Abstract
ABSTRACT We derive a consistent set of moment equations for cosmic ray (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 DF f, number density n, total energy e, kinetic energy ϵ, 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 moment 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 time-step limitations: In both, we carefully address where the RSOL (versus true c) must appear for the correct behaviour to be recovered in all interesting limits, and show how current implementations of CRs with an RSOL neglect some additional terms.
Highlights
Cosmic rays (CRs) could play a potentially crucial role in the interstellar and circum-galactic medium, star and galaxy formation, and our understanding of high-energy astro-particle and plasma physics
These applications have generally relied on moment-based approaches, where one begins by assuming that the CR distribution function (DF) f is gyrotropic, averages over the micro-scale Lorentz forces and scattering processes, considers moments of the distribution function in terms of the remaining momentum direction, the pitch angle μ
Just like with radiation hydrodynamics (RHD), it is important that the reduced speed of light (RSOL) appear only in the dynamical equations for the CRs, not in the terms that couple to the gas that are written in terms of physical quantities
Summary
Cosmic rays (CRs) could play a potentially crucial role in the interstellar and circum-galactic medium, star and galaxy formation, and our understanding of high-energy astro-particle and plasma physics. In planet/star/galaxy formation models the resolution scales are vastly larger than CR gyro-radii for CRs with energies TeV (which contain most of the energy/pressure, and dominate the interactions with the non-relativistic matter) As such, these applications have generally relied on moment-based approaches, where one begins by assuming that the CR distribution function (DF) f is gyrotropic (symmetric around the magnetic-field direction), averages over the micro-scale Lorentz forces and scattering processes, considers moments of the distribution function in terms of the remaining momentum direction, the pitch angle μ. These applications have generally relied on moment-based approaches, where one begins by assuming that the CR distribution function (DF) f is gyrotropic (symmetric around the magnetic-field direction), averages over the micro-scale Lorentz forces and scattering processes, considers moments of the distribution function in terms of the remaining momentum direction, the pitch angle μ The simplest of these – “zeroth moment methods” – correspond to pure diffusion models. We define the pitch-angle-averaging operations, pitch-angle moments of f , and DF-weighted pitch angle moments: X μ
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