Analytical solutions for energy spectra of electrons accelerated by nonrelativistic shock-waves in shell type supernova remnants

Context. Recent observations of hard X-rays and very high energy gamma-rays from a number of young shell type supernova remnants indicate the importance of detailed quantitative studies of energy spectra of relativistic electrons formed via diffusive shock acceleration accompanied by intense nonthermal emission through synchrotron radiation and inverse Compton scattering. Aims. The aim of this work was derivation of exact asymptotic solutions of the kinetic equation which describes the energy distribution of shock-accelerated electrons for an arbitrary energy-dependence of the diffusion coefficient. Methods. The asymptotic solutions at low and very high energy domains coupled with numerical calculations in the intermediate energy range allow analytical presentations of energy spectra of electrons for the entire energy region. Results. Under the assumption that the energy losses of electrons are dominated by synchrotron cooling, we derived the exact asymptotic spectra of electrons without any restriction on the diffusion coefficient. We also obtained simple analytical approximations which describe, with accuracy better than ten percent, the energy spectra of nonthermal emission of shock-accelerated electrons due to the synchrotron radiation and inverse Compton scattering. Conclusions. The results can be applied for interpretation of X-ray and gamma-ray observations of shell type supernova remnants, as well as other nonthermal high energy source populations like microquasars and large scale synchrotron jets of active galactic nuclei.