Self-Similar Evolution of Cosmic-Ray Modified Shocks: The Cosmic-Ray Spectrum

TL;DR

This paper uses kinetic simulations to analyze the self-similar evolution of cosmic-ray modified shocks, revealing how the CR spectrum develops over time and depends on shock parameters, with implications for astrophysical shock modeling.

Contribution

It introduces a self-similar analytical framework for the evolution of CR spectra in shocks, based on detailed kinetic simulations with various diffusion models.

Findings

CR spectrum at the subshock is a sum of two power-laws with an exponential cutoff.

Shock precursor structure evolves self-similarly, depending on x/(u_s t).

Analytic forms for shock structure parameters are proposed for different stages.

Abstract

We use kinetic simulations of diffusive shock acceleration (DSA) to study the time-dependent evolution of plane, quasi-parallel, cosmic-ray (CR) modified shocks. Thermal leakage injection of low energy CRs and finite Alfv\'en wave propagation and dissipation are included. Bohm diffusion as well as the diffusion with the power-law momentum dependence are modeled. As long as the acceleration time scale to relativistic energies is much shorter than the dynamical evolution time scale of the shocks, the precursor and subshock transition approach the time-asymptotic state, which depends on the shock sonic and Alfv\'enic Mach numbers and the CR injection efficiency. For the diffusion models we employ, the shock precursor structure evolves in an approximately self-similar fashion, depending only on the similarity variable, x/(u_s t). During this self-similar stage, the CR distribution at the subshock maintains a characteristic form as it evolves: the sum of two power-laws with the slopes determined by the subshock and total compression ratios with an exponential cutoff at the highest accelerated momentum, p_{max}(t). Based on the results of the DSA simulations spanning a range of Mach numbers, we suggest functional forms for the shock structure parameters, from which the aforementioned form of CR spectrum can be constructed. These analytic forms may represent approximate solutions to the DSA problem for astrophysical shocks during the self-similar evolutionary stage as well as during the steady-state stage if p_{max} is fixed.