# Non-linear diffusive shock acceleration with free escape boundary

## TL;DR

This paper introduces a semi-analytical method for modeling non-linear shock acceleration with a free escape boundary, enabling efficient predictions of particle spectra and escape fluxes relevant to cosmic ray origins.

## Contribution

A new semi-analytical solution for non-linear shock acceleration with free escape boundary, capturing particle spectra and escape fluxes efficiently.

## Key findings

- Accurately predicts particle spectra including cutoff shapes.
- Determines escape fluxes of particles from shocks.
- Provides a fast computational approach for shock structure analysis.

## Abstract

We present here a semi-analytical solution of the problem of particle acceleration at non-linear shock waves with a free escape boundary at some location upstream. This solution, besides allowing us to determine the spectrum of particles accelerated at the shock front, including the shape of the cutoff at some maximum momentum, also allows us to determine the spectrum of particles escaping the system from upstream. This latter aspect of the problem is crucial for establishing a connection between the accelerated particles in astrophysical sources, such as supernova remnants, and the cosmic rays observed at the Earth. An excellent approximate solution, which leads to a computationally fast calculation of the structure of shocks with an arbitrary level of cosmic ray modification, is also obtained.