Analytic study of 1D diffusive relativistic shock acceleration

TL;DR

This paper derives an analytical expression for the particle energy spectrum resulting from relativistic diffusive shock acceleration in one dimension, revealing its dependence on the equation of state and relativistic effects.

Contribution

It provides the first exact analytical solution for 1D relativistic DSA spectra, highlighting differences from higher dimensions and the influence of the equation of state.

Findings

The spectral index p is explicitly related to upstream and downstream velocities and Lorentz factors.

In 1D, the spectrum depends on the equation of state even at ultra-relativistic speeds.

The spectrum hardens with increasing Lorentz factor before converging to p=2.

Abstract

Diffusive shock acceleration (DSA) by relativistic shocks is thought to generate the $dN/dE\propto E^{-p}$ spectra of charged particles in various astronomical relativistic flows. We show that for test particles in one dimension (1D), $p^{-1}=1-\ln\left[\gamma_d(1+\beta_d)\right]/\ln\left[\gamma_u(1+\beta_u)\right]$, where $\beta_u$ ($\beta_d)$ is the upstream (downstream) normalized velocity, and $\gamma$ is the respective Lorentz factor. This analytically captures the main properties of relativistic DSA in higher dimensions, with no assumptions on the diffusion mechanism. Unlike 2D and 3D, here the spectrum is sensitive to the equation of state even in the ultra-relativistic limit, and (for a J{\"u}ttner-Synge equation of state) noticeably hardens with increasing $1<\gamma_u<57$, before logarithmically converging back to $p(\gamma_u\to\infty)=2$. The 1D spectrum is sensitive to drifts, but only in the downstream, and not in the ultra-relativistic limit.