Energy transfer by the scattering of resonant photons

A formal derivation is presented of the energy transfer rate between radiation and matter due to the scattering of an isotropic distribution of resonance line photons. The derivation is developed in the context of the two-level atom in the absence of collisions and radiative transitions to and from the continuum, but includes the full angle-averaged redistribution function for photon scattering. The result is compared with previous derivations, all of which have been based on a Fokker–Planck approximation (FPA) to the radiative transfer equation. A new FPA, including a Kramers–Moyal extension to higher (post-diffusive) orders, is derived to solve the radiative transfer equation, and time-dependent numerical solutions are found. The relaxation of the colour temperature to the matter temperature is computed as the radiation field approaches statistical equilibrium through scattering. The results are discussed in the context of the Wouthuysen–Field mechanism for coupling the 21-cm spin temperature of neutral hydrogen to the kinetic temperature of the gas through Lyα scattering. The evolution of the heating rate is also computed, and shown to diminish as the gas approaches statistical equilibrium.