Unsaturated Comptonization of isotropic photon spectra by relativistic electrons

The multiple scattering approach for evaluating the transformation of an arbitrary soft photon spectrum as a result of unsaturated Compton scatterings in a medium of relativistic electrons is explored. The medium is assumed to be infinite and spatially homogeneous but may be time-dependent, and the photons are isotropic. It is shown that the distortion of a radiation spectrum can be described analytically in a compact form using the Fourier transform of the single-scattering probability. In the nonrelativistic case, the validity of the known analytical results derived from the Kompaneets (1957) equation is extended to arbitrary electron distribution functions and photon spectra. For relativistic electrons, simple expressions are obtained for the total energy that is transferred from the electrons to the photons and for the distortion in the Rayleigh-Jeans regime of a blackbody spectrum. It is demonstrated that the treatment applies to Comptonization in a relativistic jet and that Comptonization of very soft trapped photons by semirelativistic electrons in an expanding medium leads naturally to a log Gaussian spectrum, of the form observed in blazars.