3.3. Submillimeter Continuum Slopes

The slope of the submillimeter continuum emission from a circumstellar disk is empirically well-described by a simple power law in frequency: {tex}F_v\propto v^\alpha{/tex}.  If the emission is assumed to be optically thin and roughly isothermal, Equation 2 can be written {tex}F_v\propto B_v(T)\small T\normal_v\propto v^{2+\beta}{/tex} (in the Rayleigh-Jeans limit). However, the submillimeter continuum has a significant contribution from optically thick emission originating in the dense, inner disk which causes a substantial deviation from the {tex}\alpha = 2 + \beta{/tex} relationship inferred for the optically thin case (Beckwith et al. 1990; Beckwith & Sargent 1991). We have combined the data presented here with additional flux densities from the literature (Adams, Emerson, & Fuller 1990; Beckwith et al. 1990; Beckwith & Sargent 1991; Mannings & Emerson 1994; Osterloh & Beckwith 1995; Motte & Andr´e 2001) to determine the values of {tex}\alpha{/tex} given in Table 1. For objects with more than 2 submillimeter flux densities, {tex}\alpha{/tex} was measured from a linear fit in the {tex}log v - log F_v{/tex} plane.  When only 1 or 2 flux densities were available, values or {tex}3-\sigma{/tex} upper limits of {tex}\alpha{/tex} were determined from a simple spectral index.

Figure 6 compares the submillimeter continuum slopes from 450 to 850 µm and 850 µm to 1.3mm. For the full sample, the best-fit slopes are {tex}\alpha = 2.06 \pm 0.02 and 1.93 \pm 0.01{/tex} for the two wavelength regions, respectively. Because the submillimeter continuum emission is generated in the outer disk where temperatures are low, the Rayleigh-Jeans limit criterion is not satisfied (because {tex}hv \sim kT{/tex}) and the continuum slope at shorter wavelengths (nearer to the peak of the thermal emission) could be systematically smaller than at longer wavelengths. However, such an effect is not seen in Figure 6: in fact, the shorter wavelength slope is slightly steeper than at longer wavelengths. This implies that the shape of the submillimeter continuum is not set by the grain temperature distribution alone, but also by the amount of optically thick emission and/or the spectral behavior of the opacity function. Within the uncertainties, it does not significantly matter where in the submillimeter continuum the slope is measured (at least between 350 µm and 1.3mm).  The slightly shallower best-fit slope for the longer wavelength data may simply be noise, or could be caused by a small fraction of the 1.3mm data which are contaminated by non-disk emission: e.g., free-free or gyrosynchrotron radiation from a wind or outflow (e.g., Chiang, Phillips, & Lonsdale 1996). Another possibility is a real concavity to the long-wavelength SEDs: the models of Pollack et al. (1994) predict a steeper opacity function shortward of ~650 µm.

The cumulative distribution of  is shown in Figure 7, constructed using the Kaplan-Meier estimator to incorporate {tex}3-\sigma{/tex} upper limits. The median value of {tex}\alpha{/tex} is 2.0, while only 6% of the sample has {tex}\alpha \geq 3{/tex}, a typical value adopted in the literature due to the (incorrect) assumption of optically thin emission in the Rayleigh-Jeans limit with {tex}\beta =1{/tex}.  A normal distribution of {tex}\alpha{/tex} with a mean of 1.97 ± 0.01 and a variance of 0.22 ± 0.02 provides a decent fit to the data, but there is a slightly enhanced probability of larger {tex}\alpha{/tex}.  Because some of the emission is optically thick, there is no straight-forward means of associating these values of the continuum slope with power law indices of the opacity function ({tex}\beta{/tex}).  One approach is to allow {tex}\beta{/tex} to vary in the disk SED models and fit it as an additional parameter (e.g., Beckwith & Sargent 1991; Mannings & Emerson 1994; Dutrey et al. 1996). However, in many cases this severely limits the number of degrees of freedom in the fits (see Table 2), which already make a number of assumptions. Beckwith & Sargent (1991) provide a means of relating {tex}\alpha{/tex} and {tex}\beta{/tex} analytically from other parameters in the SED model fits which essentially indicate that {tex}\beta\propto\alpha{/tex}, although there is a constant offset (see the Appendix).

Figure 8 shows the measured values of  {tex}\alpha{/tex} as a function of log {tex}M_d{/tex}.  The shaded region on this diagram marks the functional form of {tex}\alpha(M_d){/tex} for {tex}\beta =2{/tex} which is representative of the complete range in the measured radial temperature distributions (see Figure 3). The disk mass values are those for {tex}\beta =1{/tex}, and the error bar shown to the lower left demonstrates the systematic uncertainty introduced by varying {tex}\beta{/tex} 0 and 2.  All else being equal, the disk mass is roughly inversely proportional to the opacity, and since larger values of {tex}\beta{/tex} give lower opacities for a fixed frequency, a larger {tex}\beta{/tex} will also give a larger {tex}M_d{/tex} (assuming the normalization of the opacity function is fixed). Even allowing for such uncertainties in the disk mass and the range of temperature profiles, Figure 8 indicates that many of the disks in the sample have {tex}\beta <2{/tex}  (the curves showing the relationship between {tex}\alpha{/tex} and {tex}M_d{/tex} for lower values of {tex}\beta{/tex} always fall in or below the shaded strip: see the Appendix). This suggests that the large optical depths in the disk do not completely explain the shallow measured continuum slopes, but that there is also an evolutionary change in the typical opacity properties of dust grains from the ISM (where {tex}\beta =2{/tex}) to a disk. This result has also been noted in other studies (Beckwith & Sargent 1991; Mannings & Emerson 1994; Dutrey et al. 1996; Dent, Matthews, & Ward-Thompson 1998) for various different sizes and types of samples. It is tempting to conclude that the apparently diminished values of {tex}\beta{/tex} in these disks are due to the collisional growth of dust grains, a necessary condition in any planet formation model (Mizuno 1980; Pollack et al. 1996).  Models of the process indicate that grain growth can decrease {tex}\beta{/tex} to values as low as zero (e.g., Miyake & Nakagawa 1993; Henning, Michel, & Stognienko 1995). The data in Figure 8 also clearly show that there is no correlation between the disk mass and submillimeter continuum slope.

The coupling of {tex}M_d{/tex}, {tex}\beta{/tex}, and the opacity normalization makes it difficult to definitively associate low values of the submillimeter continuum slope with decreased opacity indices (Beckwith & Sargent 1991). The actual value of {tex}K_v{/tex} is the main uncertainty in the conversion of a submillimeter flux density into a disk mass. Aside from the effect of the grain size distribution (thus the interest in grain growth), both the normalization and {tex}\beta{/tex} depend strongly on the mineralogical composition of the grains (Pollack et al. 1994; Henning & Stognienko 1996) and their physical shapes (e.g., spherical, fractal, etc.; see Wright 1987). Further discussion of these uncertainties is given by Beckwith et al. (1990) and Beckwith, Henning, & Nakagawa (2000). As an example, Wright (1987) indicates that {tex}K_v{/tex} can be roughly an order of magnitude higher for fractal grains compared to spheres at a wavelength of 1mm (see his Figure 6).

Observational and theoretical uncertainties obfuscate the relationship between a measured submillimeter continuum slope and the functional form of the opacity in a disk. Overcoming these difficulties to pursue evidence of the collisional agglomeration of dust grains in the earliest stages of planet formation will at least require better observations, including resolved images at wavelengths extending beyond ~1mm (where the emission is more optically thin; e.g., Testi et al. 2001), flux measurements near the SED turnover point (in the 100 to 300 µm range), and studies of solid-state dust emission features in the mid-infrared (e.g., van Boekel et al. 2004). However, our results leave no doubt that the measured submillimeter continuum slopes for YSOs are significantly less than those noted for molecular clouds, where {tex}\alpha\approx 4{/tex}.