THE ENERGETICS OF MOLECULAR GAS IN NGC 891 FROM H₂ AND FAR-INFRARED SPECTROSCOPY

The low-lying rotational lines of H₂ are optically thin and easily thermalized so that the lines and their ratios probe the temperature and column density of the emitting gas. The S(2) and S(0) lines both arise from the para species of H₂ so that it in the high-density limit (n$_{\rm H_2} >$ n_crit(J = 4) ∼10³ cm⁻³; Le Bourlot et al. 1999), the I_S(2)/I_S(0) ratio is given by the para excitation temperature T_ex:

$$\begin{equation} \frac{I_{S(2)}}{I_{\rm S(0)}} = \frac{A_{S(2)}}{A_{S(0)}} \frac{\nu _{S(2)}}{\nu _{S(0)}} \frac{g_{S(2)}}{g_{S(0)}}e^{-\Delta E/T_{\rm ex}}, \end{equation} \tag{ 1 }$$

where A_i is the Einstein A coefficient for spontaneous emission of level i (A_S(2) = 2.75 × 10⁻⁹, A_S(0) = 2.94 × 10⁻¹¹, A_S(1) = 4.76 × 10⁻⁹; Wolniewicz et al. 1998), ν_i is the frequency of transition, g_i is the statistical weight of the level (g_S(2) = 9, g_S(0) = 5), and ΔE is the energy difference between the emitting levels (= 1172 K). We plot the J = 4 level excitation temperature so derived in Figure 6 (marked as S(2)/S(0)). The excitation temperature is remarkably uniform over the disk at T_ex(2,0) = 204 ± 8.5 K.

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Between the para and ortho species, the line intensity ratios have the additional factor of the relative abundance, the ortho-to-para ratio, o/p. For a high temperature (T > 200 K) gas in local thermodynamic equilibrium, this ratio is given by the ratio of the statistical weights of the two species, o/p = 3. Assuming the o/p ratio is 3, then averaged over the 11 positions for which the S(2) line is detected (hereafter, the "S(2) region"), the average extinction-corrected S(1)/S(0) and S(2)/S(1) line intensity ratios (1.92 and 0.59) yield level excitation temperatures of T_ex(1,0) = 125 K, T_ex(2,0) = 204 K, and T_ex(2,1) = 388 K, respectively. The level excitation temperatures are monotonically increasing with J, which is permitted if there is more than one emitting components within the beam, each with a different gas excitation temperature. For temperatures less than ∼200 K, the equilibrium o/p ratio is less than 3. Burton et al. (1992) calculate the equilibrium o/p ratio as a function of gas temperature. Making use of this work, we arrive at a self-consistent solution for T_ex(1,0) = 134 K with o/p = 2.30. Since the S(1)/S(0) line intensity ratios are fairly uniform across the galaxy, we assume this o/p ratio applies everywhere, and use it to compute T_ex(1,0). Notice that T_ex(2,0) is not affected by the o/p ratio, and T_ex(2,1) is much greater than 200 K, so that o/p ∼ 3. Figure 6 shows the computed excitation temperatures across the disk of the galaxy. As we found for T_ex(2,0), T_ex(1,0) is also remarkably uniform across the plane with a mean value of T_ex(1,0) = 132 ± 6 K. T_ex(2,1) shows more variation, but this is largely not statistically significant. Excluding the data point 6 kpc to the NE of the nucleus, the mean value is T_ex(2,1) = 372 ± 41 K. The position at 6 kpc NE of the nucleus is statistically higher excitation, with T_ex(2,1) = 556 +127/−87 K.

Since the line emission is optically thin, the line intensity yields the gas column of the emitting level, N_u: I = hν_S(n) A_S(n) N_u/(4π), where A_S(n) is the Einstein A coefficient for the S(n) transition at frequency ν_S(n) for n = 0, 1, 2. Averaged over the 11 positions from which the S(2) line is detected, the column densities are N₂ = 1.22 × 10²⁰ cm⁻², N₃ = 9.02 × 10¹⁸ cm⁻², and N₄ = 6.81 × 10¹⁷ cm⁻². The total column density of molecular hydrogen is given by N_tot = (N_u/g_u) Z(T) e$^{-(E_{u}/(kT))}$ where g_u = (o/p)/(2J + 1), and Z(T) is the partition function:

$$\begin{equation} Z(T) = \left(\frac{o}{p}\right) \sum _{{\rm odd}-J}^\infty g_{J} e^{- \Delta E_{J}/T} + \sum _{{\rm even}-J}^\infty g_{J} e^{- \Delta E_{J}/T}. \end{equation} \tag{ 2 }$$

The symbols have the same meanings as mentioned before. Since the level excitations are not all the same, there must be more than one warm molecular gas component. The simplest model has two gas components, a "warm" component, which will emit most of the observed S(0) line, and a "hot" component that will emit most of the observed S(2) line. Both components will emit in the S(1) line. With a two-component model there can be no unique solution as there are four variables (two components, each with a T_ex and N(H₂)), and just three observables (the lines). We model the line emission averaged within the "S(2) region." The minimum mass solution will have a high gas temperature for the warm component. The largest gas temperature permitted by the data is the observed 1–0 excitation temperature. For this temperature, all of the observed S(0) and S(1) line emission arises from the warm component. Therefore, there can be no self-consistent solution for all three H₂ lines since the warm component is too cool to emit much in the S(2) line, and the hot component, which will emit most of the S(2) line, will also emit in the S(1) line. Using the partition function, and the relationship between line intensity and column density (I_J,J−1 = hν_J,J−1A_J,J−1N_J/(4π)), we have explored the gas temperature, column density parameter space for self-consistent solutions, and find the minimum mass solution has gas temperatures of T_warm = 127 K (o/p ratio shifts to 2.2), and T_hot = 1700 K, and H₂ column densities of N(H$_{2,\rm warm}$) = 3.8 ×10²¹ cm⁻², and N(H$_{2,\rm hot}$) = 8.2 × 10¹⁸ cm⁻². Within this model, 75% and 25% of the observed S(1) line emission arises from the warm and hot components, respectively. The warm component emits less than 5% of the observed S(2) line, while the hot component emits less than 1% of the S(0) line. The case where 25% and 75% of the observed S(1) line emission arises from the warm and hot components, respectively, is obtained for gas temperatures of T_warm = 107 K (o/p ratio shifts to 1.65), and T_hot = 465 K, and H₂ column densities of N(H$_{2,\rm warm}$) = 5.8 × 10²¹ cm⁻², and N(H$_{2,\rm hot}$) = 3.4 ×10¹⁹ cm⁻². For this case, less than 1% of the observed S(2) (S(0) line) arise from the warm (hot) component. For all cases, the warm component completely dominates (more than 99%) the H₂ column density.

We use the minimum mass self-consistent model (75/25 split of S(1) line flux into warm and hot components) to calculate the total column density in H₂ line emitting molecular gas (essentially the warm component from above) across the disk of NGC 891. The results, along with the column traced in CO(1–0) are plotted in Figure 7(a). The warm molecular gas column peaks at the nucleus at a value of ∼5.26 × 10²¹ cm⁻², and is above 2 × 10²¹ cm⁻² out to ±8 kpc from the nucleus. Figure 7(b) shows the ratio of the warm molecular gas to that traced in its CO(1–0) line emission (within the Spitzer beam), i.e., the fraction of the total molecular gas that is emitting in the H₂ lines. This fraction is modest (∼13%) and nearly constant in the inner galaxy from 3 kpc NE of the nucleus to 7 kpc SW of the nucleus, but rises to ∼30% in the outer galaxy. This could be a true enhancement of the warm molecular gas mass in the outer galaxy, or it could be due to a change in the conversion factor, X between CO intensity and H₂ column density with distance from the nucleus. If the X factor were larger by a factor of two in the outer galaxy compared with the inner galaxy, at first glance, this could explain our apparent change in the warm/cold molecular gas fraction. However, we do not believe this is the case, as there is a significant NE–SW asymmetry in the warm/cold molecular gas fraction apparent within the molecular ring (r = 3–7 kpc). Averaged over the ring positions, the NE portion of the ring has twice the warm molecular gas fraction (∼28%) as the SW portion of the ring (∼14%). The total molecular gas as traced by CO is very close to the same in both sides of the ring (Scoville et al. 1993): it is the warm molecular gas column that is enhanced (Figure 7, top). To cancel out this effect, one would need to invoke an X factor twice as large for the NE ring as for the SW ring. This enhanced warm molecular mass fraction is more likely the result of enhanced star formation in the NE ring, which is reflected in the stronger [C ii] and [O i] line emission observed in the NE as well (see Figure 8). Integrated over the galaxy, the warm molecular gas totals ∼6.1 × 10⁸ M_☉, or about 16% of the total molecular gas mass as traced by CO (3.9 × 10⁹ M_☉) within our Spitzer beams (which extend ±55 of the galactic plane).

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Figure 8 provides insight into the source for the heating of the molecular gas. In this figure, we plot the "cold" molecular gas mass (traced in its CO emission) the "warm" molecular gas mass (traced in its H₂ line emission), and the [C ii] 158 μm and the [O i] 63 fine-structure lines obtained with the ISO LWS spectrometer (Brauher et al. 2008). We also include weak detections of the [O i] 146 μm at six positions along the galactic plane that we obtained through close inspection of the LWS L02 scans of the line obtained within the ISO data archive. These data were calibrated as described in Brauher et al. (2008) so as to be consistent with their [C ii] and [O i] 63 μm fluxes, and their [N ii] 122 μm fluxes which we use below. The cold and warm molecular gas distributions have been smoothed to the significantly coarser spatial resolution (∼75'') of the ISO LWS beam for proper comparisons, and the tracers are normalized to their nuclear values (excepting the [O i] lines which are normalized to their peaks) for ease of comparison. The cold molecular gas mass is centered on the nucleus and has a significantly narrower distribution along the galactic plane than the warm molecular gas mass. In addition, the warm molecular gas mass distribution is asymmetrical, with near nuclear values extending to the molecular ring, 4.6 kpc to the NE of the nucleus.

The [C ii] line predominantly arises from PDRs formed on the surfaces of molecular clouds exposed to the far-UV (6 eV <hν < 13.6 eV) radiation from nearby OB stars or the general interstellar radiation field. The [C ii] line is also an important coolant for low-density ionized gas, and "atomic clouds," but most (> 70%; Stacey et al. 1991; Abel et al. 2005; Oberst et al. 2006) of the [C ii] flux from external galaxies likely arises from PDRs, so that the line is an indicator of OB star formation activity (Stacey et al. 1991). The very close correspondence between the [C ii] distribution and that of the warm molecular gas is strongly suggestive of a PDR origin for the H₂ line emission.

The ionization potential for O (13.62 eV) is essentially the same as that of H, so that the FIR [O i] 63 and 146 μm line emission also arises from neutral gas within PDRs. The [O i] lines have significantly higher critical densities (∼4.7 × 10⁵ and 9.4 × 10⁴ cm⁻³ for the 63 and 146 μm lines, respectively) than that of the [C ii] line (∼2.8 × 10³ cm⁻³) so that the [O i] lines trace denser PDRs. The strong peaking of the [O i] 63 μm line on the northern molecular ring is strongly suggestive of enhanced gas densities in this region.

The remarkable spatial correlation between the warm H₂ mass and the [C ii] line including the extension to the NE suggests a common origin in PDRs. Indeed, warm molecular gas is expected within PDRs. The penetration of carbon ionizing photons into neutral gas clouds is limited by the extinction of these photons by dust to visual extinctions, A_V ∼3 mag (corresponding to an N_H ∼ 6 × 10²¹ cm⁻²), and along half this column, hydrogen is typically molecular in form (Tielens & Hollenbach 1985).

We have argued for a PDR origin for much of the H₂ line emission. As such, one might expect a correlation between the observed H₂ line emission and that of an independent tracer of PDRs, the polycyclic aromatic hydrocarbon (PAH) 7.7 μm emission feature (e.g., Habart et al. 2004; Peeters et al. 2004). In Figure 9, we compare the PAH 7.7 μm emission with the H₂ line emission, by plotting the H₂-to-PAH emission ratio along the plane of NGC 891. The PAH emission is measured from IRAC 8 μm imaging data (Whaley et al. 2009) using the same apertures than those used for the H₂ spectroscopy. To compute the PAH emission from the 8 μm flux, we use the same effective bandwidth (Δν = 13.9 × 10¹² Hz) as Roussel et al. (2007). The H₂ line emission is the sum of the S(0), S(1), and S(2) lines. All tracers were corrected for extinction (see Section 3.3). The extinction at 7.7 μm is similar to that at 12.3 and 17 μm (Draine 2003), so that the H₂-to-PAH emission ratio is not very sensitive to the specific choice of the extinction curve. Rigopoulou et al. (2002) and Roussel et al. (2007) have shown that the H₂ emission of star-forming galaxies is tightly correlated with the PAH emission. Figure 9 extends this correlation to the distribution of both tracers within a galaxy. The H₂-to-PAH emission ratio is remarkably constant within the NGC 891 disk, with a mean value only slightly larger than that measured for the SINGS galaxies.

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A striking outcome of the SINGS data analysis is that the H₂/PAH emission ratio is independent of the dust heating as measured by the 24 μm to FIR dust emission ratio. Roussel et al. (2007) argue that this result is easiest to understand if the H₂ and dust emission both come from PDRs, and are powered by a common energy source, the UV light from young stars. However, from their discussion it is not fully clear how this interpretation fits with the modeling of the dust spectral energy distribution in Draine et al. (2007), where the PAH emission comes mainly from interstellar matter in a low UV radiation field (typically G <10). Here, we quantify the constraint set by the H₂/PAH emission ratio on the UV field. Using the Draine & Li (2007) models available on line, we computed that the PAH emission, as it appears in Figure 9, is

$$\begin{equation} {L_{\rm PAH} = 0.083 \, G \left(\frac{q_{\rm PAH}}{3.6\%} \right) \left(\frac{M_{\rm H}}{M_\odot } \right) L_\odot }, \end{equation} \tag{ 3 }$$

where the dust-to-gas mass ratio is assumed to be Galactic and q_PAH is the fraction of the dust mass in PAHs (Draine & Li 2007). The default value of q_PAH corresponds to the median value derived by Draine et al. (2007) for metallic galaxies in the SINGS sample. The model output may be combined with the observed value of H₂/PAH (∼0.012), to estimate the effective radiation field, G, within the warm H₂ gas. For NGC 891, we find L(H₂)/M(H₂,warm) ∼1/20, so that putting this all together we have

$$\begin{equation} {G < 50 \left(\frac{3.6\%}{q_{\rm PAH}} \right) \left(\frac{M_{\rm H_{2}Warm}}{6.1\times 10^8\, M_ \odot } \right)^{-1}}. \end{equation} \tag{ 4 }$$

Here, we have left in an explicit scaling factor for the warm molecular gas mass. Clearly, this calculation only provides a rough upper limit for G, since it assumes that the PAH emission arises exclusively from PDRs containing the warm H₂ gas, while some PAH emission is expected from the H i and cold H₂ gas. However, within the current understanding of PAH emission, the H₂-to-PAH comparison therefore suggests that the bulk of the warm H₂ mass (the S(0) emitting gas) is located in PDRs exposed to modest far-UV radiation fields. We use this as a constraint in our PDR models that we develop below. It is important to note that this constraint on G only applies for the gas that radiates in the S(0) and S(1) lines, not the gas radiating in the S(2) line. The S(2) line emission could arise from a much higher G environment without increasing the PAH flux very much since the mass of this component will be so small.

Kaufman et al. (2006) have made available a grid of PDR model outputs, which include the expected rotational line emission from H₂ as a function of G, the strength of the local far-UV (6 <hν < 13.6 eV) radiation field normalized to the Habing field (G₀ = 1.6 × 10⁻³ erg s⁻¹ cm⁻²), and the cloud number density n. The model computes the local heating and cooling, chemistry, and radiative transfer as a function of depth from the cloud surface into its dust-shielded core. Diagnostic line ratios and line intensities appropriate for a single face-on slab geometry are posted online, and available interactively in the "Photodissociation Region Toolbox (PDRT)."¹³ Table 2 lists our observed H₂ S(0), S(1), and S(2) luminosities and those of the [C ii], and [O i] lines integrated over the galaxy. We estimate that the errors for the H₂ luminosities are roughly given by their corrections for extinction, and high latitude flux missed by the Spitzer beam. Figure 10 (left) plots our observed galactic average H₂ line ratios as a function of G and n from PDRT. The model results do not give a good fit to the NGC 891 line ratios. The best fit is for high density (n ⩾6 × 10⁴ cm⁻³) and moderate far-UV radiation fields (G ∼ 200–400).

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Figure 10 (center) plots the [O i], [C ii], and ([O i] 63 μm + [C ii])/FIR continuum ratios as averaged over the galaxy onto a PDR diagnostic diagram from PDRT. For this FIR data set, we make corrections to the observed luminosities of our tracers. The first correction is for optical depth and geometry. The PDRT models are appropriate for face-on single slab geometry. Since the ISO beams encompass regions ∼2.5 kpc in size at the distance to NGC 891, there is an ensemble of clouds within our beam. For a spherical cloud, the projected surface area is only 1/4 the total surface area, so that a beam filled with spherical clouds will have four times the intensity as a unit filling factor single slab if the lines are optically thin. If all the tracers are optically thin, taking the line ratios eliminates this geometric factor. Although it is almost certainly true that all the H₂ lines are optically thin, optical depth may be an issue for the FIR lines. In fact, the 63 μm [O i] is most likely optically thick (Stacey et al. 1983; Tielens & Hollenbach 1985) in star formation regions, so that the observed [O i] line intensity should be corrected for geometry, multiplying by as much as a factor of four for high optical depth in the line in spherical geometry. For our study, we make a more modest correction, multiplying the 63 μm line by a factor of two. We take the [O i] error boundary to be 50% due to this factor of two correction, and note that this error bound will encompass geometry effects up to a factor of three. The [C ii] and [O i] 146 μm lines are likely optically thin, or thinnish as is the FIR continuum, so we make no optical depth correction to their fluxes.

Next, we correct the observed [C ii] line emission for that fraction expected to arise from diffuse ionized gas. This correction is perhaps best done by comparing the observed [C ii] line flux to that of the ground state fine-structure lines of N⁺ at 205 and 122 μm. With an ionization potential of 14.5 eV, these lines only arise from within ionized gas regions, and one can show that the [C ii]/[N ii] line ratios are a measure of the observed fraction of the [C ii] line that arises from the ionized ISM (cf. Petuchowski & Bennett 1993; Heiles 1994; Oberst et al. 2006). We use the [N ii] 122 μm lines observed by the ISO LWS (Brauher et al. 2008), averaged over the plane of the galaxy. The [N ii] 122 μm line to [C ii] line ratio in NGC 891 is ∼1:9: somewhat smaller than that of the Milky Way galaxy (1:6; Wright et al. 1991). Assuming Galactic C/N abundance ratios the line ratio indicates that between 50% (low-density limit) and 4% (high-density limit) of the observed [C ii] comes from ionized gas regions (Oberst et al. 2006). Here, we will assume 27% of the [C ii] line arises from H ii regions as found for the low-density (n_e ∼ 30 cm⁻³) gas in the Carina nebula (Oberst et al. 2006). We take the error bound for the [C ii] luminosity to be equal to the fraction (27%) we have subtracted off.

Finally, not all of the FIR continuum luminosity from galaxies will arise from dense PDRs. A significant fraction can arise from the diffuse ISM. Alton et al. (1998) and Popescu et al. (2004) have fit "warm" (PDR) and "cool" (diffuse) fractions of the total FIR luminosity from NGC 891. The warm dust likely arises from H ii regions and PDRs, while the cool dust arises from the cold interiors of molecular clouds and the diffuse ISM (Draine et al. 2007). We therefore take ∼50% of the total FIR luminosity of NGC 891 to arise from PDRs. Finally, we correct L_FIR to L_{30–1000 μm} which is appropriate for the PDRT models (see footnote to Table 2). Considering these corrections, we take the error bound for the FIR luminosity to be 50%. Putting the FIR line and continuum corrections together, the PDR cooling line/cooling continuum ratio integrated over NGC 891 is ([O i] + [C ii])/FIR ∼1.7% (Table 2).

Let us now examine the overall properties of NGC 891, integrated along the galactic plane. Taken within themselves, the S(2)/S(0) and S(1)/S(0) line ratios (1.08 and 1.69, respectively) indicate G ∼ 200–400, and n ⩾ 6 × 10⁴ cm⁻³. However, the [O i] 146 μm/[C ii], [O i] 63 μm/[C ii], and ([O i]+[C ii])/FIR ratios indicate lower far-UV fields with best fit near G ∼ 100, and a significantly lower density with best fit near n ∼ 3 ×10³ (Figure 9, center). The solution spaces for the H₂ and FIR line models do not overlap. The closest approach for the two models is obtained for similar Gs (∼100–200), but the FIR lines require a lower gas density (n ∼ 10⁴–4 ×10³ cm⁻³, as G ranges from 100 to 200) than the H₂ lines (n >6 × 10⁴ cm⁻³). If we imagine the PDRs to be the surfaces of externally illuminated molecular clouds, then since the H₂ lines arise from deeper within the PDR than the FIR lines, it is likely they will arise from denser gas regions, so perhaps this is an acceptable solution. If so, then the beam filling factors of the two components should match as well. For the FIR component, the average [C ii] line intensity (less 27% for ionized gas) referred to an 11'' region along the plane of the galaxy is 2.8 × 10⁻⁴ erg s⁻¹ cm⁻² sr⁻¹. At the closest approach solution between the H₂ and FIR models (n ∼4 × 10³ cm⁻³ and G ∼ 200), the [C ii] intensity predicted by PDRT is ∼2.2 × 10⁻⁴ erg s⁻¹ cm⁻² sr⁻¹ so that the beam filling factor for the FIR lines is ϕ_FIR lines ∼ 1.3. For n ∼6 × 10⁴, G ∼ 200, PDRT predicts an S(0) line intensity ∼1.0 × 10⁻⁵ erg s⁻¹ cm⁻² sr⁻¹. The extinction-corrected value within our 11'' beam averaged over the plane of the galaxy is ∼1.7 × 10⁻⁵ erg s⁻¹ cm⁻² sr⁻¹, so that the beam filling factor is $\phi _{\rm H_2\, lines}$ ∼ 1.7, in fair agreement with the FIR value.

The H₂ lines, the FIR lines, and the FIR continuum are consistent with PDR models but only if there is a rather large (factor of ∼10) density gradient between the (denser) H₂ emitting region and the (less dense) FIR line emitting regions. In addition, the requisite far-UV field strength is significantly larger than the value we derive using the observed PAH distribution in Section 4.3.2. These difficulties motivate us to investigate the possibility that a fraction of the observed H₂ line emission arises from the dissipation of turbulence in molecular clouds. This possibility has been proposed to account for diffuse H₂ line emission in the Milky Way (Falgarone et al. 2005) and the galaxy-wide shock in Stephan's Quintet (Guillard et al. 2009).

The S(1)/S(0) line ratio is fully consistent with the far-UV fields and densities as derived from the FIR lines and continuum (see Figure 10, left). What drives the H₂ solution is the S(2) line, which is a factor of six too bright. The C-shock models of Draine et al. (1983) and, more recently Flower & Pineau-Des-Forêts (2010) are in good agreement, and indicate that the S(2) line is brighter than the S(1) line for v_shock ⩾ 15 km s⁻¹. For example, using the Flower & Pineau-Des-Forêts (2010) model, for shock velocities, v_shock∼ 20–30 km s⁻¹, with preshock density n_H = n(H) + 2n(H₂)= 2 × 10⁴ cm⁻³, the S(2) line is ∼1.6 × as bright as the S(1) line, and 75 times brighter than the S(0) line. If 80% of the observed S(2) line flux were to arise from such C-shocks, and the remainder from PDRs, then 30% of the S(1) line would also arise from shocks, and the PDR line ratios would shift to S(2)/S(1) ∼ 0.17, and S(2)/S(0) ∼ 0.22. The allowed values for G and n for this PDR plus shocks solution are shaded blue in Figure 10 (right). Clearly, there is now a large region of overlap between the PDR solution for the FIR lines, and that for the H₂ lines. This solution space is centered at about G ∼ 60 ± 40, n ∼ 5 +5/−3 × 10³ cm⁻³ (Figure 10, center). The revised filling factors are reasonably self-consistent with ϕ_FIR lines ∼ 1.9, and $\phi _{\rm H_2\, lines}$ ∼ 2.8. With v_shock∼ 25 km s⁻¹, there is significant cooling in other molecular hydrogen lines: the S(3) and S(4) lines are predicted to be four to seven and four times brighter, respectively, than the S(2) line. The total luminosity in the molecular hydrogen lines from the shocked gas would be very large: L$_{\rm H_2} \sim 10^8$ L_☉. For a 25 km s⁻¹ C-shock, the common cooling lines of [O i] and CO are more than 100 and 15 times less important energetically than those of H₂, and H₂O is predicted to have a total cooling ∼that of H₂ (Flower & Pineau-Des-Forêts 2010). A similar analysis of the region centered on the NE ring has a solution centered on modestly enhanced excitation, G_NE ring∼80, n_NE ring ∼ 8 × 10³ cm⁻³, and requires somewhat higher area filling factors ϕ_FIR lines ∼ 2.7, and $\phi _{\rm H_2\,lines}$ ∼ 3.6 to account for the enhanced [O i] 63 μm line emission there.

Leaving aside the question of the H₂ excitation, we can assess the contribution of mechanical energy dissipation to the H₂ emission, from the point of view of energetics. Is the dissipation rate of the ISM turbulent kinetic large enough to provide a significant contribution to the H₂ emission? Various groups have discussed the dissipation of turbulence using numerical magneto-hydrodynamic simulations (e.g., Stone et al. 1998; Mac Low 1999). As in Bradford et al. (2003), one can rearrange Equation (7) from Mac Low (1999) to parameterize the energy released by the dissipation of turbulence per unit mass in terms of the dispersion of the turbulent velocity, v_rms, and the scale length of energy injection, Λ_d:

$$\begin{equation} \frac{L}{M} = 2.5\times 10^{-3} \left(\frac{v_{\rm rms}}{10 \,\rm km\, s^{-1}} \right) ^{3} \left(\frac{\rm 100\,pc}{\Lambda _{d}} \right) \frac{L_{\odot }}{M_{\odot }}. \end{equation} \tag{ 5 }$$

The default values for v_rms and Λ_d correspond to the upper end of the line width–size relation for GMCs in the Galaxy (Solomon et al. 1987). We invoke shocks to account for 80% of the S(2) luminosity, and 30% of the S(1) line luminosity. By dividing the sum of these line luminosiites 1.0 × 10⁷ L_☉, by the molecular mass 4.4 × 10⁹ M_☉, derived from the CO(1–0) luminosity, we obtain $L_{\rm {H_2}}/M_{\rm {H_2}}$ = $2.3 \times 10^{-3} \frac{L_{\odot }}{M_{\odot }}$. Therefore, the expected turbulent dissipation rate is greater than the luminosities of the lines, so that a significant contribution from turbulent energy dissipation is likely.

In summary, the observed H₂ line emission from NGC 891 is well fit by a model that combines a low excitation PDR (G ∼ 60, n ∼ 5× 10³ cm⁻³) with dissipation of turbulent energy within molecular clouds. The (adjusted) [O i] (63 and 146 μm) and [C ii] 158 μm fine-structure line fluxes, (half) of the observed FIR continuum flux, the H₂ S(0) line flux and 70% of the S(1) line flux arise from the PDR component, while the H₂ S(2) line and 30% of the S(1) line flux arise from gas heated by turbulent dissipation. It is interesting to note that based on observations of the H₂ lines along a line of sight within the Milky Way galaxy, Falgarone et al. (2005) reached much the same conclusion: the S(0) line emission may be accounted for by UV heated H₂ in the diffuse ISM and low excitation PDRs, but the UV emission cannot be the unique energy source, because their PDR models fall short of the observed S(1) and S(2) lines by large factors (∼2–10). As we outline here, they propose that the S(1) and S(2) H₂ line emission is powered by the localized dissipation of turbulent kinetic energy, albeit in a low-density gas (n_H ∼ 50 cm^-3) within the diffuse ISM. The presence of a diffuse component of H₂ line emission has been confirmed by additional Galactic observations (Hewitt et al. 2009; Goldsmith et al. 2010).

It should be clear that the solutions presented here are by no means unique. On galactic scales, there will certainly be a range of PDRs along the line of sight with varying degrees of excitation, that could mimic the observed line ratios, perhaps negating the need to invoke shocks. Furthermore, a single parameter shock model is also simplistic: there will very likely be a superposition of shocks of various parameters only any given line of sight. What we present here is intended to be only a representative solution.