Power-law spectra as a result of comptonization of the soft radiation in a plasma cloud

We show that power-law spectra are the exact solutions of the radiative kinetic equation with an assumption of low-frequency photon injection for a wide range of the electron distributions. We derive the dispersion equation for the determination of the power-law index. The whole problem of the determination of power-law spectra is separable. The radiative transfer in the photon energy space and in the configuration space can be considered independently. This is a result of there being no preferred energy scale for the radiation in this Comptonization problem. We confirm Sunyaev & Titarchuk's (1985) result that the angular distribution of the hard radiation depends only on the plasma cloud optical depth and it does not depend on photon energy. Also, the power index for the monoenergetic electron distribution is derived, allowing the appropriate indices to be determined for a wide range of electron distributions. As an example we give the power-law indices as a function of the plasma optical depth and the plasma temperature for the Maxwellian distribution. The exact analytical solutions of this equation are given without any limitation on the optical depth and plasma temperature for two geometries of plasma cloud: disk and spherical. The analytical models, verified by Monte Carlo calculations, make possible a more efficient spectral analysis of data obtained from X-ray and gamma-ray sources.