On the escape of particles from cosmic ray modified shocks

TL;DR

This paper investigates how particles escape from cosmic ray modified shocks during supernova remnant expansion, affecting cosmic ray spectra observed at Earth, by analyzing non-linear shock acceleration models including magnetic field effects.

Contribution

It provides a self-consistent analysis of particle escape flux in non-linear diffusive shock acceleration during supernova remnant evolution, linking theoretical models to cosmic ray observations.

Findings

Escape flux increases as maximum particle energy decreases over time.

The model predicts the evolution of maximum energy and escape flux during the Sedov-Taylor phase.

Results connect shock acceleration processes to the cosmic ray spectrum at Earth.

Abstract

Stationary solutions to the problem of particle acceleration at shock waves in the non-linear regime, when the dynamical reaction of the accelerated particles on the shock cannot be neglected, are known to show a prominent energy flux escaping from the shock towards upstream infinity. On physical grounds, the escape of particles from the upstream region of a shock has to be expected in all those situations in which the maximum momentum of accelerated particles, $p_{max}$, decreases with time, as is the case for the Sedov-Taylor phase of expansion of a shell Supernova Remnant, when both the shock velocity and the cosmic ray induced magnetization decrease. In this situation, at each time $t$, particles with momenta larger than $p_{max}(t)$ leave the system from upstream, carrying away a large fraction of the energy if the shock is strongly modified by the presence of cosmic rays. This phenomenon is of crucial importance for explaining the cosmic ray spectrum detected at Earth. In this paper we discuss how this escape flux appears in the different approaches to non-linear diffusive shock acceleration, and especially in the quasi-stationary semi-analytical kinetic ones. We apply our calculations to the Sedov-Taylor phase of a typical supernova remnant, including in a self-consistent way particle acceleration, magnetic field amplification and the dynamical reaction on the shock structure of both particles and fields. Within this framework we calculate the temporal evolution of the maximum energy reached by the accelerated particles and of the escape flux towards upstream infinity. The latter quantity is directly related to the cosmic ray spectrum detected at Earth.