The Scattering of Lyman-series Photons in the Intergalactic Medium

We re-examine scattering of photons near the Lyman-alpha resonance in the intergalactic medium (IGM). We first derive a general integral solution for the radiation field around resonance when spin diffusivity is ignored. Our solution shows explicitly that recoil sources an absorption feature, whose magnitude increases with the relative importance of recoil compared to Doppler broadening. This spectrum depends on the Lyman-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 T<1000 K in the IGM. This approximation allows us to obtain simple analytic formulae for the total scattering rate of Lyman-alpha photons and the accompanying energy exchange rate. Our power series solutions converge rapidly for photons that redshift into the Lyman-alpha resonance as well as for photons injected at line center. 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.