Asymmetric Absorption Profiles of Ly\alpha\ and Ly\beta\ in Damped Lyman
  Alpha Systems
  Damped Ly\alpha\ systems (DLAs) observed in the quasar spectra are
characterized by a high neutral hydrogen column density $N_{HI}>2\times
10^{20}{\rm\ cm^{-2}}$. The absorption wing profiles are often fitted using the
Voigt function due to the fact that the scattering cross section near resonant
line center is approximately described by the Lorentzian function. Since a
hydrogen atom has infinitely many $p$ states that participate in the electric
dipole interaction, the cross section starts to deviate from the Lorentzian in
an asymmetric way in the line wing regions. We investigate this asymmetry in
the absorption line profiles around Ly\alpha\ and Ly\beta\ as a function of the
neutral hydrogen column density $N_{HI}$. In terms of
$\Delta\lambda\equiv\lambda-\lambda_\alpha$ we expand the Kramers-Heisenberg
formula around Ly\alpha\ to find $\sigma(\lambda)\simeq (0.5
f_{12})^2\sigma_{T}(\Delta\lambda/\lambda_\alpha)^{-2}[1+3.792
(\Delta\lambda/\lambda_\alpha)]$, where $f_{12}$ and $\sigma_T$ are the
oscillator strength of Ly$\alpha$ and the Thomson scattering cross section,
respectively. In terms of $\Delta\lambda_2\equiv\lambda-\lambda_\beta$ in the
vicinity of Ly\beta, the total scattering cross section, given as the sum of
cross sections for Rayleigh and Raman scattering, is shown to be
$\sigma(\lambda)\simeq \sigma_T(0.5f_{13})^2(1+R_0)
(\Delta\lambda_2/\lambda_\beta)^{-2}[1-24.68(\Delta\lambda_2/\lambda_\beta)]$
with $f_{13}$ and the factor $R_0=0.1342$ being the oscillator strength for
Ly\beta\ and the ratio of Raman cross section to Rayleigh cross section,
respectively. A redward asymmetry develops around Ly\alpha\ whereas a blue
asymmetry is obtained for Ly\beta. The absorption center shifts are found to be
almost proportional to the neutral hydrogen column density.
