Non-thermal bremsstrahlung from supernova remnants and the effect of Coulomb losses

I investigate the shape of the electron cosmic ray spectrum in the range up to ~1000 keV, assuming that the acceleration process at the shock results in a power law in momentum, and that downstream of the shock the spectrum is affected by Coulomb interactions with background electrons only. In the non-relativistic regime one can analytically determine the energy of an electron starting with a certain energy, and use this result to produce an electron cosmic ray spectrum, modified by Coulomb losses. An analytic expression for the electron spectrum is obtained that depends on the parameter n_e t, which can be estimated from a similar parameter used to characterize the line spectra of supernova remnants. For the brightest supernova remnants n_e t > 1e11 cm^-3 s,and most of the electrons accelerated to < 100 keV have lost their energy. Because of its high radio flux, Cas A is the most likely candidate for non-thermal bremsstrahlung. Although it has n_e t ~ 2e11 cm^-3 s, one may expect to pick up non-thermal bremsstrahlung above 100 keV with current hard X-ray detectors.