FOOTNOTES:
[408] H. C. 258, 1924.
[409] M. N. R. A. S., 84, 499, 1924.
[410] M. N. R. A. S., 84, 499, 1924.
CHAPTER X
EFFECTS OF ABSOLUTE MAGNITUDE UPON THE SPECTRUM
DIFFERENCES between the spectra of stars of the same spectral class have long been recognized. The empirical correlation of relative line intensities with absolute magnitude was made the basis for the estimation of spectroscopic parallaxes.[412] Such differences within a class were later related in a qualitative way to differences of pressure, in conjunction with the theory of thermal ionization, and have been regarded as corroborative evidence that the type of process contemplated by that theory actually represents what goes on in the atmospheres of the stars.
In the present chapter the theory of the various effects will first be discussed, and later the predictions from the theory will be compared with observational data.
INFLUENCE OF SURFACE GRAVITY ON THE SPECTRUM
The first theoretical discussion of the effects of absolute magnitude upon the stellar spectrum seems to have been made by Pannekoek,[413] who pointed out that “stars of the same spectral class ... will show differences depending solely on ...
,” where
is the surface gravity, and
the absorption coefficient. Pannekoek considered all stars of the same spectral class to have the same temperature, and for the purposes of his argument the differences in temperature between giants and dwarfs can be neglected, although actually they may for other reasons have a noticeable effect on the spectrum. If
be regarded as constant, a plausible assumption for various reasons,[414] “the physical quantity, directly given by the spectra used for the determination of spectroscopic parallaxes is the gravitation at the surface of the star.”[415] The relation between the surface gravity and the pressure is given by
where
is the “homogeneous depth.” The pressure is then directly proportional to the surface gravity.
INFLUENCE OF PRESSURE ON THE SPECTRUM
Lowered pressure increases the degree of ionization. The tendency of the atoms to lose electrons by thermal ionization should depend solely on their energy supply, and should thus be independent of the pressure. The total absorbing power of the gas will, however, depend on the number of suitable atoms that it contains, not upon their rate of formation. The number of suitable ionized atoms present at any moment in the atmosphere is a function not only of the rate at which ionization proceeds, but also of the rate of recombination. The more readily recombination takes place, the larger is the number of effective neutral atoms, and the smaller the number of effective ionized atoms, when a steady state is attained. The rate of recombination, which depends upon the probability of a suitable encounter between an ionized atom and a free electron, will increase with the pressure—more accurately, with the partial pressure of free electrons.
The higher the pressure, therefore, the greater the number of neutral atoms, and the smaller the number of ionized atoms. This argument explains at once the strength of the neutral (arc) lines in the spectra of stars of low luminosity (high surface gravity), and the predominance of ionized (spark) lines for absolutely bright stars (low surface gravity, resulting chiefly from large radius). Low surface gravity, then, increases the number of ionized atoms present by discouraging recombination.
It should be noted that any tendency to extensive ionization will increase the concentration of free electrons and tend to encourage recombination, thus counteracting the effect of low surface gravity. The effect of an increased concentration of free electrons will not, however, attain the magnitude of the surface gravity effect, since even for the hottest stars examined, three electrons appear to be the largest number that can be thermally removed under reversing layer conditions.
The theoretical effect of lowering the pressure has been discussed by Stewart,[416] who, after alluding to the importance of the surface gravity, suggested that the ultimate lines of neutral atoms easier to ionize than the average should be weakened by low pressure, and that the corresponding enhanced lines should be strengthened. For atoms harder to ionize than the average the reverse should be the case for the two classes of lines. From this standpoint he showed that the absolute magnitude effects might be qualitatively accounted for. The “average ionization potential” was the average for the lines used in the estimates; Stewart adopted the value of six volts for Classes
to
.
EFFECT OF TEMPERATURE AND DENSITY GRADIENTS UPON THE SPECTRUM OF A STAR
There is another respect, recently analyzed by Stewart,[417] in which the spectrum of a giant may be expected to differ from that of the corresponding dwarf. He points out that “in a giant, owing to the small density, there is more material overlying the photosphere than in a dwarf having the same effective temperature; while at the same time the density in the photospheric region is less in the giant, owing to the low gravity.” These conditions furnish an interpretation of the increased blackness and sharpness of the lines in giant stars, as compared with the corresponding dwarfs. The absorption lines in giants are blacker because there is more matter above the photosphere than in dwarfs; they are sharper because the effective level at which the lines originate is at a lower pressure in the giant than in the dwarf, owing to the smaller pressure gradient in the giant star, and to its lower surface gravity. The difference in line quality between a giant and a dwarf is at once obvious from the spectra, and this effect renders direct comparisons of estimated line-intensities a matter of extreme difficulty. It is an effect that must be taken into account in examining the agreement between the observations and the theory.
Stewart’s argument also suggests the answer to an important question raised by Pannekoek[418] in the course of his discussion of the absolute magnitude effect. The latter remarks that “the general decrease of luminosity with advancing type for the same value of relative line-intensity, which is shown ... by most reduction curves ... corresponds to the decrease in
, as for the same
and smaller
smaller surface brightness means smaller luminosity. If we take account, however, of the direct influence of temperature on ionization, which acts much more strongly in the opposite direction, we must expect equal ionization in the more advanced types for much smaller g and higher luminosities, contrary to the empirical reduction curves. It looks as if this effect is compensated by some other direct influence of temperature on the spectrum.”[419]
The influence suspected by Pannekoek may be found, at least in part, in the “theoretical decrease with increasing temperature and density in the quantity of material overlying the photosphere. Thus the contrast between line and continuous background tends to become less along the giant series
(since, furthermore, for the same abundance of active material, a given line is formed always at the same depth).”[420] This suggestion was advanced by Stewart to account for the observed displacement, towards cooler classes, of the maxima of absorption lines discussed in [Chapter X]. It is certain that some such factor will be operative in the reversing layer, but it is believed that the burden of the shift of maxima should be borne by the effective level, which has been discussed in more detail in the preceding chapter. It would be of interest to compare the two effects quantitatively, but the effect of temperature gradient has not yet formed the basis of numerical predictions.
PREDICTED EFFECTS ON INDIVIDUAL LINES
The discussion involving the average ionization potential appears to permit of more rigorous treatment. Suppose the “average ionization potential” of Stewart’s discussion to be replaced by the ionization potential corresponding to the atoms whose lines are at maximum for the class in question. It then follows directly from theory that the effects of lowered pressure on the different classes of lines will be as below:
| Atom | Line | Effect of lowered pressure | |
|---|---|---|---|
| Hotter than class | Cooler than class | ||
| for maximum | for maximum | ||
| Neutral | Ultimate | Weakened | .... |
| Neutral | Subordinate | Weakened | Weakened |
| Ionized | Ultimate | Weakened | Strengthened |
| Ionized | Subordinate | Weakened | Strengthened |
It is especially to be noted that all lines should theoretically be weakened in passing from dwarf to giant, excepting the lines of an ionized atom at temperatures lower than those required to bring them to maximum. This leaves out of account the effect of photospheric depth, which will be introduced later as a correcting factor.
The case of the ultimate lines of the ionized atoms is of especial interest. At their maximum, if the Fowler-Milne theory is correct, ionization is almost complete, and more than 99 per cent of the element is giving the ionized ultimate lines. At a temperature higher than that required for maximum, lowered pressure can “increase” the ionization only by the removal of the second electron. By this process the intensity of the ionized ultimate lines is decreased, since the number of singly ionized atoms is thereby reduced. The fall from maximum towards the hotter stars, which is displayed by the ionized lines of Ca+, Sr+, and Ba+ can be due only to the progress of second ionization, and there seems to be no escape from the conclusion that the ultimate lines of the ionized atom should theoretically decrease in strength, with lowered pressure, for stars hotter than those required to bring the lines to maximum. The point is made increasingly clear when it is recalled that, at the maximum, all of the substance is presumably at work giving the lines in question. It is not therefore possible to increase the number of active atoms by any process whatever that involves merely a change in pressure.
For ionized subordinate lines the theoretical effect should be the same as for the ultimate lines, for the fall after maximum is here again caused by the increase in the number of doubly ionized atoms, and the consequent decrease in the number of those singly ionized. Thus, although the subordinate lines are not already using all the available atoms at maximum, so that increased intensity with lowered pressure is possible, it would still appear that they should be weakened at temperatures higher than that corresponding to maximum intensity in the spectral sequence.
The pure pressure effects just discussed will be superposed upon the Stewart effect, which depends upon the photospheric depth. The latter will cause a general increase in the strength of all lines from dwarf to giant, as a result of the greater amount of matter lying above the photosphere in the giant. The two effects are observed together when direct intensity measures are employed, such as the estimates embodied in [Chapter VIII], while the pressure effect is given almost purely when differential estimates of intensity for the same spectrum are used, as in most investigations of spectroscopic parallax. The observational evidence from both sources will now be put forward, in order to examine the sufficiency of the theories that have been advanced to account for the absolute magnitude effects.
The empirical relations used in the estimation of spectroscopic parallax should provide material for examining the simple pressure effect, as they are derived from the ratio of two lines in the same spectrum. Unfortunately the line ratios actually in use were selected because they were convenient to measure, and gave (empirically) consistent results, not for reasons of theoretical tractability. Fourteen line ratios are used, for example, by Harper and Young,[421] but only four of these consist of pairs of unblended lines with known series relations. It is only for such lines that a useful test of theory can be made.
[TABLE XXIII]
| M | = | +7 | +8.8 | +9.2 | +11.0 | +13.2 | ||||
| +6 | +13.3 | |||||||||
| +5 | +3.5 | +5.0 | +7.4 | +9.6 | ||||||
| +4 | +10.0 | +10.8 | ||||||||
| +3 | -1.8 | +0.5 | +3.8 | +6.7 | ||||||
| +2 | +6.7 | +7.0 | +7.6 | +10.0 | ||||||
| +1 | -7.2 | -3.8 | +0.2 | +3.2 | ||||||
| 0 | +3.5 | +3.4 | +3.2 | +3.0 | ||||||
| 1 | -12.9 | -8.2 | -3.2 | -0.3 | ||||||
| -2 | +0.3 | -0.2 | -1.4 | -3.7 | ||||||
| -3 | -12.1 | -7.3 | -3.8 | |||||||
| -4 | -2.8 | -4.2 | -5.8 | -10.0 |
| M | = | +6 | -2.0 | -1.4 | ||||||
| +5 | -1.5 | -0.5 | 0.0 | +1.0 | ||||||
| +4 | +1.2 | +1.3 | ||||||||
| +3 | +1.8 | +3.3 | +4.4 | +4.5 | ||||||
| +2 | +5.0 | +3.0 | ||||||||
| +1 | +4.8 | +6.5 | +7.8 | +7.6 | ||||||
| 0 | +6.5 | +4.7 | ||||||||
| -1 | +8.6 | +9.3 | +11.0 | +11.0 | ||||||
| -2 | +12.0 | +8.6 | +8.6 | +8.2 |
| M | = | +6 | -18.1 | |||||||
| +5 | -3.7 | -5.8 | -9.6 | -14.7 | ||||||
| +4 | -15.3 | |||||||||
| +3 | -1.0 | -3.3 | -6.6 | -11.0 | ||||||
| +2 | -12.4 | |||||||||
| +1 | +1.7 | -0.6 | -3.7 | -7.5 | ||||||
| 0 | -9.4 | |||||||||
| -1 | +4.5 | +2.0 | -0.9 | -4.0 | ||||||
| -2 | -6.6 | |||||||||
| -3 | +7.2 | +4.7 | +2.0 | -0.2 |
| M | = | +6 | +7.4 | +6.2 | ||||||
| +5 | +6.2 | |||||||||
| +4 | +4.3 | +3.0 | ||||||||
| +3 | +3.0 | |||||||||
| +2 | +1.3 | 0.0 | ||||||||
| +1 | 0.0 | |||||||||
| 0 | 0.0 | -2.0 | ||||||||
| 1 | +2.6 | |||||||||
| -2 | +2.0 | +0.7 | ||||||||
| -3 | +5.2 |
The preceding table contains a transcription of the reduction-curve material given by Harper and Young for the four pairs of lines mentioned. Tabulated quantities are the “step differences” for the classes at the heads of the columns, and the absolute magnitudes contained in the first column.
Presumably the irregularities of the observed curves have been smoothed out in forming the reduction table, but the figures will certainly give an indication of the direction in which a given line is affected by absolute magnitude.
The predicted effect of lowered pressure upon the lines involved is contained in the table that follows:
| Line | Source | Max. | Effect of lowered pressure |
|---|---|---|---|
| 4071 | Fe (sub) | weakened throughout | |
| 4077 | Sr+ (ult) | strengthened in , weakened in and | |
| 4215 | Sr+ (ult) | strengthened in , weakened in and | |
| 4247 | Sc+ (?) | strengthened throughout range | |
| 4250 | Fe (sub) | weakened throughout | |
| 4455 | Ca (sub) | weakened throughout | |
| 4494 | Fe (sub) | weakened throughout |
The predicted changes in the line ratios with lowered pressure are therefore as follows:
The ratio
behaves in exact accordance with prediction, and
which decreases and then increases again, offers no evidence for or against the theory. The two remaining ratios, involving the two Sr+ lines, display a lack of agreement with theory for the
and
classes, apparently owing to the strengthening of the Sr+ lines with high luminosity, even at temperatures higher than those at which they attain maximum intensity. The strengthening of Sr+ with high luminosity is one of the best-attested facts of observational astrophysics, and it is a serious deficiency in theory if the observed behavior of the lines in the hotter stars cannot be explained. The question will be further discussed presently.
The material obtained by the writer, and summarized in a preceding chapter,[422] may be used in making a test of the predicted pressure effects by means of direct estimates. As was pointed out above, the lines of a giant are stronger than those of a dwarf, owing to the greater photospheric depth in the former. The practical difficulty of making comparable estimates upon sharp and somewhat hazy lines must also be considered in the discussion of the results. Clearly some numerical correction is required, in order to allow for the Stewart effect, and this has been done in a somewhat arbitrary manner in forming [Table XXIV]. It is assumed that the mean increase in intensity for such lines as are strengthened will be equal to the mean decrease in intensity for such lines as are weakened. For each spectral class this assumption provides a correcting factor, which never exceeds one scale unit.
The table that follows contains the material derived from the measures enumerated in [Chapter VIII], and from other sources, bearing on the intensity differences between giants and dwarfs of the same spectral class. All the available estimates have been used. Successive columns give the line, the atom, the predicted behavior, and the observed difference in the sense giant-dwarf, for the Classes
,
,
,
,
, and
. The symbols
,
, and
, following the atom, denote ultimate, neutral, and enhanced lines, respectively. The number of stars contributing to each entry will be seen from the list on [p. 119], Chapter VIII. The notation is as follows: 0 = no change; ± 0 = between 0 and 1; ± 1 = between 1 and 2; ± 2 = between 2 and 3; and so on. The values for
are taken from Menzel’s measures[423] of
Indi and
Tauri, “the scale of intensities being (0) no difference, (1) a little stronger (2) much stronger, (3) very much stronger.” The signs from Menzel’s table are reversed, in accordance with the notation used in the present table. In the column headed
are the signs indicating the direction in which the corresponding lines are affected in that class,[424] for which quantitative measures have not been published. The letters “
” and “
” in the column headed
refer to strengthening or weakening of lines, as observed by Baxandall[425] in a comparison of the solar spectrum with that of Capella. Baxandall’s estimates are inserted to supplement the present material. The numerous gaps in the table result from the difficulty of seeing the fainter lines in the dwarf spectrum.
[TABLE XXIV]
| Line | Element | Predicted Effect | Observed Effect | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| - | 0 | + |  |  |  |  |  |  |  | |||
| 3933 | Ca+ | * |  | | | +1 | .. | .. | .. | .. | 0? | + |
| 3944 | Al | u |  | .. | .. | +0 | + | -2 | .. | .. | .. | - |
| 3953 | Fe | n |  | ( ) | .. | +0 | + | -2 | .. | .. | .. | .. |
| 3961 | Al | u | | .. | .. | +0 | + | -2 | .. | .. | .. | - |
| 3968 | Ca+ | * | | | | +1 | .. | -1 | .. | .. | 0? | + |
| 3999 | Ti | u | | .. | .. | +0 | .. | -1 | .. | .. | .. | .. |
| 4005 | Fe | n | | .. | .. | +0 | .. | 0 | .. | .. | .. | .. |
| 4031 | Mn | u | | .. | .. | -2 | -1 | -1 | .. | .. | 0 | .. |
| 4041 | Mn | n | | .. | .. | -1 | -1 | -2 | .. | .. | -0 | .. |
| 4046 | Fe | n | | .. | .. | -1 | -1 | 0 | -1 | -1 | -2 | .. |
| 4064 | Fe | n | | .. | .. | -1 | -2 | .. | +2 | 0 | -2 | .. |
| 4068 | FeMn | n | | .. | .. | -1 | .. | .. | +1 | -1 | .. | .. |
| 4072 | Fe,- | n | | .. | .. | 0 | +4 | .. | .. | -1 | -2 | .. |
| 4077 | Sr+ | * | | |  | 0 | 0 | +1 | s | 0 | +3 | + |
| 4084 | Fe | n | | .. | .. | -1 | +1 | 0 | 0 | .. | .. | .. |
| 4101 | H | n | | .. | .. | -1 | -1 | +1 | -4 | 0 | -1 | + |
| 4132 | Fe | n | | .. | .. | 0 | +2 | +1 | .. | +0 | -1 | .. |
| 4135 | Fe | n | | .. | .. | -1 | .. | 0 | .. | +0 | .. | .. |
| 4144 | Fe | n | | .. | .. | -1 | 0 | +2 | 0 | +0 | -1 | .. |
| 4167 | ? | -1 | .. | -1 | -1w | 0 | .. | .. | ||||
| 4172 | Fe+ | * | |  | +1 | +1 | +3 | +1 | +0 | .. | .. | |
| 4177 | Fe+ | * | | | +1 | .. | +2 | .. | .. | .. | .. | |
| 4215 | Sr+ | * | | | | 0 | 0 | +2 | +3 | +0 | +1 | + |
| 4227 | Ca | u | | .. | .. | -1 | -1 | 0 | -2 | -0 | -2 | - |
| 4247 | Sc+ | * | .. | | | 0 | .. | +1 | .. | +1 | .. | .. |
| 4250 | Fe | n | | .. | .. | 0 | -2 | -1 | -2 | -1 | -2 | .. |
| 4254 | Cr | u | | .. | .. | -1 | .. | -1 | -1 | -1 | -1 | .. |
| 4260 | Fe | n | | .. | .. | -1 | -2 | -1 | -2 | .. | -1 | - |
| 4272 | Fe | n | | .. | .. | -1 | -2 | .. | 0 | +0 | -1 | .. |
| 4275 | Br | u | | .. | .. | -1 | .. | .. | 0 | +0 | -1 | .. |
| 4290 | Cr | u | | .. | .. | 0 | -2 | 0 | .. | +0 | -1 | .. |
| 4298 | Ti, Ca | | .. | .. | -1 | -2 | 0 | .. | 0 | .. | .. | |
| 4308 | Fe | n | | .. | .. | 0 | +1 | 0 | .. | .. | -1 | .. |
| 4315 | Fe+ | * | .. | | | 0 | +1 | +2 | .. | .. | .. | .. |
| 4321 | Sc+ | * | .. | | | +1 | .. | .. | s | .. | .. | .. |
| 4326 | Fe | n | | .. | .. | 0 | +1 | +1 | -2w | 0 | -2 | .. |
| 4340 | H | n | | .. | .. | -2 | -7 | -1 | -4 | -0 | -1 | + |
| 4352 | CrMg | ? | .. | .. | 0 | +2 | 0 | +1 | +0 | -3 | .. | |
| 4360 | Cr | n | | .. | .. | 0 | .. | +1 | .. | .. | .. | .. |
| 4370 | Fe | n | | .. | .. | +1 | .. | 0 | .. | +0 | .. | .. |
| 4376 | Y+ | * | .. | | | -1 | +2 | 0 | .. | +1 | +2 | .. |
| 4383 | Fe | n | | .. | .. | 0 | +3 | +1 | -2 | -4 | -2 | .. |
| 4405 | Fe | n | | .. | .. | -1 | .. | .. | 0 | -1 | -2 | .. |
| 4415 | Fe+ | * | .. | | | +1 | 0 | +3 | 0 | 0 | -1 | .. |
| 4435 | Ca | n | | .. | .. | -1 | .. | +1 | .. | 0 | -2 | - |
| 4444 | Ti+ | * | .. | | | 0 | 0 | +1 | s | -3 | .. | .. |
| 4455 | Ca | n | | .. | .. | -2 | .. | -1 | w | -1 | -3 | - |
| 4476 | Fe | n | | .. | .. | -1 | .. | +1 | -1 | .. | .. | .. |
| 4481 | Mg+ | * | .. | .. | | -1 | 0 | 0 | 0 | +0 | +1 | .. |
| 4490 | Fe | n | | .. | .. | 0 | .. | .. | +1 | -0 | +2 | .. |
It is seen from the table that the general agreement with the anticipations of theory is satisfactory, and that the deviations, when they occur, rarely exceed one unit. The agreement is not less good than would be expected of the material, since the measures are here used differentially. The majority of the discrepancies are apparently accidental; for example, the deviations shown by the first six entries in the first column are almost certainly the result of better definition in the giant spectrum. There remains, however, the same discrepancy for the lines of Sr+ that was noted in the earlier part of the chapter. There can be no doubt that these lines are stronger in giants than in dwarfs.
The strengthening of the ionized lines of the alkaline earths is explained, when the spectra are examined, by the fact that the neutral lines are still fairly strong long after the ionized lines have passed their maximum—neutral strontium[426] is found at
and neutral calcium[427] at
. The lowered pressure, then, must increase the concentration of singly ionized atoms at the expense of the residual neutral atoms. There is, however, apparently no satisfactory theoretical explanation of the survival of large quantities of neutral calcium long after the ionized atoms have passed their maximum. The effects predicted above would appear to be the only ones that can be anticipated if the theory holds rigidly. Clearly some factor such as effective level must be further considered.
THE STRONTIUM LINES
The strontium problem is perhaps one that will lead to more comprehensible results when it is treated as a whole. It is impossible to resist the feeling that there is some definite abnormality associated with strontium. The “strontium stars” in the still earlier classes, where the lines 4215, 4077 appear with great intensity, and the
stars
Circini and
Equulei, as well as the apparently erroneous absolute magnitudes obtained by the spectroscopic method for several other stars of low intrinsic luminosity, all point in some such direction.
It may be that these phenomena are a result of an abnormal abundance or distribution of the element. It is not, therefore, entirely necessary to assume that the theory is here at fault, although until the behavior of strontium has been satisfactorily interpreted, that possibility cannot be rejected. It is significant that calcium and barium show similar absolute magnitude behavior. In any case, the ionized strontium lines cannot be cited, as has sometimes been done, in demonstrating that the absolute magnitude effect is due to pressure. What is actually shown is that the concentration of singly ionized atoms is more greatly increased at the expense of the neutral atoms than it is reduced by the formation of doubly ionized atoms. Since a pressure effect operates by the discouragement of recombination, it would be inferred that the recombination of singly ionized atoms with electrons to form neutral atoms is less readily encouraged than the recombination of doubly ionized atoms with electrons to form singly ionized atoms. Evidently the problem is a complex one. If the maximum of the strontium lines were at
(where theory first predicted it, and where the earlier measures actually placed it) there would be no anomaly to explain; but two independent observers[428] place it definitely at
or
, and there can be little doubt that this is actually the correct position of the maximum.
The result of the study of absolute magnitude effects is disappointing. It appears that the observed phenomena are qualitatively explained in a satisfactory manner, as due to lowered pressure, or, more accurately, to low surface gravity. There is, however, a serious discrepancy in the case of the lines whose variation with absolute magnitude is perhaps best established, and upon which the most important results have been based. The results, being empirical, are of course unimpaired, and it would seem that the theory requires to be amended. Furthermore, it does not yet appear to be possible to use the observed changes of intensity for the direct estimation of pressure differences, because of the large number of variables involved and particularly because of the superposition of the pure pressure effect upon the effect of photospheric depth.