On particle acceleration around shocks II: a fully general method for arbitrary shock velocities and scattering media

The probability that a particle, crossing the shock along a given direction, be reflected backwards along another direction, was shown to be the key element in determining the spectrum of non--thermal particles accelerated via the Fermi mechanism around a plane--parallel shock in the test--particle limit. Here an explicit equation for this probability distribution is given, for both the upstream and downstream sections. Though analytically intractable, this equation is solved numerically, allowing the determination of the spectrum in full generality, without limitation to shock speed or scattering properties. A number of cases is then computed, making contact with previous numerical work, in all regimes: Newtonian, trans--relativistic, and fully relativistic.