Diffusive Shock Acceleration in Test-Particle Regime

TL;DR

This paper analyzes the test-particle solution for diffusive shock acceleration, focusing on injection rates, spectral slopes, and cutoff energies, with implications for cosmic ray pressure in various astrophysical environments.

Contribution

It provides a detailed examination of the test-particle regime, including conditions for its validity and modifications due to Alfvénic drift and exponential cutoff modeling.

Findings

Critical injection rate depends on Mach number and temperature.

Postshock CR pressure remains below 10% of shock ram pressure.

CR spectrum can be softer with an exponential cutoff at high energies.

Abstract

We examine the test-particle solution for diffusive shock acceleration, based on simple models for thermal leakage injection and Alfv'enic 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 < 10^{-3} for shocks with M < 5, while \xi_c ~ 10^{-4}(T_1/10^6 K)^{1/2} for shocks with M > 10. For T_1=10^6 K, for example, the test-particle solution would be valid if the injection momentum, p_{inj} > 3.8 p_{th}. This leads to the postshock CR pressure less than 10% of the shock ram pressure. If the Alfv'en 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.