Strong Far-ultraviolet Fields Drive the [C ii]/Far-infrared Deficit in z ∼ 3 Dusty, Star-forming Galaxies

The two galaxies analyzed in this paper were identified by Hodge et al. (2013) as a part of the ALESS survey. The ALESS survey was an ALMA Cycle 0 870 μm imaging campaign targeting all 126 sources discovered in the LABOCA Extended Chandra Deep Field South Submillimeter Survey (LESS; Weiß et al. 2009). With ALMA Cycle 0 observations providing a significant improvement over the LABOCA map in both resolution (beam area reduced by a factor of ∼200) and sensitivity (increased by a factor of ∼3), the ALESS survey identified 99 distinct submillimeter-bright galaxies in its primary sample (Hodge et al. 2013). Out of these, at the time of the proposal (2015 April) only four—ALESS 49.1, ALESS 57.1, ALESS 67.1, and ALESS 122.1—had robust redshifts that allowed for ALMA observations of both the low-J CO and [C ii] lines. The spectroscopic redshifts were acquired using VLT-FORS2/VIMOS and Keck-DEIMOS and are based on multiple line detections (Danielson et al. 2017).

In this paper, we present the ALMA Band 8 observations targeting the [C ii] line and rest-frame 160 μm continuum. The corresponding Band 3 observations, targeting the CO (3–2) (ν_rest = 345.795 GHz) emission, were recently presented by Calistro Rivera et al. (2018).

The observations were carried out as part of the ALMA Cycle 3 Project #2015.1.00019.S (PI: J. Hodge) on 2016 August 12. Only ALESS 49.1 and ALESS 57.1 ([C ii] line in ALMA Band 8) were observed; ALESS 67.1 (z = 2.12) and ALESS 122.1 (z = 2.02) have the [C ii] line in ALMA Band 9 and were not observed. The total time including calibration and overheads was 72 minutes, with an on-source time of 11 minutes per target. The array configuration consisted of 38 12 m antennas, with baselines extending up to 1400 m. The largest angular scale¹³ of the observations is ∼19 for both sources. The primary beam FWHM is 141. Synthesized beam sizes and σ_rms for the 160 μm continuum and the [C ii] emission are listed in Table 1. The target elevation range was 66°–73° for ALESS 49.1 and 67°–76° for ALESS 57.1.

Table 1.  Source Positions (Corresponding to the 160 μm Continuum Surface Brightness Maximum), Redshifts, ALMA Band 8 Naturally Weighted Beam Sizes and Noise Levels, Observed 160 μm Continuum/[C ii] Flux Densities, and the FWHM of the [C ii] and CO (3–2) Lines

Source		ALESS 49.1	ALESS 57.1
R.A. (J2000)		3:31:24.71	3:31:51.94
Decl. (J2000)		−27:50:46.9	−27:53:27.0
za		2.943 ± 0.001	2.943 ± 0.002
Beam FWHM	(arcsec)	0.16 × 0.12	0.16 × 0.12
Beam PA	(deg)	53	56
${S}_{160\mu {\rm{m}}}$	(mJy)	10.7 ± 1.0	8.2 ± 0.5
${I}_{160\,\mu {\rm{m}}}^{\max }$	(mJy beam−1)	3.42 ± 0.16	1.53 ± 0.17
S[C ii]a	(mJy)	15.4 ± 2.8	8.8 ± 2.5
${I}_{[{\rm{C}}\,{\rm{II}}]}^{\max }$ b	(mJy beam−1)	2.0 ± 0.4	3.0 ± 0.4
FWHM[C ii]	(km s−1)	600 ± 130	390 ± 70
FWHMCOa	(km s−1)	610 ± 30	360 ± 90

Notes. The redshifts are based on CO (3–2) observations (Calistro Rivera et al. 2018). The spatially integrated 160 μm and [C ii] flux densities are measured within a circular aperture with a 10 diameter, centered on the continuum surface brightness maximum.

^aAdopted from Calistro Rivera et al. (2018). ^bIntegrated over 800 and 710 km s⁻¹ for ALESS 49.1 and ALESS 57.1, respectively.

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The frequency setup was configured in four spectral windows (SPWs) in Band 8. The individual SPWs were centered at 481.953, 483.183, 493.506, and 495.386 GHz. Each SPW was split into 480 frequency channels 3.906 MHz wide, giving a total bandwidth of 1.875 GHz per SPW. The radio-velocity resolution was 2.42 km s⁻¹. Both the Stokes XX and YY parameters were observed.

The data were calibrated using the standard ALMA pipeline, with additional flagging necessary to remove atmospheric features (see Section 3.1). All the visibility data processing apart from imaging was performed using Casa versions 4.7 and 5.0 (McMullin et al. 2007).

The spectral structure of the [C ii] line overlaps with atmospheric absorption features at 481.2 and 481.6 GHz. Consequently, the noise level in the affected channels is raised by a factor of ∼2. This issue is particularly severe for ALESS 49.1. Another atmospheric feature is located in a line-free SPW at 496.35 GHz; here the affected channels were flagged to improve the continuum signal-to-noise ratio (S/N). As will be outlined in Section 3.1, we use a channel-dependent threshold for the deconvolution process to avoid introducing noise features from the affected channels.

To increase the S/N, each line was split into frequency bins ∼120 km s⁻¹ wide. Additionally, we time-averaged the data using a 30 s bin; this corresponds to an average intensity loss of <0.5% at 5 arcsec from the phase-tracking center, which we consider negligible.¹⁴ The time-bin size was chosen so as to prevent significant time-averaging smearing. The two linear polarizations were combined into the Stokes intensity I.

ALMA Band 3 observations of the CO (3–2) and rest-frame 1.0 mm continuum in ALESS 49.1 and ALESS 57.1 were presented by Calistro Rivera et al. (2018). These consisted of Cycle 2 observations of ALESS 49.1 and ALESS 57.1 (Project #2013.1.00470.S; PI: J. Hodge) at 0.34–067 resolution and additional Cycle 4 observations of ALESS 49.1 (Project #2016.1.00754.S; PI: J. Wardlow) at 11 resolution. The naturally weighted Band 3 synthesized beam size is 0.69 × 063 for ALESS 49.1 (after concatenating the Cycles 2 and 4 data) and 0.67 × 060 for ALESS 57.1, with a continuum σ_rms of 17.6 and 19.5 μJy beam⁻¹, respectively. For a detailed description of the data and the resulting analysis, we refer the reader to Calistro Rivera et al. (2018).

3.1.1. Imaging

3.1.2. ALMA 160 μm Continuum

The rest-frame 160 μm continuum is detected in both sources at >10σ confidence (Figure 1). In both sources, the continuum emission shows only a single brightness peak at 015 resolution. In ALESS 49.1, the continuum emission is almost circularly symmetric, with a minor extension in the east–west direction. In ALESS 57.1, the continuum is noticeably extended in the east–west direction (axis ratio ∼ 2:1). The spatially integrated 160 μm continuum flux density for ALESS 49.1 and ALESS 57.1 is given in Table 1. The 160 μm flux density is calculated from an aperture with a 10 diameter, centered on the 160 μm continuum surface brightness maximum. The aperture size was chosen based on the FIR continuum and [C i] sizes determined from the u-v plane fitting (Section 3.2), as we do not expect a significant surface brightness contribution ≥05 from the center of the source.

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3.1.3. [C ii] Emission

We estimate the size of the continuum and [C ii] emission regions by directly fitting the observed visibility function. Given the relatively low S/N of the data, we assume the surface brightness distribution to follow a circularly symmetric Gaussian profile.

The size is measured using the spectrally averaged continuum and line data sets. To correct the offset between the phase-tracking center and the centroid of the surface brightness distribution given in Table 1, each data set is phase-shifted to center the FOV on the centroid of the source. The data are then radially binned into bins of equal width. To test the robustness of the u-v plane fitting against the u-v bin size, we vary the u-v bin size from 5 to 50 kλ.

We fit each u-v plane data set with (1) a single Gaussian profile; (2) two Gaussian components to investigate the possibility of having compact, bright [C ii] emission embedded in an extended, low surface brightness component; and (3) a combination of a single-Gaussian profile and a constant term, corresponding to a point source in the image plane. To determine whether the two-component model significantly improves the goodness of fit compared to the one-component model, we compare the two models using the F-test (Bevington & Robinson 2003). The single-component model is preferred for the continuum emission in ALESS 49.1 and [C ii] emission in ALESS 57.1, independently of the u-v bin size. We will address the robustness of the inferred R_1/2 in Section 3.2.1. The two-component model is strongly (p > 0.95) preferred for the [C ii] emission in ALESS 49.1 and the continuum emission in ALESS 57.1. We do not find the Gaussian + constant-term (point-source) model to be preferred over the single-Gaussian model for any data set considered. The best-fitting values for the two-component model are listed in Table 2. Figure 2 shows the visibility function for the [C ii] and 160 μm continuum, as well as the CO (3–2) observations from Calistro Rivera et al. (2018) and the corresponding best-fitting profiles.

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Table 2.  Inferred Source Properties for ALESS 49.1 and ALESS 57.1

Source		ALESS 49.1	ALESS 57.1
SED Fitting
${L}_{3-2000\mu {\rm{m}}}$	(1012 L⊙)	${7.1}_{-0.9}^{+0.8}$	${7.4}_{-0.9}^{+0.9}$
SFR	(M⊙ yr−1)	${490}_{-60}^{+30}$	${480}_{-60}^{+70}$
Tdust	(K)	${46}_{-2}^{+6}$	${51}_{-4}^{+7}$
M⋆	(1010 M⊙)	${4.4}_{-0.3}^{+1.8}$	${4.3}_{-0.8}^{+1.7}$
Mgasa	(1010 M⊙)	5 ± 2	5 ± 2
Mdust	(108 M⊙)	${4.4}_{-0.7}^{+0.5}$	${4.1}_{-0.6}^{+0.5}$
Line Luminosities
L[C ii]	(109 L⊙)	3.0 ± 0.8	1.1 ± 0.4
${L}_{[{\rm{C}}\,{\rm{II}}]}^{{\prime} }$	(1010 K km s−1 pc2)	14 ± 4	5.1 ± 1.7
${L}_{\mathrm{CO}(3-2)}$ b	(109 L⊙)	0.070 ± 0.005	0.062 ± 0.016
${L}_{\mathrm{CO}(3-2)}^{{\prime} }$ b	(1011 K km s−1 pc2)	0.51 ± 0.04	0.05 ± 0.01
${L}_{\mathrm{CO}(1-0)}$ b	(106 L⊙)	2.1 ± 0.2	2.6 ± 0.7
Source sizes—Single Gaussian
${R}_{1/2}^{[{\rm{C}}\,{\rm{II}}]}$	(arcsec)	0.163 ± 0.013	0.101 ± 0.010
S[C ii]	(mJy)	10.9 ± 0.9	8.6 ± 0.9
${R}_{1/2}^{160\,\mu {\rm{m}}}$	(arcsec)	0.173 ± 0.009	0.128 ± 0.006
${S}^{160\mu {\rm{m}}}$	(mJy)	11.2 ± 0.4	7.6 ± 0.4
${R}_{1/2}^{\mathrm{CO}}$ b	(arcsec)	0.33 ± 0.5	0.39 ± 0.06
Source sizes—Two Gaussians
${R}_{1/2}^{[{\rm{C}}\,{\rm{II}}]}$ (compact)	(arcsec)	0.128 ± 0.015	⋯
${R}_{1/2}^{[{\rm{C}}\,{\rm{II}}]}$ (extended)	(arcsec)	1.1 ± 0.3	⋯
${S}^{[{\rm{C}}{\rm{II}}]}$ (compact)	(mJy)	7.4 ± 1.1	⋯
${S}^{[{\rm{C}}{\rm{II}}]}$ (extended)	(mJy)	29 ± 8	⋯
${R}_{1/2}^{160\,\mu {\rm{m}}}$ (compact)	(arcsec)	⋯	0.109 ± 0.007
${R}_{1/2}^{160\,\mu {\rm{m}}}$ (extended)	(arcsec)	⋯	0.67 ± 0.11
${S}^{160\mu {\rm{m}}}$ (compact)	(mJy)	⋯	5.8 ± 0.5
${S}^{160\mu {\rm{m}}}$ (extended)	(mJy)	⋯	7.6 ± 1.4
[C ii] and CO (3–2) Kinematics
${i}_{[{\rm{C}}{\rm{II}}]}$	(deg)	39 ± 5	58 ± 5
${M}_{\mathrm{dyn}}^{[{\rm{C}}\,{\rm{II}}]}$ (R ≤ 2 kpc)	(1010 M⊙)	6.2 ± 5.5	2.7 ± 1.6
${M}_{\mathrm{dyn}}^{\mathrm{CO}}(R\lt {R}_{1/2}^{\mathrm{CO}})$	(1010M⊙)	11 ± 2	11 ± 5

Notes. For the global source properties derived from the SED fitting (Section 3.4), we provide the median value of the posterior probability density function. The [C ii] and CO (3–2) line luminosities are spectrally integrated over the full extent of the [C ii], rather than over FWHM only (as in Calistro Rivera et al. 2018), as the [C ii] line exhibits a non-Gaussian profile. The source sizes are inferred from the u-v plane fitting (Section 3.2); we list the best-fitting one-component Gaussian model parameters, as well as the two-component Gaussian models if preferred by evidence. The source inclinations and [C ii]-based dynamical masses are inferred from the GalPak3D modeling (Section 3.3).

^aCalistro Rivera et al. (2018). M_gas estimated assuming α_CO = 1.0. ^bCalistro Rivera et al. (2018). L_line integrated over the entire line width as opposed to integrating only over FWHM as in Calistro Rivera et al. (2018).

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In physical units, for the single-Gaussian models, the [C ii] and 160 μm emission is rather compact (R_1/2 = 0.8–1.4 kpc) in both sources. The [C ii] and 160 μm continuum sizes from the single-Gaussian fitting agree within 1σ–2σ uncertainty. For the two-Gaussian models, the compact and extended [C ii] components in ALESS 49.1 have half-light radii of 1.01 ± 0.12 kpc and 8.7 ± 1.6 kpc, respectively. The compact and extended 160 μm components in ALESS 57.1 have half-light radii of 0.86 ± 0.07 kpc and 5.3 ± 1 kpc, respectively.

For the two-component [C ii] model in ALESS 49.1, the extended [C ii] component accounts for up to 80% of the total [C ii] luminosity. Note that the systematic uncertainty on this estimate might be significant, as it is unclear how much the extended component departs from the assumption of a circular Gaussian profile. For the [C ii] emission in ALESS 57.1, the single-Gaussian model is preferred. However, if we speculate that ALESS 57.1 has an extended [C ii] component with the same size as in ALESS 49.1 (${R}_{1/2}^{\mathrm{ext}}\,=\,$101), the 3σ upper limit on the total flux density contributed by this hypothetical component is 80%. Similarly, if we speculate that the 160 μm continuum in ALESS 49.1 has an extended component identical in size to that in ALESS 57.1 (${R}_{1/2}^{\mathrm{ext}}\,=\,$067), the 3σ upper limit on the flux contained in this hypothetical component is 50%.

How will the extended components in ALESS 49.1 [C ii] emission and ALESS 57.1 160 μm continuum contribute to the observed [C ii]/160 μm surface brightness distribution? The half-light radius of the ALESS 57.1 160 μm extended component is well below the maximum recoverable scale (19) and therefore should be fully accounted for in the synthesized images. This is supported by the results from the spectral energy distribution (SED) modeling (Section 3.4), which indicate that the 160 μm continuum is not significantly resolved out. For the [C ii] emission, the half-light radius of the extended component in ALESS 49.1 is comparable to the maximum recoverable scale, and some emission is likely resolved out in the synthesized images. However, while the extended [C ii] component in ALESS 49.1 dominates the total [C ii] luminosity, it contributes only between 5% and 20% of the surface brightness across the inner R < 2 kpc region. It is for this reason that our analysis in Sections 4.2–4.4 focuses on the central regions (R ≤ 2 kpc) of ALESS 49.1 and ALESS 57.1, including the uncertainty from the extended components in further analysis.

3.2.1. How Robust Are the Source Sizes Determined from the u-v Plane Fitting?

Before analyzing our resolved [C ii] and 160 μm continuum observations, we assess the reliability of the inferred source sizes. In particular, we investigate a possibility that the [C ii]/160 μm emissions follow CO (3–2) surface distribution and that the source sizes inferred in Section 3.2 are an artifact of the limited short-spacing coverage of our Band 8 observations. We estimate this uncertainty using simulated ALMA observations.

The mock observations are created as follows. First, we calculate the mean and rms scatter for the [C ii] and 160 μm continuum visibilities within a given u-v bin. We then subtract the mean signal from the data, which gives us a u-v plane coverage corresponding exactly to a given observation, along with a realistic noise measurement for each baseline. We choose this approach to account for the different noise levels for each data set and the effect of the atmospheric lines on our [C ii] data. We then inject an artificial source into the data, generating 1000 data sets with different noise realizations for each source. We consider sources with a half-light radius of 0.1–10 and peak surface brightness of 0.05–4.0 mJy beam⁻¹. Finally, we bin the mock visibilities in the u-v plane using exactly the same procedure as applied to real data in Section 3.2.

Figure 3 shows the inferred radius as a function of the surface brightness maximum alongside the 1σ uncertainty from the u-v plane fitting. The measured [C ii] and 160 μm continuum sizes in ALESS 49.1 and ALESS 57.1 are all smaller than sizes inferred for input sources with R_1/2 ≥ 02 in the relevant peak surface brightness range. In other words, given the observed peak surface brightness, ${R}_{1/2}^{[{\rm{C}}\,{\rm{II}}]}\gt 0.2\,\mathrm{arcsec}$ source sizes would be recovered within 10% uncertainty for both ALESS 49.1 and ALESS 57.1. For comparison, the CO (3–2) half-light radii in ALESS 49.1 and ALESS 57.1 are ${R}_{1/2}^{\mathrm{CO}}=0.33\pm 0.06$ arcsec and 0.39 ± 006, respectively.

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Therefore, we consider it unlikely that the [C ii]/160 μm continuum follow a single-Gaussian surface brightness distribution with the same size as CO (3–2) emission (0.33 ± 005 and 0.39 ± 006 for ALESS 49.1 and ALESS 57.1, respectively). However, we cannot exclude a combination of a bright compact and faint extended [C ii] and continuum components. In the following analysis, we will focus on the center of the sources where the extended [C ii] component is not expected to contribute significantly.

Figure 4 presents the [C ii] moment-1 (intensity-weighted velocity) maps and the comparison of [C ii]/CO (3–2) line profiles in ALESS 49.1 and ALESS 57.1. The moment-1 maps reveal a clear velocity gradient across both ALESS 49.1 and ALESS 57.1. The spectra were extracted from the naturally weighted channel maps, using an aperture 1 arcsec (∼8 kpc) in radius for CO (3–2) and 05 (∼4 kpc) in radius for [C ii], given the compact size of the [C ii] emission.

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The [C ii] line profile in ALESS 49.1 largely traces the CO (3–2) profile, exhibiting an increased brightness in the blue channels. In ALESS 49.1, we find a tentative (2.5σ–3σ) increase in the [C ii]/CO (3–2) ratio between the center (±200 km s⁻¹) and the wings (±(200–600) km s⁻¹) of the lines. This might be due to (1) a significant fraction of the [C ii] emission in the reddest and bluest channels being very extended and thus resolved out by our Band 8 imaging or (2) a spatial variation in the gas conditions. The [C ii]/CO (3–2) ratio in ALESS 57.1 is consistent with being constant across the full velocity range.

We model the velocity fields using the GalPak3D software (Bouché et al. 2015). GalPak3D uses a Monte Carlo approach to extract the kinematic and morphological parameters from three-dimensional image cubes, accounting for both the spatial and spectral response of the instrument, assuming a parametric model for a rotating disk. For our simulations, we assume an exponential disk profile—an appropriate choice for ALESS SMGs, which show a mean Sérsic index of n = 0.9 ± 0.2 (Hodge et al. 2016; n = 1 corresponds to an exponential profile). To improve the S/N and the speed of the calculations, we resampled our data onto cubes with a pixel size of 25 mas, using natural weighting.

Figure 5 shows the input and reconstructed moment-0 and moment-1 maps for ALESS 49.1 and ALESS 57.1. At the S/N of our Band 8 observations, velocity fields in both sources are consistent with an ordered, disk-like rotation. The deconvolved [C ii] rotational curves are shown in Figure 6, which also lists the FWHM velocity measurements obtained from the fitting of the spatially integrated CO (3–2) spectra (Calistro Rivera et al. 2018). The source inclinations are inferred by fitting the moment-0 map and agree with those derived from CO (3–2) imaging by Calistro Rivera et al. (2018) within 1σ–2σ uncertainty.

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We infer the global stellar and ISM properties of ALESS 49.1 and ALESS 57.1 from the spatially integrated SEDs using the MagPhys package (da Cunha et al. 2008), specifically its high-redshift extension (da Cunha et al. 2015). These differ from the previously published SEDs (da Cunha et al. 2015) by using the CO (3–2)-derived spectroscopic redshifts, compared to the photometric ones from da Cunha et al. (2015), and inclusion of ALMA Band 3/4/8 continuum flux densities. Namely, in addition to the Band 8 continuum measurements from Table 1, we include Band 4 continuum measurements (S_{2.1 mm} = 380 ± 100 μJy and 65 ± 50 μJy for ALESS 49.1 and ALESS 57.1, respectively; E. da Cunha et al. 2019, in preparation), as well as the ALMA Band 3 continuum measurement for ALESS 49.1 (S_{3.3 mm} = 37 ± 5 μJy; Wardlow et al. 2018).

The MagPhys-inferred source properties are listed in Table 2; the observed multiwavelength photometry and MagPhys SED models are provided in Appendix B. The estimated dust temperatures ${T}_{{\rm{d}}{\rm{u}}{\rm{s}}{\rm{t}}}={46}_{-2}^{+6}\,{\rm{K}}$ and ${51}_{-1}^{+7}\,{\rm{K}}$ in ALESS 49.1 and ALESS 57.1, respectively, are warmer than the median T_dust = 42 ± 2 K of the ALESS SMGs inferred from MagPhys modeling (da Cunha et al. 2015). The more precise spectroscopic redshifts and additional ALMA photometry result in a temperature increase compared to values reported by da Cunha et al. (2015), ${T}_{\mathrm{dust}}={46}_{-3}^{+0}\,{\rm{K}}$ and ${43}_{-14}^{+15}\,{\rm{K}}$, respectively. Elevated T_dust in intensely star-forming z ∼ 4.5 galaxies was reported by Cooke et al. (2018), who interpret the inferred median T_dust = 55 ± 4 K as evidence for high Σ_SFR at high redshift. Note that the da Cunha et al. (2015) and Cooke et al. (2018) models use Herschel SPIRE and ALMA Band 7 photometry, while our models include ALMA Band 3, 4, and 8 observations, thus better sampling the Rayleigh–Jeans tail of dust thermal spectrum. Compared to the MagPhys models of the entire ALESS sample (da Cunha et al. 2015), ALESS 49.1 and ALESS 57.1 have SFRs higher by a factor of ∼2 (da Cunha et al. 2015: median SFR = 280 ± 70 M_⊙ yr⁻¹) and stellar masses a factor of 2 lower (da Cunha et al. 2015: M_* = (8.9 ± 0.1) × 10¹⁰ M_⊙ yr⁻¹); the dust mass is consistent with the median da Cunha et al. (2015) value ((5.6 ± 1.0) × 10¹⁰ M_⊙). The gas depletion timescale M_gas/SFR is 100 ± 40 Myr in both ALESS 49.1 and ALESS 57.1, in line with z ≃ 2–4 SMGs (Bothwell et al. 2013; Huynh et al. 2017), and a few times lower than claimed for z = 1–3 massive main-sequence galaxies from the PHIBBS survey (0.7 ± 0.1 Gyr; Tacconi et al. 2018).

Using the SED models, we can estimate the fraction of the 160 μm continuum that is resolved out by comparing the observed rest-frame 160 μm continuum fluxes with SED modeling predictions, assuming constant T_dust and optical depth across the source. Namely, we use MagPhys to perform SED modeling using all the photometry points apart from the rest-frame 160 μm continuum. The predicted 160 μm flux densities are 11.8 and 7.6 mJy for ALESS 49.1 and ALESS 57.1, respectively; the observed flux densities match the predicted ones within <10%. Therefore, we conclude that our observations recover the bulk of the 160 μm emission, in line with the compact continuum sizes inferred in Section 3.2.

Based on the u-v plane analysis in Section 3.2, which was tested on mock ALMA data in Section 3.2.1, we found the [C ii]/160 μm continuum surface brightness distribution to be dominated by a compact component, embedded within a low surface brightness, extended emission. We now compare these morphologies to the CO (3–2) surface brightness profiles and other low- and high-redshift observations and simulations.

4.1.1. 160 μm Continuum Size

4.1.2. [C ii] Emission Size

4.1.3. Inferring FIR Source Sizes Using the Stacey et al. (2010) Relation

Given the limited angular resolution of many high-redshift [C ii] detections, the source size cannot be inferred directly from unresolved/marginally resolved data. However, parallel [C ii]/CO/FIR continuum observations have been used to constrain the source size, assuming that the bulk of the line and continuum emission originates in PDRs.

This technique, introduced by Stacey et al. (2010), compares the observed [C ii]/CO/FIR fluxes with predictions from the PDR models of Kaufman et al. (1999, see Section 4.4) to infer the far-UV (FUV) field strength G₀ and the density n. The observed L_FIR and inferred G₀ are then used to infer the FIR size R_FIR using the Wolfire et al. (1990) relations. In particular, Wolfire et al. (1990) distinguish two main regimes:

• 1.  
  $\langle {\lambda }_{\mathrm{FUV}}\rangle \lt {R}_{\mathrm{FIR}}$

  $$\begin{eqnarray}\begin{array}{rcl}\langle {G}_{0}\rangle & = & 3\times {10}^{4}\left(\displaystyle \frac{{L}_{\mathrm{FIR}}}{{10}^{10}{L}_{\odot }}\right)\left(\displaystyle \frac{\langle {\lambda }_{\mathrm{FUV}}\rangle }{100\,\mathrm{pc}}\right){\left(\displaystyle \frac{100\mathrm{pc}}{{R}_{\mathrm{FIR}}}\right)}^{3}\\ & & \times \,(1-\exp (-{R}_{\mathrm{FIR}}/\langle {\lambda }_{\mathrm{FUV}}\rangle )),\end{array}\end{eqnarray} \tag{ 1 }$$
• 2.  
  $\langle {\lambda }_{\mathrm{FUV}}\rangle \geqslant {R}_{\mathrm{FIR}}$

  $$\begin{eqnarray}&&\langle {G}_{0}\rangle =3\times {10}^{4}\left(\displaystyle \frac{{L}_{\mathrm{FIR}}}{{10}^{10}{L}_{\odot }}\right){\left(\displaystyle \frac{100\mathrm{pc}}{{R}_{\mathrm{FIR}}}\right)}^{2},\end{eqnarray} \tag{ 2 }$$

where G₀ is given in the units of the Habing field (1.6 × 10⁻³ erg cm⁻² s⁻¹) and R_FIR and the FUV photon mean free path $\langle {\lambda }_{\mathrm{FUV}}\rangle$ are in parsecs. Stacey et al. (2010) assume that $\langle {\lambda }_{\mathrm{FUV}}\rangle$ in z = 1–2 star-forming galaxies is the same as in the nearby starburst M82. For M82, Stacey et al. (2010) assume L_FIR = 2.8 × 10¹⁰ L_⊙, G₀ = 10³, and R_FIR = 150 pc.

With high-resolution [C ii], CO (3–2), and FIR observations and robust source sizes in hand, we now investigate the applicability of the Stacey et al. (2010) relations to high-redshift SMGs. Assuming that $\langle {\lambda }_{\mathrm{FUV}}\rangle$ is smaller than R_FIR (an appropriate choice given the heavily obscured, dusty environment), we infer R_FIR = 410 ± 50 pc and 420 ± 70 pc for ALESS 49.1 and ALESS 57.1, respectively, a factor of 2.5–3.5 smaller than the actual FIR half-light radii measured from the u-v plane fitting (Section 3.2). We note that the L_FIR, G₀, and R_FIR estimates for M82 from the literature show a considerable scatter; alternatively, the $\langle {\lambda }_{\mathrm{FUV}}\rangle$ in SMGs might be somewhat longer than that in the central region of M82. Crucially, if the FIR sizes of high-redshift SMGs are systematically underestimated by similar factors, the SFR surface density ${{\rm{\Sigma }}}_{\mathrm{SFR}}\propto {L}_{\mathrm{FIR}}/{R}_{\mathrm{FIR}}^{2}$ will be overestimated by 0.5–1.0 dex—a shift that might affect a number of unresolved observations in, e.g., the [C ii]/FIR—Σ_SFR plane (Figure 7).

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The [C ii] and CO (3–2) lines provide two independent measurements of the molecular gas velocity structure. Due to their different spatial extents (Section 3.2), the [C ii] and CO (3–2) emissions trace the velocity field at different radii. Figure 6 shows the line-of-sight velocity as traced by the [C ii] and CO (3–2) emissions and the GalPak3D model of the [C ii] disk. In particular, the [C ii] emission probes the velocity field within the inner 2 kpc region. Typically, rotational curves of disk-like galaxies are decomposed into contributions from the dark matter halo, and baryons in the form of a galactic bulge and disk. Given that the baryonic mass M_bar = M_* + M_gas (Table 2) is comparable to the dynamical mass enclosed within twice the CO (3–2) half-light radius (5.2 and 6.0 kpc in ALESS 49.1 and ALESS 57.1, respectively; Table 2), we conclude that the inner rotation curves are baryon dominated.

In both ALESS 49.1 and ALESS 57.1, the line-of-sight velocity v_los flattens at 2–4 kpc radius. As a dominant bulge would cause the v_los to flatten rapidly on scales of a few hundred parsecs (e.g., Sofue et al. 1999), whereas the v_los profiles in ALESS 49.1 and ALESS 57.1 are still rising at R ≥ 1 kpc, we speculate that the inner (R ≤ 2 kpc) rotational curves in ALESS 49.1 and ALESS 57.1 do not yet have a significant bulge component and hence are disk dominated. However, higher-S/N data are necessary to confirm this hypothesis.

The rotation curves have been studied at a comparable resolution in only a handful of SMGs. In this respect, ALESS 49.1 and ALESS 57.1 velocity fields are most directly comparable to those in z = 2.4 twin hyperluminous SMGs H-ATLAS J084933 (Ivison et al. 2013) and strongly lensed SMGs SMM J2135-0102 (z = 2.03; Swinbank et al. 2011) and SDP.81 (z = 3.04; Dye et al. 2015; Rybak et al. 2015b), which flatten out at ∼200 km s⁻¹ within the inner 2 kpc radius. The dynamical mass enclosed within the central 2 kpc radius region of ALESS 49.1 and ALESS 57.1 is M_dyn (R ≤ 2 kpc) = (6.2 ± 5.5) × 10¹⁰ M_⊙ and (2.7 ± 1.6) × 10¹⁰ M_⊙, respectively; although CO (3–2) observations provide mass estimates at 5–6 kpc radius, the large uncertainties prevent us from investigating the mass profiles of the two sources. The limited extent of the bright [C ii] component and the possibility that an extended [C ii] emission in ALESS 49.1 is resolved out highlight the difficulties of using very high angular resolution observations to trace the kinematics of the cold gas reservoir.

The high resolution of ALMA Band 8 data allows us to investigate the L_{[C ii]}/L_FIR ratio at z ∼ 3 on 1 kpc scales. Here we focus on the central region (R ≤ 2 kpc), in which the contribution of any extended [C ii] component is below 20%. This discussion does not include the global tracer ratios, as the Band 8 observations might resolve out a significant fraction of the total [C ii] flux from an extended component.

The SFR surface density Σ_SFR is estimated by adopting the global SFR from our SED fits in Section 3.4, assuming (1) a linear mapping between the rest-frame 160 μm emission and SFR, (2) no AGN contribution to the rest-frame FIR luminosity, and (3) a universal initial mass function (c.f. Zhang et al. 2018b). The linear mapping between the FIR continuum and SFR density assumes a constant dust temperature and opacity across the source. Further high-resolution, multiband observations of the dust continuum and spatially resolved SED modeling would be required for a more precise Σ_SFR estimate for different parts of the source. The [C ii]/FIR ratio was extracted by binning the [C ii] and 160 μm continuum maps (Figure 1) into pixels 1 beamwidth large; only pixels for which both the [C ii] and 160 μm continuum have S/N ≥ 2 are considered.

Figure 7 shows the resolved L_{[C ii]}/L_FIR for ALESS 49.1 and ALESS 57.1 as a function of the SFR surface density Σ_SFR, compared to resolved and unresolved measurements from both local and high-redshift sources (Díaz-Santos et al. 2017; Smith et al. 2017; Gullberg et al. 2018 and references therein). In both ALESS 49.1 and ALESS 57.1, the resolved ratio decreases sharply with Σ_SFR, confirming the [C ii]/FIR deficit on 1 kpc scales.

Comparing our measurements with an empirical relation between [C ii]/FIR ratio and Σ_SFR proposed by Smith et al. (2017),

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{SFR}}=(12.7\pm 3.2){\left(\displaystyle \frac{{L}_{[{\rm{C}}{\rm{II}}]}/{L}_{\mathrm{FIR}}}{0.001}\right)}^{-4.7\pm 0.8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\,\,{\mathrm{kpc}}^{-2}.\end{eqnarray} \tag{ 3 }$$

We find that the bulk of the ALESS 49.1 and ALESS 57.1 [C ii]/FIR measurements are below the 1σ scatter of the Smith et al. (2017) relation, with Σ_SFR almost 1–2 dex lower than those predicted by the Smith et al. (2017) relation. This suggests that the Smith et al. (2017) Σ_SFR—L_{[C ii]}/L_{[C ii]} might not be directly applicable in the high-Σ_SFR regime in ALESS 49.1 and ALESS 57.1.

We note that the ALESS 49.1 and ALESS 57.1 [C ii]/FIR measurements fall below the redshift z ∼ 4.5 resolved measurements of Gullberg et al. (2018). This can be attributed to several factors: (1) different source selection; (2) aperture-averaging effects in Gullberg et al. (2018), as the [C ii]/FIR ratio is calculated for apertures several kiloparsecs wide; and (3) systematic uncertainties such as Gullberg et al. (2018) assuming ${T}_{\mathrm{dust}}=50\pm 4\,{\rm{K}}$ for all their sources. Regarding source selection, ALESS 49.1 and ALESS 57.1 have L_FIR more than 2× higher than Gullberg et al. (2018) sources, while being more compact in the rest-frame FIR continuum; consequently, our measurements might probe a higher-Σ_FIR regime, which would correspond to lower [C ii]/FIR ratio and potentially higher T_gas (see Section 4.4).

The radial variation of L_{[C ii]}/L_FIR in ALESS 49.1 and ALESS 57.1 is shown in Figure 8. Fitting the resolved [C ii]/FIR data with a power law L_{[C ii]}/L_{[C ii]} ∝ R^α, we find strong evidence for a radial variation of the [C ii]/FIR ratio in ALESS 49.1 (α = 0.41 ± 0.06), while in ALESS 57.1 the slope is consistent with being flat (α = 0.05 ± 0.05). A decrease in the [C ii]/FIR ratio toward the center of the source is seen in both local star-forming galaxies (e.g., Madden et al. 1993; Smith et al. 2017) and high-redshift sources (Gullberg et al. 2018) and indicates that the [C ii]/FIR deficit is driven by local processes, as opposed to global properties of the sources. In particular, Smith et al. (2017) found the [C ii]/FIR ratio to be suppressed by on average 30% ± 15% in the central R ≤ 400 pc regions of galaxies without an AGN. On the other hand, the [C ii]/FIR ratio drops by a factor of a few over the inner 2 kpc in nuclear starbursts in M82 and M83 (Herrera-Camus et al. 2018). While the Chandra-detected obscured AGN in ALESS 57.1 (Wang et al. 2013) might be expected to suppress the [C ii] emission in the circumnuclear region, we do not detect any strong [C ii] suppression in ALESS 57.1 on 1 kpc scales. The difference in the [C ii]/FIR radial profiles in ALESS 49.1 and ALESS 57.1 is driven by the larger scatter in [C ii]/FIR ratio for a given Σ_SFR in ALESS 57.1 (see Figure 7), which is a result of the complex [C ii] and 160 μm morphology in that source (Figure 1). Note that the limited S/N of our data at R ≥ 2 kpc prevents us from studying the [C ii]/FIR radial dependence at larger radii.

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The relative intensities of the [C ii], CO (3–2), and FIR emission from hot dissociation regions (PDRs) depend on the ionizing FUV field strength G₀ and the gas density n(H), which determine the depth of the outer C⁺ layer. We now use our resolved [C ii]/CO (3–2)/FIR observations to infer the G₀ and n using the PDR models from the PDRToolbox library (Kaufman et al. 2006; Pound & Wolfire 2008). We focus on the central regions of the source (R ≤ 2 kpc), adopting the best-fitting models from Section 3.2.

For a proper comparison, several corrections need to be applied. First, the [C ii] emission from the ionized ISM needs to be subtracted from the observed [C ii] signal. The contribution from the ionized gas can be estimated from [N ii] 122 μm/205 μm lines, which have similar critical density for collisions with electrons (300 cm⁻³/32 cm⁻³) to the [C ii] line (50 cm⁻³; Goldsmith et al. 2012) but ionization energy >13.6 eV and hence trace only the ionized gas. Croxall et al. (2017) carried out a systematic study of the [N ii] 122 μm/205 μm lines in a sample of 21 nearby star-forming galaxies, estimating the fraction of [C ii] in PDRs as ${f}_{[{\rm{C}}\,{\rm{II}}]}^{\mathrm{PDR}}\geqslant 0.8$ for sources with Σ_SFR ≥ 10⁻² M_⊙ yr⁻¹ kpc⁻². At high redshift, Zhang et al. (2018a) used [N ii] 122 μm line observations in z = 1–3 SMGs to derive 10%–15% ionized gas contribution to [C ii] luminosity, assuming Galactic diffuse gas N and C abundances. Consequently, we adopt a conservative (i.e., low) estimate of ${f}_{[{\rm{C}}\,{\rm{II}}]}^{\mathrm{PDR}}=0.8$.

Second, PDRToolbox models are derived for a one-dimensional, semi-infinite slab, illuminated from the face side only. In the intensely star-forming SMGs, we expect the clouds to be illuminated from multiple sides, and the optical thickness of individual tracers needs to be considered. Namely, for optically thick emission, only the emission from the side of the cloud facing the observer is observed; for the optically thin emission, the distant side of the cloud adds to the observed fluxes, and the line intensities expected from PDR models need to be multiplied by a factor of 2 (Kaufman et al. 1999). The 160 μm continuum is generally assumed to be optically thin. Based on the models of Narayanan & Krumholz (2014), the CO (3–2) line is expected to be optically thick with a median optical depth ≥10 for Σ_SFR = 10⁻¹ to 10³ M_⊙ yr⁻¹. The [C ii] emission is optically thin in most environments, although there is evidence for moderately optically thick [C ii] (optical depth ∼ 1) in both Galactic PDRs (Graf et al. 2012; Sandell et al. 2015) and high-redshift sources (Gullberg et al. 2015). Therefore, for an optically thin [C ii], the predicted [C ii]/CO (3–2) ratio has to be increased 2×; for an optically thick [C ii], the predicted [C ii]/FIR ratio has to be reduced by 1/2. We consider both optically thin and optically thick [C ii] scenarios; however, as the evidence for optically thick [C ii] is limited, we adopt the values derived for the optically thin [C ii] for the rest of this paper.

Figure 9 shows the contours in the G₀/n space for the central R ≤ 2 kpc region of ALESS 49.1 and ALESS 57.1, with best-fitting values listed in Table 3. The combination of the three tracers provides orthogonal constraints on G₀ and n. In particular, G₀ is largely determined by the [C ii]/FIR ratio, and n by [C ii]/CO (3–2). Table 3 lists the inferred G₀ and n values for the optically thin and optically thick [C ii] scenarios, along with the PDR surface temperature. For the optically thin [C ii] case, the conditions in ALESS 49.1 and ALESS 57.1 are almost identical, with G₀ ∼ 10⁴ and n ∼ 10⁴–10⁵ cm⁻³, implying a PDR surface temperature T_PDR of ∼700 K. Increasing the fraction of [C ii] emission from the PDRs from 0.8 to 1.0 causes the inferred G₀ and n values to decrease by ≤0.25 dex. For the optically thick [C ii] case, the inferred G₀ and n decrease by up to 0.5 and 1.0 dex, respectively; T_PDR is reduced to 400–500 K.

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Table 3.  Comparison with PDRToolbox PDR Models: Observed [C ii]/CO (3–2) and [C ii]/FIR Continuum Ratios, along with Inferred FUV Field Strength G₀, Density n(H), and PDR Surface Temperature T_PDR for ALESS 49.1 and ALESS 57.1

Source		ALESS 49.1	ALESS 57.1
[C ii]/FIR		(3.3 ± 1.5) × 10−4	(3.4 ± 2.8) × 10−4
[C ii]/CO (3–2)		310 ± 100	600 ± 240
[C ii] Optically Thin
log G0		${4.1}_{-0.3}^{+0.3}$	${4.2}_{-0.4}^{+0.9}$
$\mathrm{log}n$(H)	(cm–3)	${4.7}_{-0.2}^{+0.4}$	${4.1}_{-0.4}^{+0.5}$
Tgas	(K)	720 ± 200	670 ± 170
[C ii] Optically Thick
log G0		${3.8}_{-0.2}^{+0.3}$	${3.7}_{-0.3}^{+1.0}$
$\mathrm{log}n$(H)	(cm−3)	${3.9}_{-0.2}^{+0.4}$	${3.4}_{-0.3}^{+0.4}$
Tgas	(K)	430 ± 30	480 ± 90

Note. We consider both optically thin and thick [C ii] scenarios. The [C ii] luminosities are given before subtracting the contribution from non-PDR sources.

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The G₀, n values in the central region of ALESS 49.1 and ALESS 57.1 are comparable to G₀ = 10³–10^4.5, n = 10²–10⁴ cm⁻³ inferred from unresolved observations of larger SMG samples, such as the [C ii]/CO study of Stacey et al. (2010), the [C i]/CO study of 14 z ≥ 2 SMGs by Alaghband-Zadeh et al. (2013), and FIR spectroscopy of lensed SMGs (Wardlow et al. 2017; Zhang et al. 2018a). Using [C ii] and low-J CO observations in a sample of strongly lensed SMGs, Gullberg et al. (2015) found a larger scatter of FUV strength (G₀ = 10²–8 × 10³) and density (n = 10²–10⁵ cm⁻³), although the effect of differential magnification might be substantial. For nearby star-forming galaxies, a comparison of observed spatially integrated [C ii], [O i] 63 μm, and FIR luminosities with the PDRToolbox models was carried out by Malhotra et al. (2001) and Díaz-Santos et al. (2017). ALESS 49.1 and ALESS 57.1 are consistent with the high-density Malhotra et al. (2001) sources; however, G₀ and n in ALESS 49.1 and ALESS 57.1 are higher than in the densest ultraluminous infrared galaxies (ULIRGs) from the Díaz-Santos et al. (2017) sample, which have G₀ ∼ 10³, n = 1–10³ cm⁻³. Note that the globally averaged G₀ and n in ALESS 49.1 and ALESS 57.1 might be lower than those inferred from the R ≤ 2 kpc region. Finally, compared to the resolved kiloparsec-scale observations of local starburst galaxies NGC 6946 and NGC 1313 with inferred G₀ = 10³–10⁴, n = 10^3.0–10^3.5 cm⁻³ (Contursi et al. 2002), the central regions of ALESS 49.1 and ALESS 57.1 show similar G₀ and somewhat higher n.

What drives the strong FUV fields in ALESS 49.1 and ALESS 57.1: a central AGN, or star formation? Although Chandra X-ray observations (Wang et al. 2013) revealed an obscured AGN in ALESS 57.1 (no emission from ALESS 49.1 was detected), it is unlikely that an obscured AGN would be driving a strong FUV field on few-kiloparsec scales. On the other hand, the G₀ in the vicinity of H ii regions is of the order of ${10}^{3}\mbox{--}{10}^{5}$ (e.g., Tielens & Hollenbach 1985; Hollenbach & Tielens 1999), comparable to the values inferred from our PDR models. Similarly, typical G₀ and n values for Galactic star-forming regions are of the order of G₀ = 10³–10⁵, n = 10³–10⁶ cm⁻³ (Stacey et al. 1991, 2010). We therefore conclude that the strong FUV field in ALESS 49.1 and ALESS 57.1 is due to star formation, rather than a central AGN.

Having estimated G₀ and n in the central regions of ALESS 49.1 and ALESS 57.1, we now turn to the mechanism driving the [C ii]/FIR deficit. We focus on the thermal saturation model proposed by Muñoz & Oh (2016) and the reduction of the photoelectric heating of the gas by small dust grains (e.g., Bakes & Tielens 1994; Malhotra et al. 2001); we briefly discuss other potentially relevant mechanisms—AGN contribution and dust-bounded H ii regions—in Section 4.5.3. For a more exhaustive list of proposed mechanisms for the [C ii]/FIR deficit, we refer the reader to Smith et al. (2017).

4.5.1. Thermal Saturation of the [C ii] Line

Muñoz & Oh (2016) have proposed the thermal saturation of the upper level of the C⁺ fine-structure transition as the main driver of the [C ii]/FIR deficit. In other words, when T_gas exceeds the C⁺ transition temperature (92 K), the upper-level/lower-level population ratio (and the [C ii] luminosity) depends only weakly on T_gas, while the FIR luminosity keeps on increasing.

Our PDRToolbox models imply high FUV fields strength (G₀ ∼ 10⁴) and densities (n = 10⁴–10⁵ cm⁻³) with gas surface temperatures larger than 500 K (Table 3). The [C ii] transition is saturated in this regime. Following Muñoz & Oh (2016), the thermal cooling rate per hydrogen atom via the [C ii] line depends on T_gas via

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{[{\rm{C}}{\rm{I}}{\rm{I}}]}=2.3\times {10}^{-6}\,{k}_{{\rm{B}}}\,{T}_{[{\rm{C}}{\rm{I}}{\rm{I}}]}\frac{2}{2+\exp ({T}_{{\rm{C}}{\rm{I}}{\rm{I}}}/{T}_{{\rm{g}}{\rm{a}}{\rm{s}}})}\frac{C}{H}\end{eqnarray} \tag{ 4 }$$

where k_B is the Boltzmann constant, T_{[C ii]} is the [C ii] ionization temperature, and C/H are the relative abundances of the carbon and hydrogen atoms. Following Equation (4), the [C ii] cooling rate increases by only ∼40% between T_gas = 100 and 500 K, whereas the L_FIR increases by a factor of a few hundred, assuming that T_dust scales proportionally with T_gas and ${L}_{\mathrm{FIR}}\propto {T}_{\mathrm{dust}}^{4+\beta }$, where the dust opacity β is typically assumed to range from 1.5 to 2.5 (e.g., Casey et al. 2014).

For a more direct comparison with the Muñoz & Oh (2016) model, we compare the resolved [C ii]/FIR observations in ALESS 49.1 and ALESS 57.1 (Figure 7) to the predicted Σ_SFR-[C ii]/FIR slope. According to Muñoz & Oh (2016), L_{[C ii]}/L_FIR ratio scales as

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{[{\rm{C}}{\rm{II}}]}}{{L}_{\mathrm{FIR}}}=2.2\times {10}^{-3}\displaystyle \frac{{f}_{{\rm{C}}{\rm{II}}}}{0.13}{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{\mathrm{FIR}}}{{10}^{11}{L}_{\odot }{\mathrm{kpc}}^{-2}}\right)}^{-1/2},\end{eqnarray} \tag{ 5 }$$

where f_{C ii} is the fraction of gas emitting in [C ii]. Fitting our data points with a power law following Equation (5), we obtain a best-fitting slope of γ = −0.53 ± 0.12. This is in agreement with the thermal saturation model slope of γ = −0.5 (Equation (5)).

We note that Díaz-Santos et al. (2017) discount the thermal saturation of the [C ii] line as a source of the [C ii]/FIR deficit in local star-forming galaxies. Namely, comparing the [O i] 63 μm and [C ii] line ratios with a statistical equilibrium radiative transfer model, they obtain a scaling between dust and gas kinetic temperature T_gas = (1.6–2.1) × T_dust. Given T_dust of 21–48 K, they find T_gas ≤ 92 K, i.e., below the thermal saturation regime. However, ALESS 49.1 and ALESS 57.1 show relatively high global ${T}_{{\rm{d}}{\rm{u}}{\rm{s}}{\rm{t}}}={46}_{-2}^{+6}$ and ${51}_{-4}^{+7}\,{\rm{K}}$, respectively (Table 2). Given the conversion factors from Díaz-Santos et al. (2017) and evidence for an increase in dust and gas temperature toward the center of SMGs (Calistro Rivera et al. 2018), we conclude that T_gas in the central regions of ALESS 49.1 and ALESS 57.1 likely exceeds 92 K.

4.5.2. Suppression of the [C ii] Emission due to Positive Grain Charging

In addition to the thermal saturation, another potentially important effect of the G₀, n values inferred from our PDR analysis is the reduced photoelectric heating of the gas by electrons ejected from the small dust grains by the FUV photons (e.g., Bakes & Tielens 1994; Malhotra et al. 2001; Wolfire et al. 2003). At high G₀/n ratios, the grains become positively charged, thus raising the potential barrier for the electrons to escape.

Qualitatively, a reduced photoelectric heating will manifest in moderate T_gas/T_dust ratios. Although ALESS 49.1 and ALESS 57.1 have elevated dust temperatures compared to other high-redshift SMGs (Swinbank et al. 2014; da Cunha et al. 2015) and local ULIRGs (Díaz-Santos et al. 2017), the high PDR surface temperatures indicate that the gas is already heated to high temperature, at which point the [C ii] line becomes saturated.

Quantitatively, following Wolfire et al. (2003), the photoelectric heating rate per hydrogen atom is given as

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{\mathrm{PE}} & = & \displaystyle \frac{1.1\,\times \,{10}^{-25}G^{\prime} {Z}_{d}}{1\,+\,3.2\,\times \,{10}^{-2}{\left[G^{\prime} {\left({T}_{\mathrm{gas}}/100{\rm{K}}\right)}^{1/2}{n}_{e}^{-1}{\phi }_{\mathrm{PAH}}\right]}^{0.73}}\\ & & \times \,\mathrm{erg}\,{{\rm{s}}}^{-1},\end{array}\end{eqnarray} \tag{ 6 }$$

where G' = 1.7 × G₀, Z_d is the dust-to-gas ratio (normalized to the Galactic value), n_e is the electron density, and ϕ_PAH is a factor associated with the polycyclic aromatic hydrocarbon molecules. The second term in the denominator corresponds to the positive grain charging. We follow Muñoz & Oh (2016) by adopting n_e = 1.1 × 10⁻⁴n, ϕ_PAH = 0.5. Note that Equation (6) assumes T_gas ≤ 1000 K, which is satisfied for both ALESS 49.1 and ALESS 57.1. While for the ALESS 49.1 and ALESS 57.1 values of G₀, n the second term in the denominator, corresponding to the reduction in photoelectric heating, becomes dominant, the numerator is also proportional to G₀. Comparing a typical nearby star-forming galaxy with G₀ = 10², n = 10⁴ cm⁻³ (see Muñoz & Oh 2016) and ALESS 49.1 and ALESS 57.1 with G₀ = 10⁴, n = 10⁴ cm⁻³, the second term in the determinant increases from ∼0.3 to ∼18, indicating a significant reduction in the gas heating due to grain charging. However, at the same time, the overall Γ_PE increases by a factor of ∼6.

Therefore, we attribute the pronounced [C ii]/FIR deficit in the central regions of ALESS 49.1 and ALESS 57.1 to the high gas temperature, which results in a quantum-level saturation of the C⁺ fine structure.

4.5.3. Other Mechanisms for [C ii]/FIR Deficit