Asymmetric Deviation of the Scattering Cross Section around Ly$\alpha$ by Atomic Hydrogen

We investigate the asymmetry of the scattering cross section of radiation around Ly$\alpha$ by atomic hydrogen, which may be applied to analyses of scattering media with high column neutral hydrogen densities including damped Ly$\alpha$ absorption systems of quasars. The exact scattering cross section is given by the Kramers-Heisenberg formula obtained from the fully quantum mechanical second-order time dependent theory, where, in the case of hydrogen, each matrix element is given in a closed analytical form. The asymmetric deviation of the scattering cross section from the Lorentzian near the line center is computed by expanding the Kramers-Heisenberg formula in terms of $\Delta\omega /\omega_{Ly\alpha}$,where $\omega_{Ly\alpha}$ is the angular frequency of the Ly$\alpha$ transition and $\Delta\omega$ is the deviation of incident radiation from $\omega_{Ly\alpha}$. To the first order of $\Delta\omega/\omega_{Ly\alpha}$, we obtain $\sigma(\omega) = \sigma_T (0.5 f_{12} \omega_{Ly\alpha}/\Delta\omega)^2 (1-1.79\Delta\omega / \omega_{Ly\alpha})$, where $\sigma_T$ is the Thomson scattering cross section and $f_{12}=0.4162$ is the oscillator strength for the Ly$\alpha$ transition. With this deviation, the line center of the damped wing profile apparently shifts blueward of the true Ly$\alpha$ line center. In the case of a dampedLy$\alpha$ system with a H I column density $5\times 10^{21}{\rm cm^{-2}}$, the apparent line center shift relative to the true center amounts to $0.2{\rm \AA}$ resulting in an underestimation of redshift by $\Delta z\sim 10^{-4}$. A measurable underestimation by an amount of $\Delta z\sim 10^{-3}$ is expected for absorbing systems with $N_{HI}\ge 4\times 10^{22}{\rm cm^{-2}}$.