RESOLVED IMAGES OF LARGE CAVITIES IN PROTOPLANETARY TRANSITION DISKS

The model structure includes two distinct radial zones: a dust-depleted "cavity" (R ⩽ R_cav) and an "outer disk" reservoir (R > R_cav). We define a global surface density profile, appropriate for a simple accretion disk with a power-law radial distribution of time-independent viscosities (ν∝R^γ; see Lynden-Bell & Pringle 1974; Hartmann et al. 1998),

$$\begin{equation} \Sigma _g = \Sigma _c \left(\frac{R}{R_c} \right)^{-\gamma } \exp \left[ {-} \left(\frac{R}{R_c} \right)^{2-\gamma } \right], \end{equation} \tag{ 1 }$$

where R_c is a characteristic scaling radius. This global profile is applied to the material in the outer disk: Σ(R > R_cav) = Σ_g. The densities in the cavity are scaled down by a constant depletion factor, δ_cav, such that Σ(R ⩽ R_cav) = δ_cavΣ_g. To facilitate a more homogeneous comparison of δ_cav values for a sample with a range of cavity sizes (R_cav), we decomposed the cavity into a depleted "inner disk" of fixed extent (out to R = 10 AU) and an empty "gap" between 10 AU and R_cav (the arbitrary choice of the inner disk extent is discussed in Section 3.5). Unless noted otherwise, the inner disk is truncated at the dust sublimation radius, R_sub ≈ 0.07(L_*/L_☉)^1/2 AU (assuming T_sub = 1500 K; Dullemond et al. 2001). Figure 2(a) shows a generic surface density model for reference.

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The vertical distribution of dust has been modified from our previous models, in an effort to more realistically treat the irradiation heating of the disk atmosphere. We have introduced a vertical gradient in the dust size distribution, such that larger grains are concentrated toward the midplane and smaller grains are distributed to larger vertical heights. In practice, this parameterization crudely mimics dust sedimentation (e.g., Dullemond & Dominik 2004b; D'Alessio et al. 2006). At each radius, a population of "small" grains represents a small fraction of the total column density, (1 − f)Σ, and is distributed vertically like a Gaussian with scale height h = h_c(R/R_c)^ψ. Conversely, most of the total column (fΣ) is composed of a "large" grain population, condensed toward the midplane with a scale height χh (where χ ⩽ 1). In a spherical coordinate system with azimuthal and mirror symmetry, the two-dimensional density structures for each dust population are

$$\begin{equation} \rho _s = \frac{(1-f)\Sigma }{\sqrt{2\pi } R h} \exp \left[{-} \frac{1}{2} \left(\frac{\pi /2 - \Theta }{h} \right) ^2 \right] \end{equation} \tag{ 2 }$$

$$\begin{equation} \rho _l = \frac{f\Sigma }{\sqrt{2\pi } R \chi h} \exp \left[{-} \frac{1}{2} \left(\frac{\pi /2 - \Theta }{\chi h} \right) ^2 \right], \end{equation} \tag{ 3 }$$

where the subscripts "s" and "l" denote the small and large dust populations, respectively, Θ is the (vertical) latitude coordinate measured from the pole (Θ = 0) to the equator (the midplane, Θ = π/2), and the value of Σ depends on the radial location as outlined above. Note that in this coordinate system, the scale height h is an angle: the scale height in physical (distance) units is H ≈ hR. A generic vertical density profile with this parametric form is shown in Figure 2(b).

The distinctive infrared spectra of transition disks require small, but important, local modifications to the vertical structure near the disk edges at R_sub and R_cav (Natta et al. 2001; Calvet et al. 2002; Espaillat et al. 2010). At those locations, a large surface area of material is exposed directly (or with relatively low attenuation; see Section 3.5) to stellar irradiation. That exposure substantially heats a thin layer of material and "puffs" up the local dust distribution (Dullemond & Dominik 2004a; D'Alessio et al. 2005). To mimic the emission signatures of these structures—referred to as the inner "rim" (at R_sub) and cavity "wall" (at R_cav)—we permit the local value of h to be scaled up parametrically (to h_rim and h_wall, respectively). Those perturbations are joined to the global scale height distribution exponentially over a small radial width (fixed here to ΔR = 0.1 AU).

This structure model has 11 parameters: 5 describe the surface density profile {Σ_c, γ, R_c, δ_cav, R_cav} and 6 others characterize the vertical distribution of dust {h_c, ψ, χ, f, h_rim, h_wall}. Some of these parameters cannot be constrained with the available data and were therefore fixed to representative values. The surface density gradient was kept at γ = 1, in line with the results for "normal" disks with continuous dust distributions (Andrews et al. 2009, 2010b). We set the settling parameters to χ = 0.2 (the large grains are distributed to 20% of the scale height, h) and f = 0.85 (85% of the total column is composed of the large grain population). The remaining parameters are allowed to freely vary, although substantial degeneracies remain (see Section 3.5).

A parametric density structure must be populated with dust grains of a given size distribution and composition. For the sake of homogeneity, we assume dust grains with mineral abundances as in the interstellar medium (ISM; see Weingartner & Draine 2001), despite some evidence in individual cases for different compositions. Each dust population has a power-law distribution of sizes (s), n(s)∝s^−p, from s_min = 0.005 μm to a specified s_max. The dust properties in the outer disk are fixed: "small" grains have s_max = 1 μm, "large" grains have s_max = 1 mm, and p = 3.5. The opacity spectrum for a given size distribution was derived from Mie calculations. Since most of the mass is in "large" grains, the selection of s_max ∼ λ = 1 mm is significant because it tends to maximize the millimeter-wave dust opacity. Therefore, the densities inferred from the optically thin emission we have observed with the SMA can be considered lower bounds if much larger particles are present (D'Alessio et al. 2006; Draine 2006). The 880 μm dust opacity for the large grain population is 3.6 cm² g⁻¹. The dust size distribution in the inner disk, inner rim, and cavity wall are assumed to be identical, but are characterized by free parameters for their size distribution, {s_max, p} (see Section 3.4 regarding the selection of these parameter values). Some representative opacity spectra are shown in Figure 3.

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Passive irradiation is the only heating mechanism considered here, and therefore the properties of the central star are pivotal in setting the thermal structure of a model. Careful selection of an input stellar spectrum is particularly important for modeling transition disks, as the infrared disk-to-star contrast is often low. Therefore, the details of the stellar spectrum can have a pronounced impact on how the small amount of dust inside the cavity is interpreted. To derive stellar input spectra, we first assigned effective temperature (T_eff) values based on spectral classifications in the literature, using the conversions advocated by Kenyon & Hartmann (1995) and Luhman (1999) for types earlier or later than M0, respectively. Luminosities (L_*) were determined by matching reddened, scaled spectral synthesis models (for a fixed T_eff; Lejeune et al. 1997) to the optical/near-infrared photometry (see the Appendix). We utilized the extinction curve derived by McClure (2009), which at these wavelengths is identical to the Mathis (1990) law for a large total-to-selective extinction value, R_V = 5. For cases without dynamical mass constraints, this process was iterated for different stellar masses (M_*) estimated from the Siess et al. (2000) pre-main sequence models. The stellar parameters used to generate input spectra are compiled in Table 3.

Table 3. Stellar and Accretion Properties

Name	SpT	Teff	AV	d	L*	R*	M*	$\dot{M}$	log LX	Ref
		(K)		(pc)	(L☉)	(R☉)	(M☉)	(M☉ yr−1)	(erg s−1)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)
MWC 758	A8	7580	0.0	200	15	2.25	1.8	$\hphantom{<}1\times 10^{-8}$	...	1, 2, 3,  ⋅⋅⋅
SAO 206462	F4	6590	0.3	140	7.8	2.15	1.6	$\hphantom{<}6\times 10^{-9}$	29.7	4, 5, 6, 6
LkHα 330	G3	5830	3.1	250	15	3.75	2.2	$\hphantom{<}2\times 10^{-9}$	...	7, 5, 8,  ⋅⋅⋅
SR 21	G3	5830	6.3	125	10	3.15	2.0	<1 × 10−9	30.0	9, 5, 10, 11
UX Tau A	G8	5520	1.9	140	3.5	2.05	1.5	$\hphantom{<}1\times 10^{-8}$	30.3	12, 5, 12, 13
SR 24 S	K2	4990	7.0	125	4.0	2.70	2.0	$\hphantom{<}1\times 10^{-8}$	30.0	14, 5, 10, 15
DoAr 44	K3	4730	2.2	125	1.4	1.75	1.3	$\hphantom{<}9\times 10^{-9}$	29.9	12, 5, 12, 16
LkCa 15	K3	4730	1.7	140	1.2	1.65	1.01	$\hphantom{<}2\times 10^{-9}$	<29.6	12, 17, 18, 19
RX J1615−3255	K5	4350	0.4	185	1.3	2.00	1.1	$\hphantom{<}4\times 10^{-10}$	30.4	20, 5, 5, 21
GM Aur	K5	4278	1.1	140	1.0	1.85	0.84	$\hphantom{<}1\times 10^{-8}$	<29.7	22, 23, 18, 19
DM Tau	M1	3705	0.6	140	0.3	1.25	0.53	$\hphantom{<}6\times 10^{-9}$	<29.7	24, 17, 18, 25
WSB 60	M4	3270	3.7	125	0.2	1.40	0.25	$\hphantom{<}2\times 10^{-9}$	...	26, 5, 10,  ⋅⋅⋅

Notes. Column 1: name of host star. Column 2: spectral type. Column 3: visual extinction. Column 4: effective temperature. Column 5: luminosity. Column 6: estimated distance to star-forming region. Column 7: stellar radius. Column 8: stellar mass, determined either directly from CO spectral images or indirectly from pre-main sequence evolution models (see Section 3.2). Column 9: accretion rate. Column 10: X-ray luminosity or 3σ upper limit. Note that the value for UX Tau A may include contributions from the additional stars in the multiple system. To our knowledge, no X-ray measurements of WSB 60, LkHα 330, or MWC 758 have been published to date. Column 11: literature references for spectral classifications, M_*, $\dot{M}$, and L_X: (1) Beskrovnaya et al. (1999), (2) Chapillon et al. (2008), (3) Eisner et al. (2009), (4) Dunkin et al. (1997), (5) This paper (see Section 3.3), (6) Grady et al. (2009), (7) Cohen & Kuhi (1979), (8) Salyk et al. (2009), (9) Prato et al. (2003), (10) Natta et al. (2006), (11) Grosso et al. (2000), (12) Espaillat et al. (2010), (13) König et al. (2001), (14) Luhman & Rieke (1999), (15) Pillitteri et al. (2010), (16) Montmerle et al. (1983), (17) Piétu et al. (2007), (18) Ingleby et al. (2009), (19) Neuhauser et al. (1995), (20) Wichmann et al. (1999), (21) Krautter et al. (1997), (22) White & Ghez (2001), (23) Dutrey et al. (1998), (24) Kenyon & Hartmann (1995), (25) Damiani et al. (1995), (26) Wilking et al. (2005).

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For a given set of parameters, a model structure was defined on a 200 × 90 cell grid in spherical coordinates, {R, Θ}. The radial grid was logarithmically spaced from the sublimation radius to 1500 AU, with local refinements near R_sub, 10 AU (the fixed edge of the inner disk), and R_cav to sample the optical depth gradients at those locations with higher resolution. The latitude grid was linear, with three distinct resolution levels that depend on the vertical structure of the model: high near the condensed midplane, coarse at high latitudes near the pole, and intermediate in the disk atmosphere. Simulations of the stellar irradiation of that model density grid were conducted to determine the dust temperature structure, using the two-dimensional axisymmetric Monte Carlo radiative transfer code RADMC (see Dullemond & Dominik 2004a). Those calculations explicitly accounted for the finite size of the star (R_*; see Table 3) to properly treat any shadowing effects from the inner rim or cavity wall (stellar limb darkening is not considered). For a specified viewing geometry (a disk inclination, i, and major axis position angle, P.A.), a raytracing algorithm was used on the results of the radiative transfer calculations to compute a synthetic SED and set of 880 μm continuum visibilities (sampled at the same spatial frequencies observed with the SMA).

Aided by a series of refined grid searches in parameter-space, this initial modeling effort was primarily conducted manually. A confirmation of fit quality was performed by comparing the synthetic data for a given parameter set with the observed SED and the deprojected, elliptically averaged 880 μm visibility profile (for details, see Andrews et al. 2009). For this model setup and such a large sample, parameter estimation with automated minimization algorithms would be computationally prohibitive: the free parameter-space is too large (eight density parameters, two dust parameters, and two viewing geometry parameters) and degenerate. Moreover, there are non-trivial technical obstacles: for example, great care is required in defining the model grid for a given parameter set, to ensure proper treatment of large optical depth gradients and small-scale features like the rim and wall (see also Andrews et al. 2009). Regardless of these challenges, good (albeit not unique, best-fit) matches to the data were found with a simple and systematic approach.

In most cases, the viewing geometry was fixed based on models of CO spectral images in the literature (Dutrey et al. 1998; Piétu et al. 2007; Isella et al. 2010b; A. R. Lyo et al. 2011, in preparation) or the new J = 3–2 data from the SMA (for LkHα 330, UX Tau A, and RX J1615−3255; details will be presented elsewhere). No such information is available for the Ophiuchus disks (SR 21, SR 24 S, DoAr 44, and WSB 60): estimates of {i, P.A.} in those cases were inferred from the aspect ratio and orientation of the continuum, and are considerably uncertain (see Andrews et al. 2009, 2010b). In practice, the cavity size, R_cav, was determined from the null position in the SMA visibility profile (see Section 3.5). The dust parameters in the cavity, {s_max, p}, were set based on the strength of the 10 μm silicate feature, and the density contrast (δ_cav) and rim height (h_rim) were scaled to match the weak infrared excess from the inner disk. The height of the cavity wall (h_wall) was adjusted to reproduce the shape and rise of the mid-infrared spectrum, and the scale height parameters, {h_c, ψ}, were modified based on the far-infrared SED. The characteristic radius, R_c, and density normalization, Σ_c, were inferred from the shape and amplitude of the 880 μm visibility profile.

To reinforce the intuition used to model these data and provide an overview of the individual parameter effects, it is helpful to decompose the disk structure into four essential elements: the inner rim, dust-depleted inner disk, cavity wall, and outer disk. Each element has a distinctive impact on the dust emission spectrum, and together they are capable of producing a remarkable diversity of observational signatures. To illustrate those signatures, we define a fiducial reference model and show separately the contributions from each structural element on the SED in Figure 4 (fiducial parameters are listed in the caption). Figure 5 explores how the SED and 880 μm visibility profile respond to modest changes in the parameters of specific interest for transition disks: the rim and wall heights (h_rim and h_wall, respectively), the inner disk mass (set by δ_cav), and the cavity size (R_cav). The effects of more global structure parameters were discussed by Andrews et al. (2009).

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The emission from the directly illuminated dust in the inner rim dominates the near-infrared SED (see Figure 4). The luminosity of that hot component is set by the size (and optical depth) of the rim, such that a more "puffed-up" rim (larger h_rim) produces a brighter excess. A sufficiently high (and dense) rim can intercept enough starlight to shadow material at larger radii. The attenuated irradiation field that reaches those components produces less heating and therefore a fainter infrared spectrum (see Figure 5(a)). While this shadowing effect has been explored for normal disks (Dullemond 2000; Dullemond et al. 2001), its effects can be pronounced for transition disks because much of their infrared spectrum is produced by the directly illuminated portion of the cavity wall (Espaillat et al. 2010). Espaillat et al. (2011) have shown that the Spitzer IRS spectra for many transition disks exhibit remarkable "see-saw" variability, such that an increase in the near-infrared emission is correlated with a decrease at mid-infrared wavelengths (and vice versa). They make a compelling argument that this variability is produced by changes in the height of the inner rim, as demonstrated in Figure 5(a) (see also Dullemond et al. 2003; Flaherty & Muzerolle 2010).

Beyond the rim, the tenuous inner disk emits an infrared spectrum corresponding to the temperature range between R_sub and 10 AU. In cases where this material is optically thin, the luminosity scales directly with the density (parameterized here by δ_cav). However, small amounts of dust in the cavity can be optically thick. Even for substantial optical depths, the infrared luminosity still partly scales with density: for a given h, a denser disk has more dust at larger heights above the midplane, increasing the height and temperature of the infrared photosphere. The result is a brighter (and hotter) excess spectrum. The same is true for the inner rim, and therefore δ_cav can also have an impact on shadowing. Provided a sufficiently high rim height, h_rim and δ_cav are degenerate (see Figures 5(a) and (b)): higher rims can compensate for lower densities (at least partially). Moreover, the inner disk size (fixed here to 10 AU) is also degenerate with δ_cav: a more extended inner disk (a smaller gap, or even no gap) can usually be accommodated by decreasing δ_cav. Other fixed assumptions about the inner disk structure can also have comparable effects (e.g., the width of the rim, shape of the density profile, local grain properties, etc.). The important point to be made is that even a small amount of (∼μm-sized) dust inside the cavity can have a substantial impact on the SED, depending rather sensitively on its detailed spatial distribution and grain properties. Therefore, without spatially resolved observations, the detailed characteristics of the material inside transition disk cavities are strongly model dependent and highly uncertain.

Despite our ignorance of the detailed structures of the inner disk and rim, the cavity size can be measured from the SMA data with little ambiguity. Past efforts to indirectly infer R_cav relied on estimating the wall temperature from the shape of the mid-infrared spectrum (e.g., D'Alessio et al. 2005; Calvet et al. 2005; Espaillat et al. 2007). Those models have become increasingly sophisticated to try and account for the irradiation gradients caused by shadowing (Espaillat et al. 2010), but still suffer from the implicit uncertainties in the inner disk properties (not to mention the detailed characteristics of the wall itself). However, a direct measurement of R_cav is possible with a resolved, optically thin tracer like the millimeter continuum. The sharp emission contrast between the faint dust in the cavity and the bright material in the outer disk produces a clear "ring" morphology at 880 μm. The Fourier transform of that emission profile (i.e., the visibility profile) has distinctive nulls at discrete spatial frequencies, such that the location of the first null (where the real part of the visibilities change sign) is a relatively unambiguous measure of the inner edge of the ring (R_cav; see Hughes et al. 2007). Figure 5(c) demonstrates the sensitivity of this null position to R_cav; modifications on the order of 10% are readily detectable for typical noise levels. However, we should note that these null positions also have a roughly linear dependence on the radial gradient of the surface brightness profile, which is set by the product of Σ and the midplane temperature profile. Since the latter is approximately a power law in these models (T_mid∝R^−q; typically q ≈ 0.5), the R_cav value inferred from a fixed null position is proportional to the sum γ + q. Since γ = 1 is fixed in these models, we underestimate the uncertainty on R_cav: allowing γ to vary by ±0.5 would contribute an additional ∼10%–15% uncertainty to the cavity sizes.

In a simple model with an empty cavity, the directly irradiated wall emits a thermal spectrum corresponding to the equilibrium temperature at its location (R_cav) with a luminosity proportional to its vertical extent (Calvet et al. 2002; D'Alessio et al. 2005). To first order, the behavior expected from such a model is qualitatively reproduced in Figure 5(d): smaller cavities have hotter walls and narrower infrared SED dips. As mentioned above, there are slight deviations in the details because shadowing can induce vertical temperature gradients in the wall. In principle, a high wall can shadow the outer disk if the dust is sufficiently settled (low h_c and/or ψ). The detailed properties of the cavity wall (e.g., shape, thickness) are not well constrained by the SED alone; no current observations have sufficient resolution to isolate such a (presumably) small feature.

Although all disk modeling suffers from degeneracies between structure parameters and dust properties, it is important to explicitly highlight one key issue in these transition disk models. The standard mass-opacity degeneracy for optically thin emission (e.g., Beckwith & Sargent 1991; Andrews & Williams 2005b) and the intricate relationship between irradiation, heating, and grain properties in the disk atmosphere (D'Alessio et al. 1999, 2001, 2006; Chiang et al. 2001) are important, but will not be reviewed again here. The degeneracy of interest is related to how our method of inferring densities in the inner disk (i.e., δ_cav) depends on the dust opacities inside the cavity. Since we lack any resolved millimeter signal inside R_cav (although see Section 4 regarding LkCa 15), the inner disk densities are determined solely from the infrared spectrum. In most cases, the relatively strong silicate emission features we observe suggest that the cavity is populated by "small" grains (i.e., s_max ≈ 1–10 μm or less). To be conservative, we do not assume that any additional material is present. This small dust becomes optically thick even at the relatively low column densities (∼10⁻⁴ g cm⁻², give or take an order of magnitude; see Figure 3) that are present in some non-negligible part of the inner disk. The sizes of these high optical depth regions are effectively responsible for setting δ_cav: for a given structure, more mass would overproduce the infrared luminosities. In some sense, this aspect of the modeling tends to minimize δ_cav (or maximize the depletion of dust densities inside R_cav). However, more material could easily be accommodated in the inner disk if it has lower infrared opacities (e.g., s_max larger than a few μm; see Section 4).

Finally, a few notes about the outer disk parameters are in order (for a more complete discussion, see Andrews et al. 2009). It is important to emphasize that the nature of transition disk structures makes it difficult to constrain radial gradient parameters (γ or ψ). For smaller disks, the same is true for the characteristic radius, R_c. The issue is limited dynamic range; only a few radial resolution elements can probe the millimeter emission in a ring (which is why we chose to fix γ), and the far-infrared SED that is less affected by the cavity is only sparsely sampled. While we have a reasonable constraint on the vertical distribution of dust (h_c) from the infrared luminosities, the current lack of far-infrared data means that we do not have a good handle on the scale height gradient, ψ, for these transition disks with large cavities. Linking these models with scattered light observations that specialize in constraining the flaring angle (i.e., ψ) will be beneficial in the future.

Although they were fixed, the settling parameters {χ, f} merit a brief discussion. The fraction of large dust grains, f, helps set the height where stellar photons are absorbed: smaller f means an increased optical depth (more abundant small grains) at a fixed height, and therefore a slightly warmer atmosphere that emits more infrared radiation. The effects of χ, the fractional scale height of the large grain population, are subtle. In practice, χ modifies the vertical temperature gradient at intermediate heights in the disk atmosphere: a larger χ leads to a cooler temperature in a fixed vertical layer. Although potentially important for more direct tracers of the disk atmosphere (e.g., scattered light or molecular line emission), these effects are indistinguishable from adjustments to the scale height parameters, {h_c, ψ}, with the data used here.

The previous sections describe in detail an effort to model the SEDs and 880 μm continuum visibilities for this sample of transition disks in a homogeneous, parametric framework. Before presenting those results, we first summarize some fundamental aspects of the modeling process.

• 1.  
  Using two-dimensional, axisymmetric Monte Carlo radiative transfer simulations, a parametric dust density structure is irradiated by a model star to determine an internally consistent temperature structure. The results of the simulation are used to generate synthetic data products to compare with observations. The underlying structure model is motivated by a simple viscous accretion disk model, with significant dust depletion in a central cavity.
• 2.  
  Given the relatively faint infrared excesses for transition disks, the adopted stellar input spectrum factors significantly into inferences about material inside the cavity.
• 3.  
  Small quantities of (small) dust in the cavity can produce large amounts of infrared emission. However, the lack of sufficient angular resolution in those regions means that the detailed structure of that material is unknown; considerable uncertainties remain for the mass content, spatial structure, and dust properties inside the cavity (R ⩽ R_cav). Our estimates of density depletion tend to be maximal (δ_cav is minimized): more material could reside in the inner disk if it has relatively lower infrared opacities (e.g., grain sizes larger than a few μm).
• 4.  
  Small, local modifications to the vertical distribution of material near the sublimation radius (the inner rim) and outer edge of the disk cavity (the cavity wall) are needed to explain the distinctive infrared spectra of the transition disks. Because those same features can effectively shadow material at larger radii, their properties are at least partially entangled with one another as well as more global structure parameters.
• 5.  
  Because the dust-depleted cavities are spatially resolved at 880 μm, their sizes are determined directly and with reasonable accuracy (typically within ±10% for a fixed γ).
• 6.  
  Inherent model degeneracies and limited observational information make it difficult to quantify several model parameters for an individual disk with sufficient confidence. However, because the modeling analysis was performed in a homogeneous framework for the entire sample, intra-sample (and qualitative) comparisons of the inferred structures are informative.

5.1.1. Nomenclature and Our Definition of a "Transition" Disk

5.1.2. The Frequency of Transition Disks with Large Cavities

Despite the narrowed scope, transition disks with large cavities are surprisingly common when viewed in the proper context. Consider the millimeter-wave luminosities from disks in the nearby Tau and Oph associations (Andrews & Williams 2005b, 2007a; Cieza et al. 2010). Removing non-detections, sources with likely envelope contamination (including flat-spectrum objects) and disks around early-type (A/B) stars, and scaling 1.3 mm photometry when necessary by a conservative factor 1/λ³ (steeper than the typical slope found by Andrews & Williams), we constructed the distribution of 880 μm luminosities (L_mm) shown in Figure 10 for 91 disks. Out of that sample, large cavities (R_cav ≳ 15 AU) have been directly resolved for nine disks (10%, including RY Tau; see Isella et al. 2010a). That fraction is roughly consistent with expectations from infrared-selection methods (Muzerolle et al. 2010), especially if the upper limits for millimeter-faint disks are included. But current disk imaging surveys are sensitivity-limited and have only probed the brightest disks. In the upper half of the L_mm distribution (⩾80 mJy at 140 pc), the transition disks with large cavities comprise a larger fraction, ∼20% (9/46 disks). And in the upper quartile of the distribution (⩾200 mJy at 140 pc), that fraction is slightly higher still (26%; 6/23 disks). These transition disk fractions are lower bounds; there could very well be more disks with large cavities that have yet to be imaged at sufficient resolution. The maximum fraction of this type of transition disk in the upper half (or quartile) of the L_mm distribution is ∼70%, based on the known absence of large cavities in some of the remaining disks (Andrews et al. 2009, 2010b).

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To reiterate those remarkable statistics at least 1 in 5 and possibly up to ∼two-thirds of the known millimeter-bright disks have large dust-depleted cavities (with R_cav ≳ 15 AU). We do not know how high such fractions might be for smaller cavities, or disks with even fainter millimeter emission, or young disk populations in other environments (if these results for Tau/Oph are to be applied to other clusters, one must keep in mind the counting uncertainties that correspond to ±10% errors on the transition disk fractions quoted above). Regardless, the implication is that a large fraction of the most millimeter-bright—and therefore most massive—disks show unequivocal evidence for substantial dust evolution in the regions most relevant for planet formation on timescales of less than ∼1–2 Myr (for reference, the median and quartile L_mm values correspond to disk masses of 0.005 M_☉ and 0.011 M_☉, respectively, using the standard Andrews & Williams assumptions). This reinforces a similar conjecture made by Najita et al. (2007) based on a group of unresolved Taurus disks, although see Section 5.2.3 regarding their corollary claim of low accretion rates.

5.1.3. Comments on Evolution

Up to this point, the discussion has been intentionally generic with respect to the physical origins of the dust-depleted cavities in transition disks. In a broad sense, there are two types of depletion mechanisms: those that physically remove or "clear" material from the inner disk (decrease Σ) and those that modify the emission properties of the dust (decrease κ). Various inner disk evolutionary processes have been assessed in previous studies of transition disks (D'Alessio et al. 2005; Najita et al. 2007; Alexander & Armitage 2009). Here, we review several of these proposed dust depletion mechanisms in the context of the specific type of transition disk in our sample.

5.2.1. Dispersal by Winds

5.2.2. Material Evolution—Particle Growth

5.2.3. Tidal Interactions with Companions

A large fraction of stars have companions (Abt & Levy 1976; Duquennoy & Mayor 1991), with stellar multiplicity fractions approaching 75% in the diffuse young clusters near the Sun (e.g., White & Ghez 2001; Kraus et al. 2011). Circumstellar material is expected to be strongly affected by dynamical interactions with these stellar components (see Lin & Papaloizou 1993). The interplay of resonant and viscous torques can open a gap in a circum-system disk, truncate individual disk edges, and potentially expedite disk dissipation (via accretion or ejection), depending on the orbital separation (a), eccentricity (e), and mass ratio (q) of the companion. Based on the calculations of Artymowicz & Lubow (1994), a binary with a large separation can retain individual disks similar to those around single stars, but with their outer edges truncated at R_o ∼ 0.2–0.5a. For smaller separations, a circumbinary ring structure may also be present, with a truncated inner edge at R_cb ∼ 1.8–2.6a. Pichardo et al. (2008) find similar results, but generalize to arbitrary eccentricities such that R_cb ∼ 1.9(1 + e^0.3)a. Their calculations suggest that the inner radius of the circumbinary ring has only a weak dependence on the mass ratio, at least for q ⩾ 0.1. It is natural to associate the 880 μm emission rings observed for the transition disks with these circumbinary structures. In fact, stellar companions have been detected inside the cavities inferred (indirectly) for a few notable transition disks, including CoKu Tau/4 (Ireland & Kraus 2008) and the older HD 98800 and Hen 3-600 systems (Uchida et al. 2004; Furlan et al. 2007; Andrews et al. 2010a).

If the transition disk cavities were created by tidal interactions with stellar companions, their expected orbital separations would be a ∼ 0.3–0.6R_cav, for high eccentricities (e ∼ 0.9) down to circular orbits. The range of cavity sizes measured here would correspond to companion separations of a ∼ 4–40 AU. The multiplicity fraction for similar primary star masses in this separation range is ∼10%–20% (Duquennoy & Mayor 1991; Kraus et al. 2011), which is similar to, but slightly below, the minimum frequency of millimeter-bright transition disks with large cavities we estimated in Section 5.1.2. Given how small those separations are projected on the sky, ∼004–020, direct searches for companions have been limited by both resolution and contrast. Kraus et al. (2011) have overcome such restrictions in Taurus, using a near-infrared aperture-masking interferometry technique coupled with the adaptive optics system on the Keck telescope. For the four Taurus disks in our sample (as well as RY Tau; see Isella et al. 2010a), Kraus et al. definitively rule out the presence of any companions with q ⩾ 0.01 (⩾0.02 for DM Tau) in the separation ranges implied by R_cav (see also Pott et al. 2010). The same applies to the LkHα 330 and SR 21 disks (A. L. Kraus 2011, private communication). For at least half of the known transition disks with large cavities, there are no companions with M_s ≳ 20–30 M_jup that could be responsible for the observed disk structures.

Until similar measurements are available for the other transition disks, their multiplicity status will remain uncertain. However, there are some stark differences between the transition disks and a = 4–40 AU multiples. Tidal interactions substantially diminish disk masses in multiples with this separation range (Jensen et al. 1994, 1996; Osterloh & Beckwith 1995; Andrews & Williams 2005b; Cieza et al. 2009). To illustrate that point, we compare the millimeter-wave luminosities (L_mm, see Section 5.1.2) with the projected separations of the known multiples in Ophiuchus and Taurus (Barsony et al. 2003; Kraus et al. 2011) in Figure 11(a). The transition disks are marked with red lines, representing the companion separation ranges implied by their cavity sizes. More directly, Figure 11(b) compares the cumulative L_mm distributions of the transition disks and the known 4–40 AU multiples (the latter includes upper limits and was constructed using the Kaplan–Meier product limit estimator; see Feigelson & Nelson 1985). The two populations have remarkably different L_mm (or equivalently, M_d) distributions: the median transition disk is at least 30× brighter (more massive) than the 4–40 AU multiples. Of the 44 known multiples in that separation range, only one—GG Tau—is brighter than 30 mJy at 880 μm. GG Tau is exceptionally rare, hosting a circumbinary ring not unlike the transition disk structures (it technically meets our definition of a transition disk in Section 5.1.1), but with an enormous R_cb ≈ 180 AU (Guilloteau et al. 1999). In any case, the absence of millimeter-bright 4–40 AU multiples is striking: it suggests that the specific type of transition disk studied here has not suffered from the same destructive interaction mechanism.

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Some caution is warranted in overinterpreting the differences between these two populations, as selection biases, unknown orbital parameters (e, i), and perhaps uncertainties in the tidal truncation models (e.g., see Beust & Dutrey 2005) should also be considered. One notable distinction is related to the mass ratio. There are stringent limits on q (⩾0.01–0.02) for at least half of the transition disk sample, but the 4–40 AU multiples are preferentially equal-mass (median q ≈ 0.7; Kraus et al. 2011). Such high mass ratios could impart more severe tidal perturbations to the circumstellar environment at an early evolutionary stage. Since that seems unlikely for the transition disks, it is worthwhile to explore the effects of companions with much lower q values. Such low-mass companions that form in the disk around the primary—including giant planets and brown dwarfs—can modify the disk structure without dispersing too much of the total mass (∝L_mm).

Resonant torques induced by a low-mass companion set up spiral waves that repel local disk material (Goldreich & Tremaine 1978, 1980; Lin & Papaloizou 1979a, 1979b). Above a mass ratio threshold, q ≳ q_gap ≈ 40α(H/a)² (where α is the viscosity coefficient; Shakura & Sunyaev 1973), these tidal interactions open a gap in the disk (Lin & Papaloizou 1986, 1993; Bryden et al. 1999; Crida et al. 2006). In this scenario, the companion is located closer to the outer edge of the cavity than for the high-q case. A typical separation for a circular orbit might be a ∼ 0.8R_cav, although the exact location depends on some complicated issues that influence the gap width (Winters et al. 2003; Crida et al. 2006). For our model structures and α = 0.01, the transition disks in this sample would need companions with a ∼ 10–60 AU and mass ratios at least q_gap ≈ 0.001–0.03 to open gaps near their cavity edges. Using the M_* values in Table 3, those limits translate to companion masses M_s ≳ 1–70 M_jup (the high end corresponding to larger M_*; note that q_gap and M_s scale linearly with the assumed α). For the relevant cases, the mass ratios implied from current infrared contrast limits (Pott et al. 2010; Kraus et al. 2011) exceed those needed to open gaps. Although reassuring, there is little to be learned there, given the uncertainties on the disk viscosities (α) and the luminosities of such low-mass companions (e.g., Marley et al. 2007).

An individual gap is too narrow to be responsible for the observed properties of the transition disks. However, the disk-companion interactions can also regulate the densities in the inner disk, interior to the companion orbit. As in the case for a photoevaporative flow (Section 5.3.1), the gap effectively isolates the inner disk from its replenishment reservoir at larger radii. Without a viscous flow of material inward, the inner disk material would drain onto the star on a relatively short timescale, producing a large and empty cavity. Unlike the photoevaporation mechanism, some material from the outer disk is expected to seep through a gap created by a low-mass companion in accretion streams, but only at decreased flow rates compared to a normal disk (e.g., Artymowicz & Lubow 1996; Kley 1999; Lubow et al. 1999). Lubow & D'Angelo (2006) estimate that the mass flow across such a gap is diminished by ∼75%–90%, depending on q: a more massive companion reduces the flow across the gap, leading to a lower-density inner disk. A quantitative association between the companion mass, the accretion flow, and the depletion of the inner disk is not yet clear in simulations. Some have suggested that a companion with q ≈ 0.001 is sufficient to deplete a disk cavity like those observed for the transition disks (Rice et al. 2003a; Quillen et al. 2004; Varniére et al. 2006), while others have argued that higher masses are required (Crida & Morbidelli 2007).

Najita et al. (2007) suggested that transition disks exhibit low accretion rates compared to their normal counterparts with similar disk masses, consistent with a large reduction in the mass flow across a gap produced by a low-mass companion (see also Alexander & Armitage 2007, 2009). To verify that result, we constructed the distribution of $\dot{M}$ for normal disks in Taurus and Ophiuchus that lie in the bright half of the L_mm distribution and compared it with the accretion rates for the transition disks with large cavities in those regions (including RY Tau; see Table 3). Figure 12 shows the cumulative distributions of $\dot{M}$ for both of these populations. A two-sided Kolmogorov–Smirnov test (or its survival analysis equivalents if we include the $\dot{M}$ upper limit for SR 21; see Isobe et al. 1986) indicates that the probability these $\dot{M}$ populations are drawn from the same parent distribution is low (⩽1%–2%), with the transition disks having about ∼3× lower accretion rates on average. That relative $\dot{M}$ decrease (∼70%) is similar to the expected reduction in flow rates across a gap opened by a q ≈ 0.001 (i.e., M_s ≈ 1 M_jup) companion (Lubow & D'Angelo 2006). However, the samples are small, incomplete, and likely biased. Moreover, the individual accretion rates have large uncertainties and are estimated from different diagnostics. So, although this preliminary comparison (and that by Najita et al. 2007) provides a tantalizing hint of reduced mass flow rates for transition disks, we caution that a more rigorous and homogeneous set of comparisons is warranted before any premature conclusions are drawn.

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Regardless of those comparisons, the mass flow rates for the transition disks are still high enough to be seemingly inconsistent with the low dust densities we infer inside their cavities. One potential way to reconcile those measurements is to include additional low-mass companions inside the observed disk cavity. Zhu et al. (2011) have suggested that multi-planet systems can dynamically clear large cavities while also rapidly transporting gas across the large distances between the remnant outer disk and the star. Alternatively, a single low-mass companion might be responsible for creating a cavity that is depleted in dust, but not in gas. Any companion that can open a gap will generate wakes just outside its orbit, creating a pressure bump that may act like a dust filter on the flow across the gap (Rice et al. 2006; Paardekooper & Mellema 2006). If that filter removes a substantial fraction of the dust from the flow, the inner disk can be preferentially depleted of dust (producing the thermal deficits we observe) while still maintaining high gas accretion rates.

The tidal interactions between disk material and low-mass companions offer a compelling explanation for the transition disk structures. However, the observations are only starting to provide some tangible constraints that can be incorporated into the numerical simulations. As the data improve—both for the disk structures and companion searches—we should be able to make a more nuanced assessment of whether the mechanism responsible for transition disk cavities is tied directly to the formation and early evolution of low-mass brown dwarfs and/or exoplanets.

To summarize the discussion in Section 5.2, we have argued that the dust-depleted cavities in the specific type of transition disks studied here—those with large L_mm and R_cav—were not likely produced by photoevaporative winds or tidal truncation by stellar-mass companions. The possibility that the transition disk cavities are the manifestation of substantial particle growth in large, low-ionization dead zones remains, but requires a more detailed modeling effort to determine if some significant challenges could be mitigated. While these mechanisms all play non-trivial roles in disk evolution, the weight of the observational evidence for the kind of transition disks studied here suggests that the cavities are likely the byproduct of dynamical interactions with long-period low-mass companions—either brown dwarfs or giant planets. If that is the case, it is of great interest to understand how those companions can form at (or be moved to) such large separations from their host stars (a ∼ 10–60 AU) on such a relatively short timescale (at most ∼1–2 Myr).

Giant planets or low-mass brown dwarf companions could be formed relatively rapidly from a fragmentation process in a gravitationally unstable disk (e.g., Boss 1997, 1998; Rice et al. 2003b). Moreover, this disk instability mechanism seems to naturally favor formation at wide separations from the primary (Boss 2006; Boley 2009; Rafikov 2009; Clarke 2009; Dodson-Robinson et al. 2009; Kratter et al. 2010). Of course, it is not clear that the transition disks initially had sufficiently high densities (and efficient cooling) to be unstable, and in most cases (8/12 in this sample) the orbits of any companions would be small enough (a ⩽ 35 AU) that the prospects for fragmentation are debatable (but see Boss 2009). If the companions are not too massive, the more standard core accretion mechanism for giant planet formation may be applicable. In that scenario, the collisional agglomeration of solids must produce a massive core to accrete a massive atmosphere before the gas disk disperses (e.g., Pollack et al. 1996; Hubickyj et al. 2005). Traditionally, that process has been considered too inefficient to make long-period planets (a ≳ 10 AU) in situ within the disk lifetime. However, Rafikov (2011) has argued that an improved treatment of planetesimal accretion shows expedited core growth at low surface densities, thereby enabling the capture of a massive atmosphere in the outer disk. His estimate of the ⩾0.1 g cm⁻² of solids needed for that process is strikingly similar to (or below) the Σ values we derive near R_cav (see Figure 8). Others have suggested that long-period companions could be produced relatively easily by the core accretion mechanism in the inner disk, and then subsequently moved outward via planet–planet scattering (Scharf & Menou 2009; Veras et al. 2009) or a resonant migration process (Masset & Snellgrove 2001; Crida et al. 2009).

5.3.1. Future Prospects

This appendix highlights some aspects of the modeling results for individual disks and compares with previous efforts to characterize their disk cavities whenever available. Table 5 is a compilation of the original sources used to construct the SEDs displayed in Figures 6 and 7.

MWC 758. This Herbig Ae star is relatively isolated, located east of the main Taurus–Auriga clouds. Without a firm cluster association, its distance is uncertain but assumed here to be the nominal Hipparcos value, d = 200 pc (Perryman et al. 1997). We adopt the A8 spectral classification advocated by Beskrovnaya et al. (1999), but note that others have utilized an earlier type (A5; van den Ancker et al. 1998). The SMA observations from the E configuration are the same as presented by Isella et al. (2010b), although they have been re-calibrated with an amplitude scale that amounted to a ∼30% decrease in flux levels. That modification was required to reconcile those data and our new SMA observations with short antenna spacings (C configuration); the combined data are now in excellent agreement with single-dish photometry measurements. The 880 μm visibility null calls for the largest cavity in the sample, R_cav = 73 AU, despite the bright infrared excess. The standard infrared signatures of a transition disk are heavily muted, washed out by only a small amount of dust inside the cavity. Although that dust has densities that are comparable to the other disks in the sample (and only modest rim and cavity heights), it produces a significantly brighter infrared spectrum for two reasons: (1) the central star is extraordinarily hot and luminous, substantially heating more dust and (2) the scale heights are large, increasing the irradiated surface area and therefore the emitted flux from the disk atmosphere.

Isella et al. (2010b) also modeled the MWC 758 disk structure with some of these SMA data (and supplementary 3 mm observations), choosing not to modify the accretion disk model to include a cavity (i.e., Σ = Σ_g at all radii in their work) but to allow the gradient of the viscosity profile (γ) to vary freely. They can reproduce the data with a large negative viscosity gradient, γ ≈ −4, which results in a surface density profile that increases like R⁴ inside ∼70 AU, peaks near 100 AU, and then drops off like 1/exp (R⁶) at larger radii. Roughly speaking that density distribution is similar to the narrow ring employed here (see Figure 8), even peaking at essentially the same location and surface density. Not unexpectedly, the Isella et al. model requires an additional dust component near the star to account for the infrared excess (which their R⁴ density profile alone vastly underproduces), similar to the inner disk material utilized here. Finally, the sharp density cutoff at large radii that is inherent to models with negative γ values means that the Isella et al. model has effectively no mass outside R ≈ 150 AU. On the contrary, our model has gas densities in excess of 10⁻⁵ g cm⁻² out to R ≈ 400 AU, despite the small characteristic radius (because γ = 1). We expect that this difference in the effective disk size would help explain the strong positive residuals at large radii noted by Isella et al. when applying their dust model to the CO emission.

SAO 206462. Commonly referred to as HD 135344 B, this star is also isolated, perhaps associated with the outskirts of the Sco–OB2 association. We adopt the distance estimate of van Boekel et al. (2005), d = 140 pc, and the F4 spectral type determined by Dunkin et al. (1997). The SMA data were first presented by Brown et al. (2009), and only include observations in the V configuration with few short baselines. Our models imply a large cavity, R_cav = 46 AU, with a small amount of dust near the star. The super-heated inner rim near the dust sublimation radius is responsible for the substantial near-infrared bump in the SED. To minimize the strength of the 10 μm silicate feature, comparatively large (s_max = 3 μm) dust grains were used to populate the inner disk. Our model tends to overpredict the flux densities at wavelengths longward of 1 mm, a not uncommon feature due to our decision to fix the grain size distribution in the midplane of the outer disk (see Section 3.2). The issue might also be alleviated if new data with short antenna spacings become available, enabling a more robust disk mass estimate. The SMA image of the SAO 206462 disk is notably asymmetric, with enhanced emission to the southeast and a pronounced deficit to the north. But the imaged residuals indicate that those asymmetries are only marginally significant: given the challenges of observing such a southern target, those features are not inconsistent with noise (but note that a similar pattern is present in the scattered light images presented by Grady et al. 2009).

Brown et al. (2007) also used RADMC calculations to reproduce the SED alone, and later combined with these SMA data (Brown et al. 2009), but with a different density model (a power-law Σ profile, truncated at the edges). They estimated roughly the same cavity radius, 39–45 AU, consistent within the expected ∼10% mutual uncertainties. Although their lower R_cav value was determined from the SMA data, the difference with the cavity size derived here is likely due to small offsets in the adopted disk centroids and viewing geometries. Regarding the latter, we utilize the inclination determined by A. R. Lyo et al. (2011, in preparation) from CO spectral line images, in good agreement with the scattered light constraints of Grady et al. (2009). Those scattered light images suggest a substantially settled vertical distribution of dust with an effective ψ = −0.3, unlike the more common ψ = 0.15 used here. Part of that difference is related to shadowing from the inner rim/disk and cavity wall, which can effectively steepen the scattered light surface brightness profile. But there is so little radial dynamic range available from this disk that it should be possible to find anti-flaring models that are also commensurate with the SMA data.

LkHα 330. Associated with the Perseus molecular clouds (d ≈ 250 pc; see Enoch et al. 2006), this young star was classified as spectral type G3 by Cohen & Kuhi (1979). The SMA data in the V configuration were first presented by Brown et al. (2008); new SMA data in the C configuration are also included here. Note that there is considerable disagreement between the IRAS and Spitzer data in the far-infrared, perhaps due to molecular cloud or cirrus contamination in the former. When re-calibrating the long-baseline SMA data used by Brown et al. (2008, 2009), it was noted that the observations of 2006 November 11 had such poor phase stability that they could not be used. With their removal and the addition of new data with short antenna spacings, a much more symmetric and sensitive map of the 880 μm continuum emission is available.

We measure a large cavity, R_cav = 68 AU, ∼35%–60% larger than estimated by Brown et al. (2008, 2009) from (some of) the same data. Most of that discrepancy is caused by an inaccurate estimate of the 880 μm visibility null location, due to the poor quality of one of the SMA tracks in the V configuration. Using the CO J = 3–2 spectral line images from the new data in the C configuration, we also use an updated viewing geometry (with a significant difference in the major axis P.A.; the full CO data sets will be presented elsewhere). The faint 10 μm silicate feature was difficult to reproduce with the typical grain size index (p = 3.5), but we found some success using a more top-heavy distribution (p = 2.5) for the dust inside the cavity. To reproduce the relatively bright infrared excess with that grain population, a substantial increase in the inner rim height was required (the dust size distribution parameters and rim height are strongly degenerate; see Section 3.5).

SR 21. This young star with spectral type G3 is still embedded in the L1688 dark cloud in the nearby Ophiuchus star-forming region, d ≈ 125 pc (e.g., Lombardi et al. 2008; Loinard et al. 2008). The extinction estimate of Prato et al. (2003), A_V ∼ 9, is too large to be appropriate for both the optical photometry of Vrba et al. (1993) and the near-infrared magnitudes in the 2MASS catalog (Cutri et al. 2003). Assuming variability is not responsible for that discrepancy, we adopt a lower extinction (A_V = 6.3) and therefore a substantially lower L_* compared to Brown et al. (2007, 2009). The SMA data are the same as described by Andrews et al. (2009), although the 880 μm continuum map has been synthesized with slightly improved resolution. The 880 μm continuum emission is distributed in a nearly face-on ring, with R_cav = 36 AU. The narrow ring width (R_c is only 15 AU) makes it difficult to determine the viewing geometry, and there is no uncontaminated CO emission or scattered light image available to facilitate an independent measurement. The lack of a clear silicate feature in the mid-infrared leads us to adopt a large s_max (10 μm) for the dust inside the cavity. Despite the different modeling setups, the disk structure parameters inferred here are essentially the same as those determined by Andrews et al. (2009). With (some of) the same SMA data, Brown et al. (2009) also find a comparable cavity size (33 AU). We find positive 3σ residuals for this model, suggesting excess emission to the southeast and near the stellar position. Those discrepancies lead to a poor prediction of the secondary visibility null (see Figure 6), which may indicate that there is a small amount of ∼mm-sized dust emitting from inside the cavity.

UX Tau A. The primary star of a wide multiple system in Taurus (d ≈ 140 pc; see Torres et al. 2009), UX Tau A is the only component with evidence for remnant disk material (e.g., White & Ghez 2001; McCabe et al. 2006). No 880 μm emission above 4.5 mJy (3σ) is detected around either the UX Tau B or C components (in agreement with Jensen & Akeson 2003). The SMA data presented here are the first resolved observations of the UX Tau A disk. This emission is quite compact: we use a small characteristic radius (R_c = 20 AU), presumably produced by the truncation of the outer disk by UX Tau C (located ∼25 to the west). Unlike the compact continuum, we find a bright, resolved CO J = 3–2 disk orbiting UX Tau A, providing the first direct estimates of the disk inclination and P.A. As discussed in Section 5.1, the 880 μm image shows a resolved cavity that is substantially smaller, R_cav = 25 AU, than initial estimates from the SED alone (56 and 71 AU by Espaillat et al. 2007, 2010, respectively). Like LkHα 330, we assume a top-heavy dust size distribution (p = 2.5) in the cavity to avoid overproducing the 10 μm silicate feature.

SR 24 S. Embedded in the L1688 dark cloud in Ophiuchus, SR 24 is a hierarchical triple system with a close binary (02; Simon et al. 1995) located ∼5'' north of a spectral type K2 primary (SR 24 S). The SR 24 N binary exhibits some thermal excess presumably associated with a low-mass circumbinary disk, although it is not detected at millimeter wavelengths (despite evidence for associated CO line emission; Andrews & Williams 2005a). Due to potential confusion with the SR 24 N component, we did not use a Spitzer IRS observation of this system (although one apparently exists). The 880 μm image of the SR 24 S disk shows an elongated ring with a significant brightness asymmetry along the major axis (brighter to the north). We measure a moderate cavity size, R_cav = 29 AU, essentially the same as our previous effort (32 AU; Andrews et al. 2010b). Naturally, the model does not reproduce the observed asymmetry, instead overpredicting the emission to the southwest (although marginally, based on the imaged residuals). In general, the fit quality here is superior to our initial work, thanks to the modification of the inner rim/disk structure. However, without a detailed infrared spectrum the dust properties and structure of the cavity are uncertain. Unfortunately, the asymmetry of the narrow emission ring makes a clear viewing geometry estimate difficult, and the CO emission from SR 24 S is contaminated by local cloud material. Substantial improvements in the structure constraints are feasible with a more complete SED, if the disk inclination can be measured independently (e.g., with a tracer like ¹³CO).

DoAr 44. Sometimes referred to as ROXs 44, this young star is located in the densely populated L1688 dark cloud in Ophiuchus. A preliminary analysis of the DoAr 44 disk structure was presented by Andrews et al. (2009), using the same SMA data. As in that work, we find a moderate cavity size for the DoAr 44 disk, R_cav = 30 AU. However, our new model is in much better agreement with the infrared SED, thanks to the inclusion of an inner rim and cavity wall in the model structure. To match the relatively faint far-infrared emission, the dust in the outer disk is substantially concentrated toward the midplane. Unfortunately, there are no independent constraints on the disk inclination or orientation; the adopted inclination could be uncertain by ∼20°. The small extent of the 880 μm emission suggests that the disk is quite small (the ring is narrow), with R_c = 25 AU. While the western peak of the 880 μm emission appears slightly brighter, the model residuals show that the asymmetry is not significant. Espaillat et al. (2010) modeled the DoAr 44 SED and derived a slightly larger cavity size (36 AU), but within the uncertainties.

LkCa 15. Located in the Taurus star-forming region, this transition disk has been studied extensively since its cavity was first resolved at 1.3 mm by Piétu et al. (2006). Because of the large spatial extent and prominent depression in the 880 μm emission, these new SMA observations make for a spectacular example of a resolved transition disk image. The cavity size derived here, R_cav = 50 AU, is in excellent agreement with the initial measurement (46 AU) by Piétu et al. (2006). The SED modeling by Espaillat et al. (2010) argues for a slightly larger cavity (58 AU), although that would be difficult to reconcile with the well-defined 880 μm visibility null (see Figure 7). The SEDs for the LkCa 15 and DoAr 44 disks are quite similar, with both calling for relatively settled vertical dust distributions. An examination of the modeling results in this case reveals a striking discrepancy with the data: the model predicts far too little millimeter emission inside the cavity (leaving 6σ residuals at the stellar position). The integrated residual flux density is small, ∼10 mJy, but potentially important because it can provide a more direct estimate for the dust mass inside the cavity. For the sake of homogeneity with the rest of this sample, we postpone a more detailed modeling of the LkCa 15 disk cavity until higher resolution images are available.

RX J1615−3255. Perhaps less well known than the other disks in this sample, this system has been kinematically tied to the Lupus association, with d ≈ 185 pc (Makarov 2007). Although it had been identified as a weak-line T Tauri star (Henize 1976; Krautter et al. 1997), infrared observations by Spitzer have only recently claimed RX J1615−3255 as a transition disk candidate (Merín et al. 2010). The new SMA data presented here represent the first spatially resolved images of thermal emission from this disk. A bright, rotating CO J = 3–2 emission structure is also detected, providing a preliminary estimate of the viewing geometry. Although much of the outer disk emission is spatially filtered in the synthesized 880 μm maps shown in Figures 1 and 7, the visibilities clearly indicate that RX J1615−3255 hosts a rather large disk, with R_c = 115 AU (in good agreement with the extent of the CO emission). A particularly low-density cavity is present out to R_cav = 30 AU. In fact, we were forced to empty the inner disk inside a radius of 0.5 AU to avoid producing a near-infrared emission signature above photospheric levels (therefore, no inner rim was employed in this model; see Table 4). To reproduce the comparatively faint infrared excess beyond the dip in the SED, the scale heights were decreased to values consistent with substantial dust settling. The resulting low dust temperatures, coupled with the large disk size, lead to a relatively high estimate for the total disk mass (∼0.13 M_☉, almost 12% of the stellar mass).

GM Aur. This star in the Taurus star-forming region is perhaps the canonical example of a transition disk, with many detailed discussions of the implications of its distinctive infrared SED (e.g., Strom et al. 1989; Skrutskie et al. 1990; Koerner et al. 1993; Chiang & Goldreich 1999; Rice et al. 2003a). Using some of the same SMA data, Hughes et al. (2009a) first resolved the central dust cavity. We include new 880 μm SMA observations here, with short (C) and intermediate (E) antenna spacings that provide a substantially improved Fourier sampling near the visibility null. The cavity size derived here, R_cav = 28 AU, is larger than the preferred Hughes et al. (2009a) value (20 AU) because the new SMA observations provide a more accurate location for the 880 μm visibility null (at a slightly shorter deprojected baseline). However, the alternative disk model discussed by Hughes et al. (2009a) employed a larger cavity (26 AU) that is in good agreement with our model. The long-baseline SMA data are quite noisy in this case; improved sensitivity would help provide a more accurate cavity size estimate. Like the Hughes et al. (2009a) results, our model successfully matches a high resolution 1.3 mm data set from the Plateau de Bure interferometer. The GM Aur disk is also found to be quite large, with a characteristic size R_c = 120 AU (although most of that emission has been filtered out in the images displayed here; see Table 2).

DM Tau. Although the gas disk around this cool, young star in the Taurus clouds has been studied extensively (e.g., Guilloteau & Dutrey 1998; Dartois et al. 2003; Piétu et al. 2007), our new SMA observations permit the first sub-arcsecond resolution examination of the 880 μm dust continuum emission. Based on the shape of its Spitzer IRS spectrum, Calvet et al. (2005) suggested that the DM Tau disk has a relatively small cavity of radius ∼3 AU. Since such a small cavity cannot be resolved on the longest SMA baselines, it was a surprise to measure a clear 880 μm emission ring and visibility null indicative of a much larger dust-depleted cavity, R_cav = 19 AU. Reconciling this large cavity size in the SMA data with the morphology of the IRS spectrum was a challenge, and our current model should be considered preliminary until the latter data are better matched. To account for the absence of an infrared excess shortward of ∼8 μm, we were forced to remove all dust inside a radius of 1 AU (there is no inner rim in this case; see Table 4). Like the cases of GM Aur and LkCa 15, we find that the DM Tau disk is rather large, with R_c = 135 AU, in good agreement with the extended CO emission studied elsewhere.

WSB 60. The dust-depleted cavity in the disk around this very cool star (spectral type M4) in the core of the Ophiuchus star-forming region was first discovered by Andrews et al. (2009). Compared to that initial study, we find a slightly smaller cavity here (R_cav = 15 AU, compared to 20 AU), partly related to our choice to fix the surface density gradient to γ = 1 in the new models. The cavity is just barely resolved with our SMA observations, and it shows no obvious signatures of dust depletion at infrared wavelengths. The latter point drives us to infer the smallest dust depletion factor inside the cavity for this sample, corresponding to only a factor of ∼50× lower densities in the inner disk than the nominal continuous model. Considering the "normal" appearance of the infrared SED, it is natural to speculate that there are potentially many other such disks that have similarly large cavities but will only be identified when high-resolution radio images are available.