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The original Kompaneets equation fails to describe down-Comptonization, which is the most important radiative transfer process in hard X-ray and γ-ray astronomy, compared with up-Comptonization. In this paper, we improve our previous derivation of the extended Kompaneets equation and present it more clearly. The new equation can be used to describe a more general Comptonization process, including up- and down-Comptonization, suitable for any case, hv̄ ≪kTe, hv̄≫ kTe and hv̄ ∼ kTe. The condition of the original Kompaneets equation hv̄ ≪kTe, is no longer necessary. Using the extended equation, we give some typical solutions in X-ray astronomy, and compare them with those of the prevailing Monte Carlo simulations and the Ross-McCray equation. The excellent consistency between the extended Kompaneets equation and Monte Carlo simulation or Ross-McCray equation confirms the correctness of our extended Kompaneets equation. The numerical solution of the extended Kompaneets equation is less expensive in terms of computational time than the Monte Carlo simulation. Another advantage of the equation method is the simplicity and the clarity in physics. The potential applications in X-ray and γ-ray astronomy are also emphasized.

astronomy and astrophysics

Evolution of X-ray spectra in down-Comptonization. A comparison of the extended Kompaneets equation with Monte Carlo simulation and the Ross-McCray equation