Rayleigh Scattering Cross Section Redward of Ly$\alpha$ by Atomic Hydrogen

Abstract

We present a low energy expansion of the Kramers-Heisenberg formula for atomic hydrogen in terms of $(\omega /{\omega }_l)$, where ${\omega }_l$ and $\omega$ are the angular frequencies corresponding to the Lyman limit and the incident radiation, respectively. The leading term is proportional to $(\omega /{\omega }_l{)}^4$, which admits a well-known classical interpretation. With higher order terms we achieve accuracy with errors less than 4 % of the scattering cross sections in the region $\omega /{\omega }_l\le 0.6$. In the neighboring region around Ly$\alpha$ ($\omega /{\omega }_l>0.6$), we also present an explicit expansion of the Kramers-Heisenberg formula in terms of $\Delta \omega \equiv (\omega -{\omega }_{Ly\alpha })/{\omega }_{Ly\alpha }$. The accuracy with errors less than 4 % can be attained for $\omega /{\omega }_l\ge 0.6$ with the expansion up to the fifth order of $\Delta \omega$. We expect that these formulae will be usefully applied to the radiative transfer in high neutral column density regions, including the Gunn-Peterson absorption troughs and Rayleigh scattering in the atmospheres of giants.

5 pages, 2 figures