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 elementin determining the spectrum of non--thermal particles accelerated via the Fermimechanism around a plane--parallel shock in the test--particle limit. Here anexplicit equation for this probability distribution is given, for both theupstream and downstream sections. Though analytically intractable, thisequation is solved numerically, allowing the determination of the spectrum infull generality, without limitation to shock speed or scattering properties. Anumber of cases is then computed, making contact with previous numerical work,in all regimes: Newtonian, trans--relativistic, and fully relativistic.Comment: 23 pages, 9 figures. To appear in The Astrophysical Journa