DIFFUSIVE COSMIC RAY ACCELERATION AT RELATIVISTIC SHOCK WAVES WITH MAGNETOSTATIC TURBULENCE. II. INFLUENCE OF A FINITE DOWNSTREAM MEDIUM

Abstract

The diffusive acceleration of relativistic cosmic rays at parallel shock waves with magnetostatic turbulence and a finite size of the downstream medium is investigated. For ultrarelativistic shock speeds with Lorentz factor , both the differential momentum spectrum at the shock and the volume-integrated momentum spectrum are power-law distribution functions with different spectral indices as compared to the case of an infinitely extended downstream medium. However, the spectral differences are only modest as compared to the case of nonrelativistic shocks. The behavior of the momentum spectrum of shock-accelerated particles depends sensitively on the relativistic shock wave Peclet number , i.e., the ratio between the diffusion and convection timescales of cosmic rays to propagate from the shock position to the downstream boundary z 0 . For large values of the free-escape boundary has no influence on the effectiveness of particle acceleration, still providing a flat momentum power-law spectrum of the accelerated particles. In the opposite case of small Peclet numbers at all momenta, the momentum spectrum at the shock steepens to the greater spectral index , whereas the volume-integrated momentum spectrum flattens by the same factor for its power-law spectral index, where s denotes the spectral index of the downstream power spectrum of magnetostatic turbulence. This effectiveness of relativistic shocks in generating flat power-law momentum spectra irrespective of the Peclet number differs completely from the behavior of nonrelativistic shocks.