Lyman-$\alpha$ photons through rotating outflows

Outflows and rotation are two ubiquitous kinematic features in the gas kinematics of galaxies. Here we introduce a semi-analytic model to quantify how rotating outflows impact the morphology of the Lyman-$\alpha$ emission line. The model is contrasted against Monte Carlo radiative transfer simulations of outflowing gas with additional solid body rotation. We explore a range of neutral Hydrogen optical depth of ${10}^5\le {\tau }_{\mathrm{H}}\le {10}^7$, rotational velocity $0\le v_{\mathrm{r}\mathrm{o}\mathrm{t}}\mathrm{/}{\mathrm{k}\mathrm{m}\mathrm{s}}^{-1}\le 100$ and outflow velocity $0\le v_{\mathrm{o}\mathrm{u}\mathrm{t}}\mathrm{/}{\mathrm{k}\mathrm{m}\mathrm{s}}^{-1}\le 50$. We find three consequences of rotation. First, it introduces a dependency with viewing angle; second it broadens the line and third it increases the flux at the line’s center. For all viewing angles, the semi-analytic model reproduces the radiative transfer results for the line width and flux change at the line’s center within a $7\mathrm{\%}$ and $50\mathrm{\%}$ precision for an optical depth of ${\tau }_{\mathrm{H}}={10}^5$, respectively, and within $2\mathrm{\%}$ and $1\mathrm{\%}$ for an optical depth of ${\tau }_{\mathrm{H}}={10}^7$. Using this model we also show that the peaks of integrated spectra taken from opposite sides of an edge-on rotating gas distribution should have a separation of $\frac{1}{2}{v}_{\mathrm{r}\mathrm{o}\mathrm{t}}$. The semi-analytic model presented here is a convenient tool to introduce rotational kinematics as a post-processing step of idealized Monte Carlo simulations; it provides a framework to interpret \lya spectra in systems where rotation is expected or directly measured through kinematic maps.