Leading relativistic corrections to the Kompaneets equation

We calculate the first relativistic corrections to the Kompaneets equation for the evolution of the photon frequency distribution brought about by Compton scattering. The Lorentz invariant Boltzmann equation for electron–photon scattering is first specialized to isotropic electron and photon distributions, the squared scattering amplitude and the energy–momentum conserving delta function are each expanded to order v4/c4, averages over the directions of the electron and photon momenta are then carried out, and finally an integration over the photon energy yields our Fokker–Planck equation. The Kompaneets equation, which involves only first- and second-order derivatives with respect to the photon energy, results from the order v2/c2 terms, while the first relativistic corrections of order v4/c4 introduce third- and fourth-order derivatives. We emphasize that our result holds when neither the electrons nor the photons are in thermal equilibrium; two effective temperatures characterize a general, non-thermal electron distribution. When the electrons are in thermal equilibrium our relativistic Fokker–Planck equation is in complete agreement with the most recent published results, but we both disagree with older work.