Diffusive cosmic ray acceleration at relativistic shock waves with magnetostatic turbulence

TL;DR

This paper generalizes the theory of diffusive cosmic ray acceleration at shock waves to include relativistic speeds, deriving a steady-state solution that predicts particle spectra with a power-law tail and spectral indices depending on shock velocity.

Contribution

It extends the analytical framework for cosmic ray acceleration to relativistic shocks using diffusion approximation in a mixed coordinate system, providing new spectral index formulas.

Findings

Spectral index approaches 1 at ultrarelativistic speeds.

Distribution function is Lorentzian-type at injection.

Results are independent of microphysical scattering details for symmetric pitch-angle scattering.

Abstract

The analytical theory of diffusive cosmic ray acceleration at parallel stationary shock waves with magnetostatic turbulence is generalized to arbitrary shock speeds $V_s=\beta_1c$, including in particular relativistic speeds. This is achieved by applying the diffusion approximation to the relevant Fokker-Planck particle transport equation formulated in the mixed comoving coordinate system. In this coordinate system the particle's momentum coordinates $p$ and $\mu =p_{\parallel }/p$ are taken in the rest frame of the streaming plasma, whereas the time and space coordinates are taken in the observer's system. For magnetostatic slab turbulence the diffusion-convection transport equation for the isotropic (in the rest frame of the streaming plasma) part of the particle's phase space density is derived. For a step-wise shock velocity profile the steady-state diffusion-convection transport equation is solved. For a symmetric pitch-angle scattering Fokker-Planck coefficient $D_{\mu \mu }(-\mu )=D_{\mu \mu }(\mu )$ the steady-state solution is independent of the microphysical scattering details. For nonrelativistic mono-momentum particle injection at the shock the differential number density of accelerated particles is a Lorentzian-type distribution function which at large momenta approaches a power law distribution function $N(p\ge p_c)\propto p^{-\xi }$ with the spectral index $\xi (\beta_1) =1+[3/(\Gamma_1\sqrt{r^2-\beta_1^2}-1)(1+3\beta_1^2)]$. For nonrelativistic ($\beta_1\ll 1$) shock speeds this spectral index agrees with the known result $\xi (\beta_1\ll 1)\simeq (r+2)/(r-1)$, whereas for ultrarelativistic ($\Gamma_1\gg 1$) shock speeds the spectral index value is close to unity.