Individual particle approach to the diffusive shock acceleration. Effect of the non-uniform flow velocity downstream of the shock

Abstract

The momentum distribution of particles accelerated at strong non-relativistic shocks may be influenced by the spatial distribution of the flow speed around the shock. This phenomenon becomes evident in the cosmic-ray modified shock, where the particle spectrum itself determines the flow velocity profile upstream. However, what if the flow speed is not uniform downstream as well? Hydrodynamics indicates that its spatial variation over the length scales involved in the acceleration of particles in supernova remnants (SNRs) could be noticeable.} {In the present paper, we address this issue, initially following Bell's approach to particle acceleration and then by solving the kinetic equation. We obtained an analytical solution for the momentum distribution of particles accelerated at the cosmic-ray modified shock with spatially variable flow speed downstream.} {We parameterized the downstream speed profile to illustrate its effect on two model cases, the test particle and non-linear acceleration at the shock.The resulting particle spectrum is generally softer in Sedov SNRs because the flow speed distribution reduces the overall shock compression accessible to particles with higher momenta. On the other hand, the flow structure in young SNRs could lead to harder spectra. The diffusive properties of particles play a crucial role as they determine the distance from the shock, and, as a consequence, the flow speed that particles encounter downstream. We discuss the effect of the plasma velocity gradient to be (partially) responsible for the evolution of the radio index and for the high-energy break visible in gamma rays from some SNRs. We expect that the effect from the gradient of the flow velocity downstream could be prominent in regions of SNRs with higher diffusion coefficient and lower magnetic field, i.e. where acceleration of particles is not very efficient.