A Consistent Reduced-Speed-of-Light Formulation of Cosmic Ray Transport
  Valid in Weak and Strong-Scattering Regimes
  We derive a consistent set of moments equations for CR-magnetohydrodynamics,
assuming a gyrotropic distribution function (DF). Unlike previous efforts we
derive a closure, akin to the M1 closure in radiation hydrodynamics (RHD), that
is valid in both the nearly-isotropic-DF and/or strong-scattering regimes, and
the arbitrarily-anisotropic DF or free-streaming regimes, as well as allowing
for anisotropic scattering and transport/magnetic field structure. We present
the appropriate two-moment closure and equations for various choices of evolved
variables, including the CR phase space distribution function, number density,
total energy, kinetic energy, and their fluxes or higher moments, and the
appropriate coupling terms to the gas. We show that this naturally includes and
generalizes a variety of terms including convection/fluid motion, anisotropic
CR pressure, streaming, diffusion, gyro-resonant/streaming losses, and
re-acceleration. We discuss how this extends previous treatments of CR
transport including diffusion and moments methods and popular forms of the
Fokker-Planck equation, as well as how this differs from the analogous M1-RHD
equations. We also present two different methods for incorporating a reduced
speed of light (RSOL) to reduce timestep limitations: in both we carefully
address where the RSOL (versus true c) must appear for the correct behavior to
be recovered in all interesting limits, and show how current implementations of
CRs with a RSOL neglect some additional terms.
