Cosmic Ray Transport with Magnetic Focusing and the ``Telegraph'' model

Cosmic rays (CR), constrained by scattering on magnetic irregularities, are believed to propagate diffusively. But a well-known defect of diffusive approximation, whereby not all the particles propagate at realistic speeds, causes attempts to justify an alternative approach based on the “telegraph” equation. However, its derivations often lack rigor and transparency leading to incorrect results. The classic Chapman-Enskog method is applied to the pitch-angle averaged spatial CR transport. We show that the convective term arises only from the magnetic focusing effect and no “telegraph” (second order time derivative) term emerges in any order of the proper asymptotic expansion with systematically eliminated short time scales. However, this term may formally be converted from the fourth order hyper-diffusive term in the Chapman-Enskog expansion. But, within the method’s validity range, it may only be important for a short relaxation period associated with either strong pitchangle anisotropy or spatial inhomogeneity of the initial CR distribution. However, in neither of these two cases a treatment based on merely the isotropic CR component is sufficient, regardless of the correction type (telegraph or hyper-diffusive). Moreover, in a long-time asymptotic regime these corrections are generally insignificant, as the angular anisotropy decays rapidly. 1. Preliminary Considerations The problem addressed here is fundamental but not new to the cosmic ray (CR) transport studies. It can be formulated very plainly: How to describe CR transport by only their isotropic component, after the anisotropic one has been suppressed by scattering on magnetic irregularities? Suppose the angular distribution of CRs is given by the function f (μ, t,z) obeying an equation from which the rapid gyro-phase rotation is already removed (drift approximation, e.g., Vedenov et al. 1962; Kulsrud 2005) ∂ f ∂ t + vμ ∂ f ∂ z = ∂ ∂ μ ( 1−μ ) D (μ) ∂ f ∂ μ . (1) Here z is the local coordinate along the ambient magnetic field, μ is the cosine of the particle pitch angle, and D is the pitch angle diffusion coefficient. Now, we make the next step in simplifying the transport description and seek an equation for the pitch-angle averaged distribution