A New Kinetic Equation for Compton Scattering
  A kinetic equation for Compton scattering is given that differs from the Kompaneets equation in several significant ways. By using an inverse differential operator this equation allows treatment of problems for which the radiation field varies rapidly on the scale of the width of the Compton kernel. This inverse operator method describes, among other effects, the thermal Doppler broadening of spectral lines and continuum edges, and automatically incorporates the process of Compton heating/cooling. It is well adapted for inclusion into a numerical iterative solution of radiative transfer problems. The equivalent kernel of the new method is shown to be a positive function and with reasonable accuracy near the intitial frequency, unlike the Kompaneets kernel, which is singular and not wholly positive. It is shown that iterates of the inverse operator kernel can be easily calculated numerically, and a simple summation formula over these iterates is derived that can be efficiently used to compute Comptonized spectra. It is shown that the new method can be used for initial value and other problems with no more numerical effort than the Kompaneets equation, and that it more correctly describes the solution over times comparable to the mean scattering time. 
George B. Rybicki