Empirical relations in thermal Comptonization

In many astrophysical systems, photons interact with matter through thermal Comptonization. In these cases, under certain simplifying assumptions, the evolution of the photon spectrum is described by an energy diffusion equation such as the Kompaneets equation, having dependencies on the seed photon temperature, θ ᵢ, the electron temperature, θ ₑ, and the Compton y-parameter. The resulting steady-state spectrum is characterized by the average photon energy and the Compton temperature, which both lack analytical dependencies on the initial parameters. Here, we present empirical relations of these two quantities as functions of θ ᵢ, θ ₑ, and y, obtained by evaluating the steady-state solution of the Kompaneets equation accounting for energy diffusion and electron recoil. The relations have average fractional errors ~ 1~\ per\ cent across a wide range of the initial parameters, which make them useful in numerical applications.