Generalized Special Solutions to Modified Kompaneets Equation

摘要

In the stationary regime, configurations of corrected Kompaneets equation have been analyzed within a theoretically approach and the spectrum evolution for the scattered photons has been derived. A particular attention has been granted to the impact of the nonlinearity on the spectrum distorsions, mainly to the closed form special solutions expressed in terms of Heun functions. In the nonrelativistic limit, stationary solutions of the Kompaneets equation bring into attention the Heun double confluent functions. On the other side, when various type of corrections come into play, the photon distribution functions are governed by the Heun triconfluent functions. In magnetar's magnetosphere, where the resonant Compton scattering is more important than the Compton one, the photon number density can be written in terms of Hennite and confluent hypergeometric functions.