The two-phase pair corona model for active galactic nuclei and X-ray binaries: How to obtain exact solutions

We consider two-phase accretion disk-corona models for active galactic nuclei and some X-ray binaries. We describe in detail how one can exactly solve the polarized radiative transfer and Comptonization by using the iterative scattering method while simultaneously solving the energy and pair balance equation for both the cold and hot phases. We take into account Compton scattering, photon-photon pair production, pair annihilation, bremsstrahlung, and double Compton scattering, as well as exact reflection from the cold disk. We consider coronae with slab geometry, as well as coronae consisting of one or more well-separated active regions of cylindrical or hemispheric geometry. The method is useful for determining the spectral intensity and the polarization emerging in different directions from disk-corona systems. The code is tested against a Monte Carlo code. We also compare with earlier, less accurate, work. The method is more than an order of magnitude faster than applying Monte Carlo methods to the same problem and has the potential of being used in spectral fitting software, such as XSPEC.