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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 GPO PRICE $. CSFTI PRICE(S) $ . Hard copy (HC) . Microfiche (MF) ff653 July 65 ^,^^ ,^ N68-3'?^9'rJ N68-3742 % (ACCESSION NUMBER) %, I (NASA CK OR TMX OR AD NUMBER) (THRU) L (CODE) (category:f 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 — - 1000 If tt »■ * * * 9 % * \ « * * '** * « I m i f V • « » • * ^ * ** * *• 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 t lliki'i' "■r "Ai, t "* " ' 1 "* 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 Jl^ 5 ui ^ 3 s O 1- 2 i 1- >- IE 1- 5 ? § o g S3 U C t > in t- > >o > CO o CM m >- CO N 8°= J I I I ^ R I- tu 5? K Ill S <n a: >- Ul ^ o <n «.> <n 2 fE. Srr _i I Y POWE YSEOU Y TIME ftTlONA ROSE i UJ u «T oca a: n- ^ 2 t- !-»-►- UJ % w <l Z 2 Z ft O Q. <> o ui uj uj o a. tn «) UJ Ul Ifl UJ en rcLwHmov^o oiijLjujoa</)<n E o Q V e 2 "3 e e w e 3 [TTfT o a B lU ^ ^ g UJ H W UJ (0 0) 3 o (n O CO O O Ul *"' UJ UJ a. a. ■^ S o u. fe § g s n' UJ rr Ql tC K ^ 2 (E UJ ? UJ ^ ^ t- ^ 2 1- >- rr 1- ly. z UJ lU 2 UJ Ul UJ UJ on (0 en a Ui z ^ U; Ul o 2 u. u. o z IE O .2 fe fe h- 3 I- z K 2 £ 1- 3 fi J> Ul 0. OT cn U- UJ n a UJ UJ .'I UJ 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 3 a 3 <• I s il S <» s • i » ■= o ■» k o • ^ o. • I o •- • M JC 3 *• o "5 .E 3 M Jl C O P c '1 s o. < 11 il 11 + + I ^- + I + 4 t I + + 4- -I' + -h I + + I + ST I q I + -I- + -h - H- -h -h ^i-l I Jl o ft ,2 o •o • o e 2 g E M o i* E Q. g + • 9 t J> o c 1 1 '■i II .5 o '26 S E c 2 8-2 § 1 I a .; =5 s - s o o ■f a 3 o ■o D a S £ c 2 S .2 S -o ? i §1 E > » a *- -D - £ • + -To I H + P I + I C I H ^ o * o o >--!- ° E O i c o „ C *^ Q '£ Ol o 3 O *• ■C O " .: 5. E ■o - S • 8 "^ oi 1 1 § > 5 , S « |S=S + 6 ^ s II i >= 2 a ' + + = I I I 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