FOOTNOTES:
[403] Payne, H. C. 256, 263, 1924.
[404] Menzel, H. C. 258, 1924.
[405] Harper and Young, Pub. Dom. Ap. Obs., 3, 3, 1925.
[407] Menzel, H. C. 258, 1924.
CHAPTER IX
THE IONIZATION TEMPERATURE SCALE
A preliminary application of the observed maxima of absorption lines, in the formation of a stellar temperature scale, was given at the end of the preceding chapter. The temperatures were obtained on the assumption that
, the partial electron pressure in the reversing layer, was constant for all lines and equal to
. Striking inconsistencies appear in this preliminary table of temperatures. As Menzel[408] has remarked, the maxima of most of the metallic arc lines occur in stars cooler than the ionization theory, on the stated assumptions, would predict. The ultimate lines of the ionized atoms of calcium, strontium, and barium show especially large inconsistencies. The temperatures of the maxima for these atoms, deduced from the ionization formula on the assumption that
, are about 3000° higher than the measured temperatures of the classes at which the maxima occur, as deduced from the color indices.
The following suggestion has been advanced by Fowler and Milne[409] to account for the observed deviations of Ca+, Sr+, and Ba+. “For the maximum of the principal line of an ionized atom, the fraction of atoms in the required state is almost unity.... On the other hand ... at the maxima of subordinate lines the fraction of atoms in the required state is from
to
.... Thus atoms in the required state are
to
times as abundant for intense principal lines as for intense subordinate lines. It follows that principal lines must originate at much higher levels in the stellar atmosphere than subordinate lines, and consequently at much smaller pressures.”
It appears that the behavior of the ionized atoms of the alkaline earths can be satisfactorily explained in this way. The further suggestion was made that a similar effect might be expected for atoms of low excitation potential, such as manganese and magnesium.
The possibility of varying
as well as
in the formula for the theoretical maximum places the investigation on a rather different footing. Any temperature (within wide limits) may now be obtained for the theoretical maximum of a line by appropriately varying the partial pressure. The stellar temperature scale cannot, in such a case, be fixed merely from a knowledge of the critical potentials and the observed maxima, without introducing other considerations. It is necessary to find a way of determining the appropriate partial pressures.
The procedure that will here be followed consists essentially in a calibration and an extrapolation. The temperature scale from
to
is regarded as known from spectrophotometric data. Within this range, the theoretical and observed maxima are compared. The possibility of finding a value of
appropriate to a given atomic state is next examined. Finally, a method of estimating
will be justified for the cooler stars, within the limits of accuracy permitted by the data, and will be extended by simple extrapolation to the formation of a temperature scale for the hotter stars, where the temperatures cannot be safely estimated from the color indices.
The salient point is that complete absorption will occur for any line at a depth that is inversely proportional to the abundance of the corresponding state of the atom. No light in this wave-length reaches the exterior from any lower level, and the deepest level from which the line originates therefore forms a lower boundary to the effective portion of the atom in question. The “effective level” from which a line comes is probably best regarded as the level at which the effective atoms above the “lower boundary” have their median frequency. Clearly the partial pressure will differ at different effective levels, and thus abundance has a direct influence on the appropriate value of the partial pressure.
The theory with which we have so far been concerned deals with the excited fraction of the total amount of the element which is present. A knowledge of this quantity suffices for specifying the variation of intensity for the lines of any one element. But the absolute abundance of a given atomic state varies jointly with the fractional concentration of the appropriate state and the total amount of the element present. Now, for the first time, the absolute abundance of different atomic species becomes of possible importance, as a factor affecting the depth from which radiation corresponding to the given atom will penetrate. Fowler and Milne[410] rightly claimed that their method of maxima eliminated questions of relative abundance, “if
can be regarded as known ... [and constant]. The proper value of
must be a function of the abundance of the atom in question relative to free electrons.”
The question of relative abundances of elements in the reversing layer is discussed[411] in [Chapter XIII]. It may be mentioned that the abundances there deduced depend upon estimates of marginal appearance. Probably all lines are unsaturated at marginal appearance, that is, there are not enough suitable atoms present completely to absorb all the incident light of the appropriate wave-length. Hence all suitable atoms present, as far down as the photosphere, where general opacity begins to render the gas hazy, are actually contributing to the line. At marginal appearance, then, all the intensity phenomena are probably due to pure abundance, and considerations of level are eliminated. The deduced abundances are therefore independent of effects such as are discussed in the present chapter, and the results of [Chapter XIII] may be cited as giving evidence that the stellar abundances, for all the atoms here to be considered except barium, have a range with only a factor of ten, which is negligible in comparison with the quantities to be discussed. The relative abundance of different atomic species will therefore be neglected in what follows, although, with more accurate data than are now available, it should become a factor of importance.
Fractional concentrations, as derived from the ionization formula, govern the effective level at which absorption takes place. Fowler and Milne, as was pointed out earlier, suggested that the higher the fractional concentration at maximum, the higher the level and the lower the partial pressure from which the line originates. They suggested that the pressure for a principal line at maximum is from
to
of the corresponding value for a subordinate line.
The assumption now introduced is, in effect, that the absorbing efficiency of individual atoms is the same. The partial pressure at the level from which a line originates should then vary inversely as the fractional concentration at maximum. In other words, the product
should be constant, when
is deduced from the class at which the observed maximum occurs.
The quantity
depends primarily on the excitation potential, and varies but slowly with
. It is given by the expression[iii]
[iii] For notation, see Chapter VII, [p. 106].
For subordinate lines,
is given by the expression
and this quantity is extremely sensitive to change in
.
For ultimate lines, where the excitation potential is equal to zero, and
accordingly reduces to unity, the value of
should be equal to the constant product predicted in a previous paragraph. Fowler and Milne suggested a partial electron pressure of
to
for Ca+ on the basis of a maximum at
, assumed temperature 4500°. This is the effective temperature of the class, deduced spectrophotometrically, and “the reversing layer should be at a lower temperature—its average temperature should be in the neighborhood of, or somewhat lower than, the Schwarzschild boundary temperature,[[iv]] which is some 15-20 per cent lower than the effective temperature.” The value 4000° is therefore adopted here for
. For this value
becomes
for Ca+; for Sr+ (
, 35000°),
, and for Ba+ (Ma? 3000°),
. The maximum for Sr+ is the best determined of the three, as the Ca+ lines are too strong and too far into the violet for an accurate estimate among the cooler stars, and the Ba+ line is rather faint, and is heavily blended. The constant product may then be expected to be of the order of
.
The prediction is examined in the table that follows. The temperature of the class at which the lines attain maximum is assumed from spectrophotometric data, and is expressed to the nearest five hundred degrees.
[TABLE XX]
| Atom | Ionization Potential | Excitation Potential | Max. | Sum | |||
|---|---|---|---|---|---|---|---|
| Mg+ | 14.97 | 8.83 | 9000° | 5.32 | |||
| Ca | 6.09 | 1.88 | 3000 | 3.64 | |||
| Ti | 6.5 | 0.84 | 3500 | 2.7 | |||
| Cr | 6.75 | 0.94 | 3000 | 2.39 | |||
| Mn | 7.41 | 2.16 | 3500 | 3.77 | |||
| Zn | 9.35 | 4.01 | 5600 | 3.0 | |||
| Ca+ | 11.82 | 0.00 | 4000 | 0.00 | |||
| Sr+ | 10.98 | 0.00 | 3500 | 0.00 | |||
| Ba+ | 9.96 | 0.00 | ? | 3000 | 0.00 | ||
| Mg | 7.61 | 2.67 | ? | 4000 | 3.25 |
[[iv]: The Schwarzschild approximation to the boundary temperature is given by the expression
where
is the effective temperature and
the boundary temperature.]
Successive columns give the atom, the critical potentials in volts, the spectral class at which maximum occurs, the assumed
,
calculated from the theory,
, and the sum of the quantities in the two preceding columns. The only quantity that is not fixed by the laboratory data is
, which is derived from the data presented in [Chapter II]. It will be seen that the quantity entered in the last column is sensibly constant, and equal to about -10, in accordance with prediction. All available maxima have been used.
It appears that the foregoing evidence constitutes a fair and satisfactory test of the Fowler-Milne equations, and that, in the region in which the test can be applied, the agreement with theory is as close as can be expected from the material. It also appears that the “serious and undoubtedly real” discordance of theory and observation, quoted by Menzel in the discussion of the maxima observed by him, is removed by introducing these considerations of level.
When the theory has been applied and justified for the classes where the temperature scale is well determined by other methods, it may be extrapolated to fix the temperature scale for the hotter stars. As before, the fractional concentration at maximum varies but slowly with
, and
is determined mainly by
. If now
be so chosen that
is always approximately equal to
, the value of
derived from the equations will be the appropriate one for the class in question. This value of
has to be found by trial. It so happens that the temperatures thus obtained are not very different from those originally predicted without entering into considerations of effective level. The excitation potentials of the highly ionized stages of the lighter elements are invariably large, and all lead to values of
of the order of
. It is to be noted that values of
greater than
are not indicated.
The following tabulation represents the resulting temperature scale for the hotter stars. It must be remembered that
is here the derived quantity, whereas in [Table XX] it was the known quantity used for calibration.
[TABLE XXI]
| Atom | Ionization Potential | Excitation Potential | Max. | |
|---|---|---|---|---|
| He+ | 54.2 | 48.2 | 35000° | |
| C+ | 24.3 | 18.0 | l6000 | |
| He | 24.7 | 21.1 | 10000 | |
| Si++ | 31.7 | 4.8 | 18000 | |
| Si+++ | 45.0 | 24.0 | 25000 |
The values given in the preceding table constitute the only contribution that can be made by this form of ionization theory to the formation of a stellar temperature scale. Values assigned to intermediate classes must be conjectural. From the observed changes of intensity from class to class, temperatures may be interpolated roughly, and a temperature scale, formed on these general grounds, is reproduced in [Table XXII]. Values not derived from observed maxima are italicized.
[TABLE XXII]
| Class | Temperature | Class | Temperature | |
|---|---|---|---|---|
| 3000° | 9000° | |||
| 3000 | 10000 | |||
| 3500 | 13500 | |||
| 4000 | 15000 | |||
| 5000 | 17000 | |||
| 5600 | 18000 | |||
| 7000 | 20000 | |||
| 7500 | 25000 | |||
| 8400 | to | 35000 |