An eigenfunction method for the comptonisation problem. Angular   distribution and spectral index of radiation from a disk
  We present a semi-analytic approach to solving the Boltzmann equation describing the comptonisation of low frequency input photons by a thermal distribution of electrons in the Thomson limit. Our work is based on the formulation of the problem by Titarchuk & Lyubarskij (ApJ 450 (1995) 876), but extends their treatment by accommodating an arbitrary anisotropy of the source function. To achieve this, we expand the eigenfunctions of the integro/differential eigenvalue problem defining the spectral index of comptonised radiation in terms of Legendre polynomials and Chebyshev polynomials. The resulting algebraic eigenvalue problem is then solved by numerical means, yielding the spectral index and the full angular and spatial dependence of the specific intensity of radiation. For a thin (\tau_0 < 1) plasma disk, the radiation is strongly collimated along the disk surface - for an optical thickness of \tau_0 = 0.05, the radiation intensity along the surface is roughly ten times that along the direction of the normal, and varies only slightly with the electron temperature. Our results for the spectral index confirm those of Titarchuk & Lyubarskij over a wide range of electron temperature and optical depth; the largest difference we find is roughly 10% and occurs at low optical depth. 
U.D.J. Gieseler and J.G. Kirk, (Max-Planck-Institut f"ur Kernphysik,   Germany)