Particle acceleration at shocks in the presence of a braided magnetic field

Abstract

The theory of first order Fermi acceleration at shock fronts assumes charged particles undergo spatial diffusion in a uniform magnetic field. If, however, the magnetic field is not uniform, but has a stochastic or braided structure, the transport of charged particles across the average direction of the field is more complicated. Assuming quasi-linear behaviour of the field lines, the particles undergo sub-diffusion ( 〈x 2(t)〉 ∝ t 1 2 ) on short time scales. We investigate this process analytically, using a propagator approach, and numerically, with a Monte-Carlo simulation. It is found that, in contrast to the diffusive case, the density of particles at the shock front is lower than it is far downstream which is a consequence of the partial trapping of particles by structures in the magnetic field. As a result, the spectrum of accelerated particles is a power-law in momentum which is steeper than in the diffusive case. For a phase-space density f ∝ p − s , we find s = s diff [1 + 1 (2ϱ c ) ] , where ϱ c is the compression ratio of the shock front and s diff is the standard result of diffusive acceleration: s diff = 3ϱ c (ϱ c − 1) .