# Photon propagation direction prediction possible after interacting with neutral hydrogen?

My current line of research deals a lot with hydrogen's Lyman-alpha emission and subsequent interactions of the Lyman-alpha photons with the surrounding hydrogen gas. My question is whether (stimulated emission aside) there is any way to predict (or more likely find a probability distribution for) the subsequent propagation direction of a photon after being absorbed and emitted by a neutral atom?

I am also curious to know if there is way to predict (or find a probability distribution for) the propagation direction of a photon after a Compton scattering.

Probably too late now, but I have to answer this. There is indeed a way to predict the direction of a scattered Lyman $\alpha$ photon. The answer depends on whether the scattering takes place in the core or the wings of the line.
In the core (i.e. closer to the line center than about 3 Doppler widths), we can use the dipole approximation, so the phase function describing the direction of scattering is given by $$W(\theta) \propto 1 + \frac{R}{Q}\cos^2\theta,$$ where $R/Q$ is the degree of polarization for 90º scattering. It turns out (Hamilton 1940) that with a probability $1/3$, this results in an isotropic distribution, while with a probability $2/3$, $$W(\cos\theta) = \frac{7}{16} \left( 1 + \frac{3}{7} \cos^2\theta \right).$$
In the case of wing scattering, Stenflo (1980) found that the scattering behaves like Rayleigh scattering, i.e. it follows a dipole distribution with 100% polarization at 90º, so $$W(\cos\theta) = \frac{3}{8} \left( 1 + \cos^2\theta \right).$$
For a more thorough explanation, I cannot resist referring to chapter 3.2.2 of my thesis "Interpreting Lyman $\alpha$ Radiation from Young, Dusty Galaxies", where you will also find a numerical implementation of the above, as well as a more exact functional form of the transition between the core and line.