Rayleigh scattering cross-section redward of Lyα by atomic hydrogen

Abstract

We present a low-energy expansion of the Kramers-Heisenberg formula for atomic hydrogen in terms of ω/ω1, where ω 1 and ω are the angular frequencies corresponding to the Lyman limit and the incident radiation, respectively. The leading term is proportional to (ω/ω1)4, which admits a well-known classical interpretation. With higher-order terms, we achieve accuracy with errors less than 4 per cent of the scattering cross-sections in the region ω/ω1 ≤ 0.6. In the neighbouring region around Lyα (ω/ω1 > 0.6), we also present an explicit expansion of the Kramers-Heisenberg formula in terms of Δω≡ (ω - ωLyα)/ω Lyα. The accuracy with errors less than 4 per cent can be attained for ω/ω1 ≥ 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.