On the transfer of resonant-line radiation in mesh simulations

The last decade has seen applications of adaptive mesh refinement (AMR) methods for a wide range of problems from space physics to cosmology. With the advent of these methods, in which space is discretized into a mesh of many individual cubic elements, the contemporary analog of the extensively studied line radiative transfer (RT) in a semi-infinite slab is that of RT in a cube. In this study we provide an approximate solution of the RT equation, as well as analytic expressions for the probability distribution functions (PDFs) of the properties of photons emerging from a cube, and compare them with the corresponding slab problem. These PDFs can be used to perform fast resonant-line RT in optically thick AMR cells where, otherwise, it could take unrealistically long times to transfer even a handful of photons.