Comptonization Models and Spectroscopy of X-Ray and Gamma-Ray Sources: A Combined Study by Monte Carlo and Analytical Methods

We compare the analytical generalized Comptonization models, which take into account the relativistic effects, with Monte Carlo calculations for photon Comptonization by relativistic plasma clouds. We show that the new analytical models extend the previous work to a much wider range of plasma temperatures and optical depths. In general, the emergent spectra from a hot plasma cloud depend upon the spectral and spatial distributions of source photons as well as the plasma temperature and geometry. Based on the comparison between the theoretical and Monte Carlo calculations, we determine quantitatively a range of plasma geometry parameters and temperatures for which the emergent upscattering spectra are insensitive to the spectral and spatial distribution, of source photons. Within this parameter range, we show that the shape of the emergent spectrum depends on two parameters only, namely the plasma temperature and β, the parameter which characterizes the photon distribution over the number of scatterings which the soft photons undergo before escaping from the plasma cloud. We find the exact solution of the Kompaneets equation for sub-relativistic energies and plasma temperatures in the optically thick regime. The solution covers both the low- and high-energy asymptotic forms studied by Titarchuk Also, we modify the Sunyaev & Titarchuk formula to obtain a fairly good sub-relativistic analytical approximation. This new formula, verified by Monte Carlo calculations, contains as partial cases the analytical results for optically thick and thin cases, respectively. Our Monte Carlo calculations and analytical models allow us to obtain the power-law indices of the photon spectra emergent from hot plasmas as a function of the plasma optical depth at various temperatures. The analytical models, examined by the present Monte Carlo calculations, make possible a more efficient spectral analysis of data obtained from X-ray and gamma-ray sources.