ALMA Survey of Lupus Protoplanetary Disks. II. Gas Disk Radii

We extract continuum images from the calibrated visibilities by averaging over the continuum channels and cleaning with a Briggs robust weighting parameter of +0.5; we use −1.0 when sources exhibit resolved structure (e.g., for transition disks). The average continuum beam size is 0.25 × 0.24 arcsec (36 × 38 au at 150 pc) for our Cycle 3 observations and 0.25 × 0.21 arcsec (36 × 32 au at 150 pc) for our Cycle 4 observations.

We primarily measure continuum flux densities by fitting elliptical Gaussians to the visibility data with uvmodelfit in CASA. The elliptical Gaussian model has six free parameters: integrated flux density (F_cont), FWHM along the major axis (a), aspect ratio of the axes (r), position angle (PA), R.A. offset from the phase center (Δα), and decl. offset from the phase center (Δδ). We scale the uncertainties on the fitted parameters by the square root of the reduced χ² value of the fit. If the ratio of a to its scaled uncertainty is less than five, a point-source model with three free parameters (F_cont, Δα, Δδ) is fit to the visibility data instead.

For disks with resolved structure, such as transition disks, flux densities are measured from continuum images using circular aperture photometry. The aperture radius for each source is determined by a curve-of-growth method, in which successively larger apertures are applied until the measured flux density levels off. Uncertainties are then estimated by taking the standard deviation of the flux densities measured within a same-sized aperture placed randomly within the field of view, but away from the source.

Table 1 gives our measured 1.33 mm continuum flux densities and associated uncertainties. The uncertainties are statistical errors and do not include the 10% absolute flux calibration error (Section 3). Of the 95 sources, 71 are detected with >3σ significance; the continuum images of these sources are shown in Figure 2. Table 1 provides the fitted source centers output by uvmodelfit for the detections, and the phase centers of our ALMA observations (based on 2MASS source positions) for the nondetections. The image rms for each source, derived from a 4'' to 9'' radius annulus centered on the fitted or expected source position for detections or nondetections, respectively, is also given in Table 1.

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We extract ¹²CO, ¹³CO, and C¹⁸O J = 2–1 channel maps from the calibrated visibilities by subtracting the continuum and cleaning with a Briggs robust weighting parameter of +0.5. Zero-moment maps are created by integrating over the velocity channels showing line emission above the noise. The appropriate velocity range is determined for each source by visual inspection of the channel map and spectrum. If no emission is visible, we sum across the average RV and its dispersion ($\overline{\mathrm{RV}}=2.8\pm 4.2$ km s⁻¹), as derived for Lupus protoplanetary disks in Frasca et al. (2017).

We measure ¹²CO, ¹³CO, and C¹⁸O J = 2–1 integrated flux densities and associated uncertainties (F_12CO, F_13CO, and F_C18O, respectively) from our ALMA zero-moment maps, using the same aperture photometry method described above for structured continuum sources (Section 4.1). For nondetections, we take upper limits of three times the uncertainty when using an aperture of the same size as the typical beam (025).

Of the 95 targets, 48 are detected in ¹²CO, 20 in ¹³CO, and 8 in C¹⁸O with >3σ significance. All sources detected in C¹⁸O are also detected in ¹³CO, all sources detected in ¹³CO are also detected in ¹²CO, and all sources detected in ¹²CO are also detected in the 1.33 mm continuum. Table 2 gives our measured integrated flux densities or upper limits for ¹³CO and C¹⁸O. We do not provide integrated fluxes for ¹²CO because cloud absorption is commonly seen in the spectra (see Figure 11 in Appendix A). Moreover, for 9 sources located nearby on the sky in Lupus III, cloud emission is also seen in the channel maps.

Table 2.  Gas Properties

Source	${F}_{13\mathrm{CO}}$	${F}_{{\rm{C}}18{\rm{O}}}$	${M}_{\mathrm{gas},\mathrm{mean}}$	${M}_{\mathrm{gas},\min }$	${M}_{\mathrm{gas},\max }$
	(mJy km s−1)	(mJy km s−1)	(${M}_{\mathrm{Jup}}$)	(${M}_{\mathrm{Jup}}$)	(${M}_{\mathrm{Jup}}$)
Sz65	<102	<60	<1.0	⋯	⋯
Sz66	<87	<60	<1.0	⋯	⋯
J15430131-3409153	<84	<51	<1.0	⋯	⋯
J15430227-3444059	<72	<60	<1.0	⋯	⋯
J15445789-3423392	<78	<54	<1.0	⋯	⋯
J15450634-3417378	<84	<57	<1.0	⋯	⋯
J15450887-3417333	395 ± 109	<54	0.4	⋯	3.1
Sz68	<120	<69	<1.0	⋯	⋯
Sz69	<81	<45	<1.0	⋯	⋯
Sz71	<78	<60	<1.0	⋯	⋯

Note. See Section 5.2.2 for details on the derivations of ${M}_{\mathrm{gas},\mathrm{mean}}$, ${M}_{\mathrm{gas},\min }$, and ${M}_{\mathrm{gas},\max }$.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Disk size is a fundamental property that has been difficult to measure for large samples because of observational constraints. Early measurements of disk radii from submm/mm observations focused on the largest and brightest disks, perhaps resulting in the common misconception that protoplanetary disks are typically hundreds of au in radius. The recent ALMA surveys of unbiased disk populations have revealed that typical disks are actually closer to a few tens of au in radius, at least in the submm/mm dust (Barenfeld et al. 2017; Tazzari et al. 2017).

Measuring gas disk sizes is particularly important because the gas dominates the dynamics of the disk. Gas disk radii are much more difficult to measure, however, because the line emission is faint, especially in the outer regions of the disk. Moreover, the submm/mm dust radius is not a reliable proxy for the gas disk size because dust grain growth has a radial dependnce and growing dust grains decouple from the gas and drift inward, resulting in the smaller dust disks seen at submm/mm wavelengths (e.g., Andrews et al. 2012; de Gregorio-Monsalvo et al. 2013; Hogerheijde et al. 2016).

Here we use our ALMA Band 6 data to measure the dust and gas radii of 22 Lupus protoplanetary disks. This is the first large sample of dust and gas radii for disks within a single star-forming region. These disks are listed in Table 3 and are selected because they have clearly resolved continuum emission (Tazzari et al. 2017) and exhibit unambiguous ¹²CO line emission in multiple velocity channels without significant cloud contamination.

Table 3.  Disk Radii

Source	PA	i	${R}_{\mathrm{dust}}$	${R}_{\mathrm{gas}}$
	(degree)	(degree)	(au)	(au)
Sz 65	108.6	61.5	64 ± 2	172 ± 24
Sz 68	175.8	−32.9	38 ± 2	68 ± 6
Sz 71	37.5	−40.8	94 ± 2	218 ± 54
Sz 73	94.7	49.8	56 ± 3	103 ± 9
Sz 75	169.0	60.2	56 ± 2	194 ± 21
Sz 76	65.0	−60.0	116 ± 9	164 ± 6
J15560210-3655282	55.6	53.5	56 ± 2	110 ± 3
Sz 82	144.0	−48.0	226 ± 4	388 ± 84
Sz 84	168.0	65.0	80 ± 3	146 ± 18
Sz 129	154.9	−31.7	68 ± 2	140 ± 12
RY Lup	109.0	68.0	134 ± 3	250 ± 63
J16000236-4222145	160.5	65.7	112 ± 3	266 ± 45
MY Lup	60.0	73.0	110 ± 3	194 ± 39
EX Lup	70.0	−30.5	62 ± 2	178 ± 12
Sz 133	126.3	78.5	142 ± 6	238 ± 66
Sz 91	17.4	51.7	154 ± 4	450 ± 80
Sz 98	111.6	−47.1	190 ± 4	358 ± 52
Sz 100	60.2	45.1	82 ± 2	178 ± 12
J16083070-3828268	107.0	−74.0	182 ± 4	394 ± 100
V1094 Sco	110.0	−55.4	334 ± 20	438 ± 112
Sz 111	40.0	−53.0	134 ± 2	462 ± 96
Sz 123A	145.0	−43.0	74 ± 2	146 ± 12

A machine-readable version of the table is available.

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The dust radii (R_dust) are measured from the 1.33 mm continuum images using a curve-of-growth method, in which successively larger photometric apertures are applied until the measured flux is 90% of the total flux. We use elliptical apertures based on the position angle (PA; measured east of north) and inclination (i) of the source; these values are mostly taken from Tazzari et al. (2017), who derived these parameters using two-layer disk models of the Band 7 continuum visibilities for the full Lupus disks in our sample. For the resolved transition disks with large inner dust cavities (Section 5.4.1), we use the PA and i values from van der Marel et al. (2018), who derived these parameters from by-eye comparisons of the Band 7 first-moment ¹³CO maps to model Keplerian velocity fields. The resulting R_dust values are given in Table 3 in units of au. The errors on R_dust are calculated by taking the range of radii within the uncertainties on the 90% flux measurement. Comparing our R_dust values to the R_out values derived in Tazzari et al. (2017) for the 12 sources common to both samples shows good agreement despite the very different analysis methods: the average ratio is 1.06, with a standard deviation of 0.37.

The gas radii (R_gas) are measured from the ¹²CO zero-moment maps using the same curve-of-growth method described above for the R_dust measurements (we note that this method is robust against the effects of cloud absorption only affects the blue- or redshifted side of the disk emission). The same PA and i values used for measuring R_dust are used again for measuring R_gas, which is important because applying different i values can lead to significantly different radii when implementing elliptical apertures in a curve-of-growth analysis.

Before measuring R_gas, we also reconstruct the zero-moment maps using a Keplerian masking technique to increase the signal-to-noise ratio (S/N), especially in the fainter outer disk regions (see the Appendixes in Salinas et al. 2017 and L. Trapman et al. 2018, in preparation, for detailed descriptions of the Keplerian masking technique). In short, Keplerian masking takes advantage of the fact that the gas disk is in Keplerian rotation, and thus will only emit in certain regions of the sky in a given velocity channel. Masking the pixels outside of these regions in each velocity channel therefore enhances the S/N in the final integrated zero-moment map. We show the Keplerian masking technique applied to Sz 111 in Figure 3, and also the improvement in the S/N in the outer disk regions in Figure 4 (plots for all 22 disks are shown in Figure 14 in Appendix B). The resulting R_gas values are given in Table 3.

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We find that the gas disks are universally larger than the dust disks, by an average factor of 1.96 ± 0.04 (where this is the mean and standard error on the mean). We note that this result holds even when using a 68% (rather than 90%) flux threshold for the radius measurements and also when using circular (rather than elliptical) apertures (see Figures 12 and 13 in Appendix B). Although previous observations of large individual disks have shown gas disks extending beyond dust disks by similar factors (e.g., Isella et al. 2007; Panić et al. 2009; Andrews et al. 2012; de Gregorio-Monsalvo et al. 2013; Cleeves et al. 2016), this is the first indication of systematically larger gas radii in a coherent population of disks. We note that our results differ from those of Barenfeld et al. (2017), who found in a sample of seven Upper Sco disks only four with larger gas radii, and no clear overall trend (see their Figure 6); however, their sample is much smaller and they did not apply Keplerian masking. We discuss the implications in Section 6.2.

5.2.1. Dust Masses

Dust emission at submm/mm wavelengths is typically optically thin in most regions of a protoplanetary disk, in which case estimates of dust mass can be directly obtained from measurements of the submm/mm continuum flux. Ribas et al. (2017) calculated the spectral index from far-IR to mm wavelengths for 284 protoplanetary disks, showing that the spectral index distributions become remarkably similar from 880 μm to 5 mm, despite the significant range in wavelength, which indeed suggests that the overall disk dust emission is generally optically thin at these longer wavelengths.

Assuming dust emission at submm/mm wavelengths is also isothermal, the submm/mm continuum flux from a protoplanetary disk at a given wavelength (F_ν) can be directly related to the mass of the emitting dust (M_dust), as shown in Hildebrand (1983):

$$\begin{eqnarray}&&{M}_{\mathrm{dust}}=\displaystyle \frac{{F}_{\nu }{d}^{2}}{{\kappa }_{\nu }{B}_{\nu }({T}_{\mathrm{dust}})}\approx 7.06\times {10}^{-7}\,{\left(\displaystyle \frac{d}{150}\right)}^{2}{F}_{1.33\mathrm{mm}},\end{eqnarray} \tag{ 1 }$$

where B_ν(T_dust) is the Planck function for a characteristic dust temperature of T_dust = 20 K, the median for Taurus disks (Andrews & Williams 2005). We take the dust grain opacity, κ_ν, as 10 cm² g⁻¹ at 1000 GHz and use an opacity power-law index of β_d = 1.0 (Beckwith et al. 1990). Distances, 1.33 mm continuum flux densities, and associate uncertainties are from Table 1. The calculated M_dust values are given in Table 1, and the M_dust distribution is shown in Figure 5. The median fractional difference between the dust masses derived here from our Band 6 data compared to the values derived in Paper I from our Band 7 data is 15%.

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As in previous works (e.g., Andrews et al. 2013; Ansdell et al. 2016, 2017; Barenfeld et al. 2016; Pascucci et al. 2016), we also fit the M_dust–M_⋆ relation using the Bayesian linear regression method of Kelly (2007) to take into account upper limits, error bars on both axes, and intrinsic scatter in the data. Using the same Monte Carlo method as in Paper I to account for the 19 sources with unknown stellar masses (Section 2), we find the relation:

$$\begin{eqnarray}&&\mathrm{log}({M}_{\mathrm{dust}})=1.3(\pm 0.2)+1.8(\pm 0.3)\times \mathrm{log}({M}_{\star }),\end{eqnarray} \tag{ 2 }$$

with a dispersion of 0.8 ± 0.2 dex. These fitted parameters are nearly identical to (and well within the errors of) those derived from our Band 7 observations in Paper I. We note that our assumption of an isothermal disk temperature could effect the fitted M_dust–M_⋆ relation, if there is a dependence of T_dust on stellar parameters. Although Andrews et al. (2013) derived the relation ${T}_{\mathrm{dust}}=25\,{\rm{K}}\times {({L}_{\star }/{L}_{\odot })}^{0.25}$ using two-dimensional continuum radiative transfer models, recent ALMA observations suggest that T_dust is actually largely independent of stellar parameters. In particular, Tazzari et al. (2017) used detailed modeling of 36 resolved Lupus disks to show a lack of correlation between T_dust and L_⋆ or M_⋆, at least for their sample. Thus applying model-derived relations runs the risk of introducing artificial correlations or increasing the dispersion of true correlations. Indeed, applying the T_dust–L_⋆ relation derived by Andrews et al. (2013) to ALMA disk surveys results in a shallower slope when compared to assuming an isothermal disk temperature of T = 20 K (Pascucci et al. 2016).

5.2.2. Gas Masses

We perform a stacking analysis to constrain the average dust and gas masses of the individually undetected sources. Before stacking the nondetections, we center each image on the expected source location and normalize the flux to 150 pc. The flux densities are then measured in the stacked images using circular aperture photometry, as in Section 4.1. We confirm that the source locations are known to sufficient accuracy for stacking by measuring the average offset of the detected sources from their phase centers: we find $\langle {\rm{\Delta }}\alpha \rangle =-0\buildrel{\prime\prime}\over{.} 11$ and $\langle {\rm{\Delta }}\delta \rangle =-0\buildrel{\prime\prime}\over{.} 18$, both smaller than the beam size.

We first stack the 24 continuum nondetections, but do not find a significant mean signal in the continuum, ¹²CO, ¹³CO, or C¹⁸O stacks. The lack of line emission is expected given the undetected continuum, but the absence of continuum emission is surprising given the sensitivity of the stacked image. Using an aperture the same size as the beam, we measure a mean signal of 0.00 ± 0.04 mJy; we can confirm this by calculating the mean and standard error on the mean from the continuum fluxes reported in Table 1, which similarly gives −0.06 ±0.03 mJy. This translates into a 3σ upper limit on the average dust mass of individually undetected continuum sources of ∼5 lunar masses (0.06 M_⊕), comparable to debris disk levels (Wyatt 2008). The stark contrast between the detections and stacked nondetections, illustrated in Figure 5, suggests that protoplanetary disks evolve rapidly to debris disk levels once disk clearing begins (Alexander et al. 2014).

We then stack the 12 sources detected in the continuum and ¹³CO, but not C¹⁸O. We measure a continuum mean signal of 42.10 ± 0.64 mJy and a ¹³CO mean signal of 1030 ±150 mJy km s⁻¹. The stacking also reveals a marginally significant mean signal for C¹⁸O of 270 ± 90 mJy km s⁻¹. The stacked continuum flux corresponds to M_dust ∼ 28 M_⊕ and the stacked line fluxes correspond to M_gas ∼ 0.36 M_Jup, giving an average gas-to-dust ratio of only ∼4 for sources detected in the continuum and ¹³CO, but not C¹⁸O.

Finally, we stack the 51 sources detected in the continuum, but undetected in ¹³CO and C¹⁸O. We measure a continuum mean signal of 19.06 ± 0.12 mJy. The stacking also reveals marginally significant mean signals for ¹³CO and C¹⁸O; the stacked gas fluxes are 190 ± 50 mJy km s⁻¹ and 40 ± 10 mJy km s⁻¹, respectively. Note that these stacked line fluxes of the nondetections are significantly lower than the line fluxes of the detections, similar to what is seem for the continuum. The stacked continuum flux corresponds to M_dust ∼ 13 M_⊕, while the stacked line fluxes correspond to M_gas ∼ 0.14 M_Jup for an average gas-to-dust ratio of just ∼3 for disks detected in the continuum but undetected in ¹³CO and C¹⁸O.

5.4.1. Transition Disks and Asymmetric Disks

5.4.2. Secondary Sources

We detect seven secondary sources that are not accounted for in our target list. The coordinates and 1.33 mm fluxes (F_cont) of these secondary sources are given in Table 4. Additionally, we provide the position angle (PA; measured east of north) and projected angular separation (ρ) of each secondary source relative to its primary source. The F_cont values are estimated by fitting point-source models to the visibility data with uvmodelfit, as in Section 4.1, and are consistent with values obtained with aperture photometry. We do not provide F_cont values for the secondary sources to Sz 74 and Sz 88A, as they are too close to their primary source to allow reliable individual flux measurements.

Table 4.  Secondary Source Properties

Source	R.A.${}_{{\rm{J}}2000}$	Decl.${}_{{\rm{J}}2000}$	${F}_{\mathrm{cont}}$ (mJy)	${F}_{\mathrm{cont}}$ (mJy)	PA (degree)	ρ (arcsec)
(primary)	(secondary)	(secondary)	(secondary)	(primary)
Sz 68	15:45:12.646	−34:17:29.768	3.35 ± 0.10	66.38 ± 0.20	297.32	2.84
Sz 74	15:48:05.212	−35:15:53.032	⋯	11.51 ± 0.34	355.96	0.31
Sz 81A	15:55:50.317	−38:01:32.262	1.45 ± 0.10	4.24 ± 0.11	18.61	1.93
J16070384-3911113	16:07:03.585	−39:11:12.022	0.38 ± 0.10	0.98 ± 0.29	267.54	2.84
Sz 88A	16:07:00.567	−39.02.20.202	⋯	3.72 ± 0.11	212.61	0.34
J16073773-3921388	16:07:37.562	−39:21:39.218	0.45 ± 0.10	0.52 ± 0.09	266.61	1.72
V856 Sco	16:08:34.390	−39:06:19.310	8.21 ± 0.10	23.03 ± 0.11	112.68	1.45

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Sz 68, Sz 74, Sz 81A, and V856 Sco are known binary stars from the literature (Reipurth & Zinnecker 1993; Leinert et al. 1997; Woitas et al. 2001), and here we detect the disks of their secondary companions. The PA and ρ values measured from our millimeter detections match those reported in the literature for the stellar binary systems.

Sz 88A shows a very close (038) secondary millimeter component, which is not immediately visible in Figure 2 due to the relative brightness of the primary target. Although this star has a known companion at 15 (Sz 88B; Reipurth & Zinnecker 1993), which we do not detect in our data, follow-up with VLT/NACO did not reveal any closer companions (Correia et al. 2006) that could be the possible host star for this potential disk.

J16070384-3911113 shows a weak companion at a projected separation of 285, but no known stellar sources near this location are reported in the literature. Although this source also appears to be a close binary in our Band 6 data (see Figure 2), the two millimeter continuum points are likely just the bright limbs of an edge-on disk. This interpretation is supported by its publicly available optical image taken by the Advanced Camera for Surveys on board the Hubble Space Telescope (Hubble/ACS; Proposal ID: 14212; PI: Stapelfeldt), which reveals a flared edge-on disk with a mid-plane that is aligned with the two millimeter continuum points, as shown in Figure 6. Additionally, the detected ¹²CO emission in our Band 6 data and ¹³CO emission in our Band 7 data both show Keplerian rotation encompassing the two continuum points. The edge-on nature of the disk also explains its apparently flat IR excess.

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J16073773-3921388 has a visual companion detected in the optical at 32 to the north (Merín et al. 2008). However, the submm/mm component we detect is at 176 to the west, making it unlikely to be the same object (if bound), given the time elapsed between the observations.

5.4.3. Outflow Sources

One of the largest uncertainties in converting submm/mm continuum flux into disk dust mass (e.g., Equation (1)) is the power-law index of the dust opacity spectrum, β_d, where ${\kappa }_{\nu }\propto {\nu }^{{\beta }_{{\rm{d}}}}$. If the disk dust emission is optically thin and in the Rayleigh–Jeans regime, its submm/mm SED has a power-law dependence on frequency, such that ${F}_{\nu }\propto {\kappa }_{\nu }{\nu }^{2}\propto {\nu }^{2+{\beta }_{{\rm{d}}}}$. In this case, we can fit the observed submm/mm SED between two frequencies with ${F}_{{\nu }_{1}}/{F}_{{\nu }_{2}}={({\nu }_{1}/{\nu }_{2})}^{{\alpha }_{\mathrm{mm}}}$, where α_mm is the submm/mm spectral index, and then derive the dust opacity index using β_d = α_mm–2. For interstellar medium (ISM) dust, β_d ≈ 1.7 (Li & Draine 2001), a value that should decrease (i.e., become more gray) as grains grow (e.g., Draine 2006).

Here we derive α_mm between 890 μm and 1.33 mm for the 70 Lupus disks detected in both Band 6 (this work) and Band 7 (Paper I; S. E. van Terwisga et al. 2018, in preparation). We use the standard equation for deriving α_mm described above, and calculate uncertainties by propagating the errors on the flux measurements, which include both the statistical error and the 10% flux calibration uncertainty (Section 3). Our results are given in Figure 7, which shows α_mm as a function F_1.33mm (normalized to 150 pc) as well as the cumulative distributions of the α_mm values. The median α_mm value is 2.25 (when excluding transition disks with resolved cavities; see below), similar to what is seen in other young regions at these wavelengths (Ribas et al. 2017) and also at slightly longer wavelengths of 1–3 mm (Testi et al. 2014). Moreover, these α_mm values translate into β_d values much lower than that of the ISM, implying significant grain growth in Lupus disks.

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We note that for blackbody emission, F_ν ∝ ν², thus α_mm > 2 is the limit for optically thin, graybody emission in the Rayleigh–Jeans regime. However, disks may exhibit α_mm < 2 when they deviate from these conditions, for example, in the case of exceptionally cold disks that no longer fulfill the Rayleigh–Jeans criteria, or when there is significant contamination from nonthermal sources such as stellar winds. Nonetheless, all of our Lupus sources in Figure 7 are consistent with α_mm > 2.0 when considering uncertainties, as expected from graybody emission in the Rayleigh–Jeans regime.

Figure 7 also shows α_mm as a function of M_dust (translated from F_{1.33 mm} using Equation (1)). Contrary to previous studies of young disks that found no correlation between α_mm and M_dust (e.g., Andrews & Williams 2005; Ricci et al. 2012), we find with our much more sensitive observations that low-mass disks follow an anti-correlation, followed by a flattening after M_dust ∼ 5M_⊕ as disks approach α_mm ≈ 2. When considering only the full disks (see below), we fit the data with a piecewise linear relation:

$$\begin{eqnarray}&&{\alpha }_{\mathrm{mm}}\\ =\,\left\{\begin{array}{ll}-0.55(\pm 0.17)\mathrm{log}{F}_{\mathrm{mm}}+2.62(\pm 0.09) & \mathrm{log}{F}_{\mathrm{mm}}\leqslant 0.81\\ +2.19(\pm 0.05) & \mathrm{log}{F}_{\mathrm{mm}}\gt 0.81.\end{array}\right.\end{eqnarray} \tag{ 3 }$$

To test the significance of the anticorrelation, we apply a Spearman rank test to the data where F_{1.33 mm} ≤ 0.84, which gives a probability of no correlation of p = 0.002. We note that using a simple linear relation also gives a statistically significant fit to the data, although an F-test is unable to identify which parametrization is more statistically significant. Regardless, this anticorrelation is not an observational bias: the gray regions in Figure 7 show where we are not sensitive based on 3σ limits, illustrating that our observational sensitivity does not influence the correlation.

The decrease in α_mm for brighter disks may be due to more efficient grain growth in higher-mass disks, which also tend to be around higher-mass stars (due to the M_dust–M_⋆ relation; Section 5.2.1) and thus have faster dynamical timescales. If true, this rules out one of the scenarios proposed by Pascucci et al. (2016) to explain the steepening of the ${M}_{\mathrm{dust}}$–M_⋆ relation with age, which is that grain growth is more efficient in disks around lower-mass stars. However, the decrease in ${\alpha }_{\mathrm{mm}}$ for higher-mass disks may also simply reflect larger optically thick regions, which would serve to artificially decrease ${\alpha }_{\mathrm{mm}}$ and thus mimic grain growth (e.g., Tripathi et al. 2017; S. E. van Terwisga et al. 2018, in preparation). Higher resolution data that can resolve ${\alpha }_{\mathrm{mm}}$ as a function of radius are needed to distinguish between these scenarios.

Figure 7 shows that resolved transition disks tend to have higher ${\alpha }_{\mathrm{mm}}$ values than the general protoplanetary disk population in Lupus. This is consistent with the findings of Pinilla et al. (2014), who showed that ${\alpha }_{\mathrm{mm}}$ (between ∼1 mm and ∼3 mm; see their Table 2) is larger for transition disks than for full protoplanetary disks in the Taurus, Ophiuchus, and Orion star-forming regions. They calculated a weighted mean and standard error on the mean of ${\bar{\alpha }}_{\mathrm{TD}}=2.70\pm 0.13$ and ${\bar{\alpha }}_{\mathrm{PPD}}=2.20\pm 0.07$. Here our wavelength range is smaller (890 μm–1.33 mm), but our disk population is from a single star-forming region and our observations are from uniform surveys at higher sensitivity. We find consistent results with ${\bar{\alpha }}_{\mathrm{TD}}=2.70\pm 0.10$ and ${\bar{\alpha }}_{\mathrm{PPD}}=2.27\pm 0.05$. Pinilla et al. (2014) explained the higher ${\alpha }_{\mathrm{mm}}$ values of transition disks in terms of the inner disk cavity. Namely, assuming transition disks had the same grain population as full disks before the inner disk clearing, and also that ${\beta }_{{\rm{d}}}$ increases with radius (e.g., Guilloteau et al. 2011; Tazzari et al. 2016), then disks with large inner cavities should lack large grains and therefore appear to have higher ${\alpha }_{\mathrm{mm}}$ values than full disks.

Figure 8 compares the gas radius (${R}_{\mathrm{gas}}$) to the dust radius (${R}_{\mathrm{dust}}$) for the 22 disks in our Lupus sample with constraints on both parameters, revealing that ${R}_{\mathrm{gas}}$ is universally larger than ${R}_{\mathrm{dust}}$. Although previous observations of individual disks have shown that the gas radius can extend beyond the dust radius, these were limited to the largest and brightest disks (e.g., TW Hya and HD 163296; Panić et al. 2009; Andrews et al. 2012; de Gregorio-Monsalvo et al. 2013). Here we show that this is a population-wide feature among young disks, and that ${R}_{\mathrm{gas}}$ is consistently $\approx 1.5$–3.0 × ${R}_{\mathrm{dust}}$, with an average ratio of ${R}_{\mathrm{gas}}/{R}_{\mathrm{dust}}=1.96\pm 0.04$ (Section 5.1).

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The smaller dust disk sizes relative to the gas disks may be explained by dust grain growth and radial drift. Grain growth timescales are much shorter at smaller radial distances from the host star, resulting in grain size segregation with the larger grains preferentially located closer to the host star. In addition, as dust grains grow, drag forces from gas in sub-Keplerian rotation cause the larger solids to spiral inward toward the host star on short timescales (Weidenschilling 1977). Both of these mechanisms can cause the submm/mm continuum emission to appear smaller than the gas emission, because this continuum emission primarily traces larger (submm/mm) grains, while the gas emission primarily traces smaller (sub-μm) grains. Moreover, because dust growth and radial drift both produce similar particle size segregations in disks, it can be difficult to identify an unambiguous signature of radial drift (distinct from just grain growth) based on disk sizes alone.

However, numerical and analytical models also predict that a unique signature of radial drift is the shape of the submm/mm continuum intensity profile (e.g., Birnstiel & Andrews 2014). This is because radial drift in the early phases of disk evolution, before grain growth in the outer disk has begun, naturally sets a distinct outer radius beyond which the disk is essentially devoid of dust; moreover, this sharp outer radius is preserved for submm/mm grains in later phases of disk evolution, when grain growth and viscous spreading have set in. This reasoning has been used to invoke radial drift to explain the differences in the submm/mm dust and gas radii for TW Hya (Andrews et al. 2012; Hogerheijde et al. 2016) and HD 163296 (de Gregorio-Monsalvo et al. 2013). The resolution of our data is insufficient to derive detailed continuum intensity profiles capable of fitting sharp outer edges, although higher-resolution ALMA observations of Lupus disks will be able to test for this signature of radial drift in the future. Nevertheless, radial drift is expected to come hand-in-hand with dust growth, as described above.

An alternative explanation for the larger gas disk radii is that the ¹²CO emission is optically thick, while the continuum emission is optically thin (Dutrey et al. 1998; Guilloteau & Dutrey 1998). Because the ¹²CO emission is optically thick, it is simply easier to detect small amounts of gas at large radii, whereas this is not the case for the optically thin emission from the dust. These optical depth effects can therefore produce similar observational signatures to dust grain growth and radial drift. Indeed, Facchini et al. (2017) combined the dust evolution models from Birnstiel et al. (2015) with the thermo-chemical code DALI (Bruderer et al. 2012; Bruderer 2013) to show that, at least for the case of the massive HD 163296 disk, the bulk of the difference between the gas and dust radii is due to the optical depth of the CO lines, with grain growth and radial drift having a more subtle effect on the steepness of the millimeter dust emission profile (see their Figures 16 and 17).

To simulate more "typical" Lupus disks, we update the Facchini et al. (2017) models using ${M}_{\star }=0.5$ M_⊙, ${T}_{\star }\,=4700$ K, ${\dot{M}}_{\mathrm{acc}}={10}^{-9}$ M_⊙ yr⁻¹, ${M}_{\mathrm{disk}}={10}^{-4}$ M_⊙, and a tapered surface density profile with $\gamma =1$ and ${R}_{{\rm{c}}}=50\,\mathrm{au}$. The simulated images are then convolved with a $0\buildrel{\prime\prime}\over{.} 25$ beam, and we use a distance of 150 pc. To test whether the disk size differences could be due solely to optical depth effects, we use a model with a uniform mix of small and large grains throughout the entire disk. Three other models then simulate grain growth, fragmentation, and radial drift by setting the maximum grain size as a function of radial distance from the host star based on fragmentation and radial drift limits with ${\alpha }_{\mathrm{visc}}={10}^{-2},{10}^{-3},{10}^{-4}$.

Using the same curve-of-growth method as described in Section 5.1 to measure disk radii, we find that ${R}_{\mathrm{gas}}/{R}_{\mathrm{dust}}\approx 1.5$ for the model with uniform grain sizes, and ${R}_{\mathrm{gas}}/{R}_{\mathrm{dust}}\approx 3.0$ for the models including grain growth and radial drift. Therefore, based on these models, optical depth effects could explain the lower range of the measured ${R}_{\mathrm{gas}}/{R}_{\mathrm{dust}}$ values in Lupus, but grain growth and radial drift may also be needed to explain the higher values of ${R}_{\mathrm{gas}}/{R}_{\mathrm{dust}}$ seen in the data. We caution that these models only consider the collisional growth and fragmentation of the dust grains and do not include their kinematics within the disk, which could increase the differences in the modeled gas and dust radii. Moreover, the models currently do not include simulated observational noise, which can affect the measured radii, especially for the gas due to low S/N in the outer disk.

Additionally, Figure 8 (lower left panel) shows a tentative correlation between ${R}_{\mathrm{gas}}$ and ${F}_{1.33\mathrm{mm}}$, analogous to the continuum size–luminosity relations seen previously in young disk populations (Tazzari et al. 2017; Tripathi et al. 2017). The Bayesian linear regression method of Kelly (2007) gives the correlation $\mathrm{log}{F}_{1.33\mathrm{mm}}=1.00(\pm 0.45)\mathrm{log}{R}_{\mathrm{gas}}-0.66(\pm 1.04)$ with a correlation coefficient of 0.5 ± 0.2 and a dispersion of 0.42 ± 0.08. To test the significance of the correlation, we use a Spearman rank test, which gives $\rho =0.54$ and a p-value of 0.009. However, we caution that our sample is biased toward disks with both resolved continuum and gas emission; some Lupus disks exhibit faint and unresolved continuum emission, but bright and extended gas emission, and therefore may not follow this correlation.

We do not see a correlation between ${R}_{\mathrm{gas}}$ and M_⋆ (lower right panel of Figure 8), although this is likely due to the bias of our sample toward the highest-mass disks around the highest-mass stars (upper right panel of Figure 8). More sensitive and higher-resolution ¹²CO line observations can probe the gas disks around lower-mass stars to provide better constraints on these possible relations between fundamental disk parameters.

Protoplanetary disks are traditionally thought to evolve through viscous evolution (Lynden-Bell & Pringle 1974; Hartmann et al. 1998). According to viscous evolution theory, turbulence in the disk redistributes angular momentum by transporting it outward to larger radii over time, which in turn drives the accretion of disk material inward through the disk and onto the central star. The viscosity of the disk cannot be easily quantified, but it can be characterized by the so-called α prescription (Shakura & Sunyaev 1973), where

$$\begin{eqnarray}&&{\alpha }_{\mathrm{visc}}=\displaystyle \frac{{\dot{M}}_{\mathrm{acc}}}{{M}_{{\rm{d}}}}\displaystyle \frac{\mu }{{k}_{{\rm{B}}}{T}_{{\rm{d}}}}{\rm{\Omega }}{r}_{{\rm{d}}}^{2}.\end{eqnarray} \tag{ 4 }$$

In this framework, ${\alpha }_{\mathrm{visc}}$ is a dimensionless parameter that is constant and ≲1, ${T}_{{\rm{d}}}$ is the disk temperature, ${r}_{{\rm{d}}}$ is the outer disk radius, μ is the mean molecular weight, ${k}_{{\rm{B}}}$ is the Boltzmann constant, and Ω is the Keplerian angular frequency where ${\rm{\Omega }}={({{GM}}_{\star }/{r}_{{\rm{d}}}^{3})}^{1/2}$. The ratio of the total disk mass (${M}_{{\rm{d}}}$) to the stellar mass accretion rate (${\dot{M}}_{\mathrm{acc}}$) gives the "disk lifetime," which should be on the order of the age of the disk, but only for disk ages older than the viscous timescale, t_ν (e.g., Jones et al. 2012; Rosotti et al. 2017).

Placing observational constraints on ${\alpha }_{\mathrm{visc}}$ is important because this parameter is thought to be directly related to the angular momentum transport in disks, and thus critical for understanding disk evolution. However, obtaining these constraints is complicated by difficulties with measuring the masses and radii for large samples of disks. Hartmann et al. (1998) used the average observed properties of protoplanetary disks in Taurus and Chamaeleon I, including a handful of disk sizes derived from 2.7 mm continuum observations, to estimate ${\alpha }_{\mathrm{visc}}\approx {10}^{-2}$. However, recent observations of TW Hya and HD 163296, which provided the first tentative measurements of disk turbulence, suggest lower values of ${\alpha }_{\mathrm{visc}}\lesssim {10}^{-3}$ (Hughes et al. 2011; Flaherty et al. 2015; Teague et al. 2016).

For our Lupus sample, Rafikov (2017) calculated ${\alpha }_{\mathrm{visc}}$ with Equation (4) using the $890\,\mu {\rm{m}}$ continuum properties derived in Paper I. He assumed ${M}_{{\rm{d}}}=100\,{M}_{\mathrm{dust}}$, took ${r}_{{\rm{d}}}$ from elliptical Gaussian fits to the continuum emission, and used isothermal disk temperatures of ${T}_{{\rm{d}}}=20$ K. Additionally, he took M_⋆ and ${\dot{M}}_{\mathrm{acc}}$ from Alcalá et al. (2014, 2017). Rafikov (2017) found a wide range of ${\alpha }_{\mathrm{visc}}$ values spanning over two orders of magnitude, from 10⁻⁴ to 0.04, with no clustering around a particular value. The lack of any correlations between ${\alpha }_{\mathrm{visc}}$ and other global disk parameters (${M}_{{\rm{d}}}$, ${r}_{{\rm{d}}}$, ${{\rm{\Sigma }}}_{{\rm{d}}}$) or stellar parameters (M_⋆, R_⋆, L_⋆) also led him to suggest that angular momentum transport may actually be performed non-viscously, for example, via MHD winds.

However, there are two main shortcomings of the analysis by Rafikov (2017): the small sample size of 26 sources (due to the need for well-resolved disks), and the use of Gaussian-fit estimates of the continuum emission for disk sizes (since the gas radii were not yet available). Our gas disk radii measured in Section 5.1 solve the latter issue, thus we repeat these ${\alpha }_{\mathrm{visc}}$ calculations using instead ${r}_{{\rm{d}}}={R}_{\mathrm{gas}}$ in Equation (4). As shown in the left panel of Figure 9, we still find a large range of ${\alpha }_{\mathrm{visc}}$ values spanning over two orders of magnitude, from 0.0003 to 0.09, which is not surprising given the tight relation between ${R}_{\mathrm{gas}}$ and ${R}_{\mathrm{dust}}$ seen in Figure 8.

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Similar to Rafikov (2017), we also find no correlations between ${\alpha }_{\mathrm{visc}}$ and other disk or stellar properties. The reason for this is our small sample size of 22 disks, which does not allow us to overcome the first shortcoming of the work by Rafikov (2017) described above. Importantly, these sub-samples of resolved Lupus disks are not only small, but also heavily biased toward the highest-mass disks, as shown in the right panel of Figure 9. This means that they lack sufficient leverage to exhibit the ${\dot{M}}_{\mathrm{acc}}$–${M}_{{\rm{d}}}$ correlation, which is predicted by viscous evolution and seen in larger samples of disks in Lupus (Manara et al. 2016) and the similarly young Chamaeleon I region (Mulders et al. 2017). The lack of correlations between ${\alpha }_{\mathrm{visc}}$ and other disk or stellar properties is thus dominated by the lack of correlation between ${\dot{M}}_{\mathrm{acc}}$ and ${M}_{{\rm{d}}}$ in the considered subsamples. Interestingly, Lodato et al. (2017) and Mulders et al. (2017) have shown that the large scatter in the observed ${\dot{M}}_{\mathrm{acc}}$–${M}_{{\rm{d}}}$ relation seen in Lupus and Chamaeleon I can be reproduced by viscous evolution theory, if the age of the star-forming regions is younger than the viscous timescales of the disks. If this is true, the assumption made in Equation (4) of ${\dot{M}}_{\mathrm{acc}}/{M}_{{\rm{d}}}\gtrsim {t}_{\nu }$ no longer holds, and the derived values of ${\alpha }_{\mathrm{visc}}$ must be considered with great caution.

Ultimately, estimates of disk radii for systems with lower stellar and disk masses are needed to extend the sample to the point where the ${\dot{M}}_{\mathrm{acc}}$–${M}_{{\rm{d}}}$ relation can be recovered. Similar studies in older star-forming regions (e.g., Upper Sco) are also needed to insure that the ${\dot{M}}_{\mathrm{acc}}/{M}_{{\rm{d}}}\gtrsim {t}_{\nu }$ assumption is valid.

Submm/mm continuum emission is generally optically thin in most regions of a protoplanetary disk, which is why it is the most commonly used tracer of dust mass (Section 5.2.1). However, if the dust surface densities are high enough at the center of a disk—above roughly an Earth mass of dust spread over ∼10 au in diameter—the emission can become optically thick. In this case, the observed submm/mm continuum flux is more a measure of the dust disk size than of the dust disk mass:

$$\begin{eqnarray}&&{F}_{\nu }={\int }_{{R}_{\mathrm{sub}}}^{{R}_{\mathrm{thick}}}{B}_{\nu }(T)2\pi r\,{dr}\cos i/{d}^{2},\end{eqnarray} \tag{ 5 }$$

where r is the radial distance in the disk, d is the distance to the disk, ${R}_{\mathrm{sub}}$ is the inner radius of the disk defined by the dust sublimation temperature $T({R}_{\mathrm{sub}})=1500$ K, and i is the inclination of the (geometrically thin) disk to the line of sight. The dust temperature is

$$\begin{eqnarray}&&T={\left(\displaystyle \frac{{L}_{\star }}{16\pi {\sigma }_{\mathrm{SB}}{r}^{2}}\right)}^{1/4},\end{eqnarray} \tag{ 6 }$$

where L_⋆ is the stellar luminosity and ${\sigma }_{\mathrm{SB}}$ is the Stefan–Boltzmann constant (e.g., Dullemond & Monnier 2010). For disks that are large enough to be resolved in our data, we derive i from the aspect ratio of the disk; for the unresolved disks, we use a mean value of $\langle \cos \,i\rangle =2/\pi$. We then integrate outward until we match the observed central peak flux density in a beam centered on the stellar position, or for nondetections, the corresponding 3σ upper limit. After removal of the transition disks (van der Marel et al. 2018) and unresolved binary system V856 Sco, the derived ${R}_{\mathrm{thick}}$ ranges from 0.7 to 13 au with a median of 3.8 au for the detected disks, and 0.7–1.0 au with a median of 0.8 au for the nondetections (see upper panel of Figure 10).

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For the nondetections, the limits on the dust content are very stringent, whether in terms of mass if the emission is optically thin or in radius if the emission is optically thick. We estimate that any optically thick central component would have to be less than 1 au in radius for the disk to escape detection. For the detections, our analysis provides an upper limit to the size of any central optically thick component, as we are matching the central peak flux density in the ∼20 au beam and this likely includes some contribution from an extended optically thin component.

The derived limits on the thick disk radii of a few au are at an interesting scale, as this is comparable to where we expect the water snowline to reside. We expect the emission properties to change here due to the loss of ice and evolution of the grain size distribution (Banzatti et al. 2015), as observed at larger radii in the much more luminous FU Orionis object V883 Ori (Cieza et al. 2016). We therefore replot these results in terms of the temperature of the outer edge of the optically thick disk (lower panel of Figure 10). The range for the detected disks is 31–193 K with a median of 104 K and 123–223 K with a median of 104 K for the nondetections.

If a substantial fraction of the observed emission does indeed come from a compact and optically thick region, then many of the Lupus disks should remain detectable in relatively short ALMA integrations in much more extended configurations. Small and bright optically thick cores are found in the ultra-high resolution ALMA observations of TW Hya (Andrews et al. 2016) and HL Tau (ALMA Partnership et al. 2015). If future imaging surveys show that such features are common, it would indicate the existence of large reservoirs of material in the innermost regions of disks as required by models of in situ formation for short-period planets (Chiang & Laughlin 2013).