Anisotropic photon and electron scattering without ultrarelativistic approximation

Interactions between photons and electrons are ubiquitous in astrophysics. Photons can be down scattered (Compton scattering) or up scattered (inverse Compton scattering) by moving electrons. Inverse Compton scattering, in particular, is an essential process for the production of astrophysical gamma rays. Computations of inverse Compton emission typically adopts an isotropic or an ultrarelativistic assumption to simplify the calculation, which makes them unable to broadcast the formula to the whole phase space of source particles. In view of this, we develop a numerical scheme to compute the interactions between anisotropic photons and electrons without taking ultrarelativistic approximations. Compared to the ultrarelativistic limit, our exact results show major deviations when target photons are down scattered or when they possess energy comparable to source electrons. We also consider two test cases of high-energy inverse Compton emission to validate our results in the ultrarelativistic limit. In general, our formalism can be applied to cases of anisotropic electron-photon scattering in various energy regimes, and for computing the polarizations of the scattered photons.