Diffusive shock-acceleration: breakdown of spatial diffusion and isotropy

TL;DR

This paper challenges the traditional understanding of particle acceleration spectra at shocks, showing that anisotropic scattering can cause significant deviations from the standard spectral index, with implications for astrophysical shock models.

Contribution

It demonstrates that the standard spectral index derivation relies on unjustified assumptions and shows how anisotropic scattering alters the expected particle spectra.

Findings

Anisotropic scattering can significantly deviate the spectral index from the standard value.

The standard spectral index derivation is valid only in isotropic scattering conditions.

Spectral modifications can occur due to scattering mode motions at intermediate optical depths.

Abstract

We point out that particles accelerated in a non-relativistic shock of compression ratio $r$ attain the standard, $p=(r+2)/(r-1)$ spectral index only under certain conditions. Previous derivations of the spectrum, based on the approximations of spatial diffusion or negligible anisotropy, are shown to be unjustified for a general scattering function. We explain and demonstrate numerically that in contrast to previous claims, $p$ can substantially deviate from the standard result for anisotropic scattering. We prove analytically that the standard approach is nevertheless valid in the limit of an isotropic medium. Additional spectral modifications, for example by motions of scattering modes at intermediate optical depths from the shock, are discussed.