Non-thermal pair production in compact X-ray sources: first-order Compton cascades in soft radiation fields

Non-thermal pair cascades, where injected relativistic electrons (or pairs) cool by inverse Compton scattering on soft photons in the Thomson limit and where the resulting hard photons pair produce on other up-scattered photons, are described by an integral equation. A general integral equation for all relevant optical depths is obtained by using a simple prescription for the radiative transfer effects due to photon–photon absorption and to Compton scattering by the cooled pairs. The integral equation is solved analytically in the limit of large injection compactness L_i/R (L_i is the injected particle power and R the size of the injection region), where every γ-ray photon produces a pair (saturated pair production). Each photon generation is described separately and has a well-determined spectral index at small photon energies, being α=1/2 and 3/4 for the first and second generation, respectively. Analytical and numerical solutions of the general integral equation show how reprocessing by the pair cascade becomes increasingly important with increasing injection compactness and how the saturated limit is approached. The efficiency of converting injected power into pair rest mass can reach levels of about 10 per cent. Solutions close to the Klein–Nishina limit show extreme sharp transitions to pair dominated states as L_i/R increases past $$2\times {10}^{29}\,\text{erg}\,\text{s}^{-1}\,\text{cm}^{-1}$$. A parameter space study shows that the observed ‘universal’ X-ray index, 0.7, and the observed (almost universal) 2–10 ke V compactness in Seyfert galaxies are obtained for unsaturated cascades with a pair Thomson depth of a few tenths. Radiation pressure effects may be responsible in limiting the Thomson depth of pair cascade clouds to be less than unity. Feedback mechanisms causing rapid variability are briefly discussed.