Using [C ii] 158 μm Emission from Isolated ISM Phases as a Star Formation Rate Indicator

Table 1 provides a brief description of the properties of the galaxies included in this study. This work uses the subset of the galaxies in the KINGFISH sample with Photoconductor Array Camera and Spectrometer (PACS) $[{\rm{N}}\,{\rm{II}}]$ 205 μm and $[{\rm{C}}\,{\rm{II}}]$ 158 μm spectral maps (Kennicutt et al. 2011). The overall KINGFISH survey studied 61 galaxies in the local universe (D < 30 Mpc) spanning a wide range of morphological types, luminosities, metallicities, and levels of star formation activity. Of these 61 galaxies, 54 were observed at the $[{\rm{C}}\,{\rm{II}}]$ 158 μm line. Within this smaller sample, Herschel PACS far-infrared spectral cubes targeting the $[{\rm{N}}\,{\rm{II}}]$ 205 μm line were acquired for galaxies with the highest far-infrared surface brightnesses. This subsample contains 31 regions in 28 galaxies, with 24 centered on the brightest, central nuclear region of the galaxy and seven centered on extranuclear star-forming regions in the disk of the galaxy. All of the targeted galaxies in this sample are normal star-forming galaxies in terms of infrared luminosity, with no LIRGs included, i.e., all KINGFISH galaxies have ${\mathrm{log}}_{10}({L}_{\mathrm{IR}}/{L}_{\odot })\leqslant 11$. The galaxies included in this work cover a range of nebular metallicities (12 + log(O/H)) spanning ∼8.1–8.7 as measured by Moustakas et al. (2010) using the calibration of Pilyugin & Thuan (2005), and a distance range of 3–30 Mpc (Kennicutt et al. 2011). The physical scales of the $[{\rm{C}}\,{\rm{II}}]$ 158 μm ∼11'' and the $[{\rm{N}}\,{\rm{II}}]$ 205 μm ∼145 point-spread functions (PSFs) lie in the ranges 180–1700 pc and 200–2100 pc, respectively, across the sample. The proximity and properties of the galaxies included in this sample allow us to determine the nature of the $[{\rm{C}}\,{\rm{II}}]$ emission in galaxies with a wide array of spectral coverage and compare our results to the U/LIRGS studied in Díaz-Santos et al. (2017). Further, by only targeting normal, star-forming galaxies we are able to explore the behavior of the $[{\rm{C}}\,{\rm{II}}]$ deficit without the complicating effects of AGNs or other extreme conditions. Establishing an understanding of the processes occurring in these well-studied galaxies will lay the groundwork for understanding measurements of more extreme cases.

In addition to the KINGFISH measurements, 20 of the 28 galaxies in this sample were included in the BtP survey. The BtP survey used the Spectral and Photometric Imaging Receiver (SPIRE) on Herschel to obtain $[{\rm{N}}\,{\rm{II}}]$ 205 μm maps with a larger area, extending this study to the more quiescent areas surrounding the bright regions of star formation included in the KINGFISH PACS 205 μm survey. The larger BtP maps introduce 127 additional 20'' regions within these 20 galaxies. For more information about the choice of 20'' regions, see Section 3. As the BtP regions cover a wider range of conditions, we split them into those centered nearest the galaxy nuclei and those further removed from the nuclei. We distinguish these "Inner" and "Outer" regions as those within one-quarter of R₂₅ and those outside of one-quarter of R₂₅. The numbers of each type of detection are listed in Table 1. Although the BtP regions do cover a wider range of properties, the centers of only ∼14% fall outside of one-quarter of R₂₅.

This work uses the KINGFISH program's Herschel/PACS far-infrared mapped spectral observations of the 158 and 205 μm lines. All PACS spectral maps were obtained in the unchopped mapping mode and reduced using the Herschel Interactive Processing Environment (HIPE) version 11.2637 (Smith et al. 2017). Standard data reductions were applied to all images and are summarized in Croxall et al. (2012). The resulting line maps cover a 47'' by 47'' square field of view with 26 pixels in both 158 and 205 μm lines and have calibration uncertainties of 20% and 30%, respectively (Beirão et al. 2010; Croxall et al. 2012). The detection of the $[{\rm{N}}\,{\rm{II}}]$ 205 μm line in nearby galaxies at a spectral resolution of 150 km s⁻¹ is unique to the PACS instrument, making this data set invaluable for understanding the far-infrared fine-structure lines in the nearby universe (Beirão et al. 2010). For two of the galaxies in this sample, NGC 5457 and NGC 6946, multiple star-forming regions were targeted; all others were observed only at the central nuclear region or at a single extranuclear star-forming region. As an example of the data from a typical nuclear pointing in this sample, the $[{\rm{C}}\,{\rm{II}}]$ 158 μm and $[{\rm{N}}\,{\rm{II}}]$ 205 μm maps of IC 342 can be seen in Figure 1. The flux measurements from the PACS line maps are shown in Table 2.

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Table 2.  PACS Line Measurements

Galaxy	Region	L($[{\rm{C}}\,{\rm{II}}]$ 158 μm)	L($[{\rm{N}}\,{\rm{II}}]$ 205 μm)
Name	Type	(erg s−1)	(erg s−1)
NGC 0337	Nuclear	774 (±77) × 1038	<6.44 × 1038
NGC 0628	Extranuclear	491 (±49) × 1037	52.3 (±16) × 1037
NGC 1097	Nuclear	277 (±28) × 1039	165 (±50) × 1038
NGC 1266	Nuclear	116 (±12) × 1039	108 (±33) × 1038
NGC 1377	Nuclear	142 (±15) × 1038	97.0 (±29) × 1038
IC 342	Nuclear	196 (±20) × 1038	178 (±53) × 1037
NGC 1482	Nuclear	759 (±76) × 1039	275 (±83) × 1038
NGC 2146	Nuclear	178 (±18) × 1040	48.0 (±14) × 1039
NGC 2798	Nuclear	406 (±41) × 1039	236 (±71) × 1038
NGC 2976	Extranuclear	425 (±43) × 1037	106 (±32) × 1036
NGC 3049	Nuclear	604 (±60) × 1038	24.4 (±7.7) × 1038
NGC 3077	Nuclear	759 (±76) × 1038	159 (±48) × 1036
NGC 3351	Nuclear	443 (±44) × 1038	33.8 (±10) × 1038
NGC 3521	Nuclear	395 (±40) × 1038	274 (±83) × 1037
NGC 3627	Extranuclear	765 (±77) × 1038	52.5 (±16) × 1038
NGC 4254	Nuclear	103 (±10) × 1039	84.5 (±25) × 1038
NGC 4321	Nuclear	107 (±11) × 1039	125 (±38) × 1038
NGC 4536	Nuclear	300 (±30) × 1039	39.5 (±12) × 1038
NGC 4569	Nuclear	198 (±20) × 1038	<1.80 × 1038
NGC 4631	Nuclear	670 (±67) × 1038	223 (±67) × 1037
NGC 4736	Extranuclear	104 (±10) × 1038	27.7 (±8.4) × 1037
NGC 4826	Nuclear	179 (±18) × 1038	151 (±45) × 1037
NGC 5055	Nuclear	227 (±23) × 1038	227 (±68) × 1037
NGC 5457	Extranuclear	194 (±19) × 1038	5.29 (±2.9) × 1037
NGC 5457	Nuclear	530 (±53) × 1037	101 (±30) × 1037
NGC 5713	Nuclear	349 (±35) × 1039	151 (±45) × 1038
NGC 5866	Nuclear	224 (±22) × 1038	43.5 (±13) × 1038
NGC 6946	Nuclear	606 (±61) × 1038	302 (±91) × 1037
NGC 6946	Extranuclear	216 (±22) × 1038	109 (±33) × 1037
NGC 6946	Extranuclear	965 (±97) × 1037	157 (±47) × 1037
NGC 7331	Nuclear	760 (±76) × 1038	91.9 (±28) × 1038

Note. Luminosities are measured for regions of 20'' radius smoothed to a 20'' FWHM PSF (see Section 3 for more information). 10% and 30% calibration uncertainties are included for $[{\rm{C}}\,{\rm{II}}]$ 158 μm and $[{\rm{N}}\,{\rm{II}}]$ 205 μm luminosities respectively.

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In addition to the KINGFISH PACS $[{\rm{N}}\,{\rm{II}}]$ 205 μm observations, the SPIRE-FTS on Herschel mapped the $[{\rm{N}}\,{\rm{II}}]$ 205 μm line in 20 of the KINGFISH galaxies with PACS $[{\rm{N}}\,{\rm{II}}]$ 205 μm detections as part of the Beyond the Peak program. These observations were obtained using the SPIRE-FTS intermediate mapping mode, which is a four-point dither (Croxall et al. 2017). These maps were calibrated using the extended-flux HIPE v10 package. The spectral resolution for these data is ∼300 km s⁻¹ at the 205 μm wavelength. More information about the observations and data processing can be found in Pellegrini et al. (2013) and A. Crocker et al. (2019, in preparation). The SPIRE-FTS beam size at 205 μm is 166 (Makiwa et al. 2013), and the maps produced by this survey are significantly larger than the PACS $[{\rm{N}}\,{\rm{II}}]$ 205 μm maps from the KINGFISH survey (2' × 2' versus 1' × 1' in most cases). The inclusion of these larger spectral maps extends the range of ISM conditions covered to cooler quiescent material surrounding the star-forming regions included in the KINGFISH sample. Some of the important properties of these maps are listed in Table 3.

As part of the KINGFISH survey, PACS images at 70, 100, and 160 μm were obtained in scan mode for each galaxy in this sample. These data provide valuable information about the dust temperatures of our sample and were used to determine the far-infrared colors and total infrared luminosities for our regions. These measurements were then used to quantify the $[{\rm{C}}\,{\rm{II}}]$ deficit for our sample (see Section 4.1). Uniform surface brightness sensitivities of σ_sky ∼ 5, 5, and 2 MJy sr⁻¹ at 70 μm, 100 μm, and 160 μm respectively, were achieved for each region (Dale et al. 2012). The KINGFISH PACS images have a calibration uncertainty of ${\epsilon }_{\mathrm{cal},\nu }/{f}_{\nu }\sim 5 \%$. For more information about these far-infrared images, see Dale et al. (2012). A summary of the details of these images can be found in Table 4.

Most regions in the KINGFISH sample were also included in the Spitzer Infrared Nearby Galaxies Survey (SINGS) (Kennicutt et al. 2003; Dale et al. 2006; Smith et al. 2007b). This survey used the Spitzer Space Telescope to obtain infrared imaging and spectroscopy for 75 nearby galaxies. The Infrared Spectrograph (IRS) on board Spitzer obtained low-resolution (R ∼ 50–100) spectral maps in both the short–low (SL) (5–14 μm) and long–low (LL) (14–38 μm) modules for most of the nuclear regions in the KINGFISH sample (Smith et al. 2007b). This wavelength range includes several prominent PAH emission features. The availability of these data allows comparisons between the PAH emission and the $[{\rm{C}}\,{\rm{II}}]$ emission for the regions in our sample (see Section 4.2). The extranuclear regions targeted by the KINGFISH survey were observed by Spitzer/IRS in only the SL module. In most cases, the larger BtP maps were covered with Spitzer/IRS by only the LL module. All observations were reduced with the CUBISM program to create spatially resolved spectral cubes and processed using the IRS pipeline version S14, producing an absolute flux calibration uncertainty of ∼25% (Dale et al. 2006). For more information on these observations and data processing, see Smith et al. (2007a).

As part of the SINGS survey, 3.6, 8.0, and 24 μm images were also obtained for each galaxy using the Infrared Array Camera (IRAC) and the Multi-band Imaging Photometer (MIPS) (Rieke et al. 2004). These images were then processed using the MIPS Instrument Team Data Analysis Tool (Gordon et al. 2005). Calibration errors for these images are 10% at all three wavelengths (Dale et al. 2005). Four of the galaxies in this work, IC 342, NGC 2146, NGC 3077, and NGC 5457, were not included in the SINGS sample. For these galaxies 3.6, 8.0, and 24 μm images were obtained from other archival Spitzer surveys, namely the Local Volume Legacy (LVL, Dale et al. 2009) and MIPS GTO (Pahre et al. 2004). The 3.6 and 8.0 μm images provide alternative measurements of PAH feature strength for regions without both SL and LL coverage. For the purposes of this study, the 24 μm data were incorporated into the determination of the total infrared luminosities as well as in our measurements of SFR. More information about these images can be found in Table 4.

In order to determine SFR, FUV maps were obtained from GALEX. 26 of the 31 regions in this sample were imaged as part of the GALEX Nearby Galaxy Survey (NGS) (Gil de Paz et al. 2005), and those that were not covered by NGS were imaged as part of the GALEX All-Sky Imaging Survey (AIS) or by other programs, with the exception of NGC 1377 which has no GALEX imaging. Priority was given to long-exposure data when available. Exposure times for this sample range from 110 to 21,177 s, with a median exposure time of ∼1700 s. The GALEX images have a diffraction-limited FWHM of ∼6'' (Gil de Paz et al. 2005). For more details on the GALEX images, see Table 4.

As multiple studies have found the $[{\rm{C}}\,{\rm{II}}]$ deficit to be a local effect (Smith et al. 2017; Gullberg et al. 2018), we use the extended SPIRE coverage to determine whether there are significant differences between the behavior of the phase-separated $[{\rm{C}}\,{\rm{II}}]$ deficit in the inner nuclear regions of these galaxies and in the more extended disks. In order to test these differences, we divide the BtP regions into "Inner" and "Outer," with Inner regions centered within 0.25R₂₅ and Outer regions as any falling outside this limit. As the $[{\rm{N}}\,{\rm{II}}]$ 205 μm emission is faint, only ∼14% of the BtP regions with $[{\rm{N}}\,{\rm{II}}]$ 205 μm detections fall into our definition of Outer regions. To avoid biasing our data toward the conditions within galaxies with more regions, an average of the SPIRE detections for the Inner and Outer regions of each galaxy included in the BtP survey was performed to produce the BtP Inner and Outer measurements used throughout this study.

To determine the fraction of the $[{\rm{C}}\,{\rm{II}}]$ emission originating in the neutral phase of the ISM, the following relation from Croxall et al. (2012) and Oberst et al. (2006) was used:

$\begin{eqnarray}&&{f}_{[{\rm{C}}{\rm{II}}],\mathrm{Neutral}}=\displaystyle \frac{[{\rm{C}}\,{\rm{II}}]158-({R}_{\mathrm{Ionized}}\times [{\rm{N}}\,{\rm{II}}]205)}{[{\rm{C}}\,{\rm{II}}]158}.\end{eqnarray} \tag{ 2 }$

In this equation,  R_Ionized is the expected ratio of $[{\rm{C}}\,{\rm{II}}]$ 158 μm to $[{\rm{N}}\,{\rm{II}}]$ 205 μm emission in the ionized gas where both C⁺ and N⁺ are present, as represented by the solid cyan line in Figure 2. This ratio is derived using the collision rates of Tayal (2008) for $[{\rm{C}}\,{\rm{II}}]$ and Tayal (2011) for $[{\rm{N}}\,{\rm{II}}]$ and assuming Galactic gas-phase abundances of carbon and nitrogen (1.6 × 10⁻⁴ per hydrogen atom and 7.5 × 10⁻⁵ per hydrogen atom respectively) (see Croxall et al. 2017  for further information). It has been found that there is a slight dependence of R_Ionized on gas-phase abundances, but this dependence only results in shifts of ≤5% on measurements of ${f}_{[{\rm{C}}{\rm{II}}],\mathrm{Neutral}}$ across the range of metallicities included in this sample, and therefore does not affect our results (Croxall et al. 2017). R_Ionized is determined for each region individually based on measurements of electron number density made using ratios of the [S iii] 18.7 and 33.4 μm lines (Dale et al. 2006) or the $[{\rm{N}}\,{\rm{II}}]$ 122 and 205 μm lines (Herrera-Camus et al. 2016). Over the range of conditions covered by our sample, these two ratios both provide reliable measurements of n_e (Rubin et al. 1995). Any slight differences in calculations of n_e between these two methods will not affect our results,  because R_Ionized is nearly independent of n_e. Studies of local galaxies with high specific SFR (sSFR) have found that there is a correlation between n_e and sSFR that could potentially affect the ionization parameter in the most dense regions in our study (Kewley et al. 2015; Bian et al. 2016; Holden et al. 2016), but we have not modified our calculation of R_Ionized to account for this.

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The values of ${f}_{[{\rm{C}}{\rm{II}}],\mathrm{Neutral}}$ for the KINGFISH and BtP samples are shown in Figure 3, plotted against $\nu {f}_{\nu }(70\,\mu {\rm{m}})$/$\nu {f}_{\nu }(160\,\mu {\rm{m}})$, a proxy for dust temperatures. Also shown in Figure 3 are the values determined using smaller subregions measured using Herschel SPIRE data in Croxall et al. (2017). The ${f}_{[{\rm{C}}{\rm{II}}],\mathrm{Neutral}}$ values found for this sample follow a similar trend to those in the study of Croxall et al. (2017), with an average neutral fraction of ∼67% and a decreasing dynamic range in ${f}_{[{\rm{C}}{\rm{II}}],\mathrm{Neutral}}$ for the more actively star-forming galaxies that have increased $\nu {f}_{\nu }(70\,\mu {\rm{m}})$/$\nu {f}_{\nu }(160\,\mu {\rm{m}})$ values. In this and all subsequent figures, the KINGFISH regions, shown as magenta squares, cover only higher $\nu {f}_{\nu }(70\,\mu {\rm{m}})$/$\nu {f}_{\nu }(160\,\mu {\rm{m}})$ ratios because all KINGFISH regions are centered on warmer star-forming regions, while the BtP data with a wider field of view extend to the quiescent environments surrounding these star-forming regions.

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The PAHFIT program was used to determine the strength of the PAH emission features from the fluxes extracted from the IRAC spectral maps (Smith et al. 2007b; Gallimore et al. 2010). This program separates emission from PAH features, far-infrared emission lines, warm dust, and stars. For the KINGFISH nuclear regions with IRAC spectral maps in both SL and LL large enough to cover the $[{\rm{C}}\,{\rm{II}}]$ 158 μm and $[{\rm{N}}\,{\rm{II}}]$ 205 μm emission, the total PAH power was determined by summing the emission from all the PAH emission features measured by PAHFIT (Croxall et al. 2012). For regions covered only in SL (KINGFISH extranuclear regions) or only in LL (BtP regions), the total PAH power was determined using a combination of the IRAC spectral maps and the 8.0 and 3.6 μm IRAC imaging. The bandpass used for the 8.0 μm images contains one of the strongest PAH features (Croxall et al. 2012). Stellar contributions can be removed from the 8.0 μm flux through the use of the 3.6 μm IRAC images. The total PAH power can thus be calculated photometrically using the result determined from the LVL survey (Marble et al. 2010):

$\begin{eqnarray}&&{{\rm{PAH}}}_{{\rm{Phot}}}^{\ast }=\nu {S}_{\nu }(8.0)-0.24\times \nu {S}_{\nu }(3.6).\end{eqnarray} \tag{ 3 }$

With the added information from either the SL or LL bands, this photometrically determined PAH power can be improved using the results from Croxall et al. (2012):

$\begin{eqnarray}&&{\mathrm{PAH}}_{\mathrm{SL}}^{* }=0.497\times [{\mathrm{PAH}}_{\mathrm{Phot}}^{* }+3.59\times \nu {S}_{\nu }(11.3)]\end{eqnarray} \tag{ 4 }$

$\begin{eqnarray}&&{\mathrm{PAH}}_{\mathrm{LL}}^{* }=0.7472\times [{\mathrm{PAH}}_{\mathrm{Phot}}^{* }+3.25\times \nu {S}_{\nu }(17)].\end{eqnarray} \tag{ 5 }$

With the Spitzer IRAC imaging and spectral maps, estimates of the total PAH emission for most regions in our sample were determined, allowing for comparisons of the PAH emission and $[{\rm{C}}\,{\rm{II}}]$ emission in both actively star-forming regions and the more quiescent environments surrounding these regions. For our sample, the total PAH emission strength of the KINGFISH extranuclear regions is determined using Equation (4) and that of the BtP regions is determined with Equation (5). These three methods produce similar measurements of total PAH emission power with 1σ scatters of 0.15 dex and 0.09 dex compared to the summation of all PAH features for Equations (5) and (4) respectively (Croxall et al. 2012).

The $[{\rm{N}}\,{\rm{II}}]$/TIR measurements for our sample are displayed as a function of the far-infrared color as measured by the $\nu {f}_{\nu }(70\,\mu {\rm{m}})/\nu {f}_{\nu }(160\,\mu {\rm{m}})$ ratio in Figure 4. This ratio is used because it is a proxy for dust temperature, and increased dust temperatures often indicate increased star formation activity. TIR luminosity was determined for each individual region using Equation (4) from Dale et al. (2014) (Equation (6) in this work) with the Spitzer 24 μm luminosities and Herschel PACS 70 and 160 μm luminosities:

$\begin{eqnarray}\begin{array}{rcl}{L}_{\mathrm{TIR}} & = & 1.548\nu {L}_{\nu }(24\,\mu {\rm{m}})+0.767\nu {L}_{\nu }(70\,\mu {\rm{m}})\\ & & +\,1.285\nu {L}_{\nu }(160\,\mu {\rm{m}}).\end{array}\end{eqnarray} \tag{ 6 }$

Similar to the findings of Díaz-Santos et al. (2017), we find that the $[{\rm{N}}\,{\rm{II}}]$ 205 μm line ratio with TIR shows a clear decreasing trend in warmer regions, and this trend holds irrespective of whether the inner or outer portions of the galaxies are sampled. Using the form

$\begin{eqnarray}&&{\mathrm{log}}_{10}\displaystyle \frac{[{\rm{N}}\,{\rm{II}}]}{\mathrm{TIR}}=m\nu {f}_{\nu }(70\,\mu {\rm{m}})/\nu {f}_{\nu }(160\,\mu {\rm{m}})+b\end{eqnarray} \tag{ 7 }$

we perform a linear regression for all the individual regions in our sample, the averaged inner BtP measurements and the KINGFISH nuclear regions, and the averaged outer BtP measurements and the KINGFISH extranuclear regions. For these and all following fits, the KINGFISH extranuclear region from NGC 5457 has been ignored because it is a faint source with significant noise contamination at the edges of the image, making it an outlier in each plot (see the cyan diamond with ${\mathrm{log}}_{10}\tfrac{[{\rm{N}}\,{\rm{II}}]}{\mathrm{TIR}}$ just below −3.0 in Figure 4). The slopes and intercepts for each fit are listed in Table 5 and displayed in Figure 4. Each fit has an equivalent slope within error, showing that the location of the region within the galaxy does not significantly affect our results.

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In Figure 5, the $[{\rm{C}}\,{\rm{II}}]$/TIR measurements are displayed as a function of $\nu {f}_{\nu }(70\,\mu {\rm{m}})/\nu {f}_{\nu }(160\,\mu {\rm{m}})$. In this figure, the left panel is the combined ionized and neutral $[{\rm{C}}\,{\rm{II}}]$ emission ($[{\rm{C}}\,{\rm{II}}]$ ${}_{{\rm{I}}+{\rm{N}}}$), the middle panel is the $[{\rm{C}}\,{\rm{II}}]$ emission from only the ionized ISM ($[{\rm{C}}\,{\rm{II}}]$_Ionized), and the right panel is the $[{\rm{C}}\,{\rm{II}}]$ emission from only the neutral ISM ($[{\rm{C}}\,{\rm{II}}]$_Neutral). The trend we notice in our $[{\rm{C}}\,{\rm{II}}]$_Ionized measurements is a scaled version of the $[{\rm{N}}\,{\rm{II}}]$ 205 μm measurements by nature of our method for determining $[{\rm{C}}\,{\rm{II}}]$_Ionized. Each $[{\rm{C}}\,{\rm{II}}]$/TIR measure is fit by a linear regression of the form

$\begin{eqnarray}&&{\mathrm{log}}_{10}\displaystyle \frac{[{\rm{C}}\,{\rm{II}}]}{\mathrm{TIR}}=m\nu {f}_{\nu }(70\,\mu {\rm{m}})/\nu {f}_{\nu }(160\,\mu {\rm{m}})+b.\end{eqnarray} \tag{ 8 }$

These fits are displayed in Figure 5 and the properties of each fit are listed in Table 6. For our sample, we find an average of ${\mathrm{log}}_{10}\tfrac{{[{\rm{C}}{\rm{II}}]}_{{\rm{I}}+{\rm{N}}}}{\mathrm{TIR}}=-2.53$, ${\mathrm{log}}_{10}\tfrac{{[{\rm{C}}{\rm{II}}]}_{\mathrm{Ionized}}}{\mathrm{TIR}}=-3.07$, and ${\mathrm{log}}_{10}\tfrac{{[{\rm{C}}{\rm{II}}]}_{\mathrm{Neutral}}}{\mathrm{TIR}}=-2.73$. These results match well with other studies, which find the $[{\rm{C}}\,{\rm{II}}]$ line emission to account for approximately 1% of the total infrared emission (Smith et al. 2017). As expected from previous work (Croxall et al. 2012; Smith et al. 2017), the ratio of combined ionized and neutral $[{\rm{C}}\,{\rm{II}}]$ luminosity to TIR luminosity shows a decline with warmer far-infrared color. As there are no extreme cases in this study, the decreasing trend in the combined ionized and neutral $[{\rm{C}}\,{\rm{II}}]$/TIR ratio is slight, as shown by the slope of −0.127 for our line of best fit. While the $[{\rm{C}}\,{\rm{II}}]$ deficit is pronounced for the $[{\rm{C}}\,{\rm{II}}]$ emission from the ionized ISM, it disappears when only $[{\rm{C}}\,{\rm{II}}]$ emission from the neutral ISM is considered, as shown in the middle and right panels of Figure 5.

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Table 6.  Linear Fits from Figure 5

$[{\rm{C}}\,{\rm{II}}]$ Component	m	b	Rms Scatter
All Individual Regions
Ionized + Neutral	−0.127 [±0.03]	−2.33	0.175
Ionized	−0.360 [±0.03]	−2.49	0.173
Neutral	−0.024 [±0.03]	−2.69	0.270
All Inner Regions
Ionized + Neutral	−0.113 [±0.05]	−2.36	0.197
Ionized	−0.356 [±0.04]	−2.49	0.171
Neutral	−0.024 [±0.06]	−2.69	0.287
All Outer Regions
Ionized + Neutral	−0.050 [±0.08]	−2.30	0.083
Ionized	−0.304 [±0.26]	−2.50	0.250
Neutral	0.018 [±0.16]	−2.57	0.147

Note. Properties of the lines of best fit for our $[{\rm{C}}\,{\rm{II}}]$ deficit measurements. A line of best fit is displayed for each component of the $[{\rm{C}}\,{\rm{II}}]$ emission (combined ionized and neutral, ionized, and neutral). We further divide our fits by region type, with a fit for all individual regions, a fit for the averaged BtP inner regions and the KINGFISH nuclear regions, and a fit for the averaged BtP outer regions and the KINGFISH extranuclear regions.

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By separating our detections by ISM phase, we are able to narrow down the possible causes of the $[{\rm{C}}\,{\rm{II}}]$ deficit in our sample. Using this method, we determine that the decreasing trend in $[{\rm{C}}\,{\rm{II}}]$/TIR for warmer, more actively star-forming environments is greatly reduced when only the $[{\rm{C}}\,{\rm{II}}]$ emission from the neutral phases of the ISM is considered. On the other hand, the ratio of $[{\rm{C}}\,{\rm{II}}]$ emission from the ionized phases of the ISM to TIR luminosity shows a steep decrease as a function of far-infrared color. This can also be seen in the combined ionized and neutral $[{\rm{C}}\,{\rm{II}}]$/TIR measurements, where the slight decrease observed is driven by the regions from the BtP survey, which have lower ${f}_{[{\rm{C}}{\rm{II}}],{\rm{Neutral}}}$ values and therefore more emission from the ionized phases of the ISM (see Figure 5). This trend suggests that the cause of the $[{\rm{C}}\,{\rm{II}}]$ deficit occurs predominantly in the ionized phases of the ISM, a conclusion that is supported by the work done in the GOALS survey (Díaz-Santos et al. 2017).

This decreasing trend in $[{\rm{C}}\,{\rm{II}}]$ from the ionized phases of the ISM holds both for the star-forming regions targeted in the KINGFISH study, shown as magenta squares, and for the more extended coverage from the BtP survey, shown as green crosses, and is true for both regions within 0.25R₂₅ and those outside this boundary, as seen in Figure 5 and in the consistency of our measured slopes within error for each location. The decreasing trend measured for the ${[{\rm{C}}{\rm{II}}]}_{{\rm{I}}+{\rm{N}}}$/TIR luminosity ratio is much shallower than the steep trend found in the $[{\rm{N}}\,{\rm{II}}]$ 205 μm and $[{\rm{C}}\,{\rm{II}}]$_Ionized measurements, shown in Figure 4. This difference indicates that the lack of a measured deficit in the $[{\rm{C}}\,{\rm{II}}]$ emission from the neutral ISM is not solely caused by identical decreases in the $[{\rm{C}}\,{\rm{II}}]$ 158 μm and $[{\rm{N}}\,{\rm{II}}]$ 205 μm fluxes. If this were the case we would then expect similar slopes in our line fits to the $[{\rm{C}}\,{\rm{II}}]$_I+N and $[{\rm{N}}\,{\rm{II}}]$ deficits. Instead, there must be some differing physical processes in the neutral and ionized ISM driving the observed deficit.

With this new insight into the nature of the $[{\rm{C}}\,{\rm{II}}]$ deficit, we can narrow down the possible causes of the deficit in our sample. We suggest that the $[{\rm{C}}\,{\rm{II}}]$ deficit is caused by changes in the fraction of UV light absorbed by dust within ${\rm{H}}\,{\rm{II}}$ regions and PDRs. Compact/nuclear regions with warmer far-infrared colors have fractionally higher UV absorption by dust. $\nu {f}_{\nu }(70\,\mu {\rm{m}})/\nu {f}_{\nu }(160\,\mu {\rm{m}})$ traces TIR/FUV for centrally concentrated distributions of dust (Dale et al. 2007); therefore higher $\nu {f}_{\nu }(70\,\mu {\rm{m}})/\nu {f}_{\nu }(160\,\mu {\rm{m}})$ indicates regions where UV light is being proportionally more quenched by dust in PDRs and ${\rm{H}}\,{\rm{II}}$ regions. The resulting dearth of UV emission leaking into the diffuse ISM leads to a decrease in $[{\rm{C}}\,{\rm{II}}]$ emission from the ionized phases of the ISM. Previous studies have suggested that a majority of the $[{\rm{C}}\,{\rm{II}}]$_Ionized emission must originate in the diffuse ISM because there should be little $[{\rm{C}}\,{\rm{II}}]$ emission originating in ${\rm{H}}\,{\rm{II}}$ regions where the $[{\rm{C}}\,{\rm{II}}]$ line is often thermally quenched by high temperature and densities, and the availability of photons with energies above the 24.38 eV necessary to ionize C+ limits the emission of the 158 μm line (Díaz-Santos et al. 2017; Herrera-Camus et al. 2018a). Thus, the decrease in $[{\rm{C}}\,{\rm{II}}]$_Ionized as a fraction of the total infrared emission may be due to (1) a smaller fraction of the ultraviolet radiation from stars being above the Lyman limit (i.e., a deficiency in recent star formation) and (2) a smaller fraction of the H-ionizing radiation being absorbed in low-density (n_e < 30 cm⁻³) ${\rm{H}}\,{\rm{II}}$ where $[{\rm{C}}\,{\rm{II}}]$ emission is not collisionally quenched, i.e., the diffuse ionized ISM. Under these conditions, $[{\rm{C}}\,{\rm{II}}]$_Neutral should remain unaffected because it primarily originates in PDRs, where UV photons with energies above 11.3 eV have not been significantly quenched and are still available to ionize the carbon atoms present. This interpretation is consistent with the explanations of the $[{\rm{C}}\,{\rm{II}}]$ deficit described in Abel et al. (2009) and Graciá-Carpio et al. (2011) and suggests that the third and fourth explanations of the deficit listed in Section 1 (i.e., the thermalization of $[{\rm{C}}\,{\rm{II}}]$ in ${\rm{H}}\,{\rm{II}}$ regions and increased FUV absorption leading to $[{\rm{O}}\,{\rm{I}}]$ becoming a major coolant in the diffuse ionized ISM) are the most likely for our sample. It is also possible that the other mechanisms described in Section 1 have an effect on the deficit measurements for our sample, but do not explain the changes in trends between measurements of the ionized-phase $[{\rm{C}}\,{\rm{II}}]$ and the neutral-phase $[{\rm{C}}\,{\rm{II}}]$.

PAHs are the dominant source of the photoejected electrons that heat the neutral ISM, which is in turn cooled through channels such as the $[{\rm{C}}\,{\rm{II}}]$ 158 μm line (Bakes & Tielens 1994; Weingartner & Draine 2001). The mid-infrared PAH emission features are a result of vibrational and bending transitions that have been excited by the absorption of far-ultraviolet photons. We thus compare the strength of the PAH emission features to the $[{\rm{C}}\,{\rm{II}}]$ emission from both the neutral and ionized phases of the ISM to better understand the microphysics underlying the gas heating by photoelectric ejection of electrons from PAHs and gas cooling by $[{\rm{C}}\,{\rm{II}}]$ 158 μm emission. Previous works have found that while the $[{\rm{C}}\,{\rm{II}}]$ 158 μm/TIR decreases in warmer regions, the ratio of $[{\rm{C}}\,{\rm{II}}]$/PAH emission is more constant (Helou et al. 2001; Croxall et al. 2012). By measuring this ratio in the separated ISM phases we can test the relationship between gas heating by small grains and gas cooling by $[{\rm{C}}\,{\rm{II}}]$ emission.

The ratio of $[{\rm{C}}\,{\rm{II}}]$ luminosity to the strength of PAH features for the combined ionized and neutral $[{\rm{C}}\,{\rm{II}}]$ emission and the $[{\rm{C}}\,{\rm{II}}]$ emission from only neutral and ionized phases of the ISM can be found in Figure 6. KINGFISH nuclear regions where both SL and LL IRS cubes are available are shown as magenta squares, KINGFISH extranuclear regions where only SL IRS cubes are available are shown as blue squares, and BtP regions where only LL IRS cubes are available are shown as green crosses. It should be noted that the slight separation between the different region types is likely driven by the different methods for determining PAH emission strengths, and not by any differences between the regions themselves (see Section 3.3 for more information).

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The middle and right panels of Figure 6 show the $[{\rm{C}}\,{\rm{II}}]$ to PAH emission ratio when only the $[{\rm{C}}\,{\rm{II}}]$ emission from the ionized and neutral ISM are considered, respectively. Similar to results of the deficit observed when comparing the $[{\rm{C}}\,{\rm{II}}]$ and TIR luminosity, the ratio of $[{\rm{C}}\,{\rm{II}}]$ emission from only the ionized ISM to PAH emission feature strength shows a clear decrease as a function of far-infrared color, while the ratio of $[{\rm{C}}\,{\rm{II}}]$ emission from only the neutral ISM to PAH emission feature strength remains fairly constant across the range of far-infrared color included in this sample. This holds for both the warmer KINGFISH nuclear and inner BtP regions as well as the slightly cooler outer BtP regions and the KINGFISH extranuclear star-forming regions.

We find that the neutral $[{\rm{C}}\,{\rm{II}}]$ emission traces the total PAH emission well (Figure 6, right panel). We interpret this result as naturally arising from the ubiquity of PAHs in the neutral ISM (e.g., PDRs) and their comparative paucity in ionized portions of the ISM such as ${\rm{H}}\,{\rm{II}}$ regions (e.g., Helou et al. 2004). This result is consistent with $[{\rm{C}}\,{\rm{II}}]$ being a major cooling channel, and PAHs providing a majority of the heating through the photoelectric effect, for the neutral ISM. Therefore, in a scenario where ISM heating is balanced by ISM cooling, $[{\rm{C}}\,{\rm{II}}]$_Neutral emission should trace PAH emission. As a majority of the $[{\rm{C}}\,{\rm{II}}]$ emission in our sample originates in the neutral ISM (Figure 3), the ratio of combined ionized and neutral $[{\rm{C}}\,{\rm{II}}]$ emission to PAH feature strength (Figure 6, left panel) shows a similar trend with a slight decrease in the ratio of $[{\rm{C}}\,{\rm{II}}]$ to PAH emission in regions with the highest $\nu {f}_{\nu }(70\,\mu {\rm{m}})/\nu {f}_{\nu }(160\,\mu {\rm{m}})$ values. The ratio of $[{\rm{C}}\,{\rm{II}}]$ emission from the ionized phases of the ISM to PAH emission feature strength shows a decrease with respect to far-infrared color (Figure 6, middle panel). This observed decrease is due to an increasing $[{\rm{C}}\,{\rm{II}}]$_Neutral fraction in the warmer, more actively star-forming regions (i.e., higher $\nu {f}_{\nu }(70\,\mu {\rm{m}})/\nu {f}_{\nu }(160\,\mu {\rm{m}})$), making $[{\rm{C}}\,{\rm{II}}]$_Ionized lower in these regions. Such a decrease could be due to a fractionally higher absorption of UV photons within ${\rm{H}}\,{\rm{II}}$ regions for warmer, more actively star-forming environments as described in Section 4.1.

The top panel of Figure 7 shows star formation surface densities (Σ_SFR) as a function of $[{\rm{C}}\,{\rm{II}}]$ surface brightness (${{\rm{\Sigma }}}_{[{\rm{C}}{\rm{II}}]}$) for the $[{\rm{C}}\,{\rm{II}}]$ emission from both the neutral and ionized phases of the ISM (left), the $[{\rm{C}}\,{\rm{II}}]$ emission arising from only the ionized phase of the ISM (middle), and $[{\rm{C}}\,{\rm{II}}]$ emission arising from only the neutral phases of the ISM (right). The SFRs were determined using the hybrid FUV+24 μm local SFR indicator (Hao et al. 2011; Liu et al. 2011; Calzetti 2013):

$\begin{eqnarray}&&\mathrm{SFR}({M}_{\odot }\,{\mathrm{yr}}^{-1})=4.6\times {10}^{-44}\left[\displaystyle \frac{L(\mathrm{FUV})}{\mathrm{erg}\,{{\rm{s}}}^{-1}}+6.0\displaystyle \frac{L(24\,\mu {\rm{m}})}{\mathrm{erg}\,{{\rm{s}}}^{-1}}\right].\end{eqnarray} \tag{ 9 }$

After calculating the SFR using Equation (9), Σ_SFR was calculated by dividing by the deprojected area of the 20'' regions. The area was deprojected by dividing by $\cos i$ where i is the inclination of the galaxy disk. $\cos i$ was determined using

$\begin{eqnarray}&&{\cos }^{2}i=\displaystyle \frac{{(1-\epsilon )}^{2}-{q}^{2}}{1-{q}^{2}}\end{eqnarray} \tag{ 10 }$

from Dale et al. (1997), where is the disk's ellipticity as measured by de Vaucouleurs et al. (1991) and q is an adopted intrinsic axial ratio (i.e., the ratio of the minor axis to the major axis), with a value of q = 0.13 for galaxies of morphological class Sbc and later and q = 0.2 for galaxies earlier than Sbc (Dale et al. 1997; Murphy et al. 2018). The same deprojected area was used to determine ${{\rm{\Sigma }}}_{[{\rm{C}}{\rm{II}}]}$, which is the luminosity of the $[{\rm{C}}\,{\rm{II}}]$ emission from each region and each separated phase divided by the deprojected area.

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As found in previous studies (Stacey et al. 1991; Boselli et al. 2002; De Looze et al. 2011, 2014; Sargsyan et al. 2012; Herrera-Camus et al. 2015; Díaz-Santos et al. 2017; Smith et al. 2017), there are clear trends with increasing $[{\rm{C}}\,{\rm{II}}]$ surface brightnesses indicating increasing star formation. The lines in Figure 7 represent the lines of best fit to our data for all individual regions as well as the combination of the KINGFISH nuclear regions and the inner BtP regions and the combination of the KINGFISH extranuclear star-forming regions and the outer BtP regions, determined using the method described in Kelly (2007). This method uses a Bayesian linear regression that takes into account both detections and upper limits. Between 5000 and 10,000 Markov chain Monte Carlo (MCMC) steps through the parameter space are then tested to determine the best-fit relationship. The relationships found for each component of the $[{\rm{C}}\,{\rm{II}}]$ emission are described using Equation (11) and the values displayed in Table 7:

$\begin{eqnarray}&&\begin{array}{l}{\mathrm{log}}_{10}{{\rm{\Sigma }}}_{\mathrm{SFR}}({M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2})\\ \quad =\,m{\mathrm{log}}_{10}{{\rm{\Sigma }}}_{[{\rm{C}}{\rm{II}}]}(\mathrm{erg}\ {{\rm{s}}}^{-1}\,{\mathrm{kpc}}^{-2})+b.\end{array}\end{eqnarray} \tag{ 11 }$

Table 7.  Linear Fits from Figure 7

$[{\rm{C}}\,{\rm{II}}]$ Component	m	b	Rms Scatter
All Individual Regions
Ionized + Neutral	1.04 [±0.053]	−42.74	0.230
Ionized	0.91 [±0.085]	−37.11	0.332
Neutral	0.90 [±0.050]	−37.01	0.269
All Inner Regions
Ionized + Neutral	1.11 [±0.112]	−45.15	0.270
Ionized	0.88 [±0.180]	−35.65	0.405
Neutral	1.02 [±0.104]	−41.78	0.271
All Outer Regions
Ionized + Neutral	1.23 [±0.185]	−50.00	0.152
Ionized	0.90 [±0.400]	−36.68	0.314
Neutral	0.96 [±0.198]	−39.55	0.235

Note. Properties of the lines of best fit for our Σ_SFR–${{\rm{\Sigma }}}_{[{\rm{C}}{\rm{II}}]}$ relationships determined using the method of Kelly (2007). A line of best fit is displayed for each component of the $[{\rm{C}}\,{\rm{II}}]$ emission (combined ionized and neutral, ionized, and neutral). We further divide our fits by region type, with a fit for all individual regions, a fit for the averaged BtP inner regions and the KINGFISH nuclear regions, and a fit for the averaged BtP outer regions and the KINGFISH extranuclear regions.

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To test our measurement of SFR, the SFR indicator of Hao et al. (2011) determined using a lower dust attenuation coefficient, which is identical to Equation (9) but with a proportionality constant for the 24 μm luminosities of 3.89 instead of 6.0, was also applied to each region in our sample. We that find our linear fit parameters have no dependence on the coefficient we use.

The bottom panels of Figure 7 show the differences between Σ_SFR measured with the FUV and 24 μm hybrid star formation indicator (Equation (9)) and Σ_SFR measured using the relationships we determined with all the individual regions and using the different components of the $[{\rm{C}}\,{\rm{II}}]$ emission (Equation (11) and Table 7). The median value of the difference for Σ_SFR measured by the summation of the ionized and neutral $[{\rm{C}}\,{\rm{II}}]$ emission is −0.024 dex with a range of 0.80 to −0.36 dex. The range for the differences in Σ_SFR measured by the $[{\rm{C}}\,{\rm{II}}]$_Ionized surface brightness is 1.87 to −0.54 dex with a median value of −0.058 dex, and for the Σ_SFR measured by $[{\rm{C}}\,{\rm{II}}]$_Neutral surface brightness it is 0.025 dex with a range of 0.80 to −0.39 dex. We plot these differences (logarithmic ratios) to better illustrate the scatter about our best-fit lines.

The $[{\rm{C}}\,{\rm{II}}]$ luminosity is plotted against the SFR for the combined ionized and neutral $[{\rm{C}}\,{\rm{II}}]$ emission and the $[{\rm{C}}\,{\rm{II}}]$ from the isolated ionized and neutral ISM phases in the top panels of Figure 8. In these plots, the LIRGS from the Great Observatory All-sky LIRG Survey (GOALS) are included to expand the range of parameter space covered (Díaz-Santos et al. 2017). The LIRGS in this survey were similarly covered at the $[{\rm{C}}\,{\rm{II}}]$ 158 μm line with PACS on Herschel and at the $[{\rm{N}}\,{\rm{II}}]$ 205 μm line with SPIRE-FTS on Herschel. More information about the observations and processing of these maps can be found in Díaz-Santos et al. (2013) and Zhao et al. (2013, 2016). The inclusion of this sample extends our study to include the more extreme infrared (L_IR  ≥ 10¹¹ L_⊙) LIRGS that were part of the GOALS sample. In addition to the GOALS sample, the handful of high-redshift (z ≥ 4) galaxies with $[{\rm{N}}\,{\rm{II}}]$ 205 μm detections are plotted in Figure 8 (Pavesi et al. 2016, 2019; Lu et al. 2017). As measurements of n_e were unavailable for these sources, we used an average value of R_Ionized = 4.0 to determine f_Neutral for these sources. None of the high-redshift sources were included in the line fitting. Similar linear fits were performed on these data and are shown in Figure 8. These trends are described using Equation (12) and the values listed in Table 8:

$\begin{eqnarray}&&{\mathrm{log}}_{10}\mathrm{SFR}({M}_{\odot }\,{\mathrm{yr}}^{-1})=m{\mathrm{log}}_{10}L([{\rm{C}}\,{\rm{II}}])(\mathrm{erg}\ {{\rm{s}}}^{-1})+b.\end{eqnarray} \tag{ 12 }$

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Table 8.  Linear Fits from Figure 8

$[{\rm{C}}\,{\rm{II}}]$ Component	m	b	Rms Scatter
All Individual Regions
Ionized + Neutral	0.96 [±0.036]	−39.46	0.229
Ionized	0.92 [±0.057]	−37.32	0.332
Neutral	0.92 [±0.035]	−36.03	0.237
All Inner Regions
Ionized + Neutral	1.05 [±0.079]	−42.91	0.272
Ionized	1.10 [±0.145]	−44.32	0.414
Neutral	0.96 [±0.075]	−39.42	0.270
All Outer Regions
Ionized + Neutral	0.97 [±0.176]	−39.86	0.167
Ionized	0.63 [±0.182]	−25.81	0.235
Neutral	0.88 [±0.227]	−36.31	0.228
With GOALS
Ionized + Neutral	1.02 [±0.022]	−41.64	0.309
Ionized	1.02 [±0.030]	−41.11	0.406
Neutral	0.98 [±0.022]	−40.05	0.313

Note. Properties of the lines of best fit for our SFR–L( $[{\rm{C}}\,{\rm{II}}]$) relationships determine using the method of Kelly (2007). A line of best fit is displayed for each component of the $[{\rm{C}}\,{\rm{II}}]$ emission (combined ionized and neutral, ionized, and neutral). We further divide our fits by region type, with a fit for all individual regions, a fit for the averaged BtP inner regions and the KINGFISH nuclear regions, and a fit for the averaged BtP outer regions and the KINGFISH extranuclear regions.

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The bottom panels of Figure 8 show the differences between the SFR measured by the FUV+24 μm hybrid star formation indicator (SFR(FUV+24), Equation (9)) and the SFR determined using the relationships found for the $[{\rm{C}}\,{\rm{II}}]$ luminosity, labeled SFR($[{\rm{C}}\,{\rm{II}}]$) (Equation (12) and Table 8). We find a median value of −0.036 dex in the differences between the SFR measured using the hybrid FUV+24 μm indicator and the combined ionized and neutral $[{\rm{C}}\,{\rm{II}}]$ luminosity, −0.051 dex in the differences between SFR(FUV+24) and the SFR measured by only $[{\rm{C}}\,{\rm{II}}]$_Ionized luminosity, and 0.031 dex in the differences between SFR(FUV+24) and the SFR measured by only the $[{\rm{C}}\,{\rm{II}}]$_Neutral luminosity. The high-redshift sources seem to follow similar trends with greater scatter, potentially due to the large uncertainties in the $[{\rm{N}}\,{\rm{II}}]$ 205 μm detections.

We find that the combined ionized and neutral $[{\rm{C}}\,{\rm{II}}]$ emission and the $[{\rm{C}}\,{\rm{II}}]$ emission from only the neutral phases of the ISM trace SFR as measured by the FUV+24 μm hybrid SFR indicator with a scatter of ∼0.23 dex. The measured slope of 0.96 [±0.036] for the combined ionized and neutral $[{\rm{C}}\,{\rm{II}}]$ luminosity–SFR relation is consistent with the relationship found by De Looze et al. (2014), where a slope of 1.01 [±0.02] was found for a sample of dwarfs, ULIRGs, AGNs, and starburst galaxies, and the relationship found by Pineda et al. (2014), with a slope of 0.98 [±0.07] for $[{\rm{C}}\,{\rm{II}}]$ luminosity within the Milky Way. Using Equation (2), and our result for the SFR measured by the $[{\rm{C}}\,{\rm{II}}]$_Neutral luminosity, we can write an equation for SFR measured by $[{\rm{C}}\,{\rm{II}}]$ 158 μm and $[{\rm{N}}\,{\rm{II}}]$ 205 μm luminosities:

$\begin{eqnarray}&&\begin{array}{l}{\mathrm{log}}_{10}\mathrm{SFR}({M}_{\odot }\,{\mathrm{yr}}^{-1})=0.99{\mathrm{log}}_{10}\left[L([{\rm{C}}\,{\rm{II}}]158\ \mathrm{erg}\ {{\rm{s}}}^{-1})\right.\\ -\,\left.{R}_{\mathrm{Ionized}}L([{\rm{N}}\,{\rm{II}}]\,205\ \mathrm{erg}\ {{\rm{s}}}^{-1})\right]-40.49.\end{array}\end{eqnarray} \tag{ 13 }$

This equation has potential to be used in both the local and high-redshift universe without the need for dust corrections.

As the $[{\rm{C}}\,{\rm{II}}]$ emission from the neutral phases of the ISM accounts for most of the $[{\rm{C}}\,{\rm{II}}]$ emission from these regions, it is expected that the combined ionized and neutral $[{\rm{C}}\,{\rm{II}}]$ emission and $[{\rm{C}}\,{\rm{II}}]$ emission from only the neutral phases of the ISM follow similar trends, as shown by the similar slopes measured by our lines of best fit. Although both components of $[{\rm{C}}\,{\rm{II}}]$ emission rise with higher star formation surface densities, the $[{\rm{C}}\,{\rm{II}}]$ emission from the neutral ISM shows a more tightly constrained relationship than the $[{\rm{C}}\,{\rm{II}}]$ emission from the ionized ISM. This increased rms of 0.33 for $[{\rm{C}}\,{\rm{II}}]$_Ionized, 0.1 dex above the rms for $[{\rm{C}}\,{\rm{II}}]$_Neutral, is likely due to the sharp decrease in $[{\rm{C}}\,{\rm{II}}]$_Ionized/TIR as a function of far-infrared color. As described in Section 4.1, this result could indicate that a large fraction of the $[{\rm{C}}\,{\rm{II}}]$ emission from ionized phases of the ISM is not coming from star-forming ${\rm{H}}\,{\rm{II}}$ regions, but originates instead in the diffuse ionized ISM (Díaz-Santos et al. 2017; Herrera-Camus et al. 2018a). The large scatter in the $[{\rm{C}}\,{\rm{II}}]$_Ionized–SFR relationship indicates that any attempt to use $[{\rm{C}}\,{\rm{II}}]$ emission as a tracer of SFR must be treated with caution in galaxies that will have a large fraction of $[{\rm{C}}\,{\rm{II}}]$ emission from ionized phases of the ISM, such as high-redshift Lyα emitter galaxies. This conclusion is supported by analysis of the kiloparsec-resolution $[{\rm{C}}\,{\rm{II}}]$ detections from the SHINING survey (Herrera-Camus et al. 2018a). We also find no difference in the slopes within error when we separate our regions by location within the galaxy, suggesting that these results hold in a variety of conditions. The inclusion of the GOALS sample raises the slope slightly, which we believe is due to the elevated SFRs of these U/LIRGS, which causes them to fall above the galaxy main sequence (Elbaz et al. 2011; Murphy et al. 2013).