First Results from the Herschel and ALMA Spectroscopic Surveys of the SMC: The Relationship between [C ii]-bright Gas and CO-bright Gas at Low Metallicity^*

The HS³ (Pilbratt et al. 2010) maps the key FIR lines of [C ii] 158 μm, [O i] 63 μm, [O iii] 88 μm, and [N ii] 122 μm with the PACS spectrometer (Poglitsch et al. 2010) and obtain Spectral and Photometric Imaging Receiver (SPIRE; Griffin et al. 2010) Fourier Transform Spectrometer (FTS) observations (which include [N ii] 205 μm) in five regions across the SMC with varying star formation activity and ISM conditions. These targets were covered using strips oriented to span the range from the predominantly molecular to the presumably atomic regime. The strips are fully sampled in [C ii] and [O i], while only a few pointings were observed for [N ii] and [O iii].

The HS³ targeted regions span a range of star formation activity, overlapping with the Spitzer Spectroscopic Survey of the SMC (S⁴MC; Sandstrom et al. 2012) whenever possible, and cover a range of "CO-faint" molecular gas fraction using dust-based molecular gas estimates (Bolatto et al. 2011) going from the peaks out to the more diffuse gas. The main survey covers five star-forming areas, which we refer to as "N83" (also includes N84), "SWBarN" (covers N27), "SWBarS" (covers N13), "N22" (also includes N25, N26, H36, and H35), and a smaller square region called "SWDarkPK" that covers a region with a dust-based peak in the molecular gas without any associated CO emission as seen in the NANTEN ${}^{12}\mathrm{CO}$ map (Mizuno et al. 2001). The "N" numbered regions refer to H ii regions from the catalog by Henize (1956), and the "H" numbered regions are from the catalog of Hα structures by Davies et al. (1976).

The [C ii] and [O i] maps are strips that encompass the peaks in CO, star formation, and "CO-faint" ${{\rm{H}}}_{2}$ as traced by dust. Using the PACS spectrometer $47^{\prime\prime} \times 47^{\prime\prime}$ field of view, the strips were sampled using rasters with sizes $33^{\prime\prime} \times 11$ ([C ii]) and $24^{\prime\prime} \times 15$ ([O i]) by $23\buildrel{\prime\prime}\over{.} 5\times 3$. Both [O iii] and [N ii] observations were targeted at the location of the main ionizing source in each region and sampled with $23\buildrel{\prime\prime}\over{.} 5\times 2$ by $23\buildrel{\prime\prime}\over{.} 5\times 2$ raster. The PACS maps used the unchopped scan mode with a common absolute reference position placed south of the SMC "Wing" and observed at least once every 2 hr. The PACS spectrometer has a beam FWHM of $\theta \sim 9\buildrel{\prime\prime}\over{.} 5$ at the wavelength for [O i] (63 μm) and [O iii] (88 μm), $\theta \sim 10^{\prime\prime}$ at [N ii] (122 μm), and $\theta \sim 12^{\prime\prime}$ at [C ii] (158 μm), which have corresponding spectral resolutions of ∼100, 120, 320, and 230 km s⁻¹ (Poglitsch et al. 2010). Tables 1 and 2 list the positions and uncertainties for all the PACS spectroscopy line images. The SPIRE FTS observations were "intermediate sampling" single pointing (with a $2^{\prime}$ circular field of view) at high resolution at the star-forming peak, which is typically close to the peak in ${}^{12}\mathrm{CO}$, for the N83, SWBarN, SWBarS, and N22 regions. In addition to the main survey region FTS observations, one single pointing covered the brightest H ii region N66, which has PACS [C ii] and [O i] observations as part of Guaranteed Time Key Project SHINING and is included in the Herschel Dwarf Galaxy Survey (DGS; Madden et al. 2013).

Table 1.  HS³ [C ii] and [O i] Map Properties

					$1\sigma$ Uncertainty
	Center Position				(10−9 W m−2 sr−1)
Region	R.A. (J2000)	Decl. (J2000)	Size	P.A.	[C ii]	[O i]
SWBarS	00h45m2710	−73d21m0000	17 × 61	10°	1.2	4.1
N22	00h47m5810	−73d16m5213	17 × 61	20°	1.4	3.5
SWBarN	00h48m2688	−73d06m0436	17 × 61	145°	1.5	3.9
SWDarkPK	00h52m2370	−73d14m4900	12 × 12	48°	1.3	3.4
N83	01h14m1928	−73d15m0904	17 × 80	30°	1.4	3.5

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Table 2.  HS³ [N ii] and [O iii] Map Properties

						1σ Uncertainty
	Center Position			P.A.		(10−9 W m−2 sr−1)
Region	R.A. (J2000)	Decl. (J2000)	Size	[N ii]	[O iii]	[N ii]	[O iii]
SWBarS	00h45m2185	−73d22m4936	15 × 15	75°	10°	0.34	2.8
N22	00h47m5435	−73d17m2769	15 × 15	65°	40°	0.36	2.8
SWBarN	00h48m2630	−73d06m0428	15 × 15	75°	55°	0.27	2.4
SWDarkPK	00h52m5611	−73d12m1725	1' × 1'	55°	40°	0.33	2.8
N83	01h14m0328	−73d17m0681	15 × 15	50°	65°	0.29	3.0

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2.1.1. Data Reduction

We mapped four regions in the Southwest Bar of the SMC in ${}^{12}\mathrm{CO}$, ${}^{13}\mathrm{CO}$, and ${{\rm{C}}}^{18}{\rm{O}}$ $(2-1)$ using Band 6 of the ALMA Atacama Compact Array (ACA; 7 m array consisting of 11 antennas) and Total Power array (TP; 12 m single-dish) during Cycle 2. Three of these regions were previously mapped in ${}^{12}\mathrm{CO}$ and ${}^{13}\mathrm{CO}$ $(2-1)$ using SEST (Rubio et al. 1993a, 1993b, 1996), but at a resolution of $22^{\prime\prime}$. The ACA maps were observed using a mosaic with 221 spacing of 47 pointings for the N22, SWBarS, and SWBarN regions and 52 pointings for the SWDarkPK with 25.5 s integration time per pointing. Both ${}^{12}\mathrm{CO}$ and ${}^{13}\mathrm{CO}$ were observed with 117.2 MHz (152 km s⁻¹) bandwidth and 121.15 kHz (0.2 km s⁻¹) spectral resolution. We chose a somewhat broader bandwidth for ${{\rm{C}}}^{18}{\rm{O}}$ of 468.8 MHz (642 km s⁻¹) and corresponding 0.24 MHz (0.12 km s⁻¹) spectral resolution, and we used the fourth spectral window for continuum (1875.0 MHz bandwidth, 7.81 MHz resolution). During the early ALMA cycles, the fast-mapping capabilities of the array were fairly limited, and so we decided to cover only half of the strips mapped by HS³. The coverage of the maps overlaps approximately with the main CO emission known to be present in the strips, except for SWDarkPK, where the PACS map is small and we covered its entire area.

We used the Common Astronomy Software Applications (CASA; McMullin et al. 2007) package to reduce, combine, and image the data. The ACA data were calibrated with the pipeline using CASA version 4.2.2, and no modifications were made to the calibration script. We used the calibrated delivered TP data, which were manually calibrated using CASA version 4.5.0 (described in the official CASA guide), with the exception of the SWBarN ${}^{12}\mathrm{CO}$ spectral window (SPW 17). For the SWBarN SPW 17 we modified the baseline subtraction in the calibration script to avoid channels with line emission (the delivered calibration script included all channels when fitting the baseline). We created the reduced measurement set using the script provided with the delivered ACA data and use the imaged TP SPWs as part of the delivered data.

We cleaned each spectral window of the ACA data and imaged it using clean, and then we used feather to combine the ACA images with the corresponding TP image regridded to match the ACA data (using CASA version 4.7.0). We used Briggs weighting with a robust parameter of 0.5 and cleaned to $\sim 2.5\times \mathrm{rms}$ found away from strong emission in the dirty data cube. As there was no noticeable continuum emission, the effect of continuum subtraction was negligible, and we did not include any continuum subtraction for the final imaged cubes. Due to the short integration times and the arrangement of the 7 m array, we used conservative masks for cleaning to reduce the effects of the poor u − v coverage of the ACA-only data. The data were imaged at 0.3 km s⁻¹ spectral resolution with synthesized beam sizes of $\sim 7^{\prime\prime} \times 5\buildrel{\prime\prime}\over{.} 5$ (2.1 pc × 1.7 pc) for ${}^{12}\mathrm{CO}$ $(2-1)$. The combination of the ACA and TP data make the observations sensitive to all spatial scales. The previously published single-dish SEST ${}^{12}\mathrm{CO}$ $(2-1)$ data from Rubio et al. (1993a) overlap with the SWBarN region, which they refer to as LIRS49. They found a peak temperature of 2.56 K in a 43'' beam at α(B1950) = 00^h46^m33^s, δ(B1950) =−73^d22^m00^s, and we find a peak temperature of 2.55 K in the same aperture in the SWBarN ACA+TP data when convolved to 43'' resolution. The positions, beam sizes, and sensitivities of the observations are listed in Tables 3 and 4.

Table 3.  ALMA ACA+TP Map Properties

	Map Center
Region	R.A. (J2000)	Decl. (J2000)	Map Size	P. A.
SWBarS	00h45m2454	−73d21m4263	23 × 35	10°
N22	00h47m5428	−73d17m4676	23 × 35	20°
SWBarN	00h48m1553	−73d04m5641	23 × 35	145°
SWDarkPK	00h52m5607	−73d12m1708	3' × 3'	48°

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The neutral atomic gas data come from 21 cm line observations of H i. We use the H i map from Stanimirović et al. (1999) that combined Australian Telescope Compact Array (ATCA) and Parkes 64 m radio telescope data. The interferometric ATCA data set the map resolution at $1\buildrel{\,\prime}\over{.} 6$ ($r\sim 30$ pc in the SMC), but the data are sensitive to all size scales owing to the combination of interferometric and single-dish data. The observed brightness temperature of the 21 cm line emission is converted to H i column density (${N}_{{\rm{H}}{\rm{I}}}$) assuming optically thin emission. The observed brightness temperature of the 21 cm line emission is converted to H i column density (${N}_{{\rm{H}}{\rm{I}}}$) using

$$\begin{eqnarray*}&&{N}_{{\rm{H}}{\rm{I}}}=1.823\times {10}^{18}\displaystyle \frac{{\mathrm{cm}}^{-2}}{{\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}}\int {T}_{B}(v)\,{dv}.\end{eqnarray*}$$

The SMC map has an rms column density of $5.0\times {10}^{19}$ cm⁻². While most of the H i emission is likely optical thin, some fraction will be optically thick, and the optically thin assumption will cause us to underestimate ${N}_{{\rm{H}}{\rm{I}}}$. Stanimirović et al. (1999) produced a statistical correction to account for optically thick H i line emission in the SMC; however, the correction is based only on 13 H i absorption measurements, with only two in the Southwest Bar. We choose not to apply the correction since it has little effect on our ${{\rm{H}}}_{2}$ estimate from [C ii] (see Section 4.2).

We use mid-infrared Spitzer IRAC and MIPS data from the SMC-SAGE (Gordon et al. 2011) and S³MC (Bolatto et al. 2007) surveys and spectroscopic IRS data, particularly ${{\rm{H}}}_{2}$ rotational lines, from the S⁴MC (Sandstrom et al. 2012) survey. The maps of the ${{\rm{H}}}_{2}$ rotational line images were produced by fitting and removing the baseline near the line and then calculating the total line intensity. We also use a velocity-resolved [C ii] spectrum from the GREAT heterodyne instrument (Heyminck et al. 2012) on board the Stratospheric Observatory for Infrared Astronomy (SOFIA; Temi et al. 2014) from the SMC survey presented in R. Herrera-Camus et al. (2017, in preparation).

Since the ALMA Survey focuses on the Southwest Bar of the SMC, there is no comparable map of CO from ALMA for N83. However, there are new APEX²¹ maps of ${}^{12}\mathrm{CO}$ $(2-1)$ that overlap the N83 HS³ region (PI: Rubio). We use the APEX data for the N83 region to be able to make similar comparisons to the HS3³ data, but note that the lower resolution ($\sim 25^{\prime\prime}$) limits the analysis. To do this, we take the additional step of convolving and re-gridding the Herschel spectroscopic maps ([O i], [C ii]) to match that of the APEX ${}^{12}\mathrm{CO}$ $(2-1)$ map.

We determine the total-infrared (TIR) intensity (from 3 to 1100 μm) using the Spitzer 24 and 70 μm (no Herschel 70 μm map exists) from SMC-SAGE (Gordon et al. 2011) combined with Herschel 100, 160, and 250 μm images from HERITAGE (Meixner et al. 2013). All of the images are convolved to the lowest resolution of the Spitzer 70 μm image ($\sim 18^{\prime\prime}$) using the convolution kernels from Aniano et al. (2011). The TIR intensity is calculated following the prescription by Galametz et al. (2013):

$$\begin{eqnarray}&&{S}_{\mathrm{TIR}}=\sum {c}_{i}{S}_{i},\end{eqnarray} \tag{ 1 }$$

all in units of W kpc⁻², where the coefficients (c_i) are 2.013, 0.508, 0.393, 0.599, and 0.680 for 24, 70, 100, 160, and 250 μm, respectively.

We investigate the structure of the PDR and molecular cloud by using the visual extinction (A_V) as an indicator of the total column through the cloud and to gauge the depth within the cloud associated with the observations. To match the high resolution of the [C ii], [O i], and ALMA CO data, we use the optical depth at 160 μm (${\tau }_{160}$) and the HERITAGE 160 μm map of the SMC (Meixner et al. 2013) as the basis for producing a map of A_V. Lee et al. (2015) fit a modified blackbody with $\beta =1.5$ to the SMC HERITAGE 100, 160, 250, and 350 μm data for the SMC. We re-sample their map of fitted dust temperatures at the lower resolution of the 350 μm Herschel map ($\sim 30^{\prime\prime}$) to the higher-resolution 160 μm map ($\sim 12^{\prime\prime}$) in order to estimate ${\tau }_{160}$ at a resolution comparable to the [C ii] and ALMA CO maps. We convert from ${\tau }_{160}$ to A_V using ${A}_{V}\sim 2200{\tau }_{160}$ from Lee et al. (2015), which is based on measurements in the Milky Way and provides similar A_V values to those found using UV/optical and NIR color excess methods (see Figure 1 in Lee et al. 2015). We stress that these values of A_V are estimates that include uncertainties associated with the assumptions made in the dust modeling (e.g., the assumption of a single dust temperature) and the conversion from ${\tau }_{160}$ to A_V. While the extinction at $\sim 11$ eV (ionization potential of carbon) would be a more relevant quantity to our study of [C ii] and CO emission, the conversion from ${\tau }_{160}$ to ${A}_{11\mathrm{eV}}$ is highly uncertain.

The [C ii] 158 μm line dominates the cooling of the warm ($T\sim 100$ K) neutral gas because of the high carbon abundance, its lower ionization potential of 11.26 eV, and an energy equivalent temperature of 92 K. Ionized carbon, ${{\rm{C}}}^{+}$, exists throughout most phases of the ISM except in the dense molecular gas, where most of the carbon is locked in CO. The [O i] 63 μm line also contributes to the gas cooling, but with an energy equivalent temperature of 228 K and a high critical density ($\sim {10}^{5}$ cm⁻³); this line dominates over the [C ii] emission only in the densest gas. Indeed, the bright "blue" knots of [O i] emission in Figure 1 are coincident with bright knots of Hα emission (in black contours) associated with very recent massive star formation in dense and presumably warm structures bathed by intense radiation. Oxygen can remain neutral in regions with ionized hydrogen and Hα emission owing to its slightly higher ionization potential of 13.62 eV. In warm PDRs subjected to radiation fields larger than 10³ Habings and due to the difference in critical densities, the [O i]-to-[C ii] ratio is a good indicator of density, but in colder gas and particularly below n ≲ 10⁴ cm⁻³ it is mostly sensitive to temperature and the incident radiation field (e.g., Kaufman et al. 1999).

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Figure 1 shows the [O i] and [C ii] integrated intensity images in combination with the 160 μm PACS image showing dust continuum emission. The differences in the local star formation, shown by Hα in black contours, produce different structures and varying intensities of [C ii], [O i], and dust emission. In many faint [C ii] regions in the diffuse gas the [O i] line is detected (see inset spectra in Figure 1). In principle, it is possible for [O i] 63 μm to be a very important coolant for the warm neutral medium (WNM) of the ISM (Wolfire et al. 1995, 2003). Despite the high critical density of this transition, the high temperature of the WNM excites [O i], making it an efficient coolant even in $n\sim 1$ cm⁻³ gas.

3.1.1. [O i] Self-absorption

3.1.2. [O i]-to-[C ii] Ratio

What is the origin of the observed [C ii] emission? Figure 2 shows the integrated intensity ratio of [O i] to [C ii]. The observed ratio is approximately constant, with a value of [O i]/[C ii] ∼ 0.3. One of the main departures from the mostly flat trend in [O i]/[C ii] with ${A}_{V}$ is a cluster of higher ratio values in the SWBarS, which are found in the H ii region N13, indicating the presence of warm, dense gas. This is also the typical value observed in the disks of the KINGFISH sample of nearby galaxies (Herrera-Camus et al. 2015). Using the [C ii] and [O i] cooling curves calculated for diffuse gas under SMC conditions (M. Wolfire et al. 2017, in preparation), a ratio [O i]/[C ii] ∼ 0.3 is indicative of densities of $\sim {10}^{2}\mbox{--}{10}^{3}$ cm⁻³, which are consistent with dense cold neutral medium (CNM) and/or molecular gas. Figure 3 shows the similarity between the mid-infrared ${{\rm{H}}}_{2}$ S(0) quadrupole rotational line at 28.8 μm and the [C ii] emission. This clearly demonstrates that molecular gas is associated with the [C ii]-emitting material, strongly suggesting that most of the [C ii] emission in our mapped regions has a PDR origin and arises from the surfaces of molecular clouds.

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3.1.3. Photoelectric Heating Efficiency at Low Metallicity

The dominant heating source is the photoelectric effect where a dust grain absorbs an FUV photon and ejects an electron that heats the gas through collisions. The [C ii] and [O i] FIR line emission dominates the cooling of the diffuse atomic and molecular gas, as well as PDRs. By combining the [C ii] and [O i] line emission, we can account for most of the gas cooling that is attributed to gas heated by the photoelectric effect. Taking the ratio of [C ii] and [O i] intensities to the TIR intensity indicates the fraction of the power absorbed by the grains that goes into heating the gas through the photoelectric effect. Figure 4 shows the mean [C ii]+[O i]/TIR ratios for each of the HS³ regions as a function of the mean dust temperature from Lee et al. (2015). The ratio of [C ii]+[O i]/TIR ranges from ∼0.01 to 0.018 in the SMC. This is similar to the LMC, where Rubin et al. (2009) find that the [C ii] in emission from BICE observations accounts for $\sim 1 \%$ of the TIR emission. These ratios are on the high end of the range observed for the Milky Way and galactic nuclei of 0.1%–1% using KAO observations (Stacey et al. 1991), $\lt 0.1 \% \mbox{--}1 \%$ for the nearby KINGFISH galaxies using Herschel observations (Smith et al. 2017), normal galaxies using ISO observations (Malhotra et al. 2001), and M31 using Herschel observations (Kapala et al. 2015). We also see a trend of decreasing ratios with high dust temperatures, also observed by Malhotra et al. (2001), Croxall et al. (2012), and Kapala et al. (2015), which is commonly attributed to the increased grain charging at warmer dust temperatures that increases the energy threshold for the photoelectric effect ejection of electrons and decreases the energy, and therefore the amount of gas heating, per ejected electron.

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The HS³ data set includes sparsely sampled FTS spectra for the SWBarN, SWBarS, N22, and N83 regions, as well as a pointing toward the most active star-forming region in the SMC, the giant H ii region N66. In Figure 5 we show the long-wavelength array (SLW) spectra in red and the short-wavelength array (SSL) in blue, with the positions of the ${}^{12}\mathrm{CO}$, [C i], and [N ii] lines indicated. We see clear detections of the lower rotational transitions of ${}^{12}\mathrm{CO}$ and the [N ii] 205 μm lines, as well as weak detections of [C i].

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We do not detect the [N ii] 122 μm line in any of the regions, whereas the [N ii] 205 μm line is detected in all regions in the FTS spectra (see Figure 5). Because the ionization potential of nitrogen of 14.5 eV is higher than that of hydrogen, ionized nitrogen traces the ionized gas. The [N ii] 122 and 205 μm lines result from the fine-structure splitting of the ground state of ionized nitrogen and are primarily excited by collisions with electrons.

The critical densities of the 122 and 205 μm lines differ, and the ratio can be used to estimate the electron density (${n}_{{\rm{e}}}$). The [N ii] 122 μm line has a higher critical density for collisions with electrons (${n}_{{\rm{e}}}\sim 300$ cm⁻³) compared to the 205 μm line (${n}_{{\rm{e}}}\sim 40$ cm⁻³), which has a critical density similar to that for exciting the [C ii] 158 μm line with collisions with electrons. Thus the ratio of [C ii]/[N ii] 205 μm in ionized gas is independent of density, and it depends only on the relative abundances of the ions, which are likely similar to the elemental abundances.

The sensitivity of the [N ii] 122 μm observations is $\sim 3\times {10}^{-10}$ W m⁻² sr⁻¹, while the range of detected [N ii] 205 μm intensities is $\sim (2\mbox{--}5)\times {10}^{-10}$ W m⁻² sr⁻¹. Based on these measurements, the [N ii] 122 μm/205 μm ratio is ≲1 in regions where the [N ii] 205 μm line is detected. Using the electron collision strengths from Tayal (2011), this translates to an upper limit to the electron density of ${n}_{e}\lesssim 20$ cm⁻³ since this is approximately the [N ii] 122 μm/205 μm ratio diagnostic lower density limit. In other words, our measurements are consistent with a relatively low density for the ionized material, somewhat lower than the mean ionized gas density observed in the KINGFISH sample of galaxy disks of ${n}_{e}\sim 30$ cm⁻³ (Herrera-Camus et al. 2016).

In Figure 6, we show the [N ii] 205 μm/[C ii] ratio for all of the pointings where [N ii] 205 μm is detected at $\gt 3\sigma$ and where the [C ii] intensity is found for the central FTS bolometer position after convolving the map to the FTS resolution ($\sim 17^{\prime\prime}$). We see that the [N ii] 205 μm emission ranges from 0.2% to 1.2% of the [C ii] emission. For carbon emission arising from ionized gas with a ratio ${{\rm{C}}}^{+}/{{\rm{N}}}^{+}\approx \ {\rm{C}}/{\rm{N}}$ similar to Galactic we would expect [N ii] 205 μm/[C ii] ∼ 0.2, mostly independent of density owing to the similarity in the critical densities (Tayal 2008, 2011). Because the [N ii] emission can only arise from ionized gas, the fact that we measure over ∼20 times fainter [N ii] relative to [C ii] suggests that the contribution of ionized gas to [C ii] is at most 5%.

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These ratios represent the maximum ratios throughout the regions since the FTS observations targeted the bright CO emission, which tends to be near H ii regions, and as such will have the highest fraction of [C ii] emission arising from ionized gas. The observed [N ii] 205 μm/[C ii] ratios are lower than the typical values found in the KINGFISH survey of 0.057 (Croxall et al. 2017). Similarly, Cormier et al. (2015) find depressed [N ii] 122 μm/[C ii] in dwarf galaxies, suggesting that ionized gas only produces a small fraction of the [C ii] emission in low-metallicity environments.

The [O iii] 88 μm line traces ionized gas, as the second ionization potential of oxygen is ∼35 eV, much higher than hydrogen. H³S obtained [O iii] observations toward the dominant H ii region in each strip. We observed bright [O iii] emission from all of these pointings. Van Loon et al. (2010) also detected [O iii] toward a subset of their sample of compact sources in the SMC using ISO spectra. Figure 7 shows the [O iii]/[C ii] ratios for the SMC regions, which reach as high as [O iii]/[C ii] ∼ 4. Observations of the [O iii]/[C ii] ratio for higher-metallicity galaxies based on ISO data presented by Brauher et al. (2008) found lower ratios in the range of [O iii]/[C ii] ∼ 0.1–1.5. The higher values in the SMC are similar to the ratios found for dwarf galaxies observed with Herschel PACS as part of the DGS with a median and range of $[{\rm{O}}\,{\rm{III}}]/[{\rm{C}}\,{\rm{II}}]={2.0}_{-0.52}^{+13.0}$ (Cormier et al. 2015). These comparisons to measurements in other galaxies should be tempered somewhat by the fact that the SMC observations are pointings toward H ii regions obtained at high spatial resolution, while the comparison work typically samples larger scales and therefore a mix of ionized and neutral material.

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While the critical density for the [N ii] 122 μm line is similar to that for [O iii], the [O iii] ionization potential is much higher, as is the energy above ground for excitation. Cormier et al. (2015) suggest that the hard radiation fields found at lower metallicity in dwarf galaxies could explain the high [O iii] emission and high [O iii]-to-[N ii] 122 μm ratios, with a median ratio of 86 found for the DGS sample. Note that we failed to detect [N ii] 122 μm emission toward these same pointings. A combination of hard radiation fields and low-density ionized gas may explain the high [O iii]-to-[N ii] 122 μm ratios present in the SMC, with the a lower limit of [O iii]/[N ii]122 μm ∼ 25 (calculated using $3\sigma$ of the [O iii] intensity and the $1\sigma$ sensitivity of the [N ii] intensity). We note that the O/N abundance ratio in the SMC is similar to that found in the solar neighborhood (Russell & Dopita 1992), and it is unlikely that a difference in abundance ratios would explain the relatively high [O iii] emission. Understanding the ISM conditions that produce the [O iii] line emission in low-metallicity environments is critical for interpreting new and future observations of FIR cooling lines in high-redshift galaxies using ALMA (e.g., Inoue et al. 2016).

We mapped and detected ${}^{12}\mathrm{CO}$ and ${}^{13}\mathrm{CO}$ $(2-1)$ in all of the regions targeted by ALMA, shown in Figure 8. Table 5 summarizes the properties of the detected CO and the sensitivities of the integrated intensity maps. We do not detect ${{\rm{C}}}^{18}{\rm{O}}$ with our current observations, which had the minimum integration time allowed per pointing to increase the coverage of the mosaics. Despite the lower metallicity and less dust shielding, ${}^{12}\mathrm{CO}$ and ${}^{13}\mathrm{CO}$ form and emit brightly in small clumps. Our high-resolution ALMA ACA data show that the bright CO emission is found in small structures, which we will quantify in a forthcoming paper (K. Jameson et al. 2017, in preparation), which were unresolved by previous observations. The clumpy nature of the CO emission at low metallicity has also been observed in the N83C star-forming region in the wing of the SMC (Muraoka et al. 2017), in the dwarf galaxies WLM (Rubio et al. 2015) and NGC 6822 (Schruba et al. 2017), and in 30 Doradus of the LMC (Indebetouw et al. 2013).

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For most of the regions, a large fraction of the flux is recovered by the high-resolution ACA maps. In the SWBarS and SWBarN regions, the ACA flux represents ∼60% of the flux in the combined ACA+TP ${}^{12}\mathrm{CO}$ maps. In SWDarkPK, nearly 100% of the flux is from the high-resolution imaging, whereas in the N22 region most of the emission is diffuse, with only ∼30% of the flux found at high resolution. The higher fraction of diffuse ${}^{12}\mathrm{CO}$ emission in N22, less in SWBarS and SWBarN, and none in SWDarkPK are likely due to their varying evolutionary stages: N22 is the most evolved region, with multiple large H ii regions around the CO emission; SWBarS and SWBarN both are actively forming stars and have one prominent H ii region; and SWDarkPK has no signs of active star formation. The higher UV fields likely increase how deep the PDR extends into the molecular cloud and the amount of diffuse CO emission associated with the PDR. In the more evolved regions (particularly N22), the densest peaks of molecular gas may have already been dispersed by star formation, leaving less clumps of molecular gas and increasing the fraction of diffuse CO emission.

Figure 9 shows the comparison between the ${}^{12}\mathrm{CO}$ and ${}^{13}\mathrm{CO}$ $(2-1)$ emission. We find average ${}^{12}\mathrm{CO}$/${}^{13}\mathrm{CO}$ $(2-1)$ ratios of ∼5–7.5 (in units of K km s⁻¹). These ratios are consistent with previous measurements in the SMC toward emission peaks (Israel et al. 2003) and in nearby galaxies (e.g., Paglione et al. 2001; Krips et al. 2010). In the Milky Way the ratios are similar to what we obtain for the SMC, with an average of ∼5 (Solomon et al. 1979) in the inner Galaxy and somewhat higher ratios of ∼7 for large parts of the plane (Polk et al. 1988) and in the outer Galaxy clouds (Brand & Wouterloot 1995).

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The linear trend with no turnover observed between ${}^{12}\mathrm{CO}$ and ${}^{13}\mathrm{CO}$ indicates that while the ${}^{12}\mathrm{CO}$ $(2-1)$ transition is optically thick where there is ${}^{13}\mathrm{CO}$, ${}^{13}\mathrm{CO}$ likely remains optically thin for these observations. The ${}^{13}\mathrm{CO}$-to-${}^{12}\mathrm{CO}$ ratio gives the optical depth of ${}^{13}\mathrm{CO}$ $(2-1)$, and from that we can estimate the optical depth of the ${}^{12}\mathrm{CO}$ $(2-1)$ emission. The Rayleigh–Jeans radiation temperature of the CO line emission is

$$\begin{eqnarray}&&{T}_{R}={J}_{R}({T}_{\mathrm{ex}})(1-{e}^{-\tau }),\end{eqnarray} \tag{ 2 }$$

where T_ex is the excitation temperature and τ is the optical depth of the line. The observed intensity is

$$\begin{eqnarray}&&{J}_{R}({T}_{\mathrm{ex}})=\displaystyle \frac{h\nu }{k}\left(\displaystyle \frac{1}{{e}^{(h\nu /{{kT}}_{\mathrm{ex}})}-1}-\displaystyle \frac{1}{{e}^{(h\nu /{{kT}}_{\mathrm{bg}})}-1}\right),\end{eqnarray} \tag{ 3 }$$

where T_bg is the background temperature (taken to be the cosmic microwave background of 2.73 K). If we assume that both lines share the same excitation temperature, then the ratio of the line brightness temperatures for the two isotopic species can be used to estimate the optical depth of the more abundant species. This is strictly correct only in the high-density regime ($n\gg {n}_{\mathrm{cr}}$, where ${n}_{\mathrm{cr}}\sim {10}^{4}$ cm⁻³ is the critical density of the ${}^{13}\mathrm{CO}$ 2 − 1 transition that is assumed to be optically thin), where the level populations will follow a Boltzmann distribution at the kinetic temperature of the gas. For ${}^{12}\mathrm{CO}$ and ${}^{13}\mathrm{CO}$ $(2-1)$,

$$\begin{eqnarray}&&\displaystyle \frac{{T}_{R{,}^{12}\mathrm{CO}(2-1)}}{{T}_{R{,}^{13}\mathrm{CO}(2-1)}}=\displaystyle \frac{1-{e}^{-{\tau }_{{}^{12}\mathrm{CO}}}}{1-{e}^{-{\tau }_{{}^{12}\mathrm{CO}}/X}},\end{eqnarray} \tag{ 4 }$$

where X is the abundance ratio of ${}^{12}\mathrm{CO}$/${}^{13}\mathrm{CO}$. Assuming that ${\tau }_{{}^{12}\mathrm{CO}}\gg 1$, we can solve for the optical depth of ${}^{12}\mathrm{CO}$:

$$\begin{eqnarray}&&{\tau }_{{}^{12}\mathrm{CO}}=-X\mathrm{ln}(1-{T}_{R{,}^{13}\mathrm{CO}(2-1)}/{T}_{R{,}^{12}\mathrm{CO}(2-1)}).\end{eqnarray} \tag{ 5 }$$

We adopt an abundance ratio of X = 70, which is appropriate for the Milky Way (Wilson & Rood 1994), since it is intermediate between the two values found in the SMC using radiative transfer modeling of the ${}^{12}\mathrm{CO}$ and ${}^{13}\mathrm{CO}$ lines from Nikolić et al. (2007). Using this isotopic abundance ratio, the average ${}^{12}\mathrm{CO}$/${}^{13}\mathrm{CO}$ values (where ${I}_{{}^{13}\mathrm{CO}}\gt 3\sigma$) indicate optical depths of ${\tau }_{{}^{12}\mathrm{CO}}=7.8$, 13.8, 12.5, and 19.0 for the SWBarN, SWBarS, N22, and SWDarkPK regions, respectively. In practice, ${}^{12}\mathrm{CO}$ is likely to be more highly excited than ${}^{13}\mathrm{CO}$ owing to radiative trapping, which would result in smaller ${}^{13}\mathrm{CO}$/${}^{12}\mathrm{CO}$ ratios and somewhat underestimate the optical depth by this method. At these size scales ($\sim 2$ pc), the beam may include high- and low-density gas, with the low-density gas more likely to have more excited ${}^{12}\mathrm{CO}$ and also to decrease the ${}^{13}\mathrm{CO}$/${}^{12}\mathrm{CO}$ ratio. However, the density in the [C ii]-emitting gas (see Section 4.3.2 and Figure 14) is already high, and there is unlikely to be a strong contribution from ${}^{12}\mathrm{CO}$-emitting low-density gas.

The ${}^{12}\mathrm{CO}$ emission is likely to be optically thick under most conditions found in a molecular cloud, and indeed we estimate high average optical depths of the ${}^{12}\mathrm{CO}$ $(2-1)$ line emission. Assuming, however, that ${}^{13}\mathrm{CO}$ is optically thin is reasonable throughout much of a cloud, given the high isotopic ratio and the lower C abundance in the SMC. We can then use the ${}^{13}\mathrm{CO}$ intensity to estimate a molecular gas column density and from that infer a ${}^{12}\mathrm{CO}$-to-${{\rm{H}}}_{2}$ conversion factor (${X}_{\mathrm{CO}}$).

We use Equations (2) and (3) and the assumption that the excitation temperature is the same for both the ${}^{12}\mathrm{CO}$ and ${}^{13}\mathrm{CO}$ emission to calculate the optical depth of the ${}^{13}\mathrm{CO}$ emission. The total column density of ${}^{13}\mathrm{CO}$ (${N}_{{13}_{\mathrm{CO}}}$) as a function of the excitation temperature (T_ex) and optical depth of ${}^{13}\mathrm{CO}$ (${\tau }_{{}^{13}\mathrm{CO}}$) for the $J=2\to 1$ transition is given by (Garden et al. 1991; Bourke et al. 1997)

$$\begin{eqnarray}&&{N}_{{}^{13}\mathrm{CO}}=1.12\times {10}^{14}\displaystyle \frac{({T}_{\mathrm{ex}}+0.88){e}^{5.29/{T}_{\mathrm{ex}}}\int {\tau }_{{}^{13}\mathrm{CO}}{dv}}{1-{e}^{-10.6/{T}_{\mathrm{ex}}}}.\end{eqnarray} \tag{ 6 }$$

We then make the approximation that the integral of ${\tau }_{{}^{13}\mathrm{CO}}$ is taken to be the line center optical depth (${\tau }_{{}^{13}\mathrm{CO},0}$) multiplied by the ${}^{13}\mathrm{CO}$ FWHM (${\rm{\Delta }}{v}_{{}^{13}\mathrm{CO}}$) (Dickman 1978). To convert from ${N}_{{13}_{\mathrm{CO}}}$ to ${N}_{{{\rm{H}}}_{2}}$, we scale by the ${}^{12}\mathrm{CO}{/}^{13}\mathrm{CO}$ abundance of 70 (Nikolić et al. 2007) and the ${}^{12}{\rm{C}}/{\rm{H}}$ abundance of $2.8\times {10}^{-5}$ (for a justification see Section 4). This results in ${{\rm{H}}}_{2}{/}^{13}\mathrm{CO}\,=1.25\times {10}^{6}$.

Figure 10 shows the distribution of ${X}_{\mathrm{CO}}$ for lines of sight with ${{\rm{I}}}_{{}^{13}\mathrm{CO}}\gt 3\sigma$ and ${A}_{V}\gt 1$, where we expect the molecular gas to be primarily traced by ${}^{12}\mathrm{CO}$ emission. We estimate the ${A}_{V}\gt 1$ by converting ${N}_{{13}_{\mathrm{CO}}}$ to ${A}_{V}$ using the empirically determined relationship ${A}_{V}\approx 4\times {10}^{-16}{N}_{{}^{13}\mathrm{CO}}$ cm mag⁻¹ from Dickman (1978) for Milky Way dark clouds, which should be appropriate if the ${}^{13}\mathrm{CO}$ abundance scales the same as the dust abundance. The distributions have median values of $(1.3\mbox{--}2.3)\times {10}^{20}$ cm⁻² (K km s⁻¹)⁻¹, which are consistent with a Milky Way ${X}_{\mathrm{CO}}$.

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Estimating ${X}_{\mathrm{CO}}$ using ${}^{13}\mathrm{CO}$ involves a number of assumptions that are not well constrained. We assume that the ${}^{13}\mathrm{CO}$ and ${}^{12}\mathrm{CO}$ lines share the same excitation temperature, that this temperature can be inferred from the brightness of the ${}^{12}\mathrm{CO}$ emission at our spatial resolution (typically ∼2.3 pc; see Table 4), and that the gas is in LTE. We do not account for potential beam dilution, which could decrease our measurements of T_ex from ${}^{12}\mathrm{CO}$ and cause us to overestimate ${\tau }_{{}^{13}\mathrm{CO}}$ and ${N}_{{13}_{\mathrm{CO}}}$ and overestimate ${X}_{\mathrm{CO}}$. Wong et al. (2017) make similar estimates of the molecular gas mass using ${}^{12}\mathrm{CO}$ and ${}^{13}\mathrm{CO}$ ALMA observations of molecular clouds in the LMC, which suffer from similar uncertainties, and find evidence that their ${N}_{{{\rm{H}}}_{2}}$ estimates were biased toward higher values. Conversely, the assumption of LTE could potentially cause ${N}_{{13}_{\mathrm{CO}}}$ to be underestimated by a factor of ∼1.3–2.5 based on the comparison to simulations of molecular clouds by Padoan et al. (2000) for the typical ${N}_{{13}_{\mathrm{CO}}}$ in our regions ($\sim {10}^{15}$ cm⁻² ). Both the ${}^{12}\mathrm{CO}{/}^{13}\mathrm{CO}$ and ${}^{12}{\rm{C}}/{\rm{H}}$, which we assume also traces ${}^{12}\mathrm{CO}/{\rm{H}}$, are uncertain in the SMC. The existing ${}^{12}{\rm{C}}/{\rm{H}}$ measurements vary by ±30%. Nikolić et al. (2007) constrain the ${}^{12}\mathrm{CO}{/}^{13}\mathrm{CO}$ abundance ratio to $\pm \sim 30 \%$, but recent radiative transfer modeling by Requena-Torres et al. (2016) finds ${}^{12}\mathrm{CO}{/}^{13}\mathrm{CO}=50$, which is at the lower end of the range found by Nikolić et al. (2007). If we are overestimating the abundance of ${}^{13}\mathrm{CO}$, which could be by up to a factor of 2 based on the metallicity scaling of the Milky Way measurement of ${{\rm{H}}}_{2}{/}^{13}\mathrm{CO}$ from Dickman (1978), then we would also underestimate ${X}_{\mathrm{CO}}$ by the same amount. It is unclear how all of these uncertainties would balance out in these specific regions. Ultimately, the data are consistent with a conversion factor similar to that of the Milky Way, but we caution the reader that there is a large degree of uncertainty on this estimate.

Table 4.  ALMA ACA+TP Map Properties (continued)

	${\theta }_{\mathrm{maj}}(^{\prime\prime} )\times {\theta }_{\min }(^{\prime\prime} )$			rms (K)
Region	${}^{12}\mathrm{CO}$	${}^{13}\mathrm{CO}$	${{\rm{C}}}^{18}{\rm{O}}$	${}^{12}\mathrm{CO}$	${}^{13}\mathrm{CO}$	${{\rm{C}}}^{18}{\rm{O}}$
SWBarS	6.60 × 6.06	6.92 × 6.05	6.76 × 6.18	0.17	0.16	0.10
N22	6.39 × 5.56	6.59 × 5.83	7.01 × 6.41	0.24	0.22	0.13
SWBarN	7.08 × 5.74	7.25 × 5.33	7.40 × 5.66	0.12	0.16	0.08
SWDarkPK	6.95 × 5.55	8.39 × 5.63	7.42 × 5.72	0.18	0.16	0.10

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Table 5.  ALMA CO Properties

		Tpeaka (K)		Map Sensitivityb (K km s−1)
Region	${\rm{\Delta }}{v}_{\mathrm{LSR}}$ (km s−1)	${}^{12}\mathrm{CO}$	${}^{13}\mathrm{CO}$	${}^{12}\mathrm{CO}$	${}^{13}\mathrm{CO}$	${{\rm{C}}}^{18}{\rm{O}}$
SWBarS	104–133	18.6	4.9	2.1	1.7	1.2
N22	113–133	15.9	2.6	2.1	2.4	1.0
SWBarN	104–140	14.6	4.3	1.8	2.2	1.2
SWDarkPK	137–160	12.1	3.0	1.9	1.8	1.3

Notes.

^aIn 0.3 km s⁻¹ channels. ^bDefined as $3\times \mathrm{rms}$ in regions away from line emission.

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[C ii] emission can arise from neutral atomic gas, molecular gas, and ionized gas. Our comparison of [C ii] to [N ii] shows that only a small fraction of the [C ii] emission arises from ionized gas. While [C ii] emission is an important coolant of the CNM (Dalgarno & Black 1976; we explore this further in Section 4.2), PDRs are expected to produce bright [C ii] emission (Hollenbach & Tielens 1999, and references therein). The carbon will exist as CO in the dense parts of the molecular gas, but at the edges of the cloud the lack of shielding will dissociate CO and much of the carbon will exist as ${{\rm{C}}}^{+}$, while there can still be molecular (${{\rm{H}}}_{2}$) gas (e.g., Israel et al. 1996; Bolatto et al. 1999; Wolfire et al. 2010). This leads to a layer of molecular gas surrounding the molecular cloud associated with [C ii] but no ${}^{12}\mathrm{CO}$ emission. Studying the ${}^{12}\mathrm{CO}$-to-[C ii] ratio can determine whether the [C ii] emission is primarily associated with the PDR and provide insight into the structure of the molecular cloud and its associated faint CO gas.

What is the origin of the [C ii] emission in our SMC regions? Figure 11 shows the [C ii] integrated intensity maps, with the ALMA ${}^{12}\mathrm{CO}$ $(2-1)$ integrated intensity shown with contours. There is a striking similarity between the structures, at both high and low intensity, but the [C ii] extends throughout the mapped region, whereas the CO emission is more localized. The similarity in integrated intensity emission structure suggests that the [C ii] emission predominantly traces the molecular gas. This is reinforced by the velocity-resolved observations. Recent observations of the [C ii] 158 μm line using the GREAT instrument on SOFIA for individual pointings throughout some of the HS³ regions show that the velocity ranges covered by the [C ii] and ${}^{12}\mathrm{CO}$ $(2-1)$ emission are similar, although in an FWHM sense the [C ii] line can be up to ∼50% wider than the ${}^{12}\mathrm{CO}$ line (R. Herrera-Camus et al. 2017, in preparation). Other SOFIA GREAT observations of SMC star-forming regions N66, N25/26 (covered in the HS³ "N22" region), and N88 show that the [C ii] profile is similar to the ${}^{12}\mathrm{CO}$ profile, with the [C ii] profile being up to ∼50% wider (Requena-Torres et al. 2016). SOFIA GREAT observations of [C ii] in M101 and NGC 6946 also show that the velocity profile of [C ii] has a better correspondence with ${}^{12}\mathrm{CO}$ than H i (de Blok et al. 2016). We show one example GREAT [C ii] spectrum for one pointing in the SWBarN region compared with the beam-matched ALMA ${}^{12}\mathrm{CO}$ $(2-1)$ profile and the H i emission (at a resolution $\sim 1^{\prime}$) toward the same point in Figure 12. As mentioned above, the CO and [C ii] cover a very similar velocity range, which is much more restricted than that of the H i emission. This strongly suggests that toward these molecular regions [C ii] is dominated by emission arising from molecular gas rather than atomic gas.

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Furthermore, there is a clear similarity between the [C ii] map and the map of the ${{\rm{H}}}_{2}$ rotational line emission at 28.2 μm (see Figure 3). This line requires warm gas ($T\gtrsim 100$ K) to be excited, so it is a poor tracer of the bulk of the molecular mass. But the spatial coincidence between these two transitions, together with the [C ii]–${}^{12}\mathrm{CO}$ agreement above, provides strong evidence that a majority of the [C ii] emission arises from molecular gas.

In Figure 13 we show the ratio of ${}^{12}\mathrm{CO}$/[C ii] emission as a function of A_V estimated from dust emission (see Section 2.6), which indicates the total column density through the molecular cloud along the line of sight. The ratios are typically much lower at low A_V and increase with A_V. At higher A_V and deeper within the cloud, there is enough shielding from FUV radiation for significant amounts of ${}^{12}\mathrm{CO}$ to exist, with an increasing fraction of the carbon in the form of CO as opposed to C or ${{\rm{C}}}^{+}$. The values of this ratio in the outskirts of the clouds are typically $\sim 1/5$ the fiducial value for the Milky Way of ${}^{12}\mathrm{CO}$/[C ii] ∼ 1/4400 (Stacey et al. 1991), which is consistent with the recent global measurements at low metallicity from Accurso et al. (2017). Toward the centers of the clouds, at high A_V, the ratio tends to reach the Milky Way value or higher, as in the case of the SWBarN and SWBarS regions, which show compact, bright peaks in the ${}^{12}\mathrm{CO}$ and ${}^{13}\mathrm{CO}$ emission. The lower ${}^{12}\mathrm{CO}$-to-[C ii] ratios at lower A_V suggest that there is a reservoir of ${{\rm{H}}}_{2}$ gas that is not traced by bright CO emission.

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The low range of A_V in the SWDarkPK region and the high ${}^{12}\mathrm{CO}$-to-[C ii] values seen at low A_V (e.g., in the SWBarS region) are probably in part due to the geometry of the sources, as well as physical differences in density and radiation field (see Section 4.3.2). Limitations in how we estimate A_V likely also play a role. In order to derive a dust temperature from the dust SED, we need to include the longer-wavelength data, which have poorer resolution. To obtain our A_V estimate, we have thus applied the dust temperature fit on larger ($\sim 40^{\prime\prime}$) scales, which smooth out any dust temperature variations on small scales, particularly those likely present toward smaller CO clouds or cores. Associated with this, the simple one-dust-temperature modified blackbody fit is also biased toward the higher dust temperatures that will dominate the emission. A high dust temperature results in underestimating ${\tau }_{160}$ and A_V. This combination of inability to resolve small structures and bias toward underestimating A_V should be taken into account when interpreting Figure 13.

We estimated a very low contribution to the total [C ii] emission from ionized gas based on our [N ii] 205 μm measurements (Section 3.2). Here we also estimate the possible contribution using the narrowband Hα data for the SMC to estimate the electron volume density (${n}_{{\rm{e}}}$) and column density (${N}_{{\rm{e}}}$) and assuming an ionized gas temperature to calculate ${I}_{[{\rm{C}}{\rm{}}\mathrm{ii}]}$ using Equation (9). The Hα observations provide the emission measure (EM), which can be used to estimate ${n}_{{\rm{e}}}$:

$$\begin{eqnarray}&&\mathrm{EM}=\int {n}_{{\rm{e}}}^{2}{dl}=2.75{\left(\displaystyle \frac{T}{{10}^{4}{\rm{K}}}\right)}^{0.9}\left(\displaystyle \frac{{I}_{{\rm{H}}\alpha }}{1R}\right)\mathrm{pc}\ {\mathrm{cm}}^{-6},\end{eqnarray} \tag{ 10 }$$

which we then solve for ${n}_{{\rm{e}}}$ and get

$$\begin{eqnarray}&&{n}_{{\rm{e}}}={\left(\displaystyle \frac{2.75}{l}{\left(\displaystyle \frac{T}{{10}^{4}{\rm{K}}}\right)}^{0.9}\left(\displaystyle \frac{{I}_{{\rm{H}}\alpha }}{1R}\right)\right)}^{1/2}{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 11 }$$

where ${I}_{{\rm{H}}\alpha }$ is the Hα intensity (1R = ${10}^{6}\,{(4\pi )}^{-1}$ photons cm⁻² s⁻¹ sr⁻¹ = 2.409 × 10⁻⁷ erg s⁻¹ cm⁻² sr⁻¹ at $\lambda =6563$ Å), T is the electron temperature, and l is the length over which n_e² is integrated (Reynolds 1991). This procedure is uncertain and known to be biased toward overestimating n_e since the highest densities dominate the EM, but it provides an independent check on the conclusions in Section 3.2, and it also gives us the ability to correct the [C ii] maps at higher resolution.

We assume ${T}_{{\rm{e}}}=8000$ K, appropriate for Galactic H ii regions and the warm ionized medium (WIM). While the temperature of H ii regions is a strong function of metallicity and is likely higher in the SMC (Kurt & Dufour [1998] measures ${T}_{{\rm{e}}}\approx \mathrm{12,000}$ K in N66), Goldsmith et al. (2012) calculate the [C ii] critical density for collisions with electrons only up to 8000 K and report ${n}_{\mathrm{crit},{\rm{e}}}=44$ cm⁻³. The [C ii] emission from ionized gas is not expected to dominate the total [C ii] emission (see Section 3.2), so the effect of assuming a lower temperature will be minimal. The most uncertain quantities are the assumed lengths: the distance l over which ${n}_{{\rm{e}}}^{2}$ is integrated, and the distance used to convert the volume density to a column density. For simplicity, we assume both to be 20 pc, which is approximately the diameter of the largest H ii region (N22) found within the survey regions and likely to be the size scale that produces the majority of the measured integrated ${n}_{{\rm{e}}}^{2}$. The ${n}_{{\rm{e}}}$ we estimate from the Hα emission is ∼20–25 cm⁻³, which is consistent with the upper limit on ${n}_{{\rm{e}}}$ determined from the [N ii] 122 μm/205 μm values (see Section 3.2).

The overall change after correcting for the contribution from ionized gas is small. This is consistent with the finding of Croxall et al. (2017) that the neutral fraction of gas traced by [C ii] increases with decreasing metallicity in the Beyond the Peak subsample of KINGFISH galaxies. We list the average percentages of [C ii] intensity estimated to originate from the ionized gas in Table 6. The amount of emission from ionized gas can be as high as $\sim 50 \%$, but this is only found within the H ii region, where the [C ii] surface brightness is low. The effect of correcting for the ionized gas mainly affects the structure of [C ii] emission in and near the H ii regions, where we expect little to no molecular gas.

Table 6.  Properties of ${{\rm{H}}}_{2}$ Estimates from [C ii]

	Average % I[C ii]		${\bar{N}}_{{{\rm{H}}}_{2},[{\rm{C}}{\rm{II}}]}$ (1021 cm−2)		${M}_{{{\rm{H}}}_{2},[{\rm{C}}{\rm{II}}]}$ (104 ${M}_{\odot }$)
Region	from H i	from ${{\rm{H}}}^{+}$	Fixed $T,\,n$	W16	Fixed $T,\,n$	W16	${M}_{{{\rm{H}}}_{2}{,}^{12}\mathrm{CO}}$ (104 ${M}_{\odot }$)
SWBarS	3%	6%	1.2	0.97	5.05	4.15	1.3
N22	2%	15%	1.6	1.1	5.64	4.13	1.0
SWBarN	2%	7%	2.9	2.5	8.60	7.21	1.7
SWDarkPK	7%	9%	0.48	0.63	0.90	1.16	0.10

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We estimate the amount of [C ii] emission from H i by using the typical temperature and density for the conditions of the gas and assume that the column density of ionized carbon scales with the carbon abundance such that ${N}_{{{\rm{C}}}^{+}}={({{\rm{C}}}^{+}/{\rm{H}})}_{\mathrm{SMC}}{{\rm{N}}}_{{\rm{H}}{\rm{I}}}$. The [C ii] emission from atomic hydrogen gas arises from a combination of the WNM, with typical temperatures of 6000–12,000 K, and the denser CNM, which has temperatures of 50–120 K. Some or all of the CNM component may be directly associated with the molecular gas as a shielding layer surrounding and mixed with the outer part of the molecular cloud in the PDR. Given the differences in conditions, we make separate estimates for the [C ii] emission coming from the WNM and CNM and remove both contributions from I_{[C ii]}.

To estimate the possible H i associated with the [C ii] emission, we want to select only the H i with velocities in the observed range of the [C ii] emission. However, the spectral resolution of the PACS spectrometer is ∼240 km s⁻¹, which encompasses almost all of the H i emission and is a much larger range of velocities than the ∼15–50 km s⁻¹ widths of the ${}^{12}\mathrm{CO}$ line observations. We use the velocity-resolved observations of [C ii] from the SOFIA GREAT receiver (R. Herrera-Camus et al. 2017, in preparation) as guidance to estimate the relationship between the velocity profile of the [C ii] emission and the ${}^{12}\mathrm{CO}$ throughout all our regions.

Figure 12 shows an example [C ii] spectrum from a pointing in the SWBarN region with the ALMA ACA ${}^{12}\mathrm{CO}$ $(2-1)$ and H i line profiles for the same region. The [C ii] line is clearly resolved and shows a profile similar to the ${}^{12}\mathrm{CO}$ and much narrower than the H i, but the full extent of the [C ii] profile is $\sim 50 \%$ wider than the full extent of the ${}^{12}\mathrm{CO}$ line profile owing to more extended wings where there is low or no ${}^{12}\mathrm{CO}$ emission. Requena-Torres et al. (2016) present new velocity-resolved [C ii] GREAT observations of the SMC star-forming regions N66, N25/N26, and N88 that also show similarity between [C ii] and ${}^{12}\mathrm{CO}$ line profiles and find the [C ii] emission to be at most 50% wider than ${}^{12}\mathrm{CO}$, and significantly narrower than that of H i. Similarly, comparison of the dynamics of [C ii], H i, and CO on larger scales shows that the [C ii] line widths agree better with the CO than H i (de Blok et al. 2016). The width of the H i line profile is due to emission from multiple kinematic components that are likely spatially separate, and the narrow profiles of the ${}^{12}\mathrm{CO}$ and [C ii] appear to originate from only one of those kinematic components. To select the H i associated with the strong [C ii] emission and the molecular cloud, we created integrated intensity maps of H i for each of the ALMA regions using a velocity range that is centered on the ${}^{12}\mathrm{CO}$ line velocity, but ±25% wider than the range of velocities with observed ${}^{12}\mathrm{CO}$ throughout the region (not for individual lines of sight) to account for the possibility of ∼50% wider extent of the [C ii] line profile (mostly found in the wings of the line). We use these H i maps to estimate the amount of [C ii] emission associated with neutral atomic gas.

To estimate the [C ii] emission attributable to atomic gas, we assume a single temperature and density for each of the WNM and CNM. The H i emission will be dominated by the WNM component, which we assume to have a temperature of $\approx 8000$ K (Wolfire et al. 2003) and a typical density range of 0.1–1 cm⁻³ (Velusamy et al. 2012). We set ${n}_{\mathrm{WNM}}=1$ cm⁻³ for our estimate to ensure that we do not underestimate the [C ii] contribution from the WNM. The ${{\rm{C}}}^{+}$ excitation critical density for $T\approx 8000$ K of the WNM from hydrogen atoms is ${n}_{\mathrm{crit}}({H}^{0})=1600$ cm⁻³ (Goldsmith et al. 2012). For the CNM, we assume ${n}_{\mathrm{CNM}}=100$ cm⁻³ and T = 40 K based on the values found from the H i absorption line study by Dickey et al. (2000). For collisions with electrons, we take ${n}_{\mathrm{crit},{\rm{e}}}=6$ cm⁻³. We calculate the critical density for collisions with hydrogen atoms for the CNM temperature using ${n}_{\mathrm{crit}}={A}_{{ul}}/{R}_{{ul}}$, where A_ul is the spontaneous emission rate coefficient and R_ul the collisional de-excitation rate coefficient. Using ${A}_{{ul}}\,=2.36\times {10}^{-6}$ s⁻¹ and the fit to R_ul for 20 K $\leqslant {T}^{\mathrm{kin}}\leqslant 2000$ K from Barinovs et al. (2005), we find ${n}_{\mathrm{crit}}({{\rm{H}}}^{0})\approx 3000$ cm⁻³.

We calculate I_{[C ii]} using Equation (9) for collisions with both hydrogen atoms and electrons in the WNM and CNM and collisions with ${{\rm{H}}}_{2}$ for the CNM and add them to produce our estimate. We take an ionization fraction in neutral gas to be ${n}_{{e}^{-}}/{n}_{{\rm{H}}}=4.3\times {10}^{-4}$ (Draine 2011). In order to calculate this, we need the fraction of WNM and CNM in our lines of sight, which is unknown. We assume that 50% of ${N}_{{\rm{H}}{\rm{I}}}$ associated with the CO emission is associated with the CNM and the other 50% with the WNM when calculating I_{[C ii]}. Dickey et al. (2000) find the fraction of cool H i to be $\sim 15 \%$ of the total H i in the SMC along a few lines of sight; however, locally it may be much higher. Assuming a higher fraction of CNM will generate a higher estimate of the associated [C ii] emission, decreasing the likelihood of underestimating the [C ii] emission from H i. Table 6 shows the average amount of total [C ii] emission for each region. We find that H i typically only accounts for ≲5%. When we apply statistical correction for optically thick H i from Stanimirović et al. (1999) to ${N}_{{\rm{H}}{\rm{I}}}$ used to estimate the [C ii] emission from H i, the fraction of the total [C ii] emission coming from H i only slightly increases (e.g., the average increases from 2% to 3% for the SWBarN region). The SWDarkPK region has the highest fraction of [C ii] emission coming from H i, likely because it is not actively forming stars and is more atomic dominated (as also suggested by the lower CO emission indicating less molecular gas at ${A}_{V}\gt 1$ than the other regions). These low fractions of [C ii] emission from atomic gas are consistent with the theoretical study by Nordon & Sternberg (2016), who find that most of the ${{\rm{C}}}^{+}$ column would be associated with ${{\rm{H}}}_{2}$ gas.

After removing the contributions from ionized and neutral atomic gas, we assume that the remainder of the [C ii] emission originates from molecular gas. We estimate the amount of ${{\rm{H}}}_{2}$ gas from the remaining [C ii] emission in two ways:

• 1.  
  Assuming a fixed T and n representative of PDR regions and converting I_{[C ii]} to ${N}_{{{\rm{H}}}_{2}}$ using Equation (9).
• 2.  
  Using PDR models tuned to the SMC conditions (see Appendix A) to estimate n and the FUV radiation field intensity from the observed combination of [C ii], [O i], and FIR continuum, determining ${N}_{{{\rm{H}}}_{2}}$ from the cooling curve.

We explain both methods in the following sections.

4.3.1. Method 1: Fixed $T,\,n$

Using Equation (9), we can solve for the ${{\rm{H}}}_{2}$ column density given the [C ii] intensity:

$$\begin{eqnarray}&&{N}_{{{\rm{H}}}_{2}}=\displaystyle \frac{4.35\times {10}^{23}}{{({\rm{C}}/{\rm{H}})}_{\mathrm{SMC}}}\left[\displaystyle \frac{1+2{e}^{-91.2/T}+{n}_{\mathrm{crit}}({{\rm{H}}}_{2})/n}{2{e}^{-91.2/T}}\right]{I}_{[{\rm{C}}{\rm{II}}],{{\rm{H}}}_{2}}.\end{eqnarray} \tag{ 12 }$$

To estimate ${N}_{{{\rm{H}}}_{2}}$, we need only assume a temperature and volume density and calculate the appropriate critical density for collisions with ${{\rm{H}}}_{2}$ using the latest collisional de-excitation rate fits by Wiesenfeld & Goldsmith (2014) as a function of temperature, which includes ${{\rm{H}}}_{2}$ spin effects in LTE:

$$\begin{eqnarray}&&{n}_{\mathrm{crit}}=\displaystyle \frac{{A}_{{ul}}}{{R}_{{ul}}}=\displaystyle \frac{2.36\times {10}^{4}}{4.55+1.6{e}^{(-100.0{\rm{K}}/T)}}.\end{eqnarray} \tag{ 13 }$$

Our assumption is that all of the [C ii] emission associated with ${{\rm{H}}}_{2}$ comes from the PDRs, where the gas is warm and moderately dense. We can then make the simplifying assumption that this region has a single average temperature of T = 90 K and density of n = 4000 cm⁻³ (consistent with the PDR modeling results in Section 4.3.2). These are chosen as similar to the excitation temperature and critical density of the transition, and thus at about the point where the gas emits most efficiently. For reference, the exact critical density for collisions with ${{\rm{H}}}_{2}$ at T = 90 K is ${n}_{\mathrm{crit}}({{\rm{H}}}_{2})=4648$ cm⁻³. This choice of temperature and density results in a conservative estimate of the amount of molecular gas associated with the [C ii] emission. Assuming a lower temperature or density would decrease the amount of [C ii] emission per ${{\rm{H}}}_{2}$ molecule and increase the amount of molecular gas needed to explain the observed emission, while increasing the temperature or density will not change our results significantly. The benefit of this methodology is its simplicity and straightforward calculation. The drawback is that it over simplifies the conditions of the [C ii]-emitting gas by assuming only one constant temperature and volume density throughout each region and across all regions.

4.3.2. Method 2: PDR Model

We can constrain the physical conditions of the gas in the PDRs by modeling the [C ii], [O i], and total FIR emission from the Spitzer and Herschel imaging. We use new PDR models generated using the elemental abundances and grain properties for SMC conditions that are presented in Appendix A. The models solve the equilibrium chemistry, thermal balance, and radiation transfer through a PDR layer characterized by a constant H nucleus volume density (n in units of cm⁻³) and incident FUV (6 eV <  E <  13.6 eV) radiation field (G₀ in units of the local Galactic interstellar FUV field found by Habing, $1.6\times {10}^{-3}$ erg cm⁻² s⁻¹). The models are based on Kaufman et al. (2006) but are updated for gas and grain surface chemistry as described in Wolfire et al. (2010) and Hollenbach et al. (2012). We also modify the grain properties and abundances as appropriate for the SMC (see Appendix A), which makes it tailored to modeling the emission from the HS³ regions.

We use the [C ii], [O i], and TIR intensity images as inputs to determine the nearest volume density and FUV radiation field strength in the model grids using a version of the PDR Toolbox (Pound & Wolfire 2008) that is updated with the SMC models. While the PDR Toolbox and models call for FIR intensity as the input, the term FIR is meant to represent all of the FUV and optical emission absorbed by dust and re-radiated in the infrared, which makes the current definition of TIR the most appropriate to compare to the models. We convolve the [C ii] and [O i] images to the resolution of the TIR. We run the PDR Toolbox for the intensities at each matched pixel in the images and produce maps of n and G₀. The images are then sampled with $\sim 1$ pixel per beam, matching the [C ii] beam-sampled images. While the limiting resolution of the TIR map is lower ($\sim 18^{\prime\prime}$) than the [C ii] ($\sim 12^{\prime\prime}$), it is not significantly lower, and we expect n and G₀ to vary smoothly. The initial images of n and G₀ show artifacts from the FIR image (due mostly to the strong beam patterns from the point sources). We take one final step and mask out regions in the images with unphysical values of n or G₀ that are due to the artifacts and use a linear interpolation to replace the masked values. The results of model fits are shown in Figure 14. The PDR models suggest that the typical volume densities are $n\sim {10}^{3}\mbox{--}{10}^{4}$ cm³, which is consistent with most of the gas being molecular in order to reproduce the observed FIR emission.

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Using the maps of n and G₀, we convert the [C ii] emission to ${N}_{{{\rm{H}}}_{2}}$. We use [C ii] cooling rates as a function of n that are adapted from Wolfire et al. (2003) to account for the conditions of the SMC, which we show in Figure 15 (M. Wolfire et al. 2017, in preparation). We use the same abundances as for the PDR model. The [C ii] rates are calculated for a fiducial FUV radiation field strength of ${G}_{0}=5.5$, which is approximately the average in diffuse gas (outside of the regions we mapped) in the SMC Southwest Bar based on the maps from Sandstrom et al. (2010), and the cooling rates were primarily intended for use in diffuse gas. While the [C ii] cooling rate is only determined for a single G₀, the effect of different G₀ is to change the photoelectric heating rate, which to first order is proportional to G₀ (Bakes & Tielens 1994, Equation (42)). If [C ii] dominates the cooling, then the [C ii] line intensity is proportional to G₀. The appropriate [C ii] cooling rate per ${{\rm{H}}}_{2}$ molecule for each pixel is determined by taking the [C ii] cooling rate appropriate for the specific value of n and then scaling it linearly by the ratio of estimated G₀ to the fiducial G₀ for diffuse gas. We then use this local [C ii] cooling rate to calculate an ${{\rm{H}}}_{2}$ column density and produce the final map. The benefit of this method is that it can account for the variations in density and radiation field in the [C ii]-emitting gas throughout the regions. The weakness is that the gas conditions are estimated using "single-component" models based on plane-parallel PDRs that do not account for complex geometries, or for distributions of n or G₀ within a spatial resolution element. This is alleviated by the fact that our physical resolution is 4 pc; thus, we expect the conditions in a beam to be relatively uniform and simple "single-component" models to be more applicable than they would be if applied to data on much larger scales.

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Once enough shielding from dissociating radiation is built up at high A_V, carbon will mostly exist in the form of CO. The exact amount of ${{\rm{H}}}_{2}$ traced by the ${}^{12}\mathrm{CO}$ emission (and not [C ii]) cannot be easily calculated theoretically and relies on empirical calibrations of the conversion factor from ${}^{12}\mathrm{CO}$ intensity to ${N}_{{{\rm{H}}}_{2}}$, ${X}_{\mathrm{CO}}$, typically calibrated in terms of ${}^{12}\mathrm{CO}$ $(1-0)$ emission. The exact definition of ${X}_{\mathrm{CO}}$ varies depending on the study, where it can refer to the total amount of ${{\rm{H}}}_{2}$ gas associated with the CO emission, including the ${{\rm{H}}}_{2}$ associated with [C ii] or [C i] (e.g., unresolved CO observations), or it can refer to the molecular gas only associated with gas where the dominant form of carbon is CO. For this work, we use the latter as our definition since our observations resolve the CO emission and we are separately accounting for the ${{\rm{H}}}_{2}$ gas, where carbon is mostly [C ii]. To convert from ${}^{12}\mathrm{CO}$ $(2-1)$ to the equivalent ${}^{12}\mathrm{CO}$ $(1-0)$ integrated intensity, we assume thermalized emission, which is supported by SEST observations in this region of the SMC that find ${}^{12}\mathrm{CO}$ $(1-0)$/${}^{12}\mathrm{CO}$ $(2-1)$ $\sim 1$ (Rubio et al. 1993a). This is equivalent to assuming constant integrated intensity in K km s⁻¹ units, or to dividing the 2–1 emission in units of W m⁻² Hz⁻¹ by ${J}_{u}^{3}=8$. Numerical simulations of low-metallicity molecular clouds show that the conversion factor reaches Galactic, high-metallicity values in the CO-bright regions (Shetty et al. 2011; Szűcs et al. 2016), suggesting that, regardless of metallicity, CO traces a similar amount of ${{\rm{H}}}_{2}$ at high A_V. Given the simulation results and our estimates of ${X}_{\mathrm{CO}}$ (see Section 3.6), we use a Galactic CO-to-${{\rm{H}}}_{2}$ conversion factor ${X}_{\mathrm{CO}}=2\times {10}^{20}$ cm⁻² (K km⁻¹ ${{\rm{s}}}^{-1}{)}^{-1}$ (Bolatto et al. 2013) to estimate the amount of molecular gas present in regions at high A_V, where most of the carbon is in the form of CO. This is consistent with an underlying picture where most of the difference in the conversion factor observed in low-metallicity systems is due to the shrinking of the high-A_V CO-emitting cores and the growth of the outer layer of ${{\rm{H}}}_{2}$ coextensive with C⁺ (e.g., Wolfire et al. 2010; Bolatto et al. 2013).

Overall, the two different methods to estimate ${N}_{{{\rm{H}}}_{2}}$ based on [C ii] emission produce similar results. Table 6 shows that average column densities and total ${{\rm{H}}}_{2}$ masses for both methods are different by at most a factor of ∼1.5. The total mass estimates from Method 1, assuming T = 90 K and n = 4000 cm⁻³, are higher than the estimates from Method 2, using the SMC PDR models and M. Wolfire et al. (2017, in preparation) [C ii] cooling rates, except for the SWDarkPK region. The fixed temperature and density produce higher estimates because they fail to account for the presence of higher-density regions that have a higher emission of [C ii] per unit ${{\rm{H}}}_{2}$ mass, as we show in the example in Figure 16. The opposite is true for SWDarkPK, where the densities from the PDR modeling are somewhat lower than those in other regions, and the assumed n = 4000 cm⁻³ in Method 1 appears to be too high. This is consistent with the observed lower level of CO emission in that very quiescent region, which is suggestive of less dense molecular gas.

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The Method 1 results produce structure that is similar to the [C ii] emission, with the peak in the ${N}_{{{\rm{H}}}_{2}}$ occurring at the peaks in [C ii] emission, which are nearly coincident with the peaks in the CO emission. When the variations in density and radiation field strength are accounted for in the estimate in Method 2, the peaks in the ${N}_{{{\rm{H}}}_{2}}$ maps occur around the peak in the CO emission (see Figure 16), consistent with our understanding that [C ii] emission should trace the ${{\rm{H}}}_{2}$ gas in the PDRs surrounding the dense molecular gas traced by bright CO. We prefer the ${N}_{{{\rm{H}}}_{2}}$ estimates from Method 2 because it incorporates additional information, and we use them for the remaining analysis and discussion.

Figure 17 shows the ${{\rm{H}}}_{2}$ maps derived from [C ii] alone, from CO alone, and the total corresponding to their sum for each region. The ${{\rm{H}}}_{2}$ estimated from [C ii] is more extended than that derived from CO and wraps around it, which agrees with the idea that the [C ii] should be primarily tracing the ${{\rm{H}}}_{2}$ in the low-A_V portion of the molecular cloud. The CO emission traces between ∼5% and 60% of the total molecular gas in this methodology, with a typical percentage of only 20%. Note, however, that there are no ${{\rm{H}}}_{2}$ column density peaks without CO emission.

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4.5.1. Estimating Uncertainties

Another method that can trace "CO-faint" molecular gas is to use dust emission and a self-consistently determined gas-to-dust ratio in atomic regions using H i emission (Israel 1997; Dame et al. 2001; Leroy et al. 2009; Bolatto et al. 2011). We compare our molecular gas estimates using [C ii] and CO with the recent dust-based molecular gas estimates for the SMC presented in Jameson et al. (2016). Table 8 shows the [C ii]+CO and dust-based molecular gas estimates over the area with both [C ii] and ALMA ${}^{12}\mathrm{CO}$ coverage, where we make a cut at the rms level before calculating the total mass from CO.

Table 8.  Total Gas Masses

	${M}_{{{\rm{H}}}_{2}}$ (105 ${M}_{\odot }$)		MH i (105 ${M}_{\odot }$)
Region	[C ii]+CO	Dust	${\rm{\Delta }}{v}_{\mathrm{CO}}$ a	${\rm{\Delta }}{v}_{\mathrm{total}}$
SWBarS	${0.52}_{-0.26}^{+0.52}$	${2.33}_{-1.17}^{+2.33}$	3.42	6.53
N22	${0.48}_{-0.24}^{+0.48}$	${1.23}_{-0.62}^{+1.23}$	3.15	7.30
SWBarN	${0.86}_{-0.43}^{+0.86}$	${1.61}_{-0.81}^{+1.61}$	4.64	9.00
SWDarkPK	${0.11}_{-0.06}^{+0.11}$	${1.79}_{-0.90}^{+1.79}$	1.46	3.08
N83	${0.48}_{-0.24}^{+0.48}$	${2.96}_{-1.48}^{+2.96}$	4.67	6.16

Note.

^aUsing the H i integrated intensity only around the velocity channels with CO emission.

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The [C ii]+CO estimates are not inconsistent with the dust-based estimates from Jameson et al. (2016) in the N22, SWBarN, and possibly the SWBarS regions, given the factor of $\sim 2$ uncertainty in either method. The [C ii]+CO estimates for the SWDarkPK and N83 regions, on the other hand, are significantly lower than the result of the dust method. All of the [C ii]+CO estimates are systematically lower than the dust-based ${{\rm{H}}}_{2}$ mass estimates. The systematically lower [C ii]+CO estimates can be the result of our [C ii] methodology underestimating the amount of ${{\rm{H}}}_{2}$, or the dust methodology overestimating the total molecular gas mass.

When estimating the amount of ${{\rm{H}}}_{2}$ associated with [C ii], we have been conservative in our estimates of the contribution to [C ii] from the ionized and neutral gas and the conditions of the gas (n and G₀), which may cause us to underestimate the total ${{\rm{H}}}_{2}$. The dust-based method has a number of systematic uncertainties (for more details, see Leroy et al. 2007, 2009, 2011; Bolatto et al. 2011; Sandstrom et al. 2013; Jameson et al. 2016), with the two main uncertainties being the assumptions surrounding the gas-to-dust ratio and the FIR dust emissivity. In particular, the method from Jameson et al. (2016) relies on self-consistently calibrating the gas-to-dust ratio in the atomic gas, assuming that it only varies smoothly on large scales (500 pc). This methodology could overestimate the molecular gas if the gas-to-dust ratio is an overestimate of the actual gas-to-dust ratio, which could be the case if the gas-to-dust ratio decreases in molecular regions. Similarly, the dust properties may vary from the diffuse to dense gas, which could produce a similar effect to a decrease in the gas-to-dust ratio.

There is also a systematic uncertainty of $\gtrsim 0.3$ dex in the dust-based molecular gas estimate. Bot et al. (2004) compared IRAS observations of the FIR dust emissivity in the diffuse gas with extinction studies in the SMC and found evidence for a factor of 2–3 decrease in the gas-to-dust ratio in dense gas. Roman-Duval et al. (2014) found that at the resolution of the NANTEN observation ($2\buildrel{\,\prime}\over{.} 6$), the degeneracy between the gas-to-dust ratio, effect of dust emissivity, and X_CO is too large to provide definitive evidence of a change in the gas-to-dust ratio between the diffuse and molecular gas in the SMC, but allows for up to a factor of $\sim 2-3$ decrease in the gas-to-dust ratio between the diffuse and dense gas. Using depletion measurements based on FUV spectra to estimate the gas-to-dust ratio in the SMC, Tchernyshyov et al. (2015) find evidence for up to a factor of $\sim 5$ decrease in the gas-to-dust ratio from the diffuse to dense gas. However, these results are based on difficult measurements and rely on a number of assumptions. Jameson et al. (2016) found that scaling down the gas-to-dust ratio by a factor of 2 in the dense gas leads to a similar factor of $\sim 2$ decrease in the total molecular gas mass estimate. If the dust-based molecular gas estimates were lower by a factor of $\sim 2$, which is a possibility given the existing measurements, then all of the dust estimates would be consistent with the [C ii]+CO estimates, and there would be one region (SWBarN) where the [C ii]+CO molecular gas estimate would be higher than the dust-based estimate. Similarly, if we used the density and radiation field estimates from the higher-metallicity PDR models, the [C ii]+CO estimates would be $\sim 2\times$ higher and would no longer be systematically below the dust-based estimates.

On larger scales and in high-metallicity environments, [C ii] acts as a calorimeter, as it is the main coolant of the ISM. The heating comes from FUV photons causing electrons to be ejected from dust grains owing to the photoelectric effect. Most recently De Looze et al. (2014) and Herrera-Camus et al. (2015) have shown that [C ii] emission correlates well with other star formation rate (SFR) tracers, specifically Hα and 24 μm, on kiloparsec scales in nearby galaxies. On smaller scales Kapala et al. (2015) found that the correlation between [C ii] and SFR holds, although much of the [C ii] emission in M31 comes from outside of star-forming regions, as the FUV photons that heat the gas travel long distances diffusing throughout the gas disk. When the M31 observations are averaged over kiloparsec scales, Kapala et al. (2015) recover the relationship between [C ii] and SFR observed in other galaxies.

Both De Looze et al. (2014) and Herrera-Camus et al. (2015) show that the scatter in the correlation between [C ii] and SFR increases as the metallicity decreases. The breakdown in the tight [C ii]–SFR relationship can be understood in terms of a decrease in the photoelectric heating efficiency and/or an increase in the FUV photon escape fraction. In a low-metallicity environment there is less dust, allowing FUV photons to propagate farther through the gas and potentially escape the region or even the galaxy. Cormier et al. (2015) find evidence of an increased UV escape fraction based on the observed ${\rm{O}}\,{\rm{III}}/{L}_{\mathrm{TIR}}$ for the DGS. The increase in the propagation distance causes the [C ii] emission to be less co-localized with the recent massive star formation. If the FUV propagation distance is longer than the scale on which the [C ii] emission is compared to other star formation indicators, that increases the scatter in the [C ii]–SFR relationship. When the propagation distance is larger than the scale of the gas disk, a fraction of the FUV photons escape the galaxy. The [C ii] emission will then underestimate the SFR even when considering integrated properties. This is consistent with the lower [C ii] emission per unit SFR (estimated from combined FUV and 24 μm emission) observed at lower metallicities by De Looze et al. (2014).

On small scales, [C ii] reflects the cooling of the gas and can trace the ${{\rm{H}}}_{2}$ by accounting for the amount of emission attributed to collisions with ${{\rm{H}}}_{2}$ molecules. This requires removing the contribution to the emission due to the cooling of the atomic gas and understanding the conditions (temperature and density) that account for the [C ii] excitation. In this context, the SFR is only important in that it is the source of FUV photons that ionize the carbon and heat the PDR. Because this takes place on the scale of individual H ii regions, the fraction of FUV photons that ultimately escape the galaxy does not enter in the estimates. The fact that at lower metallicity the PDR region at ${A}_{V}\lesssim 1$ is physically very extended makes [C ii] an excellent tracer of "CO-faint" molecular gas. The excellent correspondence between the ${{\rm{H}}}_{2}$ S(0) line emission and [C ii] emission (see Figure 3) and the general correspondence between the [C ii] and ${}^{12}\mathrm{CO}$ emission (see Figure 11) demonstrate that [C ii] traces the structure of the molecular gas at low metallicity.

We estimated the amount of molecular gas using [C ii] to trace the "CO-faint" molecular gas at low ${A}_{V}$ ("[C ii]-bright" molecular gas) and ${}^{12}\mathrm{CO}$ to trace the molecular gas at high ${A}_{V}$ using a Galactic X_CO to convert the observed emission to molecular gas ("CO-bright" molecular gas). Figure 18 shows the fraction of the total molecular gas coming from [C ii] as a function of A_V estimated from dust. We see that at the low metallicity of our mapped SMC regions most of the molecular gas is traced by [C ii], not CO, with ∼70% of the molecular gas coming from [C ii]-bright regions on average. The data also show the expected trend of a higher fraction of the total molecular gas being traced by CO as ${A}_{V}$ increases, although with a lot of scatter. The scatter is in part due to the fact that our A_V estimate uses data with lower resolution than the [C ii] or the CO observations, and in part due to our limitations at estimating the physical conditions from the available data. But more fundamentally, much of the scatter must arise from the fact that the relevant A_V for photodissociation is not the line-of-sight extinction that we measure, but the A_V toward the dominant sources of radiation. Note that, given the methodology, the molecular gas estimate from [C ii] is more likely to be underestimated rather than overestimated (see Section 4.5.1), suggesting that the "[C ii]-bright" molecular fractions may be preferentially higher.

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In the SWBarN and N22 regions CO traces at most $\sim 50 \%$ of the molecular gas at our maximum ${A}_{V}$. The trend in the SWBarS region is considerably steeper, and around ${A}_{V}\sim 1$ it reaches "CO-bright" fractions of $\sim 70 \%$. This is consistent with the fact that this region hosts the brightest ${}^{12}\mathrm{CO}$ and ${}^{13}\mathrm{CO}$ emission. The range of observed "[C ii]-bright" molecular gas fractions is consistent with the estimated fractions for regions in the SMC with velocity-resolved [C ii] spectra by Requena-Torres et al. (2016), who find that 50% to ∼90% of the molecular gas is traced by [C ii] emission. The fractions of "CO-faint" molecular gas in the SMC are much higher than fractions found in local Galactic clouds (${f}_{\mathrm{CO} \mbox{-} \mathrm{faint}}=0.3;$ Grenier et al. 2005) and fractions in dense molecular clouds estimated from [C ii] emission (${f}_{\mathrm{CO} \mbox{-} \mathrm{faint}}=0.2;$ Langer et al. 2014). The fractions in the SMC are more similar to those found using [C ii] emission in diffuse Galactic clouds (${f}_{\mathrm{CO} \mbox{-} \mathrm{faint}}=0.75$ for clouds with no detectable ${}^{12}\mathrm{CO}$ emission; Langer et al. 2014). Overall, the high fractions of [C ii]-bright gas suggest that most of the ${{\rm{H}}}_{2}$ gas in the SMC is found within PDRs or in regions with PDR-like conditions.

Figure 19 shows that, despite the fact that most molecular gas is related to [C ii] rather than CO emission, our estimates for ${N}_{{{\rm{H}}}_{2}}$ are correlated with the ${}^{12}\mathrm{CO}$ integrated intensity. The black dashed line in Figure 19 shows the estimated amount of ${N}_{{{\rm{H}}}_{2}}$ associated with CO using a Milky Way conversion factor of ${X}_{\mathrm{CO}}=2\times {10}^{20}$ cm⁻² (K km s⁻¹)⁻¹. There is a clear offset between the results assuming a Galactic X_CO and our estimate for ${N}_{{{\rm{H}}}_{2}}$. This offset is due to the presence of molecular gas traced primarily by [C ii] emission (at low A_V). Interestingly, we see that the relations are steeper than that in the Milky Way. The steeper slopes indicate that molecular gas builds up faster with CO intensity than it does in the Milky Way. This is due to the presence of "[C ii]-bright" ${{\rm{H}}}_{2}$ along the line of sight toward CO emission—simply the result of the fact that "CO-bright" regions still have substantial [C ii] emission associated with them at the resolution of our measurements.

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We use our ${N}_{{{\rm{H}}}_{2}}$ determination to estimate the values of X_CO throughout the mapped regions, accounting for both "[C ii]-bright" and "CO-bright" molecular gas. Figure 20 shows our estimates of X_CO as a function of A_V, where there is a trend of decreasing X_CO with increasing A_V albeit with significant scatter. The trend in X_CO matches the expectation of extended envelopes of "[C ii]-bright" molecular gas, and it compares well with similar trends seen in low-metallicity molecular cloud simulations (Shetty et al. 2011; Szűcs et al. 2016). The scatter can be explained as being due to the "breakdown" of a single value of X_CO on small scales in simulations and even observed clouds, due to the clumpy nature of molecular clouds (Glover & Mac Low 2011; Shetty et al. 2011; Bolatto et al. 2013).

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The values we find for our regions have on average a conversion factor of ${X}_{\mathrm{CO}}\sim 5\,{X}_{\mathrm{CO},\mathrm{MW}}$. For the entire SMC, and using the dust-based molecular gas estimates, Jameson et al. (2016) find a CO-to-${{\rm{H}}}_{2}$ conversion factor ∼17 times higher than the average Milky Way value. The difference between these two results can be mostly ascribed to the bias in our chosen HS³ fields. Our survey selected actively star-forming regions with bright CO emission, whereas most of the SMC is faint in CO; in fact, our regions contain some of the peaks of CO emission in the SMC. But even within our observed regions many lines of sight have high fractions of ${{\rm{H}}}_{2}$ not traced by "bright CO" emission (see Figure 18). The higher global measurement is simply a statement of the fact that the extinction in a typical molecular line of sight in the SMC is probably ${A}_{V}\sim 0.5\mbox{--}1$. Accounting for the dust-to-gas ratio of 1/5–1/7 the Galactic value, the same line of sight would have ${A}_{V}\sim 2.5\mbox{--}7$ in the Milky Way and emit brightly in CO. Even in our highly biased regions bright in CO we barely reach ${A}_{V}\sim 2.5$ in the SMC (Figure 20). Schruba et al. (2017) also found variations in the CO-to-${{\rm{H}}}_{2}$ conversion factor using dust-based ${{\rm{H}}}_{2}$ estimates in different regions of the low-metallicity dwarf galaxy NGC 6822, with the lowest values (closest to the Milky Way value) in the regions with bright CO emission. Using galaxy-integrated measurements of [C ii] and ${}^{12}\mathrm{CO}$ and isolating the [C ii] contributions from molecular gas using the results of models of [C ii] emission throughout a galaxy, Accurso et al. (2017) estimated the conversion factor and found the values to be in the range of $\sim 12.5\mbox{--}17\times$ the Milky Way value for the metallicity of the SMC, which is within the range of our estimates of X_CO.

We can compare our estimates of X_CO versus A_V with theoretical studies of the relation between CO and ${{\rm{H}}}_{2}$ at low metallicity. Wolfire et al. (2010) present a model of molecular clouds in order to understand the fraction of ${{\rm{H}}}_{2}$ not traced by bright CO emission as a function of metallicity. They assume spherical clouds with ${r}^{-1}$ density profile, predicting the fraction of "CO-faint" gas based on their mean volume density, external radiation field, mean A_V ($\langle {A}_{V}\rangle$), and metallicity relative to Galactic (${Z}^{{\prime} }$). Glover & Mac Low (2011) and more recently Szűcs et al. (2016) created simulations of molecular clouds with varying conditions, including metallicity, finding an empirical relationship for the mean X_CO value as a function of the mean A_V for the simulated clouds. The trend predicted by the Wolfire et al. (2010) calculations and formulated by Bolatto et al. (2013) is

$$\begin{eqnarray}&&{X}_{\mathrm{CO}}={X}_{\mathrm{CO},0}{e}^{(4{\rm{\Delta }}{A}_{V}/\langle {A}_{V}\rangle )}{e}^{(-4{\rm{\Delta }}{A}_{V}/(\langle {A}_{V}\rangle /Z^{\prime} ))},\end{eqnarray} \tag{ 14 }$$

where

$$\begin{eqnarray}&&{\rm{\Delta }}{A}_{V}=0.53-0.045\mathrm{log}\left(\displaystyle \frac{{G}_{0}}{n}\right)-0.097\mathrm{log}(Z^{\prime} ),\end{eqnarray} \tag{ 15 }$$

while Glover & Mac Low (2011) find

$$\begin{eqnarray}{X}_{\mathrm{CO}}=\left\{\begin{array}{ll}{X}_{\mathrm{CO},0}{(\langle {A}_{V}\rangle /3.5)}^{-3.5} & \mathrm{if}\ \langle {A}_{V}\rangle \lt 3.5\ \mathrm{mag}\\ {X}_{\mathrm{CO},0} & \mathrm{if}\ \langle {A}_{V}\rangle \geqslant 3.5\ \mathrm{mag}\end{array}\right.,\end{eqnarray} \tag{ 16 }$$

where ${X}_{\mathrm{CO},0}$ is the mean value for the Milky Way.

In Figure 20 we overplot the predicted X_CO trend from Wolfire et al. (2010) using the metallicity of the SMC ($Z^{\prime} =0.2$), an average n = 5000 cm⁻³, and ${G}_{0}=40$ based on the PDR modeling we did (see Section 4.3.2). We also overplot the empirical fit from the simulations by Glover & Mac Low (2011). Both predictions are close to the data for both individual lines of sight and the averages within a region (indicated by the star), although Wolfire et al. (2010) seem to do a better job overall. As both the Wolfire et al. (2010) model and Glover & Mac Low (2011) simulations suggest, the main effect driving the relationship between X_CO and A_V is not metallicity itself, but how metallicity and environment affect the FUV shielding in the molecular gas. In particular, the critical element is the dust-to-gas ratio in the medium. This is also consistent with the findings of Leroy et al. (2009) and Lee et al. (2015) showing that the amount of ${}^{12}\mathrm{CO}$ emission at a given value of ${A}_{V}$ in the SMC resembles that in the Milky Way.

Our new measurements present an opportunity to look at the molecular-to-atomic transition at low metallicity. We estimate the molecular-to-atomic ratio using our total molecular gas estimate, ${N}_{{{\rm{H}}}_{2}}$, and the H i column density map associated with the molecular cloud. The ${N}_{{\rm{H}}{\rm{I}}}$ map is the same that we used to determine the amount of [C ii] emission associated with H i gas (see Section 4.2), which was produced by only integrating the H i emission over a velocity range that is centered on the ${}^{12}\mathrm{CO}$ line velocity, but ±25% wider than the range of velocities with observed ${}^{12}\mathrm{CO}$ emission throughout each region. Figure 21 shows the maps of the ${N}_{{{\rm{H}}}_{2}}$/${N}_{{\rm{H}}{\rm{I}}}$ ratio with overlaid ALMA ${}^{12}\mathrm{CO}$ contours. As expected for the atomic-dominated SMC, we observe peak molecular-to-atomic ratios of ${N}_{{{\rm{H}}}_{2}}/{N}_{{\rm{H}}{\rm{I}}}\sim 1.5$ that show that the amount of molecular gas does not greatly exceed the amount of H i gas along the line of sight owing to the large amount of atomic gas. We also detect ${}^{12}\mathrm{CO}$ emission out to ${N}_{{{\rm{H}}}_{2}}/{N}_{{\rm{H}}{\rm{I}}}\sim 0.5$, which is beyond the region of the cloud where we estimate the molecular gas to dominate over the atomic gas based on the measured column densities. The observation of ${}^{12}\mathrm{CO}$ emission out to the region where we estimate that the molecular-to-atomic transition occurs indicates that the molecular gas is truly "CO-faint" and not "CO-dark." We note, however, that trying to use CO to estimate molecular masses at low metallicities rapidly runs into the problem of the fast growth in the conversion factor and its steep dependence on local conditions, nominally ${A}_{V}$ or column density.

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