A comparison of the extended Kompaneets equations with the Ross-McCray equation

Abstract

The Kompaneets and the Ross-McCray equations can be used to describe the Comptonization process. However, the classical Kompaneets equation fails to describe down-Comptonization while Ross-McCray equation can be applied to describe down-Comptonization but inappropriate for a blackbody equilibrium. Fortunately, the Kompaneets equation extended by frequency and the Kompaneets equation extended by momentum can solve both problems. The different physical connotations behind the four equations bring about formal differences that allow the spectral evolution under different conditions to show different characteristics. In order to compare the differences between the four equations, the evolution of our common radiation spectra in astrophysics is numerically calculated in this paper. Besides, the differences between the four diffusion equations are discussed from their physical significance to the evolutionary phenomena. According to the results, the four equations evolve at different rates during down-Comptonization, from fast to slow in order of the Kompaneets equation extended by frequency, the Kompaneets equation extended by momentum, the classical Kompaneets equation and the Ross-McCray equation. The regression of the four equations is consistent during up-Comptonization. The Ross-McCray equation eventually converges to a different end state at near equilibrium. These results shed light on analytical development of Comptonization process.