Extend the Kompaneets Equation to Down-Comptonization Process in a Novel Way

Comptonization is a very important phenomenon in astrophysics. Kompaneets equation describes up-Comptonization process in nonrelativistic astrophysics, while it fails for down-Comptonization Scattering which is the most important radiative transfer process in hard X-rays and $\gamma$-rays astronomy. In this study, we explore both up-Comptonization and down-Comptonization processes. A new relativistic corrections is introduced to the Kompaneets equation which is valid in both nonrelativistic energy regime with the photon energy $h\nu << m_ec^2$ and the electron temperature $KT_e << m_ec^2$. Numerical comparisons are presented which show excellent agreement between the Kompaneets equation and the new equation.