Diffusive shock acceleration in test-particle regime

Abstract

We examine the test-particle solution for diffusive shock acceleration, based on simple models for thermal leakage injection and Alfvénic drift. The critical injection rate, ζc, above which the cosmic-ray (CR) pressure becomes dynamically significant depends mainly on the sonic shock Mach number, M, and preshock gas temperature, T1. In the hot-phase interstellar medium (ISM) and intracluster medium, ζc ≲ 10-3 for shocks with M ≲ 5, while ζc ≈ 10-4(T1/106 K)1/2 for shocks with M ≳ 10. For T1 = 106 K, for example, the test-particle solution would be valid if the injection momentum pinj >3.8p th (where pth is thermal momentum). This leads to a postshock CR pressure less than 10% of the shock ram pressure. If the Alfvén speed is comparable to the sound speed in the preshock flow, as in the hot-phase ISM, the power-law slope of CR spectrum can be significantly softer than the canonical test-particle slope. Then, the CR spectrum at the shock can be approximated by the revised test-particle power law with an exponential cutoff at the highest accelerated momentum, pmax(t ). An analytic form of the exponential cutoff is also suggested. © 2010. The American Astronomical Society. All rights reserved.