A Subarcsecond ALMA Molecular Line Imaging Survey of the Circumbinary, Protoplanetary Disk Orbiting V4046 Sgr

The resulting ALMA 235 GHz and 264 GHz continuum images and moment 0 line images for the molecular species and transitions listed in Table 1, as well as a moment 1 ¹²CO(2–1) image, are presented in Figure 1 in 12'' × 12'' fields of view and in Figures 2 and 3 in 6'' × 6'' fields of view. The moment 0 images displayed in these figures have been integrated over channel ranges that span the detectable emission; for all images apart from H₂CO and N₂H⁺, the component channel maps were clipped at the 1σ level in order to isolate the signal.

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The continuum, ¹²CO, and ¹³CO images reproduce, at higher spatial resolution and signal-to-noise, the basic features seen in the best previous 1.3 mm continuum and CO imaging of V4046 Sgr, which were obtained with the SMA and presented in Rosenfeld et al. (2013). Specifically, the millimeter-wave continuum emission appears as a partially filled ring, peaking at ∼03 (∼22 au) from the star, with a sharp intensity cutoff at its outer radius of ∼08 (∼60 au), whereas the CO is strongly centrally peaked and the ¹²CO emission outer radius (∼4'', i.e., ∼300 au) is far larger than that of the continuum. As was the case for the previous SMA imaging (Rodriguez et al. 2010; Rosenfeld et al. 2013), the large-scale Keplerian rotation of the disk is readily apparent in the moment 1 image of ¹²CO(2–1).

The molecular emission-line images in Figures 2 and 3 are grouped according to their morphological properties (see also Huang et al. 2017), wherein Figure 2 displays images with centrally peaked morphologies and Figure 3 displays images with ringlike morphologies that range from distinct to somewhat diffuse. Correspondingly, in Figures 4 and 5, respectively, we display radial profiles extracted from the centrally peaked and ringlike molecular emission-line images (see Section 3.2).

• Centrally peaked morphologies (Figures 2 and 4). The images of C and O isotopologues of CO and C and N isotopologues of HCN all display centrally peaked morphologies. The line emission from the rare CO and HCN isotopologues is far more compact than that of the most abundant isotopologues (¹²C¹⁶O and H¹²C¹⁴N), as expected in light of the smaller optical depths and resulting lower fluxes in the lines of the rare isotopologues.
• Distinct ringlike morphologies (Figures 3 and 5). The HC₃N, CH₃CN, DCO⁺, and C₂H line images display ringlike emission morphologies wherein the emission rings have clearly defined central holes, relatively sharp outer edges, or both. The emission from the cyanide group molecules, HC₃N and CH₃CN, arises from compact regions whose outer radii are similar to those of the 1.1–1.4 mm continuum emission ring (see Section 3.2). Due to their low signal-to-noise ratios and small angular sizes, we find that the depths of the inner holes within the HC₃N and CH₃CN emitting regions are sensitive to the adopted visibility weighting during the image reconstruction (cleaning) process, while the degree of azimuthal asymmetry is sensitive to channel noise clipping in constructing the moment 0 images. However, the overall ringlike morphologies of HC₃N and CH₃CN—with the CH₃CN ring possibly more asymmetric and filled in than HC₃N—appear to be a robust result of these observations. The size scales of the DCO⁺ and C₂H rings are much larger than that of the cyanide group molecules, with integrated line intensities that peak far outside the continuum emission ring. The C₂H ring exhibits sharper edges than that of DCO⁺ and, as is seen more clearly in radial profiles extracted from these images (Section 3.2), the C₂H also peaks at a larger radius.
• Diffuse ringlike morphologies (Figures 3 and 5). The DCN, H¹³CO⁺, N₂H⁺, and H₂CO line images all display diffuse emission morphologies and appear as rings to a greater or lesser degree. The DCN(4–3) image presented here, which has a higher signal-to-noise ratio than the DCN(3–2) image presented in Huang et al. (2017), confirms the ringlike DCN emission morphology hinted at in that survey. Evidently, the diffuse DCN ring is more compact than the sharper C₂H or DCO⁺ rings, while the H¹³CO⁺ ring dimensions appear to more closely resemble those of DCN than those of DCO⁺. The radius of peak integrated line intensity within the bright N₂H⁺ ring is comparable to the radius of peak integrated C₂H line intensity, but with its filled in central hole and outer halo, the N₂H⁺ ring is much larger and more diffuse in appearance. The outer radius of the H₂CO emitting region appears to be similar to that of CO, but (unlike CO) the H₂CO has a central hole. As in the case of DCO⁺ and C₂H, the inner holes of the N₂H⁺ and H₂CO line emission rings, though poorly defined, appear to be similar to the outer radii of the continuum rings.

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In Figures 4 and 5 we display radial profiles extracted from unclipped versions of the moment 0 molecular emission-line images of V4046 Sgr, as well as the radial profile of the 264 GHz continuum image. These profiles were extracted after deprojecting the images assuming a disk inclination i = 335 and position angle of 67° (Rosenfeld et al. 2013). The error bars indicate statistical uncertainties, which we estimated as σ_r = σ_xy ($\sqrt{{\rm{\Delta }}V/\delta v})\,(1/\sqrt{N})$, where σ_xy is the rms flux density deviation in a channel map (as determined from an emission-free region of the map), ΔV is the line width, δv is the channel width (such that ΔV/δv is the number of velocity channels over which the line emission was integrated), and N is the number of pixels included within the radial bin.

Figure 4 provides a comparison of the radial profiles of integrated emission from isotopologues of CO and HCN, which have centrally peaked image morphologies (Figure 2). Some caution must be applied in comparing these profiles in detail, since the beam widths of the CO isotopologue images are ∼40% larger than those of the HCN isotopologues (Table 1). The radial profiles of both ¹²CO and H¹²CN show inflection points (slope changes) at ∼15 (∼100 au) that are suggestive of bright cores surrounded by larger halo structures. Such features can be produced by subtraction of optically thick continuum emission from optically thick line emission (Weaver et al. 2018), but that is unlikely to be the case here, given that the inflection points lie well beyond the outer edge of the continuum ring (∼07). The ¹²CO falls off steeply with radius within the inner continuum ring, whereas the H¹²CN profile is shallow within this same region. The radial profiles demonstrate that detectable emission extends to ∼25 (∼180 au) and ∼4'' (∼300 au) for H¹²CN and ¹²CO, respectively, with ¹²CO intensity displaying a sharp outer edge that is best seen in the comparison of radial intensity profiles of CO isotopologue emission presented in Section 4.1.

In the left and right panels of Figure 5, we compare the radial profiles of integrated emission from molecular lines exhibiting sharply ringlike and more diffuse ringlike morphologies, respectively (Figure 3). Such a direct comparison of these radial profiles is enabled by the similar beam sizes of the data collected for all of these transitions (i.e., their beam widths differ by <15%; Table 1). A sequence is apparent in the radial position of peak intensity and (hence) central hole size within these rings of molecular line emission. The smallest rings are those of the cyanide group molecules (CH₃CN and HC₃N), which peak near ∼025 (∼18 au), just inside the peak continuum emission (see Figure 6). The largest rings are the sharp-edged C₂H and diffuse N₂H⁺, both of which peak at ∼12 (∼90 au), and H₂CO, which displays a broad peak at ∼16 (∼115 au). The emission rings of the deuterated species DCN and DCO⁺, which peak at ∼06 (∼45 au) and ∼08 (∼60 au), respectively, are intermediate in size. While we do not display the radial profile of H¹³CO⁺ due to its relatively poor signal-to-noise ratio (S/N), this faint ring appears to more closely trace DCN than DCO⁺ (Figure 3; see also Figure 2 in Huang et al. 2017).

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The upper left panels of Figure 5 illustrate the close correspondence of the radial profiles of emission from the cyanide group molecules (HC₃N and CH₃CN) to each other and to that of the 264 GHz continuum continuum emission profile, which peaks at ∼03 (∼22 au) and has an outer radius (at 10% of peak) of ∼08 (∼60 au). The peak intensities of HC₃N and CH₃CN emission appear to lie just inside that of the continuum, and the lack of a central hole is apparent in the case of CH₃CN. The radial distributions of HC₃N and CH₃CN line emission both display sharp outer radial cutoffs, falling to ∼10% of peak intensity at ∼07 (∼50 au).

In Figure 7, we display line profiles extracted from the interferometric data cubes for molecular species observed by ALMA subsequent to the observations presented in Huang et al. (2017) and Guzmán et al. (2017) or whose profiles were not presented in those papers. The extraction regions were ellipses whose major and minor axes approximately correspond to the ∼3σ noise levels in the individual velocity channel images. Whereas the ¹²CO and ¹³CO line profiles display the classical double-peaked profiles characteristic of Keplerian rotation (as was already apparent in single-dish observations; Kastner et al. 2008), the profile of C¹⁸O—which is presented here for the V4046 Sgr disk for the first time—appears more rounded, with stronger wings relative to the line core. This difference reflects the smaller detectable extent of the C¹⁸O emission, which effectively suppresses the emission at the low radial velocities characteristic of the outer disk relative to the higher-velocity emission characteristic of the central disk. Similar arguments appear to pertain to the line profiles of isotopologues of HCN, wherein the line profile of the most abundant isotopologue appears double-peaked and those of the rare isotopologues are more rounded. The line profiles of N₂H⁺ and H₂CO lack high-velocity wings, as expected given their large, diffuse, ringlike morphologies. The triple-peaked C₂H and CH₃CN line profiles result from the superposition of double-peaked emission lines in neighboring hyperfine transitions (compare with C₂H and CH₃CN line profiles displayed in Kastner et al. 2014; Öberg et al. 2015, respectively).

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Given their similar beam areas (which differ by <5%) and comparable sensitivities, the images in J = 2 → 1 emission from ¹²CO, ¹³CO, and C¹⁸O (Figure 2) afford the opportunity to investigate the ratios of CO isotopologue emission-line intensity as functions of radial position across the V4046 Sgr disk. These ratios in turn can be used to assess the optical depths of ¹²CO, ¹³CO, and C¹⁸O emission. Specifically, given an assumed (constant) isotopologue abundance ratio X_1,2 within the disk, the ratio of line emission intensities in two isotopologues at radial position r can be related to optical depth in the first (more abundant) isotopologue, τ₁, via

$$\begin{eqnarray*}&&{R}_{\mathrm{1,2}}(r)=\displaystyle \frac{1-\exp (-{\tau }_{1}(r))}{1-\exp -({\tau }_{1}(r)/{X}_{\mathrm{1,2}})}\end{eqnarray*}$$

(e.g., Kastner et al. 2014; Schwarz et al. 2016 carried out such an analysis of τ(r) for the TW Hya disk). Caution must be applied in interpreting estimates for τ₁(r) as deduced from R_1,2(r), given that emission from the different isotopologues likely arises from different vertical disk layers (e.g., Zhang et al. 2017). Furthermore, the optical depth can vary significantly both spatially (within the synthesized beam), as well as across the line profile over which the intensity is integrated. Hence, we restrict the present discussion to a qualitative analysis of the radial regimes over which the various isotopologue emission lines appear to be optically thick or thin.

Comparisons of the radial profiles in the various isotopologues, as well as the radial profiles of the intensity ratios of CO isotopologue emission, are presented in Figure 8. The integrated intensity of ¹²CO emission shows a precipitous drop at ∼4'' (∼300 au), indicative of either the steep outer edge of the gas disk or a sharp decline in CO abundance; the latter would presumably be due to UV photodissociation. The maximum radii of detectable ¹³CO and C¹⁸O emission are ∼3'' (∼220 au) and ∼1'' (∼75 au), respectively. These radii set the respective limits within which we can measure the ¹²CO:¹³CO and ¹³CO:C¹⁸O (and ¹²CO:C¹⁸O) emission-line ratios. We find the ratio of ¹²CO:¹³CO line emission intensities to be essentially independent of r given the uncertainties, with a value R_12,13 ∼ 2.5, while the ¹²CO:C¹⁸O line ratio increases from R_12,18 ∼ 5 to R_12,18 ∼ 18 over the radial range $r\lesssim 25\,{au}$ to r ∼ 75 au. The ratio of ¹³CO:C¹⁸O line emission intensities R_13,18 increases from R_13,18 ∼ 2 to R_13,18 ∼ 7 over this same radial range.

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For the values of X_12,13 and X_12,18 typically assumed in the astrophysical literature—i.e., X_12,13 from ∼40 to ∼70, and X_12,18 ∼ 480 (Kastner et al. 2014; Zhang et al. 2017, and references therein)—these small values of R_12,13 and R_12,18 imply, not surprisingly, that the ¹²CO(2–1) emission is very optically thick (i.e., τ₁₂ ≫ 1) across the disk surface. Meanwhile, the observation that R_13,18 < 7 (i.e., R_13,18 < X_13,18, assuming X_12,13 = 70) within ∼75 au indicates that τ₁₃ > 1, at least in the inner disk; the small, near-constant value of R_12,13 furthermore implies that ¹³CO(2–1) is optically thick throughout the disk. Assuming that the ratio of ¹³CO to C¹⁸O optical depths is identical to their abundance ratio, the foregoing results for τ₁₃ for V4046 Sgr also imply that C¹⁸O is optically thin throughout much of the disk. These results for τ₁₃ and τ₁₈ are similar to those inferred for TW Hya by Schwarz et al. (2016). We note, however, that Zhang et al. (2017) find that C¹⁸O(3–2) is optically thick in the innermost regions (r ≤ 20 au) of the TW Hya disk.

If both ¹²CO(2–1) and ¹³CO(2–1) are optically thick, their line ratio is diagnostic of the characteristic temperatures of the regions from which the bulk of the emission in each line originates. Hence, Figure 8 indicates that the optically thicker ¹²CO(2–1) emission arises from a warmer and hence higher-lying disk layer than ¹³CO(2–1), as expected given the vertical temperature inversion that is a feature of models of irradiated disks (e.g., Zhang et al. 2017). Higher spatial resolution ALMA imaging of CO isotopologue emission from the V4046 Sgr disk, as well as observations of ¹³C¹⁸O emission analogous to those carried out for TW Hya (Zhang et al. 2017), would test the foregoing qualitative inferences concerning the radial dependencies of the optical depths of CO isotopologue emission and would provide important additional constraints on disk vertical structure, molecular mass, and ¹²C:¹³C abundance ratio.

V4046 Sgr stands out among disks surveyed thus far by ALMA for its unusually bright emission from both HC₃N and CH₃CN (Bergner et al. 2018). As in the case of MWC 480, the first disk detected in both of these complex nitrile (cyanide-bearing) species (Öberg et al. 2015), there appears to be a spatial correspondence between the HC₃N and CH₃CN line emission and the continuum emission from large (millimeter-sized) dust grains in the V4046 Sgr disk. However, the subarcsecond ALMA imaging of V4046 Sgr presented here makes apparent the ringlike morphologies of HC₃N and, possibly, CH₃CN (Figure 3). These morphologies closely correspond to those of the continuum emission as well as scattered light from small grains just interior to the continuum ring (Rapson et al. 2015a)—and they contrast with the centrally peaked emission from the rare HCN isotopologues (Figure 6)—suggesting a connection between the presence of dust rings and the production of gas-phase HC₃N and CH₃CN. Various pure gas-phase production routes for HC₃N and CH₃CN and potential grain surface chemistry production of CH₃CN are discussed by Bergner et al. (2018) in the context of their survey of complex nitriles in V4046 Sgr and several other disks. Here, we point out one possible interpretation of the apparent morphological correspondence between emission from these species and that of emission from (and scattering off) dust grains within the V4046 Sgr disk.

Specifically, based on the results of simulations (Walsh et al. 2014; Cleeves et al. 2016) and laboratory experiments (Mendoza et al. 2013), it is possible that the ringlike HC₃N and CH₃CN emitting regions are generated by stellar irradiation of ice-coated grains. In such a scenario, HC₃N, CH₃CN, and other more complex organics either form in the disk, or were inherited from the prestellar core that spawned V4046 Sgr. In either case, a significant mass of organics likely subsequently accretes (freezes out) onto grains in the present-day disk and/or at earlier phases of the protostellar (Class 0/I) disk. The intense stellar UV and X-irradiation from V4046 Sgr (e.g., Argiroffi et al. 2012) impinging on the well-defined rings of dust grains within the disk then results in efficient desorption of the organic ice coatings, generating detectable column densities of HC₃N and CH₃CN (Walsh et al. 2014). Interestingly, HC₃NH⁺ is one of the dominant products of X-irradiation of pyrimidine ice-coated grains in laboratory experiments (Mendoza et al. 2013). This suggests that the abundances of HC₃N might be enhanced in UV- and X-irradiated regions that are sufficiently rich in pyrimidine and (perhaps) other organic ices, such that both the abundance of HC₃NH⁺ and the molecular gas ionization fraction are elevated. On the other hand, it is not clear whether either HC₃N or CH₃CN would then survive long in the gas-phase, given the large photodissociation rates within such a hostile radiation environment.

Similar arguments pertain to the production of gas-phase H₂CO via UV and X-ray photodesorption of hydrogenated grain (CO) ice mantles (Walsh et al. 2014; Öberg et al. 2017). Indeed, the Walsh et al. (2014) modeling predicts that the column density of gas-phase H₂CO thus generated should peak at, and remain elevated out to, much larger disk radii than the column densities of HC₃N and CH₃CN. Qualitatively, these model predictions also seem to be borne out by the ALMA imaging presented here (Figure 5; see also Section 4.3). We note, however, that the Walsh et al. (2014) model assumes a smoothly varying disk dust component, which does not describe evolved protoplanetary disks like the one orbiting V4046 Sgr, with its relatively narrow, compact ring of large grains and interior small-grain dust ring/gap system. The HC₃N and CH₃CN images presented here hence serve as motivation for future simulations aimed at ascertaining whether and how the processes of photodesorption and photodissociation of organics in a protoplanetary disk depend on the extent of dust grain coagulation and transport within the disk. Given that thermal excitation of the ∼260 GHz transitions observed here requires gas kinetic temperatures of ∼100 K, sensitive observations of lower-lying transitions of HC₃N and CH₃CN in V4046 Sgr would also help test the hypothesis that photodesorption is responsible for production of these molecules. As noted by Bergner et al. (2018), the extant data are not sensitive to emission from these molecules within cooler disk regions at larger radii.

Beyond the ∼50 au outer radius of the continuum and HC₃N and CH₃CN emission in the V4046 Sgr disk, emission from the species DCN, DCO⁺, N₂H⁺, C₂H, and H₂CO appears as a sequence of rings with an increasing radius of peak intensity (i.e., intensity peaks at ∼45, 60, 90, 90, and 110 au, respectively) and varying sharpness in terms of inner and outer cutoff radii (Figure 5). Among these rings, the sharp-edged morphology of C₂H is perhaps most striking; this emission ring is modeled and discussed in detail in Section 4.4. Here, we briefly comment on some potential implications of this molecular ring "sequence."

The main routes proposed for formation of DCO⁺ and DCN require the presence (destruction) of H₂D⁺ and CH₂D⁺, respectively (Huang et al. 2017, and references therein). Because survival of H₂D⁺ requires temperatures $\lesssim 30$ K, whereas survival of CH₂D⁺ is energetically favorable up to ∼80 K, one expects DCN to peak in abundance at smaller disk radii than DCO⁺ (Huang et al. 2017). Such a relationship is indeed observed in the case of V4046 Sgr. As formation of DCO⁺ also depends on the presence of gas-phase CO, one further expects DCO⁺ to be confined to a disk temperature regime and hence a (midplane) radial regime where the disk is not warm enough for destruction of H₂D⁺ (which requires temperatures in excess of ∼30 K) but not cold enough that there is significant freeze-out of CO (temperatures below ∼25 K). This rather narrow range of temperatures favorable to DCO⁺ formation may explain the ringlike appearance of DCO⁺ in V4046 Sgr and other disks (e.g., HD 163296; Qi et al. 2015). On the other hand, Huang et al. (2017) caution against concluding that DCO⁺ serves to trace the specific disk regime that lies just above the CO freeze-out temperature (snow line), citing the lack of a clear relationship between the radial extents of DCN and DCO⁺ in many of the disks they surveyed.

As discussed in Qi et al. (2013a, 2013b, 2015) and Öberg et al. (2017), millimeter-wave emission lines of the species N₂H⁺ and H₂CO (particularly the former) should provide particularly effective tracers of the CO snow line, albeit for complementary reasons: N₂H⁺ is destroyed via reactions with gas-phase CO, forming HCO⁺, while, as noted, H₂CO can be efficiently generated through hydrogenation of CO ice mantles on dust grains followed by photodesorption. Hence, the largest expected N₂H⁺ and H₂CO abundance enhancements are in disk regions cold enough for CO freeze-out (although there are also warm formation pathways for H₂CO; Öberg et al. 2017). In general terms, this expectation appears to be borne out in V4046 Sgr, i.e., the N₂H⁺ and H₂CO rings both reach their peak intensities beyond the peaks of the DCN and DCO⁺ rings. However, in contrast to the TW Hya and HD 163296 disks—both of which have N₂H⁺ emission rings with sharp inner cutoffs (Qi et al. 2013b, 2015)—there is no clear central hole within the N₂H⁺ ring in the V4046 Sgr disk. As a result, the precise radial position of putative midplane CO freeze-out is difficult to ascertain for V4046 Sgr on the basis of its radial profile of N₂H⁺ line intensity. Such an interpretation is rendered even more difficult by the possibility that the N₂H⁺ column density may smoothly increase beyond the midplane CO snow line, and/or that there may be significant abundances of N₂H⁺ in disk surface layers (see discussions in Nomura et al. 2016; van't Hoff et al. 2017). Further detailed analysis of the ALMA N₂H⁺ and H₂CO imaging results for V4046 Sgr will be presented in a forthcoming paper (C. Qi et al. 2018, in preparation).

Our molecular line surveys of TW Hya, V4046 Sgr, and the disk orbiting LkCa 15 (age ∼5 Myr) established that the emission-line intensities of C₂H from these evolved disks rival or exceed those of, e.g., ¹³CO (Kastner et al. 2014; Punzi et al. 2015). Follow-up SMA and ALMA imaging of C₂H emission from TW Hya (Kastner et al. 2015; Bergin et al. 2016) revealed that the C₂H line emission exhibits a ringlike morphology. Our ALMA imaging has now established that the C₂H emission from V4046 Sgr displays the same well-defined, ringlike morphology (a result already apparent in our previous SMA C₂H imaging; Kastner et al. 2016), indicating that sharp-edged C₂H rings are a common feature of evolved disks.

Given the paucity of bright molecular line tracers of disk chemical and physical conditions, it is essential to understand the production mechanism(s) responsible for the large abundances of C₂H and its ringlike distribution in these and other disks (e.g., DM Tau; Bergin et al. 2016). Based on the SMA C₂H imaging results for TW Hya and consideration of the excitation of C₂H, we proposed that the C₂H ring traces particularly efficient UV/X-ray photodestruction of hydrocarbons derived from small grains and grain ice mantles in the low-density (n ≪ 10⁷ cm⁻³), large-grain-depleted surface layers of the outer (>45 au) regions of the TW Hya disk (Kastner et al. 2015). Subsequent ALMA C₂H imaging and accompanying modeling supports the general notion that efficient C₂H production in evolved disks is a signpost of grain size segregation and stellar irradiation (Bergin et al. 2016). However, Bergin et al. (2016) concluded that TW Hya's ringlike C₂H emission morphology is also a result of C depletion in the inner disk, and that the emission arises from disk layers with n > 10⁶ cm⁻³. If so, then the presence of a C₂H ring would most likely be the result of pure gas-phase and/or gas-grain processes deep within the disk, with little or no influence from stellar irradiation (see the discussion in Kastner et al. 2015, and references therein).

This uncertainty in the vertical location of C₂H within the TW Hya disk is a consequence of the model degeneracies inherent to its near-pole-on orientation (i = 7°; Andrews et al. 2012, and references therein). Given the more intermediate inclination of the V4046 Sgr disk (i = 335; Rosenfeld et al. 2013), this disk potentially provides a means to distinguish between the various alternative C₂H production models—i.e., surface-layer, irradiation-driven production versus midplane, pure gas-phase production—to explain the presence of bright C₂H rings in these two disks.

To investigate the potential of the ALMA C₂H imaging in this regard, we have modeled the ringlike C₂H line emission following the methodology described in detail in Qi et al. (2013b) and Kastner et al. (2015). Briefly, as in Qi et al. (2013b), we adopt a physically self-consistent accretion disk model (D'Alessio et al. 2006, and references therein) that matches the spectral energy distribution (SED), with the disk geometric (i.e., scale height and surface density profile) parameters fixed to values previously determined via SED fitting for V4046 Sgr (Rosenfeld et al. 2013). Within this framework, we then investigate a limited parameter set that describes the disk layer from which the molecular emission originates. Specifically, we invoke the presence of a molecular emission layer of constant abundance that is confined to a range of vertical column densities between ${10}^{{Z}_{1}}\times (1.59\times {10}^{21})$ cm⁻² and ${10}^{{Z}_{2}}\times (1.59\times {10}^{21})$ cm⁻². The column density of the molecular layer is then allowed to vary within radial annuli ranging from an arbitrarily small inner radius R₀ that is much smaller than the beam size (in this case, R₀ = 10 au), through an effective inner radius R_in (where the column density increases to detectable values; see below), out to an outer cutoff radius R_out. The model grid interval is 10 au in the outer disk (r > 50 au) to roughly 5 au in the inner disk (r < 50 au). The radial column density dependence is characterized by a set of radial power-law slopes p_n at breakpoints R_n−1; for our purposes we set n = 2. For a selected set of free parameters of interest—in this case, R_in, R_out, p₁, p₂, R₁, for a given Z₁, Z₂—radiative transfer calculations are carried out via the RATRAN code to determine the resulting sky-projected integrated line intensity distribution. Model-integrated line intensity distributions are then realized over a wide range for each parameter, and fitting to the interferometric molecular emission-line data is performed in visibility space via a grid search approach.

We carried out this model-fitting procedure for the 262.004 GHz hyperfine transition complex of C₂H emission from the V4046 Sgr disk (we did not model the 262.064 GHz transition complex, but given its similar excitation conditions, we would expect similar results). The resulting best-fit radial column density profile and vertical and radial distributions of the C₂H emitting region are illustrated in Figure 9. The parameters of the best-fit model are Z₁ = 0.5, Z₂ = 1.5—i.e., the emitting layer is confined to vertical disk column densities of between ∼5 × 10²¹ cm⁻² and ∼5 × 10²² cm⁻²—with inner and outer radii R_in = 30 au and R_out = 130 au. The uncertainties in the former (vertical) layer parameters (Z₁, Z₂) are of the order of 0.5 (i.e., a factor ∼3 uncertainty in vertical disk column density), and the uncertainties in the latter (radial) emission region boundaries are of the order of ∼5 au. We find a model surface density power-law breakpoint of R₁ = 100 au, with a radial power-law slope of p₁ = +0.6 between 30 au and 100 au and a (much steeper) slope of p₂ = +1.8 between 100 au and 130 au. In the model, the C₂H column density (${N}_{{{\rm{C}}}_{2}{\rm{H}}}$) behaves essentially as a step function at R_in = 30 au, increasing from ∼10⁶ cm⁻² to ∼10¹⁴ cm⁻² over the (unresolvable) span of just a few au. ${N}_{{{\rm{C}}}_{2}{\rm{H}}}$ then slowly increases to ∼2 × 10¹⁴ cm⁻² at R₁ = 100 au before sharply increasing over the outer ∼30 au in radius, to ${N}_{{{\rm{C}}}_{2}{\rm{H}}}\sim 3.3\times {10}^{14}$ cm⁻² at the outer cutoff radius R_out = 130 (Figure 9, left).

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In Figure 10, we compare the ALMA C₂H moment 0 image and line profile for V4046 Sgr with the corresponding image and line profile obtained from the best-fit model. In the latter Figure, the ALMA and model image data have been reconstructed with a robust value of 2.0, resulting in ∼08 resolution images. It is evident that the modeling procedure has yielded a reasonable match to the ALMA C₂H image, as intended. However, we caution that, as we have not employed a rigorous parameter space study, the robustness and uniqueness of this particular model fit, as well as the precise uncertainties in the various model parameters, are difficult to assess.

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The ALMA C₂H modeling results illustrated in Figure 9 (left) supercede previous extrapolations of (much larger) ${N}_{{{\rm{C}}}_{2}{\rm{H}}}$ in the V4046 Sgr disk, based on low S/N single-dish (APEX) spectroscopy (Kastner et al. 2014). Given the kinetic temperature regime in the emitting region (T ∼ 40 K; Figure 9, right), the model ${N}_{{{\rm{C}}}_{2}{\rm{H}}}$ values indicate that the C₂H emission is optically thin; this inference is consistent with the fact that the measured ratio of total line intensities in the 262.004 and 262.064 GHz hyperfine complexes, 1.37 ± 0.1, is in good agreement with the theoretical ratio of 1.41 (Ziurys et al. 1982).

In terms of disk scale height, the best-fit model C₂H emitting layer corresponds to the region between z/r ∼ 0.1 and z/r ∼0.3 over the radial range ∼30–∼130 au (Figure 9, right). This (relatively deep) vertical emitting region position contrasts with the surface-layer position we inferred for the TW Hya disk, i.e., z/r ∼ 0.5 (corresponding to far smaller vertical disk column densities, in the range ∼(5–8) × 10¹⁸ cm⁻²). However, as noted, we based our inference that the C₂H lies near the TW Hya disk surface on the apparently subthermal excitation of C₂H in that disk (Kastner et al. 2015). In contrast to TW Hya, our empirical modeling of the integrated intensity image and line profile of C₂H emission from V4046 Sgr provides more direct constraints on the scale height of its emitting C₂H. Our results for V4046 Sgr hence suggest that the C₂H emitting layer may lie deeper within the TW Hya disk than we inferred previously. However, we caution that the vertical density and temperature structures of the V4046 Sgr and TW Hya disks, and protoplanetary disks more generally, are subject to large uncertainties.

The vertical position of the emitting layer within V4046 Sgr is somewhat deeper in the disk than, but is not necessarily inconsistent with, the vertical C₂H layer position predicted by models in which production of C₂H is irradiation-driven; in particular, Walsh et al. (2010) found that the C₂H abundance should peak at z/r ≈ 0.3. On the other hand, the low temperature (∼20 K) and high densities (>10⁷ cm⁻³) of the disk layers corresponding to the region between z/r ≈ 0.1 and z/r ≈ 0.3 would imply that pure gas-phase production, perhaps enhanced by a large C/O ratio in the outer disk, is responsible for the large inferred C₂H column densities within the emission ring (see discussions in Kastner et al. 2015; Bergin et al. 2016). Indeed, whereas pure gas-phase, deep-disk-layer production mechanisms appear to be able to generate ringlike C₂H emission morphologies (Henning et al. 2010), surface-layer irradiation production mechanisms—such as photodesorption of hydrocarbon-coated grains or photodestruction of small grains—may require ad hoc assumptions such as inner disk shadowing (Kastner et al. 2015). Furthermore, under either (irradiation or pure gas-phase) production scenario, the "cusp" of high C₂H abundance at the outer edge of the C₂H ring that is required by our best-fit empirical model (in the form of an abrupt steepening of the power-law slope near R_out; Figure 9, left) appears to be very difficult to explain. Observations of C₂H emission from other disks that are viewed at higher inclinations and are similarly nearby and well-resolved by ALMA (e.g., T Cha; Huélamo et al. 2015), in combination with additional theoretical efforts aimed at better understanding C₂H production, are essential if we are to pinpoint the processes that lead to large abundances of C₂H within evolved protoplanetary disks.