Wouthuysen-Field Coupling in the 21 cm Region Around High Redshift Sources

Abstract

The 21 cm emission and absorption from gaseous halos around the first generation of star depend on the Wouthuysen-Field (W-F) coupling, which relates the spin temperature with the kinetic temperature of hydrogen gas via the resonant scattering between Lyman alpha photons and neutral hydrogen. Although the center object generally is a strong source of these photons, the transfer of these photons in the 21 cm region is inefficient, as the optical depth of the photons is large. Consequently, these photons from the source may not be able to transfer to the entire 21 cm region timely to provide the W-F coupling. This problem is important because the lifetime of first stars generally is short. The problem is investigated with numerical solution of the integro-differential equation, which describes the kinetics of these resonant photons in both physical and frequency spaces. We show that the photon transfer process in the physical space is actually coupled to that in the frequency space. Firstly diffusion in the frequency space provides a shortcut for the diffusion in the physical space. It makes the mean time for the escape of the resonant photon in optical depth \tau media roughly proportional to the optical depth \tau, not \tau^2. Secondly the resonant scattering is effective in bouncing photons with a frequency which is not equal to initial frequency back to the initial frequency. This process can restore initial frequency photons and establish the local Boltzmann distribution of the photon spectrum around the initial frequency. Therefore, the mechanism of 'escape via shortcut' plus 'bounce back' enables W-F coupling to be properly realized in the 21 cm region around first stars. This mechanism also works for photons injected into the 21 cm region by redshift.