An Analytic Approximation for Plane Parallel Compton Scattering Near Accretion Disks

Compton scattering by a locally plane parallel atmosphere is analyzed within the Kompaneets approximation. In contrast to almost all previous analytic calculations of inverse Compton scattering, we assume neither homogeneity nor isotropy of the photon distribution. However, we do adopt an assumed angular distribution of the photons. For any specified energy dependence of the flux entering the base of the atmosphere, an analytic expression is obtained for the flux at any optical depth. Only integrals over the input flux are involved. If the thickness of the atmosphere is much less than its radial position on an accretion disk, the contribution from all radii may be summed. The initial motivation (addressed in a subsequent paper) is the computation of the energy spectrum of (possibly modulated) photons originating from a specified radial region of an accretion disk, relative to the total energy spectrum.