Stochastic Acceleration of Electrons by Plasma Waves. III. Waves Propagating Perpendicular to the Magnetic Field

Abstract

In this paper we consider the interaction of particles with turbulentplasma waves propagating perpendicular to the ordered magnetic field.Our investigation of the dispersion relation for waves propagating at anarbitrary angle theta with respect to the magnetic field lines indicatesthat the qualitative behavior of the acceleration characteristics at allangles theta≠0 are similar to that of the theta=pi/2 case. We use theperpendicular case as a representative example because it leads to amuch simpler analysis. Therefore, the results presented here and theresults of our earlier papers dealing with the theta=0 case will give asomewhat complete picture of stochastic acceleration for all waves.Unlike the theta=0 case, where only the fundamental mode of each wavecontributes, for general value of theta, and in particular for theperpendicular case, many harmonics enter into the equation, making theanalysis more complicated. We develop a recipe for evaluation of thecontributions of all harmonics and derive an analytic expression for theasymptotic behavior at nonrelativistic energies. We use the formalism ofour previous papers and find that all Fokker-Planck coefficients for theperpendicular waves are of the same order of magnitude, so the ratio ofthe pitch angle to momentum (or energy) diffusion rates is of orderunity at all energies and pitch angles. This is unlike the parallelcase, where this ratio varies from being much less than 1 at lowenergies to much greater than 1 at relativistic energies. As a result,the simplifications of the transport equation arising from theseinequalities are no longer applicable. On the other hand, we find thatwhen the transport equation for the theta=pi/2 case is expressed interms of diffusion in the parallel and perpendicular components of themomentum, only diffusion in the perpendicular component is present. Thisagain yields a one-dimensional diffusion equation and implies that theparallel momentum does not change. Consequently, we expect a rapidanisotropization of the pitch angle distribution (into a pancake-likedistribution) for particles accelerated by perpendicular waves. Fromcomparison of the acceleration rate of the thermal particles with thedecay rate of the waves in hot plasmas, we find that a substantialfraction of the background plasma electrons can be accelerated throughthis process for reasonable values of the energy density and spectrum ofturbulence.