A Spatially Resolved Study of Cold Dust, Molecular Gas, H ii Regions, and Stars in the z = 2.12 Submillimeter Galaxy ALESS67.1

The ALMA Band 7 data were taken on the 2015 August 11, as part of a Cycle 1 project #2012.1.00307.S (PI: J. Hodge), which targeted 19 SMGs from the Cycle 0 ALESS survey (Hodge et al. 2013). For a detailed description of the project refer to Hodge et al. (2016).

As in the Cycle 0 ALESS program, the Band 7 data were centered on 344 GHz (∼870 μm). We used the "single-continuum" spectral mode, with 4 × 128 dual polarization channels over the 8 GHz bandwidth. The primary beam of the ALMA observations is $17\buildrel{\prime\prime}\over{.} 4$ at full width at half maximum (FWHM).

The ALMA data were obtained using 46 antennas in an extended configuration (C32-6; maximum baseline of ∼1.6 km). The bandpass, phase, and flux calibrators were J0522–3627, J0348–2749, and J0334–401, respectively, and the total integration time was approximately eight minutes. The data were taken under good phase stability/weather conditions, with a medium precipitable water vapor (PWV) at zenith of ∼0.7 mm.

The uv data were inverse Fourier-transformed using natural weighting to produce the dirty continuum image, which was later deconvolved with a synthesized beam (i.e., the dirty beam) using the clean algorithm. The image was gridded to a pixel scale of $0\buildrel{\prime\prime}\over{.} 02$ and a size of $20\buildrel{\prime\prime}\over{.} 48$ (1024 pixels) per side, covering the primary beam of our observations. The FWHM of the synthesized beam was 018 × 015 (1.5 × 1.3 kpc at the redshift of ALESS67.1), with a position angle of 648. The rms noise of the dirty map was 1 $\sigma =0.07$ mJy beam⁻¹.

The ALMA data were taken on the 2015 September 6 as part of project 2013.1.00470.S (PI: J. Hodge), which targets a sample of SMGs to obtain sub-arcsecond resolution CO maps to study the properties of molecular gas (C. Rivera et al. 2017, in preparation). The data were taken in Band 3, with the expected frequency of the redshifted CO(J = 3–2) line (${\nu }_{\mathrm{rest}}=345.7959899$ GHz) covered by the upper side band, and using three additional basebands to observe the continuum. We used the lowest-resolution frequency division mode (FDM) mode and averaged over eight channels to maintain adequate resolution while keeping the data rate reasonable. The observations were carried out with 36 antennas in an intermediate configuration (maximum baseline 1.6 km) and the maximum recoverable scale was 47. Standard calibration was used and the total integration time on the target was 29 minutes. Data were reduced using casa version 4.3.1 and the standard pipeline calibration, with some additional flags applied to address bad antennas or times. Imaging was carried out using casa version 4.7.0. The uv data were inverse Fourier-transformed using natural weighting to produce both the 3 mm continuum dirty image and the CO data cube. There was no detection in the 3 mm continuum, thus putting a 3σ constraint on the 3 mm continuum flux of ${S}_{3\mathrm{mm}}\leqslant 0.054$ mJy. The CO cube was gridded into 48 MHz per channel (∼130 km s⁻¹) and had an average synthesized beam of 057 × 049. The sensitivity of the CO cube was 0.25 mJy beam⁻¹ per 48 MHz channel.

AO-assisted, IFU observations of the strong optical lines in ALESS 67.1 were taken with the SINFONI IFU between 2013 January and October. At z = 2.12, the [N ii]/Hα lines are redshifted to λ ∼ 2.05 μm and [O iii]/Hβ are redshifted to $\lambda \sim 1.55$ μm so we used the HK-band filter and grism which has a spectral resolution of $R=\lambda /{\rm{\Delta }}\lambda \sim 5000$, sufficient to separate Hα and the two [N ii] lines. Since the low-surface brightness continuum emission is spatially extended across ∼3'' in the HST H-band imaging, we used the 8 × 8'' field of view (FOV) mode of SINFONI. To achieve high spatial resolution, we employed natural guide star AO correction exploiting a nearby bright (R = 12.9 mag) star. Each 1 hr observation block (OB) was split in to 4 × 600 s exposures, which were dithered by $4^{\prime\prime}$, thus always keeping the target in the FOV. In total, we observed the target for 7.2 ks. Data reduction was performed using the esorex pipeline, with additional custom routines applied to improve the flat-fielding, sky subtraction, and mosaicing of the cubes. The flux and astrometry calibration was calibrated from the Hawk-I K-band imaging. The AO-corrected point-spread function (PSF) had a mean Stehl ratio of 0.3, ideally corresponding to an angular resolution of ∼02 FWHM.

The optical and NIR images from the ACS and WFC3 cameras mounted on the HST were taken as part of the Cosmic Assembly Near-IR Deep Extragalactic Legacy Survey (CANDELS; Grogin et al. 2011; Koekemoer et al. 2011). The typical FWHM of the HST PSF in the optical is $\sim 0\buildrel{\prime\prime}\over{.} 1$.

A prominent source is detected in the central region of the high-resolution dirty map, with a peak flux of 0.72 mJy beam⁻¹ (corresponding to 10σ) and a location matching ALESS67.1 from Hodge et al. (2013). We clean a circular region with $1^{\prime\prime}$ radius around the source down to 2σ, and the resulting cleaned image is shown in Figure 1. As can be seen, two detections were reported at ∼1'' resolution observations by Hodge et al. (2013), ALESS67.1/67.2; however, in our data we only detect ALESS67.1. We tried tapering the map to lower spatial resolution to test the possibility that the lack of detection is due to extended structures which are resolved out in the high-resolution map. While ALESS67.2 remained undetected in the tapered maps, the sensitivity of the tapered map was not as deep as the original Cycle 0 data so the nature of ALESS67.2 remains inconclusive. It is possible that ALESS67.2 is resolved out, or that it is a false detection.

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As ALESS67.1 is clearly resolved and is the sole source detected in the map, to measure the flux and the light profile we first use the uvmodelfit algorithm to model the uv data. We find that the best-fit (reduced ${\chi }^{2}=0.5$) Gaussian profile has an intensity of 3.7 ± 0.2 mJy with a FWHM of $0\buildrel{\prime\prime}\over{.} 40\,\pm 0\buildrel{\prime\prime}\over{.} 02\times 0\buildrel{\prime\prime}\over{.} 21\pm 0\buildrel{\prime\prime}\over{.} 02$, corresponding to a physical half-light radius of $1.7\pm 0.1\times 0.9\pm 0.1\,\mathrm{kpc}$. We obtain consistent results if we instead use the imfit algorithm or sextractor on the cleaned image. The measured flux is also consistent with but marginally lower than the previous measurement (4.9 ± 0.7 mJy) based on the lower-resolution ALMA Cycle 0 data (Hodge et al. 2013), and the size is consistent with the parent sample of ALESS SMGs with $0\buildrel{\prime\prime}\over{.} 2$ high-resolution ALMA observations (Hodge et al. 2016).

A strong line detection is seen in the dirty 3 mm channel maps, and the emission appears resolved. To clean the data cube and extract the spectra, we employ the following iterative procedure. We first derive a weight-averaged map over a best-guess frequency width and choose a center based on the averaged map. We then perform a curve-of-growth analysis where we define a circular aperture centered at the chosen centroid with a radius which encompasses all the line flux. The data cube is then cleaned to 2σ within this defined aperture and the spectrum is extracted. Next, the extracted spectrum is fitted with a Gaussian profile and the frequency width used for obtaining the averaged map is updated to be the FWHM of the spectral fit. This process continues until the solution converges. Finally we find that all the line flux is contained within a circular radius of $2^{\prime\prime}$ and the results are not sensitive to the chosen position of the aperture center within a beam area.

The results are plotted in Figure 1, showing a line detection well fitted (${\chi }^{2}=1.1$) with a Gaussian profile centered at 110845 ± 30 MHz and having a FWHM of 319 ± 72 MHz, corresponding to a ¹²CO(J = 3–2) line at a redshift of $z=2.1196\pm 0.0009$, with a velocity FWHM of 862 ±195 km s⁻¹. By integrating the best-fit Gaussian we derive a total line flux of 4.2 ± 1.2 Jy km s⁻¹. The errors are estimates from the fit, and they are consistent with the errors derived from a Monte Carlo simulation. We create fake spectra by injecting the model profile into spectra extracted from randomly selected regions of the data cube with the same circular aperture used for the detection spectrum. The errors are then obtained from the standard deviations between the fit results and the input model.

Using the standard relation from Solomon & Vanden Bout (2005), ${L}_{\mathrm{CO}}^{{\prime} }=3.25\times {10}^{7}{S}_{\mathrm{CO}}{\rm{\Delta }}v{\nu }_{\mathrm{obs}}^{-2}{D}_{{\rm{L}}}^{2}{(1+z)}^{-3}$, where ${S}_{\mathrm{CO}}{\rm{\Delta }}v$ is the total line flux in Jy km s⁻¹, ${\nu }_{\mathrm{obs}}$ is the observed line frequency in GHz, and ${D}_{{\rm{L}}}$ is the cosmological luminosity distance in Mpc, we calculate a CO luminosity of ${L}_{\mathrm{CO}(3-2)}^{{\prime} }=(1.1\pm 0.3)\times {10}^{11}$ K km s⁻¹ pc². Huynh et al. (2017) have recently conducted ¹²CO(1–0) observations on ALESS67.1 using the Australia Telescope Compact Array (ATCA), and they detect strong ¹²CO(1–0) emission with a total flux of ${L}_{\mathrm{CO}(1-0)}^{{\prime} }=(9.9\pm 1.8)\times {10}^{10}$ K km s⁻¹ pc². With both the measurements we calculate a ${L}_{\mathrm{CO}(3-2)}^{{\prime} }/{L}_{\mathrm{CO}(1-0)}^{{\prime} }$ line luminosity ratio of ${r}_{\mathrm{3,1}}=1.1\pm 0.4$, consistent with previous estimates for the SMG population (Harris et al. 2010; Ivison et al. 2011; Bothwell et al. 2013; Sharon et al.2016).

The curve-of-growth analysis shown in Figure 1 suggests a CO half-light radius of $\sim 1^{\prime\prime}$. To measure the size we employ both image-based and uv-based analyses. We first conduct imfit on the averaged map and find that the best-fit two-dimensional (2D) Gaussian profile has a circularized half-light radius of 091 ± 016, consistent with the curve-of-growth analysis. As the best-fit 2D model only has a peak signal-to-noise ratio (S/N) of 3, we run the following modeling to estimate the bias and the scatter of our measurement. Random elliptical Gaussian 2D models are first convolved with the synthesized beam and then injected at random positions on the residual averaged CO map with the best-fit model of the detected signals subtracted. imfit is performed on each injected model and the output results are recorded. In total we inject 36,000 models with the peak S/N, major axis, minor axis, and positional angle all randomized, in which the peak S/N has a range of 1–10 and the half-light radius in the major and minor axes is allowed within $2^{\prime\prime}$. We collect input parameters that correspond to an output matching to the CO measurements in peak S/N and circularized radius, and we compare the input and output circularized radius by computing the fractional difference defined as (output-input)/input. We find a 3% upward bias in median (0.028 ± 0.007) and a 20% scatter. The scatter is consistent with the measurement; however, the size bias needs to be corrected. We therefore obtain a bias-corrected ¹²CO(J = 3–2) half-light radius of 088 ± 016. The deconvolved circularized half-light radius is therefore $0\buildrel{\prime\prime}\over{.} 84\pm 0\buildrel{\prime\prime}\over{.} 16$.

For the uv-based analysis we extract the averaged visibility over the FWHM channels as a function of the uv distances and then perform ${\chi }^{2}$ fitting assuming a Gaussian profile. We obtain a half-light radius of $0\buildrel{\prime\prime}\over{.} 76\pm 0\buildrel{\prime\prime}\over{.} 10$, in good agreement with the result based on the image-plane analysis. However the uv-based measurement is better constrained, with lower errors. We therefore adopt the uv-based measurement for the ¹²CO size. At the measured CO redshift, the ¹²CO half-light radius would therefore be ${r}_{1/2,\mathrm{CO}}=6.5\pm 0.9\,\mathrm{kpc}$. More details on the uv-fitting will be presented in C. Rivera et al. (2017, in preparation).

The Hα line is strongly detected in our SINFONI data with an S/N ∼ 10 at the peak, allowing us to derive 2D intensity, velocity, and dispersion maps. In the following we therefore analyze the spectra in both integrated and 2D. Weaker [N ii] and [S ii] lines are also detected, although only in the central regions of the source. We also search for [O i]6300, and [O iii]/Hβ at ∼5000 Å, but no significant detections are found.

In both 1D and 2D cases, we perform a minimizing-${\chi }^{2}$ fit to the spectra over a wavelength range of 1.9–2.2 μm, where the continuum is well described with a power-law slope and covers all the detected lines. The spectra are fit with four Gaussian models, in which all include a linear continuum component with the slope and normalization allowed to be free. We then fit different combination of lines; Hα, ${\rm{H}}\alpha +$[N ii], ${\rm{H}}\alpha +$[S ii], and ${\rm{H}}\alpha +$[N ii]+[S ii]. In all cases we assume that the [N ii] and [S ii] lines have the same line width and redshift as those of Hα, and the flux ratio of the [N ii]6583/[N ii]6548 doublet¹⁷ is fixed to a theoretical value of 3 based on the transition probabilities provided in Osterbrock (1989). The flux ratio of the [S ii]6731/[S ii]6716 doublet is sensitive to the magnetic field strength, hence it is not fixed. The fits are weighted against the sky spectrum provided by Rousselot et al. (2000) and when calculating ${\chi }^{2}$ the wavelength ranges corresponding to the skylines are masked. The velocity dispersion is corrected in quadrature for instrumental broadening. The errors are derived using Monte Carlo simulations similar to those used for measuring the errors of the CO emission. Note that by adding an extra broad Gaussian component we have also searched for broad lines with a FWHM over 1000 km s⁻¹, typical for SMGs hosting AGNs and suggesting strong outflows (e.g., Harrison et al. 2012); however, we do not find evidence of such a broad component in ALESS67.1.

The model selection is determined based on the Akaike information criterion (AIC). Specifically, we use the version that is corrected for a finite sample size (AICc; Hurvich & Tsai 1989), which is defined as $\mathrm{AICc}={\chi }^{2}+2k\,+2k(k+1)/(n-k-1)$, where ${\chi }^{2}$ is from the fit, and k and n denote the number of parameters and the number of data, respectively. Normally fits with more model parameters have lower ${\chi }^{2}$, and therefore a situation of over-fitting may not occur if one simply selects the model that produces the lowest ${\chi }^{2}$. The AICc offers a quantitative way to compare related models on the goodness of fit by penalizing the number of parameters in the model, and the model that has the lowest AICc is selected as the adopted model in most cases. However, as shown below, [N ii]6583 happens to sit on one of the bright skylines, and we find that during the curve-of-growth analyses on the integrated spectra the skyline contamination becomes significant at larger radii. Consequently we restrict the fit to ${\rm{H}}\alpha +$[N ii]+[S ii] only.

Finally, a fit with line components is considered significant if the fit, compared to a simple continuum-only model, has a lower AICc and provides a ${\chi }^{2}$ improvement of ${\rm{\Delta }}{\chi }^{2}\gt 25$, equivalent to an S/N > 5σ assuming Gaussian noise and that the noise is not correlated among wavelength channels. For models that include [S ii], given it is a separated line without skyline contamination, we require a further ${\chi }^{2}$ improvement of ${\rm{\Delta }}{\chi }^{2}\gt 9$ (3σ) compared to models without [S ii].

3.3.1. Integrated Spectrum

To determine the radius over which the total flux is measured, we again employ a curve-of-growth approach that is similar to that used for ¹²CO (see Section 3.2). We adopt the peak position of the 870 μm continuum for the centroid, which produces a converged result and lies close to the geometrical center of the 2D intensity distributions shown in the next section. We again move the centroid around within the resolution area and fold the variations into the uncertainties of the measurements. The flux is derived based on the fitting procedure outlined in Section 3.3, except at radii larger than 15, where we find that the skyline contamination significantly affects the fit and, by examining the 2D maps, we conclude that the [N ii] lines are boosted (since there is no strong emission detected in the 2D map beyond this radius). We therefore fix the peak of the [N ii] lines to be that measured at $1\buildrel{\prime\prime}\over{.} 5$ but still allow dispersion and redshift to float.

Based on the curve-of-growth approach, the integrated spectra are measured using a circular aperture with a radius of 18, at which all three line fluxes are converged. From this, we measure a redshift of $z=2.1228\pm 0.0006$, slightly higher than but still consistent with the CO redshift within 3σ. We also measure a total Hα flux of (2.6 ± 0.4) × 10⁻¹⁶ erg s⁻¹ cm⁻², a [N ii] flux of (1.1 ± 0.4) × 10⁻¹⁶ erg s⁻¹ cm⁻², and a [S ii] flux of (6.7 ± 2.9) × 10⁻¹⁷ erg s⁻¹ cm⁻², with a spectral FWHM of 670 ± 100 km s⁻¹. At the measured ${\rm{H}}\alpha +$[N ii]+[S ii] redshift we compute a Hα luminosity of ${L}_{{\rm{H}}\alpha }=(9.2\pm 1.5)\times {10}^{42}$ erg s⁻¹, a [N ii] luminosity of ${L}_{[{\rm{N}}{\rm{}}\mathrm{II}]}=(3.8\pm 1.4)\times {10}^{42}$ erg s⁻¹, and a [S ii] luminosity of ${L}_{[{\rm{S}}{\rm{}}\mathrm{II}]}=(2.4\pm 1.1)\times {10}^{42}$ erg s⁻¹.

The continuum-subtracted integrated spectra along with the best-fit Gaussian model are shown in Figure 2, in which we also show the best-fit ¹²CO(J = 3–2) profile for comparison. Because the spatial extent of [N ii] is much smaller than that of Hα, the integrated spectrum with an aperture radius of $1\buildrel{\prime\prime}\over{.} 8$ includes extra unnecessary noise and [N ii] may appear undetected. To demonstrate that [N ii] lines are indeed detected in Figure 2 we also plot the Hα/[N ii] portion of the spectrum with a smaller aperture radius. The line profiles are consistent among the molecular and atomic emission lines, suggesting that in the integrated sense the dynamics that are measured by these tracers agree with each other. We compare in more detail the spatially resolved dynamics between the two tracers in the discussion section.

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The curve-of-growth analysis suggests a half-light radius of $0\buildrel{\prime\prime}\over{.} 8\pm 0\buildrel{\prime\prime}\over{.} 1$ for Hα, which is slightly larger than but consistent with $0\buildrel{\prime\prime}\over{.} 63\pm 0\buildrel{\prime\prime}\over{.} 10$ derived from a best-fit 2D Gaussian profile on the intensity. We adopt the result from the curve-of-growth analysis since the projected Hα emitting area is non-Gaussian with clear extended structures (Figure 3) so a single Gaussian model is likely to underestimate the true size. Given the angular resolution of $0\buildrel{\prime\prime}\over{.} 2$ of the SINFONI observations, the deconvolved size of Hα is $0\buildrel{\prime\prime}\over{.} 77\pm 0\buildrel{\prime\prime}\over{.} 10$, corresponding to a Hα half-light radius of ${r}_{1/2,{\rm{H}}\alpha }=6.6\pm 0.9\,\mathrm{kpc}$ at the Hα redshift. We perform the same curve-of-growth exercise for [N ii] and [S ii], finding ${r}_{1/2,{\rm{N}}{\rm{II}}}=5.1\pm 1.7\,\mathrm{kpc}$ and ${r}_{1/2,{\rm{S}}{\rm{II}}}=5.1\pm 2.1\,\mathrm{kpc}$.

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3.3.2. Two-dimensional Kinematics

To produce 2D intensity, velocity, and dispersion maps we run our line-fitting procedures described in Section 3.3 on each spaxel. However, not every spaxel has significant line emission so we adopt an adaptive binning approach that is typically used for high-redshift IFU data (e.g., Swinbank et al. 2006). We start with one spaxel, and if the fit is not significant then we average over 3 × 3 spaxels, and if that is still not significant then we increase the binning to 5 × 5 spaxels. In regions where this adaptive binning process still fails after 5 × 5 binning to give an adequate S/N, we leave the spaxel without a fit. The caveat of this approach is that the signals are weighted toward the higher S/N pixels.

The results are plotted in Figure 3, showing the 2D intensity maps for Hα, [N ii], and [S ii]. The velocity and velocity dispersion maps are also shown.

The first and most striking feature is how most of the Hα and dust emission are not co-located, with Hα much more extended than the dust by a factor of ∼3. The sky separation between the peaks of the Hα and 870 μm continuum is 04, more than 3σ given ∼02 resolution in FWHM for both the ALMA and SINFONI observations. Although the systematic uncertainty in SINFONI astrometry could contribute to a further offset of $\sim 0\buildrel{\prime\prime}\over{.} 2\mbox{--}0\buildrel{\prime\prime}\over{.} 3$, later we show that the cold dust emission coincides with the regions with the reddest colors revealed by the WFC3 imaging, and most of the Hα emission matches the location of the brightest continuum in the rest-frame optical WFC3 maps. Therefore we conclude that the apparently disjoint nature between the cold dust as traced by the 870 μm continuum and the Hα emission is genuine. However, on the other hand, we find that the sky locations of [N ii] and [S ii] peak at the position of the dust emission, although both [N ii] and [S ii] are slightly more extended than the dust. The enhanced [N ii]-to-Hα ratio in the central regions could indicate higher gas-phase metallicity. However, the detected X-ray emission toward ALESS67.1 could also suggest that the elevated ratio is caused by the harder radiation field and/or shocks from AGNs. Deeper observations with detections including [O iii] and Hβ should help distinguish between these alternatives.

Finally, the 2D velocity map shown in Figure 3 displays a velocity gradient in Hα kinematics, with a peak-to-peak velocity difference of 750 ± 220 km s⁻¹. Given the integrated velocity dispersion of 280 ± 40 km s⁻¹ (Section 3.3.1), ALESS67.1 would be considered rotation-dominated based on the criterion of $({v}_{\max }\mbox{--}{v}_{\min })/2{\sigma }_{\mathrm{int}}=0.4$, which has been used in some works to roughly differentiate an orderly rotating disk and a merger (e.g., Förster Schreiber et al. 2009). However, the availability of the 2D velocity map allows us to conduct detailed kinematic modeling to more reliably differentiate rotating systems from mergers.

We start by modeling the velocity field assuming a rotating disc. We adopt the simplest function for the rotational curve, the arctan function (Courteau 1997), with the 1D form of

$$\begin{eqnarray}&&v(r)={v}_{0}+\displaystyle \frac{2}{\pi }{v}_{\mathrm{asym}}\arctan \left(\displaystyle \frac{r-{r}_{0}}{{r}_{t}}\right)\end{eqnarray} \tag{ 1 }$$

where v₀ is the systemic velocity, which in our case is zero as the velocity field is referenced at the systematic redshift, v_asym is the asymptotic velocity, r₀ is the central position, and r_t is the transition radius between the rising and flat part of the rotational curve. The arctan function has been found to have the flexibility to reasonably describe $z\gtrsim 1$ rotating galaxies (e.g., Miller et al. 2011; Swinbank et al. 2012). Based on Appendix A of Begeman (1989) the 1D rotational curve is projected to 2D via

$$\begin{eqnarray*}&&{v}_{p}(x,y)=v(x,y)\sin (i)\displaystyle \frac{-(x-{x}_{0})\sin \phi +(y-{y}_{0})\cos \phi }{\sqrt{{(x-{x}_{0})}^{2}+{(y-{y}_{0})}^{2}}}\end{eqnarray*}$$

where i is the inclination angle, the angle between the normal to the plane of the galaxy and the line of sight (i.e., 0° if face-on and 90° if edge-on), x₀ and y₀ is the central sky position, and ϕ is the positional angle (P.A.) of the major axis, defined as the angle taken in the anticlockwise direction between the north direction in the sky and the major axis of the receding half of the galaxy.

We fit the 2D model to the measured data based on maximum likelihood; in particular we run emcee, a Markov chain Monte Carlo (MCMC) ensemble sampler (Foreman-Mackey et al. 2013), to explore the parameter space and derive uncertainties. We limit the parameters to the range that is allowed by the data; in particular, the center position (x₀/y₀) and the turnover radius (r_t) must be within the SINFONI FOV and the P.A. lies between 0° and 180°. Because v_asym and i are essentially degenerate for our data quality we treat ${v}_{\mathrm{asym}}\sin (i)$ as a single parameter.

The best-fit model has a ${v}_{\mathrm{asym}}\sin (i)=290\pm 70$ km s⁻¹ and a P.A. of 110° ± 20°, and is plotted over the velocity field in Figure 3, in which the difference between the model and the data is also shown. The reduced ${\chi }_{\nu }^{2}=14.8$, indicating a relatively poor fit to the data and suggesting ALESS67.1 is not a pure rotating disk.

While modeling the 2D velocity field offers clues as to whether the system is well-described by a simple rotating disk, quantifying the asymmetry of both velocity and velocity dispersion provides a more complete and well-defined view of the kinematics of the system. One well-tested way to measure the asymmetry of the kinematics is to use kinemetry, originally presented by Krajnović et al. (2006) and designed to study local high-S/N stellar kinematic data such as those from the SAURON project (Bacon et al. 2001). It has been further developed into an effective tool to separate disks from mergers (e.g., Shapiro et al. 2008), although at $z\gt 1$ the effectiveness may depend on the interaction stage of the merger (e.g., Hung et al. 2015).

The basic goal of kinemetry analysis is to first decompose the 2D kinematic moment (e.g., velocity and velocity dispersion) maps into a series of concentric ellipses with increasing major axis length, which are defined by the systemic center and positional angle (Krajnović et al. 2006). For each concentric elliptical ring the kinematic moment as a function of the azimuthal angle is then extracted and further decomposed into the Fourier series, which can be described as

$$\begin{eqnarray*}&&K(r,\psi )={A}_{0}(r)+\displaystyle \sum _{n=1}^{N}{A}_{n}(r)\sin (n\psi )+{B}_{n}(r)\cos (n\psi )\end{eqnarray*}$$

where r is the length of the semimajor axis in each elliptical ring, ψ is the azimuthal angel, and A_n and B_n are nth-order coefficients. This equation can be shortened to

$$\begin{eqnarray*}&&K(r,\psi )={A}_{0}(r)+\displaystyle \sum _{n=1}^{N}{k}_{n}(r)\cos \{n[\psi -{\phi }_{n}(r)]\}\end{eqnarray*}$$

where ${k}_{n}=\sqrt{{A}_{n}^{2}+{B}_{n}^{2}}$ and ${\phi }_{n}=\arctan ({A}_{n}/{B}_{n})$.

In the case of an ideal rotating disk, the kinemetry of the velocity and velocity dispersion field would be dominated by B₁ and A₀ coefficients, respectively. Any perturbation from an ideal disk would manifest itself in the higher-order kinemetry coefficients. Therefore the ratios between the high-order and the dominant coefficients in the ideal disk case can be used to quantify the kinematic asymmetry. By using a sample of local galaxies as the training sample Shapiro et al. (2008) proposed a criterion of ${K}_{\mathrm{asym}}={({v}_{\mathrm{asym}}^{2}+{\sigma }_{\mathrm{asym}}^{2})}^{1/2}=0.5$ to separate rotating disk (${K}_{\mathrm{asym}}\lt 0.5$) and merger (${K}_{\mathrm{asym}}\gt 0.5$), where

$$\begin{eqnarray*}{v}_{\mathrm{asym}} & = & {\left\langle \displaystyle \frac{{k}_{2,v}+{k}_{3,v}+{k}_{4,v}+{k}_{5,v}}{4{B}_{1,v}}\right\rangle }_{r},\\ {\sigma }_{\mathrm{asym}} & = & {\left\langle \displaystyle \frac{{k}_{1,\sigma }+{k}_{2,\sigma }+{k}_{3,\sigma }+{k}_{4,\sigma }+{k}_{5,\sigma }}{5{B}_{1,\sigma }}\right\rangle }_{r}.\end{eqnarray*}$$

We run the kinemetry code provided by Krajnović et al. (2006) on the velocity and velocity dispersion maps shown in Figure 3. As pointed out by Krajnović et al. (2006) the dominant uncertainty of the kinemetry analyses is the choice of the center position. We therefore adopt the best-fit position based on the previous rotational curve modeling but perturb it within the error obtained from the MCMC analyses. We find a median ${v}_{\mathrm{asym}}=0.15\pm 0.01$ and a median ${\sigma }_{\mathrm{asym}}=0.48\pm 0.13$, which leads to ${K}_{\mathrm{asym}}=0.64\pm 0.15$, suggesting that, judging from the optical line kinematics, ALESS67.1 is a borderline merger (still consistent with a rotating disk given the error) based on the criterion proposed by Shapiro et al. (2008).

Measuring the kinematics of a galaxy provides an insightful view on its formation and evolution history. For SMGs in particular, kinematic measurements have mostly been used to assess the mechanisms that drive the enhancement of star formation (e.g., Bouché et al. 2007) and to investigate their evolutionary link to local massive ellipticals (e.g., Swinbank et al. 2006).

However, their high dust obscuration and low surface density means that it is difficult to obtain ∼ kpc resolution optical emission line observations, which require AO for $z\sim 2$ SMGs and are essential to complement the kinematics measured using molecular gas made with ALMA. Consequently there are still only a handful of SMGs that have AO-aided strong optical line data (e.g., Menéndez-Delmestre et al. 2013; Olivares et al. 2016). Together with the technical difficulty of obtaining sub-arcsecond resolution data for (sub)millimeter molecular lines, in particular low-J CO, most studies have relied on single tracers to assess and compare the kinematics, assuming that different tracers behave similarly. With sub-arcsecond resolution data in hand for both ¹²CO(J = 3–2) and Hα we are in a position to compare these tracers in kinematics.

Our ¹²CO(J = 3–2) detection does not have high enough S/N to produce 2D kinematic maps like Hα; we therefore taper the CO map, divide the line in half, and show the averaged maps in both the receding and approaching sides. We overlay the results on the 2D velocity map obtained from SINFONI in Figure 5.

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As revealed in Figure 5, in the outskirts the extended CO emission appears to follow the bulk rotating motion of the optical emission lines, in particular in the south-east filament. This filament links to a second stellar component revealed in the HST imaging (Figure 4) and could suggest that this second stellar component along with the connecting gas stream was falling toward the dynamical center close to the 870 μm continuum. The peak of the blueshifted part of the CO line agrees with that of the redshifted part; however, higher S/N CO observations are needed to reveal the detailed kinematics in the central regions. In general we find that the kinematics of both H ii regions as traced by the optical emission lines and molecular gas as traced by ¹²CO(J = 3–2) are in broad agreement.

The interstellar medium of the dense environments in which rapid star formation occurs is dominated by molecular hydrogen, H₂. As a result H₂ plays a central role in the formation and evolution of galaxies (Kennicutt & Evans 2012). While cold H₂ is not directly observable in emission, ¹²CO has been widely used to trace total molecular gas, with a standard conversion between ¹²CO luminosity and the total molecular gas mass, ${M}_{\mathrm{mol}}={\alpha }_{\mathrm{CO}}{L}_{\mathrm{CO}(1-0)}^{{\prime} }$, in which ${\alpha }_{\mathrm{CO}}$ is the CO-to-H₂ conversion factor. For extragalactic sources, the conversion factor depends on the properties of the galactic environments such as gas density and metallicity (Bolatto et al. 2013), resulting in a factor of ∼6 variation between local mergers and local spiral galaxies (Downes & Solomon 1998).

Previous studies have attempted to quantify ${\alpha }_{\mathrm{CO}}$ for SMGs; however, subject to a lack of constraints on dust mass and/or the size and dynamics of the CO emission, the results were generally not conclusive (e.g., Bothwell et al. 2013), though on average SMGs were found to have a low ${\alpha }_{\mathrm{CO}}$ (Danielson et al. 2011). With all the necessary measurements in hand for the case of a merger we can quantify and test the link between the merger and ${\alpha }_{\mathrm{CO}}$.

To measure ${\alpha }_{\mathrm{CO}}$ we first adopt a dynamical method, in which the gas mass is derived by subtracting stellar and dark-matter mass from the dynamical mass (i.e., ${M}_{\mathrm{dyn}}(r\leqslant {r}_{e})\,=0.5\times ({M}_{\mathrm{star}}+{M}_{\mathrm{gas}})+{M}_{\mathrm{dark}}(r\leqslant {r}_{e})$, where r_e represents the half-light radius and we assume the gas and stars have the same r_e given this is what we find (Figure 6). Within the half-light radius the dark matter contribution (M_dark) is estimated to be 10%–20% of the dynamical mass in $z\sim 2$ star-forming galaxies (e.g., Genzel et al. 2017) so we adopt 15%. Since we find that ALESS67.1 is rotation-dominated based on the optical emission lines Section 3.3.2, the dynamical masses can be estimated using Newtonian dynamics assuming a point mass, ${v}_{c}({r}_{e})=\sqrt{{{GM}}_{\mathrm{dyn}}(\lt {r}_{e})/{r}_{e}}$, in which v_c is the circular velocity and G is the gravitational constant. Based on the dynamical modeling presented in Section 3.3.2 v_c can be described by Equation (1). However in a highly turbulent environment (v_asym/$\sigma \lesssim 3$), the rotational velocity is significantly reduced due to turbulent pressure effects and needs to be corrected, with a form of ${v}_{c}^{2}({r}_{e})={v}^{2}({r}_{e})+3.36\times {\sigma }^{2}$ (Equation (11) in Burkert et al. 2010 assuming exponential surface density distribution). This correction assumes constant velocity dispersion (σ) over the spatial extent, which is supported by our data (Figure 3). We derive a median σ of 154 ± 5km s⁻¹ with a bootstrapped error. By adopting an averaged inclination $\langle {\sin }^{2}(i)\rangle =2/3$ (Tacconi et al. 2008) and a half-light radius of ${r}_{1/2,{\rm{H}}\alpha }=6.6\pm 0.9\,\mathrm{kpc}$ we derive ${v}_{c}({r}_{e})=380\pm 40$ km s⁻¹ giving a dynamical mass of ${M}_{\mathrm{dyn}}(r\leqslant {r}_{e})=(2.2\pm 0.6)\,\times {10}^{11}$ M_⊙. The stellar mass is estimated to be ${M}_{\mathrm{star}}=2\times {10}^{11}$ M_⊙ (Simpson et al. 2014; da Cunha et al. 2015). Hence we estimate a gas mass ${M}_{\mathrm{gas}}=(1.8\pm 1.0)\times {10}^{11}$ M_⊙. Given ${L}_{\mathrm{CO}(1-0)}^{{\prime} }=(9.9\pm 1.8)\times {10}^{10}$ K km s⁻¹ pc² (Huynh et al. 2017), we derive ${\alpha }_{\mathrm{CO}}=1.8\pm 1.1$.

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We can also check our ${\alpha }_{\mathrm{CO}}$ estimate by using the gas-to-dust ratio method. The idea is that by estimating the gas-to-dust mass ratio, ${\delta }_{\mathrm{GDR}}$, in a galaxy with measured molecular gas and dust masses, the conversion factor ${\alpha }_{\mathrm{CO}}$ can be derived as ${\alpha }_{\mathrm{CO}}={\delta }_{\mathrm{GDR}}({\mu }_{0}){M}_{\mathrm{dust}}/L{{\prime} }_{\mathrm{CO}(1-0)}$ assuming that molecular gas dominates the gas masses, and ${\delta }_{\mathrm{GDR}}$ can be related to gas phase metallicity (${\mu }_{0};$ Leroy et al. 2011). Assuming negligible AGN contribution to the [N ii]/Hα ratio, we derive a metallicity of 12+log(O/H) = 8.8 ± 0.1 in the central region, and 8.6 ± 0.1 over the entire galaxy, by adopting the $N2\,\equiv$ log₁₀([N ii]λ6583/Hα) empirical calibration provided by Pettini & Pagel (2004) (12 + log₁₀(O/H) = 8.90+0.57 × N2). By using a circular aperture on the CO(J = 3–2) cube centered at the 870 μm continuum peak with a deconvolved radius matching to the dusty region ($\leqslant 0\buildrel{\prime\prime}\over{.} 4$), we estimate ${L}_{\mathrm{CO}(1-0)}^{{\prime} }=(4.0\pm 1.9)\times {10}^{10}$ K km s⁻¹ pc². Based on the metallicity measurement and the best-fit linear function provided by Leroy et al. (2011), we compute ${\delta }_{\mathrm{GDR}}$ of 78 ± 6 in the central region. By adopting the dust mass of ALESS67.1 derived by Swinbank et al. (2014), we calculate a gas mass of $(5.5\pm 0.6)\times {10}^{10}\,{M}_{\odot }$, and therefore ${\alpha }_{\mathrm{CO},\mathrm{GDR}}\,=1.4\pm 0.7$ in the central dusty regions, in good agreement with the ${\alpha }_{\mathrm{CO}}$ estimated using the dynamical method. On the other hand, it is also possible that the apparent size difference between dust and CO is caused by observational bias due to the different optical depth probed by CO and 870 μm continuum. In such a case CO and dust are still well-mixed in the entire galaxy and the total CO luminosity needs to be adopted for the gas-to-dust ratio method. If we do so and adopt global integrated values for all relevant measurements, we obtain a lower ${\alpha }_{\mathrm{CO}}$ of 0.5 ± 0.3, consistent with the results derived using other methods. The results of the gas-to-dust ratio method should be treated as lower limits as the AGN contribution could lower the gas-phase metallicity and hence suggest higher gas masses.

Despite the uncertainty, our result is consistent with that of another strongly lensed SMG, SMMJ2135-0102 (Cosmic Eyelash; Danielson et al. 2011, 2013; Thomson et al. 2015), and Arp 220 (Scoville et al. 1997), a local merger that has been used to compare with $z\sim 2$ SMGs, with both having ${\alpha }_{\mathrm{CO}}\sim 1$. Confirming the results from other analyses, our results on ${\alpha }_{\mathrm{CO}}$ suggest a consistent scenario that ALESS67.1 is undergoing a merger.

Recent SMG studies of kpc-scale dust distributions using ALMA (sub)millimeter observations have found, almost unequivocally, compact sizes with an average half-light radius of 1–2 kpc (e.g., Ikarashi et al. 2015; Simpson et al. 2015; Hodge et al. 2016; Spilker et al. 2016). Above an IR luminosity of ∼3 × 10¹² ${L}_{\mathrm{IR}}$ (SFR ∼ 300 M_⊙ yr⁻¹), the dust sizes do not appear to depend on ${L}_{\mathrm{IR}}$ and redshift ($z\sim 1\mbox{--}6$), and they are on average a factor of 2–3 smaller than the NIR continuum revealed by the HST (Hodge et al. 2016).

We have presented analyses and measured sizes and spatial distributions of CO, optical emission lines, 870 μm and NIR continuum for ALESS67.1. We summarize part of the measurements in Figure 6 in a curve-of-growth style to emphasize the size difference between each tracer. All maps are convolved with the ¹²CO beam to make the spatial resolution comparable.

Besides confirming the size contrast between 870 μm and 1.6 μm continuum, we also find significantly larger sizes for all other tracers with respect to the 870 μm dust continuum. Perhaps the most striking of all is the factor of 5.2 ± 0.8 difference in half-light radius between the FIR/submillimeter dust continuum and the ¹²CO(J = 3–2) gas, which is normally found to agree within a factor of two in nearby star-forming galaxies (e.g., Sandstrom et al. 2013), as well as local mergers such as Arp220 (Scoville et al. 1991), NGC 6240 (Iono et al. 2007), and NGC 3256 (Sakamoto et al. 2006).

The measured ¹²CO(J = 3–2) size of ALESS67.1 is consistent with that of ¹²CO(2–1) from the z = 4.055 SMG GN20 (Hodge et al. 2012), as well as that of ¹²CO(1–0) in a sample of four SMGs presented in Ivison et al. (2011) and the z = 3.408 SMG SMMJ13120+4242 (Riechers et al. 2011). However, it is significantly larger than the ¹²CO sizes reported by Engel et al. (2010), in which they claimed an average half-light radius of 2.5 ± 1.3 kpc for ¹²CO lines with upper $J\leqslant 6$. The difference could be a reflection of true scatter in ¹²CO sizes, which is a factor of ∼8 in local mergers (Ueda et al. 2014) or, as claimed by Ivison et al. (2011) that low-J lines are more spatially extended than high-J ones, or both.

On the other hand, in terms of spatial extent and surface brightness, we find that Hα follows a similar distribution as the stellar components traced by the NIR emission. When comparing the Hα/NIR continuum to ¹²CO we find similar sizes.

When comparing to the measurements in the literature, the NIR size of ALESS67.1, which is 6.4 ± 0.5 kpc using the curve-of-growth method (Figure 6), lies within the average size of the ALESS parent sample (Chen et al. 2015) and, given the stellar mass estimates, is consistent with the size census on mass-selected samples of star-forming galaxies at $z\sim 2$ (van der Wel et al. 2014). The NIR size of ALESS67.1 is, however, significantly larger than that of 500 μm selected DSFGs reported by Calanog et al. (2014), in which they attribute the difference to the selection bias caused by the strong gravitational lensing in their sample.

The Hα size of ALESS67.1 lies at the higher end but still consistent with respect to other SMGs reported in Alaghband-Zadeh et al. (2012) (average 3.7 ± 0.8 kpc) and main-sequence SFGs (Förster Schreiber et al. 2009), but significantly larger than the Hα emitters (HAEs) at similar reshifts (Molina et al. 2017). The difference between ALESS67.1 and HAEs, apart from small samples, could be caused by the fact that the HAE sample presented in Molina et al. is on average a factor of $\gt 3$ less massive than ALESS67.1.

Finally, the geometrical discrepancy among different tracers, which is also seen in other studies of high-z galaxies (e.g., Spilker et al. 2015; Decarli et al. 2016; Ginolfi et al. 2017; Koprowski et al. 2016), may have some impact on theoretical modeling. As already discussed in Simpson et al. (2017), geometrical differences in SMGs between dust and UV-to-NIR emissions leads to drastically different estimations for dust extinction, with Simpson et al. deriving an ${A}_{{\rm{V}}}={540}_{-40}^{+80}$ using a hydrogen column density method based on direct measurements of dust column density, in contrast to just ${A}_{{\rm{V}}}\sim 1\mbox{--}3$ based on SED modeling of the detectable optical emissions either using simple dust screen modeling (Simpson et al. 2014) or an energy-balance approach (da Cunha et al. 2015). For SMGs, because of this geometrical discrepancy, it may be more sensible to model optical-to-NIR SEDs and FIR/submillmeter/radio SEDs separately.

In the next two sections we discuss the impact of this geometrical discrepancy on the IRX–β and Schmidt–Kennicutt relationships.

The relationship between the ratio of IR and UV luminosity at 1600 Å (IRX) and the UV spectral slope at 1600 Å (β) offers a potential route to estimate the total SFR when only the rest-frame UV observations are available. This IRX–β relationship has therefore been widely used to estimate the total star formation rate density for UV-selected populations at $z\gt 3$ and up to the epoch of reionization (e.g., Bouwens et al. 2015).

Given its fundamental implications for the measurements of SFR density at high redshifts, the IRX–β relationship and its deviation has been extensively studied both in the local universe and at high redshifts (e.g., Meurer et al. 1999; Kong et al. 2004; Buat et al. 2005; Howell et al. 2010; Overzier et al. 2011; Reddy et al. 2012; Takeuchi et al. 2012; Casey et al. 2014b; To et al. 2014). Among the many factors that affect the IRX–β relationship, such as star formation history and internal attenuation curve, the geometrical effect has been proposed to explain the deviation seen in samples of ULIRGs, both in the local universe (Howell et al. 2010) and at $z\sim 2$ (Casey et al. 2014b), in a sense that the different or completely decoupled geometry between dust and UV could explain the increase of the IRX at a fix β. Recent theoretical models have also confirmed this hypothesis (e.g., Narayanan et al. 2017; Popping et al. 2017). With ∼ kpc resolution data in rest-frame UV, optical, and FIR, we are set to examine this proposal.

We first compute the UV luminosity and spectral slope at 1600 Å. ALESS67.1 is covered by CANDELS imaging (Figure 4) as well as 3D-HST grism spectroscopy (Brammer et al. 2012), and the multi-wavelength photometry and rest-frame UV flues at 1400, 1700, 2200, 2700, and 2800 Å are provided in Skelton et al. (2014). We adopt the values in the 3D-HST catalog and derive the spectral slope by fitting the data with a functional form of $F(\lambda )=A{\lambda }^{\beta }$. We then use this best-fit function to compute the rest-frame UV flux at 1600 Å in units of erg s⁻¹ cm² Å⁻¹ and calculate the UV luminosity using ${L}_{\mathrm{UV}}=4\pi {D}_{{\rm{L}}}^{2}F(1600){\lambda }_{1600}/(1+z)$, in which ${D}_{{\rm{L}}}$ is the luminosity distance at redshift z. The IR luminosity is adopted from Swinbank et al. (2014), which uses deblended Herschel data. The results are plotted in Figure 7, in which we also show the composite image of dust, stars, and UV emission.

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Consistent with the trend found by Casey et al. (2014b), with an ${L}_{\mathrm{IR}}$ of ${10}^{12.7}$ ${L}_{\odot }$ ALESS67.1 lies significantly above the relationships found by other studies for less IR-luminous galaxies. As shown in the right panel of Figure 7 the completely decoupled geometry between rest-frame 280 μm and rest-frame 1400 Å emission confirms the hypothesis that the deviation from the IRX–β relationship is caused, at least partially, by the different distributions between dust and UV. For the individual UV-emitting regions, on the other hand, our ALMA data are not sensitive enough to put meaningful upper limits on whether or not they lie on the relationships (left panel in Figure 7). This is the same with the dusty regions, where we do not have meaningful constraints on either UV luminosity or slope.

The relationship between the surface density of star formation rate and that of gas has been studied back to the seminal work of Schmidt (1959) and Kennicutt (1989). Considering galaxies as a whole and assuming a single ${\alpha }_{\mathrm{CO}}$, it was later claimed in observations that this tight correlation, the Schmidt–Kennicutt relationship, with a scatter about a factor of two, holds valid in local star-forming galaxies over six orders of magnitude in SFR surface density (Kennicutt 1998b).

However, studies of local interacting mergers have shown a significantly lower ${\alpha }_{\mathrm{CO}}$ (${\alpha }_{\mathrm{CO}}\sim 1;$ e.g., Scoville et al. 1997; Downes & Solomon 1998), which could also be true at high redshifts based on our analyses of ALESS67.1 in Section 4.2. By adopting the lower ${\alpha }_{\mathrm{CO}}$ for (U)LIRGs, together with observations of $z\sim 2$ star-forming galaxies and SMGs, it has been claimed that the Schmidt–Kennicutt relationship becomes bimodal, in which IR-luminous galaxies have significantly higher SFR surface density at a fixed gas surface density, with a typical gas consumption time (${{\rm{\Sigma }}}_{\mathrm{gas}}/{{\rm{\Sigma }}}_{\mathrm{SFR}}$) of ∼100 Myr, in contrast to ∼1 Gyr for normal SFGs (e.g., Daddi et al. 2010; Genzel et al. 2010). Recent spatially resolved observations of SMGs on both star-forming regions as traced by dust or radio emissions and CO have found an even shorter gas consumption time of ∼10 Myr (Bothwell et al. 2010; Hodge et al. 2015). With measured luminosities and sizes on FIR, CO, and Hα we are in a position to investigate the Schmidt–Kennicutt relationship on a $z\sim 2$ merger.

We first compute the global surface density of SFR and H₂ for ALESS67.1, by adopting the measured sizes of each tracer and the conversion between ${L}_{\mathrm{IR}}$ and ${L}_{{\rm{H}}\alpha }$ to SFR from Kennicutt (1998a). We assume a Chabrier initial mass function (Chabrier 2003) and adopt 0.8 for ${\alpha }_{\mathrm{CO}}$ in order to make comparisons with other studies. The results are plotted in Figure 8, in which we also include literature values from local and high-redshift SFGs and SMGs. We only include measurements obtained from observations that spatially resolve both the star-forming regions and CO, which is the condition of our data.

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Our measurements agree with GN20, an SMG at $z\sim 4$, as well as some SMGs in the sample of Bothwell et al. (2010), suggesting that, when looking at the galaxy as a whole, some SMGs have significantly shorter gas consumption times even compared to the local mergers. However, because of the dramatic size difference between the dusty star-forming regions, which dominate the SFR surface density, and CO, the comparison of ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ and ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ on the global scale is non-physical, meaning the two parameters come from regions that are largely unrelated, and therefore the deviation from the typical relationship is expected. To make proper interpretations it is necessary to measure both parameters in resolved regions.

Motivated by the spatial distribution of the 870 μm continuum, ¹²CO(J = 3–2), and Hα, we construct concentric rings with a width of $0\buildrel{\prime\prime}\over{.} 5$ (∼4 kpc) for each ring and centered at the peak of the 870 μm continuum. The choice of width is determined by the spatial resolution of CO. We convolve the 870 μm continuum and Hα with a CO beam to match the maps in spatial resolution. We then measure ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ and ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ for each concentric ring and plot the results in the right panel of Figure 8, color coded based on their radial distances.

Unlike the global values which make ALESS67.1 almost an outlier compared to the local IR-selected galaxies, the measurements in each sub-region lie in the similar locus occupied by previous measurements. In particular, within one galaxy we obtain a similar bimodal distribution as seen between normal SFGs and mergers. The transition, which is at ∼5 kpc (∼05) in this case, occurs when there is a lack of detection at 870 μm, leaving ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ derived solely from Hα. We should stress, however, that these values are derived under the assumption of ${\alpha }_{\mathrm{CO}}=0.8$, and Hα is not corrected for dust extinction, if any. If the sub-regions in the outskirts had a Galactic ${\alpha }_{\mathrm{CO}}$ instead, the derived quantities would shift to higher ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ by ∼0.7 dex, which would make these regions significantly more gas-rich, or significantly inefficient in star formation. This result is mainly the consequence of the much more extended distribution of ¹²CO(J = 3–2) compared to the dust traced by the rest-frame 280 μm continuum.