3. Results

3.1. A Simple Disk Model

A model of the submillimeter continuum emission is needed to extract physical information (e.g., disk masses) from the data. In order to incorporate some non-negligible optical depth and a radial temperature distribution, the simplistic methods outlined in §2 (e.g., Equation 1) are passed over in favor of one that fits the disk SED with a power-law structural model (see Adams, Lada, & Shu 1987; Beckwith et al. 1990). In this scheme, the SED (from the mid-infrared through the submillimeter) is assumed to be generated from thermal reprocessing of starlight by a geometrically thin dust disk, with the flux density given by

{tex}\Large F_v = \frac{cos i}{d^2}\int^{R_d}_{r_\circ}B_v(T_r)(1-e^{-T_{v,r}sec i})2\pi rdr{/tex} (2)

where {tex}i{/tex} is the inclination angle, {tex}r_\circ{/tex} the inner radius, {tex}R_d{/tex} the outer radius, {tex}B_v(T_r){/tex} the Planck function at a radius-dependent temperature, and {tex}T_{v,r}{/tex} the optical depth of the disk material.¹ In essence, the flux density is computed by summing the thermal emission from a continuous set of dust annuli weighted by the radiative transfer properties of the material. The radial temperature distribution is taken to be a power law

{tex}\Large T_r=T_1(\frac{r}{1 AU})^{-q}{/tex}  (3)

where {tex}T_1{/tex} is the temperature at r = 1 AU.  The optical depth is the product of the disk opacity, {tex}K_v{/tex}, and the radial surface density profile, {tex}\Sigma_r{/tex}, which is also taken to be a power law:

{tex}\Large\Sigma_r = \Sigma_\circ(\frac{r}{r_\circ})^{-p}{/tex} (4)

We assume that the opacity is a power law in frequency with index {tex}\beta{/tex} and a normalization of 0.1 cm² g^-1 at 1000GHz (Beckwith et al. 1990). This value assumes a 100:1 mass ratio between gas and dust.

Because a given disk typically has relatively few SED datapoints, fitting the SED with the model described above requires that some of the remaining 8 parameters ({tex}i{/tex},{tex}r_\circ{/tex},{tex}R_d{/tex},{tex}\Sigma_\circ{/tex},p,{tex}T_1{/tex},q,{tex}\beta{/tex}) be fixed.  Fortunately, the precise values of {tex}i{/tex},{tex}r_\circ{/tex}, and {tex}R_d{/tex} do not significantly affect the determination of interesting physical parameters as long as they lie in a realistic range. A fiducial set of fixed parameters is adopted here: {tex}i=0^\circ{/tex}, {tex}r_\circ = 0.01 AU{/tex}, {tex}R_d= 100 AU{/tex} and p = 1.5.²  The inner and outer disk radii are typical values based roughly on the dust sublimation temperature (e.g., Dullemond, Dominik, & Natta 2001; Muzerolle et al. 2003) and direct disk size measurements (e.g., Dutrey et al. 1996; Kitamura et al. 2002; Akeson et al. 2005). The surface density index, p, is the most difficult parameter to constrain observationally. The value selected here is obtained when the compositions of the planets in the solar system are augmented to cosmic abundances and smeared out into annuli: the Minimum Mass Solar Nebula (MMSN; Weidenschilling 1977). The inclination value is set merely as a computational convenience. The remaining parameters ({tex}\Sigma_\circ{/tex},{tex}T_1{/tex},q,{tex}\beta{/tex}) must be determined from the data.