On the width of a collisionless shock and the index of the cosmic rays it accelerates

TL;DR

This paper derives an analytical relationship between the width of collisionless shocks and the spectral index of cosmic rays, revealing a transition point where shock structure and acceleration efficiency change significantly.

Contribution

It introduces a new instability analysis that links shock width to cosmic ray spectral index and acceleration fraction, identifying a critical transition in shock behavior.

Findings

Shock width increases by a factor of 8-10 at a critical CR acceleration fraction.

A new instability dominates when the accelerated particle fraction exceeds ~30%.

The analysis applies to strong, non-relativistic, unmagnetized shocks.

Abstract

Despite being studied for many years, the structure of collisionless shocks is still not fully determined. Such shocks are known to be accelerators of cosmic rays, which, in turn, modify the shock structure. The shock width $\lambda$ is known to be connected to the cosmic rays (CRs) spectral index, $a$. Here, we use an instability analysis to derive the shock width in the presence of CRs. We obtain an analytical expression connecting the shock width to the CRs index and to the fraction of upstream particles that are accelerated. We find that when this fraction becomes larger than $\sim$~30\%, a new instability becomes dominant. The shock undergoes a transition where its width increases by a factor $\sim 8- 10$, and the CRs acceleration effectively ends. Our analysis is valid for strong, non-relativistic and unmagnetized shocks. We discuss the implication of these results to the expected range of CRs spectra and flux observed, and on the structure of non-relativistic collisionless shocks.