The diffusion approximation and transport theory for cosmic rays in relativistic flows

Abstract

Equations describing the transport of cosmic rays in relativistic flows in the diffusion approximation are obtained. The analysis is based on the zeroth, first, and second differential moment equations of the relativistic Boltzmann equation with a BGK collision term (the equations are differential with respect to the magnitude of the particle momentum p' in the co-moving or scattering frame). A perturbation solution of the moment equations in the diffusion approximation yields both the co-moving frame particle current and viscous stresses. The resultant cosmic-ray continuity equation contains three readily recognized energy change terms: the adia-batic energy change term; the viscous shear energy change term; and a term proportional to the scalar product of the acceleration vector w a of the scattering frame £' and the heat flux q a (the latter energy change is positive if g a « a < 0). The heat flux q* consists of the diffusive particle current plus a further acceleration vector term proportional to ù a df 0 /dp\ where f' 0 (x, p') is the isotropic part of the particle phase space distribution function in X'. The latter component of the heat flux corresponds to the relativistic heat inertia term. The relation of these results to the cosmic ray continuity equation obtained by Earl, Jokipii, and Morfill in the nonrelativistic flow regime is elucidated.