Self-consistent computation of gamma-ray spectra due to proton-proton interactions in black hole systems

In the inner regions of an accretion disk around a black hole, relativistic protons can interact with ambient matter to produce electrons, positrons and $\gamma$-rays. The resultant steady state electron and positron particle distributions are self-consistently computed taking into account Coulomb and Compton cooling, $e^-e^+$ pair production (due to $\gamma-\gamma$ annihilation) and pair annihilation. While earlier works used the diffusion approximation to obtain the particle distributions, here we solve a more general integro-differential equation that correctly takes into account the large change in particle energy that occur when the leptons Compton scatter off hard X-rays. Thus this formalism can also be applied to the hard state of black hole systems, where the dominant ambient photons are hard X-rays. The corresponding photon energy spectrum is calculated and compared with broadband data of black hole binaries in different spectral states. The results indicate that the $\gamma$-ray spectra ($E > 0.8$ MeV) of both the soft and hard spectral states and the entire hard X-ray/$\gamma$-ray spectrum of the ultra-soft state, could be due to $p-p$ interactions. These results are consistent with the hypothesis that there always exists in these systems a $\gamma$-ray spectral component due to $p-p$ interactions which can contribute between 0.5 to 10% of the total bolometric luminosty. The model predicts that {\it GLAST} would be able to detect black hole binaries and provide evidence for the presence of non-thermal protons which in turn would give insight into the energy dissipation process and jet formation in these systems.