THE SPATIALLY RESOLVED $[{\rm{C}}\,{\rm{ii}}]$ COOLING LINE DEFICIT IN GALAXIES

The flow of radiative energy through the gas and dust comprising the interstellar medium (ISM) of galaxies strongly impacts their observational and physical properties. Indeed, the process of star formation itself is controlled in part by the balance between the radiative heating and cooling in star-forming clouds of gas and dust (e.g., Krumholz et al. 2011). With high ultraviolet (UV) opacity, dust grains absorb and re-radiate a significant fraction of a galaxy's radiative energy (from a few percent up to nearly 100 percent in the most infrared-luminous objects, Sanders & Mirabel 1996). The smallest of these dust grains have small heat capacities, and so are stochastically heated to high non-equilibrium temperatures (Draine & Li 2001). With their modest ionization energies (∼6–9 eV, Witt et al. 2006) and large cross-sections, these small grains, including polycyclic aromatic hydrocarbons (PAHs), are also the dominant source of the photoelectrons that energetically couple the radiation field to the gas outside of hydrogen-ionized zones (Bakes & Tielens 1994).

Neutral gas cools predominantly through collisional excitation of abundant ions with low-lying excited states. With a low first ionization potential (11.26 eV) well below the Lyman limit, atomic carbon is found in singly ionized form in a wide range of environments. Together with its high abundance and modest first excitation energy, this makes ionized carbon a highly effective coolant of neutral atomic and even diffuse molecular gas. Indeed, for this reason, the most luminous emission line of a galaxy is typically [C ii] 158 μm (e.g., Stacey et al. 1991). This high line luminosity (up to several percent of the total infrared output) also makes [C ii] a promising diagnostic tool for mapping the key physical properties of galaxies in the early universe, including star formation rate, kinematics, size, virial mass and molecular content. One of ALMA's primary science drivers involves imaging [C ii] in Milky-Way-type galaxies at redshifts greater than 3, and interferometers like IRAM/PdBI and ALMA are now detecting [C ii] in galaxies beyond redshift z = 6–7 (e.g., Venemans et al. 2012; Capak et al. 2015; Maiolino et al. 2015; Willott et al. 2015).

Helou et al. (2001) used the near constancy of [C ii] relative to PAH emission in support of models that stressed the importance of very small dust grains to photoelectric heating of neutral gas. But for stellar populations drawn from realistic star formation histories, the small grains that heat gas via photoelectrons, and the larger grains that re-process the bulk of absorbed UV/Optical radiative energy into the infrared, do not differ substantially in their contributions to absorption of starlight (e.g., Crocker et al. 2013). Thus, the rate of gas cooling via [C ii] might be expected to closely track the total infrared (TIR, 3–1100 μm) emission of galaxies—[C ii]/TIR should remain approximately constant. Indeed the suitability of [C ii] as a direct star formation tracer, which follows from this expectation, has been given considerable attention (Boselli et al. 2002; Farrah et al. 2013; De Looze et al. 2014; Herrera-Camus et al. 2015; Kapala et al. 2015).

And yet, relative to the heating rate inferred from bolometric infrared luminosity, the line luminosity of this principal coolant is found to vary by orders of magnitude among different galaxy types and environments (Malhotra et al. 1997, 2001; Luhman et al. 2003; Brauher et al. 2008; Stacey et al. 2010; Graciá-Carpio et al. 2011; Díaz-Santos et al. 2013). This so-called [C ii] cooling line deficit problem, first revealed as an unexpectedly low [C ii] luminosity among local ultra-luminosity infared galaxies (ULIRGs), has proven difficult to reconcile, with possible explanations invoking physical impacts on either the coupling of radiation to the gas via dust grains, or on the nature of the subsequent gas cooling itself.

The Herschel Space Observatory has for the first time delivered the combination of sensitivity and spatial resolution necessary to map the fundamental cooling lines on sub-kiloparsec scales in nearby galaxies. Kingfish (Key Insights on Nearby Galaxies—a Far-Infrared Survey with Herschel; Kennicutt et al. 2011) has investigated a diverse sample of galaxies in the nearby universe (d ≲ 30 Mpc). Here we explore resolved maps of [C ii] across the Kingfish sample in order to delineate the principal physical drivers of the deficit itself. In Section 2 we detail the Herschel and supporting observations and data reduction and demonstrate the resulting quality of the [C ii] maps; in Section 3 we highlight the impact of surface brightness and metallicity on the deficit; in Section 4 we evaluate the role played by star formation rate density, both locally in normal and luminous infrared galaxies, and at high redshift; in Section 5 we investigate the spatially resolved impact of active galactic nuclei (AGNs), concluding with discussions in Sections 6 and 7.

2.1. PACS Observations

2.2. Ancillary Data

2.3. Resulting Line Maps

The high sensitivity realized in Herschel/PACS unchopped mapping observations translates to significant [C ii] detection in most map pixels. An example of the [C ii] maps covering radial strips and selected extranuclear regions of 54 Kingfish galaxies is given in Figure 1 for starburst-ring and low-luminosity AGN host galaxy NGC 4736 (see also van der Laan et al. 2015). Strong variation of [C ii] brightness relative to the underlying 24 μm emission map is seen in the I_{[C ii]}/$\nu {I}_{\nu }(24\,\mu {\rm{m}})$ map in the lower panel. Example extracted [C ii] line profiles from bright, intermediate, and faint areas of the spectral cube are included. In most cases, the lines are unresolved at the Herschel/PACS spectral resolution of ∼239 km s⁻¹ at the [C ii] line (see de Blok et al. 2016).

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A complete compendium of Kingfish line fluxes, spectral cubes, and related data is under preparation. Herrera-Camus et al. (2015) used the Kingfish [C ii] data set to provide updated star formation rate calibrations of the line (see Section 4). Here we investigate variations in [C ii]/TIR with important intensive (i.e., local, not global) physical properties of galaxies, including surface brightness, gas-phase metallicity, star formation rate density, and AGN proximity.

While early surveys uncovered an apparent dependence of the fractional [C ii] line power on galaxies' bulk infrared luminosity (e.g., Malhotra et al. 2001; Luhman et al. 2003), with the highest luminosity galaxies exhibiting the deepest deficits, the advent of [C ii] surveys at redshifts z ≳ 1 has completely altered this viewpoint. In the nearby universe, to good approximation, global luminosity serves as a reliable parameterization of the strongly varying local heating conditions in the ISM, which must underly the deficit, perhaps due to strong correlation between merger activity and high infrared luminosity. But this is emphatically not the case at earlier epochs, where wide ranges of ISM conditions exist even at the highest infrared luminosities. This is well summarized by Brisbin et al. (2015), who compile local and recent higher redshift results to demonstrate conclusively that the global infrared luminosity of a galaxy is a very poor predictor of its deficit. For example, at far-infrared (FIR, 40–122 μm) luminosities L_FIR > 10^12.5 L_☉, [C ii]/FIR is not uniformly low as is found in local ULIRGS, but instead ranges over more than two orders of magnitude, and can in fact exceed the value seen in local star-forming systems more than a thousand times less luminous. This fits directly with the developing evolutionary framework in which the highest luminosity systems at earlier epochs arise not exclusively from mergers but increasingly in large-scale, cooler extended starbursts driven by the direct accretion of cold gas from the richer available reservoirs (e.g., Pope et al. 2008; Menéndez-Delmestre et al. 2009; Symeonidis et al. 2013). Results from the ISO satellite provided another viewpoint—that deficits are most strongly correlated with the infrared dust color temperature—an intrinsic indirect measure of the intensity of the interstellar radiation field (e.g., I(60 μm)/I(100 μm), Malhotra et al. 2001; Brauher et al. 2008). Other recent results have extended the breakdown of infrared luminosity as a key driver of the cooling deficit to redshifts as low as z ∼ 0.2 (Ferkinhoff et al. 2014; Magdis et al. 2014; Rigopoulou et al. 2014; Brisbin et al. 2015; Ibar et al. 2015). And some have abandoned L_IR as a driving parameter in favor of intrinsic quantities such as luminosity per unit gas mass (which relates to the efficiency of star formation, Graciá-Carpio et al. 2011). This leaves open the question of which physical parameters of the star-forming ISM dominate the strong observed deficit trends. Spatially resolved samples with detailed gas and dust diagnostics on sub-kiloparsec scales are ideally suited to exploring this. Here we consider two alternative galaxy characteristics which together explain the bulk of the variation in [C ii]/TIR found in local galaxies: infrared surface brightness (which relates to the surface density of star formation), and gas-phase metallicity, 12+log(O/H).

3.1. Surface Brightness

A number of studies have tracked the [C ii] deficit on sub-kiloparsec scales in galaxies. Nakagawa et al. (1998) reported balloon-borne [C ii] observations of a large portion of the Galactic plane, finding [C ii]/FIR decreases in star-forming regions and the Galactic center. Beirão et al. (2012) found deficit variations in the starburst-ring Kingfish galaxy NGC 1097, and Croxall et al. (2012) expanded this work to an additional galaxy, NGC 4559, uncovering changes in the ratio of [C ii] to PAH emission correlated with the charge state of PAH grains. Kramer et al. (2013) and Kapala et al. (2015) found that the fractional luminosity of [C ii] drops also toward the centers of M33 and M31, respectively. On larger, but still resolved scales, Díaz-Santos et al. (2014) found that the deficit in the luminous infrared galaxies (LIRGs) of the GOALS sample was mostly confined to their starbursting nuclei, and that infrared surface brightness anti-correlated with [C ii]/FIR. Lutz et al. (2015) extended this to quasar host galaxies and also found a tight trend of [C ii]/FIR with FIR surface brightness.

Figure 2 demonstrates the trends of [C ii]/TIR in ∼15,000 regions drawn from the full Kingfish sample. Regions within 750pc of the centers of those galaxies optically classified as AGNs or composite sources are excluded from this plot (but see Section 5). The full sample yields $\langle [{\rm{C}}\,{\rm{ii}}]/\mathrm{TIR}\rangle \,=0.48\pm 0.21$% by arithmetic average, and $\langle [{\rm{C}}\,{\rm{ii}}]/\mathrm{TIR}\rangle \,={0.46}_{-0.15}^{+0.18}$% for the median and inner 68% of regions. Regions are binned in logarithmic intervals of $\nu {I}_{\nu }(24)$ and [C ii]/TIR when at least nine points fall in a bin, and are otherwise plotted individually. The density of regions falling in each bin is shown linearly scaled in the inset (418 regions maximum). The full sample median at each $\nu {I}_{\nu }(24)$ bin (black central solid line) demonstrates the strong and monotonic drop of the fractional cooling line luminosity with 24 μm surface brightness. The trend was found to be more significant with surface brightness than with infrared color temperatures such as $\nu {I}_{\nu }(70)$/$\nu {I}_{\nu }(160)$ (but see Section 4). Using an alternative measure of infrared surface brightness, $\nu {I}_{\nu }(70)$, results in a similar declining trend (not shown) with nearly identical dispersion. To test for any dependence on physical scale of the extraction region, we partitioned the region set into two subsamples—those with physical scales above and below 0.65 kpc (the median region size among the sample galaxies). The resulting median trend line of [C ii]/TIR versus $\nu {I}_{\nu }(24)$ remains unchanged between these subsamples, indicating little dependency on physical extraction scale.

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At each bin of $\nu {I}_{\nu }(24)$ above 10⁻⁷ W/m² sr, the standard deviation of regions with that surface brightness is approximately $\sigma ([{\rm{C}}\,{\rm{ii}}]/\mathrm{TIR})\simeq 0.13 \%$. Given that [C ii]/TIR drops to ∼0.2%, this scatter is not insignificant. Rather than being random scatter, however, a single physical parameter appears to be associated with deviation from the fiducial trend: metal abundance.

3.2. Metallicity

The areal density with which galaxies form new stars fundamentally impacts their physical character, altering the temperature, pressure and kinetic state of interstellar gas, the size and distribution of molecular clouds, and the efficiency of star formation. In star-forming galaxies, on scales near or above a kiloparsec, ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ varies over a remarkably wide range, from below ${10}^{-5}\ {M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$ in nearby low-efficiency dwarf galaxies (e.g., Cook et al. 2012), up to "maximal starbursts" which are Eddington-limited by the impact of radiation pressure on dust at ${{\rm{\Sigma }}}_{{\rm{sfr}}}\simeq 1000\ {M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$ (Thompson et al. 2005). Given the strong demonstrated dependency of [C ii]/TIR on 24 μm surface brightness, it is natural to consider whether or not ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ itself underlies this trend.

Figure 3 explores the dependence of the [C ii] deficit on ${{\rm{\Sigma }}}_{{\rm{sfr}}}$. It combines the resolved regions within Kingfish galaxies (on physical scales ranging from 0.2–1.6 kpc) with central values from resolved local LIRGs and ULIRGs from the GOALS sample of Díaz-Santos et al. (2013), as well as a sample of high-redshift galaxies at z = 1.8–6.4 with directly measured physical size scales from well-resolved [C ii] and continuum interferometic measurements. For the Kingfish galaxies, ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ is obtained from a hybrid indicator, which combines FUV and 24 μm surface brightness of Hao et al. (2011).²⁴ In the central GOALS regions, ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ is obtained directly from 24 μm surface brightness using the prescription of Rieke et al. (2009). In both cases, measurements are performed at a matched common resolution of the PACS 160 μm beam (112). Since GOALS sources are typically small compared to the 160 μm beam, this required substantially trimming the sample to include only the 51 galaxies with well-resolved Spitzer/MIPS 24 μm and Spitzer/IRS 15 μm continuum emission (measured along the slit, see Díaz-Santos et al. 2013).

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The resolved Kingfish regions (colored bins, omitting low-density points as well as AGN centers, as in Figure 2) and individual GOALS galaxies are color-coded by their (mean) dust color temperature $\nu {I}_{\nu }(70\,\mu {\rm{m}})$/$\nu {I}_{\nu }(160\,\mu {\rm{m}})$. In the case of GOALS, this color temperature is derived from the nearby 63 μm and 158 μm continuum fluxes underlying the [O i] and [C ii] lines, using an average Dale & Helou (2002) template to account for the slight wavelength offsets. A clear trend of increasing color temperature at higher ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ is evident, although with some mixing between the two—bins of fixed values of $\nu {I}_{\nu }(70)$/$\nu {I}_{\nu }(160)$ ratio do not lie at a single ${{\rm{\Sigma }}}_{{\rm{sfr}}}$, and the scatter of color temperature at a given bin of ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ is significant: $\sigma (\nu {I}_{\nu }(70)/\nu {I}_{\nu }(160))=0.38$. The median and inner quartile trends of the resolved Kingfish regions are shown in solid and dashed lines. The higher star formation rate density GOALS sample clearly extends the [C ii]/TIR decline set by these resolved regions.

High-redshift sources (see Figure 3 caption) were included directly from reported ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ values or computed from sizes and IR luminosities, typically based on SED-fitting of several IR continuum measurements and utilizing the resolved physical sizes from [C ii] and nearby continuum maps. Where necessary, conversion to TIR luminosities from FIR (42–122 μm, 2.4 times) were made using the M82 template of Polletta et al. (2007), and from IR (8–1000 μm, 1.06 times) using the mean template of Dale & Helou (2002). In all cases expected uncertainties are less than 20% for conversion among these broad infrared luminosity bands. The TIR luminosity was further converted to star formation rate using the updated scaling of Murphy et al. (2011). Source sizes were computed from the FWHM area of elliptical Gaussian fits to the [C ii] and/or resolved rest-frame sub-millimeter continuum intensity maps (averaging the two where both were available).

Taken together with resolved lower surface brightness Kingfish and GOALS samples, the high-redshift galaxies extend a single trend of deepening [C ii] deficit over more than six orders of magnitude in ${{\rm{\Sigma }}}_{{\rm{sfr}}}$. Of particular interest are the two galaxies which form part of the single galaxy group BRI 0952-0115 (upward gray triangles in Figure 3, Gallerani et al. 2012). These two galaxies have approximately the same infrared luminosity of a few times 10¹² L_☉, but one is approximately 10 times more extended, and thus, with its correspondingly lower ${{\rm{\Sigma }}}_{{\rm{sfr}}}$, has a much higher limiting [C ii]/TIR ratio.

A single power-law fit to the binned Kingfish regions, GOALS galaxies, and high-redshift sources is shown as the dot-dashed line in Figure 3. The relatively tight correlation permits inverting the relation to approximately estimate ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ directly from a measured value of the [C ii] fractional line luminosity:

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{{\rm{sfr}}}=(12.7\pm 3.2)\ {\left(\displaystyle \frac{{L}_{[{\rm{C}}{\rm{II}}]}/{L}_{\mathrm{TIR}}}{0.1 \% }\right)}^{-4.7\pm 0.8}\left(\displaystyle \frac{{M}_{\odot }}{\mathrm{yr}\ {\mathrm{kpc}}^{2}}\right).\end{eqnarray} \tag{ 1 }$$

This relationship applies when young stars provide the dominant energy source, and only on scales greater than a few hundred parsecs. The slope is similar whether the fit is performed to local sources only, or with high-redshift sources included. At the highest physically meaningful ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ of ∼1000 ${M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$, [C ii]/TIR ≃ 0.03%–0.045%. Since the slope is relatively steep, care must be taken to correctly account for input uncertainties in [C ii]/TIR. Even with minimal input uncertainties, the uncertainty in the relation itself contributes to over an order-of-magnitude range in the resulting ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ values (see gray band in Figure 3). The steepness of the ${{\rm{\Sigma }}}_{{\rm{sfr}}}$–L_{[C ii]}/L_TIR relation also implies that, in blind [C ii] surveys, lower ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ galaxies will be overrepresented compared to more compact star-forming systems.

An important advantage of Equation (1) is that, in addition to applying to regions within resolved galaxies, it can be used with the observed global luminosity ratio L_{[C ii]}/L_TIR to estimate the star formation surface density ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ if the [C ii] and TIR emitting size scales are similar (as they are in resolved local galaxies; de Blok et al. 2016). It is important to note that Equation (1) is not a calibration for total star formation rate in a galaxy, as galaxies with widely differing SFR can have similar ${{\rm{\Sigma }}}_{{\rm{sfr}}}$. If, however, the total star formation rate is known separately, Equation (1) permits estimation of the compactness of a given star-forming system.

AGNs can reduce the relative cooling power of [C ii] in galaxies in several ways. They are effective at producing warm central dust continuum emission (up to dust sublimation temperatures of ∼10³ K), which can substantially increase bolometric infrared luminosity and drive an apparent deficit in [C ii]/TIR. They can also directly impact the cooling line emission itself. One direct effect on [C ii] emission results from changes in the overall ionization state of the gas. Langer & Pineda (2015) modeled this effect on [C ii] emission in AGNs and found substantially reduced values (to [C ii]/TIR ∼ 0.01%) in the inner ∼10² pc of AGN hosts. Another effect is the possible photo-destruction of small dust grains by X-rays out to kiloparsec scales (Voit 1992). The loss of smaller grains then reduces the efficiency of photoelectric heating of neutral gas by UV photons (whether those photons are generated by stellar or accretion-driven sources).

While Kingfish sources were not selected to host luminous AGNs, approximately half of the sample, including 26 of the [C ii]-targeted galaxies considered here, have nuclear optical classifications consistent with AGNs or composite sources (either Seyferts or LINERs; Moustakas et al. 2010). Although none are bright Seyfert AGNs (with typical broadband X-ray luminosities up to 10⁴⁴ erg s⁻¹), the spatial resolution available in Kingfish (down to ∼200 pc) means that any AGN impact on the cooling balance can be directly resolved.

Figure 4 illustrates the striking behavior of [C ii] in the inner 1.5 kpc of several AGN hosts in the sample. The steep radial trend of increasing deficit toward the centers of these three galaxies is immediately evident. One of these, NGC 4736, is the example map shown in Figure 1, where the strong deficit of [C ii] relative to underlying infrared continuum can be readily seen in the central region, dropping in this case to [C ii]/TIR ∼0.05%. In Figure 4, the ratio [C ii]/TIR is normalized for each galaxy at a de-projected nuclear distance of 1.5 ± 0.2 kpc, and these three galaxies in particular show a similar pattern of rapidly declining [C ii]/TIR values below ∼1 kpc. While the surface brightness $\nu {I}_{\nu }(24)$ of these AGN hosts does increase toward the center (color intensity scale of the points), the relative decline in [C ii]/TIR for the modest range in brightness from disk to nucleus is substantially steeper than the general trend shown in Figure 2. To demonstrate this, we have calculated the expected change in the normalized deficit with nuclear distance due purely to changing 24 μm surface brightness. For each galaxy, we apply the median trend line of [C ii]/TIR with $\nu {I}_{\nu }(24)$ (Figure 2) to the average region-by-region surface brightness. We bin radially in nuclear distance bins of 150 pc, and normalize [C ii]/TIR as before at 1.5 ± 0.2 kpc. These expected trends are shown in Figure 4 as the dotted black lines, which clearly demonstrate how strong the observed central suppression is in these AGN hosts, compared to the modest changes that would arise from centrally increasing surface brightness.

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One more potential impact of the presence of an AGN on the heating/cooling balance is entirely indirect: especially at low fractional AGN luminosity, the optical diagnostics used to classify nuclear regions are diluted in the presence of star formation. For this reason, most local low-luminosity AGN galaxies are dominated by older stellar populations in their centers (e.g., Singh et al. 2013). In this interpretation, the AGN is relevant only by association with the high starlight intensity and softer spectrum that characterize their host's old central stellar population.

Langer & Pineda predicted deviations in [C ii]/TIR above X-ray luminosities of 10⁴³ erg s⁻¹. To investigate trends with AGN luminosity, we draw on the central 0.3–8 keV luminosities from Chandra, which Grier et al. (2011) compiled for the SINGS sample. Figure 5 shows the central suppression of [C ii]/TIR, defined as the ratio of this value in the central 400 pc²⁵ relative to its average value at 1–2 kpc. It is immediately evident that, while all AGN host galaxies show some suppression of [C ii]/TIR toward their centers, in most cases this suppression is modest, approximately 30%. The strength of [C ii] suppression does not correlate with central X-ray luminosity, and for most AGNs, it is similar to non-AGN hosts in the sample—30% ± 15% (seen as bars at left and in the marginal histogram). But a small number of AGN sources (including the three with radial profiles shown in Figure 4) show substantially higher central suppression factors, with [C ii]/TIR reduced by up to a factor of five from disk to center. The five most suppressed sources have 0.3–8 keV luminosities well below the average (L_{0.3–8 keV} = 10^40.3 erg s⁻¹). Moreover, these highly suppressed sources are in fact those with the largest central surface densities of evolved starlight, as measured by $\nu {I}_{\nu }(3.6\,\mu {\rm{m}})$ (color scale of the points). NGC 4736, the low-luminosity AGN host with the strongest central [C ii]/TIR suppression, has the highest central 3.6 μm surface brightness of the entire sample. This suggests that the high density of soft starlight may drive the enhanced central [C ii]/TIR suppressions in AGN hosts.

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Relative to integrated infrared luminosity, the fractional power emitted in [C ii]—the dominant neutral gas coolant and most luminous emission line in most galaxies—ranges over more than two orders of magnitude, from [C ii]/TIR ≳ 1% to ≲0.01%. This "cooling line deficit" has been well studied globally in galaxies ranging from normal star-forming to ultraluminous systems, with the physical origin as yet unresolved. Such global deficits have been invoked as evidence in support of distinct physical conditions controlling the star formation process in compact luminous galaxies, versus "main-sequence" galaxies forming stars at low surface densities.

A number of physical explanations for the large variations in the power of the dominant cooling line relative to dust-reprocessed emission among galaxies have been considered, including the following.

• 1.  
  [C ii] self-absorption, which could lead to absorbed velocity-resolved line profiles in deep deficit sightlines.
• 2.  
  Strong continuum extinction at 158 μm.
• 3.  
  Significant and varying contributions of [C ii] from (hydrogen-) ionized gas (e.g., Luhman et al. 2003).
• 4.  
  Collisional quenching of [C ii] emission at high density, which would result in the dense gas coolant [O i] compensating (e.g., Brauher et al. 2008), although this effect was not seen in two Kingfish galaxies (Croxall et al. 2012).
• 5.  
  H ii regions with high ionization parameter, in which dust absorbs non-ionizing UV photons before they can enter neutral gas. This should result in deficits in all cooling lines for neutral gas (see Graciá-Carpio et al. 2011).
• 6.  
  Small grain destruction or charging (e.g., Malhotra et al. 2001), which reduces photoelectric yield and should result in correlations between [C ii]/TIR or the [C ii]/PAH ratio and, e.g., q_PAH (the fractional mass in PAH grains, Draine & Li 2007), or PAH band ratios indicative of ionization such as 7.7 μm/11.3 μm (as found by Croxall et al. 2012).
• 7.  
  The collisional coupling of gas and dust grains at high density, which can "short-circuit" the heating/cooling balance via continuum cooling at high gas densities (Spaans & Silk 2000).
• 8.  
  AGN impacts on the ionization state of gas (see Section 5, and Langer & Pineda 2015), which would lead to a correlation between deficits and central X-ray luminosity.
• 9.  
  An infrared excess relative to the rate radiation couples to the gas, for example in energetically dominant AGN/quasars, at extreme dust column densities, or in the regions of high starlight intensities that are dominated by softer spectra from old stellar populations in central galaxy bulges (e.g., Groves et al. 2012; Draine et al. 2014).
• 10.  
  Other cooling channels such as rotational CO emission dominating the gas-phase cooling power, for example at low ratios of UV field to gas density, where the atomic gas layer becomes thin and the [C ii] line intensity diminishes (see Wolfire et al. 1989; Kaufman et al. 1999). CO does not, however, appear to provide substantial global cooling in ULIRGs (Rosenberg et al. 2015).

Whatever combination of the above physical effects drives the deficit, here we have demonstrated that the same varying deficit of cooling power operates within galaxies on scales as small as 200 pc. Remarkably, a single physical property controls the bulk of this variation over six orders of magnitude: the star formation rate density, with increasing ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ driving down the fractional [C ii] cooling power.

Due to its high intrinsic luminosity, and accessibility at high redshift (e.g., Carilli & Walter 2013), there is considerable interest in employing [C ii] emission as a direct tracer of star formation. The strong deficit dependence on ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ has ramifications on the use of [C ii] as a star formation indicator. Resolved studies that explore ${{\rm{\Sigma }}}_{{\rm{sfr}}}$–I_{[C ii]} surface brightness relationships effectively "calibrate out" this dependence and as a result can yield relatively unbiased indicators (e.g., Herrera-Camus et al. 2015, who found that the [C ii]–SFR relation is better behaved in surface density than luminosity space). Applying these resolved [C ii]–SFR calibrations to global luminosities is more uncertain, unless the sample exhibits a modest range of global ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ similar to the galaxies used for the calibration. This fact is easily understood when considering the extreme range of [C ii]/FIR found at fixed FIR luminosity (≳250×, Brisbin et al. 2015). The strong trend seen in Figure 2 also explains the residual dependence of global [C ii]/SFR on surface brightness and dust color temperatures (e.g., De Looze et al. 2014; Herrera-Camus et al. 2015). Even the definition of ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ itself requires some care, as different observational size scales for extended star-forming disks can yield appreciable differences. The results here apply on size scales above 200 pc and within the main [C ii]-emitting disk of galaxies.

At a given surface brightness, deficit variations are found that are not random but are correlated with the gas-phase oxygen abundance, with decreasing metallicity associated with increasing [C ii]/TIR. These variations are consistent with the large measured fractional luminosities of [C ii] in some very low-metallicity objects (e.g., Hunter et al. 2001; Cormier et al. 2010; Cigan et al. 2016). In contrast, Cormier et al. (2015) found little dependence of [C ii]/TIR on 12+log(O/H) in global measurements of a large sample of low-metallicity dwarf galaxies. Since the ${{\rm{\Sigma }}}_{{\rm{sfr}}}$ dependence is stronger than the impact of metallicity, unresolved global measures sampling widely varying local radiation field strengths might wash out these residual trends. Vallini et al. (2015) modeled [C ii] emission at high redshift, and found a second-order dependence of the predicted [C ii]–SFR relation with metallicity Z, which has the opposite sense of our observations (at a fixed SFR, L_{[C ii]} increases in their models as metallicity increases), but this offset may arise principally from the adopted correlation between baryonic overdensity and Z.

The apparent cooling line excess with decreasing metallicity at a given surface brightness (Figure 2) is difficult to reconcile with the plummeting relative abundance of small dust grains at low metallicity. Since PAHs and related small grains provide the bulk of photoelectric heating, finding a cooling line excess relative to large grain heating near the metallicity 12+log(O/H) ≃ 8.2 where PAHs appear to begin disappearing (Engelbracht et al. 2005; Wu et al. 2006; Draine et al. 2007; Smith et al. 2007; Engelbracht et al. 2008) indicates that some other process must contribute. One possibility is that the contribution of [C ii] from ionized gas simultaneously increases as metallicity drops. This has been investigated in detail by K. Croxall et al. (2017, in preparation) using [N ii] 205 μm emission (which shares [C ii]'(s) critical density in ionized gas and thus tracks it closely). They find that only a modest fraction (<∼40%) of [C ii] originates in ionized gas and that this fraction in fact decreases significantly as metallicity drops, essentially compounding the problem. It may also be the case that other small grains with abundances that are insensitive to declining metallicity and for which PAH grains do not act as observational surrogates can still contribute substantial heating (e.g., Galliano et al. 2005; Israel & Maloney 2011). Cosmic rays may also play a role, if their heating contribution (relative to dust photoelectric heating) increases at low metallicity. This could occur, for example, due to (1) an increasing fraction of energy going into cosmic rays as a result of reduced radiative cooling in supernova blastwaves, and (2) reduced absorption of FUV photons (and consequent photoelectric heating) in low-metallicity galaxies with lower dust abundances. A final explanation is that [C ii] emission at low metallicity is dominated by larger "[C ii]-regions" surrounding hydrogen ionization zones, made possible by the increased penetration depth of photons with hν ≲ 13.6 eV in the reduced dust environments, which can maintain a larger fraction of the ISM's carbon as C⁺, and/or increase the grain heating efficiency due to a more dilute radiation field heating the gas (e.g., Poglitsch et al. 1995; Madden et al. 1997; Israel & Maloney 2011). A similar argument based purely on scaling radiation density and resulting cloud structure is given by Narayanan & Krumholz (2016). In strong recombination regions, the abundance of C⁺ could also be enhanced by the reduced rate of carbon recombination on grain surfaces expected as the small grain abundance drops (Kaufman et al. 1999). Further discriminating among the physical mechanisms underlying the deficit's response to metallicity may require partitioning [C ii] emission into its ionized, neutral, dense, and diffuse emitting environments.

Some studies have suggested that the [C ii] deficit arises solely from the impact of AGNs on the cooling balance in the ISM (e.g., Sargsyan et al. 2012, but see Ibar et al. 2015 for an opposing view). The continuity of the presented deficit trend from hyperluminous high-redshift galaxies down to sub-kiloparsec scales within normal star-forming galaxies demonstrates this not to be the case generally. While there is a strong central deficit impact of some AGNs in the Kingfish sample, as discussed in Section 5, it is likely indirect, related to the high density of low-energy starlight photons, which accompany lower luminosity AGNs. As modeled by Draine et al. (2014) for M31's bulge, the softening central starlight spectrum couples less efficiently to the PAHs than to the bulk grain populations. This presumably would lead to reduced photoelectric heating efficiency of the gas, as less photo-ionization energy per photon is available when the average photon energy drops. A detailed model of this effect with varying starlight spectra could provide a valuable test of this scenario.

At low luminosity, a related issue could be the recently revealed ambiguous power source of many LINER galaxies (with resolved studies showing extended LINER-like emission inconsistent with black hole accretion, Singh et al. 2013; Belfiore et al. 2016). The optical classification scheme of Moustakas et al. (2010) we utilize did not separate Seyfert and LINER types. Whether other expected direct impacts of AGNs, for example on the kiloparsec-scale carbon ionization state or the widescale small grain population, can be disentangled from the effects of changing starlight conditions in the centers of AGN hosts is unknown, but resolved [C ii] investigations in AGNs with X-ray luminosities above 10⁴³ erg s⁻¹ would be illuminating.

At the highest luminosities, the dust continuum of [C ii]-emitting galaxies can be dominated by powerful AGNs, rendering Equation (1) inapplicable. An example of this is a galaxy at z = 4.6 with the highest currently measured infrared luminosity (L_TIR ∼ 2.2 × 10¹⁴ L_☉, Díaz-Santos et al. 2016). Like other high-luminosity systems that host extreme AGNs, it has a resolved continuum size much smaller than its [C ii] disk, and a remarkably deep [C ii] deficit ([C ii]/TIR ∼ 0.003%). For such a deep deficit, Equation (1) would imply an unphysical star formation rate density in excess of ${10}^{7}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$. Even using a star-forming SED model to compute a reduced L_TIR by scaling from the much lower FIR (42–122 μm) luminosity, which is often argued to suffer less contamination from AGN-heated dust than the bolometric infrared, leads to ${{\rm{\Sigma }}}_{{\rm{sfr}}}\gtrsim {10}^{5}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$. Very deep deficits and the resulting unrealistically high ${{\rm{\Sigma }}}_{{\rm{sfr}}}$s implied by Equation (1) likely indicate the presence of dominant AGN heating.

We have explored [C ii] emission in approximately 15,000 regions within 54 Kingfish galaxies, supplementing these measurements with well-resolved [C ii] observations of nearby luminous infrared galaxies and interferometrically resolved high-redshift [C ii] emitters. The uncovered physical trends in the deficit connect environments ranging from low surface brightness regions in moderate star-forming systems to extreme starbursts at the theoretical limits of star formation surface density. We find the following.

• 1.  
  The [C ii] deficit, in which [C ii]/TIR exhibits large variations contrary to the simple expectations of heating/cooling balance, is present within galaxies down to scales of 200 pc. This strongly implies that whatever underlying process(es) drive the deficit, they are local physical processes of interstellar gas not related to global galaxy properties like bulk luminosity or presence on the galaxy main-sequence.
• 2.  
  Within normal star-forming galaxies, [C ii]/TIR declines markedly from over 1% down to ∼0.1% as surface brightness $\nu {I}_{\nu }(24\,\mu {\rm{m}})$ increases, with an average value $\langle [{\rm{C}}\,{\rm{ii}}]/\mathrm{TIR}\rangle =0.48\pm 0.21 \% .$
• 3.  
  At fixed surface brightness, the gas-phase metallicity within galaxies, 12+log(O/H), correlates with the residual variation in [C ii]/TIR, with reduced metal abundance associated with higher [C ii]/TIR values (i.e., reducing the deficit), and increased metallicity associated with deeper deficits. Given the apparent declining abundance with falling metallicity of the small grains thought to dominate photoelectric heating of the gas, this is a surprising result.
• 4.  
  The variation of the deficit with 24 μm and 70 μm surface brightness can be most directly interpreted as a trend with ${{\rm{\Sigma }}}_{{\rm{sfr}}}$. When combining resolved [C ii] measurements of luminous infrared and high-redshift galaxies from z = 1.8–6.4, the trend found in nearby galaxies smoothly extends over more than six orders of magnitude down to the maximal starburst at ${{\rm{\Sigma }}}_{{\rm{sfr}}}\sim 1000\ {M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$ and [C ii]/TIR ∼ 0.03%.
• 5.  
  By fitting and inverting the relation between [C ii]/TIR and ${{\rm{\Sigma }}}_{{\rm{sfr}}}$, the approximate star formation surface density can be estimated using resolved or unresolved measurements of the fractional [C ii] luminosity, with the deepest [C ii] deficits corresponding to the highest densities of star formation.
• 6.  
  Unexpectedly large deficits in the resolved Kingfish sample occur in the centers of several galaxies hosting low-luminosity AGNs, with a very steep radial suppression of [C ii]/TIR inwards from de-projected distances of 1.5 kpc to the center of up to a factor of five.
• 7.  
  While all sample galaxies, including AGN hosts, exhibit deeper deficits in their centers, the typical suppression of [C ii]/TIR is modest: ∼30% below the 1.5 kpc inner disk average. Those AGN host galaxies with substantially greater central depressions do not host the most luminous AGNs (in terms of 0.3–8 keV X-ray luminosity), but instead harbor central bulges with the highest surface brightness of starlight at 3.6 μm. This can be explained if the high intensity but softer starlight from the old stellar populations heating the dust in the centers of many AGN hosts drives continuum emission with reduced radiative coupling to the gas.
• 8.  
  Galaxies with deficits considerably deeper than [C ii]/TIR ∼ 0.01% imply unphysical star formation rate densities well above several thousand ${M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$—a potential indicator of dominant AGN contribution to the infrared luminosity.

This work is based in part on observations made with Herschel, a European Space Agency Cornerstone Mission with significant participation by NASA. Support for this work was provided by NASA through an award issued by JPL/Caltech. We thank Steve Hailey-Dunsheath, T. Rawle, and Tanio Diaz-Santos for advanced access to their compiled [C ii] data sets. We also thank them, as well as Gordon Stacey, Carl Ferkinhoff, M. Kapala, and R. Decarli, for helpful discussions which improved this work. J.D.S. gratefully acknowledges visiting support from the Alexander von Humboldt Foundation and the Max Planck Institute für Astronomie as well as support from the Research Corporation for Science Advancement through its Cottrell Scholars program.