Comptonization of an isotropic distribution in moving media: higher-order effects

We consider the Comptonization of an isotropic radiation field by a thermal distribution of electrons with non-vanishing bulk velocity. We include all relativistic effects, including induced scattering and electron recoil, in the derivation of a kinetic equation which is correct to O(theta^2, beta theta^2, beta^2 theta), where beta is the bulk velocity (in units of c) and theta is the ratio of the electron temperature to mass. The result given here manifestly conserves photon number, and easily yields the energy transfer rate between the radiation and electrons. We also confirm recent calculations of the relativistic corrections to the thermal and kinematic Sunyaev-Zel'dovich effect.