Diffusion of relativistic charged particles and field lines in isotropic turbulence: II. Analytical models

Abstract

The transport of high-energy particles in the presence of small-scale, turbulent magnetic fields is a long-standing issue in astrophysics. Analytical theories on transport perpendicular to the large-scale magnetic field disagree with numerical simulations at rigidities where the particles' gyroradii are slightly smaller than the correlation length of turbulence. At the same time, extending the numerical simulations to lower rigidities has proven computationally prohibitive. We present an analytical model for the perpendicular transport, based on (1) initial particle transport along field lines, (2) the transport of field lines and (3) the eventual decorrelation of particles from field lines. Transport parallel to the large-scale field is governed by pitch-angle scattering and so for times larger than the inverse pitch-angle diffusion coefficient, particles spatially diffuse in the parallel direction. Our results suggest that perpendicular diffusion occurs when particles have displaced in the perpendicular direction by a few correlation lengths of turbulence. We have tested the analytical theory by running a large suite of test particle simulations at unprecedentedly low rigidities, making extensive use of graphical processing units (GPUs). Our numerical results exhibit a non-standard rigidity-dependence for the perpendicular diffusion coefficient at intermediate rigidities. At the lowest rigidities, the standard rigidity-dependence is recovered. The simulated diffusion coefficients are nicely reproduced by our analytical model. We have traced the non-standard rigidity-dependence to a subdiffusive phase in the field line transport. Our study confirms our understanding of the escape of cosmic rays from the Galactic halo and its rigidity-dependence. [abridged]