A. Comments on Disk Models

A simple disk structure model was presented in §3.1 and used to generate SEDs of thermal dust emission and derive a relationship between the submillimeter flux density and disk mass. A lack of SED data severely limits the number of degrees of freedom in this modeling, and therefore a number of parameters were fixed: inclination, inner and outer radii, and the power law indices of the radial surface density and frequency spectrum of the opacity. Here we determine the effects that reasonable alternative values for these fixed parameters would have on the {tex}F_v-M_d{/tex} relationship.  Readers would also benefit from the mathematical formalism discussed in detail by Beckwith et al. (1990). The analysis is done comparatively, relative to a fiducial disk model based on the median values discussed in the text: {tex}i=0^\circ{/tex},{tex}p=1.5{/tex},{tex}T_1=150 K{/tex}, q=0.6, {tex}r_\circ=0.01 AU{/tex}, {tex}R_d=100 AU{/tex}, and {tex}\beta = 1{/tex}.   A set of 850 µm flux densities was computed using Equation 2 for a grid of disk masses with this fiducial parameter set excepting one of the previously fixed variables. The flux density disk mass relationships for various values of p, q, {tex}r_\circ{/tex}, {tex}R_d{/tex}, {tex}T_1{/tex}, and {tex}\beta{/tex} are shown in Figure 17. It should be noted that these plots are only intended to illustrate the effects of changing a single parameter in these models. In reality, the parameters are usually somewhat coupled, and therefore changes in one parameter affect others: such coupled effects are not considered in this simple comparative analysis.

Larger values of q, {tex}r_\circ{/tex}, {tex}R_d{/tex}, and {tex}\beta{/tex} all result in a larger disk mass for a given flux density. The middle panels in Figure 17 indicate that the disk boundaries play only a minor role in setting the {tex}F_v - M_d{/tex} relationship. The roles of the opacity index and temperature normalization are roughly those expected from the optically thin assumption given in Equation 1, where {tex}F_v \propto K_vM_dT^{-1}{/tex}.  For low-mass disks (i.e., {tex}M_d\frac{\lt}{\sim} 10^{-2}M_\odot{/tex}), the shape of the radial surface density profile can have a significant effect on the submillimeter flux density because most of that emission is optically thin. For the same reason, the radial temperature behavior dictated by q has an opposite effect and impacts the flux densities for the more massive, optically thick disks. The largest relative deviations in the flux density disk mass relationship are set by the parameters which describe the radial temperature profile; {tex}T_1{/tex} and q.  Fortunately, the parameters of the temperature profile can be reliably determined for individual disks using observations in the mid- and far-infrared. Sensitive observations with the Spitzer Space Telescope will essentially ensure that the opacity is the single dominant uncertainty in the determination of a disk mass from submillimeter observations.

In Figure 18, the same procedure as above was used to examine the effects of different fixed parameter choices on the relationship between the submillimeter continuum slope and the disk mass (note that here we have fixed {tex}\beta = 1{/tex} as a reference value). Once again, the disk boundaries play a negligible role in the relationship. The radial surface density index has only a small impact on the {tex} \alpha-M_d{/tex} relationship in general, although a constant surface density disk (p = 0) gives a roughly constant submillimeter continuum slope (and lower than for larger values of p) for low-mass disks. The parameters of the temperature profile again show the largest deviations from the fiducial {tex} \alpha-M_d{/tex} relationship. The effects of a changing {tex}\beta{/tex} in this relationship are also large. See Beckwith & Sargent (1991) and Mannings & Emerson (1994) for a mathematical description of what is shownin these plots.

An alternative view of the relationship between the observed continuum slope ({tex}\alpha{/tex}) and the power law index of the frequency behavior of the opacity ({tex}\beta{/tex}) can be obtained directly from these models. Figure 19 shows computed continuum slopes as a function of for various disk masses. The relationship is essentially linear for {tex}M_d\frac{\lt}{\sim} 0.1 M_\odot{/tex} with slopes of nearly unity independent of disk mass, but significantly different intercept values. All of the curves fall below the nominal {tex}\beta=\alpha-2{/tex} line which is representative of optically thin emission in the Rayleigh-Jeans limit. This effect is due to the failure of the Rayleigh-Jeans limit and the fraction of optically thick submillimeter emission, which can be fairly high at the shortest wavelengths (Beckwith et al. 1990; Beckwith & Sargent 1991; Mannings & Emerson 1994). This synthetic grid illustrates the conclusion in §3.3 that the roughly gaussian distribution of  {tex}\alpha{/tex} centered around ~2 suggests that the opacity index is likely between ~0.5 and 1.5, and certainly less than the ISM value of 2 unless the disk masses are severely underestimated.

Figure 20 was generated from the fiducial model (the top panel also assumes the median {tex}M_d=0.005 M_\odot{/tex}) described above to illustrate that the assumption of optically thin emission in the submillimeter continuum is not always valid (as pointed out by Beckwith et al. 1990). The disk becomes optically thin at the radius, {tex}r_1{/tex}, where {tex}T_v=K_v\Sigma_{r1}=1{/tex}.  That criterion can be solved to determine {tex}r_1{/tex} using Equation 4 and the relationship between {tex}\Sigma_\circ{/tex} and {tex}M_d{/tex}, giving

{tex}\large r_1=(K_v\Sigma_\circ)^{1/p}r_\circ=\Large[\large\frac{(2-p)K_vM_d}{2\pi(R^{2-p}_d-r^{2-p}_\circ)}\Large]\large^{1/p}{/tex}, (A1)

when {tex}p\neq 2{/tex}; inserting the parameters fixed in §3.1 gives {tex}r_1\approx 184(M_d/M_\odot)^{2/3} AU{/tex}.  The fraction of the submillimeter flux density from optically thick emission, {tex}\Delta{/tex} (defined as the ratio of the flux density from {tex}r < r_1{/tex} to the total flux density: see Beckwith et al. 1990) increases exponentially until {tex}\Delta = 1{/tex} around 0.4 to 1 Jy, depending on the wavelength. Even faint submillimeter sources have ~15% of their flux densities generated in the innermost (radially), optically thick regions of the disk. In terms of the fraction of the disk mass which gives rise to optically thick emission (the ratio of the integrated surface density from {tex}r_\circ{/tex} to {tex}r_1{/tex} to the total disk mass), a more gradual trend with disk mass is present. Roughly 25% of the mass in a MMSN disk with the fiducial parameter set contributes optically thick emission in the submillimeter. Comparison of the right panel in Figure 20 with the information in Figures 4 and 8 demonstrate the effects these relatively high optical depths have on estimating {tex}M_d{/tex} and {tex}\alpha{/tex} (and subsequently {tex}\beta{/tex}) from observations.