Cosmic ray acceleration at weak cosmological shocks in test-particle regime

Abstract

We examine the acceleration of cosmic-rays (CRs) via diffusive shock acceleration (DSA) at weak shocks in the test-particle regime. Assuming simple models for thermal leakage injection and Alfvenic drift, we first derive analytic solutions for accelerated particle spectra from both injected and pre-existing populations. We then compare these analytic solutions with the results from DSA simulatios. If the Alfven speed is comparable to the sound speed in the preshock flow, the power-law slope of CR spectrum can be significantly softer than the canonical test-particle slope, leading to lower injection rate and softer CR spectrum. Re-acceleration of pre-existing CRs dominates over the acceleration of injected particles at weak shocks, because thermal leakage injection becomes inefficient at low Mach number shocks in the adopted model. If the CR pressure in contributes to about 5 % of the thermal pressure in the upstream flow, the postshock CR pressure can increase to a few % of the shock ram pressure at shocks with M <∼ 3, while it can reach up to 10 % for M ∼ 4 − 5