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We study the generalized Kompaneets equation (the kinetic equation for the photon distribution function taking into account Compton scattering by electrons) using a relativistically covariant formalism. Using the generalized Kompaneets equation, we derive an analytic expression for the Sunyaev-Zeldovich e ect, which takes into account up to terms, where is the relativistic expansion O(h e 5) h e \ k B T e /mc2 parameter and is the electron temperature. We carefully study the applicable region of the obtained T e analytic expression by comparing it with the result of the direct numerical integration. We conclude that the present analytic expression can be reliably applied to the calculation of the Sunyaev-Zeldovich e ect for keV, which is the highest electron temperature in the clusters of galaxies presently known. k B T e 15 Therefore, the present analytic expression can be applied to all known clusters of galaxies.
Itoh et al. (Mon,) studied this question.