Compton Scattering by Static and Moving Media I. The Transfer Equation and Its Moments

Compton scattering of photons by nonrelativistic particles is thought to play an important role in forming the radiation spectrum of many astrophysical systems. Here we derive the time-dependent photon kinetic equation that describes spontaneous and induced Compton scattering as well as absorption and emission by static and moving media, the corresponding radiative transfer equation, and their zeroth and first moments, in both the system frame and in the frame comoving with the medium. We show that it is necessary to use the correct relativistic differential scattering cross section in order to obtain a photon kinetic equation that is correct to first order in ε/me, Te/me, and V , where ε is the photon energy, Te and me are the electron temperature and rest mass, and V is the electron bulk velocity in units of the speed of light. We also demonstrate that the terms in the radiative transfer equation that are second-order in V usually should be retained, because if the radiation energy density is sufficiently large compared to the radiation flux, the effects of bulk Comptonization described by the terms that are second-order in V are at least as important as the effects described by the terms that are first-order in V , even when V is small. The systemand fluid-frame equations that we derive are correct to first order in ε/me. Our system-frame equations, which are correct to second order in V , may be used when V is not too large. Our fluid-frame equations, which are exact in V , may be used when V → 1. Both sets of equations are valid for systems of arbitrary optical depth and can therefore be used in both the free-streaming and the diffusion regimes. We demonstrate that Comptonization by the electron bulk motion occurs whether or not the radiation