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 $10^5 \leq \tau_{\mathrm{H}} \leq10^7$,
rotational velocity $0 \leq v_{\mathrm{rot}}/\mathrm{km\ s}^{-1} \leq 100$ and
outflow velocity $0\leq v_{\mathrm{out}}/\mathrm{km\ s}^{-1} \leq 50$. 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 $7\%$ and $50\%$ precision for an optical depth of
$\tau_{\mathrm{H}}=10^5$, respectively, and within $2\%$ and $1\%$ 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
$\frac{1}{2}v_{\mathrm{rot}}$. 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.
