Identifier: 2008-19
Author(s): I Roy, J.-M. Qiu, C.-W. Shu, and L.-Z. Fang
Title: A WENO algorithm for radiative transfer with resonant scattering: the time scale of the Wouthuysen-Field Coupling
Page count: 18 pp.
Date: 2008-01-01
Abstract:
We develop a numerical solver for the photon distribution in the frequency space described by radiative transfer with resonant scattering of Lyα photons by hydrogen gas in the early universe. This problem is crucial to the estimation of the time scale for the onset of the Wouthuysen-Field (WF) coupling in relation to the 21 cm emission and absorption at the epoch of reionization. The resonant scattering without recoil leads to the photon distribution in the frequency space to be piecewise smooth containing sharp changes. The weighted essentially nonoscillatory (WENO) scheme is suitable to handle this problem, as this algorithm has been found to be highly stable and robust for solving Boltzmann-like equations. We first show that the WENO solutions can precisely match the two analytic solutions of the evolution of the photon distribution in rest background. We find further that the evolution of the photon distribution due to resonant scattering without recoil generally undergoes three phases. First, the profile of the photon distribution is similar to the initial one. Second, an extremely flat plateau forms around the resonant frequency, and the width and height of the flat plateau increase with time. Finally, the distribution around the resonant frequency is stable (time-independent), as the photons from the source is balanced by the redshift of the expansion. This result indicates that the time-independent solutions of the Fokker-Planck approximation is questionable for describing the 21 cm problem, as the time scale of approaching the time-independent solution is of the order of one Myr, which might be longer than the evolutionary time scale of the 21 cm region around the first generation stars. However, the onset of the W-F coupling is not determined by the time-independent solution, but by the formation of the flat plateau of the second phase. We found that the time scale of the W-F coupling is only equal to about a few tens or hundreds of the mean free flight time of photons with resonant frequency, and it basically is independent of the Sobolev parameter if this parameter is much less than 1.
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