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

^{1}
Dipartimento di Fisica e Scienze della TerraUniversità di
Ferrara,
via Saragat 1,
44122
Ferrara,
Italy

e-mail:
chiara.ceccobello@gmail.com

^{2}
INFN, Sezione di Ferrara, via Saragat 1,
44122
Ferrara,
Italy

^{3}
ISDC Data Center for Astrophysics, Université de Genève ,
chemin d’Écogia 16,
1290
Versoix,
Switzerland

^{4}
NASA-GSFC, Greenbelt, MD
20771,
USA

Received:
13
September
2013

Accepted:
7
December
2013

*Context. *We consider the radiative transfer problem in a plane-parallel
slab of thermal electrons in the presence of an ultra-strong magnetic field
(*B* ≳ *B*_{c} ≈ 4.4 × 10^{13}
G). Under these conditions, the magnetic field behaves like a birefringent medium for the
propagating photons, and the electromagnetic radiation is split into two polarization
modes, ordinary and extraordinary, that have different cross-sections. When the optical
depth of the slab is large, the ordinary-mode photons are strongly Comptonized and the
photon field is dominated by an isotropic component.

*Aims. *The radiative transfer problem in strong magnetic fields presents
many mathematical issues and analytical or numerical solutions can be obtained only under
some given approximations. We investigate this problem both from the analytical and
numerical point of view, provide a test of the previous analytical estimates, and extend
these results with numerical techniques.

*Methods. *We consider here the case of low temperature black-body photons
propagating in a sub-relativistic temperature plasma, which allows us to deal with a
semi-Fokker-Planck approximation of the radiative transfer equation. The problem can then
be treated with the variable separation method, and we use a numerical technique to find
solutions to the eigenvalue problem in the case of a singular kernel of the space
operator. The singularity of the space kernel is the result of the strong angular
dependence of the electron cross-section in the presence of a strong magnetic field.

*Results. *We provide the numerical solution obtained for eigenvalues and
eigenfunctions of the space operator, and the emerging Comptonization spectrum of the
ordinary-mode photons for any eigenvalue of the space equation and for energies
significantly lesser than the cyclotron energy, which is on the order of MeV for the
intensity of the magnetic field here considered.

*Conclusions. *We derived the specific intensity of the ordinary photons,
under the approximation of large angle and large optical depth. These assumptions allow
the equation to be treated using a diffusion-like approximation.

Key words: acceleration of particles / magnetic fields / radiative transfer / methods: numerical / X-rays: general / stars: magnetars

*© ESO, 2014*