Cosmic ray acceleration by fast magnetosonic waves

Recently, Schlickeiser and Miller have calculated anew the acceleration rate of cosmic rays by fast magnetosonic plasma waves in a small-beta plasma, using a linear dispersion relation. They found that the transit-time damping of fast mode waves provides the dominant contribution to the stochastic acceleration rate of cosmic ray particles, both, in pure fast mode wave turbulence as well as in a mixture of isotropic fast mode turbulence and slab Alfven turbulence. Here it is shown that the use of the linear dispersion relation is fully justified for protons down to kinetic energies of a few tens of keV, whereas for electrons, the dispersive part of the dispersion relation should be included for energies below 10VA/cMeV. By retaining the largest scales of the turbulence only (for which the linear dispersion relation holds), an upper limit is computed, which shows that possible strong damping of the turbulence spectrum at high wavenumber would not reduce the efficiency of the acceleration process in a dramatic way, and that a strong modification induced by the waves dispersivity at high wavenumber could only be in the sense of an enhanced efficiency.