Temperature Dependence of the Effective Compton Scattering Cross-Section in Hot Astrophysical Electron Gases

We present a detailed analytical and numerical study of the temperature dependence of the effective Compton scattering cross section in hot astrophysical electron gases. By averaging the Klein–Nishina cross section over a relativistic Maxwell–Jüttner distribution of electron energies and accounting for Lorentz transformations of photon energy, we derive an integral expression for the thermally averaged cross section. Our results demonstrate that the effective cross section converges to the classical Thomson limit at low temperatures but decreases significantly as the electron plasma becomes relativistic. This temperature-dependent suppression affects radiative transfer processes in high-energy astrophysical environments such as accretion flows, neutron star atmospheres, and gamma-ray bursts. Moreover, we identify a subtle enhancement of the effective cross section beyond the Thomson limit at low photon energies under certain thermal conditions, explained by relativistic Doppler boosting of photons in the electron rest frame. Our findings provide a robust theoretical framework for incorporating relativistic Compton scattering effects into astrophysical radiative transfer models, improving the accuracy of opacity and spectral predictions across diverse energetic regimes.