Exact solution for the Green’s function describing time-dependent thermal Comptonization

We obtain an exact, closed-form expression for the time-dependent Green’s function solution to the Kompaneets equation. The result, which is expressed as the integral of a product of two Whittaker functions, describes the evolution in energy space of a photon distribution that is initially monoenergetic. Effects of spatial transport within a homogeneous scattering cloud are also included within the formalism. The Kompaneets equation that we solve includes both the recoil and energy diffusion terms, and therefore our solution for the Green’s function approaches the Wien spectrum at large times. This was not the case with earlier analytical solutions that neglected the recoil term and were therefore applicable only in the soft-photon limit. We show that the Green’s function can be used to generate all of the previously known steady-state and time-dependent solutions to the Kompaneets equation. The new solution allows the direct determination of the spectrum, without the need to numerically solve the partial differential equation. It is therefore much more convenient for data analysis purposes. Based upon the Green’s function, we derive a new, exact solution for the variation of the inverse-Compton temperature of an initially monoenergetic photon distribution. Furthermore, we also obtain a new time-dependent solution for the photon distribution resulting from the reprocessing of an optically thin bremsstrahlung initial spectrum with a low-energy cutoff. Unlike the previously known solution for bremsstrahlung injection, the new solution possesses a finite photon number density, and therefore it displays proper equilibration to a Wien spectrum at large times. The relevance of our results for the interpretation of emission from variable X-ray sources is discussed, with particular attention to the production of hard X-ray time lags, and the Compton broadening of narrow features such as iron lines.