FOOTNOTES:
[412] Adams and Kohlschütter, Mt. W. Contr. 89, 1914.
[413] B. A. N. 19, 1922.
[414] Milne, Phil. Mag., 47, 209, 1924.
[415] Pannekoek, B. A. N. 19, 1922.
[416] Pop. Ast., 31, 88, 1923.
[417] Pop. Ast., in press.
[418] B. A. N. 19, 1922.
[419] In Pannekoek’s notation, a is surface brightness,
is radius, and
, surface gravity.
[420] Stewart, Pop. Ast., in press.
[421] Pub. Dom. Ap. Obs., 3, 1, 1924.
[423] H. C. 258, 1924.
[424] Adams, Pub. A. S. P., 28, 278, 1916; Adams and Joy, Pub. A. S. P., 36, 142, 1924.
[425] Pub. Solar Phys. Com., 1910.
[428] Menzel, H. C. 258, 1924; Chapter VIII, [p. 126].
PART III
ADDITIONAL DEDUCTIONS FROM IONIZATION THEORY
CHAPTER XI
THE ASTROPHYSICAL EVALUATION OF PHYSICAL CONSTANTS
IN the opening chapter the statement was made that “the astrophysicist is obliged to assume [the validity of physical laws] in applying them to stellar conditions.” The astrophysical evaluation of physical constants might therefore seem, from our avowed premises, to involve a circular argument. In certain special cases, however, the process appears to be legitimate, and the results of three investigations are contained in the present chapter. The first of these investigations involves the derivation of spectroscopic constants, assuming the series formula; the second consists of an extrapolation of the results of [Chapter X] to the estimation of unknown ionization potentials; and the third constitutes a discussion made possible by the knowledge of the stellar atmosphere that has been attained with the aid of ionization theory.
THE RYDBERG CONSTANT FOR HELIUM
The wave-lengths of a series of lines can be measured in the spectrum of a star, and the series identified with a series observed in the laboratory. The occurrence in stellar spectra of series that can be identified with the series given by terrestrial atoms presumably shows that similar relations govern the atomic processes in the two sources. That series formulae of the same type are applicable to the stellar and terrestrial atom is indeed rather an observational fact than an assumption. By inserting into the appropriate series formula the observed stellar frequencies, a physical constant involved may be evaluated, and the extent of the agreement with the corresponding value from the laboratory may be determined.
H. H. Plaskett[429] has measured the wave-lengths of the lines of the Pickering series (
) of He+ in the spectra of three
stars, incidentally separating the alternate Pickering lines from the Balmer lines for the first time. The formula that connects the frequencies of the lines with the constants associated with the atom is
Plaskett discussed the theory, and derived from the measured wave-lengths of five lines the mean value of 109722.3 ± 0.44 for the constant
. The value determined in the laboratory by Paschen is 109722.14 ± 0.04. Plaskett’s comment on the agreement is as follows: “It was not to be expected that there would be any startling changes.... It is of interest, however, to note that these “stellar” determinations are in agreement with the terrestrial values, in so far as it shows that the implicit assumption of identical atomic structure, identical electrons, and identical laws of radiation on the earth and in the stars, is in some measure justified.”
CRITICAL POTENTIALS
The theory outlined in the preceding chapters was used in determining the astrophysical behavior of lines corresponding to known series relations. When the validity of the theory has been established, it is possible, as was pointed out by the writer,[430] by Fowler and Milne,[431] and by Menzel,[432] to deduce the ionization potentials of lines of unknown series relations from their astrophysical behavior. The ionization potentials were estimated in this way for the table in [Chapter I].
In general the observations show that the higher the ionization potential, the higher the temperature at which the corresponding lines attain maximum. This is in strict accordance with theory. It is not possible to predict the exact form of the relation between temperature of maximum and ionization potential. For the observed cases in which
(the ultimate lines),
. It would appear that
should approach zero as
approaches zero. But in this case
(the negative energy of the excited state, which must always be less than
) also approaches zero, and the relation becomes indeterminate. The form of the curve as
approaches zero has merely a theoretical interest, as no known element has an ionization potential of less than four volts. In the present application the relation will be treated as an empirical one. The curves given by the writer and by Menzel for the relation between ionization potential and
display a good general regularity, and the deviations, as was pointed out in a previous chapter,[433] probably arise from differences of effective level. Owing to this source of irregularity, great accuracy is not to be anticipated in the deduced ionization potentials. The effective level is at the greatest height for lines of low excitation potential. The excitation potentials corresponding to the astrophysically important lines of the once, twice, and thrice ionized atoms in the hotter stars are in all known cases highland thus the error introduced by neglecting to correct for effective level is small. The error introduced by an excitation potential of the wrong order is, moreover, a constant and not a percentage error, and thus becomes less serious in estimating high ionization potentials. Accordingly the deduced ionization potentials will probably be of the right order.
The relation connecting ionization potential and
may, for our purposes, be treated as an empirical relation between ionization potential and spectral class. This mode of regarding the question has the advantage of being quite independent of the adopted temperature scale. We merely assume that the sequence of spectral classes is a temperature sequence. The ionization potentials corresponding to lines of known maximum may then be deduced by interpolation.
[TABLE XXV]
| Element | Ionization Potential | Authority |
|---|---|---|
| C++ | 45 | Payne |
| N+ | 24 | Ibid. |
| N++ | 45? | Ibid. |
| O+ | 32 | Ibid. |
| O++ | 45 | Fowler and Milne |
| Si | 8.5 | Menzel, Payne |
| S+ | 20 | Payne |
| S++ | 32 | Ibid. |
| Sc+ | 12.5 | Menzel |
| Ti+ | 12.5 | Ibid. |
| Fe | 7.5 | Ibid. |
| Fe+ | 13 | Ibid. |
The value of
is dependent on the effective level, and hence upon the excitation potential. Without the introduction of unjustified assumptions, more than one critical potential cannot be deduced from observations of intensity maximum. The excitation potential corresponding to a line could be roughly inferred from the observed maximum, by observing the shift of predicted maximum produced by the level effect (discussed in [Chapter IX]) if the ionization potential were known. There are, however, no data as yet that could be used in drawing inferences of this kind.
DURATION OF ATOMIC STATES
The successful application of theory to the astrophysical determination of the life of an atom requires the fulfilment of special conditions. The requirements of the idea developed by Milne[434] demand that the atom shall exist in appreciable quantities in only two states simultaneously. This condition is fulfilled by the ionized atoms of the alkaline earth elements, and it is with calcium that the estimates here discussed are concerned.
The investigation relates to the calcium present in the high-level chromosphere, where, owing to remoteness from the photosphere, thermal ionization is negligible. Photoelectric ionization may be operative in removing the first electron from the calcium atom, but the sun is too deficient in light of wave-length 1040 for second stage photoelectric ionization to be appreciable. The calcium present in the high-level chromosphere is probably largely in the once ionized condition, since an atom once ionized is likely to remain so for a long time, owing to the scarcity of free electrons in the tenuous outer regions of the sun. The present investigation neglects altogether the neutral and doubly ionized calcium atoms, and furthermore assumes that the transfers corresponding to the
and
lines of the
series are the only ones that occur in appreciable quantities. The latter assumption is apparently not accurately fulfilled, as the
lines of Ca+ have recently been detected in the high level chromosphere.[435]
In the simple case of the Ca+ atom (neglecting the small number of atoms that are giving rise to the
lines) only two states of the atom are possible: the normal state, called by Milne the
state, and the excited, or
state. A given atom exists alternately in these two states. If
be the average time spent in the
state, and
the average time spent in the
state, the average time spent by an atom in traversing its possible cycle of changes is
. Now
is connected with the probability of an emission, and
with the probability of an absorption. Clearly
depends at least partly upon the energy supply, but
is an atomic constant measuring the readiness with which the atom recovers its normal state after an absorption. It is, in fact, the “average life” evaluated from Milne’s equations. The ratio
, expressing the relative tendencies of Ca+ atoms to emit and to absorb the
and
lines, is the residual intensity at their centers, with respect to the adjacent continuous background.
Einstein’s theory of radiation[436] is used in evaluating
from the relation
where
is the ratio
.
From ordinary quantum principles,
and both
and
may be derived by eliminating between the two equations.
The only measured quantity in the formula is
, and from the fact that
is the “residual intensity” within an absorption line, we know that it must lie between 0 and 1. Hence a maximum value of
may be derived for
. On the insertion of the data given by Schwarzschild[437] for the residual intensity of the
and
lines, 2.6 magnitudes fainter than the continuous background, and corresponding to a value of
equal to 0.11, the deduced value of
is
. The agreement of this value with those obtained in the laboratory for the atoms of hydrogen and mercury has been commented upon in a previous chapter.[438]