Comptonization of X-rays by low-temperature electrons

We describe a method for calculating the spectrum that results from the Compton scattering of a monochromatic source of X-rays by low-temperature electrons, both for initial value relaxation problems and for steady-state spatial diffusion problems. The method gives an exact solution of the initial value problem for evolution of the spectrum in an infinite homogeneous medium if Klein-Nishina corrections to the Thomson cross section are neglected. This, together with approximate solutions for problems in which Klein-Nishina corrections are significant and/or spatial diffusion occurs, shows spectral structure near the original photon wavelength that may be used to infer physical conditions in cosmic X-ray sources. Explicit results, shown for examples of time relaxation in an infinite medium and spatial diffusion through a uniform sphere, are compared with results obtained by Monte Carlo calculations and by solving the appropriate Fokker-Planck equation