PART IIIADDITIONAL DEDUCTIONS FROM IONIZATION THEORY

CHAPTER XITHE 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 ofChapter Xto 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 threestars, 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” determinationsarein agreement with the terrestrial values, in so far as it shows thatthe 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 inChapter I.

In general the observations show that the higher the ionizationpotential, 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 thatshould approach zero asapproaches 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 asapproaches 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 anddisplay 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 andmay, 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 linesof known maximum may then be deduced by interpolation.

The value ofis 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 inChapter 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 theandlines of theseries are the only ones that occur in appreciable quantities. The latter assumption is apparently not accurately fulfilled, as thelines 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 thelines) only two states of the atom are possible: the normal state, called by Milne thestate, and the excited, orstate. A given atom exists alternately in these two states. Ifbe the average time spent in thestate, andthe average time spent in thestate, the average time spent by an atom in traversing its possible cycle of changes is.Nowis connected with the probability of an emission, andwith the probability of an absorption. Clearlydepends at least partly upon the energy supply, butis 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 theandlines, is the residual intensity at their centers, with respect to the adjacent continuous background.

Einstein’s theory of radiation[436]is used in evaluatingfrom the relationwhereis the ratio.

From ordinary quantum principles,and bothandmay be derived by eliminating between the two equations.

The only measured quantity in the formula is,and from the fact thatis the “residual intensity” within an absorption line, we know that it must lie between 0 and 1. Hence a maximum value ofmay be derived for.On the insertion of the data given by Schwarzschild[437]for the residual intensity of theandlines, 2.6 magnitudes fainter than the continuous background, and corresponding to a value ofequal to 0.11, the deduced value ofis.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]

FOOTNOTES:[429]H. H. Plaskett, Pub. Dom. Ap. Obs., 2, 325, 1922.[430]M. N. R. A. S., 84, 499, 1924.[431]H. C. 256, 1924.[432]H. C. 258, 1924.[433]Chapter IX,p. 133.[434]Milne, M. N. R. A. S., 84, 354, 1924.[435]Curtis and Burns, unpub.[436]Phys. Zeit., 18, 121, 1914.[437]Sitz. d. Preuss. Ac., 47, 1198, 1914.[438]Chapter I,p. 21.

[429]H. H. Plaskett, Pub. Dom. Ap. Obs., 2, 325, 1922.

[430]M. N. R. A. S., 84, 499, 1924.

[431]H. C. 256, 1924.

[432]H. C. 258, 1924.

[433]Chapter IX,p. 133.

[434]Milne, M. N. R. A. S., 84, 354, 1924.

[435]Curtis and Burns, unpub.

[436]Phys. Zeit., 18, 121, 1914.

[437]Sitz. d. Preuss. Ac., 47, 1198, 1914.

[438]Chapter I,p. 21.

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