Stochastic particle acceleration at shocks in the presence of braided magnetic fields

The theory of diffusive acceleration of energetic particles 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 on short time scales. We derive the propagator for such motion, which differs from the Gaussian form relevant for diffusion, and apply it to a configuration with a plane shock front whose normal is perpendicular to the average field direction. Expressions are given for the acceleration time as a function of the diffusion coefficient of the wandering magnetic field lines and the spatial diffusion coefficient of the charged particles parallel to the local field. In addition we calculate the spatial dependence of the particle density in both the upstream and downstream plasmas. In contrast to the diffusive case, the density of particles at the shock front is lower than it is far downstream. This 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 $f\propto p^{-s}$, we find $s=\sdiff[1+1/(2\rcomp)]$, where $\rcomp$ is the compression ratio of the shock front and $\sdiff$ is the standard result of diffusive acceleration: $\sdiff=3\rcomp/(\rcomp-1)$. A strong shock in a monatomic ideal gas yields a spectrum of $s=4.5$. In the case of electrons, this corresponds to a radio synchrotron spectral index of $\alpha=0.75$.