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 stars substantially depend on the Wouthuysen-Field (WF) coupling, which relates the spin temperature with the kinetic temperature of hydrogen gas via the resonant scattering between Lyα photons and neutral hydrogen. Therefore, the existence of Lyα photons in the 21 cm region is essential. Although the center object generally is a strong source of Lyα photons, the transfer of Lyα photons in the 21 cm region is very inefficient, as the optical depth of Lyα photons is very large. Consequently, the Lyα photons ν0 from the source may not be able to transfer to the entire 21 cm region timely to provide the WF coupling. This problem is especially important considering that the lifetime of first stars generally is short. We investigate this problem with the numerical solution of the integrodifferential equation, which describes the kinetics of Lyα 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. First, the diffusion in the frequency space provides a shortcut for the diffusion in the physical space. It makes the mean time for the escape of resonant photon in optical depth τ media roughly proportional to the optical depth τ, not τ2. Second and more importantly, the resonant scattering is effective in bouncing photons with frequency ν ≠ ν0 back to ν0. This process can quickly restore ν0 photons and establish the local Boltzmann distribution of the photon spectrum around ν0. Therefore, the mechanism of "escape via shortcut" plus "bounce back" enables the WF 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. © 2009 The American Astronomical Society. All rights reserved.