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Two methods are developed for solving the steady‐state spherically symmetric radiative transfer equation for resonance line radiation emitted by a point source in the intergalactic medium, in the context of the Wouthuysen–Field mechanism for coupling the hyperfine structure spin temperature of hydrogen to the gas temperature. One method is based on solving the ray and moment equations using finite differences. The second uses a Monte Carlo approach incorporating methods that greatly improve the accuracy compared with previous approaches in this context. Several applications are presented serving as test problems for both a static medium and an expanding medium, including inhomogeneities in the density and velocity fields. Solutions are obtained in the coherent scattering limit and for Doppler RII redistribution with and without recoils. We find generally that the radiation intensity is linear in the cosine of the azimuthal angle with respect to radius to high accuracy over a broad frequency region across the line centre for both linear and perturbed velocity fields, yielding the Eddington factors  f  ν  ≃ 1/3 and  g  ν  ≃ 3/5. The radiation field produced by a point source divides into three spatial regimes for a uniformly expanding homogeneous medium. The regimes are governed by the fraction of the distance  r  from the source in terms of the distance  r  *  required for a photon to redshift from line centre to the frequency needed to escape from the expanding gas. For a standard cosmology, before the Universe was reionized  r  *  takes on the universal value independent of redshift of 1.1 Mpc, depending only on the ratio of the baryon to dark matter density. At  r / r  *  &lt; 1, the radiation field is accurately described in the diffusion approximation, with the scattering rate declining with the distance from the source as  r  −7/3 , except at  r / r  *  ≪ 1 where frequency redistribution nearly doubles the mean intensity around line centre. At  r / r  *  &gt; 1, the diffusion approximation breaks down and the decline of the mean intensity near line centre and the scattering rate approach the geometric dilution scaling 1/ r  2 . The mean intensity and scattering rate are found to be very sensitive to the gradient of the velocity field, growing exponentially with the amplitude of the perturbation as the limit of a vanishing velocity gradient is approached near the source. We expect the 21‐cm signal from the epoch of reionization to thus be a sensitive probe of both the density and the peculiar velocity fields.  The solutions for the mean intensity are made available in machine‐readable format.

The scattering of Lyα radiation in the intergalactic medium: numerical methods and solutions