Particle acceleration at shock waves: particle spectrum as a function of the equation of state of the shocked plasma

Abstract

We determine the spectrum of particles accelerated at shocks with arbitrary speed and arbitrary scattering properties for different choices of the equation of state of the downstream plasma. More specifically we consider the effect of energy exchange between the electron and proton thermal components downstream, and the effect of generation of a turbulent magnetic field in the downstream plasma. The slope of the spectrum turns out to be appreciably affected by all these phenomena, especially in the Newtonian and trans-relativistic regime, while in the ultra-relativistic limit the universal spectrum $s\approx 4.3$ seems to be a very solid prediction.