Radiative Processes in Astronomy: L13a - Saha and Boltzmann relations

This lecture, part of a course on radiative processes in astronomy, introduces the Saha and Boltzmann relations for calculating excitation and ionization balance in local thermodynamic equilibrium (LTE). The lecturer begins by explaining the complexity of computing occupation numbers in general non-LTE conditions, where radiative and collisional transitions couple many states. He then defines LTE as the regime where the photon mean free path is much smaller than the scale of variation of thermodynamic variables, allowing the use of equilibrium distributions. The Boltzmann excitation formula is derived, showing the population ratio of excited to ground states depends on degeneracies and the exponential of the excitation energy over kT. A worked example calculates that hydrogen’s first excited state would equal the ground state population at 85,000 K if only excitation were considered, but ionization dominates at lower temperatures. The Saha ionization formula is then introduced, incorporating the electron density and a factor from the Maxwell distribution, explaining why ionization occurs at lower temperatures than the Boltzmann exponential alone would suggest. The lecture highlights the Balmer jump as an application, peaking around 10,000 K in A-type stars, and concludes with a preview of using these relations to understand cosmic recombination.