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MIRA VARIABLES: AN INFORMAL REVIEW 


ROBERT F. WING 

Astronomy Department, Ohio State University 


The Mira variables can be either fascinating or frustrating — 
depending on whether one is content to watch them go through their changes 
or whether one insists on understanding them. Virtually every observable 
property of the Miras, including each detail of their extraordinarily complex 
spectra, is strongly time-dependent. Most of the changes are cyclic with a 
period equal to that of the light variation. It is well known, however, that 
the lengths of individual light cycles often differ noticeably from the star s 
mean period, the differences typically amounting to several percent. And if 
you observe Miras — no matter what kind of observation you make — your work 
is never done, because none of their observable properties repeats exactly 
from cycle to cycle. 

The structure of Mira variables can perhaps best be described as 
loose. They are enormous, distended stars, and it is clear that many differ- 
ent atmospheric layers contribute to the spectra (and photometric colors) 
that we observe. As we shall see, these layers can have greatly differing 
temperatures, and the cyclical temperature variations of the various layers 
are to some extent independent of one another. Here no doubt is the source 
of many of the apparent inconsistencies in the observational data, as well as 
the phase lags between light curves in different colors. But when speaking 
of "layers" in the atmosphere, we should remember that they merge into one 
another, and that layers that are spectroscopically distinct by virtue of 
their vertical motions may in fact be momentarily at the same height in the 
atmosphere. 


533 



I sometimes find it helpful to think of Miras as jellyfish. As they 
move through the water, their general oscillatory motion can be expected to 
continue, but it is impossible to predict all the details of their changes 
in shape and appearance; if you tweek a jellyfish on one side, you don't 
know if, when, and with what amplitude the disturbance will reach the other. 
If you think of Miras this way, you will stop worrying about their failure 
to repeat exactly in their variations. 

No one has ever called Miras a theoretician's delight. Certainly 
they have not been very useful in testing theories of stellar pulsation. 

There is just too much going on — it's hard to know which observable pro- 
perties are even relevant to pulsation. In fact the question has sometimes 
been raised as to whether the Miras are pulsating at all. Merrill (1955) 
wrote that "the evidence for volume pulsation is so meagre that skepticism 
is warranted", and Wallerstein (1977) has proposed that the apparent radius 
changes in Miras are caused not by the outward movement of the gas but simply 
by changes in the atmospheric opacity. Recent results from infrared spectro- 
scopy (Hinkle 1978) have clarified the picture considerably, basically by 
allowing us to look more deeply into the atmosphere, and I think that the 
last doubts that Mira variables are pulsating have finally been laid to rest. 

A scheme has recently been devised by Cahn and Wyatt (1978) by which 
one can estimate the masses and luminosities (and hence ages) of individual 
Miras from two readily observable quantities, the mean period and the mean 
spectral type at maximum. This scheme is based on the assumption that Miras 
are pulsating stars, more specifically that they are pulsating in the first 
overtone. Since we have few opportunities of determine masses and lumino- 
sities of individual Miras directly, it will be difficult to decide obser- 
vationally whether this picture is correct. At this stage the question is 
not whether the relations of Cahn and Wyatt will eventually need recalibra- 
ting, but whether they exist at all. However, the fact that Cahn and Wyatt 
were able to construct a self-consistent picture suggests that they may be 
on the right track, and that it may indeed be possible to understand the 
gross observable properties of individual Miras, such as surface temperature 
and mean period, in terms of their masses and evolutionary states. 


534 



In describing to you the observed behavior of Miras, I will attempt 
to select those particular observations that have a bearing on the question 
of pulsation — although, as I have already indicated, it is not always ob- 
vious which observations these are. The discussion will be centered around 
the sizes of these stars, or more particularly the evidence for changes in 
size. No mention will be made of their absolute magnitudes, ages, chemical 
compositions, population types, galactic distribution, statistical proper- 
ties, or other matters not directly related to pulsation. Rather, we will 
be concerned with the variations of individual stars, and with some spectro- 
scopic peculiarities that are related to the enormous sizes of their atmos- 
pheres. 

As soon as we try to talk about the sizes of Mira variables, we run 
into a serious problem. The reason that we can see so many different atmos- 
pheric layers at the same time is that the continuous opacity — between the 
spectral lines and bands — — is extremely low. A Mira is about as translucent 
as a jellyfish: you can practically see right through it. Some regions con- 
tributing to the spectrum are much farther from the center of the star than 
others, and this is all within the region we call the "photosphere". In 
order to speak of the size of a Mira, we must specify the wavelength pre- 
cisely. Furthermore, there may be no real discontinuity between the photo- 
sphere and the circumstellar shell that contributes zero-volt absorption lines 
and infrared emission, since presumably the shell consists of material which 
has drifted away from the star, and which may receive new contributions with 
every light cycle. In this respect, Miras are worse than jellyfish. Although 
the radius of a jellyfish is both time-dependent and angle-dependent, at least 
there is a membrane to show us where the jellyfish ends and the ocean begins. 
The size of a jellyfish is certainly difficult to measure, but at least the 
creature has a size. 

The methods that can be used to determine the sizes of Miras fall 
into three classes: 

(1) Direct . Angular diameters of Miras have been measured directly 
with Michelson interferometers, by observations of lunar occultations, and 
by speckle interferometry. The distance must be known to calculate the ab- 


535 



solute size, but useful information about changes in size can be obtained 
directly from the observed changes in angular diameter. 

(2) Photometric . If you know the luminosity L (or equivalently the 
absolute bolometric magnitude M bol> and the effective temperature T e , you 
can calculate the size of the emitting surface area and hence the stellar 
radius R, since L = 4 ttR^oT^ . If you don't know the distance and have only 
the apparent bolometric magnitude m bol , you can still calculate the change 
in radius from the observed changes in T e and m bo i. 

(3) Spectroscopic . Measurements of radial velocities of absorption 
and emission lines give information about vertical motions in the atmosphere. 
Integration of the radial velocity curve then gives the distance moved by 
the gas producing the lines. 

All of these methods are clear-cut in concept, and all of them are 
known to "work" in the sense that they give reasonable results for the sizes 
of other kinds of stars. But all of them get into trouble when they are 
applied to the Miras, and in general the results from the three methods are 
in poor agreement. 

For example. Figure 1 shows a famous illustration from the paper by 
Pettit and Nicholson (1933) . From the variations in bolometric magnitude 
(obtained from the observed radiometric magnitudes, with crude corrections 
for absorption by the earth's atmosphere) and temperature (from a very broad- 
band color index, which compares the radiation shortward and longward of 
1.3 y), they computed the variation in the angular diameter of Mira (o Cet). 
Differentiation of the diameter curve then gave the "radiometric radial 
velocity curve", which is compared to the radial velocity curve measured 
spectroscopically by Joy (1926) for the mean absorption spectrum of Mira. 

The two curves have similar amplitude but are badly out of phase — in fact, 
they are nearly mirror images of one another. Similar comparisons for three 
other Miras have recently been published by Wallerstein (1977); the higher 
quality of the data used in his analysis did not make the discrepancy go 
away. 


536 




Fig. 1 - The variations of o Cet (Mira), according to Pettit and 
Nicholson (1933). Note that the color temperature variations are in 
phase with the visual light curve but not the bolometric curve. The 
diameter curve is computed from the temperature and bolometric magni- 
tude. Its derivative, the radiometric radial velocity curve, disagrees 
with the radial velocities determined spectroscopically. 

What is the problem? It's not that the observations of Miras are so 
difficult. All three methods (including Michelson interferometry) had been 
applied to Miras by the 1920's, and in general the measurements by each method 
are reproducible to sufficient accuracy. Also, impressive advances in each 
of the three areas of observation have been made within the past decade, and 
yet the discrepancies persist. The problem, it seems to me, is simply that 
when we observe a Mira by any of these methods, we don't know what we’re 
looking at . 

For most of the remainder of my talk, I will give examples of the use 
of each method, focusing attention on the problems of interpretation. I think 
it will be clear why observers can derive a lot of pleasure from the study of 
Miras, and why theoreticians, for the most part, can not. 


537 




THE DIRECT APPROACH 


The basic problem with the direct approach to measuring the sizes 
of Miras is that it really is not very direct. We don't measure the size 
of an image with a ruler. Rather, we observe some optical, phenomenon that 
is related to the size of the star — the visibility of interference frin- 
ges at different mirror separations, or the degree of degradation of the 
diffraction pattern as the star disappears behind the moon, or the char- 
acter of the speckles in the seeing disk. In each case, a model for the 

°f light in’ the true stellar image must be assumed before the 
observed quantity can be related to image size. 

How much limb darkening (or brightening) is there? Is the star 
round? Does it have spots on it? If the answer assumed for any of these 
questions is wrong, so is the angular diameter that we get. Nevertheless, 
the results obtained by these techniques have certainly been instructive. 
The main problem with using these results is that there haven't been enough 
of them. 

The angular diameter of Mira was measured with a Michelson inter- 
ferometer in January 1925 by Pease at Mount Wilson (see Kuiper 1938). Un- 
fortunately, the method could be employed only when the variable was at 
maximum light. 

Diameter measurements of Miras by lunar occultation observations 
are a rather recent innovation, since a time resolution of a few milli- 
seconds is needed to resolve the diffraction pattern. A few years ago 
Nather and Wild (1973) succeeded in observing an occultation of R Leo at 
V = 8 on the declining branch. Its phase was estimated to be 0.27, so 
that it should have been very nearly at its maximum diameter if the dia- 
meter curve of Pettit and Nicholson (Figure 1) is valid. The light curve 
of this event is shown in Figure 2. Anyone who has seen occultation data 
for just about any other star will recognize that R Leo is enormous . The 
diffraction pattern is completely smeared out. From the slope of the de- 
cline, Nather and Wild computed a uniform-disk diameter of 0.067 arcsec. 

For stars as large as this, the occultation technique runs into a 


538 




0 100 200 300 4 00 500 

TIME (MS) 


Fig. 2 - Light curve of the occultation of R Leo on 19 May 1972, 
as measured by Nather and Wild (1973) . The computed angular diameter 
is 0.067 arcsec. 

snag, as Nather and Wild point out. Since there is no diffraction pattern, 
there is no information as to the slope of the lunar limb at the point of 
contact. One rock could spoil the result. Fortunately this problem can be 
overcome by planning simultaneous observations at different observatories. 

It does not arise if the star is, say, one-quarter to one-tenth the size of 
R Leo; there are many Miras in this range of angular size, but of course they 
are correspondingly fainter, and photon noise becomes a problem. 

Occultation observations of Miras are now being pursued vigorously 
by Ridgway and his colleagues at Kitt Peak (Ridgway, Wells, and Joyce 1977). 
Most of their measurements are being made at 2 y in the infrared, so that 
daytime observations are possible. Since it is impossible to control the 
motion of the moon, this method will never produce a true diameter curve for 
any single star. However, even a simple pair of observations of the same 
Mira could be useful in indicating whether the spectroscopic or the photo- 
metric diameter curve tends to be confirmed by the direct method. Occultation 
measurements of U Ori were recorded by Ridgway, Wells, and Joyce in two dif- 
ferent lunations, and they indicate that the diameter is several percent 
larger at phase 0.36 than at phase 0.99. This result appears to confirm the 
photometric diameter curve, but unfortunately no conclusion can be drawn. 

The two observations were made with different filters, one in the continuum 


539 




and one in an H 2 0 band, and it is quite possible that the measured change in 
diameter has more to do with wavelength dependence than with time dependence. 

Exciting results obtained by speckle interferometry have recently been 
published by Labeyrie, Koechlin, Bonneau, Blazit, and Foy (1977). R Leo and 
o Cet were found to be twice as large at wavelengths affected by strong TiO 
bands as they are at continuum wavelengths. Information about changes in dia- 
meter with phase is still very limited, but the indications are that changes 
with wavelength at a given phase are much more dramatic than changes with 
phase at a given wavelength. The acquisition of further speckle data, espe- 
cially if timed to cover a substantial portion of the light cycle of a single 
Mira, would be enormously valuable. 


THE PHOTOMETRIC APPROACH 

The formula L = 4TrR 2 oT e 4 is straightforward enough, but the Miras 
are not. One problem is that the formula assumes the stars are round — not 

shaped like jellyfish. We now examine the problems associated with determin- 
ing L and T e . 

Observationally, it takes a lot of work to determine L, or even the 
apparent quantity m bol . But at least m^ — the total radiation from the 
star reaching the top of our atmosphere — is a well-defined quantity. Since 
the time of Pettit and Nicholson (1933), much more sophisticated methods have 
been applied to the determination of m bol , including the use of Stratoscope 
scans to interpolate between the various infrared magnitudes measured from 
the ground (Smak 1966). A few observations of Miras have also been made from 
high-altitude aircraft (Strecker, Erickson, and Witteborn 1978). I am cur- 
rently collaborating with J. Smak on the determination of m bo ^ for a large 
set of Miras from extensive wide- and narrow-band photometry. My impression 
is that if you are willing to do the work and are careful about the photo- 
metric calibrations, you can determine m bo 2 to an accuracy of a few percent. 
Recent work on this problem has not changed the character of the bolometric 
light curves derived by Pettit and Nicholson. In other words, I don't think 


540 



errors in the determination of mj^ can be responsible for the discrepancies 
mentioned above. 

Light curves measured at carefully-chosen continuum points in the 
infrared can give a good approximation to the bolometric light curve. Lock- 
wood and Wing (1971) have published light curves for 25 Miras in 1(104), 
measured photoelectrically with a narrow bandpass at 10400 K, and they were 
found to have the same amplitudes and phasing as the bolometric light curves 
of Pettit and Nicholson. Since 1(104) is very much easier to measure than 
it is nice to know that it provides essentially the same information. 

The 1(104) light curves have proved quite interesting, especially 
since each measurement of magnitude has been part of a set of narrow-band 
photometry which also gives the spectral type (from the strengths of TiO and 
VO bands) and the near-infrared continuum color. When I started this work 
in 1965, my hope was that the 1(104) curve of any Mira would repeat so well 
from cycle to cycle that its characteristics could be established once and 
for all. Several Miras were followed through two or three cycles to test 
this idea, and the first results seemed promising. Figure 3 shows the visual 
and 1(104) curves for W Peg in two successive cycles. The V curves, measured 
photoelectrically with a UBV photometer, show typical cycle-to-cycle differ- 
ences: one maximum is 0.3 mag brighter than the other, and it occurred ahead 
of schedule; the slopes on the declining branches are also different. On the 
other hand, the 1(104) magnitudes followed the same curve in both cycles. I 
would like to be able to tell you that the differences in the visual maxima 
were caused by differences in blanketing of the visual region by TiO, but the 
fact of the matter is that W Peg attained the same spectral type, M7.0, at 
both maxima. 

These observations of W Peg, from 1965 and 1966, were made with a 
spectrum scanner. Since 1969 I have been using a set of eight interference 
filters to obtain similar information. In addition, Lockwood (1972) has pub- 
lished extensive photometry of Miras on a five-color system in the same near- 
infrared spectral region. These three systems have enough in common that it 
has been possible to work out the transformations between them (Lockwood and 
Wing 1971; Wing and Lockwood 1973); in particular, all three measure 1(104). 


541 




Fig. 3 - Photoelectric light curves in V and 1(104) for W Peg, a 
typical Mira variable, during the accessible portions of two consecu- 
tive cycles. In this case the differences in the visual curves are 
not reflected in the 1(104) curve. Note also that the infrared maxi- 
mum occurs well after the visual maximum. From Wing (1967). 


When Lockwood and I combined our data to form 1(104) light curves for 
several Miras over a number of cycles, it became clear that cycle-to-cycle 
differences do occur in the 1(104) curves quite commonly, and that the nice 
behavior shown by W Peg in Figure 3 is the exception rather than the rule. 
Several of these 1(104) light curves are shown in Figure 4, where different 


✓ 


542 






Fig. 4 - Light curves in 1(104), an infrared continuum point, for 
Miras of relatively short period (left) and long period (right). Below 
the name of each star is its mean spectral type at maximum light (Keenan 
1966) and the mean period used in calculating the phases. From Lockwood 
and Wing (1971). 


symbols have been used to distinguish the different cycles. The stars of 
relatively short period usually repeat fairly well, while the large-amplitude, 
400-day Miras show more substantial cycle-to-cycle differences. From an 
analysis of these differences in terms of the simultaneously measured spectral 
types and color temperatures, we were forced to conclude that cycle-to-cycle 

differences affect the bolometric curves as well. 

Several of the stars in Figure 4 show humps on the rising branch, at 
about phase 0.7. Similar humps have long been known to occur in the visual 
light curves of certain Miras, and it was not known whether they are caused 
by superficial changes in spectroscopic features or by more basic changes in 
the continuum radiation. Now we see that the humps are present in the con- 
tinuum radiation; blanketing changes do not affect the 1(104) magnitude, and 


543 



in any case the spectral types were observed to remain constant, at their 
latest value, throughout the interval from phase 0.6 to 0.8, whether or not 
a hump occurred in the light curve. The TiO and VO bands used for spectral, 
classification are evidently formed very far from the layer emitting the 
continuum; these molecules do not start to dissociate until a month or two 
after the photospheric temperature has started to rise. 

The infrared data show that humps on the rising branch are quite 
common: most stars observed in two or more cycles show a hump in at least 
one cycle. On the other hand, few stars seem to have humps in every cycle. 

It is difficult to avoid the conclusion that humps also occur in 
the bolometric light curve, whenever they occur in 1(104). If we know the 
shape of the bolometric curve, we can use the color temperatures measured 
in the infrared continuum, along with the usual formula, to inquire how the 
radius changes when a hump occurs. Interestingly, the color temperatures 
are observed to increase smoothly and monotonically, from minimum to visual 
maximum, no matter whether a hump occurs in the light curve or not; there 
is never a hump in the temperature curve. Thus the leveling-off or decrease 
in luminosity following a hump must be the result of a rapid decrease in 
radius prior to maximum light. Lockwood and I suggested that the occurrence 
(or not) of a hump of the rising branch is simply the result of the interplay 
between rising temperature and decreasing radius during this part of the 
cycle. 

This brings us to the question of what temperature is really appro- 
priate to use in the formula L = 4rR 2 aT e 4 . The fundamental problem with 
applying this formula to the Mira variables, it seems to me, is that the 
effective temperature T 0 is defined by this formula and has meaning only 
if we can attach a meaning to the radius R. Since the star has no membrane, 
we have to think of the radius as the distance from the center of the star 
at which the optical depth takes on some value, such as unity; as we have 
seen, the radius is then strongly wavelength— dependent and may vary by as 
much as a factor of two over the width of a strong spectral feature. Some 
kind of averaging is needed, but it is not clear what kind of mean opacity, 
or mean radius, corresponds to the "effective" temperature. There are 


544 



innumerable ways of estimating the temperature of a Mira spectroscopically 
or photometrically, but different methods often give substantially different 
results, in part because they refer to different layers of the atmosphere 
which really do have different temperatures, and in part because most line 
ratios and photometric color indices are not pure indicators of temperature. 

So what do we do? The usual response is to go ahead and use the for- 
mula anyway. That is, we determine m^-^ as best we can, estimate T g from a 
color index that we hope is representative (or worse, from the spectral type), 
and bravely plug them into the formula to compute the size. This gives us a 
number, but we really don't know how this number is related to the size of 
the star. 

Applications of the photometric method to Mira variables do at least 
give internally consistent results, as exemplified by Pettit and Nicholson's 
diameter curve in the center of Figure 1. The size (or rather, this number ) 
is smallest near the time of visual maximum, and it increases most rapidly 
between the times of visual maximum and bolometric maximum, which occurs one 
or two months later. Nearly all spectroscopic and photometric temperature 
indicators agree that the highest temperature occurs very close to the time 
of visual maximum; if the temperature really drops during the following month 
or so, a rapid increase in the size of the emitting region is needed to ac- 
count for the increase in bolometric flux. 

There are three further results from the narrow-band photometry of 
Miras which, although not clearly related to radius variations, do tell us 
a good deal about the structure and extent of their atmospheres: (1) the 
temperatures measured in the continuum are usually much higher than would 
be expected from the spectral type; (2) spectral types determined from dif- 
ferent TiO bands are often grossly discordant; and (3) the variations in 
spectral type are only loosely coupled to the variations in color tempera- 
ture. While each of these findings came as a surprise, I believe they are 
all manifestations of the same thing, namely the great stratification of 
these stars' atmospheres. 

Color temperatures that are abnormally high for the spectral type 
do not always occur — as a consequence of the loose coupling indicated in 


545 



the third result above, the effect can go either way — but most Miras have 
high temperatures for their spectral types at most phases. The effect is 
most conspicuous (and best established) in the early-type Miras near maximum 
light, when the near-infrared color temperatures are completely free from 
blanketing effects. For example, R Tri at its 1965 maximum attained a color 
temperature as high as that of a normal K4 giant, but its spectral type was 
never earlier than M3 (Spinrad and Wing 1969) . I interpret this as meaning 
simply that. the continuum and the absorption spectrum are formed very far 
apart, in regions of very different temperature. In other words, the atmo- 
sphere of a Mira is more stratified than that of a normal giant (Wing 1967). 

A recent study of hydrodynamical phenomena in Mira variables (Willson 
and Hill 1979) lends credence to the conclusion that their atmospheres may 
be more distended than those of non-variable M giants of the same luminosity. 
There is simply not enough time for the atmosphere to recover from the effects 
of one shock wave before the next shock starts to propagate through it. The 
atmosphere is thus never in a "normal" state, and although the star has the 
energy output of a giant, the physical characteristics of its atmosphere, 
such as density and temperature structure, may more closely resemble those of 
a supergiant. Indeed, Mira variables seem spectroscopically to have the lumi- 
nosities of supergiants, if their pressure-sensitive, line ratios are inter- 
preted in the usual way. For this reason, Keenan has always refrained from 
assigning luminosity classes to Mira variables (Keenan 1966; Keenan, Garrison, 
and Deutsch 1974). Unfortunately, not all investigators have exercised such 
restraint, and supergiant luminosity classifications have been published for 
several Miras, leading to possible confusion as to their actual luminosities. 

The second of the results from narrow-band photometry mentioned above 
refers to the assignment of temperature classes. For the Miras, temperature 
classification becomes ambiguous as soon as we consider two different classi- 
fication criteria — even if they are just different bands of the same mole- 
cule. In Figure 5, we see that the relative strengths of the TiO bands mea- 
sured by filters 1 and 3 on the eight-color system are not the same in the 
Mira as they are in the giant. For the giant we get the same spectral type 
from both TiO bands and from the continuum color, while for the Mira we get 


546 




Fig. 5 - Eight-color photometry for 56 Leo, a normal, unreddened 
giant, and U Cet, a Mira variable. The two stars have nearly the same 
TiO strength at filter 1 but different TiO strengths at filter 3 and 
very different color temperatures. From Wing (1974). 

M6 from the zero-volt TiO band, M4.5 from the excited TiO band, and M3 from 
the color. Clearly we must be careful in using spectral classifications of 
Miras; in particular, we should not use them to infer the temperature of the 
photosphere. At the same time, these results encourage me to hope that the 
infrared color temperature from the eight-color photometry may indeed be 
suitable to use in calculations of the radius, since it appears to refer to 
the same deep layer from which most of the total flux is emitted. 

The loose coupling between color temperature and band strength is 
illustrated in Figure 6. The loops executed by Miras are really enormous — 
the band strengths can differ by a factor of two or more between phases of 
the same color temperature. Because of this, bolometric corrections for 
Miras must be tabulated as two-dimensional functions of band strength and 
color, rather than as one-dimensional functions of spectral type as have 
always been used in the past. Another consequence of these loops is that 
they render the band-strength data virtually useless for abundance deter- 
minations . 


547 




Fig. 6 - An index of molecular band strength is plotted against 
the reciprocal color temperature (5040/T) , both measured in the near 
infrared with a scanner (Wing 1967). Normal giants define the heavy 
line ending in the box labeled RX Boo, whereas Mira variables execute 
large loops. See Spinrad and Wing (1969) for details. 

These loops can be interpreted in the same way as the other phenomena 
we have discussed — the band strengths and the continuum color refer to 
widely separated regions, the temperature variations in which are out of 
phase. This behavior could be modeled if all Miras showed loops that were 
at least qualitatively similar, but they're all different! In Figure 6, 

X Cyg goes clockwise while U Her goes counter-clockwise. In fact, the loops 
shown by the same star in different cycles are not necessarily any more simi- 
lar than the loops of two different stars. 

Well, what do you expect of a jellyfish? Remember that the layers 
contributing to the spectra are near-perfect vacua separated by millions of 
miles, and you will be able to excuse their pooriy-coordinated performance. 


548 




THE SPECTROSCOPIC APPROACH 


The spectra of Miras are incredibly complex. They are dominated in 
the ultraviolet and blue regions by atomic absorption lines, in the visual 
and near infrared by bands of metallic oxide molecules , and in the infrared 
beyond 1.5 y by innumerable lines from the rotation-vibration transitions of 
CO and H 2 0. In addition to these absorption features, emission lines of vari- 
ous descriptions are present. Hydrogen lines of the Balmer, Paschen, and 
Brackett series are strong in emission during more than half the cycle, from 
just before maximum to approximately the time of minimum light; since absorp- 
tion lines can be seen (and identified) within the broad emission profiles of 
the Balmer lines, it is clear that the hydrogen emission is produced in a 
deep layer of the atmosphere (Joy 1947) . Some of the weaker emission lines 
are known to be produced by fluorescence, and although I will not discuss 
these particular lines further here, I should mention that the careful study 
of fluorescence mechanisms in Miras can provide important information about 
the structures and motions of their atmospheres (Wing 1964; Willson 1976). 

In fact, the very fact that fluorescence mechanisms are operative shows that 
these atmospheres are so rarefied that the populations of excited levels in 
atoms are governed by radiative processes rather than by collisions. Other 
metallic emission lines seem to be produced by recombination, some remain un- 
identified, and still other emission lines have been found to have molecular 
origins. Interesting reviews of the line spectra of Miras have been published 

by Merrill (1960) and Willson (1976). 

Recent observations of Miras have revealed additional emission lines. 
Just six weeks ago, the first emission lines to be detected in the ultraviolet 
spectrum of a Mira variable below the atmospheric cut-off were recorded in 
R Leo with the IUE satellite (Wing and Carpenter 1978). At the other end of 
the spectrum, lines emitted by OH, H 2 0, SiO, and CO have been detected in the 
microwave region. 

Spectroscopic studies of the pulsational properties of Miras are based 
on the measurement of radial velocities. Unfortunately, an accurate measure- 
ment is not enough; there are three problems with which we must deal before 


549 



the radial-velocity data can be converted into information about the expan- 
sion and contraction of the atmosphere. First, because of projection effects, 
the measured radial velocity does not tell us immediately the motion of the 
surface of the star; we must apply a very uncertain correction for geometry 
and limb darkening. Emission lines are particularly difficult to use, since 
they may be either limb-darkened or limb-brightened, depending on the depth of 
their formation. This problem has been discussed recently by Wallerstein 
(1977). A more important problem, also discussed by Wallerstein, is that it 
is not sufficient to know the absolute motion of the surface; we must also 
know the radial velocity of the center of mass of the star before we can tell 
whether the surface is moving up or down. Finally, when we find that different 
spectral features have different radial velocities, we must somehow decide 
which feature to use as an indicator of the photospheric velocity. 

For many years, most of the radial-velocity work on Miras was done at 
Mount Wilson Observatory, mainly by Merrill and Joy. The spectroscopic 
radial-velocity curve shown in Figure 1 was taken from an early study of Mira 
by Joy (1926); 28 years later Joy (1954) published a second study of Mira, 
from which Figure 7 was taken. Quite generally, the emission lines show 
smaller radial velocities than absorption lines, i.e. the emitting regions 
are moving outward and/or the absorbing regions are falling inward. [The only 
red-shifted emission lines that have been identified in Miras are certain 
fluorescent lines that are excited by off-center coincidences (Wing 1964) ] . 
Different emission lines, however, display very different behavior: note in 
particular the curves for Joy's "standard" metallic emission lines, the Fe II 
lines, and the hydrogen lines in panels (b) and (d) of Figure 7. The be- 
havior of the absorption spectrum is also rather complex: Merrill, in several 
papers, reported that the radial velocities of absorption lines show a de- 
pendence upon excitation potential — another indication of extreme strati- 
fication. In addition, a few instances of doubling of atomic lines have been 
reported, and evidence of incipient line doubling (fuzzy or irregular line 
profiles) is rather common. 

Faced with the many different velocities that are present in the 
spectrum at any given phase, how can we decide which velocity best represents 


550 




Fig. 7 - Radial velocities measured in the spectrum of Mira, from 
Joy (1954). Absorption and emission lines, all from the blue spectral 
region, have been grouped according to their behavior. Numbers next 
to the points indicate the number of measurements entering the mean. 


the motion of the photosphere and which the motion of the center of mass? 

Many approaches to this question have been tried. Usually some kind of average 
of the atomic absorption-line velocities is taken to be the photospheric velo- 
city. Common choices for the center-of-mass velocity are the mean absorption 
velocity at the time of maximum light, the velocity of Fe II emission lines 
(which appear to be formed in the chromosphere and show relatively little 
velocity variation with phase) , and the velocities of certain molecular micro- 
wave emission lines (which are formed still farther out). Unfortunately, all 
these velocities are different. 

For 7 stars with high-quality optical and microwave velocity data, 
Wallerstein (1975) prepared Figure 8 to illustrate the differences in the 


551 




R Aql 284 doys 

170 

SiO v=l 

h 2 o 

— OH 1665,7 

Vs 'e * *«,„ V*- 0H 1612 

11 1 1 OPTICAL 

— L— 1 — 1 — 1 — 1 — 1 — 1 — 1 1 1 

U Ori 372 days 

z-\ . 

II SiO v=l 

1 h 2 o 

III OH 1665,7 

e is p OH 1612 

t 1 OPTICAL 

1 1 1 1 f 1 1 1 | 

H Leo 313 days 

'T° 2*1 si0 v = 2 

■>!«— —n; siov=| 

1 h 2 o 

E CS ' p 0H l665 . 7 

1 1 1 OPTICAL 

— ! — i — 1 1 1 » » • _ 1 1 

W Hya 382 days 
1-0 

2 I, SiO v=2 

- 1 1 SiO v= 1 

1-0 1 h 2 o 

CS e ' P OH 1665,7 

1 1 1 OPTICAL 

llttlilil 

S Cr B 360 doys 

- '+ 1 - SiO V = 1 

1 h 2 o 

1 1 OH 1665,7 

E 1 p OH 1612 

1 -S s 1 OPTICAL 

1 1 1 1 1 1 1 1 1 t 

R Hya 388 days 

CS E ^ p SiO v-l 

1 1 1 OPTICAL 

1 1 1 1 I 1 1 1 1 

U Her 406 days 

2-1 2-1 

— 1 — - -f- SiO v = l 

'•? "° h 2 0 

pi OH 1665,7 

t rq P 1 

1 « OPTICAL 

— ! — 1 — 1 1 1 1 1 1 l 

-16 -8 0 


-16 -8 0 

Fig. 8 - Radial velocities of various optical and radio lines 
plotted relative to the velocity obtained from high-excitation absorp- 
tion lines (labeled P for "photospheric") for 7 well-studied Mira var- 
iables. Characteristic velocities of optical emission (E) and circum- 
stellar (CS) lines are plotted as well as the velocities of radio lines 
of SiO, H2O, and OH. From Wallerstein (1975). 


various observed velocities which might be considered to represent the motion 
of the center of mass. Also included are the characteristic velocities of 
optical emission lines and circumstellar absorption components. The reference 
velocity in each case is that of the absorption spectrum and is labeled P 
for "photospheric". Because of the dependence of the absorption-line velocity 
upon excitation potential, Wallerstein used only high-excitation lines, which 
are formed in deeper layers than the low-excitation lines, in the determina- 
tion of P. Even with this precaution, however, it is doubtful whether this 
velocity refers to a layer as deep as the true photosphere, and in fact the 

results from the infrared CO lines discussed below seem to show that it does 
not. 


552 



Since Figure 8 was originally drawn, thermal (as opposed to maser) 

SiO emission has been detected from several Miras. Since this emission must 
arise in a large, low-density region, it should indicate fairly directly 
the center-of-mass velocity of the star. According to Reid and Dickinson 
(1976), the circumstellar emitting region has a modest velocity of expansion 
(as inferred from the SiO line profile), and the center-of-mass velocity is 
smaller than the velocity derived from atomic absorption lines at bright 
phases, i.e. the gas producing the absorption is seen falling back in. 

Once the necessary decisions have been made, the radial— velocity 
curve can be integrated to determine the distance moved by the gas producing 
the measured lines. The corresponding change in surface area can then be 
calculated and compared to that obtained by the photometric method. When- 
ever this exercise has been carried out, as for example by Pettit and Nichol- 
son (see Figure 1) and more recently by Wallerstein (1977) , it has revealed 
a disturbing discrepancy: the motion of the gas inferred from the photometry 
is simply not confirmed by the measured radial velocities. In fact, Waller- 
stein showed that a discrepancy exists no matter what value is assumed for 
the center— of —mass velocity, since part of the problem is that the amplitude 
of the radial-velocity variations — at least for atomic lines in the blue — 
is much too small to correspond to the photometric variations. 

If, like Wallerstein, we choose to assume that the atomic absorption- 
line velocities are indicating the actual motion of the photosphere, then we 
are forced to look for an error in the interpretation of the photometry. 
Wallerstein' s (1977) suggestion is that we have been fooled by an opacity 
effect: the apparent increase in size between the times of visual and bolo- 
metric maxima is not due to an actual outward movement of the gas but is 
simply the result of an increase in the atmospheric opacity as the tempera- 
ture drops and molecules and grains form. However, this explanation has a 
fatal flaw. Although an increase in opacity can indeed cause an increase 
in the apparent size as measured directly (say by an interferometer of some 
kind), there is no way that it can produce an increase in size as "seen" by 
a photometer , since an increase in opacity cannot make the star become bolo- 
metrically brighter. 


553 



If, on the other hand, we assume that there is nothing basically 
wrong with the conventional interpretation of the photometry, we must con- 
clude that the atomic absorption lines seen in the blue spectral region, 
even those of high excitation, do not arise in the photosphere, i.e. the 
layer producing most of the bolometric radiation seen by a photometer. 

This could be the case if the opacity is much greater in the blue than in 
the infrared. Evidence in favor of this conclusion has finally been pro- 
duced by studies of high-resolution infrared spectra which show lines which 
do have the radial velocities, and the large velocity amplitudes, that are 
expected for the motion of the photosphere. 

The real break-through,' it seems to me, came from investigations of 
line doubling, particularly Maehara's (1968) study of the doubling of atomic 
lines on near- infrared spectrograms of x Cyg, an S-type Mira. Instances of 
line doubling in Miras had been reported earlier — in the S star R And 
(Merrill and Greenstein 1958; Spinrad and Wing 1969) and in the carbon star 
R Lep (Phillips and Freedman 1969) — but Maehara was the first to carry out 
a spectroscopic analysis of each set of lines separately and to establish 
that the blue component is produced in much hotter gas than the other. He 
therefore was able to construct a reasonable model involving a layer of 
shock-heated gas rising through a stratum of cooler gases. 

Maehara also measured the velocities of the lines of TiO and CN on 
his near- infrared plates of x Cyg. These were not doubled, but they didn't 
have the same velocity, either. The CN lines were found to be formed in the 
warm, rising layer, while the TiO lines were formed in the cooler layer. No 
wonder it has been hard to interpret the molecular band strengths of Miras 
in terms of a single-slab model! 

With a two-component model, it is not difficult to see how the dis- 
crepancy between the photometric and spectroscopic results might be resolved. 
We must simply suppose that, during the maximum and post-maximum phases, most 
of the light comes from the deep, rising layer — which brightens bolometri- 
cally as it swells and rises above the sources of continuous and molecular 
opacity in the cooler, in-falling layer — while most of the absorption lines 
seen spectroscopically are produced in the in-falling layer. 


554 



It is only when double lines can be seen in the spectrum that we can 
measure the temperatures and motions of both layers and thus obtain the infor- 
mation we need to specify the parameters of a two-component model. Spectro- 
grams of Miras in the blue region generally do not show double lines; evident- 
ly the opacity in the blue is too great to allow the deep layer to be seen. 
Rayleigh scattering, with its X -1 * dependence, is likely to contribute to the 
opacity in the blue, along with TiO bands and the overlapping wings of atomic 
lines. The advantage gained by Maehara (1968) in using the near-infrared and 
by Spinrad and Wing (1969) in using the one-micron region is considerable. It 
is also no mere coincidence that all reported instances of atomic line doub- 
ling have been found in S- and C-type Miras, which have lower atmospheric opa- 
cities than the much more common M— type Miras. [I do not count the doubled 
lines reported in o Cet by Adams (1941), since the displaced components that 
he saw were from an expanding circumstellar shell, rather than a deep layer 
of rising gas] . 

Much more complete information about the two-component model has come 
from the study of the infrared lines of carbon monoxide. The two-micron re- 
gion where the first-overtone CO bands lie corresponds to the minimum opacity 
due to H - , which in any case is seriously depleted at the cool temperatures 
of Miras because of the shortage of free electrons. Although the CO lines 
themselves are strong, they are not very densely packed, and it is possible 
to see down to the photosphere between these lines. Furthermore, the CO mole- 
cule is very stable and can exist at the high temperatures of the shock-heated 
photosphere. Hence it is possible to see photospheric CO absorption lines 
whenever the motion of the photosphere displaces them from the corresponding 
lines formed in the cool outer envelope. 

Figures 9 and 10 show a section of the spectrum of x Cyg in the region 
of the first-overtone CO bands on two different dates. They were obtained 
with a Fourier— transform spectrometer at the Kitt Peak National Observatory. 
Wavenumber increases from upper left to lower right. Most of the absorption 
features visible in these spectra are due to CO, although telluric H 2 O lines 
are also present. For orientation I have labeled the (3,1) and (2,0) band 
heads of C 12 0 16 on the third and fourth strips, respectively, of Figure 9. 


555 




Fig. 9 - The spectrum of x Cyg from 4240 to 4360 cm -1 (2.36 to 
2.29 y) on 1976 Jan. 12,- when the star was at minimum light (phase 

°13^16 The and heads of C 12 0 16 and the (2,0) head of 

C 0 are labeled, as are the rotational quantum numbers of some of 
the lines of the (2,0) band (in the third strip). All CO features 
are sharp and single. From an unpublished Kitt Peak spectrum (cour- 
tesy D. N. B. Hall, S. T. Ridgway, and K. H. Hinkle). 


The (4,2) head is in the left half of the first strip, where the spectrum is 
messy because of H 2 0 contamination, and the (2,0) head of the isotopic molecule 
C 13 0 16 is clearly visible in the right half of the first strip. I have also 
labeled, on the third strip, the rotational quantum numbers of some of the R- 
branch lines of the (2,0) band of C 12 0 16 ; the two sequences can, of course, be 
followed into the fourth strip, but they become blended together as they ap- 
proach the band head, which occurs at about quantum number 50. 

The spectrum shown in Figure 9 was taken at minimum light when the CO 
lines are sharp and single. The one shown in Figure 10 was obtained seven 


556 



□ 


10 


20 


30 


Fig. 10- The same as Figure 9, except that the date was 1975 June 
12, when x Cyg was at maximum (phase 0.05). Here the CO features are 
double. 

months earlier when the variable was at maximum, and we see at once that all 
the CO features, including the band heads, are doubled. The doubling is par- 
ticularly obvious in the fourth strip. The weaker components, shifted to shor- 
ter wavelengths, can be identified with the deeper, warmer layer. 

The comparison of Figures 9 and 10 points out two great advantages of 
the infrared CO bands in studies of line doubling, in addition to those already 
mentioned. First, since Miras have relatively small light amplitudes in the 
infrared, spectra of the same high quality can be obtained throughout the cycle. 
Second, because the doubling is shown by a large number of lines covering a 
substantial range in excitation, it is possible to measure the velocities and 
temperatures of both atmospheric layers with high precision. 


557 



The first report of the doubling of CO lines in a Mira variable was 
published by Maillard (1974), who described the spectrum of R Leo. Two ex- 
tensive programs of infrared spectroscopy of Miras have also been undertaken: 
one the recently-completed Ph.D. dissertation of K. H. Hinkle, supervised by 
T. G. Barnes and D. L. Lambert, at the University of Texas, and the other an 
on-going project at Kitt Peak National Observatory, where D. N. B. Hall and 
S. T. Ridgway, who initiated it, have now been joined by Hinkle. The Texas 
results for R Leo, the star observed most extensively, have been published 
(Hinkle 1978 - CO and OH bands) or are in press (Hinkle and Barnes 1979 - 
H 2 0 bands). These papers are extremely illuminating, and I hope that the 
brief summary I will give here will inspire you to read them. Results from 
the Kitt Peak program have not yet been published, but since it uses even 
higher spectral resolution and involves extensive coverage of several stars, 
it may be expected to shed still more light on the Mira phenomenon. 

From the positions and strengths of lines of both CO and OH measured 
between 1.6 and 2.5 y in the spectrum of R Leo at nine distinct phases, Hinkle 
(1978) has been able to give a much clearer picture of the velocity and tem- 
perature structure and the manner in which these quantities vary through the 
cycle. Line doubling is most apparent just before maximum light, since at 
that time we begin to see the blue-shifted components from the deep part of 
the photosphere, which has just encountered the shock wave of a new cycle, 
while the mfalling material from the previous cycle has not yet faded from 
visibility. In addition, Hinkle identified a third layer, a cool outer shell 
which produces only low-excitation lines and is falling slowly back toward the 

star; thus some of the infrared lines, at certain phases, were actually seen 
as triple. 

These results are shown in Figure 11, in which velocity is plotted 
against phase. This figure is based on the first-overtone OH bands; similar 
results were obtained for the first- and second-overtone CO bands. For the 
purposes of the present discussion, two results stand out as being the most 
important. (1) The velocities of the warm, blue-shifted component near maxi- 
mum light are algebraically much smaller than any previously seen in the ab- 
sorption spectrum and are similar to those of the hydrogen emission lines; 


558 



RADIAL VELOCITY (km s' 1 ) 



Pig. 11 — Velocity variations shown by OH lines in R Leo. Near maxi- 
mum light (phase 0) the velocity is smallest as the gas rises rapidly; 
the motion can be followed for a full cycle as the gas decelerates and 
falls back in, and near phase 0.9 it is possible to see the photospheric 
layers from two successive cycles. In addition, the low— excitation OH 
lines show another component (dot-dash curve) which is identified with 
the cool circumstellar shell. From Hinkle (1978). 


559 



this is, m fact, the sort of velocity required for the expanding photosphere 
inferred from the photometry between the times of the visual and bolometric 
maxima. (2) The amplitude of the velocity variation, 27 km s -3 -, is larger 
than any previously found and is, for the first time, consistent with the 
photometric results. The same gas can be seen throughout a complete cycle 
as it rises, decelerates, and falls back in. 

Hinkle (1978) finds that a consistent pulsational model for R Leo can 
be derived if the center-of-mass velocity is 8 ± 1 km s _1 . This value is sig- 
nificantly smaller than most of the absorption-line velocities measured in 
the blue, which range from 7 to 15 km s“l as a function of phase (Merrill 
1946, 1952). This displacement of the mean absorption velocity from the 
center-of-mass velocity, which indicates that the absorbing material is 
falling in, is consistent with the suggestion of Reid and Dickinson (1976) 
discussed earlier. 

Another molecule that produces a great many lines in the infrared 
spectra of Miras is H 2 0, and its lines, too, are double throughout much of 
the cycle (Hinkle and Barnes 1979). The spectrum of H 2 0 is so complicated 
that the doubling of its lines might have gone unnoticed, were it not for the 
fact that the H 2 0 lines are sometimes sharp and single. The component that 
is always present can be identified with the cool circumstellar layer which 
also produces CO and OH lines of the same velocity. The other, which shows 
a greater range in velocity, comes from the photosphere. The behavior of the 
H 2 0 spectrum is thus similar to that of CO and OH, but there is an interesting 
difference. At maximum light, when the warm layer is moving rapidly outward 
and the doubling of the CO and OH lines is easily seen, the H 2 0 lines are 
single. Hinkle and Barnes offer a simple explanation for this difference: 
the H 2 0 molecules are dissociated at the high temperatures of the photosphere 
near maxxmum light, and they do not start to form in appreciable numbers 
until about 0.1 cycle later, when the photospheric temperature has dropped 
sufficiently. 

Figure 12 illustrates the doubling of H 2 0 lines in R Leo. Nearly 
all the absorption in this spectral interval is due to stellar H 2 0. In the 
upper spectrum, taken at maximum light, the lines are single and only the 


560 




5726 , 5733 

FREQUENCY (cm' 1 ) 


Fig. 12 - A section of the infrared spectrum of R Leo dominated 
by lines of HoO. The lines are single at maximum light (upper spec- 
trum) and double at phase 0.2 (lower spectrum). From Hinkle and 
Barnes (1979). 

cool shell component is present. In the lower spectrum, taken 0.2 cycle 
later, the shell component of each line is shifted to the left as the material 
falls back into the star, and a second component from the rising photosphere 
appears, shifted to the right. 


561 





Fig. 13 — Schematic representation of the outer atmosphere of a Mira 
variable at phases 0.6, 0.9, 0.0, and 0.3, based on infrared spectroscopy 
of R Leo. The arrows represent gas velocities. From Hinkle and Barnes 
\iy • ? ) • 

Hinkle and Barnes (1979) have summarized in Figure 13 the picture of 
R Leo that they have put together from their infrared observations. At all 
phases a cool circumstellar shell is present, contributing low-excitation lines. 
The stellar photosphere pulsates and is at all times hotter than the tempera- 
tures conventionally ascribed to Miras — a result anticipated by narrow-band 
photometry in the infrared (Wing 1967) . Intermediate temperatures occur in the 
star’s upper atmosphere, where most of the atomic and molecular absorption lines 
are formed and through which the shock wave propagates. 

I find the recent results from infrared spectroscopy extremely encour— 
aging. On the one hand they show that the spectroscopic approach can indeed 
give information about the pulsational properties of Miras. At the same time 


562 




they show us how to construct a model which, at least in its broad outline, 
is consistent with the photometric as well as the spectroscopic observations. 


CONCLUDING REMARKS 

This review has touched upon a wide variety of topics; the common 
theme has been the great extent of the observable atmospheres of the Mira 
variables. We have considered various methods for measuring the sizes of 
these atmospheres , . and more particularly the manner in which the size changes 
through the cycle. The results obtained by different methods have been com- 
pared, and the differences thus found have reminded us that observations of 
Miras are not always easy to interpret. 

I have emphasized the problems of interpretation because it seems to 
me that these are not always given sufficient attention. I hope I have dis- 
tinguished between the problems that are the star's fault (such as departures 
from spherical symmetry) and those which, dear Brutus, are our own (such as 
unwittingly combining results which refer to different parts of the star) . 
While the Miras will always be difficult objects to treat, some of the pro- 
blems that have baffled astronomers for decades have recently disappeared. 

In particular the famous discrepancy between the photometric and spectro- 
scopic diameter curves turns out not to be a real discrepancy at all, since 
the two methods are not looking at the same gas. The discrepancies between 
molecular band strength and continuum color temperature can be accounted for 
in the same way. 

A simple model for the atmospheric structure and motions of Miras, 
based on Hinkle's recent observations of the doubling of infrared molecular 
lines, has been described. This model, consisting of two atmospheric layers 
plus a circumstellar shell, has been remarkably successful in providing a 
physically plausible picture of the atmosphere which is consistent with the 
photometrically-measured magnitude and temperature variations as well as the 
spectroscopic data. However, it is of course much too simple to account for 
all the observations. For example, in Figure 13 the outer atmosphere is 


563 



represented by a large region at the uniform temperature of 1700°K, whereas 
we have long known that temperature and velocity gradients must exist since 
the measured velocities of absorption lines show a dependence on excitation 
potential. In another paper at this conference, Pilachowski, Wallerstein, 
and Willson (1979) treat the absorption-line velocities as functions of ex- 
citation potential, ionization potential, wavelength, and line strength j 
their results for the outer atmosphere should now be combined with the 
broader picture given by studies of line doubling to obtain a more complete 
and realistic model for the atmosphere. 

Much observational work on Miras remains to be done. In particular 
I would like to encourage work in three areas. First, it is important to 
find out how well the radial velocity variations of the infrared molecular 
lines repeat from cycle to cycle. In the work done to date it has been ne- 
cessary to combine observations from different cycles, and this procedure 
generally has not been very successful with other kinds of observations of 
Miras. The current program of infrared spectroscopy at Kitt Peak should 
settle this point. Second, measurements of molecular band strengths and 
photospheric color temperatures should be made around the cycle by narrow- 
band photometry, but unlike previous measurements of this kind, they should 
be accompanied by high-resolution spectroscopy so that the region of forma- 
tion of each spectral feature can be identified from its radial velocity. 

Such combined data could form the basis for a more detailed model of the 
atmosphere. Finally, direct diameter measurements through as much of the 
cycle as possible, by speckle interferometry in well-defined wavelength bands, 
badly needed. Now that it is possible to derive diameter curves from 
photometry and spectroscopy that are at least qualitatively the same, we must 
be brave and ask whether the same result can be obtained by direct measure- 
ment. 

I would like to thank Dr. Kenneth H. Hinkle for showing me his results 
prior to publication, and for providing the Kitt Peak spectra shown in Figures 
9 and 10. I also thank Dr. George Wallerstein for helpful correspondence. 


564 



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566