The profile of a narrow line after single scattering by maxwellian electrons: Relativistic corrections to the kernel of the integral kinetic equation

The photon frequency distribution that results from single Compton scattering of monochromatic radiation on thermal electrons is derived in the mildly relativistic limit. Algebraic expressions are given for (1) the photon redistribution function, K(nu, Omega --> nu', Omega'), and (2) the spectrum produced in the case of isotropic incident radiation, P(nu --> nu'). The former is a good approximation for electron temperatures kT(e) less than or similar to 25 keV and photon energies hv less than or similar to 50 keV, and the latter is applicable when hv(hv/m(e)c(2)) less than or similar to kT(e) less than or similar to 25 keV, hv less than or similar to 50 keV. Both formulae can be used for describing the profiles of X-ray and low-frequency lines upon scattering in hot, optically thin plasmas, such as present in clusters of galaxies, in the coronae of accretion disks in X-ray binaries and active galactic nuclei (AGNs), during supernova explosions, etc. Both formulae can also be employed as the kernels of the corresponding integral kinetic equations (direction-dependent and isotropic) in the general problem of Comptonization on thermal electrons. The K(nu, Omega --> nu', Omega') kernel, in particular, is applicable to the problem of induced Compton interaction of anisotropic low-frequency radiation of high brightness temperature with free electrons in the vicinity of powerful radio sources and masers. Fokker-Planck-type expansion (up to fourth order) of the integral kinetic equation with the P(nu --> nu') kernel derived here leads to a generalization of the Kompaneets equation. We also present (I) a simpler kernel that is necessary and sufficient to derive the Kompaneets equation and (2) an expression for the angular function for Compton scattering in a hot plasma, which includes temperature and photon energy corrections to the Rayleigh angular function.