Part 3 — CHAPTER XI THE ASTROPHYSICAL EVALUATION OF PHYSICAL CONSTANTS
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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.