Cosmic ray spectrum from diffusive shock acceleration

Abstract

It is now well established that cosmic rays (CRs) are accelerated at collisionless shocks through diffusive shock acceleration. However, some key physical processes, such as thermal leakage injection, self-excitation and dissipation of waves, and resonant scatterings of particles by those waves are nonlinear and not fully understood yet. Hence it is not possible to make precise quantitative predictions for the particle spectrum accelerated at shocks from first principles. If the fraction of particles injected into the CR population is smaller than 10(-4), the CR acceleration efficiency is low and so the test-particle solutions are justified. At moderately strong shocks (M (0)a parts per thousand(3)5) with higher injection fractions, the shock structure is significantly modified by nonlinear feedback of CRs. According to time-dependent kinetic simulations of CR modified shocks, the precursor and subshock transition approach a time-asymptotic state, and then evolve in an approximately self-similar fashion, depending only on the similarity variable, x/(u (s) t). During this self-similar stage, the CR spectrum at the subshock maintains a characteristic form as it evolves: the sum of two power-laws with the slopes determined by the subshock and total compression ratios, along with an exponential cutoff at the highest accelerated momentum