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We consider the radiative transfer problem in a plane-parallel slab ofthermal electrons in the presence of an overcritical magnetic field. Under suchconditions, the magnetic field behaves as a birefringent medium for thepropagating photons, and the electromagnetic radiation is split into twopolarization modes, ordinary and extraordinary, having differentcross-sections. When the optical depth of the slab is large, the ordinary-modephotons are strongly Comptonized and the photon field is dominated by anisotropic component. The radiative transfer problem in strong magnetic fieldspresents many mathematical issues and analytical or numerical solutions can beobtained only under some given approximations. We investigate this problem bothfrom the analytical and numerical point of view, providing a test of theprevious analytical estimates and extending these results introducing numericaltechniques. We consider the case of low temperature blackbody photonspropagating in a sub-relativistic temperature plasma, which allows us to dealwith a semi Fokker-Planck approximation of the radiative transfer equation. Wereport the numerical solution obtained for eigenvalues and eigenfunctions ofthe space operator, and the emerging Comptonization spectrum of theordinary-mode photons for any eigenvalue of the space equation and for energiessignificantly less than the cyclotron energy, which is of the order of MeV forthe intensity of the magnetic field here considered. We derived the specificintensity of the ordinary photons, under the approximation of large angle andlarge optical depth. These assumptions allow the equation to be treated using adiffusion-like approximation.

Comptonization in ultra-strong magnetic fields: numerical solution of the radiative transfer problem