PROTOPLANETARY DISK STRUCTURES IN OPHIUCHUS. II. EXTENSION TO FAINTER SOURCES

The model parameter values that best reproduce the observations for the full sample are compiled in Tables 4 and 5 for the disks with continuous emission distributions and those with central emission cavities, respectively. For the continuous disks, we include the (fixed) inner disk radii inferred from a simple sublimation argument (see Paper I; Table 4, Column 7). Columns 7 and 8 in Table 5 list the estimated sizes (R_cav) and density contrasts (δ_cav) used to account for the disks with central emission cavities. Both tables also include the adopted values for the disk inclination and position angle, as well as the reduced χ² statistics for the fits to the SED and visibility data sets separately. The distributions of these values for each parameter are shown together in Figure 3; hatched regions mark contributions from the disks with central cavities (around SR 24 S, SR 21, DoAr 44, and WSB 60). In Figure 4, the new observations presented here are directly compared with the synthetic data sets generated from these best-fit models (see Paper I for the other sample disks). From left to right, we display the observed SMA millimeter continuum image (as in Figure 2), the synthesized model image, the imaged residuals, the broadband SED, and the elliptically averaged millimeter visibility profile (see Paper I for details). The latter two panels have the best-fit model behavior overlaid in red and the SED panel also shows the input stellar spectrum as a dashed blue curve. Because of its large central emission cavity, the modeling results for the SR 24 S disk are shown separately in Figure 5, with inset images synthesized at higher angular resolution to facilitate a more detailed comparison.

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Table 4. Disk Structure Model Parameters: Continuous Cases

Name	Md	γ	Rc	H100	ψ	Rin	i	P.A.	$\tilde{\chi }^2_{\rm vis}$	$\tilde{\chi }^2_{\rm sed}$
	(M☉)		(AU)	(AU)		(AU)	(°)	(°)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)
AS 205	0.029	0.9	46	19.6	0.11	0.14	25	165	2.1	3.7
Elias 24	0.117	0.9	127	8.6	0.03	0.16	24	50	2.0	1.5
GSS 39	0.143	0.7	198	7.3	0.08	0.07	60	110	1.9	32
AS 209	0.028	0.4	126	13.3	0.10	0.09	38	86	1.7	2.4
DoAr 25	0.136	0.9	80	6.7	0.15	0.06	59	112	1.9	9.2
WaOph 6	0.077	1.0	153	4.4	0.06	0.12	39	171	1.8	1.8
VSSG 1	0.029	0.8	33	9.7	0.08	0.10	53	165	1.8	12
SR 4	0.004	0.8	20	19.9	0.23	0.07	50	39	1.9	8.8
SR 13	0.012	1.0	26	12.6	0.07	0.04	32	42	2.0	1.9
WSB 52	0.007	1.1	26	14.5	0.19	0.06	46	120	1.9	6.6
DoAr 33	0.007	1.1	38	2.7	0.06	0.07	43	102	1.9	22
WL 18	0.011	0.8	14	5.6	0.07	0.04	48	115	1.9	13

Notes. Column 1: disk name (those in italics were modeled in Paper I); Column 2: disk mass assuming a 100:1 gas-to-dust mass ratio; Column 3: radial surface density gradient; Column 4: characteristic radius; Column 5: scale height at 100 AU; Column 6: radial scale height gradient; Column 7: fixed inner radius; Column 8: fixed inclination; Column 9: fixed major axis position angle; Column 10: reduced χ² statistic comparing the model fit with the continuum visibilities alone; Column 11: same as Column 10, but for the SED alone.

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Table 5. Disk Structure Model Parameters: Central Cavity Cases

Name	Md	γ	Rc	H100	ψ	Rcav	δcav	i	P.A.	$\tilde{\chi }^2_{\rm vis}$	$\tilde{\chi }^2_{\rm sed}$
	(M☉)		(AU)	(AU)		(AU)		(°)	(°)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)
SR 24 S	0.042	0.8	40	4.0	0.02	32	0.0001	57	31	2.1	11
SR 21	0.005	0.9	17	7.7	0.26	37	0.005	22	110	1.7	7.2
WSB 60	0.021	0.8	31	11.0	0.13	20	0.01	25	117	1.8	3.0
DoAr 44	0.017	1.0	80	3.5	0.04	33	0.0001	45	75	1.8	⋅⋅⋅

Notes. Columns 1–6: same as for Table 4; Column 7: the cavity radius, marking the outer edge of the diminished inner disk densities (see Paper I or the discussion on SR 24 S in Section 4.2); Column 8: the density reduction scaling factor inside the radius R_cav; Columns 9–12: same as for Table 4 Columns 8–11.

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The modeling uncertainties and relationships between the observations and model parameters were already discussed in detail in Paper I, so we will not repeat that effort here. Instead, we focus on how this expanded sample enables new constraints on the spatial distribution of mass in ∼1 Myr old circumstellar disks. The inferred surface density profiles for the full sample are shown together in Figure 6, with the targets presented for the first time here highlighted in color. The disks with large central emission cavities are shown in a separate panel. The light gray band inside 20 AU marks the survey resolution limit, while the dark gray boxes are reference points representing the minimal surface densities (with uncertainties) and presumed feeding zones for Saturn, Uranus, and Neptune in the canonical model for the primordial solar disk (Weidenschilling 1977). Although the disks in this sample exhibit a wide range of masses (M_d; Figure 3(a)) and characteristic radii (R_c; Figure 3(c)), we find a very narrow distribution of values for γ, the radial gradient of the surface density profile. As shown in Figure 3(b), that distribution is roughly normal with a peak at γ = 0.9 and a standard deviation of 0.2. Since that width is comparable to the modeling uncertainties on γ for an individual source (∼0.2–0.3; Paper I), the modeling results are consistent with a uniform γ value being representative of the entire sample.³ Moreover, by extending the survey to cover much fainter targets, we have confirmed that the shape of the surface density profile does not significantly change over a wide range of millimeter luminosities (or disk masses).

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Despite the similar radial density gradients in these disks, the observations clearly show a wide variety of millimeter emission morphologies. Some of that diversity is related to how the disks are heated by their central stars, a natural outcome of the assortment of stellar properties and vertical distributions of dust present in the sample (see Paper I for details). However, the density profiles plotted in Figure 6 demonstrate that a significant range of characteristic radii and masses are also partly responsible. A close examination of those Σ profiles reveals that the brighter disks at millimeter wavelengths tend to have both higher masses and larger characteristic radii. The former is no surprise, since the vast majority of the disk volume at these wavelengths is optically thin, but there is no a priori reason to expect the disk sizes to be correspondingly smaller for fainter sources. This relationship is not a modeling artifact, as there is an analogous empirical correlation between the brightness of the millimeter emission and how well that emission is resolved. Figure 7 demonstrates this explicitly by comparing the average 880 μm visibility profiles for disks that are brighter (red) or fainter (blue) than 0.5 Jy. These profiles were calculated by averaging the visibilities for each subset of disks into annular bins, after deprojecting the visibilities for individual disks according to their viewing geometries and normalizing their real (correlated) fluxes by dividing off their integrated flux densities. This comparison confirms that the millimeter emission from the brighter disks in this sample is more resolved by our SMA data, exhibiting substantially less correlated flux on essentially all spatial scales. Although the physical origins of this empirical relationship are not obvious, some speculations about its significance for understanding the viscous evolution process in these disks are presented in Section 5.

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In an effort to identify any trends among the disk structures or the characteristics of their stellar hosts, we performed a principle component analysis on a subset of such properties for the 12 disks with continuous density distributions in this sample. The analysis included the five free disk structure parameters, {M_d, γ, R_c, H₁₀₀, ψ}, the stellar properties {T_eff, L_*}, and the accretion rates, $\dot{M}_*$ (see Tables 3, 4, and 6). Due to the additional uncertainties of the stellar evolution models used to infer them, we did not include stellar ages, masses, or radii directly (although those values are well correlated with L_* and T_eff). As alluded to above, we identified one statistically significant correlation (3.3σ; correlation coefficient of 0.85) between M_d and R_c, which will be discussed in detail in Section 5.1. The first eigenvector from the principle component analysis, accounting for 40% of the variance in the data, is dominated by a positive trend relating R_c, M_d, L_*, and $\dot{M}_*$. The trend indicates that the more massive disks in this sample are larger and orbiting more luminous—perhaps younger—stars that are accreting disk material at higher rates. While these relationships may hint at crucial information related to the viscous evolution process, a larger sample will be required to make any definitive conclusions.

Table 6. Viscous Disk Properties

Name	$\dot{M}_{\ast }$	Rt	α	Ref.
	(M☉ yr−1)	(AU)
(1)	(2)	(3)	(4)	(5)
AS 205	8 × 10−8	23	0.005	1
Elias 24	2 × 10−7	62	0.03	2
GSS 39	7 × 10−8	95	0.03	2
AS 209	9 × 10−8	61	0.08	3
DoAr 25	3 × 10−9	39	0.0005	4
WaOph 6	1 × 10−7	78	0.05	5
VSSG 1	1 × 10−7	16	0.02	2
SR 4	6 × 10−8	10	0.02	2
SR 13	3 × 10−9	13	0.0005	2
WSB 52	4 × 10−9	13	0.001	2
DoAr 33	3 × 10−10	20	0.0006	6
WL 18	9 × 10−9	7	0.003	2
SR 24 S	3 × 10−8	19	(0.006)	2
SR 21	<2 × 10−9	9	(<0.0009)	2
WSB 60	1 × 10−9	15	(0.0002)	2
DoAr 44	9 × 10−9	40	(0.01)	7

Notes. Column 1: disk name (those in italics were modeled in Paper I); Column 2: accretion rate; Column 3: radius of mass flow reversal; Column 4: viscosity coefficient; Column 5: references for $\dot{M}_{\ast }$: (1) Prato et al. 2003; (2) Natta et al. 2006; (3) Johns-Krull et al. 2000; (4) Luhman & Rieke 1999; (5) Eisner et al. 2005; (6) Cieza et al. 2010; (7) Espaillat et al. 2010.

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Elias 24. This heavily reddened classical T Tauri star is one of the brightest millimeter continuum sources in Ophiuchus. In fact, it was initially excluded from the sample out of concern that the bright emission signaled contamination from an extended envelope. However, there is no evidence for the spatial filtering of such large-scale emission; the SMA flux densities at both 880 μm and 1.3 mm are in excellent agreement with lower resolution single-dish photometry (Andrews & Williams 2007a, 2007b; André & Montmerle 1994). Moreover, the central star is optically visible (Wilking et al. 2005) and the infrared SED shape is incompatible with a substantial envelope (Bontemps et al. 2001; Evans et al. 2003; Barsony et al. 2005). The 880 μm continuum data presented here features a bright central core and a fainter, extended, and apparently asymmetric emission halo on larger scales. Although the best-fit model for this source is able to reproduce the SED and visibilities rather well, any axisymmetric model will underpredict the extended emission to the east of the disk center at the ∼3σ level (see Figure 4). The origins of that extension remain unclear.

SR 24 S. The hierarchical triple system SR 24 is composed of this K2 star and a close binary pair (02 ≈ 25 AU; Simon et al. 1995) located ∼5'' (625 AU) to the north (Reipurth & Zinnecker 1993). Both SR 24 S and the SR 24 N binary exhibit excess emission from warm dust disks, bright Hα lines indicative of substantial accretion flows, and extended emission from CO low-energy rotational transitions. However, all of the continuum emission at millimeter wavelengths is produced by SR 24 S (Andrews & Williams 2005a; Patience et al. 2008; Isella et al. 2009). The high angular resolution inset image of the SR 24 S disk in Figure 2 reveals a resolved central emission cavity with an apparent brightness enhancement to the northeast. The origins of this ring-like emission morphology are unclear, but could perhaps be generated by abrupt emissivity variations due to particle growth, a dramatic dissipation process driven by high-energy radiation from the star, or even tidal interactions with companion objects (in this last case, see Mayama et al. 2010).

Regardless of the cause, the model described in Section 3 was adjusted to account for the observed disk morphology. As detailed in Paper I for similar cases, we adopted the simple modification of artificially decreasing the surface densities inside a radius R_cav by a factor δ_cav and then attempted to fit the SMA visibilities and the component-resolved optical (Wilking et al. 2005), infrared (Jensen & Mathieu 1997; Cutri et al. 2003; Evans et al. 2003; McCabe et al. 2006), and radio photometry. However, much like the case of the DoAr 44 disk described in Paper I, it is difficult to simultaneously account for the significant infrared excess and the lack of millimeter continuum emission near the SR 24 S stellar position. Following the approach of Espaillat et al. (2007) for other "pre-transitional" disks, we proceeded by first finding a good match to the millimeter visibilities, and then artificially increasing the surface densities near the star until sufficient infrared emission was produced to match the SED (without affecting the millimeter emission). A satisfactory result was achieved with a surface density profile scaled down by δ_cav ≈ 0.05 inside R = 2 AU, a factor of 20 lower than for a continuous disk, but ∼500× higher than for the region between 2 AU and R_cav ≈ 32 AU. Obviously these adjustments are artificial and non-unique; a modeling effort focused on a more robust exploration of the detailed inner disk structures will be treated elsewhere.

SR 4. This star harbors a disk with a compact, centrally peaked millimeter emission morphology. The low foreground extinction allows for a particularly good characterization of the stellar luminosity from optical photometry, despite the low-level variability at those wavelengths (Herbst et al. 1994). The infrared SED shows a relatively small excess at short wavelengths before a pronounced flattening from ∼8 to 24 μm (Cutri et al. 2003; Evans et al. 2003). Unfortunately, there is little additional information in the far-infrared to better constrain the SED morphology, as the Spitzer 70 μm images are contaminated by local nebulosity (Padgett et al. 2008). This lack of data between the mid-infrared and millimeter regions of the SED results in only a relatively crude constraint on the vertical structure parameters (and thus temperatures) in this case.

SR 13. Another hierarchical triple system (see Schaefer et al. 2006), SR 13 includes a close binary pair (13 mas ≈1.6 AU; Simon et al. 1995) and a tertiary companion located ∼04 (50 AU) due east (Ghez et al. 1993). Lacking sufficient component-resolved photometry for this system, we made a preliminary model ignoring the tertiary and assuming the primary is a single star. The SED was constructed from the composite optical measurements of Herbst et al. (1994), infrared data from Two Micron All Sky Survey (2MASS) and the Spitzer c2d survey (Cutri et al. 2003; Evans et al. 2003), and single-dish radio observations (André & Montmerle 1994; Andrews & Williams 2007b). Despite this simplification, the SMA data can potentially resolve another disk around the tertiary. There is a modest eastern extension in the emission map, and a potential null near 400 kλ in the deprojected visibility profile (Figure 4). These features could be produced either by a cavity in a circum-system disk or by a marginally resolved additional disk around the spatially offset tertiary. With the available sensitivity, it is difficult to differentiate these scenarios. Needless to say, the model disk structure inferred in this case should be treated with appropriate caution.

WSB 52. This classical T Tauri star with spectral type M1 exhibits a compact millimeter continuum emission distribution typical of the fainter disks in the sample. The SED used in the model fits was constructed from the red-optical photometry of Wilking et al. (2005), infrared measurements from 2MASS (Cutri et al. 2003), Spitzer c2d (Evans et al. 2003), and Padgett et al. (2008), and a 1.3 mm single-dish flux measurement from Stanke et al. (2006).

DoAr 33. The infrared excess emission around this K4 star is remarkably faint compared to the typical young disk. We constructed a SED from red-optical photometry (Wilking et al. 2005), the 2MASS and Spitzer c2d programs (Cutri et al. 2003; Evans et al. 2003), and single-dish radio measurements (André & Montmerle 1994; Andrews & Williams 2007b). Only a very weak continuum excess is present shortward of ∼8 μm. At longer wavelengths where the dust excess is brighter, the spectrum is very steep and blue. Reproducing this infrared SED shape with our models required the use of a very flat vertical distribution of dust, resulting in comparatively cold midplane temperatures. Cieza et al. (2010) argue that these infrared colors are indicative of significant dust evolution. The flat structure inferred here could be interpreted in that context as evidence for advanced dust sedimentation to the midplane, or perhaps the sign of extensive shadowing of the outer disk due to some perturbed inner disk structure. In any case, the coupling of the rare infrared SED, bright millimeter emission, and weak signatures of accretion (Cieza et al. 2010) suggest that this disk is worthy of further scrutiny.

WL 18. This star harbors the faintest disk in the sample, due in part to its small size (R_c = 14 AU). Its visibility profile shows that the millimeter emission is only marginally resolved on the longest SMA baselines. The SED compiled here is sparse due to high foreground extinction, with infrared data from 2MASS and the Spitzer c2d project (Cutri et al. 2003; Evans et al. 2003), and a single-dish 1.3 mm flux density measured by Motte et al. (1998). That same extinction makes an estimate of the underlying stellar luminosity difficult, which in turn contributes an uncertainty to the thermal structure of the disk. The apparently low luminosity of the star relative to others with the same spectral type (K7) leads to an uncomfortably large age estimate (11 Myr).

DoAr 24 E. This spectral type G6 weak-lined T Tauri star has a companion roughly 2'' to the southeast (Ghez et al. 1993; Reipurth & Zinnecker 1993) that becomes substantially brighter than the primary at wavelengths longer than ∼3 μm (e.g., McCabe et al. 2006). The unresolved composite SED has been attributed to the photosphere of the primary and dust around this infrared companion. However, resolved measurements at near- and mid-infrared wavelengths (Prato et al. 2003; Barsony et al. 2005; McCabe et al. 2006) indicate that the primary does show some very weak infrared excess at least at ∼10 μm. Regardless, most of the unresolved millimeter emission, and therefore the circumstellar dust mass, was assumed to be generated by the infrared companion (Motte et al. 1998; Andrews & Williams 2007b). The resolved 880 μm image of this system in Figure 2 clearly demonstrates that this is not the case: each stellar component hosts a dust disk with roughly equal amounts of millimeter emission. Neither individual disk is clearly resolved. The composite and component-resolved SEDs for this system are shown together in Figure 8, along with a potential model for an extincted stellar photosphere from the primary. Since very little resolved information is available and the nature of the infrared companion is uncertain, we have not modeled the circumstellar material in this system. There is potentially great interest in doing so with some modifications to the standard technique presented here. It would be worthwhile to better understand the lack of accretion onto the primary despite the presence of a substantial dust mass, as well as the potential origin of the very red SED of the companion object.

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We have used a radiative transfer modeling technique to extract the dust density structures for a significant sample of protoplanetary disks based on high angular resolution observations of their millimeter continuum emission. Those data and the SEDs for each source were reproduced well using a simple model for the two-dimensional density structure of a viscous accretion disk, with a parametric surface density profile that varies with radius like a power law with an exponential taper at large radii (see Equation (1)). This specific form of Σ corresponds to the Lynden-Bell & Pringle (1974) similarity solutions for the evolution of a thin accretion disk in Keplerian rotation around a (stellar) point mass, with a viscosity ν ∝ R^γ that does not vary with time (see also Hartmann et al. 1998). This type of model—without a sharply truncated outer edge—is favored observationally, as it provides a natural explanation for the optical absorption profiles of silhouette disks in Orion (McCaughrean & O'Dell 1996) and reconciles the apparent discrepancies in the observed spatial extents of the dust and CO line emission for nearby resolved disks (Hughes et al. 2008).

In the context of these models, Figure 3(b) demonstrates that there is a relatively narrow distribution of the parameter γ that describes the spatial distribution of mass in the disk, consistent with a median 〈γ〉 = 0.9 ± 0.2 when the sample is considered together (estimates for individual disks range from γ = 0.4–1.1). These values fall at the high end of the wider distribution of Σ gradients inferred by Isella et al. (2009, where γ ranges from −0.8 to 0.8) using a 1.3 mm continuum survey of Taurus disks with slightly poorer angular resolution (07–10), and therefore probing the mass at larger disk radii. Although there is substantial overlap in the γ distributions from both surveys (particularly when the disks with large central cavities are ignored), the differences can be attributed to the distinct approaches for interpreting the data and there is no obvious means of reconciliation at present. Focusing on our results, we infer that the mass in ∼1 Myr old disks has a similar radial distribution regardless of the wide range of total masses (M_d ≈ 0.001–0.1 M_☉) probed in this sample. This implies that whatever mechanism is responsible for generating the viscosity in these disks insures that it has a roughly linear dependence on radius (γ ≈ 1).

Since the physical origin of that viscosity is unclear, we adopt a simple prescription ν = αc_sH, where c_s is the sound speed and the viscosity coefficient α describes the efficiency of the angular momentum transport (Shakura & Sunyaev 1973). Using the density structures and measurements of mass accretion rates $\dot{M}_{\ast }$ (see Table 6 for references), we can estimate the value of the viscosity coefficient $\alpha (R) \approx \dot{M}_{\ast } (R/R_c)^{\gamma }/3 \pi \Sigma _c c_s H$ (see the Appendix in Paper I for details). The resulting distribution of α at R = 10 AU is shown in Figure 9, with values for individual disks listed in Table 6. Note that for the typical midplane temperature distribution T ∝ R^−q, the viscosity coefficient is proportional to R^γ/c_sH ∼ R^z, where z ≈ γ + q/2 − ψ − 1. The model parameters and temperature distributions inferred for this sample have z ≈ 0.0 ± 0.3, meaning α does not vary by more than a factor of ∼2–4 across the disk. The inferred distribution of these viscosity coefficients appears bimodal, with peaks just below α ≈ 0.001 and above 0.01. However, given the small sample, those peaks are not statistically significant; they simply represent the clustering of the accretion rates in this sample. The range of inferred α values is in reasonable quantitative agreement with magnetohydrodynamics simulations where the viscosity is generated by turbulence from the magnetorotational instability in slightly ionized disks (Hawley et al. 1995; Stone et al. 1996; Fleming & Stone 2003; Fromang et al. 2007). However, we should caution that these α values are not yet well constrained: they suffer the combined uncertainties in the accretion rates and disk structure parameters, as well as from systematic issues with the inherent assumptions in their derivation (e.g., the dust traces the gas, viscous heating is insignificant, etc.). They represent only a first exploration of an improved empirical understanding of disk viscosities.

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The interactions of gravitational and viscous torques control the evolution of disk structure over the vast majority of the disk lifetime. This viscous evolution process has two important, observable effects on that structure. First, the coupling of the viscosity and the Keplerian orbital shear drives a net mass flow toward small radii, where material can be magnetically channeled onto the central star. That mass flow produces bright H emission lines, and the shock generated when it impacts the stellar surface gives rise to a strong ultraviolet continuum; both tracers can be used to estimate $\dot{M}_{\ast }$ (Muzerolle et al. 1998a, 1998b; Gullbring & Calvet 1998; Gullbring et al. 2000). To compensate for the angular momentum dissipated in that process, the material in the outer disk (beyond some radius R_t; see Table 6 and Paper I) is spread to larger radii. The results of that viscous diffusion—decreased average densities (i.e., M_d) and increased sizes (R_c)—can be probed with resolved observations of the millimeter continuum emission. In principle, tracking these observational signatures as a function of time could provide strong constraints on the viscous evolution process. Isella et al. (2009) claim a significant correlation between their parameterization of R_c and stellar age in a sample of Taurus disks. No such correlation is evident in our larger sample of Ophiuchus disks. More importantly, the search for such a trend within a single star-forming region is probably premature; the ages of individual young stars can not be determined with sufficient accuracy to infer such evolutionary behavior (e.g., Hillenbrand 2009 and references therein), even though it may exist. Rather, the inferred disk structures could be considered representative of the diversity of evolutionary states and/or initial conditions at a "snapshot" in the evolution sequence corresponding to some median cluster age (in this case, ∼1 Myr).

While there may not be much difference in the shapes of the Σ profiles (γ), this sample includes disks with a wide variety of masses (M_d) and characteristic size scales (R_c). As was demonstrated in Figure 7, the disks with brighter millimeter emission have systematically larger masses and sizes. This relationship is illustrated more directly in Figure 10, with a significant correlation (3.3σ; Spearman rank coefficient of 0.85) between R_c and M_d that can be approximated as a power law, M_d ∝ R^1.6±0.3_c. The physical origin of the correlation is not clear. One potential explanation is that it reflects the range of initial conditions and viscous properties inherited at the disk formation epoch. In that case, note that the sense of the correlation is almost perpendicular to the evolutionary paths for individual disks that conserve angular momentum, M_d ∝ R^−1/2_c. The spread along the correlation is then representative of the distribution of angular momenta imparted by the collapse of the parent molecular cloud cores. Following Isella et al. (2009), we can estimate the specific angular momenta in these cores that are needed to reproduce the inferred disk structures. Assuming the centrifugal radius in the disk corresponds to the radius that contained ∼90% of the disk mass at the formation epoch (here defined as approximately twice an initial scaling radius, ∼2 R₁), then the specific angular momentum can be written

$$\begin{equation} j \approx 2 \times 10^{20} \left(\frac{R_1}{25\,\rm {AU}}\right)^{0.5}\left(\frac{M_{\ast }}{{M}_{\odot }}\right)^{-1.5}\left(\frac{R_{\rm core}}{0.1\,{\rm pc}}\right)^2 {\rm cm}^2 \,\,\, {\rm s}^{-1}, \end{equation} \tag{ 2 }$$

where R_core is the core radius (see Hueso & Guillot 2005). Precise values for the initial scaling radius are uncertain (see below), but we can estimate upper limits on the j values since R₁ ⩽ R_c by definition (see Paper I). Substituting R₁ = R_c, R_core = 0.1 pc, and the stellar masses in Table 3 into Equation (2), we find log j ≈ 19.7–20.9 in units of cm² s⁻¹. Radio observations of rotating molecular cloud cores have been used to infer similar or higher values, log j ≈ 19.6–22.2 (Goodman et al. 1993; Barranco & Goodman 1998; Caselli et al. 2002). However, recent numerical simulations have demonstrated that those measurements tend to overestimate j by roughly an order of magnitude (Dib et al. 2010). If that is the case, then the range of disk angular momenta inferred here are actually in good quantitative agreement with those determined for analogs of their parental molecular cloud cores. The low end of the j distribution postulated by Dib et al. (2010) and not probed here could then be explained by a selection effect (perhaps our flux-limited sample only recovers disks with large j values), an unspecified mechanism for angular momentum loss (e.g., outflows or magnetic braking), and/or that R₁ ≪ R_c in at least some cases.

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Unfortunately, an unambiguous explanation of the shape of the M_d–R_c correlation in this scenario is not possible due to the fundamental degeneracy between the initial disk structures and the rates at which they evolve. The values of M_d and R_c change with time such that $M_d = M_{d,0}/{\cal T}^{1/2(2-\gamma)}$ and $R_c = R_1 {\cal T}^{1/(2-\gamma)}$, where M_d,0 and R₁ were the initial disk mass and characteristic radius, respectively, and ${\cal T}$ is a dimensionless parameter that tracks how many viscous timescales (t_s) have elapsed, ${\cal T} = 1 + t/t_s$ (see Paper I; Hartmann et al. 1998). That evolution is described by two equations with three unknown parameters, {M_d,0, R₁, t_s}: the evolution rate can not be uniquely disentangled from the initial conditions. However, the lack of significant outliers to the correlation may be related to the viscous evolution process itself. The "small and massive" region in the upper left of Figure 10 is depopulated because these disks spend only a short time with such compact density configurations early in their evolution (t ≪ 1 Myr). Likewise, the "large and low-mass" region to the lower right in this parameter space is empty because these ∼1 Myr old disks simply have not had enough time to evolve into it.

Alternatively, the inferred M_d–R_c correlation could be the result of coupling the viscous evolution scenario described above with an internal dissipation process. Recent theoretical calculations have suggested that the high-energy (particularly far-ultraviolet, or FUV) irradiation of the disk surface by the central star can drive significant mass loss from large disk radii (R>30 AU) in a photoevaporative flow (Gorti & Hollenbach 2009; Gorti et al. 2009). In this scenario, the smaller, low-mass disks in this sample should be associated with an advanced stage in this photoevaporation process, perhaps because their central stars have more intense FUV radiation fields. Unfortunately, direct constraints on the FUV emission from these extincted sources in the Ophiuchus clouds are rare. Instead, we can approximate the FUV luminosity as the sum of a fraction of the accretion luminosity ($\propto M_{\ast } \dot{M}_{\ast } / R_{\ast }$) and a chromospheric component (≈5 × 10⁻⁴L_*; see Gorti et al. 2009). However, we find no evidence for an expected anti-correlation between disk masses/sizes and the FUV (or X-ray) luminosities in this sample: if anything, the larger and more massive disks tend to have more intense FUV radiation fields, whereas the less massive, smaller disks show a wide range of FUV luminosities. While this certainly does not exclude a role for photoevaporation in shaping disk structures, it does suggest that diverse initial conditions and viscous timescales are more significant contributors to the observed M_d–R_c correlation at an age of ∼1 Myr.

Comparisons of disk properties in different stellar populations will be required to definitively assess the relative impacts that initial conditions, viscous evolution, and dissipation effects like photoevaporation have on shaping disk density distributions over time. The key to disentangling the initial conditions from the viscous evolution timescale is to search for and characterize structural correlations, like the M_d–R_c relationship found here, in disks around stars with a range of ages. In a scenario where viscous evolution dominates, the disks embedded in the envelopes of Class 0/I sources should be massive and compact, providing direct constraints on the distributions of initial masses (M_d,0) and sizes (R₁). Conversely, the disks in older star-forming regions are expected to be larger and less massive, unless or until FUV photoevaporation can effectively dissipate the mass reservoir in the outer disk. Tracking the evolution of disk masses and sizes in these older stellar populations can provide crucial insights into the distribution of viscous timescales, as well as into the time frame where photoevaporation may dissipate much of the disk mass reservoir. To probe disk structures in these older star clusters that are typically at larger distances, instruments with increased sensitivity to millimeter continuum emission and access to higher angular resolution will be required. Fortunately, there is great promise for a breakthrough in our understanding of viscous evolution and disk dissipation in the near future, based on extensive millimeter continuum surveys of disk populations with the Atacama Large Millimeter Array (ALMA).

Ultimately, the drive to understand this viscous evolution process lies in the desire to constrain how mass is re-distributed over time, and therefore where and when the conditions in a given disk are suitable for planet formation. All planet formation models require that the disk exceeds some density threshold as a necessary (but perhaps not sufficient) condition for making a planet. In light of the new constraints on disk densities presented here and in Paper I, it is natural to compare them with the two basic recipes for giant planet formation: disk instability (e.g., Boss 1997) and core accretion (e.g., Pollack et al. 1996). In the disk instability model, an overdense region with a sufficiently short cooling time can fragment out of the global disk structure and precipitate a bound gaseous protoplanet very rapidly (∼10³–10⁴ yr; see the recent review by Durisen et al. 2007). Typically, those conditions are only met at large disk radii (Boley et al. 2006); such protoplanets must then migrate inward to reproduce the shorter period orbits observed in the solar system and elsewhere. In the core accretion model, a giant planet is produced from a relatively slow (∼1–10 Myr) collisional growth process that first builds up a large solid core and then rapidly accretes a massive gaseous envelope (e.g., Hubickyj et al. 2005; Alibert et al. 2005).

Assuming that the dust traces 1% of the gas mass, the disk structures presented here are stable against gravitational fragmentation, with the minimum local Toomre Q values ranging from ∼5 to 50 in most cases (Q = c_sΩ/πGΣ, where Ω is the Keplerian orbital velocity; Toomre 1964). The large, massive disks around DoAr 25 and GSS 39 are potentially exceptions, with Q values approaching 2 at radii of ∼65 and 150 AU, respectively. However, we should caution that the disk structures in both cases are particularly uncertain (see Section 4.2 of Paper I). While none of the sample disks are clear candidates for the efficient operation of the disk instability mechanism for planet formation in the current epoch (i.e., Q ⩽ 1.7), their structures may have been less stable at earlier times. Even without invoking some kind of mass loading mechanism (Boley 2009), viscous evolution implies that these disks originally had denser, more compact structures, which might suggest a formerly lower minimum Q-value and/or radius where the disk is least stable to fragmentation. A quantitative exploration of the history of Q(R) in these disks grounded in our constraints on their viscous properties may well be worthwhile, but would require new radiative transfer calculations that incorporate some model for the thermal evolution of the disk material in each case.

A similar comparison with the core accretion model is more challenging, primarily because resolution limitations do not yet permit a direct observation of the disk material inside R ≈ 20 AU. Typically, core accretion model calculations impose a scaled Minimum Mass Solar Nebula (MMSN) density structure for an initial disk, where Σ ∝ R^−3/2 (Weidenschilling 1977) and a total mass set to produce a formation efficiency commensurate with the observational constraints on disk lifetimes. Although the MMSN surface density profile is too steep compared to those derived here (where Σ ∝ R⁻¹ at the relevant radii), Figure 6 demonstrates that the surface densities for the sample are generally compatible with those expected in the 20–40 AU region of the primordial solar disk. The shape of the Σ profile should not adversely affect the likelihood of planet formation if sufficient mass is available, but it is expected to have an impact on the orbital architecture and migration properties of any resulting planetary system (Kokubo & Ida 2002; Chambers & Cassen 2002; Raymond et al. 2005; Crida 2009). Some core accretion simulations have started to employ viscous disk density structures similar to those derived here (e.g., Alibert et al. 2005; Hueso & Guillot 2005), but they still must populate that density structure with solid bodies orders of magnitude larger than those responsible for the observed millimeter emission.

This last point highlights an important uncertainty in deriving densities from observations of young circumstellar disks: the conversion of emission to mass does not account for particles much larger than the observing wavelength. Since this potential for substantially underestimating the disk densities was already elaborated in Paper I, we will not dwell on it here. The problem is not likely to be severe in the regions that can currently be probed observationally (R ⩾ 20 AU), as even optimistic estimates of particle growth timescales at those radii do not predict a substantial population of large solids (e.g., Dullemond & Dominik 2005). However, at smaller disk radii those growth timescales can be considerably shorter, generating a dust emissivity that varies with radius inside the planet formation zone. More generally, it is worthwhile to emphasize that the observations are not yet sensitive to the midplane material for R < 20 AU. Inside that resolution limit, the densities could be dramatically different than inferred here if, for example, the dust size distribution is changed or the effective viscosity profile is modified (e.g., Zhu et al. 2010). New data from ALMA and the Expanded Very Large Array will help alleviate these uncertainties at small radii, and should also help empirically guide our understanding of disk viscosities.

But despite our general ignorance of the inner disk structures, a subset of young disks show unambiguous evidence for evolution in the planet formation zone. In this SMA survey of 17 disks, selected primarily for their millimeter luminosities, four of them—around SR 24 S, SR 21, WSB 60, and DoAr 44—have large central cavities with significantly diminished dust emission. The characteristic ring-like emission morphologies and distinctive visibility nulls for these disks indicate that the radial transition in the millimeter optical depth across the cavity edge is rather sharp, commensurate with our simple models that employ a density contrast of ∼10²–10⁴ within a small radial range. In three of these four disks, a small amount of micron-sized dust grains must reside inside the cavity to account for their observed infrared excess emission. Outside of their central cavities, these disks have Σ profiles comparable to the disks with continuous dust distributions. At this point, the physical mechanism responsible for the dust evolution in these disk cavities remains a subject of active debate (see, e.g., D'Alessio et al. 2005; Najita et al. 2007). They could be produced by an abrupt emissivity decrease due to grain growth in a localized region of low turbulence (e.g., an MRI-inactive "dead" zone; Ciesla 2007; Zhu et al. 2010). Alternatively, they may represent a true lack of material in the inner disk, cleared out by a photoevaporative wind and viscous draining (Alexander & Armitage 2007) or tidal interactions with faint companions—perhaps even young planetary systems (e.g., Lubow & D'Angelo 2006).