3.2. Disk Masses

Submillimeter continuum observations provide measurements of disk masses. However, the simplistic conversion of a flux density into a mass via Equation 1 masks some important complications. For example, {tex}M_d{/tex} could be uncertain to a factor of <2 due to its roughly linear relationship with {tex}T_c{/tex}. More fundamentally, the relationship between {tex}F_v{/tex} and {tex}M_d{/tex} is nonlinear due to the significant fraction of the submillimeter emission which is optically thick (e.g., Beckwith et al. 1990). By assuming optically thin emission and using Equation 1, {tex}M_d{/tex} could be underestimated (particularly for objects with larger flux densities). To avoid these problems and fit the SEDs with the model described by Equation 2, mid- and far-infrared flux densities were taken from the IRAS Point Source Catalog and the compilations of Weaver & Jones (1992) and Kenyon & Hartmann (1995). Using data at shorter wavelengths ({tex}\lambda {<\over\sim}{/tex} 5 µm) runs the risk of contamination from an extincted photosphere, and therefore introduces more parameters into the problem (e.g., effective temperature, extinction, stellar radius). The submillimeter data presented here were supplemented whenever possible with flux densities from the literature (Adams, Emerson, & Fuller 1990; Beckwith et al. 1990; Beckwith & Sargent 1991; Mannings & Emerson 1994; Osterloh & Beckwith 1995; Motte & André 2001). We adopted absolute flux calibration uncertainties of 20% in the infrared and 25% ({tex}\lambda\leq{/tex} 800 µm) or 20% ({tex}\lambda\gt{/tex}  800 µm) in the submillimeter. Systematic and statistical errors were combined for each individual flux density measurement.

The disk mass and opacity index,{tex}\beta{/tex} , are strongly coupled parameters, making it difficult to independently infer their values (see Beckwith & Sargent 1991). Observations and models of interstellar grains in the molecular ISM, where the material is still diffuse enough to safely assume optically thin thermal emission, indicate that {tex}\beta\approx{/tex} 2 (Erickson et al. 1981; Schwartz 1982; Draine & Lee 1984). However, different mineralogies or grain size distributions in a disk could decrease the index down to {tex}\beta\sim{/tex} 0 (e.g., Pollack et al. 1994). Because of the uncertainties in independently measuring and {tex}M_d{/tex}, we modeled individual SEDs for various values of {tex}\beta{/tex}(between 0 and 2), as well as the typical compromise value for disks,{tex}\beta{/tex} = 1. Values of {tex}T_1{/tex}, q, and {tex}\Sigma_\circ{/tex}. (note that for this model {tex}M_d/M_\odot\approx 2\times10^{-35}\Sigma_\circ /g cm^{-2}{/tex}) for 44 objects in the sample were determined by fitting the SEDs to Equation 2 and minimizing the {tex}X^2{/tex} statistic. Table 2 gives the results of these fits for {tex}\beta{/tex}=1, including the reduced {tex}X^2{/tex} values ({tex}X^{\sim 2}_v{/tex}), degrees of freedom in the fit ({tex}v{/tex}), and references for the infrared and submillimeter SED data from the literature. Much more sophisticated disk models (Men’shchikov, Henning, & Fischer 1999; Chiang et al. 2001; Semenov et al. 2005) predict disk masses for a few of the same sources which are within a factor of 2-3 of those presented here.  Figure 3 shows the distributions of the best-fit values of q and {tex}T_1{/tex}.  The mean values of q and {tex}T_1{/tex} are 0.56±0.08 and 178±85K, respectively (quoted errors are standard deviations of the distributions).  We define the "median disk model" to have the above set of fiducial parameters and the median values q = 0.58 and {tex}T_1{/tex} = 148K as well as {tex}\beta{/tex}=1.  Due to the high optical depths at infrared wavelengths compared to the submillimeter, the parameters of the temperature profile, {tex}T_1{/tex} and q, are often not strongly affected by changes in {tex}\beta{/tex}.

The SEDs of most of the YSOs in the survey sample were not fitted as described above because they either lack data (i.e., there were too few degrees of freedom), are undetected in the submillimeter, or have SEDs which indicate such a simple model is insufficient. However, the results of the SED fitting can be used to determine an empirical conversion between a submillimeter flux density and a disk mass. In Figure 4 we show the relationship between the 850 µm flux densities and the best-fit values of {tex}M_d{/tex} (for {tex}\beta{/tex}=1) from the SED fitting. This relationship is well described by a simple power law,

{tex}\Large\frac{M_d}{M_\odot} = (5\pm 2)\times 10^{-5}\LARGE[\Large\frac{F_v(850\mu m)}{mJy}\LARGE]\Large^{0.96\pm 0.03}{/tex}      (5)

which is shown as a solid line in Figure 4. A fit of the same data to Equation 1 (assuming the same opacity function given above, where {tex}K_v = 0.035 cm^2 g^{-1}{/tex} at 850 µm) is shown as a dashed line, and gives a best-fit characteristic temperature {tex}T_c = 20K{/tex}.  For the median disk model, this value of {tex}T_c{/tex} occurs at a disk radius of approximately 30AU. Also shown are the relationships between {tex}F_v{/tex} and {tex}M_d{/tex} for the mean and median disk models. Disk mass values and upper limits for the objects which were not fitted with these models were computed from Equation 5. For sources without 850 µm measurements which could not be fitted with a disk model, a similar analysis as above was used to derive values of {tex}M_d{/tex} from the 1.3mm flux density:  {tex}M_d/M_\odot\approx 10^{-6}(F_v/mJy)^{1.5}{/tex}.  Disk masses (or {tex}3-\sigma{/tex} upper limits) are included in Table 1 for all of the sources in the survey sample. Those masses, which use {tex}\beta =1{/tex}, will be adopted throughout this paper, unless specifically mentioned otherwise.  Our fitting results show that the systematic errors in the disk mass due to the a priori unknown value of {tex}\beta{/tex} are ±0.5 dex on average, or a factor of 3, for a reasonable range of {tex}\beta{/tex} (0 to 2).  The {tex}M_d{/tex} values inferred for Class I objects should be considered only as upper limits on the disk mass, as there is likely a flux contribution from the inner envelope.

Figure 5 shows the cumulative distributions of the 850 µm flux densities and disk masses.  This figure shows the distributions of the full sample and a subsample consisting of only those sources which have a {tex}\geq 3-\sigma{/tex} detection at a submillimeter wavelength. The ordinates in these plots are defined as the probability of an object having a value equal to or greater than the abscissae.  In both figures, the Kaplan-Meier product limit estimator is used to construct the cumulative distributions for the full sample.³  This method allows the incorporation of the {tex}3-\sigma{/tex} upper limits of the {tex}F_v{/tex} and {tex}M_d{/tex} values in the full sample. The computations of probabilities and their errors were conducted with the ASURV Rev. 1.2 software package (LaValley, Isobe, & Feigelson 1990), following the formalism introduced by Feigelson & Nelson (1985). A significant caveat with these cumulative distributions is that there is no means to account for the uncertainties in the values of {tex}F_v{/tex} and {tex}M_d{/tex}.

Based on the detections subsample distribution in Figure 5 (not incorporating upper limits), we estimate the completeness limit of the survey to be roughly 10mJy. A log-normal distribution of {tex}F_v{/tex} with mean 1.20 ± 0.02 (16mJy) and variance 1.08 ± 0.06 dex provides a good fit to the data for the full sample, whereas a mean of 1.93 ± 0.01 (85mJy) and a variance of 0.41 ± 0.02 dex are appropriate for the detections subsample. The distribution of the detections subsample in Figure 5 indicates that 37% of the YSOs have {tex}M_d \geq 0.01M_\odot{/tex}, roughly the total mass of the MMSN (Weidenschilling 1977). Approximately 79% of the same subsample have disks with masses greater than that of Jupiter. As would be expected from the relationship between {tex}F_v{/tex} and {tex}M_d{/tex} discussed above, the disk masses are also log-normally distributed: the full sample with mean 3.00 ± 0.02 ({tex}10^{-3} M_\odot{/tex}) and variance 1.31 ± 0.06 dex, and the detections subsample with mean 2.31 ± 0.01 ({tex}5\times 10^{-3}M_\odot{/tex}) and variance 0.50 ± 0.02 dex.