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 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.
