dx 

(20) var F (0) -■ 128A/-' a^ cos- 19 / ' P' (x) A^ (x) (ix (25) 



as iV-* 00, where 



y> The results of Eqs. (4), (5), and (6) wall be obtained from 

, ^ _ \ "* ifcr -_ n£ /21) Eq. (25) by asymptotic evaluation of the integral in that 

/ J T- expression. 



fc - -00 



4. Proof of Eq. (4) 

We suppose f) is a nonzero angle. Substituting f = a sin ^ secx in Eq. (10) gives 

/(ir/^')-e 
sin (tt/c sin sec x) sec x dx (26) 

Denote the zeroth-order Bessel functions of the first and second kinds by /o (z) and Y„ (z), respectively. Then Ref. 3, 
p. 30, Eq. (5) gives 

/„ (z) + iVo (z) = -7r~'«2t| "e'-'^-^secxdx (27) 

From this, we get 

A, = ;(^)/n(7r/asinfl) - ;■ / .sin (7r;a sin fl secx) sec xdx (28) 

Integration by parts gives 

/ sin (tt/o sin 6 sec x) sec xdx = (-nja cos 6)'^ cos {■nja) + (Tr/'a sin S) * / cos (tt/o sin sec y) esc" t/ dy (29) 

Furthermore, as fc -> oo , 

/o (Trfca sin 0) cos (Trfcc) = I -jr — ^ ) cos ^irfcx) cos f nka sin ^ - j ) + 0(fc-') (30) 

JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 221 



[Ref. 4, p. 364, Eq. (9.2.1)]. Hence, since A (x) ^ oo as K -*oo, we can write 

A{x)^f{x) + g{x) + hix) 



with 



A 

/ (x) = (2a sin S)'^ N Wj (;)^ cos (tt/x) cos ( Trja sin 9 ~ T) 



1 = 1 

K 



g (x) = — (irO COS SV' > U), COS (ir/x) COS (tt/o) 
K 

hix)—" (tta sin e)-^ \ tc, cos (u/x) / cos (irja sin ^ sec y) csc^ «/ dy 

/ -^ J(r/2)~e 



Now using 



we see that 



/I 1 

cos (tt/x) cos (tt^x) dx = — 8;, 



/ A- (x) dx = (4a sin 0)' \ /u;; cos- 1 7r/a sin ^ — -j ) + o( ^ /tr; | 



And, since cos^ o = (1 + cos2o)/2 and 



A' 



2 iwj - CK= 

we have 



(31) 



(32) 



(33) 



/ AHx)dx^-^^ (34) 

y„ 8a sm ^ 



Now Eq. (3) will follow from Eq. (25), Eq. (34), and The v, are non-negative, and 

rF^{x)A^x)dx^P^{asm6)r A^{x)dx (35) ^ ■ , ^ 

Jo Jo w,= 2lv,, / = 1, •••,K (38) 

On account of Eq. (34), we can prove Eq. (35) by showing 

Inserting this into Eq. (23) gives 
K' I ' A' (x) [P' (x) - P' {a sin e)]dx-^0 (36) 

Now define A(x) =. | «..4*.(x) (39) 

Vi = Wi — Witi , i — 1, • ■■ ,K~ I 

(37) where A* (x) is the same as A (x), except that the Wj do 

u^ = Wj^ not appear, and summation extends to i instead of to K. 

222 JPL SPACE PROGRAMS SUMMARY 37-51, VOL. /// 



-^ (|K% + o(K%),ifx-0 



y (;')''* COS (tt/jc) = < "^ 

i = » \0{K}^), uniformly for x bounded away from 



\0{K^), uniformly for x bounded away from 
/ cos (TT/a sin sec «/) esc"" ydy= 0{j-^) 

J {■w/2)-e 



(40) 



Thus, Eq. (36), which is what we are trying to prove, can be rewritten 

/i K 

K- y^ y^ t), I), /" ' a: (x) a; (x) [P (x) - P^ (fl sin 0)] dx -^ 
Schwarz' mequality implies that Eq. (40) will follow from 

A' 

K ' y^ vJ r A] (xy 1 P"- (x) - P' {a sin 0) I dxT ^ (41) 

And for this, it is sufficient to prove 

K^^r A* (x)^ I P' (x) - P= (a sin e) I dx -^ (42) 

We thus complete the proof of Eq. i4) oy showing Eq. (42). 
We have the following four relations: for Qi^x^l, 

JL^ (K,ifx-0 

) cos(./x)-/ (43) 

'.^^ '^ IO( 1), uniformly for x bounded away from 



(44) 



^ (0,ifx = 

> (/)'^sin(./x) = <J (45) 



(46) 



(Eq. 46 may be seen using integration by parts.) From these, it follows that 

A* (a sin 61) -- / (c sin 6) ^ f- j (a sin 6»)-W' K% 

A*(o)-g(a)-(2,rflcos(?)-^K ) (47) 

A\ (x) = 0(K'^) uniformly for x bounded av/ay from 
a sin 6 and a 

Now let t > be arbitrary and choose 8 > small enough that | P^ (x) — P^ (a sin 6) | < c when | x — a sin fl ] < 8. 
Then write the integral in Eq. (42) as the sum of integrals over the regions 

(0,asin«-8), (asin^ - 8,asin5 + 8), (asin^ + 8,a - 8), (a-8,o + 8), (a + 8,1) 

Examination of each of the five integrals separately reveals that the limit of the left side of Eq. (42) is less than t. This 
completes the proof of Eq. (4). 

JPL SPACE PROGRAMS iUMMAfiy 37-51, VOL. Ill 223 



' '«MMMaMa«MHHVHBnBHHBH«MH«H^Pi 



5. Proof of Eqs. (5) and (6) 

When 61 -^ 0, Eq. (10) becomes 



A.^jT rsin{^if)df 



(48) 



Writing this as the integral from zero to infinity minus the 
integral from a to infinity, and using integration by parts 
on the latter integral, we obtain 



Hence, 



and 



A, --^+0(1) 



A 



(49) 



(50) 



r A^{x)dx = ^Ij V^ujA^DK'-^ 

; = i 

So to prove Eq. (5), it suffices to show 

K •* /" ' A-' (x) I P- (x) - P' (0) I dx -^ 



(51) 



(52) 



Using the same argument as above to dispose of the w,, 
we find that it suffices to prove 



K-^ [ ' A;^ (x)^ I P^ (x) - P (0) I </x -^ 



(53) 



We have 



A 



tK' 



,ifx = 



/0(K), uniformly for x 

bounded away from 

(54) 

Again let e > be arbitrary, choose 8 so that 

|P(x)-P(0)| <e 

when |x| < 8, and examine Eq. (53) as the sum of inte- 
grals over (0,8) and (8,1). The limit of the left side of 



224 



Eq. (53) is thus seen to be le:.s than e; this completes the 
proof of Eq. (5). 



To prove Eq. (6), note that by Eq. (28) we have 



A -^^ 



(55) 



asymptotically and uniformly in 9, since the integral in 
Eq. (28)isO(l). Hence, 



A 

['AMx)t/x^(|^y^^u1Aj 



^DK'--^ (56) 



Thus, 



and Eq. (6) follows from Eqs. (57) and (25). 



{x)A^x)dx^ max F-(x)-DK^-^ (57) 

o^i^i 8 



6. Example 

To illustrate the estimates derived above, we use a 
Venus radar spectrogram obtained on September 30, 1967. 
On that date, the round-trip time of a radar signal from 
Venus was 398 s. Five round-trip runs were made at a 
sampling rate of 235.8 samples/s: a total of N — 469,242 
observations. From these, K = 64 estimated correlations 
r, were computed, and the backscatter function F{$) was 
estimated ior 0^6^ -k/I. The weight factors used were 
the so-called "banning window" 






Evaluating Eqs. (13) and (14) for these u, gives 



C = -^ -4 — 0.0862 



^ = £-^^0-«237 



The bandwidth of the spectrum is 34 cycles/s; expressed 
in terms of the sampling rate, we get a rotation constant 
of = 34/235.8 = 0.1442. 

The backscatter function was estimated for values of B 
between and Tr/2 in increments of 7r/128. We have com- 
puted the values of the estimates in Eqs. (4) and (6) for 
these values, using of course Eq. (5) instead of Eq. (4) 
for e = 0. 

i?l SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



iriHi 



It was found that for 6 = 7r/128, the estimate in Eq. (6) 
is smaller than that of Eq. (4); for all other nonzero values 
considered, Eq. (4) is the better estimate. 

A A 

Table 1 shows values of F(e) and 0(6) ^ [varF(fl)]'4 
for some of the above-mentioned values of 6. The func- 
tion a (0) is taken from Eq. (5) in the case of 6 = 0, and 
from Eq. (4) in all the other cases shown. 

A 

^ Figure 6 is a graph of the three functions F (6) and 
F{9)±a (e) versus the angle 0. 

References 

1. Goldstein, R. M., A Radar Study of Venus, Technical Report 
32-280. Jet Propulsion Laboratory, Pasadena, Calif., May 25, 
1962. 

2. Grenander, U., and Szego, G., Toeplitz Forms and their Appli- 
cations. University of California Press, Berkeley, Calif., 1958. 



49.00 



175 



ISO 



125 



1.00 



0.75 



0.50 



0.2s 





-(48.82) 
-(4847) 
-(48 12) 

.-—{137 

(13.' 

' (I3J 


8) 
14) 
)7) 
















































































































\ 




» 








F{e)- 


<r(8) — 


H 


^ 


.^A 


)) + o-(( 


n 




° 1- 


1 i 


r 3 

r 1 


» 1 


r 9 

r T 


I h 


f } 


r r 
P 7 



3. Luke, Y., Integrals of Bessel Functions. McGraw-Hill Book Co., 
New York, 1962. 

4. Handbook of Mathematical Func-tions. Edited by M. Abramowitz 
and I. A. Stesiin, National Bureau of Standards, Washington, 
D.C., 1964. 



Table 1 . Experimental backscatter function values 
for values of angle 



9, mulliplat 

of7r/32 


M9) 


(r(«l 


a (»J, % 
of F (9) 




48.47 


0.3495 


0.72 




13.78 


0.2051 


1.5 




1.867 


0.08534 


4.6 




0.8086 


0.04295 


5.3 




0.3628 


0.02899 


8.0 




0.1794 


0.02206 


12.3 




0.1374 


0.01752 


12.8 




0.1126 


0.01419 


12.6 




0.07880 


0.01160 


14.7 




0.05281 


0.009507 


18.0 




0.04569 


0.007764 


17.0 




0.03964 


0.006252 


15.8 




0.03232 


0.004886 


15.1 




0.02 '02 


0.003609 


17.2 




0.01074 


0.002382 


22.2 




0.004188 


0.001185 


28.3 




0.000000 


0.000000 


— 



Fig. 6. Experimental backscatter functions vs angle 



E. Corimunications Systems Development: Design 
of One- and Two-Woy High-Rate Block-Coded 
Telemetry Systems, W. C. Lir^dsey 

1 . Introduction 

Previous work (Refs. 1^) has established performance 
characteristics and trends required for the design of one- 
way and two-way, phase coherent, uncoded communica- 
tions systems. More recently, considerable interest has 
developed (SPS 37-48, Vol. II, pp. 83-91) in applying 
known techniques and theories, evolved over the nast 
few years, to the mechanization of block-coded communi- 
cations systems for deep space applications. Such words 
as "high-rate telemetry (HRT)," implying data rates in 
excess of a few thousand bits per second, and "system 
software" are becoming a part of the vocabulary of every 
communications design engineer faced with advancing the 
technology of deep space communications. For example, 
a major objective of the Mariner Mars 1969 missions is 
to obtain television pictures of Mars by applying the 
theory of block coding to the development of a 16,200- 
bit/s telemetry system. The HRT system is a modification 
of the basic digital telemetry system used on Mariners IV 
and V. The primary difference is that the data detection 
process is more efficient. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



325 



Ofl 



Discussed here is the performance of one-way and 
two-way phase-coherent communication systems which 
employ double-conversion superheterodyne phase-loclced 
receivers preceded by a bandpass limiter to track the 
modulation. Such a setup is useful in testing, predicting 
performance, and evaluating the design of such systems 
prior to and after launch. The notation and terms used 
herein are those established in Refs. 1-4. 



2. System Model 

Before we proceed with the analysis, a functional de- 
scription of the system illustrated in Figs. 7 and 8 will 
be given. Briefly, the data to be transmitted is assumed 
to be block-encoded into binary symbols. Each code word, 
say Xj (t), I = 1, • •• , N, to be transmitted, is made comma- 
free (Ref. 5) by adding an appropriate comma-free vector 



DATA 
SOURCE 



BLOCK 
ENCODER 



'*(') 



BIPHASE 
MODULATOR 



iif) 



sAt) 



PHASE 
MODULATOR 



POWER 
AMPLIFIER 



1 5U) I cU) 

k'ig. 7. Transmitter characterization 



RF AND IF 
SECTION 



BANDPASS 
LIMITER 



L^ 



rU) 



r](.f) 



/>(j) 



VOLTAGE - 
CONTROLLED 
OSCILLATOR 



\ , 




SUBCARRIER 
LOOP 




; ' 


























SYMBOL 
LOOP 






WORD 
ACQUISITION 














!'(/) 






















CROSS CORRELATOR 
OR DATA DETECTOR 


" 




DATA USER 












- 

















Fig. 8. Receiver characterization 



226 



JPL SPACE PKOGRAMS SUMMARY 37-51, VO£. Ill 



to facilitate word synchronization at the receiver. The 
code symbols, appearing at the modulator in the form of 
a binary waveform, are used to biphase-modulate a 
square-wave data subcarrier (Ref. 3), say S (t). The modu- 
lated data subcarrier (Refs. 1 and 2), in turn, phase- 
modulates the RF carrier c(t), which is then amplified 
and radiated from the spacecraft or vehicle antenna 
(Ref. 3) as 1(f). 

On the ground, a double-conversion superheterodyne 
phase tracking receiver is used to track the observed RF 
carrier component, thus providing a coherent reference 
for synchronously demodulating the subcarrier. The re- 
ceived signal is denoted by r) (t); see Fig. 8. Due to the 
fact that this reference is derived in the presence of white 
gaussian noise, a single-sided spectral density Nu-2 watts 
per cycle per second, there will exist phase jitter due to 
the additive noise on the down-link (Refs. 1-3) and, if the 
system happens to be two-way locked (Refs. 1 and 2), 
the additive white noise, which is assumed to be white 
gaussian noise with single-sided spectral density of IV,,, 
watts per cycle per second on the up-link, also exerts 
another component of phase jitter. 

In the following discussion, we shall be concerned with 
predicting system performance in both situations. The 
results are extremely useful in designing systems which 
must operate with narrow performance margins (margin 
denoting the number of decibels in excess of the sum of 
the negative tolerances in equipment performance). For 
deep space telecommunication links, the sum of the nega- 
tive tolerances is typically 4 to 6 dB. Experience has 
shown that requiring the design to exceed the sum of the 
negative tolerances is slightly conservative; hence, reduc- 
ing excess margin results in a much "tigliter" or a less con- 
servative design. 

At the receiver (Fig. 8) a subcarrier tracking loop 
(Ref. 3) is assumed to exist for the purposes of providing 
subcarrier sync. In practice, phase jitter also exists on this 
reference; however, this phase jitter may usually be made 
negligibly small by designing a very narrowband sub- 
carrier tracking loop (Ref. 3). Finally, word sync can be 
derived at the receiver by making use of the comma-free 
properties of the transmitted code (Ref. 5). Thus, the nec- 
essary timing infor' Hon is provided for triggering the 
cross-correlation detector in Fig. 8. The output data is 
the recovered bit stream and may be recorded for the 
data user. 

We assume that the code words, Xi {t}, Z = 1, 2, • • ■ , N 
representing sequences of itl's, occur with equal proba- 



bility, contain equal energies, and exist for T = kTi, ~ 2*7, 
seconds. Here, T(, is the time per bit, the reciprocal of the 
data rate <^, T, is the time per code word symbol, and n 
is the number of kits per code word. Thus, the transmitted 
waveform may be represented by 



i (t) - (2P)'^sin [ct + (cos-' m) z, (t)] 



(1) 



where P is the total radiated power, and m is the modula- 
tion factor which apportions the total power between the 
carrier component and modulation sidebands. In Eq. (1), 
the waveform Z; (t) — xi (t) S (t), Z = 1, 2, ■ ■ • ,N, where 
Xi (t) is the code word, in the form of a sequence of il's 
to be transmitted, and S (t) is the unmodulated data sub- 
carrier possessing unit power (Fig. 7). Since S (t) is a 
sequence of ±l's, zi (t) is also a sequence of il's. 

Assuming that the channel introduces an arbitrary (but 
unknown) phase shift 6 to ^ (t) and further disturbs ^ (t) 
by additive white gaussian noise n-.. (f) of s'.igle-sided 
spectral density of N„s watts per cycle single-sided, one 
observes at the input to the receiver (Fig. 8) 



7, (t) = (2P)"^sin [wt + (cos-' m) S; (t) + 6] + n-, (t) 



(2) 



when operating in a one-way locked condition (Ref. 1). 
If the receiver is operating in a two-way locked condi- 
tion (Ref. 1), then the input to the receiver of Fig. 8 is 
taken to be 

,, (t) = (2P)Vi sin [a.f + (cos-> m) z, (t) + 6, + 0] + n. (t) 

(3) 

where 9^ represents phase modulation due to the up-link 
additive noise (Ref. 1), i.e., noise introduced in the space- 
craft transponder. 

In either case, denote the output of the receiver's 
voltage-controlled oscillator by 



r(f)^2''4cos[<>.t + ^.] 



(4) 



where 6-^ is the estimate of the phase of the observed 
carrier component. Multiplying t] (t) by r (t) and neglect- 
ing double frequency terms, it can be shown (Ref. 1) that 
the output y(t) of the receiver's carrier tracking loop, 
which is the input to the data detector, is given by 



y{t) = S''f'Zi{t)cos<j> + T^{f) 



(5) 



JPl SPACE PROGRAMS SUMMARY 37-57, VOL. Iff 



227 



where S == (1 — m-) P, m- - P-/P, P< is the power remain- 
ing in the carrier component at frequency / = o>/2tt; and 
if> is the receiver's phase error, i.e., <^ = S — 62 if one-way 
lock is assumed, and <l> — 6 + 61 — 6j if two-way lock is 
assumed. The probability distribution of the phase error <j> 
is important in determining overall system performance. 
In the next two subsections, we present a model for this 
distribution when bandpass limiters precede the carrier 
tracking loop. 

3. Probability Distribution for the Phase Error 

a. One-way link. To characterize the distribution p, {<j>) 
requires considerable elaboration (beyond the scope of 
this article) on the response (signal plus noise) of a phase- 
locked loop preceded by a bandpass limiter. However, the 
distribution may be modeled on the basis of experimental 
and theoretical evidence given in Refs. 6-8. From these 
references, the distribution for p, (<^) is approximated in 
the region of interest by 



P^iv) 



exp[pLcos<f>] 

2irI„(pL) 



\'t>\<^ 



(6) 



where 



''" N„u>M, ■ r 



/ I + n, \ 

W'f) 



(7) 



and the parameters tVi,„, r„, and fi are defined from the 
closed-loop transfer function 11^ («) of the carrier track- 
ing loop. 



1 + 



HAs)^ 



\2wu, )' 






(8) 



Here, ji is taken to be the ratio of the limiter suppression 
factor oo at the loop's design point (threshold) to the 
limiter suppression, say a, at any other point, i.e., /i = a^/a. 
This assumes that the Biter in the carrier tracking loop is 
of the form (Fig. 8) 



F,(*) = 



1 + T2 « 



in which case 



OoKtI 



(9) 



(10) 



and K is the equivalent simple-loop gain (Ref. 6). The 
subscripts refer to the values of the parameters at the 



loop design point. The parametei Wi,a is defined by 

l + r„ 



WLi, 



H^-£) 



(11) 



The loop bandwidths are conveniently defined by tVi. and 
bi, through the relationship 



Wl'= 



Substitution of Eq. (8) into Eq. (12) yields 



(12) 



Wl = Wlo 



1 + 


ft 


Li + 


''o. 



= 2b,. 



(13) 



The relation Wi,„ = 2fcti, can be defined in a similar way. 
Thus, Eq. (13) becomes 



2fo^ = {2b,„) 



1 + ^ 
t 

-l+foj 



(14) 



This is the usual definition of loop bandwidth employed 
by practicing engineers. The factor T is approximated 
(Ref. 6) by 

1 + 0.345p„ 



0.862 + 0.690p„ 



(1?; 



where p^ is the signal-to-noise ratio at the output of the 
receiver's IF amplifier, i.e., 

2P, 



''" ~ N,.tv„ 



(16) 



The parameter wh is the two-sided bandwidth of the sec- 
ond IF amplifier in the double-heterodyne receiver. In 
one-sided bandwidth notation, Wh = 2fe« and 






(17) 



The parameter p« is also the signal-to-noise ratio at the 
input to the bandpass limiter. 

The remaining parameter to define is the factor /i=ao/a- 
It can be shown that limiter suppression a is given by 

«=(ir(fr-(-f)['"(f)--(f)] 



(18) 



228 



JPL SPACE PROGRAMS SUMMARY 37-51. VOL. Ill 



where /,„ (2), m - 1, 2, is the modified Bessel function of 
argument z and order. To specify a,„ the parameter p,, is 
rewritten as follows; 



Pii 



P, b,.„ P, h,„ 



where 



JV„b„ b,,„ ]V„b,,„ bi, 



Pc 



^ zy 



(19) 



Wob,,,, 






(20) 



In practice, the parameters of the carrier tracking loop 
are specified at the loop design point or threshold. If the 
design point is defined as z„ = y„ = constant, then the 
parameter o„ is given by 

••=(i)'XW-(-f)['<^)-'(f)] 

(21) 

Therefore, it is clear that system performance depends 
upon the choice of y,j. In the Deep Space Network, this 
choice is usually y,, = 2 so that 



z„ 



or, equivalently. 



Pn, 




' {kT°) (b,. 


.) 


p 





(22) 



(kT°)(2b,.,) 



at the design point. Here N„ = kT°, k is Boltzmann's con- 
stant, and T'^ equals the syjtem temperature in degrees 
Kelvin. 

b. Two-way link. In order to characterize the proba 
bility distribution p-^W for the phase error in a two- 
way link, one must consider the up-link parameters and 
the mechanization of the transponder in the spacecraft 
(Bef. 1), As before, the characterization of Pi {ij>) requires 
considerable elaboration (beyond the scope of this article) 
on the response (signal plus noise) of phase-locked loops 
in cascade. Certain theoretical and computer simulation 
results (Ref. 9) are available for explaining the nonlinear 
behavior of loops in cascade. The characterization which 
follows is predicated upon the work reported in Ref. 9 
and that contained in Ref. 1. In the following discussion, 
we introduce the following notation: a subscript "1" refers 
to up-link parameters and constants associated with the 



spacecraft transponder mechanization, while a subscript 
"2" refers to down-link parameters and to constants associ- 
ated with the mechanization of the ground receiver. 

The generic form discussed in Refs. 1 and 9 for p. (<^) is 
given by 



P:'(«^)== 



/qo [ I Pi + P, exp (;<^) I ; 

27r/o(pi)in(p2) 



(23) 



where the definitions of pi and pa follow. The parameter 
Pa equal to pt in Eq. (7) becomes, in the new notation, 



2P,., l/l + r„„N 



and 



''' N„;W;„' T.,i ^r^l 



,K,tI. 



(24) 



Tli 



where the zero subscripts refer to the parameters at the 
loop design point. The parameter Wiu replaces the design 
point loop bandwidth w^u in Eq. (11) and is defined by 



w-,„ = 



1 + r„ 



--(-1^) 



2b.., 



(25) 



when the loop filters are of the form as given in Eq. (9) 
with Ti replaced by 7,2 and T2 by T2-. The parameter Tj is 
defined in Eq. (15) by adding the subscript "2" to all sym- 
bols. Likewise, Eqs. (18) and (21) define the limiter sup- 
pression a-,2 and a„2, respectively, by adding the subscript 
"2" to all symbols and 



Pc, _ ^20 

N„,b,. ' ^'' b„. 



(26) 



In Eq. (26), we have dropped the "L" subscript on bto and 
replaced it by "2." The remaining parameter to define is 
the variable pi, which is given by (Ref. 1) 



2Pe, I 

'" N„,w,„' G'V,K{k,X,P) 



(27) 



where G, the static phase gain of the spacecraft trans- 
ponder, is determined by the ratio of the output frequency 
to the input carrier frequency. The limiter performance 
factor is defined in Eq. (15); however, the parameter p„, 
is now defined by 

^' cl "cl 

'*"' NoiU>„i Noibw, 



iPl SPACE PROGRAMS SUMMARY 37-51. VOL. //( 



229 



where w„i is the two-sided bandwidth of the second IF amphfier in the spacecraft receiver, and bm is the one-sided 
bandwidth. The function K(fc,,^2,/8) is given by 



^ " ^' '^^ " r,„ + 1 [ ki + 2k,fi + 2ik, + k,- k,k,) p' + 2hP' + kip* J 



(28) 



where 



Ml 



2ii.„ 

'•„„ 



Oon 



r„„ = 



*^n{i'^n^2n 



W 



N<i — ^&ni> ^^ 



l + r„ 



(' + — ) 



Plln 



2P,.„ 



with n — 1,2. Now o,„i is defined by either the design 
point in the carrier tracking loops of the tra isponder, 
n = 1, or ground receiver, n = 2, through 

Xexp(-^)[,„(^) + ,,(^)] 

4. System Ptrformancc 

a. Conditional word-error probability. The problem of 
evaluating system performance is described as follows: 
The output of the carrier tracking loop is given by Eq. (5). 
For fc-bit orthogonal codes, the optimum decoder consists 
of 2* cross-correlators whose outputs C (;), / = 1, 2 ■ ■ • 2*, 
are 



C(/) 



ir 



y{t)x,(t)dt 



(29) 



where Tj is the transmission time per information bit. 
Once the set {C (;)} has been determined, the most prob- 
able transmitted word corresponds to that Xy (t) for which 



C (/) is greatest. The output of the decoder will be those 
k bits which, if encoded, would produce this ij (t). 

Since 2* crors-correlators are required to decode a fc-bit 
orthogonal code, the complexity of the decoder becomes 
impractical for k of about 8 or greater. Also, the com- 
plexity of the decoder and the maximum bit rate at which 
the decoder will operate are major factors in the design 
of the decoder. This article does not outl'ne or investigate 
techniques for reducing the decoder complexity or for 
increasing the maximum bit rate at which the decoder will 
operate. The interested reader is referred to material con- 
tained in Koemer (SPS 37-17, Vol. IV, pp. 71-73) and 
Green (SPS 37-39, Vol. IV, pp. 247-252). 

The conditional probability of correct word detection, 
Pr (<t>), is shown (Ref. 3) to be given by 



Pc {k, <i,) = £ ^2;^ cxp ^- yj dx 



where 



An = {2kR„Y^cos4, 



(30) 



(31) 



Rn = 



Non iV„„/>?„ 



and <S?„ = Tj,„.k - number of bits per code word. The 
subscript n - 1 is for one-way lock, while n = 2 implies 
two-way lock. 

For biorthogonal codes of k bits per word, the proba- 
bility of correct reception of a word, conditioned upon a 
particular phase error, is given by Ref. 3 as 



:(fc,*)=£ 



exp 



(-t) 



dx 



(2.)^ 



""[LAM-'ihr ^''^ 



230 



Jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



m m 



where A„ is defined in Eq. (31). The probability of a word 
error, conditioned upon a fixed value of ■, is, of course. 



P,(k,<{,)-1 PriK^) 



(33) 



For convenience, when n = 1 we will drop the subscript 
on A. 

b. Average word- aad bit-error probability. To obtain 
the average word-error probability Pe (k), one averages 
Eq. (33) over the phase-error distribution. Thus, 



PE{k) = l~j%h,i<l>}Pc{^.4>)d^, 



n = l,2 



(34) 



where p, {<j>) is given in Eq. (6) for one-way lock, and 
Pi {<j)) is defined in Eq. (23) for two-way lock. Substitution 
of Eqs (6) or (23) and (30) o. (32) into Eq. (34) yields 
integrals which generally cannot be evaluated analyti- 
cally; however, numerical integration by an IBM 7090 
computer is possible. 

In certain cases of practical interest, the bit-error prob- 
abihty is of importance. For k-hit orthogonal codes, the 
bit-error probability is (Ref. 10) 



PB(k) 



2k- 



2*-l 



PAk) 



while for Jt-bit biorthogonal codes the total bit-error prob- 
abilit> is (Ref. 10) 

(k — l'>2'--' 
P,(^)-P.(fc) + j^(g,,/_l) P.(fc^ 

where P, (k) is given in Eq. (34) for orthogonal codes, and 
Pj (k) is given by 7.q. (34) for biorthogonal codes. 



5. Design Results 

Since the integrals in Eq. (34) cannot be evaluated 
numerically, integration by an IBM 7090 comnuter 
yielded the results, for one-way lock, illustrated in Fig. 9 
for code words containing fc = 6 bits of informatioM. These 
figures depict word-error rates versus the signal-to-noise 
ratio in the data for various values of the signal-to-noise 
ratio X in the design point bandwidth of the carrier track- 
ing loop. Clearly, system performance depends upon the 
choice of a design point yu in the carrier tracking loop. 
For purposes of presentation, the choice is taken to be 
that which corresponds to the design point in the Deep 



^^0(2*^0) 



W 4 



<lr 




2 4 6 

ff'STt/No 



6 I0» 



Fis 9. 
for 



Werd-ciTor probability vs signal-te-neis* ratio X 
various va!<i»s of tho signol-to-noiso ratio x 
\k — 6, on«>way) 



Space Network, i.e., fo = 2, yo = 2, and y = 1/400. Clearly, 
as X approaches infinity, i.e., the case of perfect RF sync, 
the deleterious effects of a noisy phaw reference dis- 
appear and perfect coherent detection is pcssible. 

In the case of two ay lock, system performance for 
/c = 6 bit orthogoiuii codes is illustrated in Fig. 10 for 
f>i = 20 and various values of x^. The same carrier track- 
ing loop design point is used for this ca>;e as was used for 
the one-way lock case. Notice that in this sequence of 
figures as the signal-to-noise ratio in the ground receiyer s 
design point loop bandwidth, x, increases without limit, 
the deleterious effects of the up-link noise introduce an 
irreducible error probabib'ty. This irreducible error de- 
pends upon the amount cf carrier phase jitter introduced 
by the vehicle's carrier tracking loop. This irreducible 



if I SMCE fJrOGJ!AMS %\ilAtAkrf 37-51. VOL. Ill 



231 



«r 



.^ 




4 6 10' 2 4 6 102 

ff2 = ^2 r^2 //Vo2 



4 6 10* 



Fig. 10. Werd-«rroi probability vs signal-to-neit* 

ratio R2 for various values of tho <ignal-to-i:c*s« 

ratio X: (k - 6,pi = 20, two-way) 

error probability can be made arbitrarily small by increas- 
ing the up-iink transmitter power. In fact, it is easy to 
show that the irreducible error probability, say Pir{k), 
is given b> 



J-ir/1 



Piryk) = lim P, 



{k)=2\ p.{i>)d4, 

J t/z 



which is the probabihty that the phase error exceeds 7r/2, 
i.e., P lb [ 1^1 > TT, :]. This says that P;, is independent 
of the code, and it depends only upon the design of the 
carrier tradcing loops, the available power in the carrier 
components, and the channel noise. Th-.., fo/ given chan- 



nel conditions and fixed loop parameters, large transmitter 
output power capability is certainly desirable. 

For fc^5, the performance of a block-coded digital 
communication system using biorthogonal codes is essen- 
tially the same as one that uses orthogonal codes (Ref. 10). 
Henc^ for fc ^ 5, the results presented can be appUed 
to the design of systems whose code dictionaries are 
biorthogonal. 

Roforoncos 

1. Lindsey, W. C, "Optimal Design of One- Way and Two- 
Way Coherent Communication Links," 1E.EE Trans. Commun. 
TechnoL, Vol. COM-14. pp. 418-431, Aug. 1966. 

2. Lindsey, W. C, "Determination of Modulation Indexes and 
Design of Two-Channel Coherent Communication Systems," 
/£££ Trans. Commun. Technol., Vol. COM-14, pp. 229-237, 
Apr. 1967. 

3. Lindsey, W. C, "Design of Block-Coded Communication Sys- 
tems," IEEE Trans. Commun. Technoi., Vol. COM-15, No. 4, 
pp. 525-534, Aug. 1967. 

4. Lindsey, W. C, Performance of Phase Coherent Receivers Pre- 
ceded by Bandpo-3 Ltmiters. Technical Report 32-1162, Jet 
Propulsion Laboratory, Pasadena, Calif., Sept. 15, 1967. Also 
to be published in fE££ Trans, on Commun. Technol., 1968. 

5. Stiffler, J. J., "Synchronization Methods for Block Codes," IRE 
Trans. I ^forrn. Theory, Vol. IT-8, pp. S 25-S 34, Sept. 1962. 

6. Tausworthe, R. C, Theory and Practical Design of Phase- 
Locked Receivers. Technical Report 32-819. Jet Propulsion 
Laboratory, Pasadena, Calif., Feb. 15, 1966. 

7. Lindsey, W. C. and Charles, F. J., A Model Distribution For 
The Phase Error in Second-Order PIiase-Locked Loops. Tech- 
nical P^port 32-1017. Jet Propulsion Laboratory, Pasadena, 
Calif., Oct. 31, 1966. 

8. Charles, F. J., and Lindsey, W. C, "Some Analytical and 
Experimental Phase-Locked Loop Results For Low Signal-to- 
Noise Ratios," Proc. IEEE. Vol. 54, pr. 1152-1166, Sept. 1966. 

9. Lindsey, W. C, and W .er, L. C, the Theory of Auto- 
matic Pha.se Control," in Stochastic Optimization and Cor.trol. 
John Wiley and Sons, Inc., New York, 1968. 

10. Golomb, S., Digital Communications With Space Applications. 
Prentice Hall, Inc., Englewood Cliffs, N. J., 1964. 



F. Communications Systems Development: A 
Digital Demonstration of Sequential Decoding 
and Comparison With Block-Coded Systems, 

p. Sfaneir 

1 . Introduction 

Sequential decoding of tree-coded data is theoretically 
a highly e£Bcient scheme on a wide variety of channeb. 
Specifically, both high information rates (bits per symbol) 
and high data rates (bits per second) may be achieved 



232 



JPl SPACF PROGRAMS SUMMARY 37-51, VOL. Ill 



with low-error probabilities and modest equipment invest- 
ment. Basic information on sequential decoding is con- 
tained in Ref. 1; the interrelations of the physical features 
of a theoretical communications system for the gaussian 
noise case are shown in I'ef. 2. To determine the feasi- 
bility of sequential decoding, using a general-purpose 
digital computer in the role of decoder, and to discover 
realistic operating parameters for such a scheme, an exten- 
sive simulation was conducted using these theoretical 
techniques for the discrete memoryless case. 

This article describes this simulation and compares 
sequential decoding with other schemes that might be 
applied to the same communications system. One such 
scheme is the maximum likelihood decoding of orthogonal 
and biorthogonal block codes at corresponding informa- 
tion rates and on a simulated channel model derived from 
a discrete time version of the gaussian channel within 
5-dB of capacity. For this case, it is shown that sequential 
decoding exhibits an undetected bit-error probability at 
least several orders of magnitude less than that of these 
optimum block codes. 

This advantage is partly offset in a real-time decoding 
system by the appearance of erasures in the output data 
at a rate entirely dependent on the decoder's speed com- 
pared to the data rate. It will be seen that such erasures 
may be recovered simply by increasing the decoder's 
speed, and that for a con.stant erasure lale, a speed in- 
crease of ten times allows a signal energy-to- noise ratio 
decrease of nominally 1 dB. Moreover, even if a reason- 
able and nearly optimum erasure strategy is adopted for 
block decoding, the undetected bit-error probability is a 
function of the block erasure probability and cannot be 
reduced to that observed for sequential decoding unless 
an erasure rate of 50X or more is allowed. The erasures 
from block decoding cannot be recovered without a cor- 
responding increase in undetected bit-error probability. 

2. Comparison of Optimum Systems 

For an arbitrary, binary-input discrete memoryless 
channel with inputs ±1, outputs t/^, 1 ^ fc ^ K, and tran- 
sition probabilities p [y*] ±1], a block code with t code 
words X\ • •• ,X' of block length n can be found for 
which an optimum decoding procedure allowing erasures 
produces errors and erasures which satisfy 



^2-»BVfi) ^p [error] ^A2-^.<«' 



and 



A-'2 "''•<«' ^ p [erasure] ^ ^'2-'' 



'.(«) 



(1) 



(2) 



An optimum decoder minimizes expected error prob- 
ability given equally likely input probabilities for each 
code word. The probabilities in these inequalities are 
given per code word, and such a code has an information 
rate R, defined by R = {\ogt)/n. The constant A does not 
depend on the code, and the exponent functions are 
defined by 



E,iR)= max {£„(p)-pR}, R^R„i, 

< p ^ 1 
-i'„(l)-R, R^^Rori, 

£*(R) = max{E.,(p)-pR} 

0<p 



(3) 
(4) 



where 



£„(p) = -log 



z[y-^ 



[y^.|l]./(..p. 



1 l'*P 



(5) 



The number £» (1) is also called R.omp and the upper and 
lower bounds satisfy £« (R) — E* (R) with equality when- 
ever R^Rerif That rate for which E^R) = E*{R) = 
is channel capacity. Further important rates are defined 
by Rp = E„{p)/p. These exponents and their interpreta- 
tion are thoroughly explained in Refs. 3 and 4. The lower 
bounds are "sphere-packing" and cannot be transgressed, 
while the upper bounds are obtained by random-coding 
arguments and merely assert the existence of codes with 
the prescribed error behavior without exhibiting any. The 
probabihties, p [error] and p [erasure], are code word 
error and erasure probabilities after symbol and word 
synchronization is achieved. 

For the same channel, a tree code of constraint length v 
bits and rate R = i /B, for integer B, can be mechanized 
with V shift registers and B adders (Ref. 2). Viterbi (Ref. 5) 
shows that, if an optimum decoding procedure is used, 
the error probability per bit in a sufiiciently long tree 
code is bounded as 



there 



p [error] ^2-'""'»'P' 



R = Ro 



(6) 



(7) 



Moreover, Yudkin (Ref. 6) has shown that, if the Fano 
algorithm (Ref. 2) is used for sequential decoding, the 
undetected bit-error probability is exponentially upper- 
bounded with the same exponent. Hence, if thvi informa- 
tion rate is Rmmp, Fano's algorithm will produce fewer 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



?33 



than N 2 '' bit errors in decoding a tree of N bits ^oi NB 
channel symbols). For a fixed infonnation rate and a fixed 
channel, Eqs. (1) and (2) show that performance of a 
block-coded system can be improved only by increasing 
block length, a situation which quickly leads to unaccept- 
able impracticalities. For a tree-coded system at the same 
rate on the same channel, performance is improved, ac- 
cording to Eq. (6), by increasing constraint length. T' ere 
is little difficulty in using long constraint lengths in prac- 
tical decoders for tree-coded systems at high information 
r^tes. As an example, doubling the constraint length would 
increase the decoding program used in this simulation 
study by an average of six machine instructions executed 
per branch; whereas for fixed constraint length, lower- 
ing the information rate requires a significantly larger 
memory, but no increase in executed instructions for 
decoding. 

As p increases, the rates Rp decrease. The exponent 
function £*(R) can be obtained as the convex hull of the 
family of straight lines of slope —p which intercept the 
R-axis at Rp. At capacity, o = 0; at R, <,..,(,, p = 1- Geomet- 
rically, it can be seen that optimum tree codes enjoy a 
significant advantage over optimum block codes of the 
same rate, on the basis of error probability per bit. For 
example, if cham:el capacity is \ an optimum block code 
of information rate Rcomp and n = 50, an unbelievably 
enormous number for a practical system, has a bit-error 
rate greater than 2 -', while an optimum tree code of the 
same rate with constraint length 35 and tree size 1024 bits, 
not at all unreasonable parameters, has a bit-error rate 
less than 2 -=. 

3. The "Optimum" Decoders 

In the event that code words are equally likely, a 
maximum likelihood decoder can be shown to be error- 
minimizing. For given code X', •• ■ ,X', each X' of length 
n, a maximum likelihood decoder will produce the code 
word X when Y is received if 



;>[Y|X]>p[Y|X']forallX'^X 



(8) 



Instead of such a decoder, it will be convenient to con- 
sider one which, for fixed a, pioduces X in case 



p[Y|X]>p[Y|X']2-'forallX'^X,a^0 



(9) 



and which has no output if no X satisfies Eq. (9). This 
latter event is called erasure and the constant A of Eqs. (1) 
and (2) becomes then 2"p/"*p*, where p is a solution of 
Eq. (7), and clearly depends on the decoding strategy. 
By using Eq. (9) instead of Eq. (8), extra reliability is 



gained by assuring that X is more likely than its nearest 
competitor by at least some fixed amount, predictable 
ui advance. 

Because ot the various probability assumptions, Eq. (9) 
can be rewritten 



n 

E 



log 4^^ > <T for all X V X, a = 



(10) 



where i/j is the ith coordinate of Y, as are Xi and x' of X 
and X', respectively. Note that if Xi = x'„ the ith summand 
is 0. Since the code words are written in the alphabet 
{±1}, the decoding rule becomes 



E 



x.log-^%^>aforf>llXVX,a^O (11) 



and a mechanization is visualized as in Fig. 11. The era- 
sure parameter a and the strategy in using it will be taken 
up later. 

Two things are of importance here. First, according to 
Eqs. (1) and (2), the only way to improve error perform- 
ance in the optimum case is to increase block size, reduce 
infonnation rate, or increase the erasure threshold. The 
erasures produced by the decoder of Fig. 11 cannot be 
recovered by other changes within that sytem. Secondly, 
the box in that figure labeled "comparison test" contains a 
number of operations which are exponential in block size 
and such a decoder, even as a special-purpose device, will 
limit data rate. Hence, the optimum decoder becomes 
less than optimum in the ordinary meaning of that word. 



/, ^ [/, I I ] 

—^ loo — = — : — r 



n- 



CODE WORD x' 



I',' log 



"[r-H 



COOE WORD X^ 



Ix,^log 



■[-.I-'] 



CODE WORD X' 

r U I 1 1 



[y.b] 



COMPARISON 
TEST 



OEQSION 



Fig. 1 1 . Maximum likelihood decoder block diagram 



234 



in SPACE PftOGKAMS SUMMARY 37-5?. VOL. If) 



To the extent, then, that practical considerations limit 
block size, information rate, and so on, a coding-decoding 
scheme which produces error and erasure statistics satis- 
fying external requirements ought to be the design goal. 



4. Th*or*tical Parameters of Sequential Decoding 

The principal problem in designing a tree-coded 
sequentially-decoded communications system relates to 
the variable decoding time per bit (or block of bits) and 
the relative persistence of long tree searches. Berlekamp 
and Jacobs (Ref. 7) have analyzed this search problem for 
any sequential decoding algorithm and estimate the dis- 
tribution function of the random variable C, which is the 
average number of branches examined in decoding a 
block, as 



p[C>L]^DLi' 



(12) 



where D is a constant depending on the decoder, and p 
is found again by solving Eq. (7). In deriving an expres- 
sion such as Ineq. (12), it is typically assumed that the 
decoder has made no previous mistakes, that searching 
continues as long as required, and that no errors are com- 
mitted. One branch examination is called a computation, 
and, since a computation time depends on the decoder's 
speed and the efBriency of its programming, the data rate 
must be chosen in light of this constraint. 

To operate effectively, a sequential decoder must have 
a si)eed advantage in computation time over data rate; 
i.e., it must be able to search several branch paths to 
modest depth during a single bit time. To accommodate 
longer searches, an effective countermeasure to the vari- 
able decoding time is a temporary storage buffer for 
incoming channel symbols. This would allow consider- 
able searching of likely paths to a significant depth, while 
the speed advantage would allow "catching-up" after a 
difficult portion of the correct path. This problem is dis- 
cussed in Ref. 8. If the decoder lags behind incoming 
symbols further than butter size while searching, subse- 
quent incoming symbols are lost and decoding cannot 
proceed. By comparing Eq. (6) with Ineq. (12), it will be 
seen that this event, viz., buffer overflow, is far more likely 
to occur than an undetected bit error. Since buffer over- 
flow terminates decoding and since such an event will 
eventually occur, some method of restarting decoding 
after overflow must be devised. 

If the data stream is divided into blocks of fixed size 
and a known sequence inserted between blocks, then 
buffer overflow will terminate decoding only within the 
block in which ov( low occurs and the decoder can be 



restarted at the beginning of the next block. The output 
of such a scheme would consist of decoded bits and occa- 
sional blocks of erasures. It is important to note that these 
erasures resulted from the inabihty of the decoder to 
search enough paths in time. Hence, if the incoming data 
were recorded for decoding later, much of the erased 
information could be recovered, nearly error free, by an 
off-Kne decoder which would be given as much search 
time as needed for decoding. No such comparable proce- 
dure is available for block decoding. 

Constraint length for tree codes is defined as the small- 
est number i' such that two branch paths, anywhere in 
the tree, which have a segment of v consecutwe branches, 
anywhere on each path, corresponding to identical seg- 
ments of V information bits will encode subsequent bits 
identically. Therefore, a sequential decoding algorithm 
such as the Fano algorithm will commit an error if it 
accepts a wrong path as most likely for at least one con- 
straint length. If this is the case, the decoder will produce, 
most likely, v bits in error. By varying v, two extreme 
situations become apparent. For very large v, essentially 
no enors are made and the decoder performs as predicted 
by Ineq. (12). For small v, errors are made, but the 
decoder accepts paths more freely and so decodes faster. 
This latter possibility can be used to advantage in the 
real-time scheme combined with off-line decoding of era- 
sures, as outlined above. Again, the design criteria is a 
reasonable output data rate maintained at low-error 
probabilities. 

5. A Channel Model 

The discrete memoryless channel chosen for this simu- 
lation is the quantized version of the binary input, con- 
tinuous output, additive normal noise channel. According 
to Ref. 1, for unquantized outputs such a channel can be 
modeled as a radio channel with inputs a function of time 
s{t) given by 

(2Ex)'^ cos [ o>„t + -^ J for input symbol '. 



s{t) = 



(13) 



I (2£.,)'^ cos I wnt — -^) for input symbol — 1 

for O^t t^- T, where t is the symbol time and £.« is the 
received energy per symbol. On the additive normal noise 
channel with noise level No, the output can be taken to be 



--m 



+ n 



(14) 



JPL SPACE PROGRAMS SMMMARY 37-51, VOL. \\\ 



235 



where n is normally distributed of zero mean and unit variance. For the quantized version, K = 8 is suggested in Ref. 2 
(and is used in this simulation), and transition probabilities p [yk\l], p [yk\ — 1] for l^k^8 are given by 



dx 



pt„,|.l=/;"(2,)-«e«p[-<Lii|^], 

./-! 5+0.5 (Jt-2) L ■^ J 



2^it^7 



(15) 



Ply.\l]^[^ (2^}-''''exp\- 

.'1 .', \_ 



(x + 2E,/No)= 



\dx 



p[!/.|-l]=p[!/..*|l]. l^fc^S 



The basic parameter of the continuous output version is 
the symbol energy-to-noise ratio Eh/Nq. The transition 
probabilities have been tabulated, and a listing for the 
range —5 to —1 appears in Table 2. For information 
rate R, the bit energy-to-noise ratio is given by 



Eb _ £a 

N„ ~ RN„ 



(16) 



and thf^ parameter p is shown as a function of Eg /No for 
fixed rates V2, %, and ¥i in Fig. 12, for transition proba- 
bilities given in Eq. (15). 

Any digital communications scheme applied to the con- 
tinuous model performs a quantization of some kind at 
some point. The point of view adopted here is that a 
quantization is performed ou the channel symbok, and 



Table 2. Transition probabilities for various signal-to-noise ratios per symbol 



ts/N«, dB 


pir ill 


pir=|ii 


Ptyilll 


ply.lii 


Pty.l" 


Piy.m 


ptr:!" 


piy.l" 


-5.0 


0.240 


0.178 


0.197 


0.171 


0.116 


0.061 


0.025 


con 


-4.8 


0.246 


0.180 


0.197 


0.169 


0.113 


0.060 


0.025 


0.010 


-4.6 


0.2S2 


0.181 


0.197 ^ 


0.167 


0.111 


0.058 


0.024 


0.010 


-4.4 


0.259 


0.183 


0.196 


0.165 


0.109 


0.056 


0.023 


0.009 


-4.2 


0.265 


0.184 


0.196 


0.163 


0.107 


0.054 


0.022 


0.009 


-4.0 


0.272 


0.185 


0.195 


0.161 


0.104 


0.053 


0.021 


0.008 


-3.8 


0.279 


0.187 


0.195 


0.159 


0.102 


0.051 


0.020 


0.008 


-3.6 


0.286 


0.188 


0.194 


0.157 


0.099 


0.049 


0.019 


0.007 


-3.4 


0.293 


0.189 


0.193 


0.155 


0.097 


0.047 


0.018 


0.007 


-3.2 


0.30J 


0.190 


0.192 


0.152 


0.094 


0.046 


0.017 


0.007 


-3.0 


0.309 


0.192 


0.191 


0.150 


0.092 


0.044 


0.016 


0.006 


-2.8 


0.317 


0.193 


0.190 


0.147 


0.089 


0.042 


0.016 


0.006 


-2.6 


0.326 


0.194 


0.189 


0.144 


0.086 


0.041 


0.01- 


0.005 


-2.4 


0.335 


0.194 


0.188 


0.1'" 


0.084 


0.039 


0.014 


0.005 


-2.2 


0.344 


0.195 


0.186 


0.139 


0.081 


0.037 


0.013 


C.005 


-2.0 


0.353 


196 


0.184 


0.136 


0.078 


0.035 


0.013 


0.004 


-1.8 


0.363 


0.196 


0.183 


0.133 


0.076 


0.034 


0.012 


0.004 


-1.6 


0.373 


0.197 


0.181 


0.130 


0.073 


0.032 


0.011 


0.004 


— 1.4 


0.383 


0.197 


0.178 


0.126 


0.070 


0.030 


0.010 


0.003 


-1.2 


0.394 


0.197 


0.176 


0.123 


0.067 


0.029 


0.010 


0.003 


-J.0 


0.405 


0.197 


0.174 


0.120 


0.065 


0.027 


0.009 


0.003 



236 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL III 



~ 






^__ . . 


' '' ' 












RATE 1/3 








— 


^ 


3/8_ 

1/2 


^^ 


\^ 


^ 


^ 


1 




^ 




1 


1 





20 24 28 32 36 40 44 

Fig. 12. Theoretical p parameter for 3-bit quantization 

the decoder looks at a discrete memoryless channel. This 
practice minimizes the amount of special-purpose equip- 
ment between the antenna and decoder. While results 
and comparisons in this article are given on the basis of 
a theoretical Eg/No, they have been derived empirically 
from the discrete memoryless channel with the given 
channel transition probabilities. Further applications of 
this work to othnr communications schemes which are 
described in terms of energy-to-noise ratios are valid only 
to the extent that they represent a discrete channel with 
probabilities matching those listed in Table 2. 



6. Simulation Results 

In addition to the channel mod I, the important system 
parameters chosen for the study are: 



Channel model 


8-level qui atized additive normal 




noise 


Information rate 


''3 information bits per channel 




symbol 


Constraint length 


24 information bits 


Block size 


2048 information bits 


Buffer size 


512 information bits 


Coding 


systematic convolutional tree code 



These parameters were chosen as a compromise be- 
tween theoretical virtues of sequential decoding and con- 
ditions imposed by the digital computer (in this case, a 
24-bit octal machine) and are somewhat variable, except 
for information rate. Various constraint lengths in mul- 
tiples of 12 and various block sizes in multiples of 512 
could be used. Just as a starting point, with these param- 
eters undetected bit-error probability for transition prob- 
abilities, for which R, = Mi, is 10*, theoretically. (From 
Fig. 12, Efl/No = 2.2 dB in this case.) 

Figure 13 is a block diagram of the receiving system 
envisioned, and it should be noted that energy-to-noise 
ratios are measured at the input to the decoder. 

Tliere are three timing problems vo solve with this sys- 
tem, viz., symbol synchronizing, block synchronizing, and 
output data formatting. The symbol problem is solved by 
the receiver which generates a timing pulse to interrupt 
the computer, causing it to process the next received 
symbol from the converter. 

In order to provide a single-channel capability, a block 
synchronization technique was added io the decoding 
program. Each tree path ends in a known sequence of 
24 bits, i.e., one constraint length. At the end of a tree, 
this ensures that the last few bits before this sequence will 
be decoded properly. For a systematic code, !hc"i, of the 
last 72 symbols, every third one is known. In addition to 
these, the initial 8 information bits of the next block were 
also fixed in advance, and consequently 48 channel sym- 
bols are known a priori at the end and beginning of con- 
secutive blocks. Synchronizing is achieved by searching 
for this pattern in the incoming symbol stream by requir- 
ing good correlation on any multiple of the 48 chaimel 
symbols. Once block synchronization is declared, decod- 
ing begins at the correct place. Following an overflow, 
the decoder moves ahead to the start of the next block 
according to its previous reference. If overflows occur in 
consecutive blocks, for example in four or five consecutive 
blocks, i^ becomes reasonable to declare a system failure 
and the program returns to the block synchronizing mode. 



CHANNEL 


DATA SYMBOLS 




















OUTPUT DEVICE 




















SYME 


iOL TIMI 


i 

MG 


1 


i 


i 





Fig. 13. Receiving system block diagram 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



237 



Block synchronizing could also be derived from the de- 
coder alone by thf» f^lloN.ing argument. If the decoder 
begins anywhere in the symbol stream with an incorrect 
block reference, it will not find the correct path because 
there isn't one, and so will overflow and not decode. Said 
contrapositively, decoding implies synchronizing. The 
converse is not true since the decoder will sometimes 
overflow even with correct block reference. The situation 
is different in the case of optimum block coding in which 
the decoder is in reality a block synchronizer and syn- 
chronizing implies decoding. 

Figure 14 shows theoretical undetected bit-error prob- 
abilities for constraint lengths 24, 36, and 48, and ob- 
served error probabilities for length 24. No errors were 
observed for 36 and 48. This part of the experiment ob- 
served bit errors in the decoder output when the decoder 
is allowed as long as required to decode. Undetected 
errors, that is, errors produced by the decoder to the 
output device, were observed at 2 energy levels, Eb/N,, 
of 1.2 and 2.2. At 1.2 dB, the bit-error rate was nearly 
0.5, and at 2.2 dB it was less than 10-^ From 2.2 dB, the 
experiment continued in increments of 0.2 dB through 
Eb/N„ = 4.0, and no further bit errors were observed. 
The sample size at each energy level was 2 million infor- 
mation bits, or 6 million channel symbols. 

As can be seen from Fig. 12 and Inequality (12), the 
performance of a sequential decoder is very sensitive to 
small changes in energy-to-noise ratio. Part of the diffi- 
culty in this experiment was finding a range of energy 





• 




i 






























— - 


1 










j 




10-* 


^ 


k ^^ 




^ 


' — 






— 


PROBABILITY 

m 01 




"^ 




hJ 


\ 


■^ 


c=21 








^^ 


\ 




\^ 




10-" 
I0rl2 














^36 


















' 
















M8 




lo-i* 





















12 16 20 24 2B 32 36 40 44 

fflA). dB 

Fig. 14. Th«<>retical error probability for constraint 

lengths (v) 24, 36, an<^48 and observed error 

probabilities ftir constraint length 24 



levels and data rates over which anything of statistical 
interest could be observed in a reasonable length of time. 
The next part of the experiment was an attempt to verify 
Inequahty (12) and show p[C> L]~ Lf. Plotted on 
log-log paper, this distribution function should be a 
straight line of slope —p. For each of the six cases 
Eb/No = 4.6, 4.0, 3.4, 2.7, 2.2, and 1.2, two million infor- 
mation bits were processed and the decoding time ob- 
served. Time during which the computer performed input 
and output functions was not recorded. The results are 
presented in Fig. 15 and show straight-line behavior in all 
cases except the last in which the decoder was in error 
in almost half of the decoded bits. The measured slopes 
show close agreement with the theoretical values of 
Fig. 12, except for Eb/N„ of 4.6, in which case the decoder 
is about twice as fast as theoretically expected. Note that 
decoding time decreases as undetected errors increase, 
as predicted. 

It is apparent that, in a real-time situation, buffer over- 
flow is much more likely than undetected error. The final 
phase of the experiment was designed to determine the 
overflow probability. Quantized channel symbols were 
transmitted to the decoder in a serial stream at a fixed 
data rate. Tree synchronization was achieved by search- 
ing for tne special interblock symbols. A buffer overflow 
caused the computer to record that event and wait for 
the start of the next block. In addition to decoding, the 



a. 



2-10 



2-12 



^ 




. 








w\ 


I \^ 


N 


\j 






\\ 


r^ 


^\ 


N 






p= \52 


\2.8 


U7 \l: 


\l. 


\ 1 


15 


\ 




\ 




2.2 


\ 


^5A = 


4.6 > 


M.0 


\J 


x.. 


I. 



0.5 



8 32 128 

COMPUTATION TIMt, s 



512 2048 



Fig. 15. Observed distribution of computation 
time for constraint length 24 



238 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



.% 



computer performed input and output functions. Two 
output devices were tested, the line printer and the mag- 
netic tape recorder. The hne printer appears worse largely 
because the computer spends extra time formatting out- 
put bits for printing, whereas in the magnetic tape case, 
the output was merely recorded and read later with 
a.iother program. 



10" 
10-1 


(a) 


--^ 


N 










X 


\ 


\ 


^ 


^ 










\ 


\ 




V 


\'000 

\ 


)its/s 








\ 




500 ^ 




10-3 






\ 
















IC 











in-" 

















10° 



(b) ^ 




^ 


V 












"^ 


X 


N 


MOOO b 

V 

500 


ts/s 
\ 








\ 
















\ 

100 







16 20 24 26 32 36 40 44 

Fig. 16. Observed erasure probability for: (a) magnetic 

tape as output device, and (b) line printer 

as output device 



< 
en 

a. 
o 
cc 



o 

UJ 

I- 
o 

UJ 
H 
Ul 

o 

z 




10"' 2 4 6 10 2 2 4 6 10"' 2 
WORD ERASURE RATE 



4 6 10° 



Fig. 17. Comparison of erasure rate vs undetected 

bit-error rate for Eb/No of: (a) 2.2, (b) 2.8, 

(c) 3.4, and (d) 4.0 dB 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



239 



Figures 16a and 16b show the results of the real-time 
simulation for magnetic-tape and line-printer output, 
respectively. The experiment was conducted over the 
dynamic range of 1.8 to 4.0 dB in steps of 0.2 dB at three 
data rates, 100, 500, and 1000 bits/s on input (300, 1500, 
3000 symbois/s) and overflow probabilities recorded. For 
each case, 2 million bits were processed, and the graphs 
are not continued beyond the point where no overflows 
were recorded. No undetected errors were observed. 



7. Comparison With Orthogonal Block Codes 

The erasure versus error performance of orthogonal and 
biorthogonal codes, using the optimum decoder of Sub- 
section 3 in place of the sequential decoding algorithm in 
the decoder of Fig. 13, can be calculated and then com- 
pared with the siniuhtion results for tree codes. It's easv 
tc verify that, for the additive normal noise channel, the 
optimum block decoder of Fig. 13 is actually a correlation 
de(.'oder. Figure 17 ..hows the relation between word era- 
sure probability and undetected bit-error probability for 
fixed Eb/N„ of 2.2, 2.8, 3.4, and 4.0 dB. In each case, the 
horizontal line represents the performance of the sequen- 
tial decrier. A fixed-speed advantage for the decoder 
over data rate results in a fixed-overflow probability, 
while increasing that speed advantage decreases overflow 
probability and maintains the same undetected bit-error 
probability. 

For the block codes and likelihood decoding with era- 
sures, the undetected bit-error rate is a function of the 
word erasure rate. In the range of 2.2- to 4.0-dB bit 
energy- to-noise ratio, this error late is inferior by many 
orders of magnitude to the performance of the tree de- 
coder unless an unrealistically high erasure rate, above 
0.5, can be tolerated. This erasure probability can be 
improved by either accepting, for a fixed code, a higher 
bit-error rate, or by operating at a lower information rate. 
[The orthogonal (8,3) and biorthogonal (16,3) have rates 
% and V2, respectively.] This latter possibility is limited 
by Eq. (16) and the performance of the symbol synchro- 
nizer. For example, a (16,4) orthogonal code has E^/Na = 
— 3.82 for Eb/Nb = 2.2 dB. The erasures resulting from 
likelihood decoding of block codes can never be recovered 
with other changes in the system, whereas erasures from 
sequential decoding, if recorded, could be decoded off- 
line with the error rate indicated on the graph. 

For high information rates, low bit energy-to-noise 
ratio, and a general-purpose digital computer, sequential 
decoding provides superior performance to optimum 
decoding of orthogonal block codes. 



References 

1. Wozencraft, J. \1., and 'acnlvs, I. M., Principles of Communica- 
tions Enfiineerinfi, Chap. 6. John Wiley & Sons, Inc., New Yori,, 
1965. 

2. Jacob.s, I. M., "Sequential Decoding for Efficient Communi- 
cation from Deep Space," IEEE Trans. Commun. Technol., 
COM-15, No. 4, pp. 492-501, Aug. 1967. 

3. Gallager, R. C, "A Simple Derivation of the Coding Theorem," 
IEEE Trans. Inform. Theory, IT-U, pp. 3-18, Jan. 1985. 

4. Shannon, C. E., GaUager, R. G , and Berlekamp, E., "Lower 
Bounds to Ertj: Probability for Coding on Discrete Memoiyless 
Channels," Inform. Contr., Vol. 10, pp. 65-103, Jan. 1967. 

5. Viterbi, A. J., "Error Bounds for Convolutional Codes and 
an Asymptotically Optimum Decoding .algorithm," IEEE Trans. 
Inform. Theonj, IT-13, pp. 260-269, Apr. 1967. 

6. Yudkin, H. L., Cliannel State Testing: in Information Decoding, 
Ph. D. thesis Department of Electrical Enginer'ing, Massa- 
chusetts Institute of Technology, Cambiidge, Mass., Sept. 1964. 

7. Berlekamp, £., and Jacobs, I. M., "A Lower Bound to the Dis- 
tribution of Computation for Sequential Decoding," IEEE Trans. 
Infom,. Theory. IT-13, pp. 167-174, Apr. 1967. 

8. Savage, J. E., "Sequential Decoding: The Computation Prob- 
lem," Bell Syst. Tech. J., Vol. 45, pp. 149-176, Ji.n. 1966. 



G. Communications Systems Development: The 
Optimum Cross-Correlation Function for a 
First-Order Tracking Loop Under Unit 
Pc wer Constraint, j. w. Layland 

1 . Introduction 

In SPS 37-41, Vol. IV, pp. 270-272, and SPS 37-43, 
Vol. IV, pp. 321-323, Stiffler proved that the uncon- 
strained optimum cross-correlation function for a first- 
order tracking loop is a square wave and developed a 
minimum mean-square-error approximation to this cross- 
correlation function under the additional constraint that 
both the received and local reference signals have unit 
power. Subsequent work, reported in SPS 37-50, Vol. Ill, 
pp. 284-287, determined the optimum unit power local 
reference signal for use when the received signal is a 
square wave. This article describes a more precise result 
obtained for the optimum cross-correlat>on function when 
both the received and local reference signals have unit 
power but are otherwise unconstrained. 



2. Problem FormuloHen 

The probability density function of the phase error in a 
first-order loop due to additive white gaussian noise has 
been shown to be (Ref. 1) 



p (^) = C exp { -« /♦ prA iv) «>?} 



(1) 



240 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



where C is a normal'/.ing constant, pi.tiri) denotes the 
normalized cross-correlation function between r{t), the 
received signal, and A (t), the local reference signal, and 
a = 4/N„K {PAi/Pry^' denotes loop signal-to-noise ratio 
(SNR). The functions r{t) and A{t) will be determined 
such that they minimize 



depend on k; final numerical results are obtained for 
fc — 2 only. 

3. General Result 

Since A(t) and r{t) are periodic with period 2it, they 
can be represented in the form 



Ei. = j kPexp<|--a / prA{v)orild<j> 



(2) 



for some integer k. If k — 0, this maximizes P (0). For 
other k, this minimizes the un-normalized fcth absolute 
central moment of the distribution. Normalized moments 
could be substituted at the expense of iicreased compu- 
tational difficulty. The initial results presented do not 



r{t)= 2 r„ £'■'"'**» 



Ait)= 2 a„c^<"'*"''" 



(3) 



where r„, a„ are real, r„ = r_„, a„ = a „,<!'<,= ~ <j> n, and 
•Pn '-= "" 'pi- If Eq. (3) is inserted in Eq. (2), the lower limit 
of the integration over tj is set to — 2tt, and the unit power 
constraint is appended, the optimization criterion becomes 



£*= rkl''exp j~«V^-^a„r„(l-eJ"*)fc''^"«"'|d,^ + \J V^r;- li + X. ly^flj - l| 



(4) 



For convenience, denote ipn — <l>n = S„. It is clear from Eq. (4) that Ek depends upon 8„, rather than <ji„ or ^b individu- 
ally. To determine the optimum S„, compute 



?8„ 



r\<j>\''expi- ay^a„r„^{l - t?^''*)e'«-| •a„r„ f^- ^)(^ ~ e'»*) (;e>S + (^Vl - €-'"*)(- ie-i'')'\d4, 



5£t 
?8„ 



/' 2ft Q T 
|^|*exp {-«•••} • ^'-^{cos8„(l - cosfu^) t- sinS„sinn</>} d^ 



(5) 
(6) 



Note first that if P ((^) is symmetric about <^ = 0, E { 1 1 "^ sin n<^} =0; hence, dE/dS„ = if cos 8„ =^ 0, i.e., if ?„ = ±7r/2. 
Furthermore, if 8, = ±ir/2 for all n, then P(<^) will be symmetric about <^ = 0. Hence 8„ = ±7r/2 for all n is a suffi- 
cient condition for an extremum of E^. 



The determination of the optimum coefficients a„, r„ can be simplified by the following argument: Whatever the opti- 
mum values of (a„} and {r„} are, there will be some fixed amount of power m the nth component of both signals. Call 
this kn= a^ + r^ and a.'ssume that, by some means, k„ has been determined without finding a„ or r„. Finding these then 
reduces to finding the {a„} which minimizes 



E, = ^"l^l^expl- „ V^la,(jfc5 - a5)'Ae^««(l - eM)\d^ 



(7) 



To do this compute 



3£t 
9o« 



- ('|<^|*exp{-o ■ • • }- — ^•-{2sin8n-2sin(8„4-n^)}d.|. 



sign {n} sign {8„}C, 



r5-a5 



(8) 



where C. is a ' ^itive constant, and sign (x) denotes the algebraic sign of x. 



SFL SMCE PROGRAMS SUMMAKY 37-51, VOL. lit 



241 



If sign {n} sign {8„} is positive, dEk/da„ > for a'i < rj and ?Ek/?a'f, < for ai > rl, which impHes that the mini- 
mum El, occurs for either aj = or nj ^ kl. If, however, c?. = kl, then rj = 0, and if either a'i, or rj is zero, then the con- 
tribution to the correlation function from the nth component is zero. Therefore, if sign {n} sign {8„) is positive, kl = 0. 

If sigri {«} sign {8„} is negative, then PE^ dal < for or, < rl and ?Ek ?a„ > for ol > rl. Therefore, a;, = rj. 

Use of these two results reduces E/. to the form 



n>o fi >o 

The optimum {a„] are the solutions to the equations 



(9) 



(10) 



for all n. It is a relatively easy matter to show that the 
functions A(t) and r{t) are band-limited. For large o, 
cos ij> is approximately 1 - •^>72 for all <^ for which p («^) 
is not essentially zero: so for n - I, 



A' = 



1 



2-p(0) 



£{|<>|''*-'}ifaislarge 



(11) 



For any n. 



1 - cos n<^ 2 



and hence for any a. 



2 



n'p{0) 
Combining these two requirements: 

1 £{kl*" ) ^2 

2 £{|^|*) ~n 



E (!</. I ^} unless o„ =0 (12) 



But 






C,a 



(13) 



(14) 



where C, depends upon fc, and p (^), and C, depends only 
on p (^) and is in t^e range 1 :^ Cj ^ 2. Therefore, 



1 ^ — o''' unless Oh = 



(15) 



For very small a, the exponential term in Eq. (10) can be 
expanded in & series and terms of higher than first order 
in a ignored. Solution of the resultant set of equations 
shows that a, = 1/(2)'^, o, = for n ^ 1 is the only solu- 
tion allowed. The band limit thus extends, as expected, to 
small o It may be noted that the solution to Eq. (10) is 
not I' ique, since a, = 1/(2)''', a, = for n^t 1 is a solu- 
tion idi any a. However, the solution to Eq. (10) with 
the maximum possible number of non-zero components 
should be unique, and should also represent the true mini- 
mum, since the resultant p,^ (-q) will have the steepest 
slope in the vicinity of of any of the possible solutions. 



4. Nbmerical Results 

Equation (10) has been subjected to an iterative numer- 
ical solution for k = 2 and for various value« of loop 
SNR a. A typical resultant power spectrum and the associ- 
ated cross-correlation function rre shown in Fig. 18. With 
the exception that tlie even harmonics are slightly sup- 
pressed, this is very similar to the main lobe of a (sin x/x)^ 
spectrum, the spectrum of a pseudc oise (PN) sequence 
of length (approximately) a/2. The variance of the result- 
ant phase error in a loop employing optimum signals is 
plotted as a function of a in Fig. 19. Ako shown, for com- 
paiison, is the phase error variance obtained using the 
first lobe of a (sinx/x)' spectrum with a/2 components. 
Since the minimum is very broad, \ ery little loss is sufiFered 
by use of this simply generated signal. The bottom line in 
this figure corresponds to the phase-t rror variance which 
would result from use of Stiffler's non- realizable optimum 
cross-correlation function. The 3-dB difference in per- 
formance appears to be due solely to the imposition of the 
realizability constraint of unit power. 



2^7 



jn SMCE noQUAm summmhy 37.51, vol. hi 




I 

I : 



Fig. 1 8< Typical optimum p»w«r spectrum and 

cro$s-corr«iation function (a = 32, 

1 1 non-zoro compontntt) 



5. Comparison to o PN Range Traclting Loop 

Since one of the main uses of a very high SNR tracking 
loop is in range measurement, it is of interest to compare 
the performance attainable by using the optimum wave- 
forms for such a loop with that obtained when using the 
binary PN sequences which are typically used. A PN 
waveform with p digits has a series expansion given by 



sin I 



(1=1 



Xcosf * + ^>] 



(16) 



The main lobe of this power spectnmd has already been 
shown to be an effective approximatioc to the optimum 
r (t). The local reference signal for a PN wave is usually 
constructed as 



PNR(0 = 



PNrt + H) PN(*-%) 



6 

4 



10- 

6 

4 






Iff 



rS 



V 



UNIT 

POWER 
. OPTIMUM 



- FIRST LOBE OF 
(»in jr/A)* SPECTRUM 
a/i TEHMS 



PN RANCE CODE 
OF LENGTH 
«/4 



Kfi 




N0NREALI2ABLE — 
OPTIMUM 



Fig. 19. Loop phaso-orror variance i^ vs loop SNR a 
fnt various cross-corroiatien functions 



This sl^al possesses the expansion 



»^ sin'f ~] 

PNR (0 = -^ [2 (p + 1)1 ^ V ^- 

ii-i 
/ 2nw , ir \ 



(17) 



(2)v4 



(2)% 



Tlie phase relationsliip of PNP (<) to ?N(t) is the 
same as that of tht; optimal reference for components in 
the rjuige 9Jc ^ n/p ^ ?Jc + 1 and phase-revened Ita 
2k + l^ n/p ^ 2fc + 2, all a.. It would appear that track- 
ing pertormanr^e could be improved by filtering to remove 
all frequency componnts above n = p. In addition, a 
factor of sin (nir/p) modtfes ftach term of PNR (t). The 
effect of this is shown in Fi((. 20, which shows « cmnpari- 
son between the phase vananc;e which results in a Hn--' 
order tracking loop using the usual FN systtm and usir.;; 



iH SMC£ nOGKAMS SUMMARY 37-51, VOL. Ill 



243 



o-' 

6 
4 



6 
4 



n-* 



CONVENTIONAL 
PN 
REFERENCE 




PHASE- 
SHIFTTOPN 
REFERENCE 



»' 



Fig. 20. Comparison of phaso-orror vorianc* a} for 

cenvonHonal PN raforonco end phaso-shifted PN 

rtferancc, for fixed cod* length p as a 

function of a 



a phase-shifted PN for local reference signal, considering 
frequency components n — p only. The curves show that 
the usual PN system is poorer than the modified one for 
a < 4p and indicate that the best usual PN system to use 
has code length p s; a/4. The upper line of Fig. 19 corre- 
sponds to the phase variance in a usual PN syst«n with 
code-length a/4. An improvement of approximately 1 dB 
in efiFective loop SNR can be obtained by use of a phase- 
shifted PN reference signal as opposed to the reference 
signal usually implemented. 



Roforonc* 

Viterfoi, A. J., "Phase Locked Loop Dynamics In the Presence of 
Noise by Fokker-Planck Techniques," IEEE Proc., pp. 1737- 
1753, Dec. 1963. 



244 



H. Information Processing: Disjoint Cycles From 
Hie de Bruijn Graph, h. Fr*dricks^n 

1 . THo do Brui|n Diagram 

a. DeKription. An n-bit shift register (Fig. 21) is a set 
of n storage registers with logic which defines their con- 
tents at any point in time. The contents of the tth storage 
register at time t is equal to the contents of the (i — l)st 
storage register at time * — 1, for 2 ^t ^n. A feedback 
function / (x,, Xj, - - - , x.) determines the contents of the 
first register x, at time ( from the contents of the n regis- 
ters X,, Xs, ■ • • , X, at time t - 1. The contents of the 
register at time t, regarded as a binary number or a binary 
vector, is called the state of the register. At the end of 
each time interval, detormined by an external clock, there 
is a transition from one state to the next. 



.\ 1 


// 


'('«.'ii-l,--. 


.'•) ' 



Fig. 21. Gonorol shift register 

Since there are 2" vectors dcaned by a register of 
length n, there pre 2" states for t^e shift register. The 
diagtam of all possible state transitions is called the 
de Bruijn diagram. The de Bruijn diagrams for n = 
1,2,3,4,5 are shown in Figs. 22 and 23. Each node in a 
diagram has two possible successors and two possible 
predecessors. These diagrams contain all possible transi- 
tion patterns for their shift registers. 

Transition patterns in the de Bruijn diagram are deter- 
mined by the feedbadc function of the shift register. For 
the function to be well defined, we require that each state 
have only one successor. Then the feedback function 
chooses exactly one path for the exit from each state of 
the diagram. If we change the feedback function so that 
a state maps into the other state possible, we say we have 
chosen the alternate successor for the state. 

b. Cycles ofthede Bndjn diagram. Let / be the feed- 
back function whidi defines the state transitions. If a suc- 
cession of k state transitions leads from state «i back to 
Si, i.e., 



Si=f(Si), Sk = f{S,) = f{Si), 



Si=f{Si) 



jn SPACE PJtOGMiMS SUMMAItY 37-51, VOL. Ill 





/> = l 



/J = 2 



001 



100 




0001 



" 0011 




= 3 /» = 4 

Fig. 22. de Bruijn graphs for n = 1,2,3,4 



we say the states Sj,/(s,),/-(Si), • • ■ ,/*' (Sj) form a q/cte 
of length k in the diagram. The cycle can be described as 
the fc-tuple of zeros and ones which are the feedback 
values of the states on the cycle. Equivalently, the cycle 
could be described in decimal notation by the decimal 
equivalent of the binary representation of the states which 
make up the cycle. It will often be convenient to use each 
of these representations in what follows. 



We would also like to restrict the truth tables so that 
each state has a unique predecessor as well as a unique 
successor. This will decompose the de Bruijn diagram in 
such a way that every state will be on a unique cycle. 
Golomb (Ref. 1, p. 115) gives a condition that insures that 
a feedback function yield pure cycles. We state that con- 
dition here. 



Theorem 1. The feedback function for a shift register 
yields pure cycles if the last variable enters linearly into 
the feedback function. 



10000 



00001 



IIOOOi 



1 1 100 




0001 1 



001 1 1 



Fig. 23. de Bruijn graph forn = 5 



Then the feedback function /(xi.x^, • • ■ ,x,) can be 
represented as g(x,,X2, • - • ,Xn-i) + x,. When the feed- 
back function is so representable, the half of the truth 
table where x„ = 1 is the complement of the half of the 
truth table where x„ = 0. The truth table of the function g 
contains all the information about the shift register. When 
we speak of the truth table of the shift register in what 
follows, we shall mean the truth table of the function g. 
The context will make it clear when we wish to discuss 
an arbitrary feedback function /. 



c. Particular cycle decompositions. The question as to 
whether a cycle can be found which contains all the states 
of the diagram has been answered in the afBrmative by 
de Bruijn (Ref. 2). The number of de Bruijn cycles is also 
given in Ref. 2. He shows there are 2^""'" de Bruijn cycles 
in the graph. Other authors (Refs. 1 and 3) give alternate 
proofs. In Subsection 2, we discuss the distribution of the 
de Bruijn cycles by their weight. 



jn SPACE PROGRAMS SUMMARY 37-5J, VOL. Ill 



245 



The existence of cycles of all lengths from length 1 to 
length 2" from a register of length n is shown in Golomb 
(Ref. 1, p. 192). 

Other feedback functions yield special cycle decompo- 
sitions wh' h are of interest. Two simple functions shall 
be discussed in Subsection 2 and in a future article. They 
are the pure-cycling register and the c-omplementing- 
cycling register. The pure-cycUng register is given by the 
feedback function / (x,, acj, ■ • , x,) = x„ (g ^ 0). The 
complementing-cycling register is given by the feedback 
function 



/ (li, Xj, • • • , X,) = x« + 1 , 



(g^l) 



Colomb (Ref. 1) shows the number of cycles determined 
by the pure-cycling register is given by 

i/n 

where the summation is over all divisors d of n and if> is 
the Euler ^ function. He shows Z (n) is even for all n > 2. 
The number of cycles determined by the complementing- 
cycling register is 

Z* (n) - I Z (n) - ^ ^0 (2d} 2"/^" 

2d/tt 

Here the summation is over only the even divisors of n. 

Colomb makes the conjecture (Ref. 1, p. 174) that the 
maximum number of cycles into which the de Bruijn 
graph can be decomposed is equal to Z (n). This conjec- 
ture will be discussed in a future article. 



2. Distribution of Truth Tables by Number of 
Cycles and Weight 

a. Boundary of the table. Consider the set of all feed- 
back functions on the shift register. We determine the 
various cycle decompositions from the de Bruijn graph. 
For a register of length n, there are 2""' truth tables. 

For n = 3,4,5, we group the truth tables according to 
their weight and to the number of cycles they generate. 
We only need consider the half of the truth table where 
Xn = since the half where x« = 1 is just its complement. 
For n = 3, there are only two free variables Xi and Xj. 
There are four possible value pairs which these two vari- 
ables can take on. 



Corresponding to each value pair we have a two-fold 
choice of or 1 for the feedback function at that posi- 
tion. This gives us a total of 16 different truth tables. The 
weight of the truth tables ranges between and 4 and 
the distribution of truth tables, by their weight and by the 
number of cycles they produce, is given in Table 3. 



Table 3. Cycle decomposition table for n = 3 



Waight 
el 



labl* 





Numb 


•r el cycles 




I 


2 


3 


4 


c 








1 








4 








s 





I 


2 





2 








1 









Fo, the pure-cycling register, is the register of weight 0. 
The four cycles generated by Fo are (0), (1), (001), and 
,011). No other feedback truth table yields more cycles 
ihtn Fo. There is one truth table which ties F„ for the 
maximum. This is Fo, where the subscript is the decimal 
representation of the truth taole given by the values that 
the variables take on. For Fg the variable pair x,, x, take 
on the values / (0, 0) = 0, / (0, 1) - 1, f (1,0) = 1, / (1, 1) = 
and the subscript is given by 

s - /(0,0)2» + /(0, 1)2= + /(1,0) 2 + /(1, 1) 

Fs yields the cycle structure (0), (1), (01), (0011). 

A change m a single position in the truth table will 
cause a change of one in the number of cycles from one 
truth table to the next, either increasing or decreasing by 
one the number of cycles (Ref. 1). For n = 3, there are 
four possible single changes which could be made in the 
truth table. If we start from Fo, the four changes are all 
0-» 1 changes resulting in the four truth tables F„ F2, F,, Fg 
all of weight 1. In every case, we find the number of cycles 
decreases when we make this change. This is the result 
of two cycles joining and forming a single cycle. From 
Fj or F,, an additional change results in Fo, which splits 
the cycle which has been formed to form two new cycles. 

For n = 4, 5, the cycle decomposition tables are given 
in Tables 4 and 5. The truth tables F,,, Ftj, Fgo all yield 
Z (4) = 6 cycles. 

The behavior in the Hecomposition tables shown is 
typical of the general behuvior of these tables. Figure 24 
is a picture of the typical decomposition table. We state 



246 



jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



Tabic 4. Cycl* decomposition tabU for n = 4 



I 2 



Waighl 

of 

hurii 

tabU 




1 

3 
3 

4 
5 

6 
7 
8 







NumlKr 


of (yclM 






1 


2 


3 


4 


5 


« 

















1 














B 














26 





■i 








44 





12 








37 





32 





1 


12 





40 





4 








22 





6 








4 





4 














I 















Table 5. Cycio docomposition toblo for n = S 



Number af cycIo 
3 4 5 































































Weight 







1736 


of 




576 





Inirii 







4056 


tabl* 




960 












2892 






448 












736 






64 









C 


52 





























2036 


6488 


6684 


2652 



368 



12 










1253 


5050 


7326 


4338 

962 

62 

I 






476 


2132 


4098 


3572 


1210 

124 

4 










1 





16 





IK 





6 





84 





552 





15 





200 





197 





25 





278 





467 





21 





224 





767 





11 





58 





121 





1 





4 





6 



























certain theorems here which relate to the character of the 
general decomposition table. 

Theorem 2. TTie line [k,Z{n)~k] defines the (achieved) 
upper border of the decomposition tabid. 



Theorem 3. The line [2"-' 
left boundary of the table. 



Jt, Z* (n) - Jt] is the lower- 



> 

if- 



There are three other border lines on the decomposi- 
tion table which are of some interest. To state the location 
of the right-hand border of the table is to answer the con- 
jectuie on the maximum number of cycles from a register. 
In what follows, we shall assume that the conjecture is 
correct. We show below that the right-hand border is at 
least as long as twice; the number k of [a, r (a)] pairs on a 
cycle which have their respective alternate successors 



NUMBER OF CYCLES - 
Att) 



^^'» 



THEOREM S 



< 



a. 

li. 
o 

I 

S2 

UJ 



2('>-l). 




Fig. 24. General qrcle decomposition table 

on one (other) cycle. (The reverse funciion r is defined 
below.) No examples are known of a truth table of weight 
greater than 2fc and Z (n) cycles. We also show that there 
are at least 2* examples of truth tables with Z (n) cycles. 

In the cycle decomposition tables for n = 3, 4, 5, we 
note that there is more than one truth table having 
Z (n) cycles. This is true for all n ^ 3. Consider the two 
cycles from the pure-cycling register (000 • • • 01) and 
(00 • ■ • Oil) each of length n. We could change tlie suc- 
cessor of 



'00 



01 and of 010 



and have a truth table of weight 2 which had Z (n) cycles. 
The two new cycles would be (0 • • • Oil) of length n + 1 
and (0 ■ • • 01) of length n - 1. 



i?l SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



247 



The n-tuples 00 • 01 and 010 • can be related 
to one another. We can say 010 • •• is the reverse of 
• • • 01, where the reverse function r is defined by 

r(«) = r{a„,an-u • • ■ .fli) = (c„,o„as, • • • ,a„-,) 

The reverse function is an order 2 function. 

Consider the half of the truth table where a, = 0. We 
can separate the positions, by the reverse operation, into 
pairs. Some positions will be self-reverse and will not pair 
with any other position. 

Suppose we can find a pair a, r (a) on the same pure- 
cycle for which the pair a*, r (a)*, the respective alternate 
successors, is on another pure-cycle. Then we can change 
the successor of a and r (a) and not alter the number of 
cycles. Suppose therr exist k such pairs. We can show 
the following: 

Theorem 4. There exist ^ 2* truth tables yielding Z (n) 
cycles. 

Theorem 5. There exist truth tables yielding Z (n) cycles 
having weight ^ 2k. 

For n = 7, there are 15 [o,r(a)] pairs. So we have at 
least 2" truth tables which have Z (n) cycles. This is from 
a truth table set of 2"* truth tables. There are examples of 
truth tables of weight 30 producing Z (n) cycles. 

Because of the unsettled nature of the Z (n) conjecture, 
the lower right-hand border is also unsettled. The left- 
hand border consists of the so-called de Bruijn sequences. 
All of the 2" nodes of the graph are on one cycle. We dis- 
cuss this border below. 

Assume the border has been established. We make the 
following statement about the interior of the table. 

Theorem 6. There are no (weight, number of cycles) 
pairs interior to the table, for which truth tables are pos- 
sible, that do not occur. (The weight and number of cycles 
must be of the same parity for n > 2.) 

In Theorems 2 and 3, we exhibited an upper and lower 
boundary. This left-hand end of the boundary in each 
case came at the place where all 2" nodes were on one 
cycle. In the first case, the weight of the truth table 
was Z (n) — 1, and in the second case, the weight was 
2"-* — Z*{n) + 1. We showed these values were the lower 
and upper limits, respectively, for the weight of a truth 



table having exactly one cycle associated with it. We can 
show that every odd weight between these limits has a 
truth table of that weight associated with: a de Bruijn 
cycle or sequence. 

Theorem 7. There exists a de Bruijn cycle for every odd 
weight between Z(n) - 1 and 2"' - Z*(n) 4- 1. 

b. de Bruijn cycles. 

Distribution by weight. We showed in Paragraph a, 
above, that de Bruijn cycles exist for every length register. 
We also showed that there is a minimum and a maximum 
weight for a truth table which produces a de Bruijn cycle. 
Finally, we showed that every value between the mini- 
mum and the maximum had a truth table of that weight 
which defined a de Bruijn cycle. The number of all such 
cycles has been given by de Bruijn to be 2^""'"" (Ref. 2). 
For n = 3, 4, 5, the number of de Bruijn cycles is 2,16,2048. 
These are the number of de Bruijn cycles we show in 
Subsection 2-a. We also classify the de Bruijn cycles V 
the weight of their truth table. 

A well-known graph-theoretic theorem can be applied 
to find the number of de Bruijn cycles of maximum and 
minimum weight. For the cycles of maximum weight, we 
form the decomposition of the space of 2" nodes by the 
complementing cycling register. Label the cycles formed 
as A,,A2, • ■ • , Az«(n). We form a labeled graph contain- 
ing Z* (n) nodes in the following way: 

Connect Aj to A, if Aj contains a vector whose alter- 
nate successor is on Ai,i^i. Label the arc from Aj to Ay 
with the number of such vectors on Aj. We form a matrix 
B with entries fcj, = label on the arc from Aj to A,. Also 
form the diagonal matrix C = (Cjj) whose entries are 
given by Cjj equal to the sum of the labels on arcs enter- 
ing Aj. The theorem states that the number of rooted trees 
of the graph is equal to the determinant of the minor of 
dn. where Ai is a root of the graph and D = (<f «,) = C — B. 
The number of rooted trees is equal to the number of 
de Bruijn cycles of maximum weight. Applying the theo- 
rem, we find the number of de Bruijn cycles of maximum 
weight for n = 3, 4, 5, 6, 7 are 2, 4, 64, 2", 3 X 2^«. The first 
four values were checked on the computer by exhaustive 
search and the truth tables listed. For n = 7, the time 
required to check all de Bruijn cycles of weight ?7 is 
prohibitive. 

The matrix B is symmetric. This can be seen by noting 
for a c A which has its alternate successor a* 6 B, there is 
a, the 2""' — 1 complement of a on B with its alternate 



248 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



successor on A. That is, i' c = ai.Cj, 
complementing cycling register, 



(fli.fli, 



, On, fll. 



, On, then by the 



,fln) 



If o has its alternate successor, a* = a-i, Ca, • ■ • ,a„, UjonB, 
then B = (02, • • , fln, a,, 57, • ■ ■ , flT, oi)- The comple- 
ment of 0, a = Ci, 57, • • • , oT is on B with its alternate 
successor (a)* = 02,- • • , 57, fli on A. 

For the de Bruijn cycles of minimum weight, we employ 
the pure-cycle decomposition. We form the graph in the 
same way as above. An equivalent detenninant is taken 
to find the number of de Bruijn cycles of minimum weight. 

We form a table with the nui..>er of de Bruijn cycles of 
minimum and maximum weight for the first few values 
of n. 





Number ol 


cycles 


of 


Number of cycles of 


n 


maximum weight 


= 


minimum weight = 




IVjmajr 


(«)] 




[Cninin)] 


1 


1 






1 


2 


1 






1 


3 


2 






2 


4 


4 






12 


5 


2« 






2««3=' 


6 


2» 






2"' 3*' 5== 


7 


£26 


3 




228.35.53.13 



For n — 1, 2, 3, the minimum weight equals the maximum 
weight. For n = 4, there are no other de Bruijn cycles. It is 
interesting to note that 

C„„ (n) I Cmj„ (n); also, Cmax (n) \ C„„ (n + 1) 

and C„i„(n)|C„j„(n-M) 

Also, from Table 5 we see that Cmax (5) divides the num- 
ber of de Bruijn cycles of any weight. 

We give examples to illustrate the method for the 
case n = 5. For the de Bruijn cycles of maximum weight, 
we form the cycle decomposition of the complementing 
cycling' register on five variables. With maximum weight 
of de Bruijn cycles, n = 5, an example of the graph is: 



Cycle 


Set of vectors 


A 
B 
C 
D 


0, 1, 3, 7, 15, 31, 30, 28, 24, 16 
2, 5, 1], 23, 14, 29. 26, 20, 8, 17 
4, 9, 19, 6, 13, 27, 22, 12, 25, 18 
10, 21 



A 
B 
C 
D 



A 


B 


C 


D 


6 


-4 


-2 





-4 


10 


-4 


-2 


-2 


-4 


6 








-2 





2 



Matrix C-B 




where D is a root of the graph. We evaluate the determi- 
nant of the minor of the D,D position 



64 



For the number of de Bruijn cycles of minimum weight, 
we form the cycle decomposition under the pure-cycling 
register for n = 5. The graph of the cycle connections is 
given in Fig. 25. With minimum weight of de Bruijn 



6 


-4 


-2 


4 


10 


-4 


2 


-4 


6 




Fig. 25. Graph of cycl« connections for 
puro-cycio rogistor 



in SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



249 



cycles, n = 5, an example of the graph is: 



Cycle 


Set ot vectors 


A 





B 


1, 2, 4, 8, 16 


C 


?., 6, 12, 24, 17 


D 


5, 10, 20, 9, 18 


E 


7, 14, 28, 25, 19 


F 


11, 22, 13, 26, 21 


G 


15, 30, 29, 27, 23 


H 


31 



A 
B 
C 
D 
E 
F 
G 
H 



A 


B 


C 


D 


E 


F 


G 


H 


1 


-1 




















-1 


5 


-2 


-2 

















-2 


5 





-2 


-1 











-2 





5 


-1 


-2 














-2 


-1 


5 





-2 











-1 


-2 





5 


-2 

















-2 


-2 


5 


-1 




















-1 


1 



Matrix C-B 



A is a root of the graph (Fig. 25). We evaluate the deter- 
minant of the minor of the ,\ , A posiL n 



5 


-2 


-2 
















2 


5 





-2 


-1 










2 





5 


-1 


-2 













-2 


-1 


5 





-2 





-576 = 2''-3=' 





-1 


-2 





5 


-2 
















-2 


-2 


5 


-1 



















-1 


-1 





The lexicographically least de Bruijn cycle. An -example 
is given in Ford (Ref. 4) to show that de Bruijn cycles 
exist for all ordi s. We start with a register of length n 
filled with zeros. Take for the vector a„,an-u • • ,fli its 
odd successor fl„_,,a„_z, ■••,«,, 1 if possible. If the odd 
successor has been used, i.e., 



fln.On 



,a. 



a„- 



,o„l 



we use the even successor. If we follow this construc- 
tion, we have a de Bruijn cycle of order n. If we list the 
de Bruijn cycles in lexicographic order, with 1 preced- 
ing 0, the cycle thus formed is the lexicographically least 
de Bruijn cycle. 



Theorem 8. The truth table of the lexicographically 
least de Bruijn cycle has weight Z (n) — 1. 

CoroUary. The truth table of lexicographically greatest 
de Bruijn cycle L is weight Z(n) — 1. 



Referencn 

1. Golomb, S. W., Shift Register Sequences. Holden-Day, Inc., 
San Francisco, Calif., 1967. 

2. Van Aardenne-Ehrentest, T., and de Bruijn, N. G., "Circuits and 
Trees in Oriented Linear Graphs," Simon Stevin, Vol. 28, p. 203, 
1951. 

3. Hall, M., Jr., Combinatorial Theory. Blaisdell Publishing Com- 
pany, Waltham, Mass., 1967. 

4. Ford, L. R., Jr., A Cyclic Arrangement of M-tuples, Report 
P-1071. Rand Corporation, Santa Monica, Calif., Apr. 23, 1957. 



I. Information Processing: Estimating the Correla- 
tion Between Two Normal Distributions When 
Only the Means are Known, /. fitenberger 

1 . Introduction 

Let X and y denote two jointly normal random variables 
distributed N (/a,, af) and N (/xz, <ri), respectively, with cor- 
relation p, and let {x„ «/,} be a set of n independent pairs 
of sample values. In SPS 37-50, Vol. Ill, pp. 287-289, a 
linear unbiased estimator of p is given by 




n 
— \ (Xi~ ^ 



i) sgn (y, - tii) 



+ 



n 



) sgn {Xi - tn) 



The estimator p has two disadvantages: 

(1) The moments of x and y must be known; a some- 
what unrealistic assumption. 

(2) Although the efficiency of p relative to the maxi- 
mum likelihood estimator is quite high when p is 
near zero, it is quite poor for p close to ±1. For 
example, for p = 0, eff (p) = 0.778, while for p = 0.8, 
eff(p) = 0.098. 

In practical situations, however, it often occurs that, 
although the variances are unknown, nevertheless the 
means are known. Under these conditions, and assuming 
without loss of generahty that ;ii = /u = 0, we propose in 






250 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



this article the asymptotically unbiased estimator of p 
given by 





- fl 


n 


. 1 

"-2 


S J^iSgni/, 

1 = 1 

2|xO 

i = l 


2 !/iSgnXj 

Sty.l 



(1) 



For p = ±1, !/< = ± i<Ti/<Tt)xu so that Xi sgn j/i = ±|xi| 
and yi sgn Xi — ±\yi\. Thus, p has the property that as 



p-* ±1, var(p) > 0. It will also be shown that although 
the asymptotic efficiency of p decreases as p increases 
(decreases) from zero, it does so at a much slower rate 
than does the efficiency of % For example, for p = 0, 
e&ip) = 0.778 and for p = 0.8, eff (?) = 0.504. 



2. Tho Asymptotic Varianc* of pond its Effici«ncy 

It is not difficult to show the following: 



where 



£(|x|) = <r.a, var(|x|) = «r?(l-a') 

E(|!/|)-a,«, var(|y|) = a?(l-a=) 

E(i|0<x< oo) = (7ia, £(i'|0<x< oo) = a? 



It is also well known that 



"I 



E{x\y) = p — y, 






To derive the mean and variance of x sgn y, we consider the conditional random variable x|0 < y < oo. One has 
£(xI0<y< oo) = £[E(x|y|0<y< oo)] =E\p~y\Q<y< oo 1 = p — 'a^a = pcr.a 

\_ "2 J <^2 

[2 T 2 2 

af (1 - p') + P=-^!/^|0 < y < 00 J = aHl - P=) + ^••^i = o 



Thus, 



Similarly, 



var (x|0 < y < oo) = (T? - pMa" = «r? (1 - pV) 

£(i|- 00 <y<0) = -pff.o 
var(x| - 00 < y < 0) = tr?(l - pV) 

It now becomes obvious that one has 

£ (x sgn y) = pvta, var (x sgn y) = a? (1 — (^V) 

£ (y sgn x) - pwja , var (y sgn x) = «r| (1 - p V) 

JPL SPACE PROGRAMS SUMMARY 37-5?, VOL. \\\ 



251 



Now let 



2 Xi sgntfi =u,, 2 «/i sgn x, ^-- u^ 



2\Xi\=-V,, 



^\yi\ = v. 



Eq. (1) can then be written as 



^ 

p 






As an approximation to the variance of p, we will take the asymptotic variance. Thus, one has 



var(?) = ^ 



l.i = iy = l 1 = 1 i = \ i = li = l 



(2) 



where each partial derivative is to be evaluated at u, = £(«;) and v, — E{Vi), for i = 1,2. Evaluation of the partial 
derivations gives 



d^ 1 



8^ 



3mi 


nai« ' 


dU', naoo 


a? 

9c, 


n<Tia ' 


dp -p 



One also has. 



var (u,) = mr? (1 - pV), var (uj) = n<r| (1 - a*), i = 1, 2 



COV (Ml, 


«.) 


= n cov (x sgn y. 


1*1) 




cov (u„ 


«.) 


- n cov (i/ sgn %, 


\v\) 




COV (u„ 


«.) 


— n cov (i sgn j/ 


\v\) 




cov (Uj 


«.) 


= n cov (t/ sgn x 


\A) 




COV (fl„ 


u.) 


= ncov(xsgnt/,j/sgn 


X) 



cov(o„Oj) = ncov(|x|,|y|) 



We will illustrate a method of computing the above covariances by deriving cov (i sgn i/, | x| ) in some de*ail. Noting 
that 



xsgny|x| = 



x=ifx,y^O 
-x'ifx^O,y$0 



252 



JPL SPACE PKOGKAMS iMNkHkhW 37-51, VOL. Ill 



one has 



-/:y>-[-^<r^a-^^s)]-'^} 



;n:--[-^(5-s^s)]-^- 



2:rcria2(l-p')^ 

since the sum of the two integrals in Eq. (3) equals a'i. 
By means of the transformation x -= ty, Eq. (4) becomes 

^("«°»-l'l'' 2>...,a-P-)-> i7>"^{-'''[s(Fr7)(5-^4)]}-^-"--' 

Integrating first with respect to y and then with respect to t results in 

£(xsgny|x!) = aMj p(1 - p=)^ + ^ + sin-' pi - ", 

cov (x sgn «/, I X I ) = a^'ff? [p (1 - p')^ + ^7 + sin-> p 1 - a? - vW = <^?«' [p ^1 " p')^ + sin' p-p] 

In a similar manner, one obtains the tollowing: 

cov (i/ sgn X. I y I ) = aW [p (1 - p')'* + sin- p-p] 

cov(xsgnt/,|i/|) = cov(ysgnx,|x|) = pv^a^il- p^) 

cov (x sgn !/. !/sgn x) = a^aiai [(1 — p^f^ + p sin*' p — p'] 

cov(|x|,|yl) - aV.aja - p')V4 + psin-'p - 1] 



"t (4) 



Substituting the above expressions in Eq. (2) and simpli- The asymptotic variance of r is 

fying finally results in 

, , , Defining the efficiency of p as 

The maximum-lii<:elihood estimator of p when the vari- 
ances are unknown and the means are both zero is 
given by eff(?)=^ 

2*i«/i 

i = l 



(il x< y t/ A^ Table 6 gives the variance of p and its efficiency for values 

iT, \Ti 7 of p between and 0.9. 

JPl SMCE PROGRAMS SUiMMARY 37-51, VOL. Iff 253 



Tabic 6. Variance end cfficiMicy of p 



p 


n vor 1$) 


•Hip) 


0.0 


1.2134 


0.771 


CJ 


1.2651 


0775 


0.2 


1.2050 


0J65 


0.3 


1.1072 


0.748 


0.4 


0.9755 


0JJ3 


0.5 


01156 


0.69C 


0.6 


0.6JS1 


0.645 


0.7 


0.4443 


0.585 


0.« 


0.2872 


0.504 


0.9 


0.09919 


0.380 



3. EtHmaling p 

Two sets of samples, {r } and {^i}, each containing 200 
sample values, were drawn from a table of random num- 
\x^% iu which the entries are independeat and distributed 
li (0, 1). The transformation 

y;=0.8xi+0.6y: 

was then performed. Coi:.>^uendy, each Xj and y^ can be 
assumed to be distributed ti (0, 1) with a anrdation of 0.8. 
Then p and r were calculated and found to be 

^ = 0.3221 
r = 0.8235 



Two new sets of 200 values each were drawn from the 
same table of random numbers and paired at random, 
so that one can assume that p = 0. The results for this 
case were 

P = 0.0828 
r = 0.0559 



J. Information Processing: The Distribution of 
tlie Ratio of Two Jointly Normal Random 
Variables, /. ^nwktvgv 

1 . Introduction 

Let X and y be randoin vanables, distributed N (m,, af) 
and N (mj, ai), respectively. The distribution of the ratio 
t = xiy is derived in Ref. 1, imder the assumption that 
X and y are independent. In that report, the hypothesis 
that ai = ffi was tested, using quantiles, sgainst the alter- 
native hypotheses that ^2 = ^^i, when «r, was unknown. 
The test statistics that were used in order to eliminate 
dependence on ai were ratios of the sums of two sets of 
quantiles and, in rader to specify a critical region, it was 
D'^^essary to determine the distribution of t. In this note, 
-. e derive the distribution of t under the assiunption that 
X and y are jointly normal with correlation p. It will be 
shown that the density function, g (t), is given by 



g(*) 



a.a.(l->.')^exp|-^[ " ,2^(1 _p2) 



IT (<^t* — 2p a^ott + <t!) 



01 



+ 



•^2^ {t»»i<T2 — p Wfi) + Oi (mjai — p WliCTj) 
(2:7)^ (ait* - 2p<r,a2t + (rf)* 



^«'^{-i[^ 



(mjt - m.) 



"j\ { _ . ^ np f '^^^ (Wig» ~ pW»2<ri) + gi (Wt2gi — pWtig;) "!) 
2p a.<r2* + <r?J / ( "^ [ <"»» (1 " p')** W^ " 2p <T,a2t + af )W J/ 



where 



F(x) 



(2^ 



l^£e-..d. 



(1) 



The distribution of the random variable t may also be a useful approximation when one is considering the distribu- 
tion of the ratio of the sums of two sets of sample values 



2u)i 



254 



iPl SFACE PROGRAMS SUMMAkY 37-51, VOL III 



where rti and nj are large. The central limit theorem, when applicable, assures us that the numerator and denominator 
of R are approximate'y normal and hence the distribution of R can be approximated by the distribution of t. It should 
be observed, however, that, whereas no moment of t of positive order is finite, the moments of R may exist. 



2. Discussion 

The joint density function of x and y is given by 

where 

1 



2t(THT2 (1 - p^)^ 

Putting x = ty, one sees that, since the Jacobian of the transformation is \y\, the joint density of t and y is given by 

g.(*,y)-K|y|exp|-2(r^L— ^^ — + -^^\\ _«<,<«, 

. ( 1 fj/^ (alt^ — 2p(T,<r2* + ffi) 2y [<T2t(mi(T2 — pn»2<^i) + oi (w»2<ri - .'>w»i0'2)] 

mlal — 2p minij ai<T2 + "tlo-i "]) 

= K|yiexp[-|y» + By + c] (2) 

where 

alt^ — p CTiiT2t + a? 



A 



B = 



(l-p^)a!ai 

(l-p=)<Tfai 



(3) 



_ — (mi<T| — 2p miTn^ 0102 4- mlvi) 
2(l-p»)<rfal 

By completing the square in y, Eq. (2) becomes 

The density function of * can now be obtained by integrating out y in Eq. (3). Accordingly, 

g(.) = Ke^(c + |l){/--,e,pr-f(,-|y]*-£,exp[-|(,-|)']*} (4) 

jn SPACE PftOGKAMS SU/MMAftY 37-51, VOL. Ill 255 



By use of the transformation Z = (A)^ (y — B/A), Eq. (4) becomes 



^<"^^^^^t;^[/i.(^^f)--^/r"((i^^f)--] 



(Ay^ 

exp (C) 



+ ■ 



7r<ria2A (1 — p^)''^ aidj, 
which, after simphfication, becomes Eq. (1). 

If m, = m.. - 0, g (t) takes on the relatively simple form 






gW = 



"■ (<''i'^ " " 2p ffio-a^ + <r;) 



(5) 



The transformation u = aj* converts g (f) in Eq. (5) to the 
density function of a Cauchy distribution of the form 



h{v) 
where, in this case. 



X - a. (1 - p=)^ 

fl = p<Ti 

Reference 



1. Eisenberger, 1., Tests of Hypotheses and Estimation of the Cor- 
relation Coefficient using Quantiles I, Technical Report 32-718. 
Jet Propulsion Laboratory, Pasadena, Calif., June 1, 1965. 



K. Astrometrics: Pulsar Observations, R M. Goldstein 

1. Introduction 

Two of the recently discovered (Ref. 1) pulsating radio 
sources, or pulsars, have been observed at the Jet Propul- 
sion Laboratory's Goldstone Deep Space Communication 
Complex (Mars deep space station). The signals from 
these pulsars are known (Refs. 1 and 2) to have t^xtremely 
regular repetition periods, although the amplitude within 
a pulse and from pulse to pulse varfes erratically. The 
radio frequency of each pulse has been observed (Ref. 3) 
to decrease with time, following the dispersion relation- 
ship of electromagnetic propagation through a medium 
containing free electrons. Presumably, the signals near 
the source contain a wide band of frequencies. Since the 
group velocity for waves in such a medium is less for 



the lower frequencies, the received signals have the form 
of a sliding tone, or whistle, with the higher frequencies 
arriving before the lower. 



2. Magnetic Field Measurement 

The familiar equation for index of refraction (Ref. 3) is 



n- = l- 



mto 



(1) 



m 



B« 



where N is the electron density, <o is 2ir times the fre- 
quency, e is the electron charge, m is the electron mass, 
«o is the permittivity of space, and B is the component of 
any magnetic field along the line of sight. The ib sign 
depends on the relation of the direction of the circularly 
polarized waves to the direction of the magnetic field. 

This fact gives us the possibility of measuring directly 
the interstellar magnetic field, averaged along the Kne of 
sight. By observing the change of the time of arrival of 
the pulses with the antenna switched from left- to right- 
handed circular polarization, a measure of the field is 
obtained. Although this method is not as sensitive as the 
utilization of Faraday rotation (Ref. 4), it does not require 
a polarized source. 



3. The Data 

The data collected is in the form of spectrograms of the 
signals. A bandwidth of 3 MHz, centered at 84 MHz, and 
with a resolution of 50 kHz was investigated. Time was 
divided into 15.4-ms slices, and an independent spectro- 
gram was taken for each slice. Because of the periodic 
nature of the pulsars, the signal-to-noise ratio can be 
enhanced greatly by averaging together corresponding 
sets of spectra from many pulses. 



256 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



82 4 85 4 

FREQUENCY, MHz 

Fig. 26. Set of spectra of CP 1 91 9 taken in successive 

1 5.4-ms time intervals and averaged 

over 1350 pulses 



A sample set of spectra from CP 1919 is given in Fig. 26. 
It shows the time-frequency history of the signals, aver- 
aged over 1350 pulses. Signals from CP 1919 are icen to 
enter the spectrograms from the high-frequency side and 
move rapidly through them towards the low. It follows 
from Eq. (1) that, if the observed eflEect is indeed caused 
by dispersion, the relationship between f and t is 



{t - to)^ 



(2) 



The data from each set of spectrograms was processed to 
determine the constants k and to by the method of least 
squares. The central frequency of the pulse in each spec- 
trum was obtained by convol'/ing the data with the ex- 
pected pulse shape— a maximum-likelihood procedure if 
the shape is perfectly known. 



The results of the least square fit is given in Fig. 27. 
As can be seen, there is a close fit to the theoretical 
curve. Note that, at 83 MHz, the pulse has been delayed 
(dispersed) by almost 7^ s. Table 7 summarizes the valu^ 
of k obtained, along with the corresponding frequency 
sweep rates and integrated electron densities. 



85.0 

84.5 

. 84.0 

8 83.5 
UJ 

U. 

83.0 

825 


N 


V 
















^ 


X 


■^ 
















nJ 


X 


















'^ 


\ 


















•s 



7.10 715 720 725 730 735 740 7,45 7.50 

TIME.S 

Fig. 27. Least square fit of the function f = k/it — fo)^ 
to the data from CP 1919 



From Eq. (1), it follows that the change in time of 
arrival, AT, that occurs when the mode of circular polar- 
ization is switched is 



AT 



_4{t-t„)eB 



o>m 



We found that the measured AT was not statistically sig- 
nificant for either source. However, an upper limit for the 
integrated magnetic field can be set. From the standard 
deviation of the time-of-arrival estimates (0.0006 s), that 
of the magnetic field measurement is found to be 

±0.62 X 10-' G 

The time jf -arrival measurements ha\3 also allowed 
us to lelertnine the repetition period of the pulses to 
surprising , accuracy. A very small difference of timing 
betw<:«n tlie pulses and the signal sampling equipment 
produces a cumulative drift of the time-frequency tra- 
jectory of the pulses. Our measurements, corrected for 
the earth's orbital velocity and rotation, are given in 
Tabb 7. Fo ■ CP 1919, the period matches very closely to 
that publisl ed in Refs. 5 and 6, in distinction to that of 
Ref. 1. 



Table 7. Values of k, sweep rates, ond integrated 
electron densities 



Pulsar 


MHi 


dt/dtai 
84 MHi, 
MHi/( 


SNdl, 
pc/cm' 


Period, ( 


CP 191V 
CP 0831 


226/(0" 
231 (»)" 


5.81 
5.S6 


12.4 
12.9 


1.3373008 ±3 
1.2737620 ±3 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ml 



257 



Using the best-fit relation (Eq. 2), we displaced each 
spectrum of a set to a common frequency origin and then 
averaged them. The results are given in Figs. 28a and b. 
These figures, then, show the spectral characteristics of 
the average pulse. 



4. Smith, F. C, Nature. Vol. 218, p. 325, 1968. 

5. Radhakrishnan. V., et al. Nature, Vol. 218, p. 229, 1968. 

e. Moffet. A. T., and Ekers, R. D.. Nature, Vol. 218, p. 227. 1968. 

7. Lyne, A. G.. and Rickett, B. J.. Nature, Vol. 218, p. 326, 1968. 



The frequency structure of these two sources is quite 
similar to the time structure already reported (Ref. 7). 
They both have a basic triangular shape and sudden onset 
and termination. There was no evidence of any power 
outside of the main pulse. 

The peak power density, average power, and average 
bandwidth of these pulsars are given in Table 8. 



TabI* 8. Peak power density, average power, 
and average bandwidth 



Pulsar 


PMk pewM dMiity, 
u/Hi/m' X 10" 


Avarog* power, 
u/in= X 10" 


Avsrog* 


CP1919 
CP0S34 


63 
53 


49 
34 


77 
69 



References 

1. Hewish, A., et al. Nature, Vol. 217, p. 709, 1968. 

2. Davies, J. G., et al. Nature, Vol. 217, p. 910, 1968. 

3. Stratton, J. A., Electromagnetic Theory, p. 329. McGraw-Hill 
Book Co., New York, 1941. 




84.2 



O 

X 



X 



60 
50 

40 
30 
20 



10 



Z 
Z} 

X 

3 



84.6 84.8 8S.0 

FREQUENCY, MHz 



85.2 



85.4 



-10 

















/ 


V. 










/ 












/ 




\ 








/ 




\ 








/ 




u 


/^ 


/-\ 


^"S 






v/ 


\y 





84.2 84.3 84.4 84.5 84.6 64.7 

FREQUENCY. MHz 

Fig. 28. Instantaneous spectrum of the average 
pulse of: (a) CP 1919, and (b) CP 0834 



64.6 



L. Astrometrics: Optimum Range Gates, A. Garsia,^ 

E. Rodemich, and H. Ramsey, Jr. 

1 . introduction 

Let 0s denote the family of functions A (x) satisfying 
the following conditions: 



A (x) is positive definite and continuous on 
the real axis 

A(x)=0 V |x|^8 

A(0) = 1 

Our problem is to calculate 



C» = 



max 



{> 



{x)\'dx 



(la) 

(lb) 
(Ic) 

(2) 



This question has arisen in trying to maximize the aver- 
age power of the received signal in JPL's planetary radar 
system. The main conclusion is that the present system is 
nearly optimum from the analytic standpoint and cer- 
tainly the best from the standpoint of equipment sim- 
plicity. In radar mapping, A (x) depends on the hardware 
used. Since 



£|AWI= 



ax 



(3) 



is proportional to the average power received, per unit 
power sent, any A (x) which maximizes this integral would 
correspond to a best possible hardware. The present ver- 
sion uses A (x), a triangle function obtained as the correla- 
tion function of a maximum-length shift-register sequence. 



'Consultant, Mathematics Department, University of California, San 
Diego, California. 



258 



jn SPACE PKOGRAMS SUMMARY 37-51, VOL. Ill 



Let (Si be the family of functions A (x) satisfying 



It can be shown that V N ^ 1 



A (x) is positive definite and periodic of period 2tt 

(4a) 

A(x) = 0for8^|x|^,r (4b) 

A(0) = 1 (4c) 

A companion to the above problem is that of finding 



Cr 



1 



N + 1' 



D«= max [ |A(x)pdx 



We shall see that the two problems are related and, 
indeed, when 8 < tt, 

C«<Da 
Furthermore, it can be shown that 



lim TT- = 1 



It can be seen that both these problems are special 
cases of a general question which can be formulated on a 
large class of abelian groups. The first arises when the 
group in question is the real line, and the second arises 
when the group is the circle. 

We shall not go deeper here into these matters, but we 
shall be guided by these considerations and refer to the 
first problem as the "line" case and the second as the 
"circle" case. When the integers or the Nth roots of unity 
(for a fixed N) are taken as the basic group, we obtain 
two problems which are closely related to the lin*; and 
circle cases, respectively. 

The case of the "integers" can be stated as follows. We 
define £ti as the family of sequences {o„} such that 



and indeed 



i'nivTT'^"^' 



(6) 



(7) 



Inequality (6) can be used to get some very sharp upper 
bounds for Ci and, therefore, since (a« it can be easily 
shown) C« = 8Ci, also upper bounds for Cj V 8 > 0. 

In this article, we shall establish, among other things, 
that for the ''"ne problem there is a unique maximizing 
function an lat this maximizing function can be calcu- 
lated to any degree of accuracy by a successive approxi- 
mation method which is suitable to use with a computer. 
We have not succeeded in finding an explicit formula for 
the maximizing function, although such a function can be 
shown to have some rather remarkable properties. In fact, 
we shall see that the extremal function for the line case 
is also the solution of several other maximum problems. 

The circle case is open. At this moment, we are not in 
possession even of an existence proof let alone uniqueness 
for the maximizing function. As we shall see, the circle 
case leads to some very interesting, and as far as we know, 
unsolved problems for the circle group. 



2. The Factorization 

A wide variety of functions of 9^ and 6« can be ob- 
tained as follows. For the line case, we start vwth a real 
or complex valued function p (x) which satisfies 

j8 (x) is defined in ( — oo , + oo ) and is 

square integrable (8a) 



{a„} is positive definite 
fln^O V|n|>N 


(5a) 
(5b) 


;8(x) = 0V|x|^| 


(8b) 


a„-l 


(5c) 


/ |;3(x)| = dx = l 

J-S/2 


(8c) 


We then seek 




then set 




n 
h = max 2 a? 
{an} €£„"-« 




\{x)=pl3(x + t)pjt)dt 


(9) 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



259 



It is easily seen that such A{x)e0i. Indeed, Proper- 
ties (lb) and (Ic) are obvious and (la) follows from the 
identity 



^A(Xi-X,)|J,-|j|^ 



/8(x. +t)l. 



dt 



Similarly, in the circle case, we start with a function 
« (x) satisfying 

a (x) is periodic of period 2ir and 



square integrable 


(10a) 


a(x)=Ofor|-^|x|^:r 


(10b) 


/ |a(x)pdx = l 

y-6/2 


(10c; 


We then set 




T{x) = j a{x + t)a{t)dt 





Again, it is easily shown that r (x) e (Sn for any such choice 

of a(x). 

It is compelling at this point to ask whether or not such 
representations are always possible for functions of ^e 
and (?a- It is clear that this would introduce a consider- 
able simplification on our maximum problem, since the 
conditions on p{x) and a{x) are very simple and easy 
to handle. 

However, the remarkable fact which distinguishes the 
line case from the circle case is that this factorization 
holds in the former but not in the latter case. 

The factorization result can be stated as follows: 

Theorem 1. Given a function a(x) which is positive 
definite on the real axis and vanishes for |x| ^8, then 
there is a function p (x) which is square integrable and 
vanishes for |x| > 8/2 such that 



'^'^'f'l 



li(x + t)p (t) dt 



(11) 



This result can be stated and proved as a theorem on 
entire functions of exponential type (Ref. 1, pp. 124-126). 
In tills form, it is also stated without proof in a paper of 
Krein (Ref. 2). However, there is no need to use such 



sophisticated tools. We give a proof in this article using 
a method which we discovered quite independently of the 
above mentioned works and which yields at the same 
time an interesting viewpoint. We also indicate briefly 
why the corresponding factorization is not, in general, 
possible in the periodic case. From these considerations, it 
follows that (when S^tt) 

Examples may also be given which show that strict in- 
equality holds. 

Some sort of substitute for the factorization can be 
established in the periodic case. It reads aa follows: every 
function of Cj can be written in the form 



/> 



r (x) = / a (x + () A (t) dt 



(12) 



where A (t) e <96,, a (x) is periodic and vanishes for 
8,^ |x| -^ir, and 8i + 82^8. Furthermore, the Fourier 
transform 



1 /■' 
«(n) = 2; c""a(x)dx 



has all its zeros on the real axis and is non-negative at 
every integer where the Fourier transform 



A(n)=5- / °°e""A(x)dx 



is di£Eerent from zero. 

These conditions are necessary and sufficient for the 
function r (x), given by Eq. (12), to belong to the class 6^; 
unfortunately, they are not easy to work with. 

3. Symmetrization 

We have seen that every function A(x)€S'« can be 
written in the form 

/+00 

I3{x + t)p{t)dt (13) 

where j8 (x) satisfies the conditions 

/?(x) = 0V|x|^4 (14a) 

/ |)8(x)|^dx = l (14b) 

y-00 



260 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



This result not only does simplify considerably our maxi- 
mization problem but can also be used to narrow down 
our search for the maximal function. 

Indeed, we show here )3 may, without loss, be restricted 
to be non-negative and symmetrically decreasing. More 
precisely, we show: 

Theorem 2. Let ^\ he tlie subclass of Sa of functions 
a(x) uhich admit a factorization of the form Eq. (13) 
with a p{x) satisfying in addition to the conditions in 
Eq. (14) also 



/3W^0 



(15a) 







p{x)=,Ji-x) 








(15b) 






fi{x)^/i (y) when ^ 


= x^y 




(15c) 


then 
















max 

A e ^7« 


ljX{x)\^-dx = 


max 

A€X7S 


f> 


.Wl = 


dx 


4. Existence 













Symmetrization yields a very quick path to existence of 
the maximizing function. Indeed, let 



A„(x) 



J- 



x + t)l3„{t)dt 



(16) 



be a sequence of functions in the class rj'f, (i.e., each j8„ 
satisfies the Conditions (14) and (15) such that 



lim / |A„(x)| = dx = C« 
Since each Pn is non-increasing for x ^ 0, 



(17) 



i: 



xplix)^ PHt)dt^l 



so, by symmetry, 



Pnix)^ 



{\x\y^ 



V x^O 



(18) 



By a well known argument, we can produce a function 
/? (x) on ( — 00 , -I- 00 ) that is symmetric and non-d '•creasing 
for x^O and a subsequence {j8„^(x)} such that 



lim/3,,(x) = i3(x) 

k~*oo 

at all points of continuity of p (x). 



Note that, by Eq. (19) and Fatou's lemma, /3(x) will 
also satisfy 



(20a) 
(20b) 



/3(x)-0V(xi= I 
r'°li= (x) dx ^ 1 
Furthermore, from Ineq. (18), we get 



So at least, for x^O, from Lebesgue's dominated con- 
vergence theorem, we get 

I3n^{x + t)li„^{t)dt= I fi{x + t)p(t)dt 

00 . ~<X 



In other words. 



lim A,.(x)-A(x) 



when 



=/, 



A (x) - / ^ (x + *) ] (t) dt 



We have | A„ (x) | ^ 1 V n, so again by dominated 
convergence 



= iim r°°[A„,(x)]^dx=r[A(x)]= 



dx 



However, this result, combined with Eq. (20) and the 
definition of C«, implies that equality must actually hold 
in Eq. (20b). Thus, A (x) must belong to 9^ and indeed 
must be a maximizing function. 

The above argument is the one by which existence of 
the maximizing function was first obtained. Later on, we 
found another path to existence which we shall present 
in the Subsection 5 since it follows a rather interesting 
and fruitful line of reasoning. 



(19) 5. The Integral Equation, Another Path to Existence 

Before proceeding with our second proof of existence, 
we should observe that the considerations at the end of 



jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



261 



Subsection 4 yield a result which will be rather crucial 
for our later considerations, namely: 

Theorem 3. If 

A(x)^rii{x + t)pit)dt, pix) = OfoT\x\^j 

is a maximizing function in 9s, then fi (x) satiijies the 
integral equation 



C,,s{x)-^r\^{x-t)p{t)dt V|x|^-5- 



(21) 



In Subsection 6, we shall show that this function is unique. 

6. Uniqueness 

It will be convenient at this point to work with 8 = 1. 
This involves no loss since we can always reduce our- 
selves to this case by a change of scale. We shall use here 
and in the discussion that follows 9 and C to mean cf?, 
and Ci. 

Our point of departure to show uniqueness of the maxi- 
mizing function will be Eq. (21). However, we shall write 
it in the form' 



Cp{x)^pxpXp{x)x{x) Vx 
where x indicates convolution and 



(22) 



xW = 



llfor|x|^2 



0for|x|>2 



Let us say that a function /3 (x) on ( - co , + oo ) is "admis- 
sible" if and only if it satisfies the conditions 



p{x) = p{-x)^0 V x 

C 

P' (x) dx^l 



I 



p{x)^p{y) XO^x^y 



|3(x) = 0V|r|^2 



(23a) 
(23b) 
(23c) 
(23d) 



'Recall that /3(x), for the maximizing function, has been shown to 
be symmetric. Thus, A(i) = /J X fi{x). 



Our uniqueness result can then be stated as follows: 

Theorem 4. There is at most one admissible function 
P (x) which sati.fies the equation 



kp(x) = pXpXp{x)x(x} 
for some A.:^2/3. 



(24) 



From this result, it follows immediately that there is 
only one function A (x) in each class (9 ^ (see Subsection 3 
for definition of 0'^) which maximizes the integral 



/(A) 



A=(x)dx 



(25) 



Uniqueness in f}i can be established by showing that 
every A e 0i for which I (A) = Cj is necessarily in f}'^ . This 
given, it is easy to show that the equation 

CiP{x)^pXpXI3ix)x{x), p{x)=p(-x) 

has a unique solution p (x) satisfying the conditions 



/; 



;8^(x)dx = l 



^(x) = 0V|x|^2 



(26a) 



(26b) 



7. The Successive Approximation Method 

In this subsection, we shall denote our extremal func- 
tion by Pn. We know that this function satisfies the 
equation 

A„;8„(x)=)8„X)3oX)3„(x)x(x), 



X(x) = 



[Iforjxl^^ 
fOfor|x|>jr 



(27) 



where Ao is the extremal constant' 

This given, we can try to obtain Po by a successive 
approximation method of the form 



knPn.l{x) = PnX PnX Pn{x)x(x) 



(28) 



This is the constant we denoted by C, in the Introduction. 



262 



jn SMCE PROGRAMS SUMMARY 37-51, VOL III 



where A„ is determined each time by the condition that 

ii/8n.,ii=rrvi..«d^]'^=i 

This is indeed the method we shall use, and we shall 
show that the iterates in Eq. (28) do converge geometri- 
cally to /?o when the initial function p^ is taken to be 
sufiBciently close to |3„ itself, in particular when /3i = x- 

This is an obvious method to use; however, the esti- 
mates needed to complete it are not so obvious and are 
rather delicate. 



(29) 
(30) 



Let us introduce for each n ^ 1 the function 

A»(«) = y3,(x)-9,i3„(i) 
where 

<>» = (i8„,j8„) = r^i3™(x)j8„(x)dx 
Since j9o is normalized, we see that for each n 

(A„,/8„)=/ A„{x)po{x)dx=0 



Our goal is to show that under suitable circumstances 
we have 



II A„ II V n 



(31) 



for some constant < p < 1. This, of course, implies that 

li )3n - /So II -* 

geometrically as n -» oo . 
For each admiss'ble j8 set 

F(j3)=/3X|8X/8(x)x(x) (32) 

Inequality (31) can then be written in the form 

II F(|8«) - (F(/3„),|9„)|3„ II ^p I! A, || || F(;8,)!| (33) 

A simple geometric argument shows thut for any a > 
and any F 



|F-(F,|3o)|8o|| ^ ||F-affo| 
II Fl! - 



(34) 



Indeed, Ineq. (34) (when it is not trivial) simply says 
that the sine of the angle between the directions of )8o 
and F is always less than the sine of the angle between 
/So and ihe tangent to the circle through F with center 
at the point a/So. 

This given, we can assure Ineq. (33) if we can find an 
a for which 



F (|3,) - a^o II ±Sap II An I 



(35) 



To this end, using the Relation (29) into (32), in view of (27), we get 

F(/8n) = (ej^o/Jo + 3eji3„ X jSo X A„ + 3«„)3o X A„ X A„ + A, X A, X A„} x 

This suggests taking o = S J Xo in Ineq. (35). With this choice of a, we get 

II F (/?„) - aiSo II ^3(?S II XiSo X /?o X A, 11 + 3», ii x,«. X A„ X A, || -t- || xA. X A„ X A« || 

We shall need to estimate the three terms on the right-hand side of this inequality as accurately as possible. 
For the first term, we use the eigenfunction expansicn 

/3„ Xpo(x-y) = Ac /So (x) /So (y) + 2 Av <^v (x) ^. (y) 



(36) 



f37) 



and obtain, using the orthogonality of A. and jSo, 

||/SoX)8oXA,|!'^\?||A.i|» 

jn SPACE PROGRAMS SUMMARY 37-51. VOL. Ill 



(38) 
263 



where \c denotes the next largest eigenvalue corresponding to an even eigcnfunction of the kernel po X /So (^ — «/) in 
[-1/2,1/2] X [-1/2,1/2]. 

For the second term, we observe that by Schwarz's inequality 

r \ r P" X^n(x- tjAn{t)dt\dx^ ["' r [/8„ X An(x - t)Vdtdx\]An f = /?o X p. X A„ X A„ (0) || A„ ||^ 

J-l/i LJ-«! J J-Vl J-'/ll 

Using again the eigcnfunction expansion Eq. (37), we then get 

||/S„XA„XA„||^(A,)V4||A„p (39) 

The last term is easiest to estimate. We get 

||A„XAnXA„||^A„||A„||' (40) 

We see that for the method to bt accomplished we need 

36% Ke li A„ II + 3e„ (A.)^ II A„ IP + \„ II A„ II' si $1 \„p II A„ II 

Simplifying and noticing that 

||A„||» = l-fiJ 
we get 

In Subsection 6, we showed that X,/Ao — ^ and we proved A,, — %. We see then that Ineq. (41) will be satisfied if 

Note now, if we do establish this relation with a p < 1, then we shall have 

1 - ei.r = II A„.. II' ^ p' II A„ II' < (1 - Bl) 

In other words, the 6„'s increase. 

However, since the function 

3 1 3 /l\'^ /SV'i 1 

decreases as increases, we see that in order for lueq. (42) to hold for all n, all we need is to assure that it holds for 
fi = l. 

From the form of g (9), it is easy to see that if )3i is sufficiently close to j3o, we shall have 

g(«i)<l 

264 JPL SPACE PROGRAMS SUMMARY 37-51, VOL III 



To verify this relation for 



;8,W = x(x) = 



ilfor|x|^^ 
[Ofor|x|>5- 



we need a careful estimate of 6 for such a function. This can be achieved by means of the following inequality, the 
proof of which is immediate. 



Lemma 1. Let <j>{x) be measurable in [a,p] and let 

0<a^4>(x)^b V xe[o,/3] 

re 



p 



-I 



^-ix)dx^l 



then 



1 fP 
-=— <i>{x)dx^ 

-" Ja 



1 +ab 
a + b 



For our extremal function j8o, the hypotheses of the 
lemma are satisfied with a = (2)''V2 and b - l/(Ao)"^'. 
Indeed, fir(x) is symmetric around zero and does not 
increase away from zero. Furthermore, we know 

and from the eigenfunction expansion Eq. (37), we get 
1= f'^ Pi (x) dx = /3„ X /3„ (0) ^ X„ PI (0) 

Simple arithmetic then gives 

e. = (A..x)-r^4x)dx^0.965 

Substituting this value of 0, in g{0), we then get 

g (00^0.831 

Thus, the convergence of the iterates in Eq. (28) for this 
choice of )3, is established. 

Remark 1. The method establishes more than the con- 
vergence of Pn to j8„. We can write 



I Pn*\ — Po 



l + <?. 



If we have 

\\^n.^\\'^p\\^n\\ 

then, since this implies 6n*\ — ^», we get 



1 + e 



II ;8„., - ;8„ IM p' Y+lt. " ^" ~ ^» II" - P' II '^" ' ^' II' 
The triangle inequality then gives 

Ili8„-i3„||^j47ll)8--)8,|| Vn 

In other words, we can tell how close we are to /3o at 
any step of the iteration by seeing how close is Pnti to p„. 

It is not difficult to see that vhe estimates presented 
in this subsection yield the following results: 

Theorem 5. Let p be any admissible functkn and let 

B = (p.,P)if 



then 



l/3-i3o||^ 



1-gW 



F{p) 



\P(P) 



-P 



Theorem 6. Let F(.r) be a (possibly nonlinear) oper- 
ator on some hilb*;ft space ^K. LH j8o be an eigerfunction 
of F (x), i.e., let 

F(p„) = XoP„ 
Assume that for some 8 > 



P- 



/||x-(x.X?i9o||^8A'a «ll*-(*.^o)^o||j 
\ xe9(f.||x||.= l f 



<1 



JPL SPACE mOGKAMS SUMMAUY 37-51, VOL III 



265 



Then the sequence of iterates 

F(Pn) 



^■♦l 



\p(p'.n 



does converge towards j8o, and indeed 



II j8, - p, II ^p-> II ,9, - /3„ II ^-f^— II /J, - ja. 



provided 



|/8.-(j8,)3„))3„||^8 



Using theorem 5, we were able to calculate our ex- 
tremal constant to 24 decimal figures. TTie result ot this 
calculation gives 

C. s 0.686981293033114600949413 

The value attainable with the present range-gated radar 
is C, = 2/3, obtained with the triangular correlation func- 
tion of a maximal-ler ,^th shaft-register sequence. Thus, 
the present system is near optimum. 

8. Some Final Remarks 

It can be shown that our extremal function j3o(x) is 
in ( — 1/2, 1/2) the restriction of an entire function. This 
follows from the integral equation by successive differ- 
entiation and a judicious estimation of the resulting 
terms. Although straightforward, this calculation is quite 
intricate and is omitted. 

It would be interesting if fi.> (x) could be expressed in 
terms of familiar functions or if d, itself turns out to be 
related to some of the classical constants. 

It is interesting to note that our final result in Sub- 
section 7 can be put in the form 



\Pn-pA^f li |8---i8. 



(43) 



where jSn (n = 1, • • • ) is the outcome of the nth itera- 
tion of the "juccessive approximation method. Pa is the 
function we want to calculate, and p is a constant we 
can explicitly estimate. Thus, we can calculate our un- 
known function and constant with any degree of accuracy. 

However, our proof of Ineq. (43) is non-constructive. 
The same holds for our existence proof for Pa. Piis says 
that it is quite possible to obtain explicit estimates 



(thereby estimates that can be used in the applications) 
by entirely non-constructive arguments. 

References 

1. Boas, R. P., Entire Functioiis. Academic Press, New York, 1954. 

2. Krein, M., Compter Rendus (Doklady) de I' Acad, des Set. de 
/' U.R.S.S., Vol XXVI, No. 1, pp. 17-22, 1940. 



M. Data Compression Techniques: Product Entropy 
of Gaussian Distributions, E. C. Posner, 
E. R. kodemich, and H. Rumsey, Jr. 

1 . Introduction 

This article is a study of the product epsilon entropy 
of mean-continuous gaassian processes. That is, a given 
mean-continuous gaussian process on the unit interval 
is expanded into its Karhiinen expansion. Along the fcth 
eigenfunction axis, a partition by intervals of length c* 
is made, and the entropy of the resulting discrete distri- 
bution is noted. The infimum of the sum over k of these 
entropies subject to the constraint that Itl^i" is the 
product epsilon entropy of the process. It is shown that 
the best partition to take along each eigenfunction axis 
is the one in which is the midpoint of an interval in 
the partition. Furthermore, the product epsilon entropy 
is finite if and only if 2 X* log l/A* is finite, where X* is 
the kth eigenvalue of the process. When the above series 
is finite, the values of e* which achieve the product 
entropy are found. Asymptotic expressions for the prod- 
uct epsilon entropy are derived in some special cases. 
The problem arises in the theory of data compression. 

The work is motivated by the problem of data com- 
pression, the efficient representation of data for the pur- 
pose of information transmission. We shall consider the 
case in which the data to be represented con'ists of a 
sample function from a mean continuous gaussian process, 
X, on the unit interval. Our basic problem is how to trans- 
mit (over a noiseless channel) information as to which 
sample function of X occurred. We assume that the re- 
cipient of the transmitted data has full knowledge of the 
statistics of the process. In particular, he knows the 
Karhiinen expansion (Ref. 1) of the process; namely 






(1) 



where the tfk are mutually independent-unit normal ran- 
dom variables (they determine which sample function of 
the processes occurred); the <}>k (t) are the ((Mthonormal) 



266 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



eigenfunctions of the process; they are known a priori, 
ds are the Xi, which are the eigenvalues of the process. 
We note that the series in Eq. (1) converges with prob- 
ability 1. If 

Ris,t) = E[Xis)Xit)] (2) 

is the covariance function of the process, then 

R(s,t) = 2Afc</.fc(s),;,fc(t) (3) 

the convergence being uniform on the unit square. Fur- 
thermore, R{s,t) is jointly continuous in s and t. The 
functions <j>k are continuous and satisfy the integral 
equation 

where the kk are non-negative and are the eigenvalues 
of this integral equation. It follows that 



SAk == / R{s,s)ds< 00 



(5) 



In the special case when all but a finite number of the Kk 
are zero, the process X is just a finite dimensional gaus- 
sian distribution. The interesting cases, from the point 
of product entropy, turn out to be the one-dimensional 
processes and the infinite-dimensional processes. 

In the data compression problem, we wish to represent 
the sample functions of the known process X. By Eq. (1) 
we can fully describe a sample function X{t) by speci- 
fying the values of the i/* which occur in Eq. (1). We 
shall call t/* the projection of the process along the fcth 
coordinate axis. 

Our final assumption concerning the nature of our 
problem is the requirement that the information which 
is transmitted must be adequate to locate the sample 
function in some set of (Lj) diameter at 'nost e. 

The data compression procedure we propose is as fol- 
lows: Observe X (t) and compute its projections, «/*, along 
the coordinate axes. Quantize the fcth coordinate axis into 
intervals of diameter at most €*. For each k, transmit the 
index of the interval which actually occurred. If the 6* 
satisfy 



2£?: 



(6) 



to within a set which is a hyper-rectangle of diameter 
at most e. 

Our main concern in this article is to study the entropy 
of the above procedure. We observe that this entropy 
does not depend on the eigenfunctions, <j>k, of the process, 
but only on the eigenvalues, A*. This is because any two 
mean-continuous gaussian processes with the same Ajt pos- 
sess measure-nreserving isometrics between the Hilbert 
spaces generated by their <f>k. It follows that assumptions 
about stationarity, band-limiting, etc., are relevant only 
insofar as they help estimate the A/t. 

A definition of epsilon entropy for mean-continuous 
stochastic processes is found in Ref. 2. The entropy de- 
fined in Ref. 2 is upper-bounded by the product epsilon 
entropy considered here; for it uses partitions by arbi- 
trary measurable sets of diameter at most e, instead of 
hyper-rectangles of diameter at most e. It can be shown° 
that the epsilon entropy of a mean-continuous gaussian 
process on the unit interval is always finite. It turns out, 
however, that product entropy is finite if and only if 
i: At log I/Afc converges. A further discussion of data com- 
pression in a general setting is in preparation.' 

Subsection 2 treats the one-dimensional case. We show 
that the best e-partition (the e-partition with least en- 
tropy) is that partition by intervals of length € which 
contains the interval ( — e/2, e/2). We treat the cases of 
large and small € separately. Techniques of analytic func- 
tion theory are necessary. 

In Subsection 3, we show that the product epsilon 
entropy, /t(X), of a mean-continuous gaussian process 
on the unit interval is finite if and only if 2 A* log 1/As is 
finite. In case /« (X) is finite, we give a product partition 
whose entropy equals /e(X). 

Subsection 4 gives an asymptotic form for /t (X) when 
the eigenvalues satisfy a relation of the form At '~' Blc". 
In particular, for the Weiner process, /« (X) ^ C/c^ as 
6 -* 0, where C is a constant between 6 and 7. 

Subsection 5 considers a general lower bound L«(X) 
for /. (X). We show that if 



2 A»-0(nA„) 

* = !> 



then, with probability 1, when the intervals of uncer- 
tainty are known, the original sample function is known 



'Posner, E. C, Rodemich, E. R., and Rumsey, H., Jr., Epstion 

Entropy of Gaussian Processes (to be published). 
'Posner, E. C, and Rodemich, E. R., Epsilon Entropy and Data 

Compression ( to be published ) . 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



267 



then the ratio Jt/Lt remains bounded as e tends to 0; 
and if 



2 Xn=o(nA,) 



then 7, ^ L, as € -> 0. 



The term "epsilon entropy" in the following lemma 
refers to the definition of Ref. 2: the epsilon entropy of 
a separable metric space with a probability distribution 
on the Borel sets is the infimum of the entropies of all 
partitions of the space by measurable sets of diameters 
at most €. 



This last result implies that, when 
i A» = o(iU,) 



product €-entropy is asymptotically as good as e-entropy 
for small c. As an application of our techniques, we show 
that for a stationary band-limited gaussian process on 
the unit interval, with well-behaved spectrum. 



h(X) 



iog4 



2 log log - 



For conciseness, the statement of the lemma neglects 
the behavior of the partition on sets of probability zero. 
More precisely, the sets of positive probability in an 
optimal partition can be intervals of length e with sets 
of probability zero omitted. 

Lemma 1. Let X be the real line with a probability 
distribution ix on the Borel sets of X such that n has a 
density p(x) which achieves its maximum value at 0, 
is monotonic on (0, oo), and even [p{—x) = p{x)]. Then 
the c-entropy of X is attained only by a partition which 
consists of consecutive intervals of length e (or one which 
agrees with such a partition on the interval supporting ^ 
if this interval is finite). 



Subsection 6 presents an application of theorem 5 to 
band-limited processes. 

2. The One-Dimensional Normal Distribution 

In this subsection, we consider a normal random vari- 
able of mean on the line. We show ihat the c-partition 
of the line with least entropy is the "centered partition 
consisting of non-overlapping intervals of length e, and 
containing the interval ( — c/2, e/2). 

We need a series of six lemmas to prove this result, 
which is theorem 1. The first lemma shows that we need 
only consider portions consisting of non-overlapping 
intervals of length e. Lemmas 2-3 show that the centered 
partition is best (has smallest entropy) if e — 3. Lem- 
mas 4-6 are devoted Jo showing that the centered parti- 
tion is best when c^tt. 

We begin by defining the entropy of a countable par- 
tition U of the real line under a probability measure: 
Let the probabilities of the sets of V be denoted by pt. 
Then the entropy H (U) oi the partition 17 is the (Shannon) 
entropy of the discrete probability disbibution {pi), 
that IS 



H{U) 



= ^p.log^ 



The hypothesis of unimcJality of the description is 
essential for the conclusion of lemma 1. ITie distribution 
need not be symmetric, however. This assumption was 
used to simplify the treatment of a partition in which the 
interval containing zero has length less than €. In the 
problem at hand, lemma 1 implies that for gaussian dis- 
tributions, the epsilon entropy is attained only for a par- 
tition by consecutive intervals of length e. We are thus 
led to the '^ollowing definition: 



Definition. Let X be the real line with the probability 
distribution of a normal random variable with mean zero 
and variance 1. Let h{€,a) be the entropy of the parti- 
tion of X by intervals of length e centered at the points 
€(it-a),it = 0, ±1, ±2, • • • : h{e,0} is denoted by 
h (e), the entropy of the centered e partition of X. 

Lemmas 3 and 6 below show that for any £ > we 
have h (e, a)^h (c), with equabty only if a is an integer. 
We first define two functions and state some of their 
properties. Let P (c, z) be the probability of the interval 
of length c centered at ez, so that 



dy 



(7) 



P{e,z)= i'iy)dy= exp(-|-)^ 



(8) 



268 



i?l SPACE PROGRAMS %\iNitM<r( 37-51, VOL. Ill 



where <^ is the normal density function. Since 



'"^'■'(7i|y;'[('"^)'] 



for large z, all the series which we encounter will converge absolutely; we need make no further mention of con- 
vergence. 



Define 



K"-i) 

F V :) = F (6, z) = log -^ ^ 

PI 






The following lemma Usts some of the properties of P and F. 

Lemma 2. The following seven properties hold for the functions P and F: 

P(e,-z)-P(e,z) 

F(-z)=-F(z) 






>z) 



0<F'(z)forz,€>0, F'(z)^-|-e' for z>i 



■(-i) 



€*{z--5r)<F(€,z)<€''z for z>0 

F{i,z)>^(%{y)dy-Ar i,(y)dy for 0<«<| 

F (c, z) is increasing in € for fixed z > 

The next lemma proves theorem 1 for large e. However, the difiBcult case is the case of small 6. 
Lemma 3. U e^3, h (e, a) assumes its minimimi value only when & is an integer. 

JFL SPACE PROGRAMS SUMMARY 37-51, ¥OL. Ill 269 



To complete the proof of theorem 1, we shall have to study the function h (e, o) very carefully. This is because for 
small e 



;h(e,a)-o[exp(-^^] 



so that h is very flat as € ^ 0. 



The rapid convergence of the series for h («, a) ensures that it is C". From the periodicity of the function in a, it 
follows that it is the sum of a convergent Fourier series: 



h (c, a) = "I Co (€) + y^C- (e) cos {2nna) 



(9) 



where 



C„ (e) = 2 / /i (e, a) cos (2n7ra) do = 2 / > P (e, ik - o) log -^7-^ r cos {2mra) da 



We interchange the order of integration and summation here; after the substitution k — a — x, we have 

C„ (€) = 2 r P (e, x) log p^ cos (2n,rx) dx (10) 



To get useful inequalities for these coefficients, we need 
to investigate the properties of P (s, z) as an entire func- 
tion of the complex variable z. 



Define 



so that 



Q{e,z)=l eKpl-zy--^jdy 






(11) 



which shows that Q (c, z) is an even entire function of z 
of exponential type. Hence, it can be expressed in terms 
of the canonical product of its zeros ±^1, dzfa, • • • as 



Q(€,Z) = Q(€,0) 



n('-s) 



(12) 



Thus, information about the zeros of Q{e,z) would be 
quite useful, and the next lemma furnishes the required 
information. 



Lemma 4. The zeros {±fjt} of Q(€, z) are all distinct 
and are on the imaginary axis for < e < Co = 4.309 • ■ • . 
Furthermore, under the appropriate indexing, we have 



27rJt<-2^<27r(fc+ 1), Jt = l,2, 



(13) 



Next, lemma 4 will be applied to get estimates for the 
Fourier coefficients €„{() of h{e,a). This is the content 
of lemma 5. 



Lemma 5. If < € < c„, 

C,(e)^-2exp^-^)fl-P(c,0)] 
and, for n ^ 2, 

I C„ (e) I ^ exp (- ^) [2 + 4P (e, 0)] 

, 2e V^ r 2n''(2nk-k') 

■'^r(2;;)^Z-«"4 ^' — 



(14) 



] 



(15) 



270 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



Lemma 6. For < € = 
an integer. 



h (e, a) >h (c) when a is not 3. The Product Epsilon Entropy Function J, (X) 



Theorem 1. The e-entropy Hi{X) of the real line X 
under a one-dimensional gaussian distribution with 
mean 0, variance a' is h (e/a). The only €-partition of the 
line with this entropy is the partition into consecutive 
intervals of length e with one interval centered at zero. 

Proof. We can assume <t = 1, since the general case 
follows by a change of scale. By lemma 1, the only 
c-partitions whose entropy can be the e-entropy of the 
space are those which subdivide the line into intervals 
of length e. We run through all these partitions by taking 
the partition into €-intervals with one interval centered 
at — eof, ^ a < 1. These partitions have entropies h (e, a), 
so that 

H,{X)= inf h{€,a) 

By lemmas 3 and 6, for any positive e, this infimum is 
assumed only at a = 0, which proves theorem 1. 

The final lemma of this subsection lists some properties 
of the function h (e). These properties are interesting in 
themselves, and they are also needed at various points 
throughout the remainder of this article. 

Lemrrm 7. For < e < oo, /«' (t) < and [h' (€)/€]' > 0. 
The function h' (e)/e varies monotonically from — oo to 
for € on (0, oo). Also, the following asymptotic formulas 
hold: 



as e- 



0, 



h(e)^log- 



€ 



(16) 



as €-» 00, 



h{e). 



2(2,r)''4"P 



h'(€}^- 



8(2,r)^ 



e.xp 






(17) 



Now that we have gotten "preliminaries" about the one- 
dimensional gaussian distribution out of the way, we can 
begin to study the case of arbitrary mean-continuous 
gaussian processes on the unit interval. 



In this subsection, we define the product e-entropy, 
/«(X), of a mean-continuous gaussian process X on the 
unit interval. The main results are contained in theorem 2. 
We find a necessary and sufficient condition for J, (X) to 
be finite. In the case when /, (X) is finite, we show how 
to construct a product e-partition with entropy equal to 

In order to define the product e-entropy function /« (X), 
we first consider the class tt, of all product e-partitions 
of Lo (0, 1). A product e-parb"tion of L, (0, 1) is the car- 
tesian product of e*-partitions of the kth coordinate axis 
in the Karhiinen expansion of the process, where 2 e| — e^. 
Thus product e-partitions consist of hyper-cubes of diam- 
eter at most e. Next define ire to be the subclass of tt, of 
partitions in which a countable collection of the sets have 
a imion with probability 1. A product partition in -rrt 
includes a denumerable partition of a subset of X of 
probability 1. By the entropy of the product partition 
we mean the entropy of this denumerable partition. 

The product epsilon is defined as 
/f (X) = 00 if n-( is empty 
/c (X) = inf H{ U) if ttj is not empty 

The entropy H (U) is defined as in Eq. (7) over the sets 
of U of positive probability. 

It turns out that ttc is empty if the series Eq. (19) 
diverges, and otherwise /e(X) is finite. 

Our first lemma shows how to compute tne entropy of 
a product partition in terms of the entropies of its one- 
dimensional partitions. 

Lpmma 8. Let the probability space X be the product 
of a sequence of probability spaces X-.Xs, ■ • • , with 
product measure. If Uk is a partition of X*, fc = 1, 2, ■ • • , 
and U the product partition of X, then 



H{U)^ 2H(t7t) 



This is to be interpreted to mean that if the union of 
countably many sets of 17 does not have probability 1, 
then H{U) is infinite. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



271 



The next two lemmas taken together show that, for a 
mean-continuous gaussian process on the unit interval, 
either v, is empty for all e > 0, or else ir, contains a par- 
tition of finite entropy for all e > 0. 

Lemma 9. Let X(t) be a mean-continuous gaussian 
process on the unit interval. Let U be a product partition 
-tf Lj (0, 1) obtained as the product of partitions t/* of 
the coordinate axes by intervals of lengths c^. Then the 
following three conditions are equivalent: 

(1) The union of countably many sets of U has prob- 
ability 1 

(2) U contains a set of positive probability 

(3) With probability 1, all but a finite number of com- 
ponents of an element of L} (0, 1) lie in the unique 
interval containing zero in the partition of that 
coordinate. 

If the partitions Ui, are centered, these conditions are also 
equivalent to 



ZH^)< 



where ^ is the unit normal density function, and {kit} are 
the eigenvalues of the process. 

Lemma 10. For fc = 1, 2, • • ■ , let t/* be a given «»- 
partition of the kth coordirate axis. Let 2^^ converge, 
and let a countable subpartition of the product partition, 
U = iricUk, cover a set of probability 1 in L2 (0, 1). Then 
for every e > there exist Cit-partitions, V*, of the kth 
coordinate axis such that 



and 



«' = 2cl 



2H(V,)<oo 



Lemma 11. For a mean-continuous gaussian process 
on (0, 1) with eigenvalues X„ = crj, n = 1, 2, • • • , the 
product e-entropy is given by 



/.(X)= inf y^ft(-) (18) 

!«»«=<» ^^ \<'*/ 



Proof. With each product c-partition of X, we can asso- 
ciate a sequence {e*} such that the partiticm of the kth 
component space Xt is an Ct-partition, and 2 e| = e'. For 
given {€it), the minimum possible entropy of the partition 



of Xic is /»(e»/at), by theorem L Hence, Eq. (18) follows 
from lemma 8. Lemma 11 is proved. 

Equation (18) reduces the problem of finding an opti- 
mal product e-partition to the problem of selecting an 
optimal set, (ck), of quantizations for the coordinate axes. 
The next theorem soVes this problem and gives a neces- 
sary and sufficient condition for /< (X) to be finite. 

Theorem 2. The product c-entropy /{(X) of a mean- 
continuous gaussian process on (0, 1) with eigenvalues 
{kk} is finite if and only if 



E 



Ajtlog-— < 00 
kk 



(19) 



If this condition is satisfied, the equations 

h'{h)=-AkkSk, fc = 1.2, ■•• (20) 

have a unique solution {8*} with A such that 

2Mll = c^ (21) 



Then 



/.(X)= 2 ft (8*) 



(22) 



On the other hand, if Eq. (19) is violated, Eqs. (20) 
and (21) have no solution. The condition Eq. (19) is also 
the condition that there be a countable subpartition of 
some product epsilon partition covering a set of prob- 
ability 1. 

Proof. Set <r* = XJ^* . We want to minimize 

'<'■•'■•■■■> =E''(«) 

subject to condition l€l = c^ Equation (20) is the con- 
dition for a minimum, by the method of Lagrange multi- 
pliers, if 8* = Ck/a*. To avoid justifying the use of this 
method in an infinite-dimensional space, we will consider 
finite dimensional subspaces of X. 

First we show that Eqs. (20) and (21) have a (unique) 
solution for any € > 0, if any, only if Eq. (19) is satisfied. 
According to lemma 7, for any A > there is a unique 
solution [iit] of Eq. (20); each 8* is a monotonic decreas- 
ing function of A, and 



lim i* = 00 , 

ii-»0+ 



lim 8* = 



272 



Jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



For a given value of A, AA* -^ as fc -♦ oo . Hence for 
k suflBciently large, 8* is so large that we can conclude 
from lemma 7 that 



where 



Then we have 



which implies 



16(27r)W' 



<Ck<l 



C*8texp 



{-!«)- 



Ax. 



81 -8 log 



At 



(23) 



We see that the series Eq. (21) is finite if and onl_> if 
Eq. (19) is satisfied. If Eq. (19) holds, then the monotone 
dependence of 8* on A shows that the series in Eq. (21) 
is a strictly decreasing function of A, taking all positive 
values as A ranges over all positive values. Therefore, 
Eqs. (20) and (21) have a unique solution. 



aries. Hence the infimum is assumed at some point, and 
we have there 



!ll). 



"k 



-A<"'£*, k = l,--,n 



where A<"' is a positive constant. Let c* = 8^"' vn be a 
solution of this system of equations, which lies on the 
sphere. Then 



and 



^^=-A<-'X.. k = l,--,n 



2 A,S<»)' = €^ 



(24) 



(25) 



For any value of A'"', the solutions of Eq. (24) are 
uniqi. by lemma 7. Furthermore, as A'"' varies from 
to 00, each 85;"' varies monotonically from oo toO. Thus, 
there is a unique value of A'"' at which Eq. (25) is satis- 
fied. We have 



/.(X)^/.(X<->)=2'»(8i-') 



(26) 



Notice also that the existence of a solution of Eqs. (20) 
and (21) implies that /« (x) is finite, for if we put e* = inh, 
then 



m = ^ 



and 



h{x)^y^h(^^)^ih(h 



This series converges, for by lemma 7, 

Now let X'"* be the product of the first n coordinate 
spaces. By lemma 11, 

n 

;.(x...)=_i„f^*(i) 



This sum is a continuoiL function over the positive 2"-tant 
of the sphere 2 cj^ = e', approaching infinity at the bound- 



This can be done for any n. In particular, for the num- 
bers A*"*" and 

s (n+1) ... » (n+1) 

are solutions of Eq. (24) with A<"> replaced by A'"*", and 

It follows that A<"*"^A'»'. Define 
A = lim A'"' 

A is either a positive real number or oo. 

First suppose A = oo . Then as n -> oo, A'"' Xi -* oo and 
8<-'-^0. From Eq. (26), 

/.(X)^/i(8<-))-* 00 

so /, (X) = 00 . It follows from above that in this case 
Eq. (19) is violated. 



jn SPACE PROGKAMS SUMMAkY 37-51, VOL III 



273 



Now let A be finite, and let {8*} be the solution of 
Eq. (20) when A = A. Since A'"> =^ A, 



hence 



* = i k-.\ 



It- 1 



This shows that there is a value A* of A for which the 
solution of Eq. (20) satisfies Eq. (21), and A* ^ A. Denot- 
ing this solution by {8*}, we have 



k 1 



Also note that Eq. (19) is the entropy of the distribution 
{Ajk}, provided the X* are normalized so that S An = 1. The 
occurrence of the entropy of the eigenvalues in this way 
appears to be fortuitous. 



4. Some Special Processes 

In this subsection, we shall consider a class of gaussian 
processes whose product £-entropies can be estimated for 
sir.all c by theorem 2. We begin with some general re- 
marks on product e-entropy. 

Let X be a finite-dimensional mean-continuous gaus- 
sian process on (0, 1). That is, X has only a finite number 
of non-zero eigenvalues, Xi, • • • ,\„, say. It is a conse- 
quence of theorem 2 and lemma 7 that 



hence A*^ A"", for all n. It follows that A - A*. 



A(X)-nlog-- 



For each k, we have 8^"' -» Sj as n-» oo. From Eq. (26), 
if m^n, 

in 

A(x)^2ft(8r) 

Letting n-* oo, then m-» oo, we obtain 

On the other hand, we have seen above that this series 
is the entropy of an c-product partition of X. Therefore, 
equality holds, and Eq. (22) is true. The last assertion of 
the theorem follows from lemmas 9 and 10. This com- 
pletes the proof of theorem 2. 

Corollary, /«(X) is a continuous function of e. 

Proof. This is a consequence of the formulas of the- 
orem 2. The asymptotic fomtmla (23) is uniform over 
any interval 0<Ai^A^A2< oo. Thus the series in 
Eqs. (21) and (22) are uniformly convergent. It follows 
that these series are continuous functions of A. Since e, 
given by Eq. (21), is a strictly decreasing function of A, 
A, and /e(X) are continuous functions of c This proves 
the corollary. 

We remark that when the A* are written in non- 
increasing order, condition (19) is equivalent to 

2A*logfc< 00 



as e -» 0. For this reason the interesting processes to now 
consider, from the point of view of product e-entropy, 
are the infinite-dimensional ones. 

The first thing we observe about an infinite-dimensional 
process X is that, as « -» 0, its product e-entropy must 
increase faster than any positive multiple of log 1/e. To 
verify this, let X*"* be the finite-dimensional process ob- 
tained from X by setting A* = for k> n. Then as e -> 



/.(X)^/.(X<'")~nlog^ 



Since n was arbitrary, this proves our assertion. 

In the final Subsection 5, we shall develop some tech- 
niques which are more generally applicable than theo- 
rem 2. For the present, however, we shall consider 
mean-continuous gaussian processes on (0, 1) whose eigen- 
values satisfy a relation of the form 

Afc ^ Bk-" as fc ^ 00 

where B > and p > 1 are constants. Special cases of 
these processes arise as solutions of the stochastic differ- 
ential equation 






+ bN 



where N (t) is white gaussian noise of spectral density 
1/2 and the a's and b's are constants with b„^0 and 



274 



JPl SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



n>m. For these processes, R{i>.t) = E[X{s)X{t)] can 
be found as well as the A^. However, for our purposes 
;t is enough to know that 

kk '-' Bk" 

where B > and p = 2 (n — m). This is true for stationary 
processes by Ref. 3 and apparently is also true for non- 
stationary processes. The most important special case is 
the Weiner process, for which 



dX/dt ^N,R (s, t) = min (s, t) 



and 



kk = 



■(-iT' 



k = l,2, 



The main result of this subsection is the following 
theorem which gives an asymptotic formula for J({X) 
as 6 -» 0. 

Theorem 3. Let X be a mean-continuous gaussian 
process on the unit interval with eigenvalues {X„} such 
that 

\„ ^ Bn-" 



B>0, p>l. Then, as €-»0, 

/ 2B \v<p-i) 



X 



/i'(3c)-li-i/P y/tr-n 



Ul-^]'"'""}^ 



(27) 



Corollary. For the Weiner process on (0, 1) 

C 



/.(X). 



as € -* 0, with 



C = — iri-xh'{x)]'^dxy =6.711 ■ 



Proof. We apply theorem 3 with B - I/ttS p - 2, and 
evaluate the integral numerically to prove this corollary. 

The «-entropy Hi (X) of llie Weiner process has been 
considered,' where Ht (X) is the infimum of the entropies 



of all countable partitions or sets of probability 1 in 
La [0, 1] by measurable sets of diameters at most e. Thus, 
Hf {X)-^}( (X). However, it has been shown'' that for the 
Weiner process 



17 



,<i/.(X)<~ 



32e 



(the notation U <V means lim sup U/V — 1). Thus, for 
the Weiner process, 

lim inf^^^ 6.711 • • • 

CO ni{X) 

This means that, for small e, the product e-partition on 
the average requires at least 6.7 times as many bits to 
transmit the outconic of the process as does the optimal 
e-partition. 

5. The Order of Magnitjde of J( (X) 

In this final subsection, a useful lower bound Lt(X) 
for /t (X) is considered. Conditions on the eigenvalues X* 
are given, which guarantee that /t(X) = 0[Le(X)], or 
even /» (X) ^ Lt (X). Since Lj (X) is a lower bound for 
the epsilon entropy He (X), these results imply that Hf (X) 
is of the same order as, or even asymptotically equal to, 
/( (X), so that not much is lost by the restrictioij to prod- 
uct partitions in these cases. Finally, these results are 
applied to a stationary band-limited gaussian process on 
the unit interval to obtain a simple asymptotic expression 
for /j (X) in that case. 

The lower bound Lt{X) derived" for the e-entropy 
Hi (X) of a gaussian process X is as follows: Assume 
e= < 2 Xic. Define the number b = b{e) by 



£=' = 2 



A* 



1 + b\k 



Then 



Since 



L.(X)-2 2log(l + b\,) 



(28) 



(29) 



L. (X) ^ H. (X) and H, (X) ^ /. (X), L, (X) 

also provides a lower bound for /« (X). 

The next lemma gives a lower bound for L« (X), which 
is actually the bound we shall be using. 



JPL SPACE PKOGRAMS SUMMARY 37-51, VOL. Ill 



275 



Lemma 12. Let X be a mean-continuous gaussian 
process on [0, 1] with eigenvalues Ai •> Aj — " ' ' • Define 
A(x),x^l, as the function such that 



A (n) = A, , 



n = l,2. 



then, as « -» 0, we have 

/c(X) = 0[L.(X)] 
If the stronger condition 



and xA (x) is hnear on each interval (n, n + 1). For e' < A,, 
define the function y = y(f} to be the smallest root on 
(1, oo) of the equation 



t/A (!/) = £' 



Then 



as e -> 0. 



[5/(0-l]y + 



(1) 



(30) 



holds, we have 



2 Ak-o(nA.) 

it = n 



/.(X)-L.(X) 



(32) 



An important consequence of theorem 4 is the next 
result, which has been proved within theorem 4. 

Theorem 5. Let X be a mean-continuous gaussian 
process on the unit interval with infinitely non-zero eigen- 
values {A„} arranged in non-increasing order. If 



The next lemma estimates the number A = A(f) given 
by Eqs. (20) and (21) in terms of the function y (c) of the 
preceding lemma. To make these estimates, certain restric- 
tions must be put on the eigenvalues A»; these restrictions 
imply that the influence of the eigenvalues far out is not 
too large. 

Lemma 13. Let A = A{€) be the number in the solu- 
tion of Eq.<i. (20) and (21). If the gaussian process X has 
an infinite number of positive eigenvalues, and 

2A»=0(nA,j 

* = !! 

when the eigenvalues are arranged in non-increasing 
order, then Eqs. (20) a:id (21) have a solution, and 
Ae' = 0[y{e)] as €-*0. If the stronger condition 



2 '\=0(nA,) 



then 



/.(X) = o(^"%(Oy) 

If the stronger condition 



holds, then 



2 Ak=o(nA,) 



/.(X)-j^ yit)~ 



Note that theorem 5 applies in the case of theorem 3, 
but gives less precise information. 



2 A» = o(nA,) 

ken 



(31) 



holds, then Ae' r-^ y (c). 



The main result of this subsection is the following 
theorem. 

Theorem 4. Let X be a mean-continuous gaussian 
process on the unit interval with infinitely many non-zero 
eigenvalues {A.) arranged in non-increasing order. If 



2 A* = 0(nA,) 

k-n 



Since 

/.(X)^H.(X)^L.(X) 

Theorem 4 can be thought of as a condition for 

/.(X)=0[//.(X)] 



or 



/.(X)-H.(X) 

In the former case, X can be transmitted by product par- 
titions with a number of bits not worse than the optimal 
system by more than a constant multiple. For processes 
with the stronger property (Eq. 32), the product partition 



276 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. HI 



sysiem is asymptotically as good as the best possible sys- 
tem as e -» 0. It can, moreover, be shown that /< (X) can 
be finite and yet not 0[W,(X)]. 

6. Application of Thoortm 5 f'9 Band-Limittd Proctitoi 

Let X be a mean-continuciis stationary gaiissian process 
on the real line whose covariance function 

p(t) = R(s,« + t) 

has Fourier transform dS (f) with support in some finite 
interval. Suppose dS (/) = a (/) df with a (f) continuous. 
Then when X is restricted to the unit interval, it is known 
[Ref. 4, lemma 2] that 

n 
for some constant C. It is seen that 



log- 



!/(«)'-'■ 



log log J 



Theorem 5 then implies 



hW 



-r&P 



so that 



IV 






(33) 



log log J 



Equation (33) shows that band-limited processes are 
not much more random than finite-dimensional distribu- 
tions, since /c(X) does not increase much more rapidly 
than a constant times log 1/e. This is tu be expected, since 
the sample functions are analytic with probability 1. 



3. Widom, H., "Asymptotic Behavior of Eigenvalues of Certain 
Integral Operators," Arrh. Ration. Mech., Vol. 17, pp. 215-229, 
1964. 

4. Widom, H., "Asymptotic Behavior of Eigenvalues of Certain 
Integral Equations," Trans. Amer. Math. Soc., Vol. 109, pp. 278- 
295, 1963. 



N. Data Compression Techniques: Estimators of the 
Parameters of an Extreme-Value Distribution 

Using Quantites, / Eisenberger 

1 . Introduction 

The statistical theory of extreme values for large sam- 
ples has been applied by Posner (Ref. 1) to the problem 
of estimation of low probability of error in threshold 
communications receivers. The extreme value distribution 
function that he considers is of the form 

G (x) = exp I - exp I ~ "o (^ " «) J|> 

-00 <x< », i8>0 

where a, the mode of the distribution, and j3, a scale 
parameter, are unknown and hence must be estimated. 
Posner, after making a change of parameters, derives the 
maximum-likelihood equations, the solutions pf which 
give thf maximum-likelihood estimators of his parameters. 
He then suggests a novel method for obtaining good first 
approximations. 

The purpose of this article is to provide optimum or 
near-optimum asymptotically unbiased estimators of a and 
p using k quantiles when the sample size is large and both 
a and p are unknown. First we estimated a for 
Jt = 1, 2, 3, • • • ,10, assuming fi unknown. Then we esti- 
mated /J for fc = 2, 3, 4, ■ ■ ■ , 10, assuming a unknown. 
Finally, since the ord'^rs of the k quantiles which give 
optimum or near-optimum estimators of a are not those 
which give optimum or near-optimum estimators of p, we 
derived estimators of a and p us'ng the same k quantile!>, 
for it = 2, 4, 6, 8, and 10. The orders of the k quantiles are 
taken to be those which minimize 



Reforoncot 

1. Lo^e, M., Probabilitv Theory— FoundaHom, Bandom Sequmcu, 
Sec. 34.5. D. van Nostrand Co. Inc., New York, 1955. 

2. Posi>er, E. C, Rodemich, E. R., and Ruinsey, H., Jr., "Epsilon 
Entropy of Stochastic Processes," Ann. Math. Utatiit., Vol. 38, 
pp. 1000-1020, 1967. 



var(a)-l-Cvar(j3), 



C = l,2 



These estimators are designated as suboptimum. The 
efiBciencies of the quantile estimators relative to the 
maximum-likelihood estimators were also determined. 



jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



277 



2. Review of Quantiles 

To define a quantile, consider n independent sample 
values, Xi.X:;, • • • ,i,, taken from a distribution of a con- 
tinuous type with distribution function H (x) and density 
function h (i). The pth quantile or the quantile of order p 
of the distribution, denoted by f * , is defined as the root 
of the equation // (4*) = p; that is 

p= f^' dHix)= P' h{x)dx 

J-x .'-X 

The corresponding sample quantile Zp is defined as fol- 
lows: If the sample values are arranged in non-decreasing 
order of magnitude 

X(.. Xf-) ■ ^=X(nt 

then x,ii is called the ith order statistic and 

where [np] is the greatest mteger ^np. 

If h (x) is differentiable in some neighborhood of each 
quantile considered, it has been shown (Ref. 2) that the 
joint distribution of any number of quantiles is asymp- 
totically normal as n-* 00 and that, asymptotically, 

p(i-p) 

"■= - Lp.(i-p.)J 

where pi2 is the correlation between z,, and Zp„, pi < p2- 
We will denote by F (x) and / (x) = F' (x) the distribution 
tunction and density function, respectively, of the stan- 
dardized extreme-value distribution; that is 



Fw»/; 



/(Otft = exp(-e-) 



where 



/(x) = exp[-x-exp(-x)] 



Thus, denoting by ^p the pth quantile of the standardized 
distribution, one has 

p= g{x)dx^ / f{x)dx= f{x)dx 

J -00 J -00 J-aa 



Hence, one sees that, asymptotica 



and, since 



1 



giCp) = jf{U 

Cp=- In(-lnp) 
fiip) - exp {In(-lnp) - exp [In(-lnp)]} 
= -plnp 
one also has 

i8^p(l-p)_)8-,l-p) 



var(2,) = 



nf(Cp) np (In p)= 



Since n is assumed to be large, the statistical analysis 
to be given will be based on the asymptotic distribution 
of the sample quantiles. 

3. Unbiased Estimators of a Using Quantiles 

Let a and J3 denote the quantile estimators of a and p, 
respectively, and let a and p denote the corresponding 
maximum-likelihood estimatOTs. We then define the effi- 
ciency of a and p as 



eff{S) = 
eSCp) = 



var( a) 
var (o) 

var(g 
vat (p) 



Using large-sample theory, a long and involved calcula- 
tion, which will be omitted, gives the asymptotic results 



var (a) 
var(;8) 



1.10867)3= 



n 

0.60793/3^ 
n 



The only linear unbiased estimator of a using one sam- 
ple quantile, when p is unknown, is given by 

A 

a = Z 

where z is of order p — e-^ = 0.3679. For, since 

f--ln(-lnp) 
one has 

E(S) = E{z) = E{pt + a) = £ [-i8ln(lne) + a] = a 



278 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL Iff 



The variance and efficiency of a are gi\ en by 



for a to be unbiased when p is unknown, the restrictions 



var (a) = var (z) 



eff(S) = 0.6452 



(l-p)|8- e-1 1.7182 18^ 
np (In p)' n n 



2 a. = 1 

2 Cif i = 



(1) 



The best Hnear unbiased estimator of a using fc > 1 
quantiles is of the form 



« = 2 «,~, 



Since 



E{a)^ 2 a, ipCi + a) - i8 2 a.C. + « 2 fl, 



must be placed on the coefficients Oj. Moreover, for maxi- 
mum efficiency, the values of the Ci and the orders of the 
quantiles should be chosen so as to minimize var (a), 
subject to the above restrictions. For fixed values of p„ 
i = 1, 2, ■ • • ,k, the first part of the optimization pro- 
cedure can be carried out using two Lagrange multipliers. 
Since 

k k 

var (a) = 2 '^ ■■Mjifij 

i = l j = l 



where a,, is the coviriance between Zi and zy, form the function 



k h k k 

Ri(a,, ■ ■ ■ ,at) = 2 2 OiOyCTi, + A., 2 Oifi + Xo 2 Oi 



DifiEerentiating R, (a,, • • • ,0/.) with respect to a„ J = 1,2, • • • ,fc, results in 



da 



= 2fl,<ri +2 2 a,uii + X,Ci + ^2, t = 1,2, ■ ,k 






Setting the k partial derivatives equal to zero and adding Eqs. (1) provides a system of fc + 2 linear equations, the 
simultaneous solutions of which give the values of the Oi that minimize var (f ), in terms of the moments of the sample 
quantiles. By varying the orders of the k quantiles, one can then determine the optimum pi and Ui for maximizing 
eff (a). This procedure was carried out for k = 2, 3 and 4. 

For fc > 4, in order to simplify the calculations, a modification of the above method for determining the optimum 
Oi for fixed values of the p, was adopted. If one assumes that the Zi are independent, one has 



K k k 

R^iot, ■ ■ ■ ,ak) '-= 2 aW, + X, 2 flif. + Xa 2 a. 

i=l 1=1 1=1 



2ai<r?+Aifi + Aj=0, i = l,2, 



aR, 

Ci = AiXi + BjX: 



(2) 



where 



Ai- 



-L 



2a? ' 



Bi = - 



2a? 



JPL SPACE PROGRAMS SUMMARY 37-57, VOL. W 



From Eqs. (1), one then has 



k k 

X, 2 A. + A, 2 B, = 1 

i =1 1 = 1 



Thus, one sees immediately that j8 cannot be estimated 
using a single quantile and when two quantiles are used 
b = — ba. The procedure .v. determining the optimum 
bj is similar to that for determining the optimum Oj when 
a is being estimated. Form 



A. 2 A.C, + A, 2 Bid = 

1=1 isl 



Solving the above equations for A, and A2 and then sub- 
stituting these values in Eq. (2) results in 



k k 
RaCfci, ■ ■ ,fc*) ~ 2 2 bibjUi; 



k k 

+ Ax 2 biCi + X, 2 b. 

4=1 <=i 



Ai 2 B,Ci - B. 2 AfC, 
i^ i^j 

^t ~ k k k k 

2 2 A,BmCm "~ 2 2 BjA,n^„ 
j -I m~l j = \ m=:l 



. = 1,2, ■■ ■ ,k 



Set dRs/dbi = 0, i = 1,2, • ,Jt, add Eqs. (3) and solve 
for the b,. Then by varying the p*, the optimum bi and p* 
will be determined. This was done f or fc = 2 and 3. For 
fc > 3, two procedures were used to determine near- 
optimum estimators. For odd values of k, the simplified 
method used to estimate a was adopted, resulting in 



This procedure was carried OMt for fc = 5,6, ■ • • , 10, 
resulting in near-optimum estimators of a. Table 9 lists 
the optimum and near-optimum estimator of a fork — 10 
and its efiBciency. The high eflBciencies (> 95%) achieved 
for fc > 5 indicate that the efficiency lost by adopting the 
simplified method of determining the Ci was not excessive. 



k k 

Bi 2 A, - As 2 Bj 

h = '— '^ — 

"• k 



2 2 AiB„^„ - 2 2 BiA„U 

J = 1 in=l j = l m = l 



4. Unbiased Estimators of p Using Quantiles 

The best linear unbiased estimator of p when a is un- 
known is of the form 



i=l,2, ■ ■ ■ ,k 



For even values of k, the estimator was formed given by 



/8 = 2 biZi 



Since 



k k 

E{fi) = P 2 hit,^<x 2 b. 

»=i i=i 



one must impose the restrictions 



j=i 



It is readily seen that 



A */2 

E(/3) = )8 2t> 



2 b*^ = 1 



2bi = 

i = l 



(3) 



so that the only restriction required is 



*/2 

2b,-l 



(4) 



280 



in SPACE PROGRAMS SUMMARY 37-5?, VOL III 



Let W, = St J,, - :;. If we assume that the Wi are inde- 
pendent, then one has, using one Lagrange multiplier 



^-,2 



R.iK 



v>^E(^ 



bjuj 



^iY 



k/2 

x-2b, 



•Zbiaj 



where 



dbi " (ft-;., - f,)= 



b, = xDi 



<,; = var(\V,) 



^ _ ~ iCk-i-n ~ Ci)' 



A = 



i<TJ 



Using Eq. (4), one obtains 



k/i k/z 

2 fc, = A 2 D, = 1 



k/2 

2D, 



optimum or near-optimum quantiles and estimate both pa- 
rameters independently. However, suppose, for example, 
one wishes to achieve maximum data compression of space 
telemetry by using the same k quantiles to estimate the 
two parameters. Which quantiles should be used? Using 
the op H mum quantiles for estimating one parameter, in 
order lo estimate the other, results in a substantial loss of 
eflBciency. For instance, for fc = 8, if one uses to estimate a 
the near-optimum quantiles for estimating /3, eff (o) drops 
from 0.9725 to 0.8263, while estimating /3 with the near- 
optimum quantiles for estimating a results in eflF(^) = 
0.4807 instead of 0.9317. 

What is required then is a method, based on a reason- 
able criterion, for determining suboptimum quantiles to 
be used to estimate both a and p. The method we propose 
here is as follows: Determine the orders of the quantiles 
which minimize var (a) + C var (/8) and form unbiased 
estimators of a and j8 using the quantiles thus specified. 
This was done, f or C = 1 and 2, for it = 2, 4, 6, 8, and 10. 
The estimators forC = 1 are given in Table 11, and the 
estimators for C = 2 are given in Table 12. A comparison 
of Tables 11 and 12 with Tables 9 and 10 showed that if 
one uses 2k suboptimum quantiles to estimate a and p 
simultaneously, the efficiencies of both estimators are 
greater than the efficiencies of the corresponding optimum 
or near-optimum k quantile estimators. 



6. Estimating Functions of a and p Using Quantiles 

The mean n and the standard deviation of the distribu- 
tion with distribution function G (x) are given by 



and, finally 



ti^Cp-^a 



h - ^' 

"> k/2 

2D, 



Table 10 lists the optimum and near-optimum estimators 
of p and its efificiency for k — 10. Efficiencies in excess of 
902 were found for fc > 6. 



5. Suboptimum Estimators of a and p Using 
the Some Quantiles 

One can see from Tables 9 and 10 that the optimum 
and near-optimum quantiles for estimating a are not opti- 
mum or near-optimum for estimating j3. For fc-quantile 
estimators of a and p, one can, of course, select the 2k 



&A 



where C = 0.5772 denotes Euler's constant. Quantile esti- 
mators of IX and <7, and their variances, are given by 



;i = C)8 4-ft 



var Q = C var (p) -F var (o) + 2C cov (a, p) 



JPL SMCE PROGRAMS SUMMARY 37-51, VOL. Ill 



281 



Table 9. Optimum and near-optimum estimators of a and 
their efficiencies when p is unknown (k ~ 10) 



k 


Etlimatera a 


•fi&\ 


1 


z (0.3679) 


0.6452 


2 


0.5370 X (0.1797) + 0.4430 z (0.6023) 


0.8156 


3 


0.3514 z (0.1041) + 4089 z (0.3705) + 0.2397 z (0.7365) 


0.8863 


4 


0.2423 z (0.0676; + 0.3306 z (0.2474) + 0.2838 z (0.5193) + 0.1433 z(0.8187) 


0.9226 


5 


0.1691 z (0.0466) + 0.2729 z (0.1735) + 0.2763 z (0.3837) + 0.1976 z (0.641 2) + 0.0841 z (0.8763) 


0.9436 


6 


0.1277 z (0.0342) + 0.2220 z (0.1294) + 0.2489 z (0.2924) + 0.2124 z (0.5051) + 0.1361 z(0.7305) + 0.0529z(0.9131) 


0.9569 


7 


0.0970 z (0.0262) + 0.1779 z (0.0969) + 0.2)74 z (0.2231) + 0.2091 '0,3959) + 0.1637 z (0.5954) 
+ 0.0984 z (0.7903) + 0.0365 z (0.9252) 


0.9660 


8 


0.0771 z(0.0208) + 0.1 462 z (0.0757) + 0.1894 z (0.1 767) + 0.1961 z(0.3188) + 0.1713 z(0.4916) + 0.1250 z (0.6746) 
+ 0.0700 z (0.8413) + 0.0249 z (0.9525) 


0.9725 


9 


0.0637 z (0.0169) + 0.1247 z (0.0622) + 0.1669 z (0.1961) + 0. 1 806 z (0.2669) + 0.1680 z (0.4164) + 0.1 362 z (0.5805) 
+ 0.0934 z (0.7427) + 0.0496 z (0.8793) + 0.0169 z (0.9650) 


0.9771 


to 


0.0547 z (0.01 46) + 0.1080 z (0.0529) + 0. 1 480 z (0.1 239) + 0.1654 z (0.2274) +0.1610 z (0.3573) + 0.1395 z (0.5041) 
+ 0.1069 z(0.656l) + 0.0695 z (0^976) + 0.0351 z(0.9088) + 0.01 19 z (0.9733) 


0.9806 



Table 10. Optimum and near-optimum estimators of /3 and 
their efficiencies when a is unknown Ik = 10) 





A 
EsKmatort ft 


•H(^l 




0.3345 [z (0.8326) - z (0.0262)] 


0.6635 




0.3440 z (0.8159) - 0.2289 z (0.041 3) - 0.1151 z (0.00624) 


0.7152 




0.1139 [z (0.9290) - z (0.00701)) + 0.2360 [z (0.7193) - z (0.0504)] 


0.8304 




0.1 167 z (0.9268) + 0.2336 z (0.7100) - 0. 1 356 z (0.0681) - 0.1448 z (0.0227) - 0.0700 z (0.00328) 


0.8509 




0.0510 [z (0.9644) - z(0.00273)l + 0.1294 [z(0.8496) - z(0.0185)] + 0.1817 [zrO.6457) - z(0.0715)] 


0.8979 




0.0517 z (0.9649) + 0.1350 z (0.8428) + 0.1720 z (0.6430) - 0.1019 z (0.0867) - 0.1 239 z (0.0397) 
- 0.0960 z (0.01 14) - 0.0369 z (0.00159) 


0.9079 




0.0264 [z (0.9798) - z (0.001 29)] + 0.0743 [z (0.91 20) - z (0.00827)] + 0.1225 [z (0.7838) - z (0.0309)] 
+ 0.1464 [z (0.5995) - z (0.0881)] 


0.9317 




0.0264 z (0.9809) + 0.0795 z (0.9089) + 0.1 258 z (0.7738) + 0.1 329 z (0.5976) - 0.0807 z (0.1014) - 0.1037 z (0.0549) 
- 0.0980 z (0.0220) - 0.0614 z (0.00589) - 0.0208 z (0.000831) 


0.9366 


10 


0.0159 [z (0.9866) - z(0.000775)] + 0.0462 [z(0.9428) - z(0.00448)] + 0.0824 [z (0.8561) - z(0.0159)] 
+ 0.1113 [z (0.7279) - z(0.0437)J + 0.1218 [z (0.5601) - z(0.1036)] 


0.9509 



282 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL 11/ 



Table 1 1 . Sub-optimum estimators of a and /3 for c = 1 



k 


Eitimalors 


•H 


2 


J = 0.5671 z (0.0865) + 0.4329 1 (0.7338) 
$ = 0.4836 (* (0.7338) - z (0.0865)] 


0.7334 
0.5661 


4 


a = 0.1067 z (0.0172) + 0.4025 1 (0.1388) + 0.3825 z (0.5548) + 0.1083 i (0.8783) 
/8 = 0.1879 [z (0.8783) - z(0.0172)] + 0.2919 [z (0.5548) - z(0.1388)] 


0.8930 
0.7632 


6 


a"- U.0512Z (0 00674) + 0.1661 z(0.0463) + 0.2836 z (0.1884) + 0.2769 z (0.4496) + 0.1663 z (0.7442) 
H- 0.0559 z (0.9331) 

/8 = 0.0930 [z (0.9331) - z (0.00674)] + 0.1939 [z (0.7442) - z (0.0463)] 
+ 0.2009 [z (0.4496) - z (0.1884)] 


0.9434 
0.8479 


8 


S = 0.0243 z(0.M343) + 0.0808 z (0.0204) + 0.1628 z (0.0769)+ 0.2252 z (0.2169) + 0.2232 z (0.3986) 
-• 0.1 659 z (0.6409) + 0.0891 z (0.8412) + 0.0287 z (0.9596) 

is = 0.0517 [z(0.9596) - z(0.00343)] + 0.1216 [z(0.8412) - z(0.0204)] + 0.1662 [z (0.6409) - z(0.0769)] 
+ 0.1497 [z (0.3986) - z(0.2169)] 


0.9647 
0.8955 


10 


a = 0.01 18 z (0.00162) + 0.0424 z (0.00962) + 0.0942 z (0.0356) + 0.1533 z (0.1010) + 0.1909 z (0.2322) 
+ 0.1 894 z (0.3746) + 0.1534 z (0.5822) + 0.1001 z (0.7681) + 0.0495 z (0.9031) +0.01 50 z (0.9760) 

is = 0.0286 [z(0.9760) - z(0.00162)] + 0.0752 [z(0.9031) - z(0.00962)] + 0.1202 [z (0.7681) - z(0.0356)] 
+ 0.1393 (z (0.5822) - z(O.IOIO)] + 0.1200 [z(0.3746) - z(0.2322)] 


0.9743 
0.9259 



Table 12. Sub-optimum estimators of a and /8 for c = 2 



k 


Estimators 


•ff 


2 


a = 0.5592 z (0.0606) + 0.4408 z (0.7569) 
is = 0.4374 [z (0.7569) - z (0.0606)] 


0.6863 
0.6077 


4 


a = 0.091 8 z (0.0136) + 0.4170z (0.1 1 17) + 0.3971 z (0.591 8) + 0.0941 z(0.8929) 
P - 0.1649 [z (0.8929) - z (0.0136)] + 0.2799 [z (0.5918) - z (0.1 117)] 


0.8678 
0.7882 


6 


a = 0.0478 z (0.00524) + 0.1 609 z (0.0368) + 0.2953 z (0.1632) + 0.2851 z(0.4792) 
+ 0.1597 z (0.7711) + 0.0512 z (0.9421) 

P - 0.0811 [z (0.9421) - z (0.00524)] + 0.1794 [z (0.7711) - z (0.0368)] 
+ 0.2003 [z (0.4792) - z (0.1632)] 


0.9307 
0.8615 


8 


a = 0.0215 z (0.00251) + 0.0766 z (0.01 61) + 0. 161 7 z (0.0633) + 0.2372 z (0.1 982) + 0.2334 z (0.4176) 
+ 0.1617 z (0.6758) + 0.0828 z (0.861 8) + 0.0251 z(0.9662j 

;8 = 0.0439 [z (0.9662) - z (0.00251)] + 0.1106 [z (0.8618) - z (0.0161)] + 0.1602 [z (0.6758) - z (0.0633)] 
+ 0.1512 [z(0.4176) - z(0.1982)] 


0.9576 
0.9046 


10 


a = 0.0108 z (0.001 29) + 0.0399 z (0.00790) + 0.0911 z(0.0299) + 0.1 533 z (0.0875) + 0.1992 z (0.2217) 
+ 0.1971 z(0.3824) + 0.1529 z (0.6077) + 0.0963 z (0 7886) + 0.0460 z (0.9141) + 0.01 34 z (0.9795) 

P = 0.0250 [z{0.9795) - z(0.00129)] + 0.0688 [z (0.9141) — z(0.00790)] + 0.1143 [z (0.7886) - z(0.0299)] 
+ 0.1379 tz (0.6077) - z (0.0875)] + 0.1208 [z (0.3824) - z(0.22i7)] 


0.9700 
0.9312 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. IH 



283 



A percentage point Xp of the distribution is defined by 

p = exp^- expl - — (Xp - a) i 



Then one has 



(Xp- a) = -ln(-lnp) 



Xp= —p\n{ — \np)+a 
A quantile estimator of Xp and its variance are given by 

Xp= - 18 In ( — In p) + a 
var(Xp) = [ln(-lnp)]2var{|3) + var (a) - 2 In ( - In p) cov (o, /3) 
One might wish to estimate the probabiHty that x will not exceed some threshold value x„. Thus, 

pr (i < x„) = S = exp < -exp - ~(x- a) V 
and a quantile estimator of S is given by 

S = expi-exp| --^(x-a) i 
The approximate variance of S is given by 

var (S) ^ ~P^ {var (a) + [In (-InS)]^ var(|8) - 2 [In(-lnS)] cov (Sj)} 



7. Estimating a and j3 From Real Data Using Quantiles 

In order to obtain a sample quantile Zp of order p from 
a sample of size n drawn from a population with distribu- 
tion function G (x), a table of random digits can be used. 
A set of n fc-digit numbers is drawn from the table and 
the sample quantile of order p, say Vp, is determined from 
this sample. Then the desired sample quantile Zp of G (x) 
is obtained by solving for Zp in the equation 

K + 0.5) 10-* = G(Zp) 

This procedure was adopted in order to obtain sample 
quantiles necessary for estimating a = and /8 = 1. Two 
sets of sample values, sample A and sample B, each of 
size 500, were drawn from a table of random digits 
(Ref. 3). For each sample, the suboptimum quantiles were 
determined for both C = 1 and C = 2, and used to esti- 
mate a and yff. TTie results are as follows (a* and Pi, will 
denote the estimates of a and p using k suboptimum 
quantiles) : 



From sample 


A, with C = 1 






a-, - 


0.0006 


A 


= 1.0059 


A 

a* — 


0.0576 


A 


= 0.9640 


ft, = 


0.0436 


A 

i8« 


= 0.9625 


Sn - 


0.0387 


A 

i8. 


= 0.9943 


A 


0.0291 


^^. 


= 1.0044 



From sample A, with C = 2 



5. = 


-0.0044 


P, =1.0018 


&. = 


0.0400 


/§: =0.9664 


&. = 


0.0333 


^, =0.9795 


A _ 

«8 — 


0.0396 


^, =0.9798 


A 


0.0257 


j^x„ = 0.9875 



284 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



From sample B, with C = 1 
a, = -0.0339 
a, = -0.0354 

5, = -0.0173 

6, = -0.0256 
S^„ = -0.0167 



p, =0.9621 
p, = 1.0589 
% =1.0443 
J8„ = 1.0309 
%o = 1.0444 



with replacement, and using these as the time location 
of 33 peaks for the mass spectrogram. At each of these 
points, the non-negative amplitude of the peak was chosen 
from a geometric Jistribution whose mean was 10. Each 
peak was then converted into a triangular pulse with 
height equal to the chosen peak value and with a base of 
width equal to 2 time units, centered on the original time 
location. The sum of these triangles resulted in the gen- 
erated data shown in Fig. 29. 



From sample B, with C = 2 



&2 = -0.0468 


% 


= 0.9949 


^4 = -0.0282 


I 


= 1.0334 


Se = -0.0286 


% 


= 1.0500 


Sg = -0.0252 


% 


= 1.0375 


a,„ = -0.0092 


%. 


= 1.0387 


References 







1. Posnet, E. C, "The Application of Extreme Value Theory to 
Error-Free Communication," Technometrics, Vol. 7, No. 4, 
pp. 517-529, Nov. 1965. 

2. Cramer, H., Mathematical Methods of Statistics. Princeton Uni- 
versity Press, Princeton, N. J., 1946. 

3. The Rand Corporation, A Million Random Digits with 100,000 
Normal Deviates. The Free Press, Glencoe, 111., 1955. 



O. Data Compression Techniques: Mass Spectro- 
gram Data Compression by the Slope 
Threshold Method, L. Kleimock 

A complete description of the slope threshold method 
of data compression is given in SPS 37-49, Vol. Ill, 
pp. 325-328. 

TTie data used for this experiment was randomly 
generated by choosing 33 integers from the set 



In Fig. 30, we show the results of compressing the mass 
spectrogram using the slope threshold method for the var- 
ious values of a shown and with fc = (see SPS 37-49, 
Vol. Ill, pp. 325-328, for details of the compression algo- 
rithm). In Fig. 31, we show the result of compression by 
periodic sampling. Table 13 lists the parameter values 
and the rms error, as well as the Posner norm d (the rms 
error is merely e,,), for the two sampling methods. This 



Table 13. Experimental results 



{1,2,3, 



,100) 















Giotf 


Figurt 





b 


Period 


fo 


fi 


cemprattion 
ratio 


29 
















1.56 


30a 







— 


O.I 


0.17 


1.67 


30b 







— 


0.28 


0.47 


1.79 


30c 







— 


0.52 


0.88 


2.0 


30d 







— 


0.75 


1.19 


2.28 


30e 







— 


1.05 


1.71 


2.44 


301 







— 


1.27 


2.1 


2.7 


30g 







— 


1.79 


2.62 


2.94 


30h 


11 





— 


3.15 


4.56 


4.16 


30i 


15 





— 


3.6 


5.3 


5.0 


30i 


20 





— 


.4.55 


6.33 


5.26 


30k 


25 





— 


6.35 


8.04 


7.7 


301 


30 





— 


22.4 


23.5 


16.7 


31a 


— 


— 


1 








1.0 


31b 


— 


— 




5.23 


9.07 


2.0 


31c 


— 


— 




5.5 


9.67 


3.0 


31d 


— 


— 




6.26 


11.14 


4.0 


31e 





— 




6.1 


10.3 


5.0 


31f 


— 


— 




6.8 


11.55 


6.0 


31g 


— 


— 




7.8 


11.7 


7.0 



/\ aAaa a ^ A 



-H2K- 



TIME UNIT 
Fig. 29. Randomly generated mass spectrogram 



jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



IBS 



(a) 



(b) 



(c) 



AA ^ 



l\ J\aa a ^ A J\ 



(d) 



(e/ 



AA ^ yy^wy^ 



^^-'—^ 



A- A/v 



A aAaa a ^A Jl 



(f) 



- »--^^>- 



A- AA r> /Vv /^V 



7\ aA^V^V_^^v. 



TIME UNIT 



-H2h- 



I 

100 



Fig. 30. Slope threshold sompling: (a) o = 2, (b) a = 3, (c) a = 4, (d) a = 5, (e) a = 6, (f) a = 7, (g) a = 8, 
(h)a= n,(ila= 15,(pa = 20, (k) a = 25,(1)0 = 30 



266 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. lU 



(g) 

^o./-^ A^ Aa «^ 



y\=:U\ A^W_A J\ Jl 



(h) 

(i) 

-^^ A ^ Aa. ^ >v^ /"N 



j\ aAaa>>^ /T ^^-^ y 1 \ J \ 



(j) 

r*r-^ A^ AiCi <^ /\r^ r\ 

(k) 

. ■r^r-. A^ A^^ rt Ar> /^ 

(I) 

- ^->-^ A ^ Aa. ^ A>^ /"x 



A. 




A^-/nAv._-a ^ 7l 

TIME UNIT 
Fig. 30 (contd) 



I JPL SPACE PROGkAMS SUMMARY 37-51, VOL. Ill 287 

i 



(a) 



l\ /sAaa a . .a 



(b) 



.A^ Aa. 



i\_x/W^ 



.^J^ 



A 



(e) 



-^ A - 



Ajw.. 



A. 



A 



A 



(d) 



AA fi^-A?<7>:»^ 



iL^^\^ 



A 



A 



A 



(e) 



,A>^ A^ 



J\sJW^,^^ 



-^^ 



A 



(f) 



-^-^^ 



A-. A^L. 



A.i;^/W 



A. 



A 



A 



(g) 



^ y\.^ /"X 



iL-JW^^t:-. ^ 



TIME UNIT 

Fig. 31. Periodic sampling: (a) period — 1, (b) period = 2, (c) period = 3, 

(d) period = 4, (e) period = 5, (f) period = 6, (g) period = 7 



100 



norm, suggested by E. C. Posner, is defined as 



c« = 



^^(/»-/>+^^(A/,- 



Atr 



I- n-l 



Figures 30 and 31 show the reconstructed function, /„, 
superimposed on the original data, /.. For example, in 
Fig. 30f, we see that /. has missed a number of peaks. 
It is interesting to observe the behavior of the Posner 
norm, which is designed to measure the mean-squared 
amplitude error plus a times the mean-squared slope 
error. Figure 30b and Table 13 show that «i is 70% larger 



than Co. indicating that the slope error is almost as sig- 
nificant as the amplitude error. The extreme case shown 
in Fig. 301 shows that the slope error is insignificant com- 
pared to the amphtude error; in Fig. 30k, the slope error 
is almost the same as in Fig. 301, but the amplitude error 
is much reduced. We conclude that the use of the Posner 
norm here is more significant as the mean-squared ampli- 
tude error decreases. 

In Fig. 32, we plot the Posner norm (for a = 0, 1) as a 
function of gross compression ratio. We observe for mod- 
erate compression ratios (less than 3) that the slope 
threshold method of sampling is far superior tr> t>eriodic 



288 



JPL SPACE PftOGMMS SUMMARY 37-51, VOL III 



o 



liJ 

z 

(0 

2 




3 4 5 

GROSS COMPRESSION RATIO 



Fig. 32. Comparison of period!', sampling and slope threshold sampling, using Posner norm (a = 0, 1) 



sampling. In this range, we observe only slight distortion 
of the peaks. However, if one were to transmit only the 
peaks themselves (of which there are less than 33), one 
obtains a gross compres'^'^n ratio of approximately 3 at 
small cost. We therefoiv. vjnclude that the slope thresh- 
old method of data compression for mass spectrogram is 
not practical. 



P. Data Compression Techniques: Estimating the 
Correlation Between Two Normal Populations 
Using Quantiles of Conditional Distributions, 

/. Eisenberger 

1 . Introduction 

The problem of estimating the parameters of a uni- 
variate normal distribution using quantiles when »hc 
sample size is large is considered in Ref. 1, where estima- 
tors of the mean and standard deviation are given using 
up to twenty quantiles. If a set of pairs of sample values 
taken from a bivariale normal distribution is given, one 
must also estimate the correlation in order to completely 
describe the distribution. The problem of estimating the 
correlation coefficient p using quantiles is considered in 
Refs. 2, 3, and 4, where asymptotically unbiased estima- 
tors of p are constructed using up to eight sample quan- 
tiles. However, before constructing the estimators, it was 
necessary to perform a linear transformation on the sam- 
pfb ^airs in order to obtain a new set of independent pairs. 
Since, from the viewpoint of data compression of space 



telemetry, this procedure is not entirely satisfactory due to 
the equipment complexity, .1 was felt that a new approach 
to the problem of estimating p was desirable. 

Ii is reasonable to conjecture that if one considers the 
quantiles of the conditional distribution of, say, i/. it 
might be possible to construct satisfactory estimators jf p 
without a transformation of variables. As a result of the 
ensuing investigation, quantile estimators of p will be 
given when the quantiles are taken from the conditional 
distribution of y given that x lies in specified intervals, 
for a large sample size. These estimators are very nearly 
unbiased, with good efficiencies relative to the maximum- 
likelihood estimator when p is not too large. 

2. Review of Quantiles 

To define a quantile, consider a sample of n indepen- 
dent values, x,,X2, ■ - - , x„, taken from a distribution of a 
continuous type with distribution function G (i) and den- 
sity function g (r\ The quantile of order p of the distribu- 
ti'^ or population, denoted by fp, is defined as the root 
of the equation G (fp) = p; that is. 



dG{x)= g{x)dx 



The corresponding sample quantile Z, is defined as fol- 
lows: If the sample values are arrange J in nondecreasing 
order of magnitude 



*(i)— «(») ^ • • ■ — X(») 



JPL SPACE PROGRAMS SUMMAKY 37-51, VOL. Ill 



289 



then X(,) is called the ith order statistic and 

where [np] is the greater integer — np. 

If g (i) is differentiable in some neighborhood of each 
quantile value considered, it has been shown (Ref. 5) that 
the joint distribution of any number of quantiles is asymp- 
totically normal as n-^ oo and that, asymptotically, 

£(Zp)=fp 

p(i-p) 



var (Zp) 






_rp -(i-p-) T 



where pu is the correlation between Z,, and 2,^, pi < pj. 
Since n is assumed to be large, the statistical analysis 



to be given will be based on the asymptotii; distribution 
of the sample quantiles. We will denote by Zi the quantile 
of order p„ and it should alwajs be assumed that p, < p, 
when i < ;. 

3. The Distribution and Moments ofv = y{a<x<b 

Given a set of n independent pairs of sample values, 
(xi, !/,), (i2, y,), • , {x„ i/n), taken from two jointly nor- 
mal standard distributions with distribution functions 
F (x) and F (y), density functions / (x) and / (y), and joint 
density h (x, y), we derive the distribution of y given that 
X lies in the interval a < x < fo, that is, we consider the 
random variable t3 = t/|o<x<b. Denoting by G («) and 
g (v) the distribution function and density function of v, 
respectively, one has 



G(«)-pr(V<tj) = 



LO"- 



t)dxdt 



rf{x)dx 



(1) 



Differentiating Eq. (1) with respect to v results in 

rh(x,v)dx 

(U) Ja 



g(^) = 



cG(t 
dv 'F{h)-F(a) 



2.(1 - p ^- y'^ --p(' h")}! ^'^ f ^[iT^lh 

F(b)-F{a) 



(2„ 



1 / 1 \ /•»-p''/(i-p=)'^ / 1 \ 



F(b)-F{a) 



= /(«) 



pf b-p^ \_p( «-p" Y 



F(b)-F(a) 



(2) 



We will derive the mean and variance of v from the moments of the truncated variable x|a < x < fc. It is shown in 
SPS 37-38, Vol. IV, pp. 252-258, that the mean /t, and variance al of this truncated variable are given by 

_ /(a)-f(b) 
"' F{b)-F(a) 

_ af{a)-bf{b) \ f{b)-f(a) Y 
"' ^^ F{b)-F'a) [F(fc)-F(o)J 



290 



JPL SPACE PROGRAMS SUMMARY 37-51. VOL III 



It is also well known that the mean and variance of the 
conditional distribution ot ij\x are given by 

E(y\x) = px 

var(!/|x) = 1 - p' 
Now, 

E (t;) = E(!/|o < X < b) = £ [£(j/|xia < X < b)] 

= E{px\a<x<b) 

Similarly, 

£ (u^) = E (t^ I a < X < b) - E [E (t/^ I x|a < X < fc)] 

= £(l-p» + pV|o<x<b) 

- I - p- + p-{oi + nl) 
Thus, one has 

p-i- = E{v) = pp.j 

<T? = var(c) = l+p'(a|- 1) 

4. Estimators of p Using Quantiles 

We divide the x-axis into the six intervals /*: c^ < x < b*, 
K = 1,2, ■ ■ • ,6, where 

Oi ~ — 00 
flj = —Ok 

04 = 

b* = 0*., for fe = 1, • • ,5 

be = 00 

This partitions the x-axis into three pairs of symmetric 
regions. For each region, we will estimate p using two 



pairs of optimum symmetric quantiles taken from the set 
of y values such that the corresponding x values fall into 
the given region. Denoting by pi the estimator of p from 
the u, of h, we then form thf estimator 

3 

Pi/ = 2 Ci (pi + p^.,) 



determining the C, so as to minimize var(p) under the 
condition that 

2 2 Ci = 1 

Thus, let Zi be a sample quantile of order pi taken from 
a set [Vk}, for i = 1, 2, 3, 4 such that p, = 1 — pi and p, — 
1 — p.. Then 

£ (Zi) = ^i = fip + OkC* = PMi- + <'»C 

where ^t is the population quantile of order pi of the stan- 
dardized distribution of v. Although strictly speaking, the 
sample size of each of the sets {«,} is a random quantity, 
we wtIU take as the variance of Z. the approximation 



var(Zi) = 



Pi(l-P.) 



where 



mi = npr(ai<x<b,) 



This means that ve are taking as the sample size the 
expected number of x*s falling in the interval a < i < b. 



Forming the estimator 

0.1918 (Z» -t- Z,) + 0.3082 (Z^ + Z3) 



P> 



^i 



(3) 



where 



one has 



Pi = 0.1068, p3 = 0.6488 

Pa = 0.3512, p4 = 0.8932 



E (^.) = 0-1918 [2p|.. + g, (C + CI)] + 0.3082 [2p^, + a, (^ + Ta)] 
^ r 0.1918 (;^ + g) + 0.3082 (g; -h C\) -\ 



(4) 



JPL SPACE PROGRAMS SUMMARY 37-5?, VOL. Ill 



291 



The orders of the quantiles and the values of the coeffi- 
cients of % were chosen for two reasons. First, when 
p — ^tSi^)~f (^) ^"^ ^^^ numerator of Eq. (3) becomes 
the best unbiased estimator of the mean of a normal dis- 
tribution using four quantiles, and hence has the smallest 
variance. Secondly, it was found after repeated trials that 
if the estimator p„ using one set of quantiles had a smaller 
variance than the estimator had using another set, when 
p = 0, then the same result held when p^O. 

The variance of py depends upon the choice of fls and Oe, 
that is, on how we partition the positive x-axis into three 
regions. It was determined that if one chooses O5 = 0.8 
and fle — 1.5, the resulting estimator py will be very nearly 
optimum. However, the optimum choice of the d for 



given values of 05 and Oc depends upon the value of p. 
If one determines the d from 



1 



Ci = 



var(^i) 



/ ^ var( 



(piO 



var(p„) will be minimized for a given value of p, but 
since p is not known in advance, one set of the Cj must 
be chosen for all possible values of p. It was found that 
by using in py the optimum values of the Cj for p = 0.5, 
very little loss in efiBciency resulted for p between and 
±9. Thus, the estimator pj, that we propose is given by 



p, = 0.2632(^1 + ?„) + 0.1920 (A + p«) + 0.0448(^3 +%) 

and, because pi is mdependent of p, for i ^ ;, the variance of pi, is given by 

var(^) = 2 [(0.2632)^ var (?0 -I- (0.1920)^ var (^2) + (0.0448)^ var (^3)] 

The value of the bias term of pt, the second term of the right-hand side of Eq. (4), depends upon the degree of sym- 
metry of g(«). If g(t3) were symmetric, then C\- -Ct' ^* - ~^t' ^"^ ^(p) = P- Fortunately, g{v) is sufficiently sym- 
metric, for p between and 0.9, that the bias term is negligible. This is shown in Table 14, which lists the mean and 
variance of pt, fc = 1, • • • , 6, and p„, for p between and 0.9. 

Table 14. Mean and variance of ?* and S, [ft, = 0.2632 {p, + ^„) + 0.1920 (^^ + p,) + 0.0448 (^3 + ft)l 



p 


E(p,) 
E(p.l 


n var (pi) 
n var ({Si) 


E(p,) 


n var (p,) 
n var (^s) 


E(pJ 
E(^) 


n vor (pi) 
n vor (p'4) 


C(p,) 


var (p,) 








4.328 





6.14;j 





26.232 





1.158 


0.1 


0.09~> 


4.289 


0.1000 


6.087 


0.1001 


25.984 


0.1000 


1.147 


0.2 


0.1999 


4.179 


0.2001 


5.910 


0.2000 


25.239 


0.2000 


1.116 


0.3 


0.2998 


3.995 


0.3001 


5.615 


0.3000 


23.995 


0.2999 


1.064 


0.4 


0.3997 


3.737 


0.4001 


5.202 


0.3999 


22.254 


0.3999 


0.991 


0.5 


0.4995 


3.405 


0.5001 


4.669 


0.5000 


20.017 


0.4998 


0.896 


0.6 


0.599) 


2.998 


0.6001 


4.020 


0.5999 


17.281 


0.5996 


0.781 


0.7 


0.6985 


2.516 


0.7000 


3.252 


0.6998 


14.050 


0.6992 


0.645 


0.8 


0.7973 


1.958 


0.8000 


2.367 


0.7997 


10.323 


0.7986 


0.487 


0.9 


0.8947 


1.321 


0.8998 


1.364 


0.8993 


6.106 


0.8971 


0.308 



It is of interest to determine the efficiency of py relative to several commonly used estimators involving £.11 the sample 
values, such as: 

(1) The maximum-likelihood estimator p*, the solution of the equation 

{p*Y -c(p*y + {a + b-l)p'-c^O 



292 



jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



where 



TabI* 15. Efficiency of py relative to p*, r, and p 



n 
C = — > X, 

» = l 

1=1 

fi 

(2) The sample correlation •^'ifficient r, given by 

(3) The easily computed estimator p, given by 



The 


asymptotic 


variances 


of the above estimators 


are 


given 


by 


var(p*) = 
var (r) = 
var(p) = 


(1 - p^y 

n(l+p^) 

(1 - p^r 

n 

1 + r 





Defining the efficiency of py relative to any other estima- 
tor ^ as 



efffe) = 



varffl 
var (ft,) 



Table 15 gives the efficiency of p„ relative to the above 
three estimators. 

By applying to the x-values the method described above 
for obtaining ft,, one also obtains px with identical statis- 
tical properties. One can then form the final estimator p 
given by 



p 


»M (p,l 1 


Rclaliv* le p* 


Rtlativ* le r 


Ralativ* to p^ 





0.864 


0.864 


0.864 


O.I 


0.846 


0.854 


0.880 


0.2 


0.794 


0.826 


0.932 


0.3 


0.714 


0.778 


1.025 


0.4 


0.614 


0.712 


1.171 


0.5 


0.502 


0.628 


1.394 


0.6 


0.386 


0.524 


1.741 


0.7 


0.271 


0.403 


2.311 


0.8 


0.i62 


0.266 


3.366 


0.9 


0.065 


0.117 


5.875 



In order to compute var (p), one must determine the cor- 
relation between a quantile Zp of order p taken from 
y\a < X < b and a quantile ZJ of order q taken from 
x\c<y<d.IiE{Zf) = r,andE (ZJ) = f , then E. Rodemich 
has shown that the asymptotic correlation p,, between Z, 
and ZJ is given by 



N 



Pn 



[pq{l - p)(l - q)pi{a <x <b)pi{c <y <d)V'' 



where 

N = pr(a <i<f,c<t/<i;)-ppr(a<x<fc,c<i/<ij) 

- <7pr (a < X < f,c < y < d) 

-I- pq pr(c <x<b,c<y<d) 

when a<C <b and c <i] <d. If f < o, the terms in N 
which contain the condition z <x<C become zero, and, 
similarly, for i> < c. If f > fo, the condition a < x < f 
should be written z < x < fc, and if 7/ > d, c < y < rj 
should be written c<y <d. 

The extensive computations necessary to compute 
var (p) for even the simplest case, p = 0, were carried out 
for this case, resulting in 

var(p|p = 0) = -j[2var(p,) + 2cov(p„ft,)] 



= 2(1158 -M.068) = 1.113 



Estimators p^ and p, with the same statistical properties 
as ft, and pi can be obtained by a somewhat simpler pro- 
cedure. To the set of y'iS such that Xj « h add the set of j/iS, 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



293 



J ■ 



with their signs changed, such that Xi € /? *, for fc = 4, 5, 6. 
Then fonn the quantile estimators % and pj, given by 

% = 0.1918 (Z. + Z,) + 0.3082 (Z, + Z.) , 
/c = 4,5,6 

% = 0.5264 $„ + 0.3840 p, + 0.0896 ^, 

Then one has 

var(^») = ivar(^,), fc = 4,5.6 

E (S) = £ (P'z) 

var(^) = var(p„) 
The estimator pi is obtained in a similar fashion. 

5. Estimating p 

Two sets of samples {x,} and {t/'}, each containing 600 
sample values, were drawn from a table of random num- 
bers in which the entries are distributed N (0, 1). The 
transformation 

t/i=0.tix, +0.8!/: 

was then performed. Consequently, each x, and {/, can be 
assumed to be distributed N (0, 1) vvith a correlation of 
p = 0.6. Using the method involving Xj sgn y, and t/i sgn x, 



to estimate p using q-aantiles resulted in the following: 
P^ = 0.5601 , py ^ 0,5909 , p = 0.5755 

The following estimates were also obtained : 

p' - 0.5771 , ■? = 0.5404 , r = 0.5644 

Two new sets of 600 values each were then drawn from 
the same table of random numbers and paired at random, 
so that one can assume that p =^ 0. The results were: 

P, - -0.0070, p„ = -0.0060, p = -0.0065 

P* = 0.0027 , pT = - 0.0041 , r = - 0.0102 

References 

1. EisenberKcr, !., and Posner, E. C, "Systematic Statistics used for 
Data Compression of Space Tclfmetry," ]. Am. Stat. Assoc, 
Vol. 60, pp. 97-13.3. Mar. 1965. Also published as Technical 
Report 32-510, Jet Propulsion Laboratory, Pasadena, Calif., 
Oct. 1, 1963. 

2. Eisenberger, I., Tests of Hypotheses and Estimation of the Cor- 
relation Coefficient using Quantiles I, Technical Report 32-718. 
Jet Propulsion Laboratory, Pasadena, Calif., June 1, 1965. 

3. Eisenberger, L, Tests of Hypotheses and Estimation of the Cor- 
relation Coefficient using Quarailes //, Technical Report 32-755. 
Jet Propulsion Laboratory, Pasadena, Cilif., Sept. 15, 1965. 

4. Eisenberger, I., Tests of Hypotheses and Estimation of the Cor- 
relation Coefficient using Six and Eight Quantiles, Technical 
Report 32-1163. let Propulsion Laboratory, Pasadena, Calif., 
Jan. 1, 1968. 

5. Cramer, H., Mathematical Methods of Statistics. Princeton Uni- 
versity Press. Princeton, N. J., 1946. 



294 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



N68- 



37418 



XXI. Communications Elements Re^^arch 

TELECOMMUNICATIONS DIVISION 



A. RF Techniques: Switching Frequency 
Determination for the Nodding Subdish 

System, T. Soto, W. V. T. Rusch, C. T. Sieizried, 
S. D. S/obin, O. 6. Parham 

The Nodding Subdish System (NSS), used in the Octo- 
ber 1967 lunar edipse measurements (SPS 37-50, Vol. Ill, 
pp. 290-295) takes the place of the microwave switch 
used in conventional Dicke radiometers. The advantages 
of the NSS are the minimization of loss and the reduction 
of atmospheric scintillation effects. 

Because the NSS is a mechanical device, the number 
of switching cycles per second is limited as excessive 
speed leads to rapid wear and possible self-destruction. 
An original switching frequency of 1.16 Hz was chosen 
to ensure NSS longevity. During the final system checks 
prior to the eclipse observation, the noise output of the 
radiometer was larger than anticipated. A series of radi- 
ometer noise output measurements were made at various 
switching rates selected to be non-harmonically related 
to 60 Hz. The general trend of these data suggested a 
noise decrease with increased switching fiequency. 



The radiometer was reconfigured into a conventional 
Dicke radiometer, a ferrite switch replacing the NSS, to 
allow switching frequencies up to 37 Hz. The results of 
this experiment are shown in Fig. 1. After measuring 
both balanced and unbalanced cases, a 2.7-Hz switching 
frequency was selected as a good compromise between 
suflBciently reduced noise and a reasonable NSS life 
expectancy. 

The noise power spectrum of the radiometer output 
was measured using the non-real time digital spectrum 
analyzer shown in Fig. 2. The radiometer was switched 
between a high-temperature and ambient load to pro- 
duce a known output at the 2.7-Hz switching frequency. 
The output spectrum, given in Fig. 3, shows that the 
noise power spectral density at 2.7 Hz corresponds to 
35°K. 

These data show that low switching rates rapidly com- 
promise radiometer performance, and that a detailed 
knowledge of the radiometer components' noise charac- 
teristics must be known to select an optimum switching 
frequency. Further study is required in this general area. 



JPL SPACE PROGRAMS SUMMAkY 37-51, VOL. /// 



295 




X.'JV 'AilSN30 1vai33dS HSNCd 3SI0N 



296 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



HIGH- 
TEMPERATURE 
LOAD 




I FERRITE 
> SWITCH 
< 2.7 Hz 

AMBIENT 
LOAD 



TOTAL 
POWER 

IF 



+40 dB 




HEWLETT - PACKARD 
MODEL 461 A 
AMPLIFIER 



DC BLOCK VIDEO 
300 pF DETECTOR 



T 



0.01 



r = IO s 
LOW FREQUENCY 
CUTOFF AT 0.1 Hz 



DC AMPLIFIER 
+ 37 dB 



SKL 

VARIABLE 

FREQUENCY 

FILTER 
0-50 Hz 



AMPLIFIER 
X|0 



SDS 930 
COMPUTER 







FREQUENCY 
SYNTHESIZER 
lOO-Hz 
SAMPLING RATE 



Fig. 2. Instrumentation for radiometer noise and gain-change measurements 



JPL SPACE PROGRAMS SUMMAkY 37-51, VOL. Ill 



297 



2.7 Hz 




2 3 

SWITCHING FREQUENCY, Hz 



Fig. 3. Radiomttcr noi** spectrum 



4" 



298 



jn 5PACB PROOItAMS SUMMAkY 37-51, VOL. Ill 



B. Precision Calibration Techniques: Microwave 
Thermal Noise Standards, C. SteUrhd 

1. Introduction 

Calibrated microwave thermal-noise standards (Ref. 1) 
are used for microwave radiometry, antenna temperature 
calibrations, loss measurements (SPS 37-41, Vol. Ill, 
p. 83), low-noise amplifier performance evaluation and 
low-level continuous-wave signal-level calibrations 
(Ref. 2), A typical thermal-noise standard consists of a 
matched resistive element thermally isolated by a uni- 
form transmission line. The transmission line is usually 
fabricated from copper-plated stainless steel and has 
distributed temperatures and transmission loss factors. 
Although thermal-noise standards have been constructed 
without the use of transmission lines by pointing an 
antenna beam directly at bulk termination material 
(Ref. 3), the calibration cf these standards is complicated 
by the antenna characteristics (side lobes, etc.). The 
present discussion is limited to the use of a transmission 
line with matched termination. 

Microwave thermaUnoise standards are usually desig- 
nated hot, ambient, or cold, depending upon whether the 
resistive element is above, at, or below ambient temper- 
ature. The construction and calibration techniques used 
in hot or cold loads are similar. The primary difference 
is the method used to obtain temperature equilibrium of 
the resistive element. Hot loads normally use electrical 
heaters or boiling liquids with a high boiling point (e.g., 
water), and cold loads nonnally use refrigeration or boil- 
ing liquids with a low boiling point (e.g., liquid helium). 
Ambient loads are the easiest to fabricate and calibrate, 
requiring only a matched termination with a suitable 
thermal heat sink and thermometer. 

2. Theory 

Nyquist's theorem (Ref. 4), including the zero-point 
energy (Ref. 5), states that the available termination 
noise power P is given by 



Assuming hf/kT « 1, 



*^ 2 ^^^ + exp (hf/kT) -1 
where 

T = termination temperature, "K 

Jt = Boltzmann's constant, 1.38054 X 10"" J-^K"' 

h = Planck's constant, 6.6256 X 10"" J-s 

B = bandwidth, Hz 

/ = frequency, Hz 



(1) 



kTB 



(2) 



Consider a thermal-noise standard, as shown in Fig. 4, 
consisting of a termination at temperature T and a trans- 
mission line with distributed temperatures and propaga- 
tion constants. The problem is to determine the noise 
power or noise temperature at the output reference point. 
Signify the propagating noise power, transmission line 
thermal temperature, and propagation constant at x by P„ 
T„ and Sa,. 









^xjx 






T 




V/a 








— » 


jg 


< — 


— dx 































T' 



Fig. 4. Thermal noise staodard with lots and temperatvr-^ 
of the transmission line as a function of position 

The propagating noise power can be separated into 
two parts: (1) from the termination, attenuated by the 
transmission line, and (2) from the noise contribution of 
the lossy transmission line. The noise power at the ref- 
erence output due to the termination is given by P/L 
(the termination noise power divided by the total line 
loss). Total line loss L is given by 



L = exp (2aO = exp f / '2ajbi J 



(3) 



The noise power generated by a transmission line ele- 
ment of length dx is 

kBT, (1 - exp (2cwfe)) ~ kBT, (2a,dx) (4) 

The contribution at the reference output is given by 
dividing by the transmission line loss from x to the out- 
put reference 



expM 2(Micj ^^ ' 



(5) 



ifl SPACE PROGRAMS SUMMARY 37-51, VOL. I// 



299 



I I 



where 



3. Calibration Errors 



L, ~ exp 



(r^*) 



is the loss from the source to x. The total noise power at 
the output reference is found by integrating end adding 
the contribution from the termination 



P' = 



2kB 



/' P 



(6) 



Dividing by kB gives the noise temperature (Ref. 6) 



T' = T" 4- 

L 



(7) 



where 



T*" = 



2 /•' 



is the contribution from the transmission line. If a^ and 
Tx are treated as constants a and Tp, then L, = exp (2ax) 
and 



^' = (i-t)^^ + t 



(8) 



The most critical measurement in the calibration of the 
noise temperature of a thermal-noise standard is usually 
the transmission line loss. For example, if the loss and 
temperature distributions are constant, the error in V 
due to loss measurement errors is [assuming a small loss 
and differentiating Eq. (10)] 



AT' ~ 0.2?026 {Tp - T) M, dB 



(11) 



To determine T' to an accuracy of better than O-l^K for 
a liquid helium rooled termination requires better than 
a 0.002-dB measurement accuracy. 

The contribution of an ambient temperature transmis- 
sion line with a 0°K termination is approximately (0.23026 
TpAL. dB), or 6.7°K/0.1 dB. As seen from Eq. (11), the 
transmission line loss has no net effect with an ambient 
termination (assuming the transmission line and termina- 
tion are at the same temperature Tp). 

For precision measurements, it is necessary to account 
for the pressure inside the dewar with cryogi nically 
cooled terminations. In this case, replace the termination 
temperature 1 with 



Cap 



(12) 



where 



A useful expansion for small losses is given by 



j- = i-x-^^j:' + 



where 



X = 2al= ,^,'.'y_ ~ 0.23026 L, dB 



(9) 



10 logioe 



Tc = cryogenic liquid boiling temperature at stand- 
ard pressure °K (approximately 77.36° K for 
liquid nitrogen and 4.216°K for liquid helium) 

C = cryogenic liquid pressure constant, "K/torr 
(approximately 0.010987 °K/torr for liquid ni- 
trogen and 0.001352 ^K/torr for liquid helium) 

AP = barometric pressure greater than standard, 
(76^ torr) 



Then 



r = T + {Tp - T) (x - Y -^"^ + ■■) (10) 



Other solutions are presented in Table 1 for various com- 
binations of transmission line temperature and propaga- 
tion constant distributions. 



In cryogenically cooled terminations, it is necessary to 
maintain the termination material in temperature equi- 
Ubrium with the boiling liquid (unless the termination 
material temperature is determined by means other than 
the boiling temperature of the liquid). This can be ac- 
coir.plished by submerging the termination material in 
the liquid, or by providing a very low thermal heat path 
to the liquid relative to the thermal heat path to the out- 
side environment. 



300 



jn SPACE PROGRAMS SUMMAKY 37-51, VOL. Ill 



■"WW 



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a 
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a X H 



JPL SPACE PROGRAMS SUMMAKY 37-5?, VOL. (If 



301 



The magnitude of the error made by assuming 
hf/kT « 1 in Eq. (2) can be estimated by considering the 
higher order terms. In this case (vahd for hf/kT < 4ir»), 
we have, from Eq. (1) 

(13) 



The correction term {hf/kT)yV2 contributes less than 
2'ib error at an operating frequency of 10 GHz for T 
greater than 1°K. Some authors have expressed doubt 
concerning the inclusion of the zero-point energy term 
(Ref. 7). It should be noted that Eq. (13) reduces to the 
same correction term if the zero-point energy is neglected 
when the calibrated terminations are used to perform 
temperature-difference calibrations in an actual radiom- 
eter. 

Other sources of error include the inaccuracies in the 
temperature and loss distribution calibrations, non- 
homogeneous transmission line ettects, and microwave 
mismatches. 

References 

1. Stelzried, C. T.,"A Liquid Helium-Cooled Coaxial Termination," 
Proc. IRE, Vol. 49, No. 7, p. 1224, July 1961. 

2. Stelzried, C. T., and Reid, M. S., "Precision Power Measurements 
of Spacecraft CW Signal Level with Microwave Noise Stand- 
ards," IEEE Tians. Inst. Meas., Vol. IM-15, No. 4, p. 318, Dec. 



3. Singer, A., Ulrich, R. R., and Naess, E., Thermal Catibrators in 
MUlimeter-Wave Radiometry, TM-67-2. Harry Diamond Lab- 
oratories, Washington, D.C., Mar. 1967. 

4. Nyquist, H., "Thermal Agitation of Electrical Charge in Con- 
ductors." Phys. Rev.. Vol. 32, p. 110, July 1928. 

5. Siegman, A. E., "Zero-Point Energy as the Source of Amplifier 
Noise," Proc. IRE, p. 633, Mar. 1961. 

6. IRE Standards on Electron Tubes: DefitUtions of Terms, 1962 
(62 IRE 7.S2), Proc. IEEE, p. 434, Mar. 1963. 

7. MacDonald, D. K. C, Noise and Fluctuations: An Introduction, 
p. 37. John Wiley & Sons, New York, 1962. 



C. RF Breokdown Studies: RF Breakdown in 
Coaxial Transmission Lines, R. Woo 

1. introduction 

A scheme for presenting breakdown data was discussed 
in SPS 37-45, Vol. IV, pp. 323-330 and SPS 37-46, 
Vol. IV, pp. 259-263. A series of breakdown experiments 
have bpen conducted for the 50-0 coaxial transmission 



line configuration in frequency range of 4-800 MHz. 
These measurements yielded breakdown data for fd 
values of 20-600 MHz-cm. 

2. Results 

The breakdown data obtained are shown in Fig. 5. 
Two experimental setups were used: (1) 10-150 MHz 
lumped-circuit test set (Ref. 1), and (2) 150-800 MHz 
transmission line test set (Ref. 2). The data are plotted 
in terms of similarity parameters and, as can be seen, the 
scaling correspondence between data obtained from both 
test sets is remarkably good (within reproducibility of 
the data). It must be pointed out that the transmission 
line test-set frequency in one case is as high as seven 
times that of the lumped-circuit test set. There is a 
spread in the results for fd = 100 MHz-cm (Fig. 5b) at 
the lower values of pd. This is not surprising since, as 
will be discussed below, this corresponds to a region of 
several transitions, and breakdown conditions are some- 
what dependent on surface conditions. 

3. Discussion 

The data of Fig. 5 can be combined with that obtained 
previously to form the composite breakdown plot shown 
in Fig. 6. S. C. Brown and A. D. MacDonald (Ref. 3) 
showed that breakdown data can be represented by a 
three-dimensional surface using similarity parameters. 
Figure 6 defines this three-dimensional surface with 
breakdown power as the vertical axis and fd and pk as 
the horizontal axes (see Fig. 7). The similarity param- 
eters of Fig. 6 are, however, more useful to a design 
engineer than those used by Brown and MacDonald. For 
a given coaxial line operating at a particular frequency, 
the engineer computes the corresponding fd, and, by re- 
ferring to Fig. 6, he hns the breakdown behavior as a 
function of pressure. In addition, he has information on 
the effects of changing either frequency or line size. 

The fd vs pA plane shown in Fig. 8 is very useful in 
understanding the breakdown processes involved. Al- 
though the various limits are indicated in the form of 
lines, it should be pointed out that these are meant to 
indicate transition rather than abrupt change. The mean 
free path limit serves to separate ionization breakdown 
from multipacting breakdown. The term "ionization 
breakdown" encompasses all breakdown processes where 
the dominant electron production mechanism is ioniza- 
tion by electron collision. This type of breakdown occurs 
when p\ is greater than the mean free-path limit be- 
cause, under these couditions, the electron mean free 



302 



JPL SPACE PROGKAMS SUMMARY 37-51, VOL. Ill 



KS470MH2(l09«Cfli) 
• IS00MHil4.0cml 



10* 

6 
4 



10* 

6 

4 



10' 

6 
4 

2 

10* 

6 
4 

f£ 10' 

UJ 

I lo' 
Q- 6 

4 
2 

10* 

6 

4 

2 

I05 

6 

4 

2 

10* 

6 

4 

2 

10' 



"T T TXT 

600 



T — nn r 

fd, MHz-cm- 



710 MHi(0 564 cm) 
100 MHz (4 cm) 



443 MHzIO 564 cm) 
62 5 MH<(4 O cm) 



282 0MH:(0.S64cil<l 
39 75MMi(4 0cm) 



X TRANSMISSION LINE TfST SET 
• LUMPEO-CPCUIT TEST SET 







» .X .•* 



I I I I 



I I I I 



I I I 



T- 



— T 1 — r 

J} TRANSMISSION LINE TEST SET 
• LUMPED-CIRCUIT TEST SET 



1 I I I 



"T 1 — TTl 1 T 

fd. MMi-cm-lOO 

X 177 4 MH<(0S64cn<) 
o 91 25 MH>(I 096 cm) 
• 25.0 MMj(4.0 cm) 



XX X fc 



I I II 



J L-L 



_L 



T — TTT" 



X TRANSMISSION LINE TEST SET 
• L'JMPEC -CIRCUIT TEST SET 



Ic) 



— I — m 1 1 — i~r 

td. MHi-cm • 70 
X 124 MHz(0S64cm) 
• 17.5 MHi(4 0cffl) 



»>" 



I I I 



I I I 



J L_L 



n — mr 



X TRANSMISSION LINE TEST SET 
• LUMPEO-CIRCUIT TEST SET 



-^ — m 1 1 — m I r 

fd, MHj-cm«50 
X e8.SMHl(0.S64aii) 
• l2 5MH«(4 0cm) 



I I 
TT" 



«<J^ 



(d) 



1. ill 



I I 1 I 



I III. 



_l 1_L. 



4 6 Itf 



4 6 10 



4 6 K3 



4 6 0* 10° 2 
p\, torr-cm 



4 6 K} 



4 6 10 



4 6 10 



6 10* 



Fig. 5. RF breakdown data plotted in terms or simiicrity parameters 



in SPACE PROGRAMS SUMMARY 37-51, VOL f» 



303 



K 
UJ 

O 

a. 



6 

4 



10' t 



-I — 1 — i—r 



-1 — I — r-r 



-1 — I — i—T 



^tf,MHz-cm = 450 



400 



350 



300 



250 



200 



159 



130 



70 



too 




J L_L 



-L J_ 



_!_ 



_l_ 



10' 



2 4 ( 

p\, torr-cm 



10^ 



I0» 



6 10' 



Fig. 6. Unified plot for RF breakdown in 50-n coaxial transmission line 



path is shorter than the gap distance. When discussing 
ionization breakdown, it is convenient to think of it in 
terms of the two ranges of fd presented below. 

a. fd > 100 MHz-cm. Under these conditions, fre- 
quency is suflBciently high and the gap distance sufiB- 
ciently large that the electrons are not s'.vept out of the 
discharge region by the field as in the case of dc break- 
down. Instead, the electrons are concentrated in the 
center of the discharge region and slowly diffuse away 
towards the electrodes. The speeds are so low that the 
electrons produce, essentially, no secondary effects at 
the electrode surfaces. Breakdown of this type is termed 
diffusion-controlled or microwave breakdown (Ref. 4). 
This, in many ways, is the simplest high-frequency t?reak- 
down since only two main processes are involved; elec- 
trons are produced through ionization by electron 
collision and are removed by diffusion to the walls. In 
certain gases, electrons are also effectively lost by at- 
tachment to gas molecules. 

The minimum of the diffusion-controlled curves occurs 
at approximately px = 30 torr-cm. The pA = 30 torr-cm 



line is called the collision frequency transition. At the 
collision frequency transition, the applied frequency 
and the electron-molecule collision frequency are ap- 
proximately equal, and energy transfer to the electrons 
from the field is at a maximum. If pressure is increased, 
the electron-molecule collision frequency increases, the 
energy gained by electrons from the field per mean free 
path decreases, and the breakdown level correspond- 
irgly increases. In a perfect vacuum, the electrons oscil- 
late with their velocity 90 deg out of phase with the RF 
field, and no energy ;; I'ained by the electrons from the 
field. The electrons gain energy from the field only by 
undergoing collisions with the gas molecules. A decrease 
in pressure from the collision frequency transition cor- 
responds to an increase in loss of energy transfer from 
die field to the electrons. Breakdown power, therefore, 
rises with decreasing pressure. 

b. fd < 100 MHz-ctn. When the applied iri^quejcy is 
sufficiently low or the gap distance sufficiently short, the 
amplitude of oscillation of the electron cloud approaches 
the gap distance and the electrodes enter the breakdown 
picture. This situation occurs when fd is less than 



304 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



o 
a. 




LECTRON 
OSCILLATION 
AMPLITUDE LIMIT 



PATH LIMIT 



-COLLISION FREQUENCY TRAiMSITION 



10' 10 



Fig. 7. Three-dimensional surface representing RF brealcdown in 50-n coaxial transmission line 



jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



305 



z 
s 






15 
I0« 


\ 


1 1 1 1 I I -Till I 

UNIFORM FIELD LIMIT 


1 r 1 1 1 1 1 1 1 1 1 


1 


6 


\ 


^. 




- 


4 


- 


\mean free path limit 


COLLISION FREQUENCY TRANSITION 


- 


2 


- 


\. 




- 


I0» 




\^ 




- 


6 


- 


\^ 




- 


4 


- 


\^ 




- 


2 


- 


\ 




- 


I0« 


. 


\^^ 




. 




- 


MULTIFACTING CUT-oFF LIMIT \"^^^ 


^'"'^^^^^ELECTRON OSCILLATION AMPLITUDE LIMIT 


- 


6 


- 


\ 


4 


- 


\ 


. ^^^^^^^ 


- 


Z 


- 




\ ^^^^^^^^ 


- 


10' 




1 1 1 1 1 1 1 1 1 1 1 


1 1 1-^1 1 1 1 i_J 1 i_: 


t>>. 1 



2 4 6 

^X,forr-cm 



I0« 



6 \0* 



Fig. 8. piK-fd plane showing limits of breakdown processes 



100 MHz-cm. Under such conditions, the loss of elec- 
trons is governed by mobility. Brown (Ref. 5) has termed 
this type of breakdown mobility-controlled breakdown. 
It must be emphasized that tli e transition from diffusion- 
controlled to mobility-controlled breakdown is gradual 
and occurs at approximately 100 MHz-cm. The oscilla- 
tion amplitude limit corresponds to the condition for 
which the amplitude of oscillation of the electron cloud 
is equal to the gap distance. At this limit, electrons are 
lost to the electrodes and the power required for break- 
down rises rapidly. This behavior is illustrated in the 
data for fd-50 and 20 MHz-cm in Fig. 6. In the case 
of fd = 20 MHz-cm, another minimum is observed if 
pressure is further decreased. This additional minimum 
appears when f d < 20 MHz-<m. This region has been 
studied extensively by Gill and von Engel (Ref. 6) who 
attribute the additional minimum to the ions. At this 
additional minimum, the amplitude of oscillation of the 
ion cloud is equal to the gap distance, and the ions 
impinging on the electrodes release secondary electrons. 
Electrons are, therefore, produced by ion bombardment 
of the electrodes. 



When pk is less than the mean free path limit, the 
electron mean free path is longer than the gap distance 
and secondary electron emission is the electron produc- 
tion mechanism. Under these conditions, secondary 
electron resonance or multipacting breakdown occurs. 
Although multipacting has been adequately covered 
elsewhere (Refs. 1, 2, 5, and 7), the following are points 
worth mentioning m connection with the multipacting 
data of Fig. 6: 

(1) The multipacting data of Fig. 6 corresponds to the 
lower breakdown boundary. The upper boundary, 
above which multipacting will not occur, is not 
shown in Fig. 6. 

(2) For fd less than the multipacting cut-off limit of 
fd ~ 70 MHz-cm, multipacting will not occur. 

(3) Multipacting is independent of pressure. 

(4) Multipacting breakdovim power levels are very 
sensitive to surface and outgassing conditions. In 
genera], this is not the case for ioniz-'tion break- 
down. 



306 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 




Fig. 9. Power handling capability in terms of fd 

(5) As id is increased, breakdown power levels rise 
more rapidly for multipacling than for ionization 
breakdown (see Fig. 9). Therefore, for a fixed 
power level, ionizatioi. breakdown will cover a 
wider range of experimental variables than multi- 
pacting. 

From the above discussion, the reason is clear for the 
spread in the data of Fig. 5b at the low values of pX. A 
transitional region between difFusion-controUed and 
mobility-controlled breakdown is represented by fd = 
100 MHz-cm. At approximately pk — 10 torr-cm, there 
is also a transition between multipacting and ionization 
breakdown. The fd = 100 MHz-cm corresponds to the 
minimum energy boundary in the case of multipacting, 
and breakdown data is especially sensitive to surface 
conditions. 

Engineers are, in general, interested in the minimum 
power-handling capability of a given component. The 
breakdown power levejs along the collision frequency 
transition are shovra in Fig. 9 as a function of fd, thus 
giving tne minima of the diffusion-controlled breakdown 
curves. The multipacting breakdown data are also in- 
cluded for comparison. As can be seen, for fd > 145 
MHz-cm, the ionization breakdown level is lower than 
the multipacting breakdown level, while the reverse is 
true for fd < 145 MHz-cm. 

4. Concluding Remarks 

Breakdown data obtained for the 50-n coaxial trans- 
mission line are summarized in Fig. 6, which is concise 
and compact and should prove to be valuable to the 
design engineer. When using Fig. 6, the design engineer 



should be aware of the various breakdown processes in- 
volved and, consequently, the accuracy to be expected 
from these curves. It must be remembered that the data 
in Fig. 6 were obtained through carefully controlled ex- 
perimental conditions. When testing a component for 
breakdown, the engineer must assure himself that he is 
measuring the pressure level in the area where break- 
down occurs. The materials of the component should 
have a low outgassing rate and be relatively clean. The 
breakdown procedures should be similar to the ones 
used in obtaining the data of Fig. 6. Figure 2 gives the 
breakdown power levels for air., Ionization breakdown 
is dependent on the type of gas while multipacting is not. 
The power levels of Fig. 6 correspond to a perfectly 
matched transmission line. If mismatches exist in the line, 
the breakdown power level must be correspondingly 
derated. 

Finally, the scheme of data presentation of Fig. 6 can 
be used for configurations other than the 50-0 coaxial 
transmission line. Similar cun'es can also be obtained 
for gases other than air. 

References 

1. Woo, R., "Multipacting Discharges Between Coaxial Electrodes," 
;. Appl. Phys.. Vol. 39, pp. 1528-1533. 1968. 

2. Woo, R., "Multipacting Breakdown in Coaxi.- '. Transmission 
Lines," Froc. IEEE (Letters), Vol. 56, pp. 776-777, 1968. 

S. MacDonald, A. D., and Brown, S. C, "Limits for the Di£Fusion 
Theory of High Frequency Gas Discharge Breakdown," Phys. 
Rev., Vol. 76, pp. 1629-1633, 1949. 

4. MacDonald, A. D., Microwave Breakdown in Gases. John Wiley 
& Sons, Inc., New York, 1966. 

5. Brown, S. C, Handbuch der Physik, Vol. 22, pp. 531-575. Edited 
by S. Flugge. Springer-Verlag, Berlin, 1956. 

6. Gill, E. W. B., and von Engel, A., "Starting Potentials of Elec- 
trodeless Discharges," Proc. Roy. Soc. London, Ser. A197, 
pp. 107-124, 1949. 

7. Woo, R., and Ishi laru. A., "A Similarity Principle for Multipact- 
ing Discharges," J. Appl. Phys., Vol. 38, pp. 5240-5244, 1967. 

D. Spacecraft Antenna Research: 400-MHz 
Coaxial Cavity Radiator, Part II, K. Woo 

1. introduction 

The power handling capability of the 400-MHz coaxial 
cavity radiator (SPS 37-48, Vol. Ill, pp. 238-240) at very 
low pressures has been determined. The ionization break- 
down of the antenna occurs at as low as 76 W in air and 
62 W in 100% COj. The multipacting breakdovm was 
not observed up to an input power level of 100 W (operat- 
ing limit of the feeding hybrid). 



JH SPACE PROGRAMS SUMMARY 37-51, VOL. (// 



307 




CAVITY 



Fig. 10, Ciiaxial cavity rodialet 



30« 



JPi SPACE PROGRAMS SUMMARY 37-Sl, VOL HI 



2. Antenna Design 

The design of the antenna is shown in Figs. 10 and 11. 
The coaxial cavity is excited by two orthogonal probes. 
The input feeds of the probes are connected to the two 
output terminals (having a 90-deg ^hase diflFerence) of a 
3-dB hybrid fed by the incoming line. For the purpose 
of preventing breakdown in the input feeds of the probes, 
and between the cavity walls and the probes, teflon 
insulators are used to fill up each input feed (between 
outer and center conductors) and they extend out into 
the cavity to wrap completely around each probe 
(see Fig. 11). With this arrangement, the voltage standing- 
wave ratio looking into each input feed with the other 
terminated is 1.25. When energized, the antenna radiates 
circularly polarized waves. 

ALL DIMENSIONS ARt IN INCHES 




3 074 

INPUT FEED I 

(TNC CONNECTOR) 



! 

INPUT FEED 1^ 
(TNC I 

CONNECTOR) >$ 



PROBE 



^^^^^^m\m\^\^\m\\^^^ 



CAVITY 



I 



I2?96r:| 



3. Test Results 

The power handling capability of the antenna was 
determined at the JPL Voltage Breakdown Facility. The 
antenna was tested in the vacuum chamber first with air, 
and then with 100% CO^. The ionization breakdown 
power level of the antenna is shown in Fig. 12 as a func- 
tion of pressure near and at where the power-handling 
capability of the antenna is least. The ionization break- 
down of the antenna occurs at as low as 76 W (at 
0.28 torr) in air, and 62 W (at 0.25 torr) in 100% CO^. 
In both ca«es, the breakdown took place at the aperture 
of the antenna. The multipacting breakdown (tested at 
10-' torr) was not observed up to an input power level of 
100 W (operating limit of the feeding hybrid). 

To increase the antenna power-handling capability, 
the following modifications are being implemented: 

(1) The apertiu-e of the existing antenna is being flared. 

(2) A new cavity having a wider slot width is being 
fabricated. 



01 

UJ 

o 
a. 



120 




All? 








COa 


100 
80 








V 








\ 
\ 

\ 
\ 


^v.^__ 


_____^.-^ 


BO 




/ 
y 
y 



Fig. 1 1 . Cavity and feed configuration 



O.i 2 OS 

PRESSURE, torr 

Fig. 12. Ionization breakdown characteristics 



4 



JPl SPACE PROGRAMS liil^lAkKi 37-51, VOL. Ill 



309 



N 68-37419 



XXII. Spacecraft Telemetry and Command 

TELECOMMUNICATIONS DIVISION 



A. Multiple-Mission Telemetry System: Bit-Sync 
Lock Detector Evaluation, N. Burow and A. Voisnys 

The multiple-mission telemetry system (MMTS) bit 
tracking and detection functions are accomplished by 
means of a mission-dependent program in the TCP com- 
puter. In the original demonstration and Mariner Mars 
1969 versions of this program, an estimate of the ratio 
of energy per bit to noise spectral density (ST/No) is 
used as an in-lock indicator. The threshold value of 
ST/No is entered via typewriter and is a function of the 
expected ST/No. 



A preliminary analysis of ST/No estimation in the bit- 
sync lor » was presented by Dr. J. Layland in SPS 37-48, 
Vol. IL, pp. 209-212. Additional analytical work, and 
suggested ST/No thresholds, are presented in Chapter 
XX-G of this volume. This article describes the approach 
used in evaluating the ST/No estimator as a lock detector. 



The overall test configuration if shown in Fi(,. 1. The 
MMTS demonstration bit-sync program was modified io 
output ST/No samples on mdgnetic tape in groups of 
1000. Measurements were made for input ST/No of 0, 



DATA CLOCK 




NOISE 
GENERATOR 


DATA + 
NOISE 




1 




' 


' 




PSEUDO MOISE 

DATA 

GENERATOR 


- 


SIGNAL 


DATA 


Ml> 


(ER 




INTEGRATOR 



H 



ANALOG-TO-DiGITAL 
CONVERT SIGNAL 



ANALOG-TO- 
DIGITAL 
CONVERTER 



NUMBER- 
CONTROLLED 
OSCILLATOR 



(BIT TIMING 
INTERRUPTS) 



DIGITIZED 
SIGNAL^ 



u u u 



(BIT PERIOD 
ESTIMATE) 



SDS 920 COMPUTER 

(MODIFIED MMTS 

BIT TIMING 

PROGRAM) 

1 



sr/No 

ESTIMAT ^ 



DIGITAL 

MAGNETIC 

TAPE 



TIMING INTERRUPT 
FOR COMPUTATION OF 
577/Vb ESTIMATE — 



Fig. 1 . ST/No lock dtttctor •valuation ttvf configuration 



311 



in SMCE PROGXAMS SUM/MARY 37-51, VOL. m 



c 


(start) 


YES 




' 






input tracking PARAMETERS 
PERCENT BANDWIDTH, BIT RATE 

1 








ENABLE BIT TIMING 
TRACK INS 




























Jno 




' 






TRACK BIT TIMING AND 
ESTIMATE BIT PERIOD 




SET PROGRAM TO 

OUTPUT NOMINAL 

BIT PERIOD 






\ 






t 








OUTPUT BIT PERIOD TO THE 
NUMBER-CONTROLLED OSCILLAT( 

i 








)R_r 








Lb 
1 ''° 


.ler data for st/n^ estimate | 
^^^'-"''^ Time ^"~""---,^_^ 








-~>^0R ST/Nq ESTIMATE__,J>" 
jTfES 


> 








COMPUTE ST/N^ 








i 








STORE 5^/^^J VALUE IN CORE 








^.,— -"'Iiave'^^^---.^^ 

^^"^1000 SAMPLES BEEN~;^ 
^^~— ..^OREDJ,,---''''^ 








■ N0~ 




? 




DISABLE TRACK OF 
BIT TIMING 








* 




* 




O'JTpuT STORED ST/N^ 
SAMPLtS TO MAGNETIC TAPE 








^ '^OUGH SAMPLES----^ 

■~~~-,^^^^ OBTAINED 1 ^^...-^ 






NO 






(terminate test) 





Fig. 2. Flow diagram of modifiod bit timing program 

2.5, 5.2, 7.5, and 10 d3, both with the bit-sync loop 
locked and cycle slipping. The data samples were filtered 
in a bandwidth equivalent lo 0.3% of the bit rate for 
input values of ST/No > 7.5 dB, and 0.1% of the bit 
rate f'^r input values of ST/No < 7.5 dB. Each test con- 
tains a minimum of 10,000 independent samples of ST/No 
estimate. This required that samples be taken at least 
[l/(band width X bit rate)] seconds apart. For conven- 



ience, a bit rate of ..oO bits/s was arbitrarily selected, 
yielding sample rates of 0.75 sampljs per second for the 
0.3% bandwidth and 0.25 samples per second for 
the 0.1% bandwidth. For the frequency offset or cycle 
slipping measurements, an offset of 3.6% of the bit rate 
was used. Figuie 2 is an abbreviated flov/ diagram show- 
ing the operation of the modified bit timing prograin. 

The data tapes were processed using a di^'^a-analysis 
computer program, and a histogram of the ST/No esti- 
mate was plotted for each value of input ST/No. The 
complete <et of plots is included in Chapter XX-G of this 
volume. Figure 3 is a summary of the results showing 
the spread of each ST/No estimate probability density 
together with the proposed lock thresholds. 



1^ 

to 



12 
10 










■ 




r 




c 

c 


I TRACK 
1 CYCLE 


SLIPPIN 


G 


T 


y 
















1) 


- r PROPOSED 
\ THRESHOLD 

\r:!-:TTiNGS 

1 








" 


(1 


i- 


■■ 














\ 




I 


n 




1 












? 






1 


c 


' 


[ 


3 


1 




'? 










• 






K 






2 ! 


) : 


■ 




) 


( 


> 


e 1 


_...... ^ 



5^A'o INPUT, 1)8 

Fig. 3. ST/No ostlmnt* distributions 

B. Rftlay Teiemetry Modulation System 
Davelcpment, c. Cwi 

The overall objective of this development effort ir the 
design a^d test of telemetry modulation systems for 
reiay-Unk applications, such as between a planetary 
enlry capsule and a i>earby orbiting or flyby spacecraft. 

The brtaHKftard evaluati'>ri of a proposed relay link 
for a Mariner 1971 type mission is continuing as pre- 
viously described in SPS 37-50, Vol. Ill, pp. 326-331. 
That article described the test results of an audio 
equivalent RF transmitter-receiver followed by .. bit 
synchronizer. 



in SPACE PHOGHAm SUMMARY 37-51, VOL. (N 



311 



PSEUOO -NOISE 
OATA INPUT- 




BANDPASS RLTER 
BANDWIDTH = 100 kHz 



IF 



r" 



sr 



HYBRID 



BANDPASS FILTER I 

aaOOO MHz 
BANDWIDTH = l90kHz 



CRYSTAL 
DETECTOR 



BANDPASS FILTER 2 

30.025 MHz 
BANDWIDTH- 176 KHz 



CRYSTAL 
DETECTOR 



FSK DEMODULATOR 
















lo' 




DECISION 

1 








■ 






|B1T 
1 cvwr 








BIT SYNi-HRONIZER 



DATA DEKODULATOR 



L. 



.J l_. 



■DATA 



Fig. 4. Experimental relay link 



The audio equivalent transmitter-receiver pair has 
been replaced by a breadboard 400 MHz FSK transmitter 
and receiver* for the purposes of running complete link 
compatibility tests. The configuration is as shown in 
Fig. 4. Random data modulates the transmitter; the 
down-c<Hiverted transmitter output, at 30 MHz, is mixed 
with broadband noise, to establish a controlled signal- 
energy to noise-density ratio (ST/No) at the receiver IF. 
After IF amplification, the signal is FSK-demodulated 
by the conventional topology consisting of crystal filtei-s, 
square-law detectors, channel-balance amplifien, and 
subtractor. Finally, the bit synchronizer and data detec- 
tor recover data and bit-sync timing from the noisy 
FSK-demodulated data stream. 



phase detector tcqwlogy and 6Q-Hz loop bandwidth 
(2Bi,) as described in the referenced SPS. 

The first bit-error and acquisition-time tests have been 
completed and are shown in Figs. 5 and 6, respectively. 
The theoretical performance curve of Fig. 5 is extracted 
from Boyd.^ The hardline bit-sync data is in excellent 
agreement with theory. Using bit sync derived from the 
bit synchronizer, a 0.3-O.4 dB loss is observed; this value 
of sync loss was also observed in the audio-equivalent 
receiver tests. The 0.9 probability of acquisition time for 
frequency offsets (A/) of 2.0 and 4.0 Hz. The values are 
also consistent with those obtained with the audio 
receiver. 



The noise bandwidths of bandpass filters 1 and 2 were 
averaged and that value (18.3 kHz) used for determining 
Sr/No and N, the IF bandwidth to bit-rate ratio 
{N = 36.6). The bit synchronizer uses the absolute-value 



"The RF equipment has been developed under NASA Code 186-68- 
04-08, Relay-Link RF Systems. 



Extensive bit-error and acquisition time tests are 
scheduled to determine the performance of this RF relay 
link as a fimction of RF limiting, square law versus linear 
envelope detectors, and channel unbalance. 



'Boyd, D. W., Performance of FSK Systems wUh Large Uncertainty 
in the Carrier Frequenq/, Apr. 3, 1967 ( JPL internal document). 



312 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Iff 



(E 
O 

IE 



m 

< 
m 
o 



10-2 

6 

4 

2 

lO-J 
6 



6 
4 



10-5 

6 
4 

2 

10"* 



>^^-T~"- -T- - 1 






i\° . 1 






1~ \~ n 








1 \ 


--^ 




1 X ' 








1 ; \ ' 






.__ 




\ 




\ 




\° 




' i Q ^ 


\ \ i ' \ 


O HARDLINE BIT SYNC \ 


D DERIVED BIT SYNC ^^--^ 


THEORETICAL, N = 36.6 - 


'^ 


\u 


\ 


\ 


' \ 


' \ 




1 1 1 - i 



12 



14 



IS 16 

ST/A/q, dB 



Fig. 5. FSK bit-error test 



17 



>- 

CD 
< 

m 
o 

(E 

0. 

<n 
o 

I 
I- 



UJ 
O 

u 

< 



llJ 



10' 



10' 



SYNC LOOP 25^= 60 Hr 
O PSEUDO-NOISEDATA.A/ = 2 Hz 
D PSEUDO-NOISE OATA,^ = 4 Hz 




17 

sr//Vo,(iB 
Fig. 6. FSK bit-cync acquisition time 



test 



JPL SPACE PROGRAMS SUMMAkY 37-51, VOL. Iff 



313 



^_ 07420 



XXIII. Spacecraft Radio 

TELECOMMUNICATIONS DIVISION 



A. Lunar Orbiler V Side-Looking Radar Expeiiment, R. L Horftor 

1 . Introduction 

For some time the Laboratory has been developing surface imaging or mapping radar systems applicable to lunar 
and planetary missions. Present spacecraft ordinarily have telecommunication elements that are very similar to the 
elements used in such radar systems. Preliminary investigation has shown that the S-band ranging transponder with 
a high-gain antenna could serve as a side-looking radar. To demonstrate this idea, an experiment was performed on 
January 24, 1968, using the S-band ranging transponder and high-gain antenna of the Lunar Orbiter V spacecraft in 
flight. A description of the experiment and the derivations of the mapping equations are presented in this article. 

2. Experiment 

The equipment used in this experiment is different from that of the usual side-looking radar, because the radar trans- 
mitter and receiver are widely .separated. As far as is known, a bistatic side-looking radar experiment has never been 
performed before. With reference to Fig. 1, the actual radar signal is transmitted from the spacecraft, reflected from the 
lunar surface, and received at the Mars DSS. In order to keep time and frequency references, the ranging modula- 
tion is actually transmitted from the Mars DSS to the spacecraft, routed through the transponder, and retransmitted 
on a different carrier frequency. 

As described, the experiment communication link contains three time-varying delay times. Proper tracking of the 
round trip delay and doppler is the crux of the data-processing problem. 

3T4 JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 






siW 



?(') 



s^(t)/^'~^ 



LUNAR 
ORBITERV 



^^ ^ -f moonJ 



CLOCK 



PN 

TTTT 



PN 



PHASE 
MODULATOR 






r,(/N 



»5l(') 






*,(/) 






MASER 



•eO 



-*G> 



WORD 



IF 
50 MHz 



•r(/) 



FREQUENCY 
SYNTHESIZER 



IF 

3.3 -MHz 

BANDWIDTH 



10 MHz 



•eC) .^'sO 



A,(/) 



1 — "x * 



60 MHz 



AMPLIFIER, 



TAPE />z(f) «„{f) 



10 MHz 




TAPE 



#I2(/) 



Fig. 1 . Block diagram of bistatic side-looking radar experiment and signal flow model 



The received signal is recorded in phase quadrature at baseband. Range resolution is achieved using a pseudonoise 
(PN) code biphase-modulated on the carrier. The received signal is multiplied by an identical locally generated PN 
sequence. The portion of the received signal whose modulation is synchronized to the local code can be separated 
frcMn the rest by a low-pass filter. This signal corresponds to a narrow strip at constant range from the spacecraft, as 
shown in Fig. 2. Each point within that strip passes through the lines of constant doppler caused by the motion of the 
spacecraft. A filter which matches that motion-induced phase behavior can resolve individual point scatterers. This is 
the basic principle of the side-looking radar. 

The surface resolution is determined by the radar beam incidence angle, the PN code bit length, and the bandwidths 
of the spacecraft transponder, the DSN receiver IF amplifier, and the tape recorder. The bit rate chosen for the experi- 
ment was a ^-MHz clock rate, allowing a 3-jus bit time. "Aiis is 1.0 km in slant range, corresponding to about 1.4 km 
on the surface. 

The code length was long enough to keep the spacecraft direct signal and the surface signal unambiguous. Also, the 
longer the code, the better the suppression of the direct signal. However, searching for the surface reflected signal 
gets more difiBcult as the code is lengthened. The code chosen was 1023 bits long. 

Signal strength calculations were based on resolving a 1-km square on the surface. Predictions showed a 3-dB 
signal-to-noise ratio. Such a noisy picture should reveal a recognizable shape, such as a large crater. Analysis of the 
data is not yet complete. 



jn SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



315 




LUNAR 
SURFACE 



RANGE 
STRIP 



LUNAR ORBITER V 



VELOCITY V 
SPACECRAFT TRACK 




ENLARGEMENT OF ILLUMINATED 
AREA MAPPING COORDINATES 



Fig. 2. XMnar Orbtter V side-iooking radar mapping coordinates 



3. Analysis 

This section presents the signal flow from the earth station to the spacecraft to the lunar surface and back tc the 
stafon. Further operations are performed, culminating in the system response to a point reflector on the lunar surface. 
A map is the superposition of many responses from the surface features. 

o. Cmnmwmca:6on link round trip time delay. Each link in the process is characterized by the time delay n (t). The 
delay is a function of time because the velocity of light is finite, and the three elements in the system are moving with 
respect to one another. Figure 1 has already identified these links. Only the time delay of the spacecraft-Moon link, 
T2 (t), is of real interest, because it is the particular nature of its variation which allows resolution along the direction 
of travel. 

The modulation of the transmitted signal is a PN code. One length of the code is denoted by x(t) and is T sec long. 
Hie modulating signal is then 

X{t)= 2 x(t-nT), x{t) = ±l (1) 

316 JPL SPACE PROGRAMS SUMMARY 37-51. VOL Iff 



Assuming a modulation index j3, the transmitted signal is 

s. (t) = Ai 2*^ cos (b)i t + )3X (*)) (2) 

The signal at the spacecraft is attenuated and delayed. 

s, (t) = Aa 2^4 cos («., [t - Ti (*)] +pX[t- T, (01) (3) 

At the spacecraft, the transponder retransmits the modulation on a different carrier frequency. 

s,(t) = A,2'AcosLSt-^n{t)] + pX[t-T^(t)]\ •> (4) 

On the lunar surface, the signal has additional delay T2{t) and a time- varying amplitude factor caused by motion 
through the antenna illumination pattern. 



S,(t) = A,{t- r, (t)) 2"^ cos (u, At - -^T.it - ra(t)) - tj(o1 + pX[t- r,(t - T,(t)) - r,{t)]\ 



(5) 



A^t the earth station the signal has additional time delay T3 (t), an attenuated power factor A5 (°), and white gaus- 
sian lunar background noise. The phase factor 9, is arbitrary but fixed for each scatterer, while 62 is uniformly random. 

e, (t) = A, [t - T2 (* - T3 (t)) ~ T3 (t)] 2^ cos L3 (t-^r,[t- Tj {t - T3 (*)) " T, (t)] " T, (t " T3 (f)) ~ T3 {A 
+ px(t- T, [t - T, /J - T3 (f)) - T3 (t)] -T,{t- Ts (t)) - T3 (A + ^tl 

+ n, (t) cos (<«3 1 + ©j) + n^ («) sin (.03 * + e^) (6) 

Let the total time delay be denoted by t (t). 

T(0 = T:(*-Ta(*)) + Ta(t) 
T*(*)=-Tat-r„(t)) + r.(*) (7) 

0)3 
Ta'r)-Ts(t-T3(t))+T,(«) 

The functions will later be calculated by the ephemeris and trajectory data and will contain the desired phase behavior 
of the spacecraft-to-surface link. The received signal is, tlierefore, 

e, (t) = A, (t - Ta{t)) 2^ cos [»3 (t - T^ (0) + j8X (f - T (*)) + Or] + n, (*) cos (a, t + ^0 -!- n, (*) sin (». t + «,) (8) 

In addition, there is the direct component from the spacecraft omni-antenna. Denote it by s^ (t), the interference 
component. 

*»/ {t) = A„ 2H cos L, Ft ~ -^ T. (t - r. (*)) - T. (Ol + pX[t- u (t - T. (0) - T, (t)]^ (9) 

JPl SPACE PJtOCicAMS SUMMARr 37-51, VOL. \\\ 317 



Let the interfjrence signal time delay be r,{t). Then 

s,,{t) - A„2''4cos [u,4t - T,*(t)) + j8X(t - T,(0)] (10) 

where 

r, (t) = T, (f - T, (t)) + T, (t) 
r,*(0=-T,((-T.(t)) + T,(t) (11) 

(1);) 

This component could be coherently tracked by the receiver, if desired. 

b. Receiver signal flow. At the receiver, signal is amplified by the mascr front end and mixed with the local oscil- 
lator frequency wlo- The receiver is not tracking, hence wlo is constant. 

Be (t) = (Ss (t) + S„ (t) + n, (t)) 2'^ COS a)LO t 

The frequency diflference (013 — wlo) is 50 MHz and is denoted by wso- Double frequency terms are dropped. 

fe (t) ^A4t- T„ (0) COS [0)50 t - <03T« (0 + /3X (t - T (*)) + e,]+ A,,; COS [«,50 t - (»3T;^ (0 + /3X (t - T, (t))] 

+ n, (t) 2^cos(<05o« + S2) - n^it) -^sin {mio t + O^) (12) 

Mixing with a 60-MHz referent e produces .signal on a 10-MHz IF. Make substitutions for the phase terms 0.3 t^ (t) 

and (<>3 T/^ {t). 

<^(t)=a,3T^(t) (13a) 

^/ {t) = «.3 T„, (f) (13b) 

^7 (0 = ^0 (t) ?i COS woo * 



e, (0 = A5 (t - 7,(0)00$ [«.,„* + ^{t)- pX{t - t(0) -<?.]+ A„cos [,otot + ^/(O - j3X(t - T,(f))] 



+ ni{t)-T^cos{o>iot - 62) - th(t)^sm(u>,at - 62) (14) 



After mixing to 10 MHz, the signal is passed through the 10-MHz IF filter, with impulse response hi (t). The filter 
bandwidth is 3.3 MHz. The convolution integral uses the dummy variable pi. 

es(t) = / dp, h, (p,) U,{t -Ta(t- p,) - p,)cos [«,o(t - pi) -H ^(t - p.) -pX(t-T(t- p,) - p,) - »,] 

+ A,, cos [a.,0 (t - p.) + ^/ (« - p,) -pX{t- J, (t - Pi) - Pi)] 

Xr>.{t- p,)^cos(a,o{t- pi) -Oz) - rh(t - pi)^sin(o>io(t - Pi) - ^2)1 (15) 

Remove the IF frequency and record the result on magnetic tape. Since the signal is now at baseband, quadrature 
compfnents must be kept. 

^» (t) = gg (t) 2 sin uio t 
eio(0 = e8(i;)2cos<*,o* 

318 jn SMCE nOGRAm SUMMARY 37-51, VOL III 



Upon substitution these signals become 

Co (t) = / dpi hi (p,) 2 sin <oio * Ia, {t - Ta{t - p.) - p,) cos [<»i„ (t - p,) + <^ (t - p,) - /JX (t - t (t - p,) - p,) - fl,] 

+ i^r„ COS [«,.,." Pi) + </>;(<- pO ~pX{t-7{t- p,) - p,)] 

+ n, (f - p,) gij cos («,,„ (t - p,) - flj) - Mj (* - p.) gij sin ("lo (« - pO - f^z) [ (16a) 

e,o(t) = / dp,/ii(p,)2coso),„f '|Ar,(<-Ta(t ~p,) - pi) COS [<uio (t - pi) + ^(t-p,) -)8JC(t-- T(t-p,) -pi) - fli] 

+ Ar, cos [a.,,, it - pi) + ,^; (* - p,) ~pX{t- T, (< - p,) - p,)] 

+ ni(f- p,)27^cos(o.,n(f-p,) - 62) -nj(<- p2)2^sin(<»,o{f- pi) - fl:i)|- (16b) 

Expanding the sinusoidal products and discarding the double-frequency terms gives 

e^{t) = dpiHi(p,)<A,-,(f-T„(f-p,)-p,)sin[(oi„p, -<^(t-p,) + )3X(f-T(t- p,) - p,) + tfi] 

+ Ar.i sin [w,o pi - <j>i(t~ p,) + jSX (t - T; (t - p,) - p,)] 

+ n, (t-p,)2^sin(<»iopi + O^)- rh{t- pt)-^cos(o>topi + 82) > (16c) 

eio(t) = j dpih, {pi)<Ar,{t - T„(t - pi) - p,)cos[«.i„p, -^(t-p,) + j8X(t - t(*- p.) - p.) + $,] 

+ Ar,/ cos [wio pi — 4>i{t- pi) + /3X (< - T; {t " pi) - pi)] 

■T f'l (t - p,) 2^ cos (uio p, + ^j) +«!(*- pi) 2V4 sin (")io pi -I ^2) > (led) 

But hi (t) is a bandpass filter function 

ft, (0 = hir (t) 2 cos («,,„ t + 6„ (t)) (17) 

where hirlt) and 9iir(t) are the amplitude and phase functions, respectively. 

Cb (*) - / dpi hiT (pi) 2 cos (uio p, + Oik (pi)) 

X iA,(t -Ta(t- pi) - p,)sin [«„opi - *(t - pi) + pX(t - t(* - pi) - pi) + fl.] 
+ A,, sin [.»io p, - <t>i{t- pi) + /5X (i - t, (t - p,) - pi)] 

X n, (t - p.) 2^ sin («.,o pj + «»)-»»»(*- pi) 2h ^o* (<*>» P> + ^s) [ (18a) 

,4^1 SPACE PROGRAMS SUMMARY 37-51, VOL III 319 



eio(<) = / dpihir(pi}cos(a)iopi ^flir(pi)) 

X <A,{t - T,(* - p,) - p.) cos [oi.opi - ^{t - pi) + j3X(f - T(t - p,) - p,) + e^] 

+ As, cos [«.,„ p, - ^/ (< - p.) + )8X (f - T, (t - pi) - p,)] 

X Til (< - pi) 2^4 cos («>io p. -.' ^2) + Hs (* - Pi) 2^5 sin ("10 pi + ^i) > (ISb) 

Expanding the sine and cosine products and discarding double-frequency terms gives 

e,(t) ~j dp,h,p(p,)|A5{t ~Ta{t- p.) - pi)sin [-^(t - p,) + BX{t~T{t-- p,) - p,) - e,r(p,) + fl,] 
+ As/sin [-^;(t - pi) + i8X (t - T,(; - p.) - pO -fl,F(pi)] 
+ ni(* - Pd^^sinie, - «,k(p,)) - »h(t - pOg^cosCfl, - «,p(p,)) i (19a) 

e.o W = j dp^hi,(pr)iA,{t ~ra{t- p,) - p,)cOS [-* (t - p,) + j8X (f - T(t - p,) - pi) - e,p{p,) + »,] 
+ A5, cos [ -^; (t - p.) + 0X (t - T, (t - p.) - pi) - fl,F (pi)l 

+ ".(*- P.)2^cos(«^ - e„(pi)) + rh{t- p,)2:;^sm{6, ~ fl,»(p,)) I (19b) 

Let hi (t) be the filter associated with the tape recorder response. 
«ii (0 = / dpi / dptK (pj) hit (pi) 

X ^^(t - TB(t - p, - p,) - pi - p2)sin [-*(t - pi - P2) + ^X(t - T(t - pi - Pi) - p, - P2) - ^if(pi) + ^ij 

+ A5,sin[-^,(t - pi - pj) + pXif - T(t - pi -pi) - pi - ps) - eir(pi)l 

+ ni (f - p, - Pa) ^ sin (9, - fl„ (p,)) - n, (t - pi - p,) ^ cos (9, - B„ (pi)) V (20a) 

/« /■« 
dpi 1 dp2 hj (pj) JiiP (pi) 

X <A,(t - T,(t - pi - pi) - pi - pOcos I-^(t - pi - pi) + j3X(t - T(t - pi - pi) - p, - Pj) - tfir(pi) + *i] 

+ As,COS [-^(< - pi - Pi) + /?X(« - t(< - p, - p,) - pi - Pj) - Cir (pi)l 

+ «i (« - Pi - Pj) 2w cos (9t - $„ (pi)) + nj (t - Pi - p.) 25J sin (^, - ff,r (pi)) >■ (20b) 

c. Range code demodulation. Rpage gating or demodulation is performed by correlating the re^^ived signal vilth 
time shifted locally generated versions of the PN code modulaticHi. 

er^it) = eu(t)X(t~To) 

etUt) = e»(t)X(t-To) (21) 

320 JPL SPACE PItOGIIAMS SUMMAKY 37-51, VOL. Iff 



Both signals are passed through a low-pass filter fw whidi X (t) is rapidly varying and ^ (t) is slowly varying. 

e.5 (t) = I dps fcj (pj) eti {t - Pi) 

— I dps hi (ps) en (t — Pi) X{t — To- Pi) 

J a 
«i« (t) = I dpihi (pi) lit (t - Pi) 

--Tdpihi (dO e,, (t - p,) X (* - r„ - p,) (22) 

Substituting the expressioi gives 

/•« Top Tm 

«i5 (0 = / dpil dp, I dpi hi (pj) fej (pj) fcir (pi) X (t - To — pj) 

X -(A., (t — T. (p, — p, — Ps) — p, — p, - Ps) sin [ — ^ (t — pi — Pj — Ps) 

+ pX{t— T^^ f.. - fb — Ps) — pi — P2 — pj)— ^if(pi) + ^i] 

+ A5, sin [ —^, (t — pi — pt — Pi) + pX{t — T,(i - pi — pt - Pi) - pi - pi- ps) - fliF (pi)] 

+ ni(* - Pi - P2 - Pi)^sm{et - 0i,(pi)) -nt{t- pi-pi- ps)2^cos(e, - fl,F(pi))| (23a) 

e.. (t) ^TdpiT dp, ("dpi hi (ps) hi (pi) h„ (p.) X (* - To - Ps) 

Jo Jo Jo 

X <Ai{t - T,{t - pi - Pi - pi) - pi - pt — Pi)cos[-<l>{t - pi - pt - Pi/ 

+ pX(t — T(t — Pi — pi — ps) — pi — Pi — ps) — ^ir(pi) + *il 

+ As/ COS [-^/(t — p, - Pi — ps) + /5X(t — T,(t - pi - p2 — ps) - Pi - Pi - Ps) " *if(pi)1 

+ n,(« - pi - p, - ps)2ViCOs(fli - «,f(pi)) + ni {* - p, - Pi - pi)^sm{ei - e,p(p,)) V (23b) 

Expand the sine and cosine products to separate ths j3X (t) terms. 

sin (-* (0 + i8X (t)) = -sin * (t) cos j8X (r) + cos ^ (t) sin )8X (*) 
cos(-*(*) + j8X(0) ^cos^(t)cosi8X(t) + sin^(«)sinj8X(t) 

But X(0 is ±1 only, so it may be removed from the arguments. 

sin(-*(*) + pX(t)) = - cos « sin (*(»)) + X(«)sinjffcos(^(«)) 
cos ( - * (t) + /SX (<)) = cos j8 cos (^ (»)) + X (*) sin /8 sin (^ (»)) 

jn SMCE PROGRAMS SUMMARY 37-51, VOL. IN 321 



Substituting into F.q. (23 a, b) gives 

eis (t) = I dps I dpi I dpi hi (pa) K (pa) Kr (pi) < As (* - Ta (* — pi — p, — pa) - pi - pj — pa) 

X Tx (t - To - ps)X(t - T(t - p, - p, - pa) - p, - P2 - Wsin)8cos(<^(t -px-p^- pa) + fl„(pi) - »,) 
- X(t - To - pa) COS /3 sin (^(« - pi - pa - pa) + flip(pi) - Oi) 

+ Ai,\ X(t —To — P3}X(t — T;(< — Pi — p2 — Pa) — Pi — Pj — p3)sinj8C0s(^;(f — pi — p2 — pa) + tfip(pi)) 
-X{t-To- pa) COS i9 sin (<j>j (* - pi - pj - pa) + ^iF (pi)) 

+ X(*-ro-p3) ni(*-pi-p2-p3)^sin(02-«,t.(pi)) - nj (t - pi - p, - pa) ^^^ cos (<*2 -flip(pi)) > 

(24a) 

«ie (t) = I dp3 I dp2 / dpi hs (pa) hi (pa) /Iif (pi) < A, (t — Ta (t - pi — Pi - pa) — pi — P2 — pa) 

X X{t- To - P3) COS P COS (<l>{t - pi - Pi- pa) + diF(pi) - 6i) 

+ X(t — To - p3)X(t — t(* — pi — p2 — pa)- pi - p2 — Pa) sinjS Sin(^(* — pi — p2 — pa) + tfip(pi) — 0i) 

+ As; X (* - To - Pa) COS j8 COS (ij.1 {t- Pi- Pi- pa) + 9ir (pi)) 

+ X (f — To — pa) X (t — T, {t — pi — Pi — pa) — pi — P2 — Pa) sin p sin (^ (* — Pi - - p? — pa) + Otr (pi)) 

+ X(*- To- Pa) n,(t-pi -p2-p3)2i;j-cos(e2- eiF(pi)) + "2 (* - pi - P2 - pi) gt^ sin (Ss - Ojf (pi)) \> 

(24b) 

It has been stated previously that ha (t) changes rapidly compared to X (t), but very slowly compared to t, (t), t (t), 
4, (t) and i>i (t). Hence, the integration over p, affects only X (t) terms. The other factors may be removed horn ihe 
pi integral. This statement implies the following: 

Prfpa K (pa) X (t - r - Pa) S X 

/ dp3fh(p3)X{t -To- p3)X{t - T(t - pj - P2 - pi) - Pl - p, - Pa) » R,(To - t(# - pi - p,) - pi - Pj) (25) 

322 JPL SPACE PROGRAMS SUMMAkY 37-51, VOL. Ill 



where X is the mean value of X (t) and R, (t) is the autocorrelation function of X (t). Equations (24a, b) are now 
eis{t) = I dpi I dpth2(p2)hiF(pi}<Ai(t — Ta(t — pi — p2) — pi-pi) 

X sinj3Rx(r„ - T(t - pi - Pa) - p, - p2)cos(<^ {t- pi- 92) + «ip(pi) - Oi) 

— X cos jtj sin (<^ {t — pi — p2^ + fliF (pi) — 61) 

+ As; sinjSRr (T„ - t/ (* - p, - ps) - pi - pj)cos (^/ (* - pi — pg) + ^ip (pi)) 

- X cos )8 sin {4, {t - p, - p^) + ^i, (p,)) 11 + n,, (t) (26a) 

eiiit) = I dp, I dpih2(p.)hif{pi)<As(t-Ta{t-pi-p2)~pi - P2) Xcos^cos(^(t - p, - pj) + eip(pi) -^1) 

+ sinj3R,(t - T(t- p, - pj) - Pi - p2)sin(^(t - p, - ps) +Oi,(pi) - fl,) 

+ As, Xcosj8cos(^i(' - p, - pa) + e,p(p,)) 

+ sinj8R,(* - T,(*- p, - pa) - p. - P2)sin(.^,(f - p, - p^) + ^ifCpO - fli) J> + n,6(t) (26b) 

The noise processes are quasi-gaussian. Multiplication by X (t) makes the process values at the PN code transi- 
tion points undefined. But averaging in hsit) applies the central limit theorem. Hence, the term "quasi-gaussian." 
Assume they are gaussian. The autocorrelation functions are equal. 

r* rao /"oo /*« /"oo Too 

R»i5 (r) = R„ig (t) = / dpel dpr. I dpt I dp3 I dpi f dpi hi (ps) hi (pa) h, (p,) K (pj) 

Jo Jo Jo Jo Jo Jo 

X hir (P4) hir (p,) X (f — To — Pa) X (t + T — To — ps) 

X E<|ni(*- pi - p2 - p3)ni(< +T - p« - p, - p6)-2-sin''(e2 - diF(pi)) 

+ n2(* - Pi - , . - p,)n2(t + T - p4 - Ps - p«)-^cos^ei - Oiriptm (27) 

The cross terms of iti (<) and fh (<) have already been eliminated, since the two are independent. But the two noise 
processes are white with correlation (*/2) 8 (t). 

raa tab /"oo Tmi /*« /*« 

^»15 ('■) ~ '^»1» ('■) ~ / ^P« / '^P" / '^Z'* / ^P^ I ^P^ I ^P' '•' (/*•) '*» (<*») ^' M ^ (P») ^W (P<) '»IF (Pl) 

JO yo Jo Jo Jd Jo 

X X(* - To - p,)X(t -f- T - r« - p,)|8 (t + Pl - p4 + P* - p. f p» - p.) (28) 

JPL SPACE PROGRAMS SU/MMAXy 37-51, VOL. Ill 323 



Integrating over pa gives 

^ ran rm /•» /"ao /•oo 

fl,j5 (t) =7 / dp-, / dpt I dp3 I dpi I dpi h, (t + pi - p, + pj - ps + ps) h^ (pa) ftj (ps) /ij (pi) 

X /iiF (p,) hiF (pi) X (* - To - Ps) X (* — To + P4 - pi + p5 - P2 - pa) (29) 

For purposes of calculating Rn^ (t) it is fair to assume that ^2 (*) and hf (t) do not aflPect X (*), since the filters are 
wideband conipared to hi (t). Treating them as unit impulses then gives 

R,„ (t) = I r dp3 /l3 (t + pa) h, (pa) X^ (* - To - pa) (30) 

But X(*) is ±:1, so X-(i) is constant. 

«-.5W = «-«W = f r<^P-^^3(T + pa)f»3(pa) -«)<T<oo (31) 

Equations (26a, b) may be simplified somewhat, since the phase and time delay functions are essentially unafiFected 
by filters hi (t) and hjr (t). Parts of the integrands may be moved outside the integrals. 

e,5 (t) = As (t - Ta (*)) < cos (^ (t) - e,) / dp^ dpi h, (p^) /i,p (p,) sin /Jfl, (To - t (t) - pi - pj) cos $„ (pi) 

- X cos j8 sin ^ik (pi) - sin (^ (t) — 6^) j dpi I dpi K (pj) hiv (p,) 
X I sin pR, (To - T (t) - pt - P2) sin e„ (pi) + X cos /8 cos e,F (p.) > 

f As, <! cos {<t>, (t)) I dp2 1 dpi K (pa) /»,p (p,) sin jSR, (T„ - t, (f) - p, - P2) cos flip (pi) 

— X cos )8 sin 9ir (pi) — sin ^, (t) j dp^ / dpi h^ (pz) /iip (pi) /' r (pi) 

X rsinj3R,(r- T/(0 - p, - p2)sine,p(pi) +Xcosj8co:i«,i ( >i) > + ni5(t) (32a) 

Similarly for eig (t) 
ei6 (0 =^ As (t - T. (t)) [cos {i> (t) - Oi) / dp; I dpi h; (pz) /»ir (pi) .sii. j3R, (!„ - t (t) - pi - pj) sin «„ (p.) 

+ X cos )3 cos 9ir (pi) + sin (^ (*) - fl,) / dp2 / dp, fn (pi) h p (pi) 

X sin j8R, (To - T (t) - pi - P2) cos e,p (pi) - X cos /3 sim fl,p (p,) > 

+ As/ < cos <^, (t) / dps / dp, hi (pj) /»ip (pi) sin jSR, (T - t/ (*) - pi - pj) sin ^ip (pj) + X cos ^ cos (9ip (pi) 
dp2 / dpih2(p2))iip(pi) 

X fsin /5R, (To - r (0 - pt ■ p.) cos fl„ (pi) - X cos )8 sin fl, , (0,) i + n„ (*) (32b) 

324 JfL SP.4CE PROGRAMS SUMMARY '7-51, VOL. Ill 



The expressions in Eqs. (32a, b) are not so formidable as they may seem. They contain both the modulated and 
carrier components of the reflected and direct signals. Observe that the carrier component may be suppressed by 
adjusting the modulation index fi or the average value X of the PN code. This is more easily observed by assuming 
hir (*) is very broad band compared to the code autocorrelation. In practice, most of the resolution degradation comes 
from the tape recorder /lo {t). These are part of the assumptions used earlier in calculating fh {t) and n^ {t). 

hir(t} = ^(t) hAt)^0 (33) 

Equations (32a, b) become 

e.5 (t) = A,{t- T„ (t)) icos (4, (t) - e,) sin j8 f dp, h, (p^) R, (r„ - t (t) - p,) 

--Xcos)3sin(^(t)- e,)i + A.„icos^,(t)sinj3 / rf/>2 /»2 (pj) R. (To - t, (t) - pz) 

- X cos )S sin {<l>, it}}\ + n„ (t) (34a) 

e.6 (t) = As (t - Ta (t)) isin (<^ (t) - »,) sin p j dp, h, (p,) R, (To - t (t) - p,) 

+ X cos p cos (<t,{t) - Oi)i + A5/<Xcosj8cos<^/(t) 

+ sin i„ (t) sinpj dp, h, (p,) R, (To - t, (t) - p,)\ + n,e (t) (34b) 

Equations (34a, b) are the operating equations for analysis defining system mapping capability and the e£Eects of 
interfering terms. 

It is now instructive to demonstrate the two-dimensional lature of Eqs. (34a, b) by showing how an array of point 
scatterers appear at e,5 (t) and eie (t). Assume that the surface is an anay of point reflectors whose complex reflection 
coefficients are characterized by an and On, the amplitude and phase angle, respectively. Furthermore, each resolution 
strip parallel to the vehicle track is associated with a time delay ry (t) and a phase variation <t,j (t). Within a strip, 
points are separated by their time occurrence ti. Time delay t> (t) does not vary along a strip limited by antenna 
beam width. Signals e^ (t) and Cie (t) are then expressed by a double summation over (t, ;'). 

e.5 (t) = 2 Is "i, A, (t - Tai (t) - U) { Tdp^h, (P2) Rx (To - r, (*) - p,) 

X sinj8cos(^,(« - u) - On) - Xcosj3sin(^>(t - U) - 6 a) i 

+ Ai, cos ^, (t) sin j8 / dpih, (pj) R, (To - t, (t) - pj) 

- X cos )8 sin <i>j {t) 1 + Oi, (*) (35a) 

JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 325 



e.6 W = 2<J2«„ A5(f- Ta,(t) - tA ^dp,h,{p^)R,{To - T,it) - p^) 
X sin p sin (4>iit- tj) - 0,,) + Xcos j8cos(<^y (t - ti) - da) \\ 
+ As, [sin </), (t) sin p / dp^ /», (p.,) R, (To - tj (t) - p,) 

+ X cos /? cos <^, (t) + «i6 (t) 



(35b) 



Because of the peaked nature of R, (t), as shown in Fig. 3, the dominant term in Eq. (35a, b) is the one for which 
To s T, (t). For this condition, the terms may be separated into signal from the /th surface strip plus interference. Thus 

ei5 (t) and eie (t) become 

9l5 (t) = 2 a,i A, {t - Tai (t) - ti) I jdp, K (p.) Rx {To - Tj (t) - p,) 

X sin /3 cos («, {t - t,) - On) - X cos j8 sin (^, {t - t,) - On) 

- X 2 2 "i. Ar, {t - Ta, (t) - U) sin {<!>, (t~ti)-IS- di,) 

It] \ 

- X A„ sin (,/.; (f) - i8) + n,5 (t) (36a) 

e,6 («) = 2 «.; A3 (f - Ta,- {t) - ti) r r dp. /l= (p..) Rx (To - T, (t) - p,) 

X sinj3sin(</.X* - U) - da) + Xcos/Jcos(^y(t - tt) - e,j) 

+ X 2 2 ai. A5 (^ - T, (t) - ti) cos (.^, (0 - i3 - «i.) 
I*; i 

+ XA„ cos (.^, (f) -p) + n,6 (0 (36b) 






THIS FUNCTION IS PERIODIC 
FROM - eo < T < to 




1 — l/2*-l 



m 



■2r^ 



(2*-i)r, 



Fig. 3. Autocorrelation of rang* codo X (f) 



326 



JPL SMCE PROGRAMS SUMMARY 37-51, VOL. Ill 



A glance at Fig. 3 indicates that R, (t) may be expressed as 

R*(t) = [H,(t)-X]+X 
= R„.(t) + X 

where Ro. (t) is nonzero only in the region t s 0, n (2* — 1) Tc. Equations (36a, b) become 

e,5 (t) = sin ;3 2 «;, A, {t - t„; (t) - U) / dp^ h^ (p^) Ro, (To - t, {t) - pj) cos {<j>, (t - U) - 0^) 

i Jo 

- X 2 2 «i> A5 (f - T„y (t) - *i) sin (^> (t - ti) - 0ii - /?) 

I • 

- X A5, sin (./.; (t) - j8) + fiis (t) (37a) 

e,6 (t) = sin /3 2 «*,- A5 (t - Tai (t) - ti) I dp^ h, (p^) Ro, (To - t,- (t) - pa) sin (<^y (« - tt) - 0^) 

i Jo 

+ X 2 2 «i; A3 (t - Tai (t) - ti) cos (^,- (* - ti) - j8 - <?„■) 

+ X A5, cos (^, (t) - /S) + n,a (t) (37b) 

Note how the return signal consists of a portion limited by the modulation to the strip and a portion from the whole 
area contributed by the carrier component caused by the code average value. 

It is important to make X as small as possible. This is done by making the PN code as long as possible. For a 
code of length (2* - 1), 

Rr{nT,) = 1 n = 0, ±(2*- - 1), ±2(2* - 1), • • • 

Tc = bit period 

^"i^) - ~2'' - 1 



X = 



The map of the surface is reproduced in the following way. Signals eis{t) and e,s{t) are combined in a single 
sideband mixer and passed through a filter matched to A.i(t) and 4>j{t). Depending on the detailed nature of 4>i{t), 
the scatterers for each /-strip are resolved. The process is repeated for each /. The results are mapped on a (To,t) 
plane. A strip parallel to the vehicle track will be reproduced along the contour To — ry (t). Because of the relative 
motion between station, spacecraft, and moon, t/ (t) is a function of time, and hence, a strip maps into a curved 
strip, in general. However, To is the time reference of the locally generated PN code. If provision is made for track- 
ing the variable portion of t^ (t) by making 

To = T{t)+T 
Ti{t)^Tit) + r, 

then 

^0 - Ty (t) = T - T> 

and the output map is fixed, because the mapping coordinate would be fixed to (t, t). The analysis of phase processing 
of 4>i {t) and the corresponding need for phase tracking will appear as convenience allows. 

JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 327 



B. Power Spectral Densities for Binary 

Frequency-Shift-Keyed Waveforms, D. W. Boyd 

1 . Introduction 

In designing a communications receiver, it is important 
to know the power spectral density of the received wave- 
form. This quantity defines the distribution of average 
signal power versus frequency and is useful primarily 
for locating the frequency bands of most interest. In this 
article we shall specialize certain general results from 
Ref. 1 for the power spectra of binary frequency-shift- 
keyed waveforms. 

2. Basic Aitumptions and Definitions 

Following Ref. 1, we assume that the transmitted wave- 
form is given by 

u(t) = u,{t), 

= U2 {t) , 



nT^t<(n+l)T 
n = 0,1,2, • ■ • 



vary like l/(f — fkY- The contributions of the 
neglected terms become appreciable only when 
the /fc are smaller tl. n the signaling frequency, 
/, = 1/T. For situations in which we shall be 
interested, this will never occur. 

3. Discontinuous Phaso FSK 

For discontinuous phase FSK, we shall consider two 
subcases: 

(/a - /i) ^ mf, (m an integer) 

(/.-/.)^^/. 



and 



(1) 



/i and /s arbitrary 



where 



u,(t) = Acos(27r/,t + e„) 

U^it) = A COS {2tt fit + <l,n) 



(2) 



We shall distinguish between two different cases for the 

u*(t),fc=l,2. 

In both cases, the choice of the Uk{t) is made inde- 
pendently and with equal probability for each interval 
of length r. In the first case, discontinuous phase 
frequency-shift-keying (FSK), the values of 6n and «/>„ 
are unconstrained from interval to interval. This corre- 
sponds to the case of switching between two independent 
oscillators. In the second case, continuous phase FSK, the 
initial values of the phase at t = are ^o = flo = ^. and 
the succeeding values ^„, On are chosen so as to make the 
phase of u{t) continuous at the transition points. This 
corresponds to the case of shifting the frequency of a 
single oscillator. 

For each case, we shall specify the power spectra to be: 

(1) One-sided, that is, specified completely in terms 
of positive frequencies. 

(2) Approximations obtained by neglecting terms of 
the order of l/(/ + /»)' compared to terms which 



+ • 



In the first case from Eq. (76) of Ref. 1, we have the 
power spectrum given by 

w. if) = (AV88) (/ - /,) + (AV88) (/ - /,) 

(3) 

We see from Eq. (3) that the spectnim consists of im- 
pulses at /i and /a with the familiar {sm'x)/x^ form 
centered about these impulses. 

For the second case of discontinuous phase FSK, we 
have to use Eq. (15) of Ref. 1. This equation is much 
more complicated and invoh es the values of ^h and On 
explicitly. However, if we assume that ^n and On are 
independent random varip.oles with uniform distributions 
over the interval [0,2? J, we can average the power 
spectrum given in Eq. (15) of Ref. 1 over ^n and On- 
Doing this, we obtp.m exactly the same expression as 
given in Eq. (3). Thus, for our purposes, Eq. (3) com- 
pletely specifies the power density spectrum for discon- 
tinuous phase FSK. If for any reason <t>n and On take on 
particular values, this statement will no longer be true, 
and we will have to go back to Eq. (15) of Ref. 1. 



328 



in SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



4. Continuous Phot* FSK 

For continuous phase FSK, we shall also corsider two 
subcases: 

fj-f,zj^ mf, (m an integer) 



and 



/a — /i ~ mf, (m an integer) 

U + fi^ mf. 



f.-fr^ 



(m+1) 



/. 



f.^f.^^^^f. 



Other cases are treated in Ref. 1, but these two should include most practical situations of interest. In the first case 
from Eq. (48) of Ref. 1, we have 



"'u(f) 



--[(^)-]-[(^)']LK^) -m. 



U 



1 — 2 cos 



24-j{24,+27rf,) 



cos 



[m-y-im-. 



(4) 



Although the behavior of uj„ (f) is not as transparent as it was before, it appears that the spectrum has peaks in the 
vicinities of /i and /j. 



For the second case, in which ^2 — /i = mf,, we use Eqs. (52), (53), and (54) of Ref. 1 to obtain 

Wu if) = w. if) + (AV8) [S (/-/.) + 8 (/ - f^)] 



where 



w. if) = (AV2/.) sin= [, (tj^^ - ^] 



«;„(/) = (AV2f.)cos^[:^(^)-^] 



-m -(^)— . 



for m even 



H^) ^(^)-^. 



for m odd 



Since m is an integer, the above expressions can be simplified: 



u).(/) = (AV2/.)sin^['r(^)] 



M^) -m-H 



for all m 
Here we see once again that impulses at /i and fi are combined with a continuous spectrum. 



(5) 



(6) 



(7) 



(8) 



JPL SPACE PItOGRAMS SUMMARY 37-51, VOL. (fl 



329 



5. G«n«ral Obitrvatient 

One of the most important characteristics to consider 
is the behavior of the spectra as a function of frequency. 
By studying Eqs. (3), (4), and (8), we conclude: 

(1) The spectra for discontinuous phase FSK fall oflE 
as l/f for large /. 

(2) The spectra for continuous phase FSK fall off as 
1/f* for large /. 

This observation is important in designing practical sys- 
tems. For example, in systems with small relative dif- 
ference frequency 



(0) 



C = 



f. 



(9) 



we would, ideally, like to use a continuous phase oscil- 
lator to minimize overlap from each of the two fre- 
quencies. Practically, it will be a question of how much 
the frequency of a real oscillator can be pulled. 

Another way to compare the behavior of the spectra 
versus frequency is to calculate the perceniage of total 
power in an arbitrary bandwidth. A convenient band- 
width to consider is 2C/„ for which we calculate P„ ^^ per- 
centage of total power within a bandwidth 2Cf, centered 
about (/, + ft)/2. Figure 4a shows the bandwidth defined 
above, and Fig. 4b gives representative values of P» for 
various C. Included for comparison are values for a phase- 




(b) 



c 


P« , PSK 


DISCONTINUOUS 
PHASE 


^»,FSK 

CONTINUOUS 

PHASE 


1 


0.903 


0.926 


0.984 


2 


0.95 


0.967 


0.995 


3 


0.966 


0.977 


0.996 


4 


0.975 






5 


0.98 







Fig. 4. (a) Bandwidth 2Cf. (b) Percentage of total 
power P„ in bandwidth Cf, 

shift-keyed (PSK) spectrum centered about (/i -I- fJ)/2. 
As can be seen from the figure, continuous phase FSK 
is by far the most eflBcient in terms of having the most 
power in the smallest bandwidth. 



Another interesting characteristic is the shape of the spectra as a function of C. It is clear from Eq. (3) that the 
spectrum for discontinuous phase FSK is just the properly separated sum of the impulses and the (sin' x)/x' terms. 
Thus for large C, when the overlap between the two terms is negligible, the shape of the spectrum in the region of /i 
and fi is a constant independent of C. The same sort of behavior for the continuous phase spectra can be deduced 
from Eqs. (4), (5), and (8). Substituting Eq. (9) into Eq. (4) and simplifying, we obtain. 






2|l - 2cOSr27r(^Y^) - ttcIcOSttC -I- COS'rrCl U (^-J^)\ ^ (^~r^) " ^ 



(10) 



It is convenient to consider the spectrum as a function of the normalized variable 



so that we have 



miflf. 



f. 



sin' (ttx) sin' (ttx — irC) 



|_TrJc(irX — irC)J 



(11) 



(12) 



330 



/, 2 { 1 - 2 cos (2irx - ttC) cos (ttC) -I- cos' (irC)} 

JPL SMCE UtOGHAMS SUMMAKY 37-51, VOL. Ml 



a function which is symmetrical about C/2. If we let 

x = C + A, |A|<1 



(13) 



and expand all the trigonometric identities, we obtain 



tM/)jf._ 



[sin irC cos ttA + cos ttC sin rA]" sin' (ttA) 



2 {1 - 2 cos (27rA) cos' (jtC) + 2 sin (2irA) sin (vC) cos (ttC) + cos= (irC)} 



7)}U(C + a)^aJ ^^^^ 



^ 



i: 



Now if C> > 1, the C over (C + A) in the last bracket 
will cancel, and we will have an expression which de- 
pends only on A and periodic functions of C with period 1. 
What this means practically is that in the region of 
interest around fi and ft {x = and x — C) the spectrum 
for e.g., C = 15.2 is approximately the same as the 
spectrum for C = 16.2, with a difiFeunt separation. Some 
of the characteristic shapes will be identified in Sub- 
section 6. Usin^' similar reasoning, we also arrive at the 
same conclusion for Eq. (8). 

The same sort of arguments also show that the fol- 
lowing properties of the spectra in the region of fi and 
fi {x-0 and x = C) hold: 

(1) The spectrum for continuous phase FSK with 
fi — ji = tnf, is approximately equal to that for 
discontinuous phase FSK for large C. 

(2) For large C, there is a rotational symmetry about 
x = and x = Co for the continuous phase spectra 
for C = C„ -H |9 and C = Co - )3, where < jS < 1. 

The practical implication of the second statement is that 
we can determine the shape of the spectrum for C = 15.2 
by looking at the spectrum for C = 14.8 and rotating that 
portion of it in the vicinity of x = 15 (or x = 0) about the 
point X = 15 (or X = 0). Further explanations of this sym- 
metry property of the examples are given in Subsection 6. 



(b) Continuous phase, shape for C is approximately 
the same as for C -I- 1. 

(c) Spechrum for continuous phase with 

U-fx = mf, 

is approximatftly equal to spectrum for dis- 
continuous phase. 

(d) Rotational symmetry about x = and x = Co 
for C = Co -i- )3 and C = Co - )3. 

The first property is the most fundamental; the others 
are pointed out to give a better insight to the behavior 
of the spectra, 

6. Plots 

Figures 5a to 5k show plots of the spectra for various 
values of C. In each case we have plotted only the con- 
tinuous portion of the spectrum as a function of the 
normalized variable x = (f — fi)/f,. The plots shown are 
symmetrical about the point x = C/2; x = ct^rresponds 
to /i, and X = C corresponds to ^2- For purposes of com- 
parison, we have included plots of the PSK spectrum 
centered on x = C/2. In each figure the numbered "-^ "es 
correspond to the following functions: 



To summarize, we list the symmetry properties which 
we hfcve outlined: 

(1) Behavior versus frequency: continuous phase FSK 
falls o£f as l/f* and discontinuous phase FSK falls 
off as l//». 

(2) Shape of spectra in region of /i and /j (x = and 
X = C) as function of C for large C: 

(a) Discontinuous phase, shape is same for all C. 



Curve 0-PSK 



iVu if) f, ^ sin' (tx - 7rC/2) 
A' "" 2(irx-7rC/2y 

Curve l—DUconHnuout phate FSK 



(15) 



Wu{f)f. ^ ir sin'(7rx) sin'(:rx-r C)-j 

A' 8L M' (ttX-ttC)' J ^^"^ 



Jn SM« PROGRAMS SUMMARY 37-51. VOL. Ill 



331 



ISO 



100 



050 



(a) 



PSK 

1 DISCONTINUCXJS PHASE 

FSK 

2 CONTMJOUS PHASE 
FSK 




1.50 
100 
0.50 


(d) 




















N 


,/ 


Vj 


, 1 







N^ 


^ 


^ 







050 



040 0.80 1.20 1.60 2.00 2.40 0.60 1.20 160 2.00 2 40 280 

150 



"I 



\ (b) 


^ 


^- 1 







1.00 



050 



0.50 0.90 1.30 1.70 



ISO 



1.00 



0.50 



(c) 












\ 


^Z 






> 


I 


^0 







210 250 090 1.30 

OSOIS^ 



(e) 




j 














\/ 


^2 






^ 




/ 


^1 





170 210 2 50 2 90 




0.60 100 1.40 1.80 2.20 2.60 100 1.40 1.80 2.20 E 60 3.00 

/ X 

Fig. 5. Power sptctra (a) for C = 0.8, (b) for C = 1 .0, (c) for C = 1 .2, (d) for C = 1 .6, (e) f or C = 1 .8, (f) for C = 2 



332 



if I SMCE PROGMMS SUMMAftf 37-51, VOL. Iff 



ISO 



100 



050 



(g) 






1 PSK 










1 DISCONTINUCX'? 










FSK 










2 CONTINUOUS 












FSK 










1 


/■ 




\ 








\ 


r 

/ 






s. 






\ 


(^0 






\ 






y 


V 






\ 


^ifflihK^ 


i..^ 


1.10 1 ■ 


so 


1 < 


» 


2 


30 Z. 


TO 310 



O50 



(i) 

2 

1 



^ 


^ 







13.90 14.30 I4.TO 1510 15.50 15.90 



M 






ISO 
100 
O50 


(h) 


















r^ 


\ 




/ 




r' 







L 


rjfll 


Bl^^ 


ro 



ISO 



100 



0.50 



120 160 2.00 2.40 2.80 320 



(k) 












^A 










'A 


i 


i 





ISO 



100 



050 



l(.) 




1 








■\ 












L 







14.00 14.40 I4.a0 15.20 1560 1600 

X 



13.60 14.20 14.60 15.00 15 40 15.80 
X 



Fi9. 5 (contd). Power spectra (g) for C = 2.2, (h) for C = 2.4, (I) for C = 14.8, (j) for C = 15, (k) for C = 15.2 



JPL SPACE PROGRAMS SUMMARY 37-51. VOL. Ill 



333 



Curve 2— Continuous phase FSK 

A^ 2 \^2ttx 27rx-27rmJ 



(17) 



for C = m = integer 



Wu if) f. ^ Sin'(7rX)sin''(7rX-7rC) 

A* ~2{l-2cos(2irx-irC)cos7rC + cos^irC} 



X 



r '^ T 

\_TrX (ttX - wC) J 



(18) 
for C ^ integer 



The total transmitted power is the same in each case. The 
(/-axis is the value of u?, (/) f,/A- and x-axis is the value 
of X. Since we have only plotted the continuous portions 
of the spectra, impulses must be added to the curves 
corresponding to F.qs. (16) and (17). 

Figures 5d to 5g in particular illustrate the efiFect on 
the continuous phase spectrum of changing C. For C less 
than 2 but greatsr than 1.5 we have a peak inside the 
point X = 2. As C approaches 2, the peak becomes more 



pronounced and moves closer to the point x = 2, until 
we obtain an impulse for C = 2. For C greater than 2, 
but less than 2.5, we observe a similar behavior, except 
that the peak is outside the point x = 2. A similar 
behavior can be expected as C varies through any integer 
value. 

The general properties discussed in Subsection 5 should 
be evident from the plots, particularly Figs. 5i and 5j. 
The PSK spectrum has decayed to a negligible level in 
these figures, and the symmetry relations discussed are 
clear. 

Another point of interest is that the spectral peaks for 
continuous phase FSK become less pronounced as C 
varies away from integer values. For example, compare 
Fig. 5d with Fig. 5f. To obtain sharp spectral peaks, 
fi — f, must be approximately an integer value. This 
property may be important in system design. 

Reference 

1. Bennett, W. R., and Rice, S. O., "Spectral Density and Auto- 
correlation Functions Associated With Binary Frequency-Shift 
Keying," BeH Si/rtem Technical Journal, pp. 2355-2385, Sept. 
1963. 



334 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



*N«8-g742l 



XXIV. Future Projects 

ADVANCED STUDIES 



A. Science Utility of Automated Rovins Vehicles, 

R. G. Brereton 

1. Introduction 

The geology of the earth has been synthesized from a 
prodigious amount of data that was contributed from 
many observations and scientific disciplines and acquired 
over several decades. There is every reason to suppose 
that knowledge about lunar geology (i.e., knowledge of 
the structure and processes of the lunar interior, the 
composition, structure, and processes of the lunar surface, 
and the history of the moon) will be unfolded in the 
same way. Although working hypotheses have matured 
through experience and the terrestrial sphere is avail- 
able as an accessible geological example, the true picture 
of lunar geology can only be formed from much new 
data that will have to be acquired from a wide range of 
surface location, structures, and physiographic provinces. 
The very nature of the lunar exploration task suggests 
that a surface mobility system will be required to acquire 
the needed data. Several types of designs for this mo- 
bility system have already been proposed. 

Previous studies have indicated that a separable rover, 
delivered as payload by a Surveyor and hence limited to 



a total mass of 100 to 200 lb, could be useful in local 
surveys; however, it is recognized that such small ve- 
hicles, with a payload capability of about 20 lb, would 
have only marginal utility for most roving missions. At the 
same time, a reasonable upper limit on size for an auto- 
mated rover would seem to be that of the local scientific 
survey module, whose mass is more than 1000 lb and 
whose size is compatible with a Saturn V launch. Be- 
tween these lower and upper size-mass limits, there is 
probably a feasible vehicle design that can perform the 
required roving vehicle mission, while still being small 
and light enough to be delivered as an integral package 
by Centaur, alternately as payload aboard a single launch, 
manned Apollo mission, or as a separable payload item 
aboard an unmanned soft-landed vehicle intermediate in 
size between Centaur and Saturn V. 

In the past, discussions and designs for roving vehicle 
systems have been constrained by specifying the size, 
.♦•e'ght, power, etc., of the roving vehicle to fit it into a 
particular launch vehicle, or the vehicle design has been 
constrained by a specified program or c^ rating mode. 
These constraints, however justified, have tended to limit 
considerations regarding full scientific utility of auto- 
mated roving vehicles. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL. Hi 



335 



The scientific instruments carried on a lunar roving 
vehicle will vary with the function of the programmed 
scientific task, although one basic vehicle design will 
probably suffice for all tasks provided reasonable range 
and mobility requirements are satisfied. This basic design 
must incorporate an imaging system and a navigation 
system that can perform both the vehicle guidance and 
navigation function, and also support the task or science 
function. It is expected that the vehicle will travel slowly 
over the surface (at the rate of a few kilometers per hour 
at most) and \t'ill be long-lived. Lifetime of the vehicle 
will, tf course, be a function of the particular science 
task that the rover is programmed for; but, in general, 
the science requirements call for a vehicle lifetime mea- 
sured in months to perhaps years. This would suggest 
that the primp scourer of power be nuclear, solar, or a 
combination ot these. Telecommunications are not criti- 
cal for nnera^'on anywhere on the front face of the moon; 
however, for backside operation, an orbiter relay link 
would be required. 

There are four basic science tasks or separate missions 
that an automated roving vehicle can be useful for in a 
program of lunar exploration. Vehicles utilized in this 
way can be expected to provide significant new data 
about the moon that may not be available through other 
cost comparable techniques. Each of these tasks has its 
place in the overall lunar exploration program; any plan 
that defines the most feasible and economical lunar ex- 
ploration program must consider a mix of these roving 
vehicle tasks with other lunar missions, both manned 
and unmanned. 

2. Imaging System 

The automated roving vehicle will require an imaging 
system for purposes of navigation and guidance and for 
terrain assessment. The system should have stereometric, 
polarimetric, and colorimetric capabilities and possibly 
telescopi. lens combinations to allow a close look at 
features with minimum amount of vehicle travel and 
shufiSing. The specific objectives of the imaging system 
on the rover are: 

(1) Provide near-real-time images that can be used to 
guide the roving vehicle. 

(2) Acquire dimensionally stable images from which 
topographic maps can be made by photogram- 
metric methods. 

(3) Provide reconnaissance-eye-type geological infor- 
mation in color. 



(4) Provide near-field information on surface structure 
and texture, with the capability to detect particle 
sizes down to at least 0.5 mm. 

A variety of sensors and camera systems could perhaps 
be adapted to the roving vehicle mission; however, the 
selected system should meet the following requirements 
that are believed to be essential to the objectives of the 
roving vehicle mission. 

(1) A stereographic baseline of the camera system of 
preferably 3 ft but no less than 1 ft. The baseline 
may be vertical or horizontal. 

(2) A measurement of the local vertical to 0.5 deg at 
each position of the roving vehicle from which an 
image is obtained. 

3. Science Tasks 

a. Sample acquisition. The Apollo sample return ex- 
periment is recognized as one of the most important in 
the entire lunar program, since it affords the opportunity 
for elaborate earth-based investigation of the isotopic 
composition, chemistry, mineralogy, and physical state 
of the lunar surface material. To extend this experiment 
beyond the Apollo landing locations appears highly de- 
sirable. Therefore, a possible mission for a small rover is 
the collection of samples along an extended (up to 
hundreds of kilometers) traverse, followed by the de- 
livery of the samples to a collection point where they 
would be returned to earth, presumably by an Apollo 
spacecraft. The traverse could be either from one 
Apollo landing point to another or from an unmanned 
vehicle landing point to an Apollo site. It is assumed 
that any special packaging requirement for samples to 
be returned to earth could be accomplished by the astro- 
naut at rendezvous. It has been suggested (Ref. 1) that 
the automated rover be capable of traverses up to 500 km 
with at least 100 stations for observation and sample 
collection, and be capable of carrying 25 kg of samples 
collected and individually packaged in 100- to 250-g 
containers. It would appear that the real limitation for 
this mission is not the weight of samples that can be 
conveniently carried by the rover to the rendezvous point 
or transported by Apollo back to earth, but rather the 
time required to acquire meaningful samples. It does not 
seem that random sampling along a profile is the most 
desirable technique; however, it may turn out to be the 
most practical one. Samples should be acquired from 
select locations (outcrops, etc.); this will require consid- 
erable observer effort and time, and much stop and go 
maneuvering for the automated roving vehicle. 



336 



JPl SPACE PROGRAMS SUMMARY 37-51, VOL. Ill 



The minimum scientific instrumentation for this type 
of mission would include an imaging and navigation de- 
vice plus techniques for acquiring lunar samples. To 
perform the latter task, two separate sampling modes 
are desirable — one for hard rock material, and another 
for sifting the particulate material that appears to form 
much of the lunar surface. Nash (Ref. 2) has given an 
excellent discussion of the strategy, principles, and instru- 
ment requirements for sampling planetary surfaces. 

The imaging device would perform several functions. 
During traverse, it would observe the general lay-of-the- 
land, its structure, stratification, and topographic form, 
and color changes and rock textures down to at least 
0.5 mm; therefore, it would indicate interesting areas for 
sampling. The device would also be used to locate 
sample stations with respect to identifiable lunar surface 
features to within 100 m on base maps or orbiter photo- 
graphs. 

It would be desirable to equip this type of rover with 
a device for elemental chemical analysis that could be 
used in a reconnaissance mode, and, in conjunction with 
the imaging device, for selecting meaningful samples. A 
number of lightweight instruments using techniques such 
as alpha scattering, neutron activation, and nondispersive 
X-ray emission spectroscopy seems to be suitable for this 
operation. 

Table 1 presents some information about a typical 
science payload for an automated rover designed for the 
sample acquisition task. 

b. In situ analysis. One of the most obvious and per- 
haps more important roles of the automated roving 
vehicle in the lunar program will be geological recon- 
naissance, or the ability to extend the local measurements 
of Surveyor or Apollo into the surrounding area. Only a 
very small area of the moon is expected to be explored 
by manned missions of the near future; therefore, a 
properly instrumented automated rover capable of prob- 
ing the environs of the moon out from Apollo sites should 
have an important mission in a lunar exploration pro- 
gram. Although an automated rover, however iiistru- 
mented, can never be expected to replace the on-site 
geologist, a properly instrumented rover can be expected 
to provide: (1) survey type data on the geochemistry of 
the moon to include information about the kind, origin, 
and distribution of lunar rocks and minerals; and (2) re- 
connaissance imagery bearing on lunar physiography, 
surface structures and stratigraphy. These data will con- 



Tablt 1 . Science inttrumcnii for tompU 
acquisition rov«r 



Imaging (yiltm 
(• lb; 2 W) 



Thl> inttrumtnl would provid* imagai for guidanc* end portioning of 
tti* rovor and for tompl* Mioction. Starao, color, and rsiolution to at l*a>l 
1 mm It dcilrobl*. 



Elamonlal anolytlt 
(■ lb; 4 W, during epwallen) 



Th* int*rumtnt (nonditponilv* X-ray •mittlon ipactrotcopy) li formed 
from a rodioactiv* excitation tource, a got fill-d proportional counter 
for detecting a lignol, an ampiifier and deployment mechonifm. In 
operation, ttie inttrument excitation source and tensor mutt be deployed 
to the lunar surface. 



Poitlculale (ampler 
(5 lb; a W) 



The suggested instrument is a so-called rigid helical conveyor with 
drill tip. It would be capable of sampling the typical lunar soil to a depth 
of perhaps 5 in. It size-sorts particles so as to diminish the content of 
Hiose over SOO iim and reject those over 1 000 /un. Device would hove 
two functions — acquire samples ond distribute them to sample containers. 



Hard reck drill 
(10 lb; 25 W) 



This is a rotary impact drill capable of sampling rock material as hard 
OS dense basalt. The instrument has a depth capability of about 1 ft. 
Device would have two functions — ocquire samples from hard rock and 
distribute them to sample containers. 



Sample container 
(50 lb, full; 2 W) 



Desire about 100 sample containers for 0.25- to 0.5-lb samples. 



tribute to the understanding of the moon and indicate 
areas of high interest for planning future missions. This 
type of rover mission can serve a useful scientific purpose 
in both regional and local studies. 

It should be realized that a chemical basis alone is 
incapable of classifying the many diverse products of 
rock-forming processes. Thus, chemical elemental analysis 
experiments will not distinguish crystalline rock from 
volcanic glass or ash with the same chemical composi- 
tion, nor a physical mixture of local debris from a 
crystalline rock. The accepted schemes of rock classifica- 
tion are based on texture (the size, shape, and geometrical 
relation of grains in a rock) and the identification of the 
minerals in the rock. From these parameters, informaticm 
regarding the nature, geologic history, and origin of the 
rock may be defined. 



Jn SPACE PROGRAMS SUMMARY 37-51, VOL. »l 



337 



On the basis of the above, the scientific instrumenta- 
tion tor this type of rover mission should include: (1) an 
imaging device, (2) an array of geochemical instruments, 
and (3) a sample acquisition preparation device. The 
imaging device would perform the same function as on 
the sample acquisition task. 

The sample acquisition preparation device would be 
the same as described for the previous task. Samples of 
lunar surface material would be obtained by the par- 
ticulate sampler or hard rock drill, and this material 
would be distributed to the geochemical instruments. 

As a minimum set, the array of geochemical instru- 
ments must include methods for elemental analysis, 
phase analysis, and study of rock textures. The suggested 
iny'Tuments here are an X-ray spectrometer (Ref. 3) for 
elemental ana^/sis, an X-ray diffractometer (Ref. 4) 
for mineral phase determination, and a petrographic 
microscope (Ref. 5) that could observe crushed rock 
samples in transmitted light. These instruments were 
previously considered for both Surveyor and rover mis- 
sions. Table 2 presents a typical science payload for an 
automated rover designed for the in situ analysis task. 

In addition to the above instruments, it may be desir- 
able to include a gas chromatograph in the payload for 
this vehicle. The chromatograph would provide an 
analysis of the volatile constituents in lunar surface 
material. 

c. Traverse geophysics. Traverse geophysics has a 
very special place in the lunar exploration program. It 
can provide data toward the solution of problems that can 
be solved only by the combined techniques of surface 
mobility and geophysical instrumentation. Traverse geo- 
physics using automated roving vehicles is not a panacea 
for all the problems of lunar exploration; however, it is 
a powerful tool for providing data on the subsurface of 
the moon and when these data are correlated with lunar 
geology and multiple working hypotheses, they can pro- 
vide an informative picture of the possible structure and 
processes of the lunar crust. The choice of scientific 
instruments for traverse task is quite large, because geo- 
physical techniques and instrumentation have become 
more diversified through the effect of space age tech- 
nology and the revolutionary growth of science that has 
taken place since 1940. For example, 10 years ago, a 
magnetic survey was usually accomplished with a field 
balance magnetometer,' which measured only one com- 



Tabu 2. Scitnc* indrumanft for in titu 
analysis rover 



Imaging lyttam 
(• lb; 2 W) 



ThU iRitrumant would provide imagat for guidanc* and petiDoning of 
Hi* rovar and alto for gooioglcut tyaboll lyp* Information. Stvoo, 
color, and lyttam rttolution to at loai' O.S mm It dciirobl*. 



X-ray dlffradomttar 
(15 lb; 4 W) 



Th* X-ray diffractomatar will b* utad to conduct minaraloglcal anolyiai 
of lunar surfoca matarlal acquirad at a numbar of fixad pointt on a roving 
vahicia Iravarw. Tha rtr'morf objactiva of thii initnimant is to Idantify 
tha typat ond ralativa abundonca of tha various crystalllna phosas 
axpactad to ba prasant in a lunar tompla. Tha instrumant will provida 
diffroction data of sufflciant quality to idantify any of tha major rock- 
forming and occassory minarals. 



X-ray ipacframatar 
(15 lb; 4 W) 



Tha X-ray spactromatar will ba usad to conduct an alamantol analysis 
t.r lunar surfoca malarial acquirad at a numbar of fixad points on a roving 
vahicia travarsa. TX- moda of analysis con datact alamants from sodium 
through uranium; howavar, oiity those alamants from sodium through 
nicltal ora axpactad to ba prasant In sufficient quantity to allow detection. 



Petrographic micraecepa 
(15 lb; 4 W) 



The petrographic microscope would provide texlurol and optical infor- 
mation on rocks and particulate malarial from the lunar surface. 



Particulate tampiar 
(5 ib; 2 W) 



Tha suggested instrument is a so-called rigid helical conveyor with 
drill Hp. It would be capable of sampling tha typical lunar soil to a depth 
of perhaps 5 in. It size-sorts particles so as to diminish tha content of 
those over 500 Mm and reject those over 1 000 ^m. Device would hove 
two functions — acquire samples and distribute them to the geochemical 
instruments above. 



Hold rack drill 
(to Ib; "I W) 



This is o rotary impact drill capable of sampling rock material as hard 
OS dense basalt. The instrument has a depth capability of about 1 ft. 
Device would hove two functions — acquire samples from hard rock and 
distribute them to the geochemical instruments above. 



'Designed by A. Schmidt; manufactured by Askania Werke, Berlin. 



porent of the earth's magnetic field. Thus, a magnetic 
survey to measure the magnitude of the geomagnetic 
field vector required two separate survey operations with 
two separate magnetometers (one survey and instrument 
to measure the horizontal component and another to 
measure the vertical component). Today, this same oper- 
ation can be carried out with one small and completely 



338 



JPL SPACE PKOGkAMS SUMMAkY 37-51, VOL. Ill 



portable instrument called a proton procession magnetom- 
eter at a fraction of the time and at perhaps greater 
accuracy. Space age technology has similarly a£Fected 
seismic, electrical, radioactive, and gravity instruments 
and their application. 

Although new technology has afiected geophysical 
instrument design and its application, particularly in the 
sense that it makes the roving vehicle traverse geophysics 
mission feasible, classical geophysical experiments in 
magnetism, gravity, and seismic prospecting appear to 
be most practical for the early traverse missions, as their 
data are more understood and interpretable in terms of 
terrestrial analogs. An imaging system and laser ranging 
experiment should also be a part of the minimum science 
package for the traverse geophysics task. 

The imaging system as previously described would be 
suitable for the traverse geophysics task. This instrument 
would serve as the eyes of the rover for navigation, 
guidance, and positioning and, in addition, support the 
geophysical experiments by providing eyeball type geo- 
logical information at each measurement site. 

It has been suggested that the present absence of a 
strong internal magnetic field for the moon may reduce 
the efiFectiveness of standard magnetic surveying tech- 
niques for understanding deep structural features; how- 
ever, the absence of this field may now enhance the 
detection of remnant magnetism that could have con- 
siderable cosmogonic significance. Also, because the dif- 
ference in measured susceptibility between acid and 
basic rocks, between nickel-iron meteorites and silicate 
rocks, and even between chondrites and silicate rocks is 
large, it is probable that they have become polarized by 
external fields, relic lunar field, or flowage. Therefore, 
magnetic survey techniques may prove to be a valuable 
tool for mapping contacts, providing criteria for dis- 
tinguishing impact and volcanic features, and, in gen- 
eral, providing new data on the structure and processes 
of the lunar surface. 

The traverse operation will require a three-component 
orthogonal magnetometer of the flux-gate, proton pro- 
cession, or optical pumping type. The last two types are 
favored because they provide absolute magnitude data. 
It is desirable that the magnetometer operate continu- 
ously; i.e., operate both during station stops and while 
the rover is in traverse \n accuracy of ±5 y is desirable. 
This suggests that the magnetometer sensor must be 
compensated for both perm and induced magnetic inter- 
ference from the roving vehicle, or else removed from 



its vicinity during measiurements. A base control for 
monitoring external fluctuations and changes in the lunar 
magnetic field is required. This could be provided by 
Apollo lunar surface experiments package (ALSEP) sci- 
ence or an emplaced science station (ESS) package 
containing a magnetometer. The base control station 
should be located in the survey or traverse area, but a 
separation up to 500 km could be tolerated. 

The surface gravity of the moon is only one-sixth that 
of the earth; therefore, gravity anomalies on the moon 
resulting from a density contract in lunar material will 
comprise a larger part of the total-field measurement 
than similar measurements on earth. Lunar gravity anom- 
alies may be caused by local near-surface density con- 
tracts in rock units, as between the regolith and basement 
rocks, or perhaps by regional isostatic phenomena where 
the moon's crust is out of isostatic equilibrium because 
of anc''>nt frozen tidal effects or crustal overloading by 
ejecta trom large meteor impacts. The Carpathians and 
Apennines are possible examples of crustal overloading 
from the Imbrium impact event. Surface gravity data 
from profiles across virtually all lunar structures and 
contacts are desirable as these data may be critical to an 
understanding of the origin and evolution of these fea- 
tures and even the moon itself. 

The best gravity instrument for the traverse task is 
probably a conventional spring-mass gravimeter in con- 
trast to a torsion balance or pendulum. The state-of-the- 
art in design of these instruments is highly advanced. 
Terrestrial gravimeters nre required to detect changes in 
gravity of the order of i part/10' The lunar instrument, 
because of the lower gravity on the moon and the result- 
ing higher ratio between anomaly and total gravity, 
should be calibrated to detect changes of the order of 
1 part/lO" over a dynamic change of 500 mgals. Con- 
siderable care in instrument design to control long term 
temperature and mechanical drift will be required, since 
it is unlikely that the vehicle on a traverse mission can 
be returned to a previous station to measure instrument 
drift. 

It will be necessary to make both free-air and Bouguer 
corrections to the gravity observations. These corrections, 
if not observed, could mask regional trends and even 
local anomalies. An integrating tiltmeter, supplemented 
with data from base map and imaging system, can pro- 
vide the information for this correction. A terrain cor- 
rection may be needed locally; a tidal correction to 
account for differential alignment between the sun, earth, 
and moon will also be required. 



JPL SPACE PROGRAMS SUMMARY 37-51, VOL III 



339 



The active seismic experiment, as defined here, can 
be considered a r^allow exploration technique for probing 
the uppermost ' ilometer of the moon, and designed to 
measure the depth of the lunar regolith and its elastic 
and phyical properties, seismic wave velocities, and rock 
contacts, and, in general, to provide subsurface data on 
the moon's structure and stratification. Both refraction 
and reflection techniques can be useful, for one phe- 
nomenon rarely occurs without the other. The geological 
picture that is emerging for the lunar surface suggests 
a low density regolith overlying a denser unchumed base- 
ment. The study of this may present an ideal problem 
for the seismograph. 

The experiment consists of a seismometer which can 
be deployed to the lunar surface under the rcver, an 
auger for shot-hole preparation, and approximately 100 
charges weighing 0.25 lb each for providing a seismic 
energy source. The charges would be activated sep- 
arately by radio command from the rover. In terrestrial 
seismic prospecting, where mobility and backtracking 
are not problems, the normal procedure is to use one 
shot point in conjunction wi