FAR-INFRARED FINE-STRUCTURE LINE DIAGNOSTICS OF ULTRALUMINOUS INFRARED GALAXIES

In most cases, the lines are reproducible by single Gaussians with widths between 250 km s⁻¹ and 600 km s⁻¹. We do not see greater widths in the higher ionization lines. We also do not see strong asymmetries or systemic offsets in the velocity (compared to the optical redshift) of any line. In a few cases, the line profiles are not reproducible with single symmetric profiles. The [O i]63 lines in two objects, IRAS 06206−6315 and IRAS 20414−1651, are consistent with significant self-absorption (see also Graciá-Carpio et al. 2011). There is weaker evidence for [O i]63 self absorption in IRAS 00188−0856, IRAS 11095−0238, and IRAS 19254−7245. Self-absorption in [O i]63 can occur when cool oxygen in foreground clouds reduces the [O i]63 flux (Poglitsch et al. 1996; Fischer et al. 1999; Vastel et al. 2002; Luhman et al. 2003; Okada et al. 2003). The [C ii] profiles in IRAS 20087−0308, IRAS 20414−4651, and IRAS 23253−5415 may show subtle asymmetries. Finally, in four cases (IRAS 06035−7102, Mrk 463, IRAS 11095−0238, and IRAS 20100−4156), [C ii] may show an additional, broad emission component with widths between 600 km s⁻¹ and 1200 km s⁻¹. For consistency with the other lines, we do not include the broad component in the line fluxes in Table 3, but instead discuss it in a future paper.

We examine the distribution of line widths in Figure 4, using only the narrow components in those cases where an additional broad one exists. The ranges in width for all six lines are consistent with each other. We see no dependence of the ranges in width on optical class. For individual objects, however, there are sometimes substantial differences between individual line widths. In Mrk 1014, for example, the [C ii] and [N ii] line widths differ by nearly a factor of two. We speculate that these differences are due to one or more of the following: (1) differences in the critical densities of the lines, (2) an individually unresolved broad component in one line is brighter than the equivalent component in the other line, and (3) in the case of [C ii] and [N ii], the [N ii] emission arises mostly from H ii regions, while at least some of the [C ii] emission arises from photodissociation regions (PDRs) or the diffuse interstellar medium (ISM). It is, however, also possible that the [N ii] profile is affected by the 4₃₂ − 4₂₃ transition of o-H₂O at 121.72 μm (Fischer et al. 2010; González-Alfonso et al. 2010). This is discussed in Spoon et al. (2013).

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We plot the luminosities of selected individual lines against each other in Figure 5. The line luminosities range from just under 10⁷ L_☉ to ∼3 × 10⁹ L_☉. The [C ii] or [O i]63 lines are usually the most luminous lines, though the [O iii] line may be the second-most luminous in 3C273 and IRAS 00397−1312. The two least-luminous lines in most cases are [N ii] and [O i]145. In some cases, [N iii] is consistent with being the least luminous, and in a few cases (e.g., IRAS 07598+6508 IRAS 23253−5415) [O iii] is consistent with being the least luminous. The range in luminosities decreases approximately with increasing wavelength; the [O iii] and [N iii] line luminosities span a factor of ∼50, whereas the [C ii] luminosity spans only a factor of five, which is less than the range in L_IR of the sample.

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To determine if the line luminosities correlate with each other, we fit the relation in Equation (2) (see Section 3.2.1 for methodology). We find positive correlations in all cases, with β values mostly between 0.5 and 1.2. In no case, however, is the correlation particularly strong (in terms of the S/N on β). We find that among these six lines, the luminosity of one cannot be used to predict the luminosity of another to better than ∼0.2 dex. If we instead plot line luminosities normalized by L_IR (Figure 6), then the correlations do not change significantly; they do not depend on optical spectral type either.

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One object, IRAS 01003−2238, may have an unusually low [N ii] luminosity. The six line measurement methods for [N ii] in this object are in reasonable agreement, despite the low S/N. We therefore do not have a good explanation for this. The only other way in which IRAS 01003−2238 is atypical is that its optical spectrum contains Wolf–Rayet star features (Farrah et al. 2005), but we do not see why this would affect the [N ii] luminosity.

Finally, we comment briefly on two line ratios. First is [O i]145/[O i]63, which is related to the physical conditions in PDRs. This ratio (note that the inverse is plotted in Figure 5) depends on both the gas temperature and density. It decreases with decreasing gas temperature, and decreases with increasing gas density if the density is above the critical density. In a diffuse PDR illuminated by intense UV radiation, the expected [O i] ratio lies below ∼0.1 (e.g., Kaufman et al. 2006). From Figure 5, we see that several of our sample have [O i] ratios above 0.1. A likely reason for this is the self absorption of [O i]63, or a higher optical depth in [O i]63 than [O i]145 (e.g., Abel et al. 2007).

Second is the [N ii]/[C ii] ratio, which can be used to estimate the relative contribution of [C ii] arising in PDRs and in the ionized gas. While [N ii] arises almost entirely from H ii regions, [C ii] can additionally arise from PDRs and the diffuse ISM (cold neutral, warm neutral, and warm diffuse ionized). The [N ii]/[C ii] ratio can therefore quantify the contribution to [C ii] from the ionized gas by comparing the observed [N ii]/[C ii] ratio with theoretical expectations from a photoionized nebula. The expected ratio in the ionized gas lies between 0.2 and 2.0 depending on density (e.g., Figure 7 of Bernard-Salas et al. 2012). The ratios in our sample, however, are nearly all below 0.25. This indicates that a significant fraction of the [C ii] emission originates from photodissociation regions (PDRs). A precise estimate, however, is hampered by our lack of knowledge on the ionized gas density.

We compare far-IR line luminosities to L_IR in Figure 7. Qualitatively, some of the line luminosities crudely correlate with L_IR, with greater scatter among the Sy1s. To determine if correlations exist between L_IR and L_Line, we assume that L_IR and L_Line are related by

$$\begin{equation} {\rm log}_{10}\left(\frac{L_{{\rm IR}}}{L_{\odot }}\right) = \alpha + \beta {\rm log}_{10}\left(\frac{L_{{\rm Line}}}{L_{\odot }}\right), \end{equation} \tag{ 2 }$$

where α and β are free parameters. We fit this relation following the method of Tremaine et al. (2002). For a correlation to exist, we require that β > 0 at >3.5σ significance. We do not claim that Equation (2) codifies an underlying physical relation, or that Equation (2) is the only relation that applies over the luminosity range of our sample.¹⁸ Finally, we do not consider more complex models as our data do not prefer them.

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Excluding objects with Sy1 spectra yields the following relations between L_IR and L_Line:

$$\begin{eqnarray} {\rm log}_{10}(L_{{\rm IR}}) & = & (4.46\pm 1.77) + (0.92\pm 0.20){\rm log}_{10}(L_{{[\rm O\,\scriptsize{III}]}})\qquad \ \end{eqnarray} \tag{ 3a }$$

$$\begin{eqnarray} &&\quad\ = (4.74\pm 1.59) + (0.94\pm 0.19){\rm log}_{10}(L_{{[\rm N\,\scriptsize{III}]}}) \end{eqnarray} \tag{ 3b }$$

$$\begin{eqnarray} &&\quad\ \ = (4.28\pm 1.89) + (0.91\pm 0.21){\rm log}_{10}(L_{{[\rm O\,\scriptsize{I}]63}}) \end{eqnarray} \tag{ 3c }$$

$$\begin{eqnarray} &&\ \ \ \ = (5.29\pm 1.57) + (0.89\pm 0.20){\rm log}_{10}(L_{{[\rm N\,\scriptsize{II}]}}) \end{eqnarray} \tag{ 3d }$$

$$\begin{eqnarray} &&\quad \ \ \ = (1.75\pm 2.11) + (1.34\pm 0.27){\rm log}_{10}(L_{{[\rm O\,\scriptsize{I}]145}}) \end{eqnarray} \tag{ 3e }$$

$$\begin{eqnarray} &&\quad \ = (6.73\pm 2.44) + (0.64\pm 0.27){\rm log}_{10}(L_{{\rm \rm [C\,\scriptsize{II}]}}), \end{eqnarray} \tag{ 3f }$$

which are significant (in terms of β) for all of the lines except [C ii]. However, there is the caveat that some of the relations are based on a small number of formal detections (Table 3). Including the Sy1s yields significant correlations only for [O iii] and [O i]145, both with flatter slopes than the above. The insignificant correlation for [C ii] suggests that it traces L_IR to an accuracy of about an order of magnitude at best. This result is consistent with the findings of previous authors, who find a crude correlation over a wider luminosity range (Sargsyan et al. 2012).

We also investigated whether sums of the lines in Equations (3a)–(3f) show stronger relations with L_IR than individual lines. We found correlations in several cases, but none that significantly improved on those for the individual lines.

The weaker correlations between L_IR and L_Line when Sy1s are included in the fits are consistent with far-IR lines being better tracers of L_IR in ULIRGs without an optical AGN. If this is true, then we may see a similar result if we include in the fits only those objects with prominent PAH features, since PAHs are probably exclusively associated with star formation (see Section 3.3.1). We test this hypothesis by repeating the fits, but this time including only those objects with 11.2 μm PAH EWs greater than 0.05 μm. We chose the 11.2 μm PAH because (1) it is bright, and (2) its value correlates well with the evolutionary paradigm for ULIRGs in Farrah et al. (2009).¹⁹ We find

$$\begin{eqnarray} {\rm log}_{10}(L_{{\rm IR}}) & = & (7.01\pm 1.07) + (0.63\pm 0.12){\rm log}_{10}(L_{{[\rm O\,\scriptsize{III}]}}) \qquad\ \end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray} &&\quad\ = (6.52\pm 1.33) + (0.71\pm 0.16){\rm log}_{10}(L_{{[\rm N\,\scriptsize{III}]}}) \end{eqnarray} \tag{ 4b }$$

$$\begin{eqnarray} &&\quad\ \ = (6.71\pm 1.30) + (0.64\pm 0.15){\rm log}_{10}(L_{{[\rm O\,\scriptsize{I}]63}}) \end{eqnarray} \tag{ 4c }$$

$$\begin{eqnarray} &&\quad = (6.43\pm 1.51) + (0.74\pm 0.19){\rm log}_{10}(L_{{[\rm N\,\scriptsize{II}]}}) \end{eqnarray} \tag{ 4d }$$

$$\begin{eqnarray} &&\quad\ \ \ \ = (2.77\pm 2.12) + (1.20\pm 0.26){\rm log}_{10}(L_{{[\rm O\,\scriptsize{I}]145}}) \end{eqnarray} \tag{ 4e }$$

$$\begin{eqnarray} &&\quad\ \ = (9.44\pm 2.20) + (0.33\pm 0.25){\rm log}_{10}(L_{{\rm \rm [C\,\scriptsize{II}]}}). \end{eqnarray} \tag{ 4f }$$

These relations are somewhat flatter than those in Equations (3a)–(3f). Again, the [C ii] line shows no significant correlation. We conclude that these far-IR lines are better tracers of L_IR in systems without type 1 AGN, but that it is unclear whether they are better tracers of L_IR in systems with more prominent star formation.

We compare L_IR to continuum luminosity densities (in units of L_☉ Hz⁻¹) near the wavelengths of the far-IR lines in Figure 8. We find a significant correlation at all six wavelengths, irrespective of whether Sy1s are included (though the continua of the Sy1s are usually not detected). For consistency with the line comparisons, we exclude the Sy1s and fit relations of the form in Equation (2), obtaining

$$\begin{eqnarray} {\rm log}_{10}(L_{{\rm IR}}) & \,{=}\, & (12.87\,{\pm}\, 0.09) \,{+}\, (0.88\,{\pm}\, 0.17){\rm log}_{10}(L_{{\rm 52\,\mu m}}) \qquad \end{eqnarray} \tag{ 5a }$$

$$\begin{eqnarray} & = & (12.86\pm 0.05) + (0.91\pm 0.11){\rm log}_{10}(L_{{\rm 57\,\mu m}}) \end{eqnarray} \tag{ 5b }$$

$$\begin{eqnarray} & = & (12.86\pm 0.08) + (0.98\pm 0.18){\rm log}_{10}(L_{{\rm 63\,\mu m}}) \end{eqnarray} \tag{ 5c }$$

$$\begin{eqnarray} &&\ = (12.95\pm 0.07) + (0.87\pm 0.11){\rm log}_{10}(L_{{\rm 122\,\mu m}}) \end{eqnarray} \tag{ 5d }$$

$$\begin{eqnarray} &&\ = (13.05\pm 0.11) + (0.90\pm 0.16){\rm log}_{10}(L_{{\rm 145\,\mu m}}) \end{eqnarray} \tag{ 5e }$$

$$\begin{eqnarray} &&\ \ = (13.08\pm 0.11) + (0.83\pm 0.11){\rm log}_{10}(L_{{\rm 158\,\mu m}}). \end{eqnarray} \tag{ 5f }$$

Including the Sy1s yields comparable relations. If we instead exclude objects with PAH 11.2 μm EW ≲ 0.05, then we obtain consistent relations.

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Several far-IR lines in ULIRGs show a deficit in their L_Line/L_IR ratios compared to the ratios expected from systems with lower values of L_IR (e.g., Luhman et al. 2003). In contrast, high-redshift ULIRGs, at least for [O i]63 and [C ii], do not show such pronounced deficits (Stacey et al. 2010; Coppin et al. 2012; D. Rigopoulou et al., in preparation). We examine if the [O i], [N ii], and [C ii] lines in our sample show such deficits in Figure 9 (we do not test the other lines as we lack archival data to compare to). We find a deficit in all four lines. Comparing the mean ratios at 10¹¹ L_☉ and 10^12.2 L_☉, we find differences of factors of 2.75, 4.46, 1.50, and 4.95 for [O i]63, [N ii], [O i]145, and [C ii], respectively.

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The four line deficits show no clear dependence on optical spectral type, the presence of an obscured AGN (as diagnosed from the detection of [Ne v]14.32, see Section 3.5), or on PAH 11.2 μm EW for any line. There are, however, trends with both S_Sil and merger stage.

• 1.  
  If S_Sil ≳ 1.4, then the [C ii] and [N ii] deficits increase with increasing S_Sil. However, there is no obvious trend of the [C ii] and [N ii] deficits with S_Sil at S_Sil ≲ 1.4 (top left panel of Figures 10 and 13). Conversely, the [O i] deficits show no trends with S_Sil.
• 2.  
  We find no evidence that the [N ii] and [O i] deficits depend on merger stage, but the [C ii] deficit is stronger, on average, in advanced mergers (classes IVb and V) than in early-stage mergers (classes IIIa through IVa, top right panel of Figure 10,²⁰ see also Diaz-Santos et al. 2013).

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Finally, in the bottom row of Figure 10, we plot L_[C ii]/L_IR against L_PAH/L_IR as a function of merger stage and S_Sil. We see consistent trends in both plots; ULIRGs in advanced mergers and with S_Sil ≳ 2 have lower L_[C ii]/L_IR and L_PAH/L_IR ratios, compared to ULIRGs in early-stage stage mergers and with S_Sil ≃ 1.4–2.

We do not believe that the line deficits arise from missing an asymmetric or broad component (see Section 3.1.1), since such components are rare, or that the deficits arise due to self-absorption, since we see no self absorption in [C ii], which shows the strongest deficit. We do, however, see self absorption in [O i]63, which shows a weaker deficit. Instead, the stronger [C ii] and [N ii] deficits in sources with higher S_Sil (at S_Sil ≳ 1.4) are consistent with the H ii regions in ULIRGs being dustier than H ii regions in lower luminosity systems. In this scenario (Luhman et al. 2003; González-Alfonso et al. 2008; Abel et al. 2009; Graciá-Carpio et al. 2011), a higher fraction of the UV photons are absorbed by dust rather than neutral Hydrogen, thus contributing more to L_IR but less to the photoionization heating of gas in the H ii regions, and thus decreasing line emission relative to L_IR. This mechanism would also produce a deficit in [C ii] and the "deficit" in the PAH emission, even if the bulk of the [C ii] and PAHs are in the PDRs, since there would also be fewer UV photons for photoelectric heating of the PDRs. Furthermore, this mechanism is consistent with the [O i] deficits. From Figure 3 of Graciá-Carpio et al. 2011, we see that the conditions consistent with the observed deficits of all four lines are n_h ≲ 300 cm⁻³ and 0.01 ≲ 〈U〉 ≲ 0.1.

However, we propose that dustier H ii regions are not the sole origin of the line deficits. This is based on three observations. First, it is puzzling that we see no strong dependence of any of the line deficits on S_Sil when S_Sil ≲ 1.4; if the deficit arises entirely in H ii regions, and if S_Sil is a proxy for the dust column in these H ii regions at S_Sil ≳ 0, then we should see a dependence. We note, however, that we have only a small sample, so we could be missing a weak dependence, and the assumption that S_Sil is a proxy for the dust column in H ii regions is not proven.²¹ Second, we see significant line deficits for some sources with S_Sil < 0, i.e., a silicate emission feature. Third, only the [C ii] deficit becomes stronger with advancing merger stage. If more advanced mergers host dustier H ii regions, then we would also see a dependence of the [N ii] deficit on merger stage.

These observations suggest that some fraction of the [C ii] deficit is not driven by increased dust in H ii regions. We lack the data to investigate this in detail, so we only briefly discuss this further. We consider three possible origins: (1) increased charging of dust grains (e.g., Malhotra et al. 2001), leading to a lower gas heating efficiency, (2) a softer radiation field in the diffuse ISM (e.g., Spaans et al. 1994), and (3) dense gas in the PDRs, making [O i] the primary coolant rather than [C ii].

The third of these possibilities is feasible, but we do not have the data to confirm or refute it as a mechanism. The second possibility is unlikely, due to the energetic nature of star formation in ULIRGs and considering the arguments in Luhman et al. (2003). For the first possibility, if the origin of the additional deficit in [C ii] is grain charging, then we would see a higher G₀ in advanced mergers compared to early stage mergers. The advanced mergers have roughly an order of magnitude higher value of G₀ for about the same n (see Section 3.6).²² We therefore infer, cautiously and with the caveat that we cannot rule out [O i] being a major cooling line, that part of the [C ii] deficit arises due to grain charging in PDRs or in the diffuse ISM. We further propose that this increase in grain charge is not driven mainly by a luminous AGN, obscured or otherwise, since we see no dependence of the [C ii] deficit on either optical spectral type or the presence of [Ne v]14.32.

We examine far-IR fine-structure lines as star formation rate indicators by comparing their luminosities to those of PAHs. PAHs are thought to originate from short-lived Asymptotic Giant Branch stars (Gehrz 1989; Habing 1996; Blommaert et al. 2005; Bernard-Salas et al. 2006), and therefore to be associated with star-forming regions (Tielens 2008). They are prominent in starburst galaxies but weak in AGNs (Laurent et al. 2000; Weedman et al. 2005). While we are still uncertain how to calibrate PAHs as star formation rate measures (Peeters et al. 2004; Förster Schreiber et al. 2004; Sargsyan et al. 2012), their luminosities are likely reasonable proxies for the instantaneous rate of star formation. Conversely, for far-IR fine-structure lines, the relation between line luminosity and star formation rate is less clear. We may expect a correlation, as the inter-stellar radiation field (ISRF) from young stars may be an important excitation mechanism. Moreover, correlations between far-IR line luminosities and the star formation rate have been observed previously (e.g., Boselli et al. 2002; de Looze et al. 2011; Sargsyan et al. 2012; Zhao et al. 2013). The extent to which correlations exist is, however, poorly constrained.

Considering the caveats that PAH luminosity depends on both metallicity (Madden et al. 2006; Calzetti et al. 2007; Khramtsova et al. 2013) and dust obscuration, neither of which we can correct for, we assume that the PAH luminosities give "true" star formation rate measures, which we compare to the far-IR line luminosities. In doing so, we further assume that there is negligible differential extinction between the PAHs and the far-IR lines. To mitigate effects from the variation of individual PAH features (Smith et al. 2007), we sum the luminosities of the PAH 6.2 μm and 11.2 μm features into a single luminosity, L_PAH (Farrah et al. 2007a).

We plot far-IR line luminosities against L_PAH in Figure 11. Unlike the plots with L_IR, there is no significant difference between the Sy1s and H ii/LINERs. This is consistent with both PAHs and the far-IR lines primarily tracing star formation. Fitting relations of the form in Equation (2) to all the objects and converting to star formation rate by using Equation (5) of Farrah et al. (2007a) yields

$$\begin{eqnarray} {\rm log}_{10}\left(\!\frac{\dot{M}_{\odot }}{M_{\odot }\ {\rm yr}^{-1}}\!\right) & \,{=}\, &(-7.02\,{\pm}\, 1.25)\nonumber\\ &&+ (1.07\,{\pm}\, 0.14){\rm log}_{10}(L_{{[\rm O\,\scriptsize{III}]}}) \qquad\qquad \end{eqnarray} \tag{ 6a }$$

$$\begin{eqnarray} & = &(-5.13\pm 0.72) + (0.91\pm 0.09){\rm log}_{10}(L_{{[\rm N\,\scriptsize{III}]}}) \ \ \ \end{eqnarray} \tag{ 6b }$$

$$\begin{eqnarray} & = &(-5.44\pm 1.79) + (0.86\pm 0.20){\rm log}_{10}(L_{{[\rm O\,\scriptsize{I}]63}}) \ \end{eqnarray} \tag{ 6c }$$

$$\begin{eqnarray} & = &(-7.30\pm 0.87) + (1.19\pm 0.11){\rm log}_{10}(L_{{[\rm N\,\scriptsize{II}]}}) \quad\ \end{eqnarray} \tag{ 6d }$$

$$\begin{eqnarray} & = & (-10.04\pm 1.34) + (1.55\pm 0.17){\rm log}_{10}(L_{{[\rm O\,\scriptsize{I}]145}}) \end{eqnarray} \tag{ 6e }$$

$$\begin{eqnarray} & = &(-6.24\pm 1.72) + (0.95\pm 0.19){\rm log}_{10}(L_{{\rm \rm [C\,\scriptsize{II}]}}). \quad \end{eqnarray} \tag{ 6f }$$

Using the criteria from Section 3.2.1, all of the lines show a significant correlation. The trends with [O i]63 and [C ii] are only barely significant (see also Diaz-Santos et al. 2013). Excluding the Sy1s and Sy2s from the fits yields consistent slopes and intercepts in all cases, though the fits are now no longer significant for [O i]63 and [C ii]. Considering only those lines detected at >3σ, the derived star formation rates for a given object are consistent to within a factor of three in nearly all cases. We present the mean star formation rates in Table 4.

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An alternative to PAH luminosity as a tracer of star formation rate is the sum of the [Ne iii]15.56 μm and [Ne ii]12.81 μm line luminosities, L_Neon (Thornley et al. 2000; Ho & Keto 2007; Shipley et al. 2013). We thus compared L_Neon to L_Line. For the whole sample, we find relations that are mostly in agreement with Equations (6a)–(6f), though the relation with [C ii] is now formally insignificant. Excluding the Seyferts and comparing L_Neon to L_Line yields similar results to those with L_PAH.

We note four additional points. First, we tested sums of far-IR lines as tracers of star formation rate but found no improvement on the individual relations. Second, if we instead consider L_Line/L_IR versus L_PAH/L_IR, then we see correlations with comparable scatter to those in Figure 11. Third, we investigated whether using L_PAH is better than using a single PAH luminosity by reproducing Figure 11 using only the PAH 6.2 μm luminosity. We found consistent relations in all cases, albeit with larger scatter for the $L_{\rm {PAH}}^{6.2}$ plots. Fourth, if we instead code the points in Figure 11 by the 11.2 μm PAH EW, then we see no dependence of the relations on the energetic importance of star formation.

Finally, we note two points about the relation between star formation rate and L_[C ii]. First, we investigated whether L_[C ii] shows an improved correlation with star formation rate if objects with a strong [C ii] deficit are excluded. Adopting a (somewhat arbitrary) boundary of L_[C ii]/L_IR = 2 × 10⁻⁴ yields only a marginally different relation:

$$\begin{eqnarray} {\rm log}_{10}\left(\frac{\dot{M}_{\odot }}{M_{\odot }\ {\rm yr}^{-1}}\right) &\,{=}\,&(-6.59\,{\pm}\, 1.58) \nonumber\\ &&+ (0.99\,{\pm}\, 0.18){\rm log}_{10}(L_{{\rm [C\,\scriptsize{II}]}}), \end{eqnarray} \tag{ 7 }$$

suggesting that the correlation does not depend strongly on the [C ii] deficit, though there is the caveat that star formation rate is derived from L_PAH (see Section 3.2.3 and Figure 10). Second, the relation between $\dot{{M}}_{\odot }$ and L_[C ii] is consistent with that given by Sargsyan et al. (2012), though their sample mostly consists of lower luminosity systems.

We examine continuum luminosity densities as star formation rate tracers in Figure 12. Employing the same method as in Section 3.2, we find that the continua near all the lines provide acceptable fits. Converting to relations with star formation rate (see Section 3.3), we find

$$\begin{eqnarray} {\rm log}_{10}\left(\!\frac{\dot{M}_{\odot }}{M_{\odot }\ {\rm yr}^{-1}}\!\right) & \,{=}\, & (3.24\,{\pm}\, 0.24) \,{+}\, (1.89\,{\pm}\, 0.46){\rm log}_{10}(L_{{\rm 52\,\mu m}})\nonumber\\ \end{eqnarray} \tag{ 8a }$$

$$\begin{eqnarray} & = & (2.95\pm 0.10) + (1.38\pm 0.19){\rm log}_{10}(L_{{\rm 57\,\mu m}}) \end{eqnarray} \tag{ 8b }$$

$$\begin{eqnarray} & = & (3.04\pm 0.17) + (1.79\pm 0.39){\rm log}_{10}(L_{{\rm 63\,\mu m}}) \end{eqnarray} \tag{ 8c }$$

$$\begin{eqnarray} &&\ = (2.83\pm 0.11) + (0.93\pm 0.16){\rm log}_{10}(L_{{\rm 122\,\mu m}}) \end{eqnarray} \tag{ 8d }$$

$$\begin{eqnarray} &&\ = (3.02\pm 0.18) + (1.09\pm 0.25){\rm log}_{10}(L_{{\rm 145\,\mu m}}) \end{eqnarray} \tag{ 8e }$$

$$\begin{eqnarray} &&\ \ = (3.36\pm 0.22) + (1.42\pm 0.30){\rm log}_{10}(L_{{\rm 158\,\mu m}}), \end{eqnarray} \tag{ 8f }$$

which are all significant using the criteria from Section 3.2.1. However, the correlations with continua at ⩽63 μm may be stronger, which is consistent with stronger correlations with warmer (star formation heated) dust. This is consistent with findings by previous authors (Brandl et al. 2006; Calzetti et al. 2007). We present the mean star formation rates derived from these relations in Table 4. In most cases, they are consistent with the line-derived star formation rates.

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