The Origin of [C ii] 157 μm Emission in a Five-component Interstellar Medium: The Case of NGC 3184 and NGC 628

With its relatively low ionization potential, C+ can be found throughout the interstellar medium (ISM) and provides one of the main cooling channels of the ISM via the [C ii] 157 μm emission. While the strength of the [C ii] line correlates with the star formation rate, the contributions of the various gas phases to the [C ii] emission on galactic scales are not well established. In this study we establish an empirical multi-component model of the ISM, including dense H ii regions, dense photon dissociation regions (PDRs), the warm ionized medium (WIM), low density and surfaces of molecular clouds (SfMCs), and the cold neutral medium (CNM). We test our model on ten luminous regions within the two nearby galaxies NGC 3184 and NGC 628 on angular scales of 500–600 pc. Both galaxies are part of the Herschel key program KINGFISH, and are complemented by a large set of ancillary ground- and space-based data. The five modeled phases together reproduce the observed [C ii] emission quite well, overpredicting the total flux slightly (about 45%) averaged over all regions. We find that dense PDRs are the dominating component, contributing 68% of the [C ii] flux on average, followed by the WIM and the SfMCs, with mean contributions of about half of the contribution from dense PDRs, each. CNM and dense H ii regions are only minor contributors with less than 5% each. These estimates are averaged over the selected regions, but the relative contributions of the various phases to the [C ii] flux vary significantly between these regions.


The [C II] Line
The [C II] 157 μm is a fine-structure line that arises from the 2 P 3 2 0  2 P 1 2 0 transition of singly ionized carbon, C + . With an ionization potential of only 11.26∼eV, C + is found throughout the interstellar medium (ISM). [C II] emission provides one of the main cooling channels in the ISM. With a relative line luminosity of typically~-[ ] L L 0.1% 1% , it is often the strongest line in the far-infrared (FIR) wavelength regime (Crawford et al. 1985;Stacey et al. 1985Stacey et al. , 1991Wright et al. 1991;Malhotra et al. 2001;Brauher et al. 2008). Observations and theoretical modeling both have indicated that [C II] is the dominant cooling channel in the cold neutral medium (CNM) (Wolfire et al. 2003), andtogether with [O I], in dense photon dissociation regions (PDRs) associated with regions of massive star formation (Tielens & Hollenbach 1985;Madden et al. 1997;Mizutani et al. 2004;Kaufman et al. 2006).
Previous studies have demonstrated that the strengthof [C II] emission correlates well with other star formation tracers (Boselli et al. 2002;de Looze et al. 2011;De Looze et al. 2014;Pineda et al. 2014;Sargsyan et al. 2014;Herrera-Camus et al. 2015), although this relation breaks under certain gas condition. As PDRs are commonly associated with H II regions, in which massive star formation occurred, it is not surprising that [C II] correlates with the star formation rate (SFR). Unlike optical lines such as a H , [C II] is much less susceptible to dust extinctionand has therefore been used as the SFR diagnostic of choice, in particular in luminous star-forming systems (Stacey et al. 1991;Pierini et al. 1999;Boselli et al. 2002). The relation between SFR and [C II] started to be heavily studied with the advent of new sensitive detectors on theKuiper Airborne Observatory (KAO), the Infrared Space Observatory (ISO), theCosmic Background Explorer (COBE), and balloon observations, and has become a common tool with the advent of the Herschel Space Telescope (Stacey et al. 2010;Sargsyan et al. 2012Sargsyan et al. , 2014Herrera-Camus et al. 2015).
Using [C II] to measure the SFR in galaxies is still problematic. In the extreme case, luminous infrared galaxies (LIRGs) and ultraluminous infrared galaxies (ULIRGs) suffer the so-called "[C II] deficit" problem where the ratio of [C II] to L FIR decreases with increasing ratio of 60/100 micron (FIR color) (Luhman et al. 1998(Luhman et al. , 2003Malhotra et al. 2001;Sargsyan et al. 2012;Díaz-Santos et al. 2013;De Looze et al. 2014;Herrera-Camus et al. 2015). An increase in the 60/100 μm ratio indicates warmer dust and more intense radiation fields. The same [C II] deficit is seen also in our Galactic Center (Nakagawa et al. 1995), local galaxies Beirão et al. 2012;Croxall et al. 2012), and in a large sample of subgalactic regions of KINGFISH data (Smith et al. 2017). The [C II] deficit suggests that caution must be taken if we wishto use [C II] as an SFR tracer in different ISM conditions. All of these studies point out that examining the gas heating-cooling processes under different conditions is necessary to better understand the SFR probed by [C II]. Several observational studies have shown that [C II] can arise from phases of the ISM different than the dense PDRs and SfMCs, these includeH II regions (Carral et al. 1994), the diffuse cold or warm neutral medium (CNM/WNM) (Bock et al. 1993;Ingalls et al. 2002), and the warm ionized medium (WIM) (Heiles 1994). The WIM is pervasive throughout the ISM and can give rise to both [N II] and [C II] emission (Heiles 1994). Given its ionization and critical density ( Table 2), the [N II] 205 line traces the WIM, the ionized ISM phase with low electron density. In particular, Bennett et al. (1994) found that the [C II] intensity correlates well with the intensity of the [N II] 205 as measured in the large beam size (  7 ) of COBE. Goldsmith et al. (2015) show based on GOTC+ data and Herschel [N II] 205that [C II] emission and [N II] 205 are correlated morphologically. On the other hand, Crawford et al. (1985) concluded from KAO observations thaton a galactic scale, [C II] emission arises from molecular clouds exposed to UV fields with 10-300 times the local interstellar radiation field. The recent study of GOTC+ (Pineda et al. 2013) revealed that [C II] in the Galactic plane is produced by dense photon-dominated regions (47%), with smaller contributions from CO-dark H 2 gas (28%), cold atomic gas (21%), and ionized gas (4%).
In this paper we examine the [C II] emission from selected regions within the galaxiesNGC 3184 and NGC 628 (see Section 1.3). The aim of this work is to quantify the relative contributions to [C II] from different ISM phases within these regions. We define five components ("phases") of the ISM as follows: (1) dense H II regions, (2) low-density WIM, (3) dense PDRs, (4) low n H and low G 0 surface of molecular clouds (SfMCs), and (5) the diffuse neutral medium (see Section 3.2 and Table 1 for more details). We use the observed emission lines, listed in Table 2, to probe the physical conditions of these phases. For a more detailed discussion on the multiphase ISM we refer the reader to Section 3.1.
This paper is organized as follows: in Section 2 we describe the observations and main data reduction steps. In Section 3 we characterize the multiple phases of the ISM and their contributions to the [C II] emission. We describe our method in Section 4and discuss the results in Section 5. We finish with a Summary and Outlook in Section 6.

Estimating the Energy Budget
In this paper we assume that the heating energy of the gas originates from the photons of massive young stars. For simplicity we do not takeother sources of mechanical heating into account, i.e., turbulence, which can also be important to the physics and chemistry of the ISM phases, as has been seen in the high-latitude Galatic diffuse clouds (Ingalls et al. 2002). The regions that we inspect are mainly active star-forming regions and donot represent the diffuse cold ISM. Hence the contribution from mechanical heating is considered to besmall. We infer the ionizing energy from the extinction-corrected a H flux, which traces photons with  n h 13.6 eV that ionize surrounding hydrogen gas, creating H II regions. Some fraction of these photons leaks into the diffuse ISM, possibly because of the clumpy structure of H II regions. These leaked photons create a low-density ionized phase called the WIM. We find that the a H luminosities of our studied regions with H II region size of 30-170 pc well exceed those of the Orion nebula or M17, but are an order of magnitude lowerthan 30 Dor in the Large Magellanic Cloud (LMC) (Kennicutt et al. 1989;Doran et al. 2013).
Photons with energy lower than the ionization energy of neutral hydrogen (13.6 eV) are able to escape the H II region and become the energy source for adjacent dense PDRs, for the surfaces of molecular clouds (SfMCs), and for the diffuse neutral medium. To calculate the incident radiation field, we convert the number of ionizing photons (NLyC) to L UV defined as stellar luminosity between 6 and 13.6 eV using Starburst99 (SB99; Leitherer et al. 1999), assuming a continuous SFR over 10 Myr following theclassical Salpeter initial mass function (Salpeter 1955).
The incident radiation field or G 0 canin principle alsobe calculated from the infrared radiation. This can be done by assuming that the L UV is fully absorbed by dust and re-emitted in the FIR. The incident radiation field G 0 can also be determined by fitting a dust model to the infrared spectral energy distribution (SED) to determine the heating radiation fields. We use the dust model of Draine & Li (2007), in particular the fitting described in Aniano et al. (2012), which provides an estimate of the minimum value of the Mathis radiation field. Mathis et al. (1983) evaluated the background radiation field from 5.04 to 13.6 eV to be 1.14 in terms of Habing fields (Habing 1968). Habing fields are defined as background radiation fields between 6-13.6 eV and havea value of´-1.2 10 4 erg --cm s sr 2 1 1 (Draine 2010;Tielens 2010). The dust model of Draine & Li (2007 adopts a two-component model for the dust emission of the region. One component is the fraction ( f PDR ) of the total infrared (TIR) emission that originates in dense PDRs due to the illumination by an enhanced radiation field, commensurate to the stellar luminosity and size of the H II region (G 0 from L UV ). This component accounts for the illumination of dense PDRs by regions of massive star formation. The second component is the fraction(f 1 PDR ), which isattributed to the low UV field. It corresponds to the CNM and the SfMCs. For our analysis we adopt the fitted values for our two target galaxies of f PDR , and the average UV field, G 0 dust, from the analysis of the dust SED by Aniano et al. (2012). These values are also listed in Table 5.

NGC 3184 and NGC 628
The improvements in spatial resolution and sensitivity of the new generation of FIR and submillimeterobservatories, Herschel (Pilbratt et al. 2010) and ALMA (Hills & Beasley 2008), offer the opportunity to study heating and cooling processes in the ISM of galaxies in great detail. In this paper we use the instruments onboard Herschel and combine them with diagnostics at shorter wavelengths that werepreviously taken with the Spitzer Space Telescope or from the ground. For this case study we have selected a total of ten regions in the two nearby galaxies, NGC 3184 and NGC 628. These galaxies have been selected on the basis of their existing multiwavelength data sets. Both galaxies are members of the Herschel open time key program KINGFISH, which stands forKey Insights into Nearby Galaxies: Far-Infrared Survey with Herschel , providing both FIR dust continuum observations and the vital FIR spectroscopy measuring the [C II] and other fine-structure lines listed in Table 4.
Within each galaxy, these regions sample the nuclei and the spiral arms, and are bright in the FIR lines of [O I]63 μm, [N II] 122 μm, [O III]88 μm, and [C II]157 μm. The ten regions have been selected based on their high signal-to-noise ratio (S/N) in the optical maps of PPAK Integral Field Spectroscopy Nearby Galaxies Survey (PINGS) (Rosales-Ortega et al. 2010;Sánchez et al. 2011). In addition to the PINGS data, a wealth of ancillary data at other wavelengths exists: FUV and NUV by GALEX (Martin et al. 2005), optical BVRI bands at the Kitt Peak National Observatory (KPNO) as part of the Spitzer Infrared Nearby Galaxies Survey (SINGS) program, and NIR-MIR from SINGS (Kennicutt et al. 2003).
We need to emphasize thatwhile modern optical and infrared observatories enable spatially resolved studies of galaxies beyond the Local Group, their spectroscopic sensitivity is still limited to the more luminous regions within the galaxies. By necessity, this introduces a selection bias toward regions of massive star formation since the "colder" non-star-forming regions provide insufficient S/N for detailed spectroscopic studies (see also Section 4.6). This general limitation also applies to our study of NGC 3184 and NGC 628 (Figure 1).
Both galaxies are shown in Figure 1. Their dust properties (dust-to-gas ratio, polycyclic araomatic hydrocarbon,PAH, mass fractions relative to the total dust mass) and UV-radio SEDs have already been studied within the SINGS project (Dale et al. 2007;Draine & Li 2007). NGC 3184 is a SAB(rs) cd type galaxy located at the distance of 11.6Mpc, while NGC 628 is an SA(s)c type galaxy located at 7.2Mpc (Kennicutt et al. 2003). At this distance, an aperture size of 12″ corresponds to a physical size of 500-600 pc. On this scale it is very likely to have multiple H II regions and dense PDRs within one beam. For comparison, 30 Dor in the LMCstellar cluster has a half-light radius of 70 pc (Shields 1990). The two galaxies, based on their averaged stellar formation rate, are categorized as normal galaxies (Kennicutt et al. 2003. We select regions with ongoing star formation (H II regions), measured from their a H flux (also see Figure 2). We list the general properties of the two galaxies in Table 3.

Observations and Data Reduction
The Herschel KINGFISH survey is an imaging and spectroscopic survey of 61 nearby galaxies thatwere chosen to cover a large range of galactic properties. It is descended from the SINGS program (Kennicutt et al. 2003), and one of its main aims is the study of the heating and cooling processes in the ISM within spatially resolved galaxies. Here we describe the KINGFISH photometry and spectroscopy, and give a summary of the ancillary data used in this analysis.

KINGFISH Photometry
The KINGFISH photometry includes data from both the Photodetector Array Camera and Spectrometer (PACS) (Poglitsch et al. 2010) and the Spectral and Photometric  Table 4. Each circle refers to a flux extraction aperture of 12″ in diameter, which corresponds to physical sizes of 600 and 500 pc for NGC 3184 and NGC 628, respectively. Imaging Receiver (SPIRE) (Griffin et al. 2010). However, we use only the three PACS broadband filters, centered on 70, 100, and 160 μm, for our analysis because ofthe low spatial resolution of the SPIRE data. The KINGFISH photometric observations were designed to reach a sensitivity of s 1 per pixel (~-1 MJy sr 1 ) at 160 μm at the optical radius of R 25 . The PACS images were taken in scanning mode with a scanning speed of  -20 s 1 . Initial data reduction was performed with the Scanamorphos package (Roussel 2013). The reader is referred to the papers by Kennicutt et al. (2011) andDale et al. (2012 for more details of the data reduction steps performed on the KINGFISH galaxies.
We used the library of convolution kernels, provided by Aniano et al. (2011), to match the different resolutions at the various wavelengths. We convolved all our broadband PACS images to match the resolution of the PACS 160 μm map, which corresponds to approximately 12″. We used this angular size to define our regions of interest (described in Section 3.1), and extract all our photometric fluxes in apertures of 12″ diameter centered on our regions (see Table 4).  Figure 1). Only the nuclei of NGC 3184 and NGC 628 were observed in the [N II]122 (cyan square in Figure  1) Figure 1), while for NGC 628 the IRS strip is orthogonal with respect to the PACS observational strip (see Figure 1). The nuclei ofNGC 3184 and NGC 628 were also observed in the [N II] 122 line (cyan square in Figure 1). Additional observations in the[N II] 122 and [N II] 205 lines have been taken in NGC 628. Unchopped-line scans were performed on both galaxies. To overcome the effect of undersampling and to minimize the gap between pointings, a dither pattern of 23 5×23 5 was imposed . For single-pointing maps, i.e., in the [N II]122 line, a 2×2 subpixel dither pattern of 4 5×4 5 was performed to overcome this problem. The Herschel Interactive Processing Environment pipeline version 8.0 has been used to reduce the PACS spectroscopy maps (Ott 2010), which have calibration uncertainties of 15%. The line map of each emission was created after removing the line continuum by third-order polynomial fitting. The line profile was then fitted with a Gaussian function. Whenthe fit failed to converge, direct integration was performed instead. The reader is referred to the paper by Croxall et al. (2012Croxall et al. ( , 2013 for more details of the spectroscopic data reduction steps performed on the KINGFISH galaxies. The line uncertainty is calculated from the calibration and line fitting process.
On average, we reached a surface brightness sensitivity of ---10 10 10 9 -W m 2sr 1 for all PACS spectroscopy lines. We extracted the line fluxes inside photometric apertures with  12 diameterand present the resulting surface brightnesses in Table 4. For each line, we present the flux derived from the maps at their original spatial resolution. When comparing lines and deriving line ratios, we convolve to a common spatial resolution.

IRS Spectroscopy
We used the Spitzer IRS (Houck et al. 2004) Long-Low (LL) data from 14 to 40 μm for our analysis. The observed IRS LL strips are overlaid in Figure 1. We extracted the LL flux inside the 12″ aperture using CUBISM (Smith et al. 2007a). We combined the LL1 and LL2 spectral segments after scaling the continuum level of LL2to match that of LL1. We then fitted the data cube using PAHFIT (Smith et al. 2007b) Table 4. We derive the uncertainty on the IRS flux based on the uncertainty of PAHFIT fitting procedure (Smith et al. 2007b).

CO (J=21) and H I Data
We used the CO ( =  J 2 1) data from the HERA CO-Line Extragalactic Survey (HERACLES) program (Leroy et al. 2009). The emission line map has a beam resolution of 13″ (Leroy et al. 2009). The H I data have been taken from The H I Nearby Galaxy Survey (THINGS) (Walter et al. 2008). The typical beam size for the H I map is 6 5-7 5 for both galaxies. We did not convolve the H I maps to a lower resolution. The uncertainty is calculated from rms scatter and the systematic uncertainty (Leroy et al. 2009).

PINGS Optical Spectroscopy
For optical spectroscopy, we used data from the PPAK Integral Field Spectrograph (Kelz et al. 2006;Rosales-Ortega et al. 2010). The data on NGC 628 have been made publicly available by the PINGS program (Rosales-Ortega et al. 2010;Sánchez et al. 2011). The data on NGC 3184, which had already been taken but not yet published, have been been kindly provided by F. Rosales-Ortega.
The PPAK integral field unit (IFU) has dimensions of ´ 74 65 , using fiber bundles with diameters of 2 7 each. Approximately 16 pointings with the IFU were obtained on NGC 3184, covering a large part of the galaxy. Since dithering was not performed on NGC 3184, we are missing the flux thatfalls in the gaps between the individual fibers. On NGC 628, an area of 34 arcmin 2 was observed, and dithering was performed for some of the pointings. For more details on the observations and data reduction, we refer the reader to Rosales-Ortega et al. (2010).
We matched the coordinates of the PINGS maps to the coordinates of the SINGS a H maps. The a H images were obtained with the KPNO and CTIO telescopesusing a set of narrowband filters centered on a H . Comparison between the PINGS and the stellar continuum-subtracted a H SINGS line intensity revealed a discrepancy in the fluxes, with PINGS a H fluxes being about 4-7 times higher. Additionally, we compared the a H PINGS flux with the a H flux inside a 2 5×2 5 aperture from Moustakas et al. (2010). The result is similar, the PINGS a H fluxes are significantly higher, while the a H fluxes from Moustakas et al. (2010) and SINGS agree within 20%. This discrepancy is most likely due to the difficulty in absolute flux calibration of the sparsely sampled fibers in the PINGS data. The PINGS survey focuses on line ratios,not on absolute fluxes, and we similarly only used line ratios. The relative flux calibration is accurate to within 5% for the whole mosaic (private communication with F. F. Rosales-Ortega). To obtain the absolute flux calibration, we scaled all optical lines in the PINGS spectra such that the a H fluxes matches those determined from the narrowband imaging SINGS data. As we uniformly scaled all lines, leaving the line ratios unchanged, this does not affect the ionized gas modeling of the optical lines in the later sections. However, the scaling affects the determination of the ionizing luminosity (NLyC), which in turn affects our predicted [C II] luminositites arising from H II regions (Section 4.1). The line intensity uncertainty was derived from the calibration and reduction step, which is ∼20% in the case of NGC 3184 and 30% for NGC 628 (F. F. Rosales-Ortega 2017, private communication).
We used the extinction law from Fitzpatrick (1999) to correct the optical emission for dust extinction. We assumed "case B" recombination with = T 10,000 e K, and an intrinsic ratio of a H over b H of 2.86 (Osterbrock & Ferland 2006). We calculated the color excess for each of the regions measuring the intensity-weighted averaged A V inside the 12″ aperture. We found that A V ranges from 0.4 to 1.2 mag. The extinctioncorrected surface brightnesses inside the 12″ apertures are presented in Table 4. The MAPPINGS models were compared to these extinction-corrected line fluxes.

A Multiphase ISM
In the following section we give a brief overview of the literature on the topic of the multiphase ISM. We emphasize that these studies investigated different objects with different physical sizes. Most of these studies focused on small scales within well-resolved (nearby) objects. Some discrepancies in their results are therefore to be expected. Madden et al. (1993) were among the first tosuggestthat the [C II] originates from a multiphase ISM (WIM, dense PDR, and CNM). Since then, many attempts have been made to disentangle the contributions of the multiple ISM phases (H II regions, CNM, WIM, dense PDRs (such as the Orion bar), or from the surface of molecular clouds (SfMCs) to the observed [C II] emission (Mookerjea et al. 2011;Beirão et al. 2012;Cormier et al. 2012;Croxall et al. 2012;Lebouteiller et al. 2012;Madden et al. 2013;Pineda et al. 2013). In this section, we describe the model used in the analysis. We find that the general picture is confusing and different studies come tosometimesdifferent conclusions on the relative importance of the various components. Heiles (1994) and Velusamy et al. (2012) advocated the importance of diffuse ionized gas (the WIM) to the [C II] from the Milky Way. Bennett et al. (1994) Carral et al. (1994) found that 30% of [C II] in NGC 253 comes from H II regions. A very different picture was developed by Vastel et al. (2001) and Mizutani et al. (2004), in which [C II] mainly arises from dense PDRs, following the theoretical work of Tielens & Hollenbach (1985) and Kaufman et al. (2006). Indeed, a number of observations demonstrated that [C II] originates from the dense PDR interfaces that separateionized gas from the surrounding molecular clouds (Crawford et al. 1985;Shibai et al. 1991;Stacey et al. 1993;Matsuhara et al. 1997;Orr et al. 2014). Finally, other studies (Bock et al. 1993;Wolfire et al. 1995;Ingalls et al. 2002) have suggested that [C II] on a galactic scale arisesfrom cold diffuse clouds (the CNM).
Several Herschel studies took into account the complexity of the ISM and simultaneouslymodeled the [C II] from multiple phases. Beirão et al. (2012) andCroxall et al. (2012 demonstrated that about 3%-50% of the [C II] arises from ionized gas, with no distinction between that arising from H II regions and WIM-like gas. A similar percentage has alsobeen found by Goldsmith et al. (2015) from Galactic Observations of Terahertz C+ (GOTC+) in arecent study of theMilkyWay: about 30%-50% of [C II] arises from ionized gas. A different picture, however, was shown by Cormier et al. (2012) for low-metallicity dwarf galaxy of Haro 11, wihere[C II] comes to40% from theionized diffuse medium and 10% comes from PDRs. Cormier et al. (2012) filled up the missing [C II] by introducing diffuse PDR components into their model. Pineda et al. (2013) showed from the GOTC+ survey that 47% of [C II] in the Galactic plane comes from PDRs with gas densities ∼ -10 10 3 4 cm 3 and G 0 in the range from 1 to 30, while the rest arises from CO-dark H 2 gas (28%), CNM, and WNM (21%), and small amounts of ionized gas (4%). Kapala et al. (2015) analyzed regions in M31 with physical sizes comparable to our regions, namely 700 pc apertures with 50 pc resolution. They found for M31 that from 20% to 90% of the [C II] comes from outside star-forming regions. The rest originates in the ISM and is related to star-forming regions (H II and PDRs).
A detailed modeling scheme is needed to determine the contribution of the different ISM components to the [C II] emissionand to understand how relative contributions change with the physical conditions in the ISM. A better understanding on the overall picture of the gas heating and cooling can be used to calibrate the use of [C II] to probe the star formation process. In this section we define a set of ISM phases (Section 3.2). We then modelthe [C II] emission from these different ISM phases independently from each other for two target galaxies(Section 4).

Definition of the Five Phases
Following the literature, we define the following five ISM phases and summarize their characteristics in Table 1: 1. Dense H II regions are the ionized gas surrounding the young stellar clusters with typical density ranges from 100 to 10 4 cm 3 and agas temperature of ∼8000 K (Osterbrock & Ferland 2006). 2. WIM is the extendeddiffuseionized phase. Some of the photons from the stellar cluster can leak and travel to large distance and ionizeless dense gas, which createsthis ISM phase. The typical densities are in the order of 0.1cm 3 (Haffner et al. 2009), as obtained from dispersion measures. 3. Dense PDRs are largely neutral, associated with (or surrounding) H II regions, and are characterized by high densities (n H ) and high incident radiation fields (G 0 ). The Orionbar provides a good example. 4. SfMCs. SfMCs are PDRs characterized by low G 0 and low n H . They represent extended regions of massive star formation where molecular clouds are exposed to the local average interstellar radiation field. The SfMCs are essentially PDRs (Hollenbach & Tielens 1997) characterized by low densities and the average interstellar radiation field. 5. The diffuse neutral mediumconsists of two components: the CNM (T∼40-100 K) and the WNM (5000-10,000 K) (Heiles & Troland 2003). We do not model the WNM here because we infer from models for the emission of the phases of the ISM (Wolfire et al. 1995) that its contribution per hydrogen is only 0.1on averagerelative toCNM gas and it does not have a distinct tracer.
We make use of CO( =  J 2 1) as a molecular cloud tracer, H I as a neutral medium tracer, andoptical emission lines as tracers of H II regions. The WIM contribution is not constrained by any specific observation. These diagnostics are then used to constrain the models and predict the [C II] emission. For a more detailed description of the methods used, we refer the reader to Figure 3. In general, we do not take into account the specific geometry that the ISM phases may have. In order to calculate the gas properties (G 0 and n H ) for the dense PDRs from the gas properties of the H II regions, we have to assume a spherical geometry. For other ISM phases, however, we assume no specific geometry with respect to the central stellar populations. The other three ISM phases are expected to extend beyond the physical size covered by one beam; this assumption implies that there is no connection between the dense PDR and the surface of molecular clouds in general. In our analysis, we attribute all the H I to the diffuse ISM. This should be seen as an upper limit as some of the H I may arise from the photodissociated surface of molecular clouds (Heiner et al. 2011(Heiner et al. , 2013.

Locations and Morphologies of the Regions
At the spatial resolution of these two galaxies, we expect multiple ISM components to overlap in the beam. These components of H II regions and PDRs typically appear as compactunresolved objects, while the CNM and WIM are more diffuse components thatmay extend well beyond the region corresponding to one beam size. Furthermore, they are seen at low inclination, which minimizes dust obscuration and line-of-sight confusion. We select threeregions in NGC 3184 and sevenregions in NGC 628. The ten regions are selected based on the availability of the data. Of the tenselected regions, two are located in the nucleus i.e., at thecenter of galaxies, while the other eight are located in the spiral arms of the galaxies (see Figure 1). The center of each aperture is selected by the peak brightness of the a H emission. As can be seen in Figure 1, there is significant emission outside the photometric aperture. We cannot tell whether this emission is physically connected to the H II regions. However, we estimated the error on the fluxes, mostly [C II], from the centering of the aperture for some cases (like for Reg5 NGC 628, where a clear peak is evident). In these cases the errors in the extracted flux can be up to 20%.
We compare the [C II] emission against several other phase tracers such as a H , [N II] 122, CO( =  J 2 1), and H I in Figure 4. The [C II] morphology in general can be divided into two categories: the first category is where we see [C II] emission associated with the region (see Figure 4). This applies to regions Nuc. N3184,Reg2 N628,Reg4 N628,Reg5 N628,and Reg6 N628.The second category is where the [C II] emission is rather weak and diffuse, as shown by Reg2 N3184,Reg3 N3184,Nuc. N628,and Reg7 N628.In four regions, the a H emission coincides with the [C II] emission. These regions are Nuc N3184,Reg4 N628,Reg5 N628,and Reg6 N628.We find three regions where the [C II] emission coincides with the CO( =  J 2 1) emission (Nuc N3184,Reg2 N628,and Reg5 N628). However, only three regions show a correlation with [N II] 122, namely Nuc. N3184,Reg4 N628,and Reg5 N628.In most regions we find that the 8 μm emission, which comes from stochastic heating process of PAHs and small-grain continuum emission (Tielens et al. 1999), correlates well with the [C II] emission. This is not surprising as the two are related through the photoelectric heating process (Helou et al. 2001). We find little or no correlation between [C II] emission and H I. Pineda et al. (2013) showed that the H I distribution in our Galaxy is smoother than the [C II] emission, given that the beam of H I is smaller than the [C II] beam. We stress thatin our analysis of individual regions, correlation or non-correlation in morphology does not determine our estimate of the contribution of a specific phase to the [C II] emission in that region.

Analysis of the Gas Conditions and the Modeled [C II] Emission
In this section we use a wide range of spectroscopic data from Table 4 to derivefor each of the ten distinct regions (Section 3.3) within NGC 3184 and NGC 628the main physical properties of the individual ISM phases, such as density, strength of the radiation field, and gas temperature. With these physical parameters in hand, we can then model the fractions of the observed [C II] flux coming from the individual ISM phases. The systematic uncertainties of the individual contributions to the [C II] emission arediscussed in Section 4.6.

The H II Region
An H II region is confined by an ionization front. Photons with energies lower than 13.6eV can easily escape the region, while most of the ionizing photons with energies  E 13.6 eV are absorbed inside the Strömgren radius to ionize hydrogen. The ionized gas cools through cooling lines, which radiate away energy. Most of these cooling lines lie in the optical wavelength range (see Table 2). These lines serve as good diagnostics of the physical conditions in the H II region.
The optical spectra of H II regions are governed by three parameters: (i) the electron density (n e ), (ii) the ratio of photon density to particle density (or ionization parameter, q, as defined in Dopita et al. 2000), and (iii) the metallicity Z. We use an analytical calculation to guide us through the parameter space (see Appendix A.1).
The optical line fluxes guide the initial parameter ranges of MAPPINGSIII to model the gas condition of the H II region to compare it to observations. H . We calculated a set of model spectra for varying n e and q, while we kept the metallicity fixed. The n e was set to several values between 1 and 10 4cm 3 , while qwas chosen to range between1 10 6 and4 10 8 . We chose optical lines rather thanmid-IR(MIR)lines ([Ne III]15 μm, [S III] 18 μm, and [S III] 33 μm), as the relative uncertainty within the optical set is smaller than the discrepancy between optical set and MIR lines. As a sanity check, we compared two cases of MAPPINGSIII modeling. First, we modeled only optical lines, and second, where we modeled all lines including the MIR lines. The second case yields gas conditions with unphysical properties where the ionization parameter is by one order of magnitude lower thanthe typical H II regions.
We assume a continuous SFR and constant pressure throughout the H II region. Since MAPPINGSIII yields the ratio between [C II] and b H , the filling factor cancels out. We consider only optical lines thathave been detected at S/N  3, and use a reduced c 2 minimization to determine the best fit, weighted by the measured uncertainty in the observed flux. The fit result is non-degenerate with n e varying from 500 to 1000 (see Figure 5), while q varies between´-1 10 4 10 7 7 . Obviously, the fit result is better constrained by the ionization parameter rather than by the electron density, except for Nuc NGC 628. As expected, the resulting n e and q values from the MAPPINGS fitting procedure are in good agreement with the preliminary analysis (see Appendix A.1). The electron density derived from the two methods agrees, except for Reg3 NGC 3184, Nuc NGC 628, Reg4 NGC 628, and Reg5 NGC 628, where n e from the line ratio is by a factor of two higher than in theMAPPINGSIII modeling.
Most of the observed line fluxes in the optical and FIR can be reproduced within the s 3 uncertainty of the model (see Figure 6). If instead we were tobase this analysis on the MIR lines, the physical characteristics would change: one order of magnitude lower q and three times higher density. Still, for almost all sources, the contribution of dense H II regions to the observed [C II] emission would be small, in the range of 0.5%-5%. The only exceptions are Nuc N3184and Reg5 N628,where the analysis of the MIR lines results in densities of 1-10cm 3 , and for such low-density gas the predicted [C II] emission would become important. However, for these two regions, we consider the results from the MIR line analysis with MAPPINGSIII as unphysical becauseionized gas of such low density could not producethe observed optical line fluxes. The MIR line fluxes may just qualitatively indicate the presence of lower density gas. We consider the emission from low-density ionized gas to the observed [C II] emission further in Section 4.2. In summary, we conclude that dense H II regions do not give an important contribution to the observed [C II] line intensity.
In our MAPPINGS calculations we assumed a temperature = T e 8000 K. We also determined the electron temperatures using CHAOS data (Berg et al. 2015 Table 6). Our findings are in good agreement with a study on NGC 253 by Carral et al. (1994) (with ∼700 pc aperture size) and theMookerjea et al. (2011) study of starforming regions in M33 (physical size ∼500 pc). A more recent study of Pineda et al. (2013) of theMilky Way found that 4% of the [C II] rises from ionized gas, while Goldsmith et al. (2015) found that a larger fraction of30%-50% of [C II] arises from ionized gas. Both studies are performed over physical sizes thatare different from our regions.

WIM
In our study, the WIM is not constrained by any specific observation. Within the available Herschel diagnostics, the electron density can be traced by the ratio of the [N II] 205 and [N II] 122 lines (Bennett et al. 1994 , is calculated based on the assumption of a three-level system, while the upper-level population of , is calculated based on a two-level system (Draine 2010). We show the emissivity ratio in Figure 7 for two metallicities. As nitrogen is a secondary nucleosynthesis element, its elemental abundance increases nonlinearly with Figure 5. c 2 fitting from the MAPPINGS model. We fixed the metallicity Z and varied the electron density n e and ionization parameter q. metallicity in environments of high metallicity (Dopita et al. 2000). The variation in the line ratio reflects the difference in critical densities for these two transitions. We use the metallicities thatwe derived in Section 4.1. For 1 Z e =´-C H 2.57 10 4 , =´-N H 6.03 10 4 and for 2 Z e =´-C H 9.10 10 4 =´-N H 2.09 10 4 ). We assume that the mixing of metals is very efficient in the regions, which may not be accurate on small scales (O'Dell et al. 2011;Lebouteiller et al. 2013 Haffner et al. (2009) quoteemission measure values of~-10 60 cm 6 pc for the WIM in galaxies. In the low-density limit, we can calculate the emission measured from the observed [N II] flux, assuming that it fills the beam and that the nitrogen abundance is representative of the metallicity. Assuming a scale length given  by the beam size, we arrive at rms densities ranging from 1.6 to 3cm 3 . These values are an order of magnitude higher than the typical WIM densities found by Monnet (1971), Reynolds (1991), and Haffner et al. (2009), which are in the range of 0.  (2016), we find that n e ranges from 5 to 10 cm 3 , althoughwith large uncertainties that aredue to the scatter in the FIR-[N II]205 relation. Again, this value is higherthanthe above-mentioned densities of -0.1 0.5cm 3 for classical WIM gasand is more characteristic of giant H II regions. Similar electron densities have been reported by the GOTC+ study of Goldsmith et al. (2015). The authors conducted a survey of several lines of sight in the Milky Way.
Following Haffner et al. (2009), we have adopted a typical WIM electron density of 0.1cm 3 in our calculations (see Table 6). Figure 7 shows that the emissivity ratio is insensitive to the assumed electron density in the range of -0.1 2cm 3 and not very sensitive even up to densities as high asñ 30 e cm 3 . We assume adensity of 30cm 3 to estimate the uncertainty (see Section 4.6).
On average, we find a wide range of the [N II] arising from the H II region rather than the WIM. In Nuc. N3184and Reg7 N628,the observed [N II] flux can even be fully reproduced by the H II region and there is no significant WIM contribution to [C II]. For regions Reg2 N3184 and Reg3 3184, for which we have no data on [N II], we assume a most likelyWIM contribution given by the average of the other eight regions.
On average, our WIM model yields 40%-but with a wide range of 10%-90%-of the observed [C II] flux. Goldsmith et al. (2015) found that about 30%-50% of [C II] arises from ionized gas, and correlates with [N II]205 emission. This ionized gas has a density between that of H II regions and WIM. The main challenging aspect comes from distinguishing the WIM from the H II region contributions to the [N II] flux.

The Dense PDR
For PDRs, the [C II] surface brightness mainly dependson two parameters: the hydrogen density n H ,and the incident radiation field G 0 .
The first parameter, the PDR density, can be obtained by assuming pressure equilibrium with the H II regionand adopting an electron temperature of 8000K for the H II region temperature. We derive the PDR gas temperature from the excitation diagram of H 2 S(0) and S(1) lines (Parmar et al. 1991;Sheffer et al. 2011), assuming an ortho-to-para ratio (OPR) of 3 (Burton et al. 1992). This adopted OPR ratio for both galaxies is in agreement with Roussel et al. 2007. The derived temperatures range from 160 to 300 K ( Table 5), typical for PDRs (Habart et al. 2011;Sheffer et al. 2011). The derived hydrogen densities are quite high, (1.6 10 4 tó 9.6 10 4 cm 3 ), as the electron densities from the H II region are also high.
One way to estimate the second parameter, G 0 , is from the total infrared (or stellar) luminosity L TIR (L UV ) and from the distance from the FUV source. The second method of estimating G 0 is by measuring the absorbed bulk UV radiation of the central star by the ISM. We explainour method of deriving G 0 in more detail inAppendix A.2. In our further analysis, we have adopted the G 0 values derived from b H . We prefer this approach, which directly yields G 0 ,over the method of Aniano et al. (2012), which adopts a power-law distribution of U for the PDR component.
We use the PDR model of Kaufman et al. (2006) and Pound & Wolfire (2008) to derive the [C II] surface brightness of dense PDRs for the derived G 0 and n H (Figure 8). However, the total contribution of dense PDRs to the observed [C II] emission depends on the beam filling factor, i.e., whichfraction of the area corresponding to one resolution element is covered by dense PDRs. This beam filling factor can be estimated in three different ways.
The first method to determine the filling factor is by comparing the total UV radiation and the intercepted UV radiation by the dense PDR. Consider a PDR cloud of radius R PDR at the distance R H II from the star (i.e., the radius of the H II region measured from the a H emission). The fraction of the UV light intercepted by the PDR and transformed into the infrared is given by If the PDR has a surface brightness in the line given by I line , then the observer will see a line flux given by line . However, under specific assumptions, f scale is also f PDR . In the study of the dust emission in KINGFISH galaxies by Aniano et al. (2012), only a fraction of the L TIR arises from dense PDRs. The authorsdefined this fraction as f PDR . For a small PDR filling factor, typically between 0.1 and 0.3 (Aniano et al. 2012), the contribution from the PDR to the total observed IR luminosity is small, but there is no discrepancy with the line-to-continuum ratio compared to models, i.e., with G 0 constrained from the stellar properties the PDR model can account for the line emission.
There are two key assumptions: first, the PDRs are at the typical distance of the size of the H II region (which is also the assumption that we use to derive the incident UV field). Second, the PDR surface seen by the star is the same as the surface area seen by the observer. An edge-on geometry will give a smaller surface area, but this reduction is compensated forby higher surface brightness. The [C II] surface brightness expected from dense PDRs has then accordinglyto be scaled down by f PDR . Hence, f PDR may be considered a lower limit becausein the optically thick environment of a dense PDR, the infrared radiation produced by absorption of UV photons can be processed by cooler dust deep inside the PDR (Hollenbach et al. 1991).
Second, given its high critical density and excitation energy, [O I] is an excellent tracer of dense PDRs (Beirão et al. 2012;Lebouteiller et al. 2012 (Kraemer et al. 1998;Vastel et al. 2001;Vasta et al. 2010). Indeed, Vasta et al. (2010) found in the study of 28 galaxies that about 20%-80% of Third, PDRs are generally bright in the H 2 S(1) and S(0) lines (Sheffer et al. 2011;Sheffer & Wolfire 2013 While these estimates suggest that shocks and turbulent heating do not play a dominant role for the physical properties of our regions of interest, we cannot rule out minor contributionseither. Hence we have decided against using the H 2 line as quantitative diagnostics. In principle, the [Si II] line can also be used as an indicator for the beam filling factor, but becausethis line can also have a large contribution from ionized gas (Abel et al. 2005;Kaufman et al. 2006), we do not consider it here.
Considering thedisadvantage of several dense PDR tracers as explained above, we decided to scale the [C II] from the model with f PDR . We emphasize that the interpretation of f PDR as a beam filling factor is justified for our selectedH II regiondominated regions where the TIR emission arises predominantly from UV flux converted by dust (see Figure 2). For regions dominated by the diffuse ISM, older stars would contribute a significant fraction to the TIR. In our regions, this contribution is very small and therefore neglected.
The [C II] emission from the dense PDR is then calculated from the [C II] surface brightness in Figure 8, multiplied by f PDR . Perusing the values in Table 6, our model can explain the majority of 68% on average (40%-100%) of the observed [C II] emission coming from dense PDRs.

Surface of Molecular Clouds
The main difference between the surface of molecular clouds and the dense PDRs lies in the adopted density n H , the strength of the illuminating radiation field G 0 , and the beam filling factor. In comparison to dense PDRs, SfMCs have amuch lower density and are irradiated by lower G 0 , but theyare more extended, leading to significantly larger beam filling factors. Since we do not have a suitable tracer of density, we assume n H =300cm 3 for the SfMCs, in good agreement with Pineda et al. (2013) andBattisti & Heyer (2014).
We emphasize that the above-mentioned density of the SfMC differs from the densities we assumed for other ISM components (see Table 1). However, the entire ISM is not in thermal pressure equilibrium, which is mainly relevant for the diffuse gas. The thermal pressure on the SfMCs can be higher than the thermal pressure in the diffuse gas because the molecular clouds are gravitationally bound. If the gas is converted into WNM, then the thermal pressure could be sufficiently high for the gas to evaporate, but this would only occur in hot spots near H II regions.
We furthermore assume that the SfMCs are irradiated by an average radiation field, whose strength we estimate from the dust model of Aniano et al. (2012). We use their dust maps for NGC 3184 and NGC 628, which have been derived from the dust model of Draine & Li (2007). This model considers two ISM components thatcontribute to the FIRcolor temperature: the first component is the diffuse medium as characterized by the Mathis field (Mathis et al. 1983) with a minimum U value (scaling factor of Mathis field). The second component has a radiation field thatis characterized by a power law as afunction ofU. This dust-fitting model was applied to all KINGFISH maps on a pixel-by-pixel basis (see Aniano et al. 2012 for details).
We use the minimum U value and convert it into G 0 by multiplication with the factor 1.14 to convert the Mathis field into the Habing field. We list these G 0 , notated as "G 0 dust," in Table 5. The observed FIR color temperatures of these regions indicate low incident radiation fields, G 0 = 1.7-3.4 ). The adopted G 0 values compare well with the finding of the GOTC+ study (Pineda et al. 2013), which derived G 0 between 1 and 30 for the surface of molecular clouds in the Milky Way.
The beam filling factor of SfMCs can be estimated using the CO emission as a tracer. Stacey et al. (1985), Shibai et al. (1991), Pineda et al. (2013), and Orr et al. (2014) showed that SfMCs are well traced by CO( =  J 1 0). On galactic scales, CO( =  J 1 0) correlates well with the CO( =  J 2 1) emission (Braine et al. 1993;Leroy et al. 2009). We note that the dense PDR component will also contribute to the observed CO( =  J 2 1) emission. Our models yield that the expected CO( =  J 2 1) surface brightness of dense PDRs is an order of magnitude higher than that of SfMCs. The emission of CO( =  J 2 1), estimated from dense PDRs after multiplication with the PDR filling factor (0.1-0.3), ranges froḿ . The model is higherby afactor of 4than the observed CO, hence the observed CO must risefrom lower G 0 and n H gas than our dense PDR. On the other hand, CO( =  J 2 1) from modeled SfMCs ison average´-1 10 10 --W m sr 2 1 . Whenwe scale the modeled CO( =  J 2 1) from SfMCs down by 1-f PDR ,we find thatthe modeled and observed CO( =  J 2 1) differs by factor of 1.5. This impliesthat CO( =  J 2 1 from theSfMC occupies asmaller beam filling factor than 1-f PDR . Our investigation scale (500-600 pc)does not guarantee that dense PDR and the SfMC occupy asimilar distance, however, hence using 1-f PDR as a beam filling factor for CO is problematic.
Unfortunately, we do not have a better tracer of SfMCs than CO( =  J 2 1) and do not have means to distinguish thedense PDR and SfMCs contribution to the observed CO( =  J 2 1). Our assumption that CO( =  J 2 1) mainly rises from the surface of molecular clouds is based on the recent study of Pineda et al. (2013) and Orr et al. (2014). The derived contribution of SfMCs to the observed [C II] emission isthereforean upper limit.
Since the range in physical and chemical conditions between dense PDRs and SfMCs overlaps, we expect some fraction of the [O I] emission and H 2 also to arise from SfMCs. We find that SfMCs can produceon average∼15% ofthe observed [O I] and H 2 .
As for dense PDRs, the physics, chemistry, and emission characteristics of SfMCs can be described by PDR models, with G 0 obtained from the dust model and the assumed n H as input. We used the PDR model of Kaufman et al. (2006) andPound & Wolfire (2008) to calculate the CO( =  J 2 1) and [C II] emission from this component (see Figure 9), and then compared it to the observed CO( =  J 2 1) to obtainthe expected [C II] emission from SfMsCs (Figure 9).
We find that the modeled contribution from SfMCs to the observed [C II] flux ranges from 4% to 60% (Table 6). This large range in the fraction of [C II] reflects the large variation in the observed CO( =  J 2 1) surface brightness (see Table 4).

CNM
The CNM is usually well traced by the H I 21cm emission. We usedata from The H I Nearby Galaxy Survey(THINGS, Walter et al. (2008) to derive the H I mass, brightness temperature T B , and hydrogen column density N H for each of our regions, following the prescription by Walter et al. (2008). The resulting H I parameters are listed in Table 5.
We note that not all of the detected H I is in the CNM phase. Following the study of H I emission in the Milky Way by Heiles & Troland (2003), we make the assumption that only one-third of the H I comes from the CNM, while two-thirds are in the WNM. We reduced the H I mass accordingly. We do not separately account for the WNM contribution to the [C II] emission, as this contribution-per H-atom-is a factor of ten smaller for the WNM than for the CNM (Wolfire et al. 1995). For the assumed CNM mass fraction, the [C II] contribution from the WNM contributes less than 20% of CNM values, which itself is a rather small contribution (see below). Therefore, we decided to neglect the contribution from the WNM.  (2008). We assume n H =300cm 3 at the surface, and we use the G 0 from Aniano et al. (2012). The contours are the ratios of [C II] over CO( =  J 2 1) for the given G 0 and n H . We selected four regions to represent the spread in parameters and the resulting variation in [C II] / CO( =  J 2 1).  Notes. The surface brightness values in brackets are lowerthan 3σ, and we include these values as upper limits in our analysis. a The extinction correction for a H .   Draine & Li (2007). e = f pdr fraction of L TIR thatcomes from U  100 ). f Gas temperature for the dense PDR derived from H 2 S(0) and S(1). g Hydrogen density for the dense PDR derived from the pressure equilibrium assumption. h Derived from L UV , see Equation (7). i Derived from L TIR , see Equation (7). j From the dust model of Aniano et al. (2012). k The atomic hydrogen column density derived from H I. l Derived using therelation in Kennicutt et al. (2009). In order to estimate the total [C II] emission, we multiplied the resulting N H with the appropriate cooling rate, taken from Wolfire et al. (1995). The cooling rate per hydrogen nucleus is a function of thermal pressure P k and G 0 . Analogous to the SfMCs, we adopted the minimum U (see Section 4.4) for each region as derived from the dust model of Aniano et al. (2012). We assumed that the ISM gas pressure in these galaxies is similar to the pressurederived for the Milky Way through UV absorption lines of [C I] (Jenkins & Tripp 2011) and in the recent GOTC+ study of the [C II] m 158 m line (Pineda et al. 2013), with~-P k 1000 3000 K cm 3 . As expected, these thermal pressures are lower than those in the H II regions and PDRs (Table 1) by several orders of magnitude. We adopted P k 2000 Kcm 3 for the extra-nuclear regions, andfollowing Wolfire et al. (2003), the higher value ofP k 10,000 K cm 3 for the nuclear regions. This results in a cooling rate of -6.2 10 26 erg s H for the extra-nuclear regions and -11 10 26 erg s H for the nuclear regions.
These values are somewhat higher than the average cooling rate observed by COBE for the Milky Way (∼  2.65 -0.15 10 26 erg s H; Bennett et al. 1994) and by ISO for the spiral arms in M31 (´-2.7 10 26 erg s H; Rodriguez-Fernandez et al. 2006). However, this apparent discrepancy merely reflects the fact that Bennett et al. (1994) did not correct for the WNM contribution when calculating the [C II] cooling rate per H-atom. Even with our higher values for the cooling rate, our model predicts rather small [C II] contributions from the CNM, ranging from 0.3% to 10% (Table 6). We found that the [C II] fraction from CNM has an uncertainty~30%, but becausethe [C II] fraction from CNM is small, this uncertainty is also small.

Quantifying the Uncertainties on the [C II] Fraction
We summarize the main uncertainties in our method on the derived [C II] fractions. Figure 10 summarizes our analysis and uncertainties.
The uncertainty of the [C II] fraction from H II region is calculated by propagating the model grid and the observational uncertainty. If the densities were much lower than the optical lines indicates, the contribution from H II regions to the observed [C II] would be concomitantly larger. For an electron density of only n e =100 cm 3 , the [C II] fractions from our selected regions would increase by factors ranging from 2 to 20, with a linear average of 8. In all cases, however, would the H II regions remaina minor [C II] contributor with fractions well below 25%. The uncertainty in the [C II] emission from the H II regions is estimated from the model grid study when we consider models with c2 within a factor of two ofthe best fit.
The uncertainty from WIM components mainly rises from the electron density assumed. If we adopt the low ionized gas density from Herrera-Camus et al. cm 3 ). The uncertainty in the [C II] fraction from dense PDR is set by G 0 and n H . The uncertainty of n H is propagated from n e in the H II regions, while the G 0 uncertainty is adopted from the difference between G 0 derived from L TIR and from L UV . We  calculate the uncertainty of the [C II] fraction by propagating the uncertainty of n H and G 0 . Another uncertainty is the radius of the H II region (R H II ). If we eliminate the dependency on R H II , i.e., using a dimensionless q parameter from MAPPINGS to derive G 0 , the derived G 0 would beon averageabout three times lower (albeit with larger scatter) than G 0 from L UV . While this would reduce the [C II] fraction from dense PDRs in somebut not allof our selected regions, the main conclusion, namely that dense PDRs are the dominating contributor to the [C II] emission, remains unaffected. For SfMCs, the uncertainty of [C II] fraction is calculated from the uncertainty of G 0 derived from the dust map and the gas density. On average, G 0 is uncertain by about ∼50%. This translates into an uncertainty in the [C II] fraction by factor of 1.4 on average. The main uncertainty comes from the dense PDR contribution to the CO( =  J 2 1) flux. We derived the uncertainty of the gas density from the density distribution of molecular clouds, which ranges from -100 500cm 3 (Battisti & Heyer 2014). The uncertainty in the density translates into anuncertainty of the [C II] fraction of ∼80%. As we explained in Section 4.4, the calculation of the [C II] fraction of SfMCs is an upper limit.
The CNM [C II] fraction is governed by the assumption of the CNM:WNM fraction and the thermal pressure. We adopted the observed average ratio of CNM:WNM=1:2 from the study of Heiles & Troland (2003). In their survey of the [C II] emission in the Milky Way, Pineda et al. (2013) also arrived at an average CNM:WNM ∼1:2. The range in the ratio is also adopted from the work of Pineda et al. (2013), and the uncertainty in the fractional [C II] contribution from the CNM is ∼0.4. The cooling rate per hydrogen atom depends weakly on the gas thermal pressure. The ISM pressure measured by Pineda et al. (2013) for the Milky Way ranges from 1000 to 3000 K. These values are in good agreement with the [C I] study by Jenkins & Tripp (2011). We have adopted a thermal pressure of 2000 -K cm 3 with an uncertainty of ±1000 -K cm 3 .

Discussion
In Section 4 we discussed how we independentlyderived the [C II] flux for each ISM phase in NGC 3184 and NGC 628. We summarize the model procedure in Figure 3 for convenience. Now we combine the results from Section 4 to obtain the total [C II] flux for each of our ten distinct regions (Section 3.3) Figure 10. Bar diagram with the contributions to [C II] from each of the five phases. For Reg2 and Reg3 in NGC 3184 no WIM tracer is available and we have adopted the [C II] fraction derived from the median value of other regions (see Table 6 for details). The blackdashed line is the observed value of [C II].
within NGC 3184 and NGC 628. We then compare the observed [C II] flux densities to the sum of the modeled contributions. Obviously, if the sum of the independently modeled contributions does not come close to the observed [C II] strength, there would be little confidence in our model. On the other hand, given the theoretical and observational complexity of our model, an agreement within a factor of two could be considered a success and provides sufficient confidence in our model.

Main Results
The modeled fractions of the [C II] emission from the five ISM phases are listed in Table 6for each of the ten regions within NGC 3184 and NGC 628. Table 6 also shows in its rightmost column the sum of the various contributions. For the subsequent discussion of the various regions, a graphical visualization of the numbers in Table 6 seems helpful. Hence, we converted them into a bar diagram, which is shown in Figure 10. Although the exact contributions vary from region to region, we can state two general findings.
First, our model reproduces the observed [C II] emission quite well. The exceptions are Nuc N3184,Reg3 N3184,Nuc N628,and Reg5 N628,and even there, the discrepancy is within a factor of two, for reasons thatare further discussed in Section 5.2. Given the good match on one handand the manylargely independentobservational and theoretical parameters ( Figure 3) thatenter our model on the other hand, we consider our approach to bequite successful. Second, although the exact contributions from each phase vary from region to region, there is an underlying trend that wediscussin Section 5.3.
Obviously, the uncertainties play an important role in the interpretation of the results. The uncertainties quoted in Table 6 include the observational errors as well as the systematic uncertainties. The lattervary in their reliability as diagnostic tracers. Generally speaking, the three phases, H II region, dense PDRs, and CNM, have relatively good tracers, leading to very well-constrained gas parameters. Unfortunately, this is not the case for the SfMC and WIM. As shown in Figure 10, the two phases have relatively large error bars and vary from region to region. More specifically, the main uncertainties for the [C II] from WIM arise from the uncertainties in electron density. The uncertainties of the [C II] fraction from dense PDR and the SfMC arise from densities and G 0 . The uncertainties of the [C II] emission from CNM come from the thermal pressure in the CNM:WNM ratio. The total uncertainty quoted in Table 6 assumes that the individual uncertainties are mutually independent. While this is likely for most cases, it may not apply to the H II regions and dense PDRs, whose gas properties are linked.
Arguably the greatest uncertainty with regard to generalizing our results lies within the selection of our ten analyzed regions. As stated in Section 1.3, the unavoidable selection on the basis of sufficient S/N for spectroscopy introduces a bias toward regions of massive star formation. This bias is visualized in Figure 2, which shows the histograms of total infrared emission for each resolution element (beam) within both galaxies in comparison to the selected regions. It is obvious that the majority of regions (beams) have significantly lower TIR emissions than our selected regions. While this may not be a significant bias with respect to the study of the ISM in galaxies at higher redshift-which underlie similar selection biases due to S/N requirements-it is clear that our results do not describe the ISM in the more quiescent regions of normal galaxies, for which we would expect relatively higher contributions from the CNM and the SfMCs.

Results Regionby Region
In regions Nuc N3184 and Reg3 N3184, we apparently overestimate the total [C II] emission by up to factor of 2, probably because ofan overestimate of the contributions from SfMCs and WIM. Nuc N3184 and Reg3 N3184 show a two to three times higher contribution from SfMCs than the other regions. As discussed in Section 4.4, it is possible that part of the CO( =  J 2 1) flux, which arises in dense PDRs, may be incorrectly attributed to SfMCs.
We underestimate the total [C II] in Reg2 of NGC 628, with our model yielding only 50% of the observed value. It might be that the missing [C II] emission comes from mechanisms thatwere not included in our model. For instance, Appleton et al. (2013) showed that the [C II] emission in Stephan's Quintet is greatly enhanced, most likely byshock-heated gas. However, we consider such a scenario unlikely for the reasons given in Section 4.3. Furthermore, the nuclei of NGC 628 and NGC 3184 host no active AGN (Moustakas et al. 2010) that could make a significant contribution, although there isa lowluminosity X-ray (XDR) region (Grier et al. 2011).
We might bemissing [C II] from Reg2 of NGC 628 becausehidden star-forming regions may be present. Reg2 is consistently bright in the 8, 24, 70, and 160 μm continuum images, but not in a H . If the UV photons are directly and efficiently absorbed by thick layers of dust surrounding the H II region, our extinction correction is insufficient. Consequently, we underestimate G 0 and the [C II] strength from the dense PDR. If G 0 were underestimated by a factor of three, we would miss ∼30% of the [C II] from dense PDRs for Reg2. In order to test this hypothesis, we checked the ratio of 24 μm over a H , which is twice as high asthe median value of the other regions, indicating that a substantial fraction of the starforming regions is deeply buried in dust clouds. We do not consider the contribution from an older stellar population to the 24 μm flux (Leroy et al. 2012) to be significantbecause of our selection bias toward regions thatare dominated by ongoing star formation.

The Dominating ISM Components
We have modeled the galactic ISM with five phases or components to estimate the [C II] emission. The results are presented in Table 6 as "plain average." Our comprehensive approach of modeling all relevant phases of the ISM has the additional advantage over studies of single ISM components of providing a cross-check: the sum of all phases should yield the observed [C II] flux. As can be seen from Table 6, this is, as expected, not exactly true. On average, the sum of the individually and independently modeled components is about 45% higher than the observed [C II] flux. Given the many observational and theoretical uncertaintiesand the fact that for both WIM and SfMCs we have only upper limits, we consider a mismatch of less than 50% as strong support for our approach.
We recallthat our main goal is to quantify the relative contributions of the various phases and identify the dominating contributor(s). In order to make such statements, we have to normalize the contributions to a total of unity. As an extreme example, it would not make physical sense to state that the strongest component produces 120% of the [C II] flux. This may lead to the incorrect conclusion that one single component would be sufficient to explain all of the observed [C II] flux, although[C II] emission from other components has been detected as well.
Since we have no exact knowledge of the systematic observational and theoretical uncertainties of the various methods, we assume that all methods suffer from the same systematic uncertainties. The resulting normalized values are also listed in Table 6 as "normalized mean." The discrepancy of 45% on average illustrates the inherent uncertainties. We therefore refrain from overstating the numbers perse, but instead focus on what the dominating, important, and minor contributors are. These conclusions do not depend critically on whetherthe results have been normalized.
In summary, we find ( Table 6) that in most regions, dense PDRs are the dominating component, contributing about twothirds of the [C II] flux on average. The second most important component s are the SfMCs and the WIM. On average, SfMCs and WIM components together account for about half of the contribution from dense PDRs. The WIM contribution shows the largest scatter between the individual regions. We emphasize again that both WIM and SfMCs are estimated from upper limits, but even for lowerrealistic estimates, our finding still holds that they are significant contributors to the [C II] flux. Finally, the contributions from dense H II regions and the CNM are rather minor, with less than 5%each. Including more phases in the model would likely not significantlyaffect the results.

Comparing Our Results to Other Studies
We now considerthe individual phases in more detail, and compare our findings to results from the literature. We caution the reader to compare the numbers at face value, since many published values have been derived with different methods, often not clearly stating the systematic uncertainties. For instance, the term "PDR" is defined by different authors with very different G 0 and n H . Nevertheless, it is interesting to review the large scatter among the results in the literature.
K. V. Croxall et al. (2017, private communication) find, based on [N II]205 μm data from the BtPand the KINGFISH samples of numerous regions in several galaxies, that 84%, with 15% scatter, of [C II] emission arises from neutral gas, which includes in our notation the dense PDRs, the SfMCs, and the CNM. The remaining 16% arise from ionized gas, which includes our H II regions and WIM components. We obtained a neutral fraction (PDRs + SfMC + CNM) of 80%, which is roughly 5% lower but well within the range of the result of Croxall et al. (84%), derived in a completely different way. For comparison, Beirão et al. (2012) found in NGC 1097 that 3%-33% [C II] arises from ionized gas, and Croxall et al. (2012) foundfor NGC 4559 and NGC 1097 thatthe average ionized gas contribution to the [C II] emission is 20%-50%. Neither study distinguished between the contributions from diffuse and dense ionized gas.
We have seen that more than half of the [C II] emission comes from dense PDRs, with G 0 ranging from 50 to 150 and n H of~( -) 2 10 10 4 cm 3 . The star-forming regions thatare bright in a H -Nuc, Reg2, Reg3 in NGC 3184, and Reg4, Reg5, Reg6, and Reg7 in NGC 628-show the largest fractional contributions from PDR gas, about 50%. For comparison, Kapala et al. (2015) investigated several starforming regions in M31 with 700 pc in physical size and 50 pc of resolution. They distinguished between star-forming and non-star-forming regions using a a H threshold, and found that between 20% and 90% of the [C II] emission rises from nonstar-forming regions, i.e regions or components other than H II regions and dense PDRs. Our selected regions are bright in a H emission, indicating very highstar-forming activity. Therefore, our finding that star-forming regions have a strong contribution from dense PDRs agrees with the conclusion drawn by Kapala et al. (2015). Chevance et al. (2016) studied 30 Dor in the low-metallicity LMC as a template for H II extragalactic regions. The starforming complex of 30 Dor hasLyα continuum photon fluxes that are comparable to our sample. Chevance et al. (2016) found that 90% of the [C II] emission in 30 Dor arises from PDRs with G 0 ranging from 100 to2.5 10 4 . Chevance et al. (2016) simulated 30 Dor at alarge distance and found that the drop in G 0 will possibly bring the [C II] fraction from PDRs close to our result. Chevance et al. (2016)  205 with CO, aiming to disentangling the emission from ionized and photodissociation regions. IC 342 was selected because of its distance and data completeness. Röllig et al. (2016) found that between 35% and 90% of the [C II] arises from ionized gasand that the central region shows a higher contribution from ionized gas. In comparison, our result that 3%-50% of the [C II] arises from WIM and H II regions is smaller by factor of two. Oftwo nucleus regions, only Nuc N628 shows a high fraction (43%) of ionized contribution to the [C II] emission. Thestudies by Röllig et al. (2016) andbyPineda et al. (2013) claimed that ionized gas might be an important [C II] contributor toward the central parsecs of IC 324 and the Milky Way, respectively. A larger sample of galactic nuclei is needed to verify this claim.
For the interior of the Milky Way, Pineda et al. (2013Pineda et al. ( , 2014 sampled 425 lines of sight and found that PDRs with n H -100 1000 cm 3 , located far away from massive star-forming regions (G 0 ∼1-30), contribute only up to 30%-47% to the total [C II]. They identified the PDR components from the presence ofCO emission and [C II], which made no distinction between denseOrion-like PDRs and less intense PDRs. In fact, their PDR gas properties are rather similar to our SfMC component, for which we assumed G 0 ∼1-3 and n H ∼1000 cm 3 . Comparing this result to Pineda et al. (2013Pineda et al. ( , 2014, keeping in mind that our estimation is an upper limit for SfMCs, our approach yields lower [C II] contributions from the SfMCs than the studies ofPineda et al. (2013,2014).
Concerning the contributions from H II regions, previous studies by Carral et al. (1994) andKramer et al. (2005 yielded about 30% from H II regions on average. On the other hand, Pineda et al. (2013) found for regions in the Milky Way that only 4% of [C II] comes from ionized gas. Our findings of contributions of 4% on average agree well with the results of Pineda et al. (2013), although the assumptions on parameters like n e affect the results considerably. We recallthat our modeling uses a comprehensive set of optical emission lines to constrain the H II region properties.
Last, we found that the CNM component, with an average contribution of 4%, is not a very significant [C II] contributor in our star-formation-dominated regions. A fraction of 10% was observed in the irregular galaxy IC 10 by Madden et al. (1997), while Pineda et al. (2013) andVelusamy & Langer (2014 found that 20% of [C II] in the Milky Way arises from the CNM. A most recent study by Fahrion et al. (2017) found thatonly 9% of [C II] arises from CNM in a dwarf galaxy of NGC 4214.

Summary and Outlook
With an ionization potential of only 11.26eV, C + can be found throughout the ISM. Its [C II] 157 μm fine-structure line provides one of the main cooling channels of the ISM. Since its strength correlates with the SFR density andis much less susceptible to dust extinction than other star formation tracers, it serves as an important diagnostic, although its accuracy is under discussion ("[C II] deficit").
The goal of this work was to establish an empirical multicomponent model of the ISM to quantify the individual contributions of the various gas phases to the observed [C II] 157 μm emission. We defined five gas phases as follows: (1) dense H II regions, (2) the WIM, (3) dense PDRs, (4) low n H and low G 0 SfMCs, and (5) the diffuse neutral medium (CNM/WNM).
To validate our model, we selected ten regions within the two nearby galaxies NGC 3184 and NGC 628 at distances of 11.6Mpc and 9.5Mpc, respectively. The main data were taken from the Herschel open time key program KINGFISH, and wecombined them with a large set of ancillary photometric and spectroscopic data that were previously taken from the ground or by the Spitzer Space Telescope. Of the ten selected regions, three are located in NGC 3184 and seven in NGC 628. They comprise the two areas around the galactic nuclei as well as eight extra-nuclear regions. Our analysis was done at the resolution of the Herschel beam size, which corresponds to a physical size of 500-600 pc.
Our five-componentISM modelreproduces the [C II] emission from the individual phases quite well. While the introduction of five ISM components and a wealth of ancillary datamay at first sight introduce additional parameters into our model, our comprehensive approach also introduces another constraint: namely that the sum of the five phases must equal the total observed [C II] emission. We find that for most regionsthe sum of the modeled fluxes is within a factor of two of the totalobserved [C II] flux. Hence, we consider our results relatively reliable and accurate.
More specifically, our findings are as follows. with fractions of less than 5% on average. 6. The relative strength of all components varies significantly, depending on the physical properties of the gas. This variation provides the important physical basis for a subsequent analysis of a much larger sample of galaxies and regions.
The main uncertainties in the modeling of the H II region, the WIM, the dense PDR, the SfMC, and the CNM come from the uncertainties in electron density, n e , G 0 , the adopted densities and radiation fields, and the thermal pressure, respectively. While these uncertainties are inherent to all studies, it is sometimes difficult to compare numbers at face value. Many published [C II] contributions have been derived with different methods, andthe systematic uncertainties are often not clearly stated. Nevertheless, our findings are in good agreement with specific studies in the literature. We should also keep in mind thatdespite the tremendous progress in this field that the Herschel Space Telescope has enabled, the angular resolution is still poor, with beam sizes corresponding to approximately 500 pc, which makes a direct comparison to studies of resolved regions within our Milky Way difficult.
As stated in Section 4.6, the main limitation with regard to generalizing our results to the entire galactic ISM is the selection bias toward regions with higher total infrared emission, i.e., ongoing massive star formation. While this bias is observationally unavoidable in spectroscopic studies of distant galaxies with the current generation of space observatories, it limits the applicability of our results to the ISM in more quiescent galactic regions. This selection bias is likely less relevant for more luminous galaxies, in particular those at cosmological distances, which suffer from similar selection biases.
In a subsequent paper we will apply our approach to a larger sample of galaxies and regions for a statistically more significant sample, making use of the comprehensive set of available ancillary data. In addition toKINGFISH, other Herschel programssuch as the "Survey with Herschel of the ISM in Nearby Infrared Galaxies (SHINING)" (Sturm et al. 2011), and the "Dwarf Galaxy Survey (DGS)" (Madden et al. 2013) provide an excellent basis. Applying our model to a much larger galaxy sample will allow us to study the variations in gas physics and the resulting [C II] emission with higher statistical significance, leading to a thorough understanding of the origin of the [C II] emission across the universe.

Appendix Deriving Gas Properties with anAnalytical Calculation
A.1. The n e , q, and Z Parameters We investigate the parameter space of the electron density (n e ), metallicity (Z), and ionization parameter (q) using an analytical calculation. The electron density n e , estimated from the doublet optical line [S II]6717,6731 ratio, ranges from 100 to 1500 cm 3 . We also used the MIR[S III]18, 33 for an independent estimate following theequation from (Watson & Storey 1980). The values of n e as listed in Table 5are calculated by averaging the n e from both [S III] and [S II] lines. The densities derived from optical lines are in good agreement with the values found for extragalactic H II regions (Kennicutt 1984).
To calculate the ionization parameter (q), we follow the definition of Dopita et al. (2000), where p = ( ) q Q R n 4 H 2 e II , the ratio of ionizing photons per unit area per second over the number density of hydrogen atoms. We calculate the analytical H II region radius from q and n e derived from MAPPINGSIII modeling. Our calculation yields radii ranging from 2 to 8 pc, which is very small relative to the radius obtained from Gaussian fitting of the a H images. This is due to our simple model assuming of all ionizing radiation arising from a single spherical H II region, while in reality, our selected regions likely comprise several H II regions (like in 30 Dor) thattogether provide the measured a H profile. The discrepancy of analytical calculation and observational a H does not affect the [C II] fraction from H II regions. However, our estimation of G 0 and consequently the [C II] emission from dense PDRs will be affected, as discussed in Appendix A.2. At any rate, the apparent discrepancy does not affect our overall conclusion that dense PDRs arethe dominating [C II] contributor.
Since the metal abundances from Moustakas et al. (2010) are measured over a drift-scanned strip of galaxy, while we need the metallicity of each individual region, we decided to measure the metallicity in a different way. We used the two line ratios of II ratio required a certain assumption of the nitrogen/oxygen elemental abundance. We followed (Dopita et al. 2000) and assumed that nitrogen is a secondary nucleosynthesis element and does not scale linearly with increasing metallicity Z. We used the MAPPINGS 1 Z e metallicity set with C/H=2.57×10 −4 , N/H=6.03× 10 −4 , and =´-O H 4.57 10 4. For the 2 Z e metallicity we set =´-C H 9.10 10 4 , =´-N H 2.09 10 4 , and = O H -9.10 10 4.
We find that Z and q vary by factors 1.7 and 1.5, respectively, whenwe change n e by a factor of 3. A factor of 3 is taken to cover the range of density of extragalactic H II regions. These variations are less than the observational uncertainty in the line ratios. The derived metallicities for the four regions range from 1 Z e to 2 Z e , in agreement with the study by Moustakas et al. (2010). The ionization parameters derived from the [O II], [O III], and [N II] lines are in accordance with the best-fit MAPPINGS model described in Section 4.1. We tabulate the metallicities and ionization parameters in Table 5.

A.2. Deriving G 0
We calculate G 0 of theISM by three methods. The first method is by using the TIR luminosity. The basic assumption is thatall UV radiation is converted by dust and re-emitted in theinfrared. L TIR is calculated following Equation (2)  and G 0 is calculated following Equation (7), where L TIR is in erg s −1 , R H II is the radius of the H II region (or, more precisely, the radius of the a H emission region) in cm, and´-1.6 10 3 is the Habing field in units of -erg cm s 2 1 (Tielens 2010). We used the size of the H II region to convert the observed L TIR to the incident radiation field. We estimated the H II region radius via Gaussian fitting for regions with adistinct circular a H structure and careful by-eye inspection for regions where the fitting fails. We found that the a H emission is resolved for all of the regions. The radii of the H II regions range from 30 to 170 pc. The G 0 derived from L TIR is in the range 30-530 (Table 5). We note that for some regions L TIR is larger than L UV (6 regions in NGC 628 out of 7 regions). Older populations of stars may substantially contribute to the dust heating, as shown by Groves et al. (2012) andDraine et al. (2014, but not to the ionizing flux. Furthermore, we have made the simplifying assumption of spherical H II regions. Observations of Galactic as well as extragalactic H II regions amply illustrate that their PDR interfaces are highly corrugated, which can easily lead to an overestimation of the distance between the star clusters and the PDR surface.
As mentioned in Appendix A.2, the assumption of the geometry of the regions, i.e., the relative location of H II regions and dense PDRs, may affect our G 0 estimates. In our model we assumesingle H II regions with radius as listed in Table 5, and dense PDRs located adjacent to the H II region. Since we have no way ofknowing the actual morphology of the regions, we adopt the simplest model as describe in Sections 4.1 and 4.3. If the the H II regionsconsist of several H II regions thatare intermingled with the dense PDR, depending on the exact geometry, the G 0 is estimated to be higher than our current adopted value becausethe distance of H II regions and dense PDRs will be smaller. Increasing G 0 in turn will increase the [C II] fraction from dense PDRs.
The second method forestimatingthe incident radiation field emitted by the stellar cluster can be obtained from the ionizing photon luminosity as inferred from the extinctioncorrected b H flux thatis converted into NLyC and L UV . We have estimated the total stellar radiation field in the -6 13.6 eV range per ionizing photon using SB99 (Leitherer et al. 1999) assuming a continuous starburst for 10million years, and derived G 0 from the above-described H II region radii. We find that G 0 derived from L UV ranges between 8 and 180 (Table 5), which is two to threetimes smaller (on average) than the G 0 derived from L TIR . The third method is by using the dust model