The double Compton emissivity in a mildly relativistic thermal plasma within the soft photon limit

Aims. We wish to provide simple and accurate analytic approximations for the low frequency double Compton (DC) emission coefficient , which are applicable in a broad range of physical situations up to mildly relativistic temperatures and may be useful for checking under which circumstances the double Compton process is important. Methods. We perform series expansions of the DC emission integral for low energies of the incident photon and electron and compare the derived analytic expressions with the results obtained by numerical integrations of the full DC cross section. Results. We explicitly derived analytic approximations to the low frequency double Compton emission coefficient for initial monochromatic photons and Wien spectra. We show that combining the analytic approximations given in this paper an accuracy of better than 5 % in a very broad range of temperatures and under various physical conditions can be achieved. Furthermore we show that the double Compton emissivity strongly depends on the ratio of the energies of the incoming photon and electron: for hard photons and cold electrons the emission is strongly suppressed as compared to the case of similar photon and electron energy, whereas in the opposite situation, i.e. hot electrons and soft initial photons, the emission is enhanced. For photons and electrons close to ther-modynamic equilibrium the double Compton emissivity increases less rapidly with temperature Lightman-Thorne approximation and the corrections exceed the level of ∼ 10% for temperatures above 4 keV.