THE FOKKER-PLANCK COEFFICIENTS OF COSMIC-RAY TRANSPORT IN RANDOM ELECTROMAGNETIC-FIELDS

We derive the three Fokker-Planck coefficients D(mu-mu), D(mu-p), D(pp) of cosmic ray particle transport in a uniform magnetic field with superposed small-amplitude electromagnetic plasma waves in the weak interaction approximation by calculating first-order corrections to the particle's orbit. The calculations are done in the full electromagnetic case avoiding the magnetostatic approximation and any restriction on the waves' propagation direction and the waves' dispersion relation. Special attention is given to low-frequency magnetohydrodynamic waves, because of their importance in space physics and astrophysics, and the Fokker-Planck coefficients D(mu-mu) and D(mu-p) are simplified by expanding the general expressions to zeroth and first order in omega(j)/kc much less than 1, respectively. Then the concept of slab and isotropic turbulence is critically re-examined leading to the introduction of the cone model in which waves can propagate with an angle psi < chi with respect to the background magnetic field. The Fokker-Planck coefficients D(mu-mu) and D(mu-p) for this cone model can be further simplified and results depend critically on the value of delta = \tan-chi\. For transverse Alfven waves we demonstrate, for small cone opening values delta much less than 1, the following. (i) For small values of the cosine of the pitch angle mu much less than delta, an additional factor \mu - j-alpha\ appears in D(mu-mu) and D(mu-p) as compared to large values of mu > delta where j = +/-1 and alpha = upsilon(A)/upsilon. A similar factor has been discovered before by numerical calculations for isotropic turbulence (where delta --> infinity) by Fisk and co-workers; we present an analytical proof for its occurrence. This extra factor plays an essential role in calculating the mean free path of cosmic rays from D(mu-mu) in the diffusion approximation. (ii) At large values of mu > delta only the fundamental resonance interaction mode n = 1 contributes to the evaluation of the Fokker-Planck coefficients, whereas at small values mu < delta all modes n greater-than-or-equal-to 1 contribute to the evaluation of the Fokker-Planck coefficients. (iii) At values mu > delta the magnetic helicity of the Alfvenic turbulence has a strong influence on the values of D(mu-mu) D(mu-p) and D(pp) While its influence becomes negligibly small for mu much less than delta. The slab and cone model arc compared and we discuss the resulting spatial and momentum diffusion coefficients. While the momentum diffusion coefficient is approximately the same in both models, the spatial diffusion coefficient can be verv different. even for verv small opening angles.