Cosmic Ray Transport with Magnetic Focusing and the “telegraph” Model

Cosmic rays (CR), constrained by scattering on magnetic irregularities, are believed to propagate diffusively. However, a well-known defect of diffusive approximation, whereby some of the particles propagate unrealistically fast, has directed interest toward an alternative CR transport model based on the “telegraph” equation. Though, its derivations often lack rigor and transparency leading to inconsistent results. We apply the classic Chapman– Enskog method to the CR transport problem. We show that no “telegraph” (second order time derivative) term emerges in any order of a proper asymptotic expansion with systematically eliminated short timescales. Nevertheless, this term may formally be converted from the fourth order hyper-diffusive term of the expansion. However, both the telegraph and hyperdiffusive terms may only be important for a short relaxation period associated with either strong pitch-angle anisotropy or spatial inhomogeneity of the initial CR distribution. Beyond this period the system evolves diffusively in both cases. The term conversion, that makes the telegraph and Chapman–Enskog approaches reasonably equivalent, is possible only after this relaxation period. During this period, the telegraph solution is argued to be unphysical. Unlike the hyperdiffusion correction, it is not uniformly valid and introduces implausible singular components to the solution. These dominate the solution during the relaxation period. Because they are shown not to be inherent in the underlying scattering problem, we argue that the telegraph term is involuntarily acquired in an asymptotic reduction of the problem.