On Particle Acceleration around Shocks. I.

Abstract

We derive a relativistically covariant (although not manifestly so)equation for the distribution function of particles accelerated atshocks, which applies also to extremely relativistic shocks andarbitrarily anisotropic particle distributions. The theory is formulatedfor arbitrary pitch-angle scattering and reduces to the well-known casefor small-angle scatterings via a Fokker-Planck approximation. Theboundary conditions for the problem are completely reformulatedintroducing a physically motivated Green's function; the new formulationallows derivation of the particle spectrum both close and far away fromthe injection energy in an exact way, while it can be shown to reduce toa power law at large particle energies. The particle spectral index isalso recovered in a novel way. Contact is made with the Newtoniantreatment.