A Consistent Reduced-Speed-of-Light Formulation of Cosmic Ray Transport Valid in Weak and Strong-Scattering Regimes

Abstract

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.

arXiv comment17 pages, 2 figures. Readers not interested in the detailed derivations can skip to Appendix A, which simply summarizes the key equations derived. Updated to match published MNRAS version

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• Standard Self-Confinement and Extrinsic Turbulence Models for Cosmic Ray Transport are Fundamentally Incompatible with Observations
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• The Impact of Cosmic Rays on the Kinematics of the Circumgalactic Medium
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• An accurate treatment of scattering and diffusion in piecewise power-law models for cosmic ray and radiation/neutrino transport
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• A Reduced-Speed-of-Light Formulation of the Magnetohydrodynamic-Particle-in-Cell Method
• Comparing energy and entropy formulations for cosmic ray hydrodynamics