Issue 
A&A
Volume 554, June 2013



Article Number  A59  
Number of page(s)  7  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201321327  
Published online  05 June 2013 
Online material
Appendix A: Particle transport in a moving medium
In the solar wind and various other astrophysical situations, cosmicray transport often occurs in a moving medium. When this is the case, the FokkerPlanck Eq. (1) holds in a comoving 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 cosmicray 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 B_{0} = B_{0}ẑ. It is convenient to transform to the mixed comoving 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 − (U^{2}/c^{2})] ^{− 1/2} is the bulk Lorentz factor of the flow.
For brevity we drop the ∗notation in what follows, keeping in mind that the phasespace coordinates are to be taken in the mixed comoving 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, λ_{0}/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 c_{0}, 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) = D_{0}(1 − μ^{2}) , 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 cosmicray phasespace distribution in the weak adiabatic focusing limit obeys the following diffusionconvection 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 cosmicray density already in the diffusion approximation. The equation above should be used for relativistic flows, such as those present in gammaray 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∂_{z}F and the adiabatic energy loss term Γ^{2}(∂_{z}U)p∂_{p}F/3. We neglect the latter term throughout the paper, assuming that ∂_{z}U is small enough.
© ESO, 2013
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