Leading Relativistic Corrections to the Kompaneets Equation

Abstract: 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 v^/c^4, 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 v^2/c^2 terms, while the first relativistic corrections of order v^4/c^4 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.