The Dust and [C ii] Morphologies of Redshift ∼4.5 Sub-millimeter Galaxies at ∼200 pc Resolution: The Absence of Large Clumps in the Interstellar Medium at High-redshift

We determine the far-infrared luminosity of the galaxies in our sample by fitting modified blackbodies to their spectral energy distributions, including (deblended) 250, 350, 500 μm flux densities (see Swinbank et al. 2014). We adopt an average dust temperature of T_d = 50 ± 4 K, a dust emissivity index of β = 1.5, and assume the dust is optically thick at λ = 70 μm. The choice of dust temperature is motivated by recent studies by Faisst et al. (2017) and E. Cooke et al. (2018, in preparation), both of which suggest that high-redshift galaxies with high specific star formation rates have higher characteristic dust temperatures than redshift z ∼ 2 SMGs (T_d ∼ 35 K e.g., Chapman et al. 2005; Swinbank et al. 2012; Weiß et al. 2013). E. Cooke et al. (2018, in preparation) stack Herschel PACs and SPIRE photometry (including from 100, 160, 250, 350, 500 μm) and ALMA 870 μm continuum measurements of thirteen z ∼ 4.5 ALMA SMGs with similar selection criteria to our sample, and thereby show that the ALMA SMGs at redshift z ∼ 4.4 have characteristic dust temperatures of T_d = 50 ± 4 K. We note that the far-infrared luminosity is sensitive to the dust temperature, where a lower dust temperature will result in a lower far-infrared luminosity. In Table 1, we provide the far-infrared luminosities from from the best-fit modified blackbodies.

We calculate the dust masses using the measured continuum flux from ALMA listed in 1 and ${M}_{{\rm{d}}}={S}_{\nu }{D}_{{\rm{L}}}^{2}/(\kappa {B}_{\nu }({T}_{{\rm{d}}})(1+z))$, where $\kappa {B}_{\nu }({T}_{{\rm{d}}})$ is the Planck function modified by the dust-absorption coefficient of 0.076 m² kg⁻¹ (James et al. 2002), which has been corrected from the rest-frame wavelength of ∼160 μm to observed wavelength of ∼870 μm, assuming β = 1.5, D_L is the luminosity distance and S_ν is the observed flux density at frequency ν. We adopt a characteristic dust temperature of 50 K and β = 1.5 (Table 3). Here, we only use a single modified blackbody; however, the dust mass for ALESS 73.1 was determined by Swinbank et al. (2014) to be 9.3 ± 0.6 × 10⁸ M_⊙ for a multi-component model. This difference in masses is likely due to the different dust temperatures and assumed β-values, combined with the fact that the multi-component model traces a larger fraction of the dust mass at multiple temperatures. None of the other three SMGs have previously derived dust masses.

Adopting a single gas-to-dust mass ratio of δ_GDR = 90 ± 25 (Swinbank et al. 2014), we estimate the gas masses (Table 3). Also listed in Table 3 are gas masses estimated using the [C ii] luminosities and the scaling relation: ${M}_{\mathrm{gas}}=10\,\pm 2\times ({L}_{[{\rm{C}}{\rm{II}}]}/{L}_{\odot })$ (Swinbank et al. 2012). The two independent methods of estimating the gas masses result in masses agreeing within the uncertainties.

Lower-resolution ALMA studies have demonstrated that these four SMGs are all bright [C ii] emitters (Swinbank et al. 2012; De Breuck et al. 2014). By using uv-tapering, we recovered 80%–100% of the continuum flux detected in ALMA cycle 0/1. However, uv-tapering recovers emission only in the image plane, and does not improve the signal-to-noise ratio of the spectral line. Therefore, we search for [C ii] emission in our 003 observations and select extraction apertures to maximize the recovered signal-to-noise ratio of the line emission. This results in the recovery of modest, significant (2.7–4.7σ) [C ii] emission lines (see Figure 2), with measured rms values for the spectra of 2.0–7.4 mJy in 130–210 km s⁻¹ channels (see Table 4). Figure 2 shows the moment-zero maps with a uv-taper of 500 kλ (017 × 016) and compares the recovered [C ii] emission to the spectra from the lower-resolution observations from ALMA Cycles 0 and 1 (Swinbank et al. 2012; De Breuck et al. 2014).

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ALESS 61.1 and ALESS 65.1 were detected in [C ii] emission in ALMA Cycle 0; we recover between 90 and 100% of the velocity integrated line flux at 003 resolution, using apertures with diameters of 06 and 072 respectively.

For ALESS 73.1 at 003 resolution (Figure 2), our observations recover only ∼20% of the peak flux emission in the 05 resolution map from De Breuck et al. (2014). To compare with the emission line profiles detected in the 05 resolution data (De Breuck et al. 2014), we simply scale the peak of the 05 resolution [C ii] line to that of the 003 emission line (i.e., multiply by 0.2). This results in the red wing of the [C ii] emission line we detect being consistent with the red wing of the [C ii] emission line from the 05 resolution data before downscaling.

Emission from [C ii] was detected for UDS 47.0 as a very broad line at ∼351 GHz in the shallower 03 data from ALMA Cycle 1 (Simpson et al. 2015b, 2017). In our deeper 003 resolution observations, we detect a broad ∼4σ [C ii] emission line. Adopting an optimized aperture size of 06, we recover the full flux seen in the shallower low-resolution observations from ALMA Cycle 1.

4.2.1. Continuum

The natural-weighted resolution 345 GHz dust continuum maps recover between 44% and 81% of the continuum flux detected at lower resolution. This suggests that around 40 ± 20% of the flux has been resolved out at 003 resolution, compared to the Cycle 0/1 maps. To estimate the size of the rest frame 160 μm emission in the SMGs, we determine the behavior of the amplitude as a function of uv-distance. We do this by first aligning the phase center of our cubes with the source position listed in Table 1. We then radially average the data in 75 kλ bins to extract the amplitudes as a function of uv-separation. For the continuum, a binning of 75 kλ is the most optimal to minimize the scatter; however, the overall trend of the amplitude as a function of uv-distance is independent of the binning. Figure 3 shows the amplitude as a function of the uv-distance for the inner 1500 kλ in each of the four SMGs.

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For a point source, the observed amplitudes will be constant as a function of uv-distance, whereas the amplitude declines at larger uv-distances for an extended source. Hence, the physical size of the source can be determined from the FWHM of a half-Gaussian profile fit to these uv-profiles. In that case, the total flux is represented by the peak value of the half-Gaussian fit. As Figure 3 shows, the amplitude declines as a function of uv-distance for the continuum emission in all four SMGs, and they are therefore consistent with a centrally peaked brightness profile, meaning that the sources are resolved. We add the low-resolution observations from Cycle 0/1 to the plots at the uv-distance corresponding to the LAS of the observations and the single dish flux at 0 kλ, as these represent our best estimate of the total integrated flux. We fit a half-Gaussian profile plus a constant (representing a point source) to the amplitudes, in order to measure the physical size of the emission and establish whether a point source is present. The fitted FWHMs converted into physical sizes are listed in Table 2.

We find that the continuum point source components of the fits are non-zero for all four sources, with flux densities of 0.4–0.7 mJy. This suggests that, on average, about ∼14% of the total continuum flux in each source is emitted from a component with a size ≲200 pc.

There is a published size for the 330 GHz continuum, reported for ALESS 73.1 of 029 ± 006 (De Breuck et al. 2014). To compare with this, we derive a size from a single Gaussian fit without a point source of 038 ± 005, which is consistent.

As already stated, we only recover the total continuum flux density from the low-resolution observations in the uv-tapered map by applying an outer taper of ∼350 kλ. Figure 3 illustrates that, because the amplitudes only diverge from the constant value of 0.4–0.7 mJy at uv-distances of ≲350 kλ, only a strong uv-taper gives enough weight to the shortest baselines to lower the resolution sufficiently to make a significant difference in the recovered flux density.

4.2.2. [C ii] Emission

From the half-Gaussian profile fits in Figure 3, we measure the median continuum size to be 032 ± 003 and the [C ii] size to be 065 ± 015 (see Tables 2 and 4). The measured size ratio for our sample suggest that, on average, the [C ii] emitting gas is 2.1 ± 0.4 times more extended than the continuum emitting dust. The fact that the continuum sizes are smaller than the LAS allows us to recover between 80 to 100% of the flux detected in ALMA cycle 0/1. The [C ii] sizes, however, are larger than the LAS, meaning it is not possible to recover the emission distributed on scales larger than the LAS. This results in low signal-to-noise [C ii] emission lines and low-significance moment-zero maps (Figure 2).

Figure 4 compares our estimated [C ii] and rest-frame 160 μm continuum sizes for our SMGs. It also shows the [C ii] and rest-frame 160 μm dust continuum sizes for four quasars at z = 4.6–7.1 (Wang et al. 2013; Kimball et al. 2015; Díaz-Santos et al. 2016; Venemans et al. 2017), a starburst galaxy at z = 3.4 (Nesvadba et al. 2016), a Lyα Blob at z = 3.1 (Umehata et al. 2017), and LBGs at z = 5.3–6.1 (Capak et al. 2015; Jones et al. 2017). The [C ii] and rest-frame 160 μm dust continuum observations have been taken at the same spatial resolution in each source, but this varies between 02 and 1''. These observations appear to support the conclusion that [C ii] emitting components are more extended than the rest-frame 160 μm dust components in a majority of the systems.

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Although the resolution of these studies is ∼6–30 times lower than our observations, the relative sizes of the [C ii] and rest-frame 160 μm dust emission still suggest that the [C ii] emitting gas is more extended than the rest-frame 160 μm continuum. The weighted mean of the [C ii] to rest-frame 160 μm dust continuum size, including our four SMGs and the comparison sample, is 1.6 ± 0.4. Only three of the 18 galaxies have apparently larger rest-frame 160 μm continuum than [C ii] sizes and therefore lie off this relation. Of these, only one is significantly different: a lensed starburst galaxy, where the relative sizes are sensitive to the details of the lens model. The fact that the majority of the galaxies follow a trend, although they are very different populations with different gas masses and active galactic nucleus (AGN) luminosities, suggests that these global physical parameters are unlikely to account for the observed size differences. This means that the nature of the dominant heating source (whether, for example, it is AGN or starburst activity) does not appear to significantly influence the relative size of the rest-frame 160 μm dust continuum and [C ii] emitting gas.

At these high redshifts (z ≃ 4.5), the temperature of the cosmic microwave background (CMB) is ≃15 K. This means that, if the star-forming dust has similar temperature to the CMB, it will not be detectable (da Cunha et al. 2013; Zhang et al. 2016). We note that, given that the dust temperature is higher than the background CMB, this means that the CMB is unlikely to be the reason why the [C ii] emission is 1.6 times more extended than the rest-frame 160 μm dust emission.

The [C ii] emission line is one of the brightest cooling lines of the interstellar medium and traces the ionized, neutral, and molecular gas. It is therefore a good tracer of the gas dynamics in high-redshift galaxies (e.g., Carniani et al. 2013; Capak et al. 2015). Only ALESS 73.1 and UDS 47.0 have low-resolution observations from Cycle 0/1 deep enough to allow us to search for possible velocity gradients. From the study of De Breuck et al. (2014), it is already known that the gas in ALESS 73.1 has a rotating configuration, and the broad line of UDS 47.0 suggests that a velocity gradient may also be present there.

To investigate the velocity gradient in these two SMGs, we make moment-zero maps (i.e., narrow-band images) in the low-resolution (from ALMA Cycle 0/1, see Table 1) continuum-subtracted cube of the channels covering the [C ii] emission. We make two independent maps; one of the redshifted half of the line and the other of the blueshifted half. These cover ±400 km s⁻¹ for UDS 47.0 and ±200 km s⁻¹ for ALESS 73.1. We find that the peak of the [C ii] emission shifts by 025 ± 004 (∼1.7 ± 0.3 kpc) between the red and blue halves of the line for UDS 47.0 and 024 ± 001 (∼1.5 ± 06 kpc) for ALESS 73.1. This implies a velocity gradient across the [C ii] emitting gas in both SMGs.

Having established the presence of a velocity gradient, and using a disk model for the dynamics, we estimate the dynamical masses (${M}_{\mathrm{dyn}}\sin (i)=R\times {v}^{2}/G$) of ALESS 73.1 and UDS 47.0 within a region given by twice the size of the [C ii] sizes listed in Table 2 corresponding to R = 5 kpc and R = 4 kpc for ALESS 73.1 and UDS 47.0, respectively. Using the line widths of the [C ii] lines detected in ALMA Cycle 0/1 (see Table 1), this yields dynamical masses of $3.7\pm 0.7\times {10}^{10}\sin (i)\,{M}_{\odot }$ for ALESS 73.1 and $20\pm 4\,\times {10}^{10}\sin (i)$ M_⊙ for UDS 47.0. Using a similar disk model, De Breuck et al. (2014) estimate an inclination angle of i = 50° ± 8 for ALESS 73.1, which is similar to the average inclination angle calculated by Law et al. (2009). By assuming the same inclination angle for UDS 47.0 and the gas masses listed in Table 3, we estimate an average gas mass fraction within the half-mass radii assumed to calculate the dynamical masses of 0.4 ± 0.2. This is in agreement with the result from Tacconi et al. (2018) for redshift ≲4 star-forming galaxies when converting to the same units.

Table 3.  Dust and Gas Masses

Source name	Mdust	Mgas	${M}_{\mathrm{gas}}^{[{\rm{C}}\,{\rm{II}}]}$
	[108 M⊙]	[1010 M⊙]	[1010 M⊙]
ALESS 61.1	2.9 ± 0.6	2.6 ± 0.9	1.5 ± 0.4
ALESS 65.1	2.8 ± 0.6	2.6 ± 0.9	3.2 ± 0.8
ALESS 73.1	4.3 ± 0.8	3.9 ± 1.3	4.9 ± 1.0
UDS 47.0	5.9 ± 1.1	5.3 ± 1.8	2.6 ± 0.7

Note. The estimated dust and gas masses based on the ALMA Cycle 0/1 observed 345 GHz dust continuum, extrapolated to rest-frame assuming β = 1.5 and [Cii] fluxes. The dust masses (M_dust) are calculated using the 345 GHz continuum flux, which are then scaled using a gas-to-dust mass ratio of 90 ± 25 to achieve the gas masses (M_gas). Gas masses estimated using the [C ii] fluxes (M${}_{\mathrm{gas}}^{[{\rm{C}}\,{\rm{II}}]}$) are likewise listed, and agree with the gas masses estimated using the dust mass.

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Figure 1 demonstrates that three SMGs in our sample (ALESS 61.1, ALESS 65.1, and ALESS 73.1) show smooth 345 GHz dust continuum morphology, while one (UDS 47.0) appears to have structure. However, the apparent structures seen for UDS 47.0 have significance levels of just 3.9–5.4σ. Similarly, a recent high-resolution (003) resolution 345 GHz dust continuum study by Iono et al. (2016) of three z ∼ 4.3 AzTEC SMGs claimed to reveal about 40 ≥ 3σ 200 pc clumps. These visually identified structures are similar to the features we see in UDS 47.0.

To test whether the structures in UDS 47.0 are likely to be real, or if they could arise from noise in smooth disk light profiles, we model a set of observations of smooth profiles. We use the casa tasks simobservations and simanalysis to create a library of simulated interferometry observations of exponential disks as they would appear if observed with ALMA in the same configuration as used for our observations and with similar noise properties (following the example of Hodge et al. 2016). Our 50 input models of smooth exponential disk models have Sérsic indices of n = 1 (comparable to what we see in our sample) and flux densities and sizes of 8.7 ± 0.6 mJy and 028 ± 003, as seen for UDS 47.0 (Simpson et al. 2015a). The results of these simulations also reveal apparently clump-like structures (Figure 5). While these structures qualitatively look similar to those seen in UDS 47.0, we attempt to quantitatively compare the flux distribution between the simulated maps and the observed map for UDS 47.0. We do this by fitting single smooth profiles (with the Sérsic index as a free parameter, to the simulated maps), and subtracting the best fit model. For the central part of the residual image, the number of pixels as a function of the flux has a Gaussian profile with a tail of excess emission at positive values. This excess flux should represent the emission seen in possible structures, and we isolate it by subtracting a one-dimensional Gaussian profile fit to the histogram.

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We apply this analysis to both the simulated smooth disks and the observation of UDS 47.0. A Kolmogorov–Smirnov test comparing the average of the residual pixel distribution from the simulated smooth disks to that of the observations reveals that the likelihood of the observed map being drawn from the simulated library of smooth disks is ∼70%. Applying the same analysis to the three other SMGs in our sample shows that they are consistent with smooth morphologies.

Our simulated library of smooth disks, in combination with the apparently smooth morphology of three out of four SMGs in our sample, illustrates that smooth disks can appear to have substructures when observed at high resolution and with sparse coverage of the inner part of the uv-plane. We conclude that it is not possible to rule out the hypothesis that all four SMGs in our sample are smooth exponential disks.

We note that the structures identified by Iono et al. (2016) in their sources have similar significances to those seen in UDS 47.0. Moreover, only ∼30% of the continuum flux detected at 07 resolution with the SMA (Younger et al. 2008; Iono et al. 2016) has been recovered in their 003 ALMA maps, with less than 20% of that in the claimed clumps. The fact that the resolution and depth of the observations in Iono et al. (2016) are similar to our 003 maps, and that their claimed structures contain only a small fraction of the total flux, casts doubt on whether their clumps are real structures either.

As noted above, one of the sources in our sample (UDS 47.0) appears to have a clumpy morphology in our high-resolution ALMA continuum maps; however, we have suggested that this is not statistically significant. Nevertheless, we can also ask if we should expect to see sub-structures at this resolution, given the estimated star formation rate surface densities of these galaxies.

The average sizes of star-forming clumps in a self-gravitating gas disk are given by the Jeans length,

$$\begin{eqnarray}&&{\lambda }_{J}\approx \displaystyle \frac{{\sigma }^{2}}{G{{\rm{\Sigma }}}_{\mathrm{gas}}},\end{eqnarray} \tag{ 1 }$$

where G is the gravitational constant, Σ_gas is the gas surface density, and σ the velocity dispersion of the gas within a clump (Toomre 1964). We estimate Σ_gas using:

$$\begin{eqnarray}&&\left(\displaystyle \frac{{{\rm{\Sigma }}}_{\mathrm{SFR}}}{{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}}\right)=A{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{\mathrm{gas}}}{{M}_{\odot }{\mathrm{pc}}^{-2}}\right)}^{n},\end{eqnarray} \tag{ 2 }$$

where A = 1.5 × 10⁻⁴ and n ≃ 1.5 (Kennicutt 1998; Swinbank et al. 2012). For clumps to have a size of ≳200 pc (and thus be observable at the resolution of our observations at the estimated gas surface density), the velocity dispersions within the gas disk have to be σ ≳ 60 km s⁻¹ for ALESS 61.1, which has the lowest estimated gas surface density of our sample, and σ ≳ 85 km s⁻¹ for UDS 47.0, which has the highest gas surface density.

A recent high-resolution (∼003) observation of the lensed SMG SDP.81 (ALMA Partnership et al. 2015; Hatsukade et al. 2015) measured the velocity dispersions in regions within the gas disk in this system to be in the range of 11–35 km s⁻¹ (Swinbank et al. 2015). The velocity dispersion of the gas disk in ALESS 73.1 was likewise estimated to be 40 ± 10 km s⁻¹ (De Breuck et al. 2014). Thus, the required velocity dispersions to observe clumps at 200 pc resolution are 1.5–2 times higher than those observed in other SMGs. Hence, if our sources have velocity dispersions comparable to that observed for other SMGs, then any clumps in their gas disks would have sizes below the resolution limit of our ALMA observations. This suggests that the clumps in UDS 47.0, if real, are unlikely to represent self-gravitating physical structures.

We now turn to the overall energetics of these systems and their cooling. As noted earlier, emission from [C ii] is a major contributor to the gas cooling, carrying 0.1%–1% of the far-infrared luminosity in luminous starburst galaxies (Stacey et al. 1991; Brauher et al. 2008; Graciá-Carpio et al. 2011). The [C ii] to far-infrared luminosity ratio (${L}_{[{\rm{C}}{\rm{II}}]}$/${L}_{\mathrm{FIR}}$) varies with far-infrared luminosity in local galaxies, such that the most far-infrared luminous galaxies have a lower ${L}_{[{\rm{C}}{\rm{II}}]}$/${L}_{\mathrm{FIR}}$ ratio. Figure 6 shows ${L}_{[{\rm{C}}{\rm{II}}]}$/${L}_{\mathrm{FIR}}$ as function of star formation rate surface density for the low-redshift KINGFISH sample (Kennicutt et al. 2011; Dale et al. 2016; Croxall et al. 2017), the GOALS sample (Díaz-Santos et al. 2013; Lutz et al. 2016), and a high-redshift sample of SMGs (Gullberg et al. 2015; Lutz et al. 2016, typically lensed).

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At the highest star formation rate surface densities (i.e., typically smaller sizes, FWHM ∼ 1–3 kpc), we see that the lowest ${L}_{[{\rm{C}}{\rm{II}}]}$/${L}_{\mathrm{FIR}}$ ratios are frequently associated with AGNs, while normal star-forming galaxies (including Lyman Break galaxies) have higher ${L}_{[{\rm{C}}{\rm{II}}]}$/${L}_{\mathrm{FIR}}$ ratios.

We see in Figure 6 that the ${L}_{[{\rm{C}}{\rm{II}}]}$/${L}_{\mathrm{FIR}}$ ratios for our sample agree with the high-redshift comparison sample, but both show a large scatter when compared to the local galaxies from KINGFISH and GOALS. Figure 6 shows that the high-redshift sources generally have higher ${L}_{[{\rm{C}}{\rm{II}}]}$/${L}_{\mathrm{FIR}}$, compared to local galaxies at a fixed far-infrared luminosity.

To investigate if the [C ii] deficit is due to a local or global process, we plot two points for each of our SMGs: a core measurement from an aperture the same size as the 345 GHz dust continuum (Table 2); and an annulus the size of the optimized [C ii] aperture listed in Table 4. The [C ii] luminosities are calculated by extracting two spectra: one within the continuum aperture and one within the [C ii] aperture. The [C ii] luminosity in the annulus is then given by the difference between the two luminosities. Note that, for ALESS 65.1, the [C ii] emission line is undetected in the core, meaning that the ${L}_{[{\rm{C}}{\rm{II}}]}$/${L}_{\mathrm{FIR}}$ ratio for the core is an upper limit—and therefore a lower limit within the annulus. We scale the far-infrared luminosities and star formation rates according to the fraction of emission we recover within the 345 GHz dust continuum apertures (see Table 2), and assume that the remaining fraction originates from the annulus. The star formation rate surface densities are then calculated using the areas of the continuum apertures for the core measurement and the difference between the [C ii] and continuum apertures for the annuli.

Table 4.  Properties of the [C ii] Emission Lines Detected at 003 Resolution

Source	rms	S/N	${{SdV}}_{[{\rm{C}}{\rm{II}}]}$	FWHM${}_{[{\rm{C}}\,{\rm{II}}]}^{\mathrm{line}}$	FWHM${}_{[{\rm{C}}\,{\rm{II}}]}^{\mathrm{uv}}$	Aperture	Recovered
	(mJy)		(Jy km s−1)	(km s−1)	(arcsec)	(arcsec)	(%)
ALESS 61.1	7.4	2.7	4.6 ± 1.7	280 ± 110	1.1 ± 0.4	0.72	180 ± 80
ALESS 65.1	4.0	4.7	4.9 ± 1.0	270 ± 70	0.6 ± 0.2	0.6	90 ± 20
ALESS 73.1	2.0	3.0	1.6 ± 0.5	270 ± 110	0.7 ± 0.1	0.72	22 ± 7
UDS 47.0	4.4	3.1	6.8 ± 1.8	590 ± 250	0.3 ± 0.1	0.6	160 ± 50

Note. Column 2: rms of the [C ii] spectra. Column 3: signal-to-noise ratio of the [C ii] emission lines. Column 4: velocity-integrated line fluxes. Column 5: FWHM of the [C ii] line velocity width. Column 6: spatial FWHM given by the Gaussian fit to the amplitude as a function of uv-distance. Column 7: diameter of the aperture used to measure the line flux. Column 8: percentage of recovered line flux.

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We use these measurements to investigate the variations of the ${L}_{[{\rm{C}}{\rm{II}}]}$/${L}_{\mathrm{FIR}}$ ratio within our SMGs. The expanded part of Figure 6 shows the trend between the core measurements and the annuli. All four of the galaxies show the same behavior: the core has lower ${L}_{[{\rm{C}}{\rm{II}}]}$/${L}_{\mathrm{FIR}}$ and higher star formation rate surface density than the surrounding annulus. This follows the relation seen by Smith et al. (2017) for local star-forming galaxies. A higher star formation rate surface density in the core regions compared with the outer annuli is expected, but the same expectation is not true for the ${L}_{[{\rm{C}}{\rm{II}}]}$/${L}_{\mathrm{FIR}}$ ratio.

The fact that our SMGs follow the same trend seen by, for example, Smith et al. (2017), implies that the [C ii] deficit in SMGs is also due to a local process, where the core regions have a higher [C ii] deficit than the regions further out from the core.

Smith et al. (2017) suggest that the [C ii] deficit is related to the metallicity of the gas, where a low metallicity results in a high ${L}_{[{\rm{C}}{\rm{II}}]}$/${L}_{\mathrm{FIR}}$ ratio. Alternatively, a recent study by Muñoz & Oh (2016) explored the possibility of [C ii] saturation. This hypotheses had been proposed before, but had not been investigated in detail (e.g., Stacey et al. 2010; Díaz-Santos et al. 2013; Magdis et al. 2014; Gullberg et al. 2015). Muñoz & Oh (2016) show that the [C ii] emission can be thermally saturated at high temperatures. At gas temperatures >92 K (the ground state temperature of [C ii]), the [C ii] cooling rate becomes constant, forcing the gas to cool through other channels (e.g., the [O i]63 μm fine structure line). This means that the [C ii] emission line saturates and a [C ii] deficit can therefore occur as a result of the further increase of the far-infrared luminosity. By considering the specific [C ii] luminosity (the [C ii] luminosity to [C ii] mass ratio, ${L}_{[{\rm{C}}{\rm{II}}]}$/${M}_{[{\rm{C}}{\rm{II}}]}$) and specific far-infrared luminosity (the far-infrared luminosity to gas mass ratio, ${L}_{\mathrm{FIR}}$/M_gas), Muñoz & Oh (2016) predict an ${L}_{[{\rm{C}}{\rm{II}}]}$/${L}_{\mathrm{FIR}}$ relation dependent on the infrared surface density (Σ_IR) and fraction of the gas mass in ionized carbon (${f}_{[{\rm{C}}{\rm{II}}]}={M}_{[{\rm{C}}{\rm{II}}]}/{M}_{\mathrm{gas}}$):

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{[{\rm{C}}{\rm{II}}]}}{{L}_{\mathrm{FIR}}}\sim 2.2\times {10}^{-3}\displaystyle \frac{{f}_{[{\rm{C}}{\rm{II}}]}}{0.13}{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{\mathrm{IR}}}{{10}^{11}{L}_{\odot }{\mathrm{kpc}}^{-2}}\right)}^{-1/2}.\end{eqnarray} \tag{ 3 }$$

Here, ${f}_{[{\rm{C}}{\rm{II}}]}$ is estimated to between 0.10 and 0.17, assuming a fixed CO(1–0) to [C ii] luminosity ratio, the CO to H₂ conversion factor (${\alpha }_{\mathrm{CO}}$), a gas density higher than the critical density of [C ii] (${n}_{\mathrm{crit}}=2.7\times {10}^{3}$ cm⁻³, e.g., Stacey et al. 2010), and a temperature higher than 92 K. This fraction does not take into account the fact that some of the carbon is in the neutral phase, meaning that the actual mass fraction is likely to be lower. The relation between the [C ii] to far-infrared luminosity ratio as a function of the star formation rate surface density found by Muñoz & Oh (2016) adopts fixed values of G = 100 and ${n}_{\mathrm{gas}}={10}^{4}$ cm⁻³ for the radiation field strength and the gas density. We plot the predicted trend line on Figure 6 for ${f}_{[{\rm{C}}{\rm{II}}]}=0.03$ and 0.14, which span the range of the data, suggesting that fraction of the total mass in ionized carbon is between 3% and 14%.

However, other studies—for example, Díaz-Santos et al. (2017) and Lagache et al. (2018)—argue the [C ii] deficit arises from other factors. In particular, they emphasize the importance of a varying radiation field strength to gas density ratio ($G/{n}_{\mathrm{gas}}$). Díaz-Santos et al. (2017) fit a power law to local ULIRGs from GOALS (see Equation (3) in Díaz-Santos et al. 2017) and find that the suppression of the [C ii] to far-infrared luminosity ratio at high far-infrared surface densities could be due to high $G/{n}_{\mathrm{gas}}$ ratios. Using semi-analytical models, Lagache et al. (2018) likewise suggest that the [C ii] to far-infrared deficit is correlated with the intensity of the interstellar radiation field. This suggests that the star formation rate surface density is dependent on the geometry of the photon dominated regions and the distribution of the gas and dust within it.

We show the relation between the [C ii] to far-infrared luminosity ratio and the star formation rate surface density with ${f}_{[{\rm{C}}{\rm{II}}]}=0.10\mbox{--}0.17$ in Figure 6, along with the power-law relation determined by Díaz-Santos et al. (2017). Both the power law from Díaz-Santos et al. (2017) and the model of Muñoz & Oh (2016) are in agreement with our observations. This means that it is not possible, with the existing data, to distinguish between the two models and determine if the [C ii] deficit can be explained by a saturation of the [C ii] emission at high temperatures and densities in the dense core regions of the SMGs or a high ratio of radiation field strength to gas density. We also note that contributions from other local processes (e.g., [C ii] self-absorption, dust extinction, and dust grain charge), may also play a role in the deficit (Smith et al. 2017).