共振散射辐射转移的WENO算法：Wouthuysen场耦合的时间尺度
A WENO algorithm for radiative transfer with resonant scattering: the time scale of the Wouthuysen-Field Coupling
摘要：
本文发展了一个描述早期宇宙中氢气对Lyalpha光子共振散射时光子在频率空间分布的辐射转移的积分-微分方程的数值求解器。这个方程的随时间变化的解决方案是至关重要的Wouthuysen场（WF）耦合的影响，在21厘米的发射和吸收的再电离的时代的估计。共振散射导致光子在频率空间的分布是分段光滑的，包含急剧的变化。加权基本无振荡（WENO）格式是适合于处理这个问题，因为这种算法已被发现是高度稳定和强大的解决玻尔兹曼方程。我们测试这个数值求解器对静止背景下的光子分布的演化的解析解，在没有共振散射的扩展背景下的解析解，以及在共振频率周围形成与原子反冲温度相同的局部玻尔兹曼分布。我们发现，光子分布的演化经历了三个阶段：轮廓与初始阶段相似，在共振频率附近形成平坦的平台（无反冲）或局部玻尔兹曼分布（有反冲），最后当来自源的光子被膨胀的红移平衡时，共振频率附近的分布饱和。这一结果表明，W-F耦合的开始不应该由第三阶段决定，而是由第二阶段的时间尺度决定。我们发现W-F耦合的时间尺度约等于共振频率光子的平均自由飞行时间的几百倍，并且当Sobolev参数远小于1时，W-F耦合的时间尺度与Sobolev参数无关。
英文摘要：
We develop a numerical solver for the integral-differential equations, which describes the radiative transfer of photon distribution in the frequency space with resonant scattering of Lyalpha photons by hydrogen gas in the early universe. The time-dependent solutions of this equation is crucial to the estimation of the effect of the Wouthuysen-Field (WF) coupling in relation to the 21 cm emission and absorption at the epoch of reionization. The resonant scattering 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 equation. We test this numerical solver against analytic solutions of the evolution of the photon distribution in rest background, analytic solution in expanding background without resonant scattering and formation of local Boltzmann distribution around the resonant frequency with the temperature same as that of atom for recoil. We find that evolution of photon distribution undergoes three phases; profile is similar to the initial one, a flat plateau (without recoil) or local Boltzmann distribution (with recoil) forms around the resonant frequency, and finally the distribution around the resonant frequency is saturated when the photons from the source is balanced by the redshift of the expansion. This result indicates that the onset of the W-F coupling should not be determined by the third phase, but by the time scale of the second phase. We found that the time scale of the W-F coupling is equal to about a few hundreds of the mean free flight time of photons with resonant frequency, and is independent of the Sobolev parameter if this parameter is much less than 1.
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