Rayleigh Scattering Cross Section Redward of Ly$α$ by Atomic Hydrogen

Abstract

We present a low energy expansion of the Kramers-Heisenberg formula for atomic hydrogen in terms of $(ω/ω_l)$, where $ω_l$ and $ω$ are the angular frequencies corresponding to the Lyman limit and the incident radiation, respectively. The leading term is proportional to $(ω/ω_l)^4$, 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 $ω/ω_l\le 0.6$. In the neighboring region around Ly$α$ ($ω/ω_l >0.6$), we also present an explicit expansion of the Kramers-Heisenberg formula in terms of $Δω\equiv (ω-ω_{Lyα})/ω_{Lyα}$. The accuracy with errors less than 4 % can be attained for $ω/ω_l \ge 0.6$ with the expansion up to the fifth order of $Δω$. 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.