ASYMMETRIC ABSORPTION PROFILES OF Lyα AND Lyβ IN DAMPED Lyα SYSTEMS

The scattering cross section is given by the Kramers–Heisenberg formula that is obtained from a second-order time-dependent perturbation theory (e.g., Sakurai 1967; Merzbacher 1970). In terms of the matrix elements of the dipole operator, the Kramers–Heisenberg formula can be written as

$$\begin{eqnarray} {d\sigma \over d\Omega }&=& {r_0^2 m_e^2 \over \hbar ^2} \left| \sum _I \omega \omega _{I1}\left({{({\bf r}\cdot {\bf e}^{(\alpha ^{\prime })})_{1I} ({\bf r}\cdot {\bf e}^{(\alpha)})_{I1}}\over {\omega _{I1}-\omega }} \right.\right. \nonumber \\ &&- \left.\left. {{({\bf r}\cdot {\bf e}^{(\alpha)})_{1I} ({\bf r}\cdot {\bf e}^{(\alpha ^{\prime })})_{I1}}\over {\omega _{I1}+\omega }} \right) \right|^2, \end{eqnarray} \tag{ 1 }$$

where m_e is the electron mass and r₀ = e²/m_ec² is the classical electron radius. The polarization vectors of incident and scattered radiation are denoted by e^(α) and ${\bf e}^{(\alpha ^{\prime })}$, respectively. Here, ω_I1 is the angular frequency between the intermediate state I and the ground 1s state. The intermediate state I that participates in the electric dipole interaction of Lyman photons consists of bound np states and free n'p states. In the atomic units adopted in this work, the bound np state has the energy eigenvalue E_n = −1/(2n²) and correspondingly ω_I1 = ω_n1 = (1 − n⁻²)/2. Similarly, for the n'p state, $E_{n^{\prime }}=1/(2n^{\prime 2})$ and $\omega _{I1}=\omega _{n^{\prime }1}=(1+n^{\prime -2})/2$. The term denoted by (r · e^(α))_I1 represents the matrix element of the position operator between the intermediate state I and the ground 1s state.

Being a two-body system, the hydrogen atom admits analytically closed expressions of the wave functions, which enables one to compute explicitly the matrix elements of the dipole operator in terms of the confluent hypergeometric function. A typical matrix element (r · ^(α))_I1 for an intermediate state with I = |np, m > is explicitly written as

$$\begin{eqnarray} ({\bf r}\cdot {\bf e}^{(\alpha)})_{I1}&=& \int _0^{2\pi }\int _0^\pi \int _0^\infty \left[R_{nl}(r)Y_l^m(\theta,\phi)\right]^* ({\bf r}\cdot {\bf e}^{(\alpha)})\nonumber\\ &&\times R_{10}(r)Y_0^0(\theta,\phi) r^2 dr \sin \theta d\theta d\phi, \end{eqnarray} \tag{ 2 }$$

where $Y_l^m(\theta,\phi)$ is a spherical harmonic function and R_nl(r) is the radial wave function given by an associated Laguerre function. The angular integration followed by summing over magnetic substates m = ±1, 0 of each np state and averaging over polarization states of incident and outgoing radiation results in a numerical factor of 8π/3, which is discussed in more detail in the Appendix.

The radial matrix elements of the dipole operators 〈r〉_n1 and $\langle r\rangle _{n^{\prime },1}$ are readily found in many textbooks on quantum mechanics (e.g., Berestetskii et al. 1971; Bethe & Salpeter 1967). The radial matrix elements for the bound np states are given by

$$\begin{equation} \langle r\rangle _{1n}=\langle 1s| r | np\rangle =\left[ {2^{8}n^7 (n-1)^{2n-5}}\over {(n+1)^{2n+5}} \right]^{1\over 2} a_B, \end{equation} \tag{ 3 }$$

where a_B = ℏ²/me² = 0.5292 Å is the Bohr radius. For the continuum n'p states, the corresponding values are given by

$$\begin{equation} \langle r\rangle _{1n^{\prime }} = \langle 1s| r| n^{\prime }p\rangle =\left[ {2^{8}(n^{\prime })^7 \exp [-4n^{\prime }\tan ^{-1}(1/n^{\prime })]}\over {[(n^{\prime })^2+1]^{5}[1-\exp (-2\pi n^{\prime })]} \right]^{1\over 2}a_B, \end{equation} \tag{ 4 }$$

which is obtained through analytic continuation into the complex plane (Bethe & Salpeter 1967; Saslow & Mills 1969).

Lee (2003) provided the expansion of the Kramers–Heisenberg formula for Rayleigh scattering in the vicinity of Lyα in terms of Δω = ω − ω₂₁. The result is summarized as

$$\begin{eqnarray} \sigma (\omega) &=& {\sigma _T} \left({f_{12}\over 2}\right)^2\left({\omega _{21}\over \Delta \omega } \right)^2 \left |1+a_1\left({\Delta \omega \over \omega _{21}}\right)\right. \nonumber \\ &&\left. +a_2\left({\Delta \omega \over \omega _{21}}\right)^2 +\cdots \right |^2, \end{eqnarray} \tag{ 5 }$$

where $\sigma _T = 8\pi r_0^2/3=0.6652\times 10^{-24}{\rm \ cm^2}$ is the Thomson scattering cross section and f₁₂ = 0.4162 is the oscillator strength for the Lyα transition. The coefficients a_k were numerically computed by Lee (2003), who gave

$$\begin{equation} a_1 = -8.961\times 10^{-1},\quad a_2 =-1.222\times 10^1. \end{equation} \tag{ 6 }$$

In Table 1, we show these coefficients up to a₅.

Up to the first order of Δω/ω₁₂, the scattering cross section can be expressed as

$$\begin{eqnarray} \sigma (\omega) &\simeq & \sigma _T \left({f_{12}\over 2}\right)^2 \left({\omega _{12}\over \Delta \omega }\right)^2 \left(1+2a_1{\Delta \omega \over \omega _{21}}\right) \nonumber \\ &=& \sigma _\alpha \left({\omega _{12}\over \Delta \omega }\right)^2 \left(1-1.792{\Delta \omega \over \omega _{21}}\right). \end{eqnarray} \tag{ 7 }$$

Here, we introduce the characteristic cross section σ_α for Lyα defined as σ_α ≡ σ_T(f₁₂/2)² = 2.880 × 10⁻²⁶ cm². The coefficient a₁, being less than zero, is responsible for the redward asymmetric deviation of the scattering cross section in frequency space.

Astronomical spectroscopy is often presented in wavelength space, which makes it necessary to express the scattering cross section in terms of Δλ = λ − λ_α, the difference in wavelength from the Lyα line center. From the following relation,

$$\begin{equation} {\Delta \omega \over \omega _\alpha } =-{\Delta \lambda \over \lambda } =\sum _{n=1}^\infty \left(-{\Delta \lambda \over \lambda _\alpha }\right)^n, \end{equation} \tag{ 8 }$$

we may notice that expansion in wavelength space will yield different coefficients from those obtained in expansion in frequency space. Substituting Equation (8) into Equation (7), we obtain

$$\begin{eqnarray} \sigma (\lambda)&\simeq &\sigma _T \left({f_{12}\over 2}\right)^2 \left({\lambda _{\alpha }\over \Delta \lambda }\right)^2 \left [1+2(1-a_1)\left({\Delta \lambda \over \lambda _{\alpha }}\right)\right. \nonumber \\ &&+\left(1-2a_1+2a_2-a_1^2\right) \left({\Delta \lambda \over \lambda _{\alpha }}\right)^2 \nonumber \\ &&\left. -2(a_3+a_1a_2) \left({\Delta \lambda \over \lambda _{\alpha }}\right)^3+\cdots \right ] \nonumber \\ &=&\sigma _\alpha \left({\lambda _{\alpha }\over \Delta \lambda }\right)^2 \left [1+3.792\left({\Delta \lambda \over \lambda _{\alpha }}\right)\right. \nonumber \\ &&\left. -20.84 \left({\Delta \lambda \over \lambda _{\alpha }}\right)^2 +83.14\left({\Delta \lambda \over \lambda _{\alpha }}\right)^3+\cdots \right ]. \end{eqnarray} \tag{ 9 }$$

Thus, up to first order, we have

$$\begin{equation} \sigma (\lambda)\simeq \sigma _\alpha \left({\lambda _{\alpha }\over \Delta \lambda }\right)^2 \left [1+3.792\left({\Delta \lambda \over \lambda _{\alpha }}\right)\right ], \end{equation} \tag{ 10 }$$

from which it is found that the coefficient 3.792 is significantly different from the coefficient −1.792 in frequency space.

The red asymmetry in the Lyα scattering cross section can be explained by noting the denominator ω_n1 − ω in Equation (1), where the dominant contribution comes from n = 2. The contributions from all the excited states with n > 2 interfere positively with the dominant contribution from n = 2 for photons on the red side whereas the interference is negative for photons on the blue side. Therefore, the red wing is strengthened relative to the pure Lorentzian profile by the positive interference of scattering from all other levels.

In a similar way, we define the difference in frequency from Lyβ by

$$\begin{equation} \Delta \omega _2 \equiv \omega -\omega _{31} \end{equation} \tag{ 11 }$$

and expand the Kramers–Heisenberg formula in terms of Δω₂/ω₃₁. We note that the terms appearing in the Kramers–Heisenberg formula include

$$\begin{equation} {\omega \omega _{31} \over \omega _{31}-\omega }=-\omega _{31} \left(1+{\omega _{31}\over \Delta \omega _2} \right) \end{equation} \tag{ 12 }$$

and

$$\begin{equation} {\omega \over \omega _{n1}-\omega } = {\omega _{31}\over \omega _{n1}-\omega _{31}} \left[ 1 +{\omega _{n1}\over \omega _{31}} \sum _{k=1}^\infty \left({\Delta \omega _2\over \omega _{n1}-\omega _{31}} \right)^k \right]. \end{equation} \tag{ 13 }$$

For n ⩾ 2, the second term in the summation in Equation (1) can be written as

$$\begin{equation} {\omega \over \omega _{n1}+\omega } = {\omega _{31}\over \omega _{n1}+\omega _{31}} \left[ 1 -{\omega _{n1}\over \omega _{31}} \sum _{k=1}^\infty \left({-\Delta \omega _2\over \omega _{n1}+\omega _{31}} \right)^k \right]. \end{equation} \tag{ 14 }$$

Similar expressions for the continuum states are obtained in a straightforward manner.

We expand the Kramers–Heisenberg formula in frequency space for Rayleigh scattering near Lyβ, which is written as

$$\begin{eqnarray} &&\sigma ^{{\rm Ray}}(\omega) = {\sigma _T\over 9} \left({\omega _{31}\over \Delta \omega _2}\right)^2 \left| \omega _{31}\langle r\rangle _{13}^2 +\omega _{31}\langle r\rangle _{13}^2 \left({\Delta \omega _2\over \omega _{31}}\right) \right. \nonumber \\ &&- \sum _{n\ne 3} {\Delta \omega _2\omega _{n1}\over \omega _{n1}-\omega _{31}} \left[ 1 +{\omega _{n1}\over \omega _{31}} \sum _{k=1}^\infty \left({\Delta \omega _2\over \omega _{n1}-\omega _{31}} \right)^k \right] \langle r\rangle _{n1}^2 \nonumber \\ &&+ \sum _{n\ne 3} {\Delta \omega _2\omega _{n1}\over \omega _{n1}+\omega _{31}} \left[ 1 -{\omega _{n1}\over \omega _{31}} \sum _{k=1}^\infty \left({-\Delta \omega _2\over \omega _{n1}+\omega _{31}} \right)^k \right] \langle r\rangle _{n1}^2 \nonumber \\ &&- \int _0^\infty dn^{\prime } {\Delta \omega _2\omega _{n^{\prime }1}\over \omega _{n^{\prime }1}-\omega _{31}} \left[ 1 +{\omega _{n^{\prime }1}\over \omega _{31}} \sum _{k=1}^\infty \left({\Delta \omega _2\over \omega _{n^{\prime }1}-\omega _{31}} \right)^k \right] \langle r\rangle _{n^{\prime }1}^2 \nonumber \\ &&+ \left. \int _0^\infty dn^{\prime } {\Delta \omega _2\omega _{n^{\prime }1}\over \omega _{n^{\prime }1}+\omega _{31}} \left[ 1 -{\omega _{n^{\prime }1}\over \omega _{31}} \sum _{k=1}^\infty \left({-\Delta \omega _2\over \omega _{n^{\prime }1}+\omega _{31}} \right)^k \right] \langle r\rangle _{n^{\prime }1}^2 \right|^2.\nonumber\\ \end{eqnarray} \tag{ 15 }$$

Here, angular integration has been performed and the atomic unit system is adopted, in which the Bohr radius a_B = 1.

An algebraic rearrangement of Equation (15) can be made to express σ^Ray(ω) as

$$\begin{eqnarray} \sigma ^{{\rm Ray}}(\omega) &=& \sigma _T\left({\omega _{31}\over \Delta \omega _2} \right)^2 \left |B_0+B_1\left({\Delta \omega _2\over \omega _{31}}\right)\right. \nonumber \\ &&\left. +B_2\left({\Delta \omega _2\over \omega _{31}}\right)^2 +\cdots \right |^2. \end{eqnarray} \tag{ 16 }$$

The coefficients B₀ and B₁ are determined through the following relations:

$$\begin{eqnarray} B_0 &=& {\omega _{31}\over 3}\langle r\rangle _{13}^2 = f_{13}/2 =0.03955\nonumber \\ B_1 &=& {1\over 2}\omega _{31}\langle r\rangle _{13}^2 -{2\over 3}\sum _{n\ne 3}{\omega _{31}^2\omega _{n1} \over \omega _{n1}^2-\omega _{31}^2}\langle r\rangle _{1n}^2 \nonumber \\ &-& {2\over 3} \int _0^\infty dn^{\prime } {\omega _{31}^2\omega _{n^{\prime }1} \over \omega _{n^{\prime }1}^2-\omega _{31}^2}\langle r\rangle _{n^{\prime }1}^2 =0.6414, \end{eqnarray} \tag{ 17 }$$

where f₁₃ = 0.07910 is the oscillator strength for the Lyβ transition. The coefficients B_k for k ⩾ 2 are given as

$$\begin{eqnarray} &&B_k =\left({-1\over 2}\right)^k {\omega _{31}\over 3}\langle r\rangle _{13}^2 \nonumber \\ &&-{1\over 3}\sum _{n\ne 3}^\infty {\omega _{n1}^2\over \omega _{31}} \left[ \left({\omega _{31}\over \omega _{n1}-\omega _{31}}\right)^k -\left({-\omega _{31}\over \omega _{n1}+\omega _{31}}\right)^k \right]\langle r\rangle _{n1}^2 \nonumber \\ &&-{1\over 3}\int _{0}^\infty dn^{\prime } {\omega _{n^{\prime }1}^2\over \omega _{31}} \left[ \left({\omega _{31}\over \omega _{n^{\prime }1}-\omega _{31}}\right)^k -\left({-\omega _{31}\over \omega _{n^{\prime }1}+\omega _{31}}\right)^k \right]\langle r\rangle _{n^{\prime }1}^2.\nonumber\\ \end{eqnarray} \tag{ 18 }$$

The numerical values of the coefficients up to k = 5 are computed as follows:

$$\begin{eqnarray} b_1=B_1/B_0 & = & 1.621\times 10^1 \nonumber \\ b_2=B_2/B_0 & = & {-}4.299\times 10^2 \nonumber \\ b_3=B_3/B_0 & = & {-}2.176\times 10^3 \nonumber \\ b_4=B_4/B_0 & = & {-}6.005\times 10^4 \nonumber \\ b_5=B_5/B_0 & = & {-}8.414\times 10^5. \end{eqnarray} \tag{ 19 }$$

In particular, the coefficient b₁ is positive due to the predominant contribution from the 2p state.

Therefore, up to the first-order approximation in Δω₂/ω₃₁, the Rayleigh scattering cross section around Lyβ is given by

$$\begin{eqnarray} \sigma ^{{\rm Ray}}(\omega) &\simeq & \sigma _T\left({f_{13}\over 2}\right)^2 \left({\omega _{13}\over \Delta \omega _2}\right)^2 \left(1+2b_1{\Delta \omega _2\over \omega _{31}}\right) \nonumber \\ &\simeq & \sigma _T \left({f_{13}\over 2}\right)^2 \left({\omega _{13}\over \Delta \omega _2}\right)^2 \left(1+32.42{\Delta \omega _2\over \omega _{31}}\right). \end{eqnarray} \tag{ 20 }$$

In wavelength space, the Rayleigh scattering cross section in the vicinity of Lyβ with the line center λ_β = 1025.722 Å can be expanded as

$$\begin{eqnarray} \sigma ^{{\rm Ray}}(\lambda) & \simeq & \sigma _T \left({f_{13}\over 2}\right)^2 \left({\lambda _{\beta }\over \Delta \lambda _2}\right)^2 \left [1+2(1-b_1)\left({\Delta \lambda _2\over \lambda _{\beta }}\right)\right. \nonumber \\ &&+\left(1-2b_1+2b_2-b_1^2\right) \left({\Delta \lambda _2\over \lambda _{\beta }}\right)^2 \nonumber \\ &&\left. -2(b_3+b_1b_2) \left({\Delta \lambda _2\over \lambda _{\beta }}\right)^3+\cdots \right ] \nonumber \\ &=&\sigma _T\left({f_{13}\over 2}\right)^2 \left({\lambda _{\beta }\over \Delta \lambda _2}\right)^2 \left [1-31.64\left({\Delta \lambda _2\over \lambda _{\beta }}\right)\right. \nonumber \\ &-& \left. 5.970\times 10^2 \left({\Delta \lambda _2\over \lambda _{\beta }}\right)^2 +1.792\times 10^4\left({\Delta \lambda _2\over \lambda _{\beta }}\right)^3\cdots \right ],\nonumber\\ \end{eqnarray} \tag{ 21 }$$

where Δλ₂ ≡ λ − λ_β is the wavelength deviation from the Lyβ center.

The negative value of the coefficient 2(1 − b₁) implies that the Rayleigh scattering cross section near Lyβ is asymmetric to the blue of Lyβ, which is in high contrast with the behavior around Lyα. In the case of Lyβ, the main contributor to the Kramers–Heisenberg formula is the 3p state and the residual contribution comes from the 2p state and all the p states lying higher than the 3p state. The contribution to σ(λ) of a given np or n'p state is measured roughly by the oscillator strength inversely weighted by the energy difference from Lyβ. Therefore, the contribution of the 2p state is more important than that from all the p states lying higher than the 3p state. Hence, in the case of Lyβ, the n = 2 contribution has lower frequency and the interference with the principal n = 3 scattering contribution is negative on the red side and positive on the blue side, which explains the blue asymmetric scattering cross section.

Interaction with a hydrogen atom of electromagnetic radiation around Lyβ has another channel, which is inelastic or Raman scattering. The scattering hydrogen atom de-excites into the 2s state re-emitting an Hα photon into another line of sight, which provides an important contribution to the absorption profile around Lyβ. The astrophysical importance of Raman scattering can be appreciated in the emission features at 6830 Å and 7088 Å that appear in the spectra of about a half of symbiotic stars. These are formed through Raman conversion of the resonance doublet O vi λλ1032,1038 (Schmid 1989; Nussbaumer et al. 1989). Another example of Raman scattering by atomic hydrogen is provided by far-UV He ii emission lines in symbiotic stars and young planetary nebulae (e.g., Birriel 2004; Lee et al. 2006; Lee 2012). It has also been proposed that broad Hα wings often found in planetary nebulae and symbiotic stars are formed through Raman scattering of far-UV continuum around Lyβ (e.g., Lee 2000; Arrieta & Torres-Peimbert 2003).

As is illustrated in Sakurai (1967), the term corresponding to the "seagull graph" is absent in the Kramers–Heisenberg formula for the case of Raman scattering. This difference allows an alternate expression of the Kramers–Heisenberg formula given in terms of the matrix elements of the momentum operator (see also Saslow & Mills 1969; Lee & Lee 1997). In a manner analogous to what is illustrated in the Appendix, taking angular integrations, summing over magnetic substates, and averaging over polarizations of incident and outgoing radiation, we arrive at an explicit expression of the Raman cross section given by

$$\begin{eqnarray} \sigma ^{{\rm Ram}}(\omega) &=&{\sigma _T\over 9}\left({\omega ^{\prime }\over \omega }\right) \left |\sum _{n=3}^\infty \langle p\rangle _{n1}\langle p\rangle _{n2} \left({1\over \omega _{n1}-\omega }+{1\over \omega _{n1}+\omega ^{\prime }} \right)\right. \nonumber \\ &&\left. +\int _0^\infty dn^{\prime } \langle p\rangle _{n^{\prime }1}\langle p\rangle _{n^{\prime }2} \left({1\over \omega _{n^{\prime }1}-\omega }+{1\over \omega _{n^{\prime }1}+\omega ^{\prime }} \right) \right |^2.\nonumber\\ \end{eqnarray} \tag{ 22 }$$

Here ω' = ω − ω₂₁ is the angular frequency of the Raman scattered radiation. The matrix element 〈p〉_n1 associated with the momentum operator between the np and the 1s states is given by

$$\begin{eqnarray} \langle p\rangle _{n1}&=&\int _0^\infty R_{n1}(r) \left[{d\over d r}R_{10}(r)\right] r^2 dr\nonumber\\ &=& \left[{ 2^6 n^3 (n-1)^{2n-3}\over (n+1)^{2n+3}} \right]^{1/2}. \end{eqnarray} \tag{ 23 }$$

Here, an atomic unit system is adopted and the reality of the radial wave functions is noted.

The matrix element 〈p〉_n2 corresponds to the transition between the np and 2s states, which is explicitly given by

$$\begin{equation} \langle p\rangle _{n2}=\left[{ 2^{11} n^3 (n^2-1)(n-2)^{2n-4}\over (n+2)^{2n+4}} \right]^{1/2}. \end{equation} \tag{ 24 }$$

The contribution from the continuum n'p states is obtained by considering the matrix elements of the momentum operator given by

$$\begin{eqnarray} \langle p\rangle _{n^{\prime }1} &=&\left[{ 2^6 n^{\prime 3} e^{-4n^{\prime }\tan ^{-1}{1\over n^{\prime }}}\over (n^{\prime 2}+1)^3(1-e^{-2\pi n^{\prime }})} \right]^{1/2} \nonumber \\ \langle p\rangle _{n^{\prime }2} &=&\left[{ 2^{11} n^{\prime 3}(n^{\prime 2}+1) e^{-4n^{\prime }\tan ^{-1}{2\over n^{\prime }}}\over (n^{\prime 2}+4)^4(1-e^{-2\pi n^{\prime }})} \right]^{1/2}. \end{eqnarray} \tag{ 25 }$$

Due to the vanishing matrix element 〈p〉_n2 for n = 2, the sum in Equation (22) begins from n = 3 for the bound np states, which implies that the 2p state does not contribute to the cross section for Raman scattering around Lyβ. This is decisively important to the behavior of the cross section, as we discuss later in more detail.

The terms involving angular frequencies can be rearranged using the following relation,

$$\begin{eqnarray} {\omega ^{\prime }\over \omega } &=& {\omega _{31}-\omega _{21}+\Delta \omega \over \omega _{31}+\Delta \omega } \nonumber \\ &=&{\omega _{32}\over \omega _{31}}- {\omega _{21}\over \omega _{31}} \sum _{k=1}^\infty \left({-\Delta \omega \over \omega _{31}}\right)^k, \end{eqnarray} \tag{ 26 }$$

where ω₃₂ = ω₃₁ − ω₂₁ is the angular frequency for Hα. Use is also made of the following relations:

$$\begin{eqnarray} {1\over \omega _{n1}-\omega } &=&{1\over \omega _{31}} \sum _{k=0}^\infty \left({\Delta \omega \over \omega _{31}} \right)^k \left({\omega _{31}\over \omega _{n1}-\omega _{31}}\right)^{k+1} \quad {\rm for}\quad {n\ge 4} \nonumber \\ {1\over \omega _{n1}+\omega ^{\prime }}&=&{-}{1\over \omega _{31}} \sum _{k=0}^\infty \left({\Delta \omega \over \omega _{31}} \right)^k \left({-\omega _{31}\over \omega _{n1}+\omega _{31}-\omega _{21}}\right)^{k+1}.\nonumber\\ \end{eqnarray} \tag{ 27 }$$

The Kramers–Heisenberg formula for Raman scattering near Lyβ can be expanded in frequency space as follows:

$$\begin{eqnarray} \sigma ^{{\rm Ram}}(\omega)&=& \sigma _T\left({\omega _{31}\over \Delta \omega }\right)^2 \left({\omega ^{\prime }\over \omega }\right)\nonumber\\ &&\times | C_0+C_1(\Delta \omega /\omega _{31})+C_2(\Delta \omega /\omega _{31})^2+\cdots |^2,\nonumber\\ \end{eqnarray} \tag{ 28 }$$

where the coefficients C_k are given by

$$\begin{eqnarray} C_{0} &=&-{\langle p\rangle _{32}\langle p\rangle _{31}\over 3\omega _{31}} ={-}{1\over 2} \left({\omega _{32}\over \omega _{31}}\right)^{1/2}(f_{13}f_{2s,3p})^{1/2} \nonumber \\ &=&{-}3^4\cdot 2^{1/2}\cdot 5^{-5} =-0.03666 \nonumber \\ C_1&=&{\langle p\rangle _{32}\langle p\rangle _{31}\over 3(2\omega _{31}-\omega _{21})}+{1\over 3}\sum _{n\ge 4} \langle p\rangle _{n2}\langle p\rangle _{n1}\nonumber\\ &&\times \left({1\over \omega _{n1}-\omega _{31}}+ {1\over \omega _{n1}+\omega _{31}-\omega _{21}} \right) \nonumber \\ &&+{1\over 3}\int _0^\infty dn^{\prime } \langle p\rangle _{n^{\prime }2}\langle p\rangle _{n^{\prime }1}\left({1\over \omega _{n^{\prime }1}-\omega _{31}} \right.\nonumber\\ && \left.+ {1\over \omega _{n^{\prime }1}+\omega _{31}-\omega _{21}} \right)=1.018 \end{eqnarray} \tag{ 29 }$$

and

$$\begin{eqnarray} C_k &=&\!\sum _{n\ge 4} {\langle p\rangle _{n2}\langle p\rangle _{n1}\over 3\omega _{31}}\left [\!\! \left({\omega _{31}\over \omega _{n1}-\omega _{31}} \right)^k\!-\! \left(\!{-\omega _{31}\over \omega _{n1}+\omega _{31}-\omega _{21}}\! \right)^k \!\right ] \nonumber \\ &&+\int _0^\infty dn^{\prime } {\langle p\rangle _{n^{\prime }2}\langle p\rangle _{n^{\prime }1}\over 3\omega _{31}}\left [ \left({\omega _{31}\over \omega _{n^{\prime }1}-\omega _{31}}\right)^k\right. \nonumber\\ &&\left. -\left({-\omega _{31}\over \omega _{n^{\prime }1}+\omega _{31}-\omega _{21}} \right)^k \right ], \end{eqnarray} \tag{ 30 }$$

for k ⩾ 2. Here, $f_{2s,3p}={(2/3)}\omega _{32}^{-1}[\langle p\rangle _{32}]^2=0.4349$ is the oscillator strength between the 2s and 3p states (e.g., Bethe & Salpeter 1967).

Therefore, in frequency space the Raman scattering cross section is written as

$$\begin{eqnarray} \sigma ^{{\rm Ram}}(\omega) &=& \sigma _T \left({\omega _{31}\over \Delta \omega _2} \right)^2 {\omega _{32}\over \omega _{31}} |C_0|^2\nonumber\\ &&\times \left[1+\left({\omega _{21}\over \omega _{31}}+{2C_1\omega _{32}\over C_0\omega _{31}}\right) \left({\Delta \omega _2\over \omega _{31}} \right)+\cdots \right] \nonumber \\ &=& \sigma _T \left({\omega _{31}\over \Delta \omega _2} \right)^2 {\omega _{32}\over \omega _{31}} |C_0|^2 \left[ 1-7.832\left({\Delta \omega _2\over \omega _{31}}\right) +\cdots \right].\nonumber\\ \end{eqnarray} \tag{ 31 }$$

In wavelength space, we obtain

$$\begin{eqnarray} \sigma ^{{\rm Ram}}(\lambda) &\simeq & \sigma _T \left({\lambda _\beta \over \Delta \lambda _2} \right)^2 {5\over 32} |C_0|^2 \left [1+\left(-{17\over 5}-2c_1\right) \left({\Delta \lambda _2\over \lambda _\beta }\right)\right. \nonumber \\ &&+\left(c_1^2+{44\over 5}c_1 -{49\over 5}+2c_2 \right) \left({\Delta \lambda _2\over \lambda _\beta } \right)^2 \nonumber \\ &&+\left(-2c_3-2c_1c_2-{27\over 5}\left(c_1^2-2c_1+1+2c_2\right) \right)\nonumber\\ &&\left. \times \left({\Delta \lambda _2\over \lambda _\beta }\right)^3+\cdots \right ] \nonumber \\ &=& \sigma _T \left({\lambda _\beta \over \Delta \lambda _2} \right)^2 {5\over 32} |C_0|^2[1+5.223\times 10^1(\Delta \lambda _2/\lambda _\beta) \nonumber \\ &&+ 9.103\times 10^2(\Delta \lambda _2/\lambda _\beta)^2 -8.267\nonumber\\ &&\times 10^3(\Delta \lambda _2/\lambda _\beta)^3+\cdots ], \end{eqnarray} \tag{ 32 }$$

where the lower case coefficients c_i are defined by c_i = C_i/C₀. The numerical values of these coefficients up to c₅ are shown in Table 1.

Unlike the case for Rayleigh scattering near Lyβ, the Raman scattering cross section shows a redward asymmetry with respect to the Lyβ center. This result can be traced to the fact that no contribution is made from the 2p state, which played a dominant role in the case of Rayleigh scattering near Lyβ. In the absence of the 2p contribution, all the perturbing p states are more energetic than Lyβ. This situation is exactly the same as the Rayleigh scattering around Lyα resulting in a redward asymmetry.

The ratio r_b(λ) of the cross sections for Raman scattering to Rayleigh scattering in the vicinity of Lyβ is given by

$$\begin{equation} r_b(\lambda)={\sigma ^{{\rm Ram}}(\lambda)\over \sigma ^{{\rm Ray}}(\lambda)}=R_0+ R_1\left({\Delta \lambda _2\over \lambda _{\beta }} \right)+R_2\left({\Delta \lambda _2\over \lambda _\beta }\right)^2+\cdots, \end{equation} \tag{ 33 }$$

where the first three coefficients are explicitly R₀ = (5|C₀|²/32)/(f₁₃/2)² = 2¹⁸5⁻⁹ = 0.1342, R₁ = 11.26, and R₂ = 535.9. This result shows discrepancy with that provided by Yoo et al. (2002), in which there is an error in their numerical calculation of the coefficients A₃ and A₄. The leading term can also be expressed as R₀ = (f_{2s, 3p}/f₁₃)(ω₃₂/ω₃₁)² = 0.1342, which implies that the branching ratio is determined by a combination of the oscillator strength and the phase-space volume factor represented by ω². It is seen that the dominant contribution is made by the phase-space volume available to scattered radiation.

In Figure 1, we show the branching ratio r_b(λ) in the neighborhood of Lyβ. The solid line shows the result from a direct numerical computation of the Kramers–Heisenberg formula. The dotted line shows the linear fit and the dot-dashed line shows the second-order fit using Equation (33). Because the coefficient R₂ is large, the nonlinearity of r_b(λ) is quite conspicuous in the figure. This behavior leads to a redward shift in broad Hα wings observed in young planetary nebulae and symbiotic stars which are also attributed to Raman scattering of Lyβ (Jung & Lee 2004).

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In this subsection, we combine the results of previous subsections to provide the expansion of the total scattering cross section around Lyβ. The total scattering cross section around Lyβ is the sum of Equations (20) and (31), which is, to the first order of Δω₂/ω₃₁, given by

$$\begin{equation} \sigma _{{\rm tot}}(\omega) \simeq \sigma _\beta \left({\omega _{31}\over \Delta \omega _2}\right)^2 \left[ 1+31.37\left({\Delta \omega _2\over \omega _{31}}\right) \right]. \end{equation} \tag{ 34 }$$

Here, we introduce another parameter σ_β defined by

$$\begin{equation} \sigma _\beta =\sigma _T (f_{13}/2)^2[1+0.1342]=1.180\times 10^{-27}{\rm \ cm^2}. \end{equation} \tag{ 35 }$$

In wavelength space, we may combine Equations (21) and (32) to express the total scattering cross section around Lyβ as

$$\begin{equation} \sigma _{{\rm tot}}(\lambda)\simeq \sigma _\beta \left({\lambda _{\beta }\over \Delta \lambda _2}\right)^2 \left[ 1-24.63\left({\Delta \lambda _2\over \lambda _{\beta }}\right) \right]. \end{equation} \tag{ 36 }$$

From this result, it is seen that the Lyβ absorption profiles tend to shift blueward of the Lyβ line center. In Figure 2, we show the total scattering cross section obtained from a numerical evaluation of the Kramers–Heisenberg formula around Lyα and Lyβ in wavelength space. The vertical axis shows the logarithm to the base 10 of σ(λ) in units of cm².

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In order to take a clear view of the asymmetry of the scattering cross section, we plot the same quantities in Figure 3 as a function of the absolute value of the wavelength deviation. The cross sections redward of Lyα and Lyβ are shown with solid lines in the upper panel and the lower panel, respectively. The dotted lines show the cross sections blueward of Lyα and Lyβ. The dotted lines are mirror images of the curves blueward of Lyα and Lyβ shown in Figure 2. In Figure 3, we see that red Lyα photons have a larger scattering cross section than blue counterparts and that the opposite is the case for Lyβ.

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In Figure 4, we show the transmission coefficient $t(\lambda,N_{{\rm H\,\scriptsize{I}}})$ defined by

$$\begin{equation} t(\lambda, N_{{\rm H\,\scriptsize{I}}}) \equiv 1-\exp [-\sigma (\lambda)N_{{\rm H\,\scriptsize{I}}}] \end{equation} \tag{ 37 }$$

for various neutral hydrogen column densities. The upper panel is for Lyα and the lower panel is for Lyβ. The solid line shows the result for $N_{{\rm H\,\scriptsize{I}}}=10^{20}{\rm \ cm^{-2}}$, the dotted line for $N_{{\rm H\,\scriptsize{I}}}=10^{21}{\rm \ cm^{-2}}$, and the dashed line for $N_{{\rm H\,\scriptsize{I}}}=5\times 10^{21}{\rm \ cm^{-2}}$. In the case of $N_{{\rm H\,\scriptsize{I}}}=5\times 10^{21}{\rm \ cm^{-2}}$, the asymmetry is quite noticeable in the scale shown in the figure.

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In this subsection, we investigate the shift of the absorption line center near Lyα and Lyβ as a function of the neutral hydrogen column density. Denoting by λ_c the line center wavelength of Lyα or Lyβ, the scattering cross section is approximated to the first order of dimensionless wavelength deviation from the line center x = Δλ/λ_c by a function

$$\begin{equation} f(x)={1\over x^2}+{a\over x}. \end{equation} \tag{ 38 }$$

The equation f(x) = k > 0 has two solutions x₁, x₂, of which the mean is x_m = a/(2k). This implies that the absorption center can be meaningfully defined when we fix the value of cross section. The sign of the coefficient a determines the direction of asymmetry, where a positive a results in a red asymmetry.

Given a value of the H i column density $N_{{\rm H\,\scriptsize{I}}}$ we define the mean wavelength $\lambda _{m1}^\alpha$ of the two wavelengths λ_{1, 2} at which $\tau (\lambda _1)=\tau (\lambda _2)=\sigma (\lambda)N_{{\rm H\,\scriptsize{I}}}=1$ around Lyα and in a similar way we define $\lambda _{m1}^\beta$ for Lyβ. We also introduce $\lambda _{m2}^{\alpha }$ and $\lambda _{m2}^\beta$ as the mean value of the two wavelengths $\lambda _{1,2}^{\prime }$ where we have $\tau (\lambda _{1}^{\prime })=\tau (\lambda _2^{\prime })=0.5$ around Lyα and Lyβ, respectively. Corresponding to these wavelengths λ_m1, we define the velocity shift ΔV₁ by the relation

$$\begin{equation} \Delta V_1^{\alpha }\equiv c\left(\lambda _{m1}^\alpha -\lambda _\alpha\right)/\lambda _\alpha \end{equation} \tag{ 39 }$$

for Lyα and in a similar way $\Delta V^\beta _1$ is defined for Lyβ. Here, c is the speed of light.

In Table 2, we show the values of $\lambda _{m1}^\alpha$ and $\lambda _{m2}^\alpha$ for various neutral hydrogen column densities. Also in Table 3 we show the quantities corresponding to the Lyβ transitions. At $N_{{\rm H\,\scriptsize{I}}}=10^{21}{\rm \ cm^{-2}}$ we obtain a redward center shift in the amount of Δλ = +16 km s⁻¹ for Lyα and a blueward shift of Δλ = −3.9 km s⁻¹ for Lyβ.

Table 2. Absorption Center Shifts Around Lyα for Various Neutral Hydrogen Column Densities

log $N_{{\rm H\,\scriptsize{I}}}$	$\lambda _{m1}^\alpha$	$\Delta V_1^\alpha$	$\lambda _{m2}^\alpha$	$\Delta V_2^\alpha$
	(Å)	(km s−1)	(Å)	(km s−1)
19.0	6.10E-04	0.151	1.34E-03	0.331
19.7	3.30E-03	0.813	6.47E-03	1.60
20.0	6.47E-03	1.60	1.32E-02	3.25
20.7	3.26E-02	8.04	6.51E-02	16.0
21.0	6.51E-02	16.0	1.30E-01	32.1
21.7	3.25E-01	80.2	6.51E-01	1.60E+02
22.0	6.51E-01	1.60E+02	1.31	3.20E+02
22.7	3.25	8.01E+02	6.48	1.60E+03

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Table 3. Absorption Center Shifts Around Lyβ for Various Neutral Hydrogen Column Densities

log $N_{{\rm H\,\scriptsize{I}}}$	$\lambda _{m1}^\beta$	$\Delta V_1^\beta$	$\lambda _{m2}^\beta$	$\Delta V_2^\beta$
	(Å)	(km s−1)	(Å)	(km s−1)
19.0	−1.22E-04	−0.03568	−2.44E-04	−0.07136
19.7	−7.32E-03	−0.214	−1.34E-03	−0.392
20.0	−1.34E-03	−0.392	−2.69E-02	−0.785
20.7	−6.59E-03	−1.93	−1.34E-02	−3.92
21.0	−1.34E-02	−3.92	−2.67E-02	−7.81
21.7	−6.64E-02	−19.4	−1.32E-01	−38.5
22.0	−1.32E-01	−38.5	−2.60E-01	−75.9
22.7	−6.22E-01	−1.82E+02	−1.16	−3.38E+02

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In Figure 5, we show $\Delta V_1^\alpha$ and $\Delta V_2^\alpha$ as a function of $N_{{\rm H\,\scriptsize{I}}}$ in the cases of Lyα (upper panel) and $\Delta V^\beta _1$ and $\Delta V^\beta _2$ for Lyβ (lower panel). The dotted line shows a fit to the data, which implies that both $\Delta V_{1}^{\alpha,\beta }$ and $\Delta V_{2}^{\alpha,\beta }$ are proportional to $N_{{\rm H\,\scriptsize{I}}}$. The linear fit shown by the dotted line for Lyα in the figure is given by

$$\begin{equation} \Delta V_1^\alpha \simeq 1.6 \left[{N_{{\rm H\,\scriptsize{I}}}\over 10^{20} {\rm \ cm^{-2}}}\right]{\rm \ km\ s^{-1}}, \end{equation} \tag{ 40 }$$

and similarly for Lyβ it is given by

$$\begin{equation} \Delta V_1^\beta \simeq -0.39 \left[{N_{{\rm H\,\scriptsize{I}}}\over 10^{20} {\rm \ cm^{-2}}}\right]{\rm \ km\ s^{-1}}. \end{equation} \tag{ 41 }$$

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Another way of quantifying the asymmetry is provided by fitting the absorption profiles. In this subsection, we compare the transmission coefficient $t(\lambda,N_{{\rm H\,\scriptsize{I}}})$ derived from the Kramers–Heisenberg formula with that obtained from the Lorentzian shifted by a finite amount. For simplicity, we fix the H i column density $N_{{\rm H\,\scriptsize{I}}}=5\times 10^{21}{\rm \ cm^{-2}}$. This procedure may illustrate an error estimate in determining the redshift of a DLA system with $N_{{\rm H\,\scriptsize{I}}}=5\times 10^{21}{\rm \ cm^{-2}}$.

In Figure 6 we show the result for Lyα. The solid line in each panel shows the transmission coefficient obtained from the Kramers–Heisenberg formula. The dotted line in the top panel shows the transmission coefficient from the Lorentzian function, which provides an excellent fit near the line center. However, a considerable deviation in the wing part is quite noticeable. With the dotted line in the bottom panel, we show the quantities obtained by shifting the Lorentzian redward by an amount of +0.8 Å. Improvement of the fitting in wing parts is achieved only at the expense of a poor approximation near the line center. In the middle panel, we show the Lorentzian shifted by +0.4 Å, in which the quality of the fit is compromised between the previous two cases.

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In the analysis by Lee (2003), the optimal wavelength shift was proposed by +0.2 Å, which is smaller than +0.4 Å suggested in this work. This discrepancy is due to the fact that the fitting procedure in Lee (2003) was confined to a rather narrow interval of |Δλ| < 34 Å excluding extreme wing parts. The procedure of fitting a DLA profile using the shifted Lorentzian tends to overestimate the line center wavelength of Lyα leading to corresponding overestimate of the redshift of the DLA. We note that Lee (2003) made a mistake in pointing out that the redshift would be "underestimated," which should be corrected to be "overestimated."

A similar analysis corresponding to Lyβ is shown in Figure 7 for the same neutral hydrogen column density $N_{{\rm H\,\scriptsize{I}}}=5\times 10^{21} {\rm \ cm^{-2}}$. In the figure, the dotted line in each panel shows the transmission coefficient from the Lorentzian function (top panel) and shifted Lorentzian functions (middle and bottom panels), whereas the solid line shows the exact transmission coefficient computed from the Kramers–Heisenberg formula. The amount of wavelength shift blueward of Lyβ is Δλ = −0.1 Å and −0.2 Å for the middle and bottom panels, respectively. As in the case of Lyα illustrated in Figure 6, the unshifted Lorentzian gives an excellent fit to the core part of the absorption profile whereas the bottom panel shows an improved fit to the wing parts with the loss of fitting quality at the core part.

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In Figure 8, we present the transmission coefficients using the Kramers–Heisenberg formula and the Lorentzian functions around Lyα and Lyβ in the wavelength interval between 980 Å and 1400 Å for a very thick H i medium with $N_{{\rm H\,\scriptsize{I}}}=5\times 10^{22}{\rm \ cm^{-2}}$. This kind of an extreme neutral hydrogen column density has been found toward the gamma-ray burst GRB080607 (e.g., Prochaska et al. 2009). For comparison, we show the transmission coefficients obtained from the Lorentzian around Lyα and Lyβ by the dotted line and the dashed line, respectively. In this highly thick medium, the deviation from the Lorentzian is quite severe due to the contribution from higher order terms, which prevents one from obtaining satisfactory results by fitting the absorption profiles by a Voigt or equivalently a Lorentzian function.

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In particular, in the wavelength range shown in Figure 8, the local peak transmission is found at λ_p = 1062 Å, for which $t(\lambda, N_{{\rm H\,\scriptsize{I}}}=5\times 10^{22} {\rm \ cm^{-2}})=0.0144$. However, the sum of two Lorentzian functions around Lyα and Lyβ admits a local maximum at λ = 1070 Å. This shows the inadequacy of using a Voigt function for fitting analyses in extended wing parts in the case of very high column density systems. Furthermore, at this high $N_{{\rm H\,\scriptsize{I}}}$, the blue wing region of Lyα overlaps with that of the red Lyβ wing, for which a full quantum mechanical formula should be invoked for an accurate analysis.