Cosmic-Ray Transport and Acceleration. II. Cosmic Rays in Moving Cold Media with Application to Diffusive Shock Wave Acceleration
We discuss the transport and acceleration of cosmic rays in a cold background medium that moves with the nonrelativistic (U/V much less than 1) bulk speed U(z) parallel to the ordered uniform magnetic field B0 = B0eZ with superposed Alfén plasma waves propagating parallel and/or antiparallel to B0. For a power-law dependence 1(K||)∝ K||-q, 1 < q < 4 of the power spectrum of magnetic irregularities on the parallel Alfvénic wavenumber K||, we calculate the respective Fokker-Planck coefficients Dμμ, Dμp, Dpp, and the implied transport parameters as the spatial diffusion coefficient κ(z, p), the momentum diffusion coefficient α2(z, p), and the rate of adiabatic deceleration α1(z, p) whose general relation to the Fokker-Planck coefficients has been established by Schlickeiser. A detailed discussion of the influence of the Alfvén wave's polarization state and propagation direction on the results is provided. The main results are the following:1. Waves of only one polarization state (left-hand or right-hand) propagating in one or both directions lead to a whole resonance gap interval where Dμμ = Dμp = Dpp = 0, implying infinitely large spatial and momentum diffusion coefficients. 2. With waves of both polarization states traveling in only one direction (either forward [parallel] to B0 or backward [antiparallel] to B0 in the fluid's rest frame), a resonance gap occurs at one point in μ-space leading to an infinitely large spatial diffusion coefficient for q ≥ 2. Only in this unphysical case (κ = ∞!) the cosmic- ray transport equation is of first order in momentum. 3. With waves of both polarization states traveling with equal or different but nonzero intensities in both directions, no resonance gaps occur, and the full transport equation (κ ≠ # 0, α1 ≠ 0, α2 ≠ 0) results. 4. In the case of right-hand and left-hand waves propagating with equal intensity in both directions, the mean free path of particles and the cosmic-ray anisotropy are calculated for different values of the power spectrum spectral index q. Unlike earlier work, finite expressions result for q ≥ 2. For q ≥ 2, the anisotropy and the mean free path are enhanced by a factor (V/U)q-2 as compared to the case q <2. The anisotropy consists of two parts: one contribution results from the momentum gradient of the isotropic distribution function (Compton-Getting effect), the second contribution is related to pitch-angle scattering and the spatial gradient of the isotropic distribution function. Within the confines of quasi-linear theory the cosmic-ray transport equations contains spatial diffusion and convection as well as momentum diffusion and convection terms. The widely used transport equation without momentum diffusion can be reproduced only in the unrealistic case of waves moving only in one direction, but this case would imply an infinitely large spatial diffusion coefficient for q ≥ 2 due to a resonance gap at μR = (U±VA)/V. Finally, two astrophysical applications of the derived transport equation are considered: (i) cosmic-ray transport and acceleration in the dynamical interstellar medium, and (ii) steady diffusive acceleration of energetic particles at astrophysical shock waves. In the latter case the full transport equation is solved analytically for a special spatial variation of the spatial diffusion coefficient but for any given velocity pattern. It is shown that the resulting particle momentum spectrum is an infinite superposition of power-law spectra approaching a single power law at momenta large compared to the injection momentum.