ABSTRACT
We model the diffusive shock acceleration of particles in a system of two colliding shock waves and present a method to solve the time-dependent problem analytically in the test-particle approximation and high energy limit. In particular, we show that in this limit the problem can be analysed with the help of a self-similar solution. While a number of recent works predict hard (E−1) spectra for the accelerated particles in the stationary limit, or the appearance of spectral breaks, we found instead that the spectrum of accelerated particles in a time-dependent collision follows quite closely the canonical E−2 prediction of diffusive shock acceleration at a single shock, except at the highest energy, where a hardening appears, originating a bumpy feature just before the exponential cut-off. We also investigated the effect of the reacceleration of pre-existing cosmic rays by a system of two shocks, and found that under certain conditions spectral features can appear in the cut-off region. Finally, the mathematical methods presented here are very general and could be easily applied to a variety of astrophysical situations, including for instance standing shocks in accretion flows, diverging shocks, backward collisions of a slow shock by a faster shock, and wind–wind or shock–wind collisions.
TIME-DEPENDENT DIFFUSIVE SHOCK ACCELERATION IN SLOW SUPERNOVA REMNANT SHOCKS