Diffusive acceleration in relativistic shocks: particle feedback

TL;DR

This paper develops a computational method to analyze how feedback between particle distribution and diffusion affects the spectral index in relativistic shock acceleration, revealing modest variations unless diffusion is highly anisotropic.

Contribution

It introduces a relaxation code to compute the spectral index and distribution function for arbitrary diffusion functions depending on particle distribution, extending previous models.

Findings

Variations in spectral index are modest even with strong diffusion dependence.

Local diffusion dependence influences particle confinement and angular distribution.

A mild softening of spectral index may occur with rapidly rising diffusion functions.

Abstract

The spectral index $s$ of particles diffusively accelerated in a relativistic shock depends on the unknown angular diffusion function $\mathcal{D}$, which itself depends on the particle distribution function $f$ if acceleration is efficient. We develop a relaxation code to compute $s$ and $f$ for an arbitrary functional $\mathcal{D}$ that depends on $f$. A local $\mathcal{D}(f)$ dependence is motivated and shown, when rising (falling) upstream, to soften (harden) $s$ with respect to the isotropic case, shift the angular distribution towards upstream (downstream) directions, and strengthen (weaken) the particle confinement to the shock; an opposite effect on $s$ is found downstream. However, variations in $s$ remain modest even when $\mathcal{D}$ is a strong function of $f$, so the standard, isotropic-diffusion results remain approximately applicable unless $\mathcal{D}$ is both highly anisotropic and not a local function of $f$. A mild, $\sim 0.1$ softening of $s$, in both 2D and 3D, when $\mathcal{D}(f)$ rises sufficiently fast, may be indicated by ab-initio simulations.