SPIRE Spectroscopy of Early-type Galaxies

The nine galaxies chosen for observation are a subset of the most far-infrared (FIR) bright members of the Atlas3D sample that additionally have the largest CO(1–0) fluxes (Cappellari et al. 2011; Young et al. 2011). This choice was made to ensure as many detections as possible for the various CO transitions, and to complement PACS observations of the [C ii], [O i], and [N ii] 122 emission lines discussed in Lapham et al. (2017).

Atlas3D was a volume-limited sample of high mass elliptical and lenticular galaxies within 40 Mpc. The sample selection was based on K-band magnitude and stellar morphology (lack of spiral arms), but not color, so the sample includes some gas-rich ETGs actively forming stars. The FIR selection for this project was done using Infrared Astronomical Satellite (IRAS) fluxes. The resulting subset of galaxies are mostly red sequence galaxies, with stellar masses 2·10¹⁰ to 2·10¹¹ M_⊙, though a couple are as blue as typical spirals (see Figure 1 in Lapham et al. 2017).

The bluest galaxies in the sample are NGC 1222, 2764, and 7465, with NUV – K and $u-r$ colors similar to spiral galaxies—clear evidence of recent star formation activity (Young et al. 2014; McDermid et al. 2015). NGC 1222 and NGC 7465 are also recent merger remnants, and they contain kinematically disturbed molecular gas that has not yet settled into relaxed disks (Alatalo et al. 2015). The interactions have been driving high levels of star formation activity and high FIR luminosities. NGC 1266 is also unusual for having an active galactic nucleus–driven molecular outflow (Alatalo et al. 2011). The remaining galaxies in the sample evidently have kinematically relaxed molecular disks, to the best of our knowledge. NGC 3665 has powerful radio jets (Nyland et al. 2017), but they are apparently not interacting with the molecular disk at a level we can detect.

All galaxies in the sample were observed by the SPIRE instrument (Griffin et al. 2010) aboard the Herschel space-borne observatory (Pilbratt et al. 2010) with eight observations coming from the project OT1_lyoung_1 and the NGC 1266 observation coming from OT1_jglenn_1. SPIRE has a three-band imaging photometer, as well as a Fourier Transform Spectrometer (FTS). Broadband photometric data is taken at roughly 250, 350, and 500 μm, while the FTS covers 194–671 μm (447–1550 GHz) with two bands (SPIRE short wavelength and long wavelength, SSW and SLW). The spectral resolution is 1.2 GHz in high-resolution mode, and the beam FWHM ranges from 17'' to 42''. Previous CO observations from Alatalo et al. (2015) suggested the molecular emission would be mostly or entirely contained within the beam of the central SPIRE detector, so observations were done as single pointings in the high spectral resolution mode. Additionally, all of the galaxies have SPIRE photometric observations available to assist with calibrating the spectroscopic observations.

The SPIRE spectra were reduced in the Herschel Interactive Processing Environment (HIPE) 14, using the built-in Single Pointing user pipeline (Ott 2010). In addition to the standard reduction, we also processed the spectra with HIPE's Semi-Extended Flux Correction Tool (SECT). Because the central detector might not actually cover the entire source, the SECT is used to create a total flux for each source, as well as set a standard reference beam, as the beam size is strongly wavelength dependent. The SECT can optimize the size of the source based on the overlap between the SSW and SLW spectra; however, the spectra have random continuum offsets (with respect to each other) on the order of 0.3–0.4 Jy (Hopwood et al. 2015), and in our sources the offsets are a large percentage of the flux in the overlapping region. Instead, we prefer to specify the source size, position angle, and eccentricity based on two-dimensional Gaussian models of the CO (1–0) emission. Additionally, we do not require that our final spectrum match the photometry perfectly, but rather use the photometry as a sanity check, along with prior knowledge of the probable emission size. The spectra before and after the SECT can be seen in Figure 4 of Lapham et al. (2017). A few examples of emission lines from the SPIRE spectra can be seen in Figure 1.

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Once the SPIRE spectra were reduced in the pipeline and corrected by the SECT, the CO and [C i] emission lines were fit simultaneously by HIPE's built-in Spectrometer Line Fitting script. We used the unapodized spectra in the fitting routines and fit a sinc model to each line simultaneously. Uncertainties were calculated using a Monte Carlo method explained in detail in the Appendix. Since the [N ii] 205 μm lines can be broader than the instrumental profile, those lines were fit with a sinc-Gauss or two sinc components in cases where the CO(1–0) spectra had sharply double-horned shapes. For NGC 3665 and NGC 4526 in particular (see Figure 8 of Lapham et al. 2017) the [N ii] 205 μm were better fit with two sinc components rather than one sinc-Gauss; however, the differences in the derived flux measurements were marginal. The fluxes and uncertainties for the [N ii] 205 μm line were previously published in Lapham et al. (2017). The SPIRE Handbook¹ version 2.5 recommends summing the absolute calibration uncertainty (4%) and relative calibration uncertainty (1.5%) directly for a conservative total estimate of the calibration uncertainty of 5.5%. We have not combined the calibration uncertainties with our Monte Carlo uncertainties, as they are typically only a small component of the error budget. Fluxes and uncertainties for the [C i] and CO emission lines can be seen in Table 1. Correction factors applied by the SECT can be seen in Table 2.

Most of the CO and [C i] lines were previously measured by Kamenetzky et al. (2016). When comparing to their work, we find our CO fluxes to be on average 21% greater. The [C i] lines are in better agreement, averaging 17% greater than those in Kamenetzky et al. (2016). We prefer to use a more direct approach to measure the line fluxes after correcting with the SECT, rather than assuming that the spatial distribution of the gas can be inferred from the FIR continuum levels as Kamenetzky et al. (2016) did. Despite these differences, we find similar results.

CO has long been recognized as a powerful tracer of otherwise undetectable molecular gas, specifically its ground state J = 1–0 transition. However, with the launch of the Herschel satellite, many mid- to high-J CO transitions previously obscured by the atmosphere could be observed. The SPIRE spectral range contains the CO transitions J = 4–3 to J = 13–12 for nearby galaxies. These higher transitions allow us to gain insight on a different phase of molecular gas, a phase that is warm (relative to J = 1–0 emitting gas) but still molecular. For instance, the CO(1–0) and (2–1) transitions have excitation temperatures of 5.53 K and 16.6 K, respectively; while the mid-J transitions (4–3), (5–4), and (6–5) have excitation temperatures of 55.32 K, 82.97 K, and 116.16 K, respectively; and the high-J lines (11–10), (12–11), and (13–12) have excitation temperatures of 364.97 K, 431.29 K, and 503.13 K, respectively. Which transition has the largest flux (where the CO SLED peaks) depends on the excitation mechanism of the gas (i.e., collisions, shocks, X-rays; Rosenberg et al. 2015).

The CO ladders for each of our galaxies can be seen in Figure 2. Observations of CO(1–0) and CO(2–1) with the Institute for Radio Astronomy in the Millimeter Range (IRAM) 30 m telescope are taken from Young et al. (2011), with a few coming from Young et al. (2008), Crocker et al. (2012), and Welch & Leslie (2003). Observations of CO(3–2) with the James Clerk Maxwell Telescope were available for three of our galaxies (NGC 3665, 4526, and 5866; Bayet et al. 2013).

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3.1.1. Excitation Based on Ladder Shape

In a study of luminous infrared galaxies (LIRGs) and ultra-luminous infrared galaxies (ULIRGs), Rosenberg et al. (2015) separate CO ladders into three distinct classes by a parameter α that they define as (CO(11–10) + CO(12–11) + CO(13–12))/(CO(5–4) + CO(6–5) + CO(7–6)). Class I are galaxies with α < 0.33, Class II have 0.33 < α < 0.66, and Class III have α > 0.66. The higher excitation classes (II and III) require some form of heating other than UV radiation, whether it be mechanical, X-ray, or other. Roughly speaking, class II require mechanical, class III require X-rays, but there are other necessary criteria (i.e., line widths) to fully characterize the heating mechanism. See Rosenberg et al. (2015) for more details.

Unfortunately, we do not have detections for all of these lines in all of our galaxies, so we can only determine upper limits for some, while others have too few detections to compute an alpha value. If we had detections for neighboring CO lines, a linear interpolation was used to define a flux value. According to this classification of our nine galaxies, NGC 1222 and 2764 have upper limits of α < 0.6 and α < 1.37, respectively, while NGC 1266 has a value of α = 0.61. This places NGC 1266 at the upper end of Class II. NGC 1266 is an unusual ETG, and has been previously studied in detail by Glenn et al. (2015) and Pellegrini et al. (2013).

The upper J level at which the CO SLEDs peak is also a crude approximation of the gas excitation, and we find that NGC 2764, 3665, 4526, 5866, and 7465 peak at an upper J of 4, while NGC 1222 and 4710 peak at J = 5, and NGC 1266 peaks at J = 8. Upper limits make it difficult to define the peak for NGC 4459, but it appears to peak at an upper J of 4 or 5.

Rosenberg et al. (2015) find that α has a positive correlation with dust temperature. This correlation suggests the presence of dense, warm molecular gas implies the presence of warm dust. Though we do not have enough detections to claim any trends, our sparse sample is consistent with their result, as seen in Figure 3.

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In addition to some of the CO transitions being used to measure star formation in galaxies, the [N ii] 205 μm line from the SPIRE spectra is being explored as a possible tracer of star formation. Nitrogen is ionized by UV emission from hot young stars with lifetimes on the order of a few to 10 Myr, and thus provides a measure of the SFR on relatively short timescales, similar to the way in which thermal radio free–free emission or the optical Balmer lines are used as a star formation tracer.

Zhao et al. (2013) find a tight correlation between IR luminosity and the [N ii] 205 μm luminosity for a sample of LIRGs, with log(L_IR) = (4.51 ± 0.32) + (0.95 ± 0.05)log(L_{[N ii]}) and a scatter of 0.26 dex in L_IR. We do not find such a relation among our ETGs with a similar analysis using FIR luminosity, but our sample size is only 9 galaxies, whereas Zhao et al. (2013) are working with a sample size of 70 galaxies, so more observations of ETGs could improve this relation. We also see a large variation in the relative strength of the [N ii] 205 μm line and the mid-J CO lines, a ratio which should be relatively constant assuming both scale with the amount of star formation. The [N ii] 205/CO(5–4) ratio varies from 0.85 to 34, while the [N ii] 205/CO(6–5) and [N ii] 205/CO(7–6) ratios range from 0.83–44.4 and 0.75–15.5, respectively. Thus, the [N ii] 205 μm line does not appear to scale with the mid-J CO lines.

However, Lu et al. (2015) find a trend for the [N ii] 205/CO(7–6) ratio to decrease with increasing dust temperature. Their results, along with discussion in Zhao et al. (2013), imply a deficit in the [N ii] 205 μm line at higher dust temperatures. Because we have more detections of the CO(6–5) line, we have recreated the middle panel of Figure 2 from Lu et al. (2015), with the [N ii]/CO(6–5) ratio using published data from Lu et al. (2017), along with data for galaxies from the KINGFISH and Beyond the Peak surveys published in Kamenetzky et al. (2016). Our ETGs and the spiral galaxies appear to continue the trend seen in the LIRGs, as seen in Figure 4. Thus their claim that the [N ii]/CO line ratio can infer dust temperature, and hence the intensity of the ISRF, is strengthened. This method would be useful for situations when dust continuum detections are unavailable, such as in high-redshift galaxies.

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In the standard onion skin PDR model, the deepest layers of a PDR only contain carbon in the form of CO. Neutral carbon exists alongside molecular hydrogen in a shell surrounding the CO core, and finally there is a layer of C+ on the exterior surface of the PDR. Since the layers of gas outside of the core cannot be traced by CO, the gas is deemed "CO dark." However, observational evidence is mounting that neutral carbon is not relegated to a simple layer outside of the CO, but exists alongside CO in the interior of molecular clouds. A newer model of clouds composed of PDR "clumps" has been introduced to explain these observations.

[C i] (1–0) emission is found to have similar spatial and velocity structure to ¹³CO emission in the Orion giant molecular cloud (Ikeda et al. 2002). Ikeda et al. (2002) also derive the ratio of [C i]/CO column densities and find that it varies from 0.2 to 2.9 on the edges of molecular clouds, while it is roughly constant (0.1–0.2) in the cloud interiors, suggesting CO and [C i] are coexistent within molecular clouds. They find [C i] column densities of a few to 10 times 10¹⁷ cm⁻². The nearly constant ratio between CO and [C i] in the cloud interiors lends credence to the clumpy PDR model, and slight discrepancies in the ratio from region to region are postulated to be the result of differing cloud size distributions. Ikeda et al. (2002) put forth an argument to constrain clump sizes between 0.03 and 0.25 pc based on the [C i]/CO column density ratio, with larger clumps corresponding to lower ratios. However, observations of [C i] with higher spatial and velocity resolution will be necessary to affirm this hypothesis.

The spatial correlation found by Ikeda et al. (2002) could be explained by low density PDRs where [C i] abundance is expected to be greater, or by FUV photons penetrating deep into molecular clouds and triggering emission from [C i], coating the many clumps comprising a cloud, but the intensity correlation shown in Keene et al. (1997), along with similar temperatures derived from CO (and other means) and [C i] by Stutzki et al. (1997), strongly suggest that [C i] and CO reside in the same molecular gas clouds and trace the same molecular hydrogen.

At low temperatures, the structure of neutral carbon effectively has three accessible energy levels. There are three fine-structure levels in the 3P ground state, where the [C i] (2–1) and (1–0) transitions arise. Because of the simple structure, the fundamental (1–0) emission line is a good tracer of the column density, as it does not have a strong dependence on excitation temperature. On the other hand, the [C i] (2–1) transition is quite sensitive to the temperature, making the ratio of the [C i] lines a good tracer of the gas conditions.

3.3.1. Carbon as a Temperature Probe

3.3.2. Measuring Gas Mass with [C i]

The [C i] lines can be used to derive the global molecular gas mass, a concept used by Papadopoulos & Greve (2004), who use just the J = 1–0 transition, and shown again by Kamenetzky et al. (2014), who use both J = 1–0 and J = 2–1 and Boltzmann population distributions to determine the total molecular gas mass. Additionally, a nearly linear relation between CO(1–0) luminosity and both [C i] (1–0) and (2–1) luminosities was found by Jiao et al. (2017). The largest uncertainties when using [C i] as a tracer of molecular gas stem from the ratio of the number density of neutral carbon atoms to the number density of H₂ molecules. Papadopoulos et al. (2004) point out many factors that affect the C/CO abundance ratio, including turbulence and the effect of cosmic rays; however, they point out these same processes affect the X_CO factor, and it is still widely used and approximately constant. The small range of N(C i)/N(CO) of 0.1–0.2 found by Ikeda et al. (2002) supports the idea that the X_{C i} factor may be approximately constant, too, but large surveys in the future will be necessary to establish an X_{C i} value the community can trust.

Papadopoulos & Greve (2004) use an abundance ratio of 3·10⁻⁵ C/H₂, which is roughly the average between the value derived in Ikeda et al. (2002) from [C i] and CO observations of the Orion A and B clouds and the value derived in White et al. (1994) for the starbursting nucleus of M82. Kamenetzky et al. (2014) assume an equivalency between masses derived from CO(1–0) and [C i], and use the relation to calculate an abundance ratio of ${10}_{-7}^{+20}\cdot {10}^{-5}$ C/H₂. Since the starbursting nucleus of M82 does not represent ETGs well, we adopt the value in Ikeda et al. (2002) of 10⁻⁵ and have carried out calculations based on Boltzmann populations as in Kamenetzky et al. (2014) using the pair of [C i] transitions, producing estimates of the total molecular gas mass for each galaxy. They are compared to masses derived from CO(1–0) in Figure 5.

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For our small sample, there is good agreement with the mass derived from CO. In galaxies with lower metallicities or higher UV radiation fields where CO is less reliable as a mass tracer, [C i] could be quite useful in providing a more accurate measure of the molecular gas content. Both of these conditions would be applicable to high-redshift galaxies, and the [C i] lines may be shifted to frequencies observable from the ground in high-redshift sources. Had we used the conversion factor derived in Kamenetzky et al. (2014), it would have given H₂ masses significantly lower than those inferred from CO. This indicates a difference between ETGs and the sources (most of which are luminous infrared galaxies) in Kamenetzky et al. (2014), in that the ETGs have somewhat weaker [C i] emission relative to CO. Further work should be done to explore whether this method extends to lower metallicity objects, as well as what differences exist between various galactic morphologies.

The ratio of [C ii]/FIR or ([C ii] + [O i])/FIR is often used as an estimate for the heating efficiency of gas by photoelectrons off of dust grains. Dust is heated by UV photons and then emits infrared photons, while photoelectrons ejected off of these dust grains heat the gas, which in turn is cooled via spectral emission lines. By examining these ratios, one can determine how large of a role these lines play in cooling the gas, as well as how that role changes in different conditions. For instance, as the dust temperature increases, the [C ii] line becomes less efficient at cooling the gas and the [C ii]/FIR ratio decreases. Another result of Rosenberg et al. (2015) is that the [C i] line experiences a similar deficit as the [C ii] line as a fraction of FIR, and they interpret this as star formation that is dominated by ultracompact H ii regions. In such a region, the dust would absorb a larger fraction of the UV photons, driving down the relative flux from all emission lines. Conversely, CO does not show a CO/FIR deficit. This could be because the CO molecules reside in a different phase of the ISM than C and C+. Additionally, CO is responsible for much more of the gas cooling than PDR models predict, reaching values of 10s of percent, while models only predict 3%–5% of the total gas cooling budget to be attributed to CO (Meijerink et al. 2011). Rosenberg et al. (2015) surmise that the CO heating mechanism is different, and is not one that leads to dissociation or ionization. There is not a deficit in our ETGs for CO/FIR nor [C i]/FIR, though much of our data consists of upper limits. The values of ratios are consistent with the values for other galactic morphologies.

Gerin & Phillips (2000) find a trend of [C ii]/[C i] (1–0) to decrease as [C i] (1–0)/FIR increases; however, the two ULIRGs in their sample do not fit the trend. They also find a correlation between [C ii]/FIR and [C i] (1–0)/FIR, barring Arp 220 and Mrk 231, which fall below the trend line. The interpretation they give is that these correlations are the result of changes in the value of G₀ (the incident far-ultraviolet [FUV] flux of photons between 6 and 13.6 eV in units of 1.6·10⁻³ ergs cm⁻² s⁻¹, the local ISRF). Around G₀ = 10–100, the efficiency of the gas heating reaches a maximum, which saturates the [C i] (1–0)/FIR or [C ii]/FIR ratios. If the value of G₀ increases from this point, the gas heating will be less efficient, while the dust is still being heated the same amount per photon. PDR models show the FIR flux to increase linearly with G₀ beyond this range, while [C ii] increases logarithmically and [C i] (1–0) stays relatively constant, leading to [C i] (1–0)/FIR and [C ii]/FIR ratios that decrease as the radiation field increases. Since [C i] (1–0) has a weak dependence on G₀ and n, the [C ii]/[C i] (1–0) will increase with stronger radiation fields. Their PDR models find that the majority of the galaxies studied lie within the parameter space G₀ = 10²–10³ and n = 10²–10⁵ cm⁻³.

We have plotted the same ratios for our galaxies, shown in Figure 6. We find a range of [C i] (1–0)/FIR values for ETGs of around 10⁻⁵ to 4.2·10⁻⁵. This is more narrow than the range of [C i] (1–0)/FIR found for spirals or irregulars/mergers by Gerin & Phillips (2000), but falls within their bounds. They find values of [C i] (1–0)/FIR between 10⁻⁵ and 10⁻⁴ for spirals, and 2·10⁻⁶ to 2·10⁻⁴ for irregulars/mergers. Our galaxies show the same tendency for [C ii]/FIR to increase with [C i] (1–0)/FIR, though not at a statistically significant level using a Kendall rank test. The same is true for the decreasing trend of [C ii]/[C i] (1–0) that they find, albeit over a much more narrow range of [C i] (1–0)/FIR than that spanned by their sample. The same plots were made using the [C i] (2–1) emission line, as well, with similar results. The [C ii]/FIR ratio increases with [C i] (2–1)/FIR, but with even less significance, and the [C ii]/[C i] (2–1) ratio decreases with respect to [C i] (2–1)/FIR. We find a range of [C i] (2–1)/FIR values of 10⁻⁵ to 3.8·10⁻⁵.

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To learn more about the conditions of the gas in our galaxy sample, we utilized the online PDR toolbox (PDRT), a set of PDR models that takes a given set of gas phase elemental abundances and grain properties and solves for the equilibrium chemistry, thermal balance, and radiation transfer through a PDR layer for a fixed hydrogen nucleus density, n, and incident FUV intensity, G₀. Observed line ratios can be compared to their theoretical counterparts to infer the possible combinations of n and G₀ capable of producing the observations. Solutions are then found through a χ² minimization routine applied to the line ratios computed from the input fluxes (Kaufman et al. 2006; Pound & Wolfire 2008). Although the ISM has a fractal structure, such that no single density or ISRF can accurately characterize the ISM of a galaxy, these volume averaged quantities can help us understand the differences from one galaxy (or morphology) to the next. For future work it would be useful to model various density distributions within a beam to explore their effects on PDR models.

To a certain degree, you get out of a PDR model what you put in. For example, using emission lines that only originate in dense gas such as high-J CO lines will likely return a solution with a high density. One approach to this issue is to explicitly consider different phases of the ISM (molecular clouds, diffuse warm neutral gas, etc.) separately, as in Abdullah et al. (2017). But our ETGs have significantly worse spatial resolution and lack much of the ancillary data needed to characterize multiple phases of the ISM. Hence we take a different approach. We effectively lump all the phases of the ISM together, but differences in the global average quantities may still illuminate differences in the mix of phases from one galaxy to the next. To explore this conundrum, we have used every possible combination of inputs for the PDRT (with a minimum of three and a maximum of six spectral lines or the FIR continuum). We use combinations of the pair of [C i] lines, [C ii] 158, [O i] 63, CO(1–0), and FIR for galaxies with SPIRE observations, and only [C ii] 158, [O i] 63, CO(1–0), and FIR for our ETGs without SPIRE data. This subset of lines spans the ionized, neutral, and molecular components of PDRs, but does not include warm molecular gas that the mid- and high-J CO lines originate from. Malhotra et al. (2001) use only the [C ii] 158, [O i] 63, and FIR continuum, so we also find values using only [C ii], [O i], and FIR for a direct comparison to the values derived by Malhotra et al. (2001). The [C ii] and [O i] fluxes for our ETGs are taken from Lapham et al. (2017).

Once a set of input fluxes has been chosen, contour plots of the G₀ and n values that reproduce the observed flux ratios are overlaid, and their intersections indicate potential solutions. A pair of contours typically has more than one solution, so additional contours can help break this degeneracy and find a favorable solution, as shown in Figure 7. However, the addition of more contours can also produce more potential solutions, as shown in Figure 8. In either case, the final solution is based on a χ² distribution for every flux ratio. The solutions for each galaxy can be found in Table 3.

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3.5.1. PDR Models Using Only [C ii] and [O i] Lines

Following the recommendations by Kaufman et al. (1999), we divide the observed FIR flux by a factor of two. This is because the model assumes emission only comes from the side of the cloud illuminated by the UV radiation, when it likely comes from both near and far sides of the cloud, as the cloud is optically thin to dust continuum IR photons. Additionally, Kaufman et al. (1999) recommend increasing the [O i] emission by a factor of two. They argue that because the line is optically thick, the variety of different PDR orientations will lead to only half of the emitted [O i] flux being observed. The extreme cases would be a cloud lit from behind or face on. No emission would be observed in the former, and all of the emission would be observed in the latter. The [C ii] line also needs a correction, as part of the total flux comes from H ii regions rather than PDRs. This contribution can be quantified if observations of the [N ii] 122 and 205 μm lines are available. Galaxies with SPIRE observations use corrected [C ii] fluxes (flux only from PDR contributions) determined in Lapham et al. (2017); otherwise a flat 30% is removed for ionized gas contributions. These corrections are applied to all of our galaxies, as well as the comparison sample from Malhotra et al. (2001) in the following analysis.

When we relegate our models to only the [C ii], [O i], and corrected FIR fluxes, five ETGs fall along a line above the comparison samples (NGC 3607, 4429, 4459, 4526, and 5866), while the remaining fourteen drift toward higher density, lower UV intensity solutions where the comparison galaxies lie. This can be seen in Figure 9. ETGs having lower ISRF values would be consistent with their older stellar populations; however, the galaxies with the lowest UV intensity solutions are from the KINGFISH and Malhotra et al. (2001) samples. Thus the majority our ETGs are similar to spiral galaxies, while some exhibit different behavior. The ([C ii] + [O i])/FIR ratio is likely what drives the unusual behavior of a handful of our ETGs, as shown in Figure 10.

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3.5.2. PDR Models Including CO and [C i]

After modeling all of the allowed combinations of PDR lines chosen, the median values for n and G₀ for our ETGs are plotted against the results from the Malhotra et al. (2001) sample found in the previous section (using only [C i], [O i], and FIR) in Figure 11. Typical uncertainties for G₀ and n are also calculated from the various allowed flux ratio combinations, and are scaled to the uncertainty of the median by a factor of 1.25 (the square root of π/2). Our ETGs occupy a similar range as the spirals from the Malhotra et al. (2001) sample. NGC 4526 lies furthest from the group, with the highest density and a low value of G₀. Curiously, NGC 4526 was also found to have the largest [N ii] 122/[C ii] 158 ratio in Lapham et al. (2017). Though these results may not be related, it could be an indication that metallicity plays a role in regulating the ISRF, a possibility that will be explored in future work. It also appears that the galaxies known to be merger-driven starbursts (i.e., NGC 1222 and NGC 7465) or blue in color (NGC 2764) have relatively high values of G₀, while the quiescent Virgo Cluster members have lower values of G₀. Galaxies with log(G₀) ≥ 2 include NGC 1222, 2764, 3665, 4710, and 7465, as well as IC 1024 and IC 0676. Galaxies below this threshold include Virgo Cluster members NGC 4429, 4435, 4459, 4526, 4694, as well as NGC 1266, NGC 3032, 3489, 3626, 4150, and NGC 5866, though NGC 1266 and 3626 are close with log(G₀) = 1.75.

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The inclusion of the [C i] and CO lines likely probes a denser phase of the ISM, possibly getting into the molecular cloud and not the PDR at the surface of the cloud. Contour plots of G₀ and n show that solutions for [C i] (2–1)/CO lie exclusively at large densities and low radiation fields for all of the galaxies in our sample, as shown in Figure 12. A similar plot for the [C ii]/CO ratio does not restrict solutions to the same high n, low G₀ regime, seen in Figure 13. Thus it appears the inclusion of the [C i] emission lines is the primary driver for pushing the models to the high n, low G₀ solutions; however, it is not solely responsible, as the addition of only the CO(1–0) also pushes solutions to this regime, as seen in Figure 11. We will revisit the [C i]/CO ratios in Section 3.6, where RADEX models will be used to recreate the observed line ratios.

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It is curious to note that values of G₀ and n that can reproduce the other line ratios (such as [O i]/[C ii] and ([O i] + [C ii])/FIR) apparently cannot reproduce the observed [C i]/CO line ratios. Comparing Figures 10 and 12, one can see the observed [C i]/CO ratios disagree with the models. The observed [C i] flux is typically at least a factor of five fainter than what the models would predict based on [O i], [C ii], and FIR.

3.5.3. Density–UV Field Scaling Relations

One of the prominent results from the PDR modeling in Malhotra et al. (2001) is a scaling relation between G₀ and n, with G₀ ∝ n^1.4. This scaling relation was not reproduced by our PDRT models. Malhotra et al. (2001) use a similar method to correct the [C ii] flux with the [N ii] 122 μm emission, and increase the [O i] flux by a factor of two. However, there is no mention of a correction to the FIR flux. Thus, in an effort to reproduce their scaling relation, we have also used the PDRT without correcting the FIR flux for and of the samples (our ETGs, the KINGFISH galaxies, and the galaxies from Malhotra et al. 2001). These results are plotted in Figure 14 and appear to be consistent with the G₀ ∝ n^1.4 trend found for the galaxies studied in Malhotra et al. (2001). However, they still lie at lower values of the UV field strength G₀ than the original result in Malhotra et al. (2001).

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There are two proposed explanations for the G₀ ∝ n^1.4 scaling relation observed in Figure 14 and Malhotra et al. (2001). The first is that it is the result of a Schmidt law on a smaller scale, with high gas densities leading to high star formation rates (Schmidt 1959). The exponential behavior agrees well with further work done on the Schmidt law by Kennicutt (1998), who find the global surface densities of star formation to scale with the surface density of gas by an exponent of 1.4. Another explanation offered by Malhotra et al. (2001) is that the relation between G₀ and n is the result of geometry. Assuming the FIR line and continuum emission is dominated by PDR contributions outside of young stars, Strömgren sphere calculations predict the FUV flux at the neutral surface outside of the sphere to have a dependence of n^4/3. Malhotra et al. (2001) prefer the Strömgren sphere explanation. First, they cite the high values of G₀ and n found by PDR models. The large values of G₀ are only seen very close to O and B stars (around 1 pc), making it clear the results are probing local behavior, rather than global averages. While some of this radiation escapes the molecular cloud and irradiates the diffuse ISM, the PDR model is weighted to the high G₀, high n component, which is exactly where the Strömgren sphere scaling is applicable. Additionally, they have pressures derived from [O iii] 88/[O iii] 52 ratios that support this conclusion.

If we assume the latter explanation is what dictates the observed trend of G₀ ∝ n^1.4, it could potentially explain why our models with [C i] or CO do not reproduce the trend. The flux scaling predicted by Strömgren spheres occurs at the surface illuminated by FUV photons, while the [C i] and CO reside deeper in the cloud where the ISRF is weaker and the gas is more dense, leading to higher density, lower G₀ solutions. However, this does not explain the galaxies that deviate from the trend even when only the [C ii], [O i], and FIR fluxes are used in the PDR models. For future work it would be useful to compute G₀ directly to test the accuracy of the PDRT solutions.

Line ratios of the [C i] transitions to their neighboring CO transitions can be used to constrain the conditions of molecular gas in their host galaxy. The [C i] (1–0) emission line lies near the CO (4–3) emission line, and the [C i] (2–1) transition is even closer to the CO (7–6) transition at 809 and 806 GHz, respectively. Their proximity makes the line ratios [C i] (1–0)/CO(4–3) and [C i] (2–1)/CO(7–6) less affected by uncertainties in the baseline and corrections applied by the SECT. We use RADEX models to produce line ratios over a range of kinetic temperatures and H₂ densities with a fixed column density for C of 5·10¹⁷ and 8·10¹⁸ for CO (Van der Tak et al. 2007). The temperatures vary from 10 to 100 K and the densities range from 10¹ to 10⁸ cm⁻³. We then use an error minimization routine to determine the conditions that best reproduce our observed line ratios. Uncertainties in the model solutions are based on 1σ variations in the line ratios. If solutions showed no change, the grid spacing is reported as the uncertainty. The RADEX results can also be compared to temperatures derived directly from the ratio of the [C i] (1–0) and (2–1) emission lines. The results of the RADEX models can be seen in Table 4, with the last column showing the temperatures derived solely from the [C i] line ratio.

For the majority of galaxies in our sample, there is good agreement between the temperatures derived from the [C i] line ratio and those inferred from the [C i]/CO ratios from RADEX models. However, one galaxy in our sample (NGC 2764) has line ratios which best match high temperature (∼80 K), low density RADEX models, while the temperature derived from the [C i] line ratio was moderate (∼20 K). The discrepancy is largely motivated by the upper limit on the [C i] (1–0) flux from NGC 2764, which in turn creates an upper limit on the [C i] (1–0)/CO(4–3) ratio of 0.28. Ratio values below this limit begin around 55 K, and at a low H₂ density RADEX can reproduce the observed [C i] (2–1)/CO(7–6) ratio of 1.63.

Additionally, the PDRT models did not have solutions that could produce the observed values of [C i] (2–1)/CO(1–0) for our ETGs. Our ratios mostly fall between atomic hydrogen densities with log(n) of 4 or 5, but far below any G₀ contours, as shown in Figure 12. However, RADEX models can reproduce our observed [C i] (2–1)/CO(1–0) with a wide variety of conditions. Ratios for our ETGs vary between about 10–60. At T = 10 K, molecular hydrogen densities with log(n) between 4 and 8 and produce ratios of 10 to 11. At temperatures of 20 to 60 K and beyond, it appears any ratio between 10 and 60 can be reproduced at some respective density. At T = 20 the densities vary between log(n) of 2.3 to 8; above that they range from 2 to 3.5 or 4. RADEX seems to be better suited for emission lines originating within molecular clouds, while the PDRT is more suited for the PDR layers on the surface of molecular clouds.