Analytical and fitting formulae for solutions to Lyman-alpha radiative transfer equations: the effects of geometry, recoil, and velocity gradients

Abstract

Lyman-alpha (Ly$\alpha$) radiative transfer (RT) is important in many astrophysical environments and governed by multiple physical processes. In this paper, we provide analytical formulae/procedures for the solutions to Ly$\alpha$ RT equations under three simple geometrical symmetries and investigate the effects of atomic recoil and gas bulk motion. We first study Ly$\alpha$ spectra by solving Ly$\alpha$ RT equations for a static, uniform gas cloud under cylindrical geometry. The solution is verified through Ly$\alpha$ Monte Carlo RT simulations, and compared to those under slab and spherical geometries in literature. Second, to characterise the recoil effect, we empirically modify recoil-free Ly$\alpha$ spectra. The method is motivated by Ly$\alpha$ RT equations with recoil and justified by simulations. Finally, we account for constant velocity gradients in Ly$\alpha$ RT equations and obtain series solutions for Ly$\alpha$ spectra. The solutions demonstrate good agreement to Ly$\alpha$ spectra from simulations for small velocity gradients (i.e. edge velocity $v_{\rm E}$ of a cloud being comparable to the thermal velocity $b$) but become less accurate for large ones. To characterise Ly$\alpha$ spectra under large velocity gradients, we empirically extend the functional form of solutions and constrain them from fitting simulated Ly$\alpha$ spectra. The resulting fitting formulae show significant improvement for large velocity gradients ($v_{\rm E}/b \sim 100$) under large optical depths. The analytical study of Ly$\alpha$ spectra in this work completes the set of solutions under simple geometries, provides physical insights for Ly$\alpha$ RT under recoil and velocity gradient, and develops analytical tools for theoretical studies that require inputs from Ly$\alpha$ RT.