Modified form of the Kompaneets equation

Possible application to the distortion of high-temperature blackbody and CMB spectra^⋆

¹ Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, PR China
e-mail: xchen@impcas.ac.cn
² University of Chinese Academy of Sciences, Beijing 100049, PR China
³ Zhiyuan College, Shanghai Jiao Tong University, 800 Dongchuan Road, 200240 Shanghai, PR China
⁴ Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan Road, 200240 Shanghai, PR China
e-mail: dbliu@sjtu.edu.cn
⁵ Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University, 800 Dongchuan Road, 200240 Shanghai, PR China

Received: 15 August 2019
Accepted: 26 May 2020

Abstract

Comptonization is a very important phenomenon in astrophysics. The Kompaneets equation describes the Comptonization process in up-Comptonization (hν̄ ≪ kT_e), while it fails in describing the down-Comptonization (hν̄ ≫ kT_e), 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 ≪ kT_e, hν̄ ≫ kT_e, and hν̄ ∼ kT_e. 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.