We present a high spatial resolution submillimeter continuum survey of 24 circumstellar disks in the Taurus-Auriga and Ophiuchus-Scorpius star formation regions using the SMA. In the context of a simple model, we use broadband spectral energy distributions and submillimeter visibilities to derive constraints on some basic parameters that describe the structure of these disks. For the typical disk in the sample we infer a radial surface density distribution Σ_r ∝ r~(-p) with a median p ≈ 0.5, although consideration of the systematic effects of some of our assumptions suggest that steeper distributions with p ≈ 1 are more reasonable. The distribution of the outer radii of these disks shows a distinct peak at R_d ≈ 200 AU, with only a few cases where the disk emission is completely unresolved. Based on these disk structure measurements, the mass accretion rates, and the typical spectral and spatial distributions of submillimeter emission, we show that the observations are in good agreement with similarity solutions for steady accretion disks that have a viscosity parameter α ≈ 0.01. Weprovide new estimates of the spectral dependence of the disk opacity κ_v ∝ v~β with a mean β ≈ 1.0, corrected for optically thick emission. This typical value of β is consistent with model predictions for the collisional growth of solids to millimeter-size scales in the outer disk. Although direct constraints on planet formation in these disks are not currently available, the extrapolated density distributions inferred here are substantially shallower than those calculated based on the solar system or extrasolar planets and typically used in planet formation models. It is possible that we are substantially underestimating disk densities due to an incomplete submillimeter opacity prescription.
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