Steady-State Solutions in Nonlinear Diffusive Shock Acceleration

TL;DR

This paper investigates stationary solutions in nonlinear diffusive shock acceleration, emphasizing particle escape mechanisms and their impact on cosmic-ray acceleration, with implications for supernova remnants.

Contribution

It introduces an alternative spatial boundary approach for particle escape and constructs stationary solutions using an iterative numerical scheme.

Findings

Stationary solutions with efficient acceleration are found at the balance point of wave growth and advection.

The energy distribution near the cutoff provides observable diagnostics.

The approach is applicable to supernova remnant conditions.

Abstract

Stationary solutions to the equations of non-linear diffusive shock acceleration play a fundamental role in the theory of cosmic-ray acceleration. Their existence usually requires that a fraction of the accelerated particles be allowed to escape from the system. Because the scattering mean-free-path is thought to be an increasing function of energy, this condition is conventionally implemented as an upper cut-off in energy space -- particles are then permitted to escape from any part of the system, once their energy exceeds this limit. However, because accelerated particles are responsible for substantial amplification of the ambient magnetic field in a region upstream of the shock front, we examine an alternative approach in which particles escape over a spatial boundary. We use a simple iterative scheme that constructs stationary numerical solutions to the coupled kinetic and hydrodynamic equations. For parameters appropriate for supernova remnants, we find stationary solutions with efficient acceleration when the escape boundary is placed at the point where growth and advection of strongly driven non-resonant waves are in balance. We also present the energy dependence of the distribution function close to the energy where it cuts off - a diagnostic that is in principle accessible to observation.