On the Width of a Collisionless Shock and the Index of the Cosmic Rays It Accelerates

Abstract

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 (CRs), which, in turn, modify the shock structure. The shock width λ is known to be connected to the CR 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 CR index and to the fraction of upstream particles that are accelerated. We find that when this fraction becomes larger than ∼30%, a new instability becomes dominant. The shock undergoes a transition where its width increases by a factor ∼8–10, and the CR acceleration effectively ends. Our analysis is valid for strong, nonrelativistic, and unmagnetized shocks. We discuss the implication of these results on the expected range of CR spectra and flux observed and on the structure of nonrelativistic collisionless shocks.