The scattering of Lyman-series photons in the intergalactic medium

We re-examine scattering of photons near the Ly alpha resonance in the intergalactic medium (IGM). We first derive a general integral solution for the radiation field around resonance within the usual Fokker-Planck approximation. Our solution shows explicitly that recoil and spin diffusivity source an absorption feature, whose magnitude increases with the relative importance of recoil compared to Doppler broadening. This spectrum depends on the Ly alpha line profile, but approximating it with the absorption profile appropriate to the Lorentzian wings of natural broadening accurately reproduces the results for a full Voigt profile so long as the IGM temperature is less than similar to 1000 K. This approximation allows us to obtain simple analytic formulae for the total scattering rate of Ly alpha photons and the accompanying energy exchange rate. Our power series solutions converge rapidly for photons that redshift into the Ly alpha resonance as well as for photons injected at line centre. We confirm previous calculations showing that heating through this mechanism is quite slow and probably negligible compared to other sources. We then show that energy exchange during the scattering of higher-order Lyman-series photons can be much more important than naively predicted by recoil arguments. However, the resulting heating is still completely negligible.