Saha’s discussion was based on the observation of “marginal appearance”—the spectral class at which a particular absorption line is at the limit of visibility. The use of this quantity as a criterion for the temperature scale has certain practical drawbacks. Marginal appearance depends directly on relative abundance, since a more abundant element will give visible lines at a lower “fractional concentration,” that is to say, when a smaller fraction of the element is contributing to the lines in question. Further, in estimating the intensities of lines in stellar spectra, difficulty is experienced when the lines are faint, and the spectral class at which they are first or last seen depends on their width and definition, the intensity [Pg 106] of the continuous background, the presence of other lines, and the dispersion used. All of these factors are subject to variation, and in particular the intensity distribution in the continuous background changes with the temperature. The statistical theory of Fowler and Milne has, therefore, a great advantage in that it leads to an estimate of the temperature at which a given line attains maximum. A maximum, unlike a marginal appearance, can be determined without ambiguity from homogeneous material, whatever the dispersion. In the cooler stars the estimates may be made difficult by blending, but the uncertainty can generally be removed by examining the maxima of several related lines. Of the observational factors enumerated above as affecting the estimation of marginal appearance, the changing intensity distribution of the continuous background with the temperature is the only one that may prove serious for the method of maxima. THEORETICAL FORMULAE The theory developed by Fowler and Milne has been exhaustively discussed by these authors in several papers, and it appears unnecessary to reproduce the analysis in detail. The ionization and excitation curves have been treated diagrammatically in the previous chapter. The detailed formulae follow. “If is the ionization potential of the atom, the (negative) energy of a given excited state, then for a given partial electron pressure of free electrons, the temperature of maximum concentration of atoms in the given excited state is given by[371] [Pg 107] “The temperature at which the concentration of singly ionized atoms reaches a maximum is given by[372] The two formulae that have been quoted assume, in effect, that at any stage of ionization the number of atoms, in stages other than those whose constants appear in the formulae, is negligible. When two ionizations occur in very close succession, this assumption no longer holds, and the equations, as modified to embody the necessary correction, are as follows.[373] For the subordinate series of a neutral atom, “This equation must be used to calculate whenever the ionization potential of the stage in question is closely followed by the ionization potential of the succeeding stage.”