Hi Andy and thanks for your encouraging post.
When I started this project (about 10 years ago) I tried to deduce the photon capture cross-sections purely from the idea that they were proportional to capture rates. The BAA review of my original paper highlighted the fact that the Einstein coefficients needed to be included. This sent me on a bit of a lengthy “wild goose chase” as, although I eventually realised the B coefficients could be expressed in units of area (the same units as cross-sections), the variation of the B’s with wavelength was far to strong to reproduce my experimental results for absorption line depths. I only had the Planck and Boltzmann functions plus the B coefficients and various physical constants to play with and I just could not reproduce the Balmer series relative absorption line depths across the various stars I had modelled.
In desperation I returned to my original approach as represented by equation A.4.10 (in my previously attached Europa paper) and deduced equation A.4.15 using my, now fairly detailed, knowledge of Einstein Coefficients. Equation A.4.15 contains Einstein A coefficients which vary much more slowly with wavelength than the B’s and they appear in a summation so are really only converting the expression to the correct units, the variation of relative line depths in the spectral series is being determined by the Boltzmann and Planck functions. This form of cross-section performed well across the stars I had measured.
I would like to know how my analysis compares to the conventional approach using “oscillator strengths and B coefficients but have not managed to find such n analysis in the literature can anyone point me in the right direction?