Modified form of the Kompaneets equation

Abstract

Comptonization is a very important phenomenon in astrophysics. The Kompaneets equation describes the Comptonization process in up-Comptonization (hν̄ ≪ kTe), while it fails in describing the down-Comptonization (hν̄ ≫ kTe), which is the most important radiative transfer process in hard X-ray and γ-ray astronomy. In this study we extend the Kompaneets equation to a new modified equation which works in a more general Comptonization process, including up- and down-Comptonization, suitable for the cases hν̄ ≪ kTe, hν̄ ≫ kTe, and hν̄ ∼ kTe. Numerical solutions for the evolution behavior of Gaussian emission lines show excellent agreement between the classical equations and our new equation for up-Comptonization, while the big differences are displayed for down-Comptonization. Based on this extended equation, the modified Sunyaev-Zel’dovich effect is investigated. Instead, some typical calculated results for up- and down-Comptonization in X-ray astronomy are compared between the new equation and the Kompaneets equation. The potential applications of the extended equation in astrophysics are also highly emphasized for further study.