A Simple Sub-Grid Model For Cosmic Ray Effects on Galactic Scales

Many recent numerical studies have argued that cosmic rays (CRs) from supernovae (SNe) or active galactic nuclei (AGN) could play a crucial role in galaxy formation, in particular by establishing a CR-pressure dominated circum-galactic medium (CGM). But explicit CR-magneto-hydrodynamics (CR-MHD) remains computationally expensive, and it is not clear whether those results can be applied to simulations that do not explicitly treat magnetic fields or resolved ISM phase structure. We therefore present an intentionally extremely-simplified sub-grid model for CRs, which attempts to capture the key qualitative behaviors of greatest interest for those interested in simulations or semi-analytic models including some approximate CR effects on galactic (>kpc) scales, while imposing negligible computational overhead. The model is numerically akin to some recently-developed sub-grid models for radiative feedback, and allows for a simple constant parameterization of the CR diffusivity and/or streaming speed; it allows for an arbitrary distribution of sources (proportional to black hole accretion rates or star-particle SNe rates or gas/galaxy star formation rates), and interpolates between the limits where CRs escape the galaxies with negligible losses and those where CRs lose most of their energy catastrophically before escape (relevant in e.g. starburst galaxies). The numerical equations are solved trivially alongside gravity in most codes. We compare this to explicit CR-MHD simulations and discuss where the (many) sub-grid approximations break down, and what drives the major sources of uncertainty.