DIFFUSIVE SHOCK ACCELERATION IN TEST-PARTICLE REGIME

We examine the test-particle solution for diffusive shock acceleration, based on simple models for thermal leakage injection and Alfvenic drift. The critical injection rate, xi(c), above which the cosmic-ray (CR) pressure becomes dynamically significant depends mainly on the sonic shock Mach number, M, and preshock gas temperature, T-1. In the hot-phase interstellar medium (ISM) and intracluster medium, xi(c) less than or similar to 10(-3) for shocks with M less than or similar to 5, while xi(c) approximate to 10(-4)(T-1/10(6) K)(1/2) for shocks with M greater than or similar to 10. For T-1 = 10(6) K, for example, the test-particle solution would be valid if the injection momentum p(inj) >3.8p(th) (where p(th) is thermal momentum). This leads to a postshock CR pressure less than 10% of the shock ram pressure. If the Alfven 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, p(max)(t). An analytic form of the exponential cutoff is also suggested.