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

Abstract Lyman-alpha (Lyα) 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 Lyα RT equations under three simple geometrical symmetries and investigate the effects of atomic recoil and gas bulk motion. We first study Lyα spectra by solving Lyα RT equations for a static, uniform gas cloud under cylindrical geometry. The solution is verified through Lyα 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 Lyα spectra. The method is motivated by Lyα RT equations with recoil and justified by simulations. Finally, we account for constant velocity gradients in Lyα RT equations and obtain series solutions for Lyα spectra. The solutions demonstrate good agreement to Lyα spectra from simulations for small velocity gradients (i.e. edge velocity vE of a cloud being comparable to the thermal velocity b) but become less accurate for large ones. To characterise Lyα spectra under large velocity gradients, we empirically extend the functional form of solutions and constrain them from fitting simulated Lyα spectra. The resulting fitting formulae show significant improvement for large velocity gradients (vE/b ~ 100) under large optical depths. The analytical study of Lyα spectra in this work completes the set of solutions under simple geometries, provides physical insights for Lyα RT under recoil and velocity gradient, and develops analytical tools for theoretical studies that require inputs from Lyα RT.