Here we give a simple estimate for the population of 2s-state hydrogen atoms by considering a three-level system (1s, 2s, and 3p). We set the rate equation for the 2s population as
(1)
\begin{eqnarray*}
n_{\rm H,1s}C_{\rm 1s,2s}+n_{\rm H,3p}A_{\rm 3p,2s}-n_{\rm H,2s}A_{\rm 2s,1s}=0,
\end{eqnarray*}
where nH, j is the number density of hydrogen atoms in the state j; Cj, k and Aj, k are the collisional excitation rate and spontaneous decay rate for the transition from j to k, respectively. For the bound states, we use the notation j = njlj, where nj is the principal quantum number of the state j. Similarly, lj = 0, 1, 2, 3, ..., nj − 1 (equivalently: s, p, d, f,...) is the orbital angular-momentum quantum number of the state j. Here we suppose that the depopulation term of 2s-state atoms is dominated by the spontaneous transition at the rate of A2s, 1s ≃ 8.2 s−1. In reality, the collisional transition from 2s to 2p can be a subdominant process for depopulation. For a collision at a velocity ∼108 cm s−1, which is a typical velocity scale for young SNR shocks, the cross-section is ∼10−13 cm2, giving a reaction rate ∼10−5 cm3 s−1 (e.g. Janev, Langer & Evans 1987; Sahal-Brechot et al. 1996). Thus, if the density is ∼106 cm−3, the collisional depopulation becomes important. Note that we assume no strong radiation field inducing the radiative transition from 2s to any other state.1 The occupation number of 3p, nH, 3p, depends on the absorption of Ly β. Here we assume an isotropic radiation field for Ly β. Then, we obtain the rate equation for 3p as
(2)
\begin{eqnarray*}
n_{\rm H,1s} \left(C_{\rm 1s,3p}{+}\int _0^{\infty } \frac{4\pi \sigma _\nu ^{\rm 1s,3p}}{h\nu }I_\nu {\rm d}\nu \right) {-}n_{\rm H,3p} \left(A_{\rm 3p,1s}+A_{\rm 3p,2s}\right){=}0, \nonumber \\
\end{eqnarray*}
where h, ν, $\sigma _\nu ^{\rm 1s,3p}$, and Iν are the Planck constant, frequency, absorption cross-section for the transition from 1s to 3p and the specific intensity, respectively. The intensity is set to be
(3)
\begin{eqnarray*}
I_\nu =S_\nu (1-{\rm e}^{-\tau _\nu }) =\frac{ \frac{h\nu }{4\pi } A_{\rm 3p,1s} n_{\rm H,3p} }{\sigma ^{\prime } n_{\rm H,1s}} (1-{\rm e}^{-\tau _\nu }),
\end{eqnarray*}
where Sν and τν are the source function and optical depth, respectively. σ′ is a combination of physical constants relevant to the radiative absorption cross-section. Thus, we derive the occupation number of 2s as
(4)
\begin{eqnarray*}
n_{\rm H,2s}=\frac{C_{\rm 1s,2s}}{A_{\rm 2s,1s}} \left[ 1+\frac{A_{\rm 3p,2s}}{ {\rm e}^{-\tau _0}A_{\rm 3p,1s}+A_{\rm 3p,2s} } \frac{ C_{\rm 1s,3p} }{ C_{\rm 1s,2s} } \right] n_{\rm H,1s},
\end{eqnarray*}
where τ0 is the optical depth at the line centre. Here we assume a narrow line profile function ϕν for which we can approximate as $\int _0^{\infty }(1-{\rm e}^{-\tau _\nu }){\rm d}\nu \approx 1-{\rm e}^{-\tau _0}$. The terms in the brackets [...] indicate the contribution of the combination of the absorption and cascades. Note that roughly say, the ratios are A3p, 2s/(A3p, 1s + A3p, 2s) ≃ 0.118, C1s, 3p/C1s, 2s ∼ 2–10, and C1s, 2s/A2s, 1s ∼ 10−9np, where np is the proton number density. Thus, if Ly β is in the optically thick limit, nH, 2s is enhanced roughly at most 10 times compared with the optically thin case. The absorption coefficient of H α at the line centre becomes
(5)
\begin{eqnarray*}
k_0({\rm H\alpha }) &amp;=&amp; \sigma _0({\rm H\alpha }) n_{\rm H,2s} \nonumber \\
&amp;&amp;{\sim} 10^{-23}\!-\!10^{-22}{\rm cm^{-1}} \!\left(\frac{T_0}{6000{\rm K}}\right)^{-\frac{1}{2}}\! \left(\frac{n_{\rm H,1s}}{ {\rm 1cm^{-3} } }\right)\! \left(\frac{n_{\rm p}}{ {\rm 1cm^{-3} } }\right)\!, \nonumber \\
\end{eqnarray*}
where σ0(Hα) is the radiative cross-section of H α at the line centre for given temperature T0. Thus, if the SNR shock interacts with somewhat dense clump with a density of ∼30 cm−3 and a size of ∼1 pc, the H α emission can be scattered. Note that the H β emission can also be scattered but its absorption coefficient is about quarter of the H α coefficient. The interaction between the shock and a dense clump is implied by the ripple of an SNR shock with a length-scale of ∼10 per cent of SNR radius (e.g. Ishihara et al. 2010; Williams et al. 2013, 2016; Miceli et al. 2014; Sano et al. 2017; Tsubone et al. 2017, and see the discussion of Shimoda et al. 2015). Note that according to magnetohydrodynamic simulations performed by Inoue, Yamazaki & Inutsuka (2009) and Inoue et al. (2012), even if the shock propagates into a simulated ISM having density contrast ranging in ∼1–30 cm−3 as a consequence of thermal instability, the scale length of rippling is ∼10 per cent of the length of sides of simulation box.
Citation: Sahal-Brechot et al. 1996
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