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Appendix A: Particle transport in a moving medium

In the solar wind and various other astrophysical situations, cosmic-ray transport often occurs in a moving medium. When this is the case, the Fokker-Planck Eq. (1) holds in a co-moving reference frame, i.e. in the rest frame of a moving plasma. Below we generalise the analysis of the paper by deriving a diffusive transport equation for cosmic-ray particles in a moving background plasma.

Suppose the plasma, supporting electromagnetic fluctuations that interact with the particles, moves with respect to the observer with a bulk speed U = Uẑ along the mean magnetic field B₀ = B₀ẑ. It is convenient to transform to the mixed co-moving coordinate system (Webb 1985; Kirk et al. 1988), in which the time t and position x are measured in the laboratory (observer) coordinate system, whereas the particle momentum p^∗ is measured in the rest frame of the streaming plasma. Equation (1) is written in these variables as follows: (A.1)where Γ =  [1 − (U²/c²)] ^− 1/2 is the bulk Lorentz factor of the flow.

For brevity we drop the ∗-notation in what follows, keeping in mind that the phase-space coordinates are to be taken in the mixed co-moving coordinate system. Moreover, for simplicity we restrict our analysis to spatially varying, but stationary flows, U = U(z), so that Eq. (A.1) becomes (A.2)We proceed to generalise the argument of Sect. 2 to the case of a moving background plasma. We substitute Eq. (2) in Eq. (A.2) and use Eq. (3) to obtain (A.3)Now averaging over μ yields (A.4)Next we subtract Eq. (A.4) from Eq. (A.2) and express the anisotropy g(μ) in terms of the isotropic distribution function F, assuming that the spatial gradients of the plasma flow velocity are small. The resulting approximate equation for g is as follows: (A.5)As in the analysis of Sect. 2, in the weak focusing limit, λ₀/L ≪ 1, this equation can be approximately integrated to give (A.6)where we used D_μμ( ± 1) = 0. Now integrating Eq. (A.6) yields (A.7)On using Eq. (3) to determine the integration constant c₀, we obtain the following generalisation of Eq. (7): (A.8)To complete the derivation of the diffusive approximation for particle transport in a moving plasma, we integrate Eq. (A.8) to determine the two integrals that appear in Eq. (A.4): (A.9)where we used Eqs. (9) and (10), and (A.10)Recall that the scattering coefficient is an even function, D_μμ( − μ) = D_μμ(μ), when scattering is isotropic, D_μμ(μ) = D₀(1 − μ²) , or more generally when the coefficient is calculated using quasilinear theory (Eq. (16)). Hence the function g(μ) is odd, and so the second moment given by Eq. (A.10) vanishes. On substituting the first moment (A.9) into Eq. (A.4) and rearranging terms, we find that the isotropic part of the cosmic-ray phase-space distribution in the weak adiabatic focusing limit obeys the following diffusion-convection equation: (A.11)which reduces to Eq. (11) if U = 0.

We conclude that the bulk flow of a background plasma leads to a modified telegraph equation for the cosmic-ray density already in the diffusion approximation. The equation above should be used for relativistic flows, such as those present in gamma-ray burst sources and jets of active galactic nuclei. For the SEP transport in the solar wind, though, the main corrections to the diffusive transport model are given by the convective term U∂_zF and the adiabatic energy loss term Γ²(∂_zU)p∂_pF/3. We neglect the latter term throughout the paper, assuming that ∂_zU is small enough.

© ESO, 2013