Dust distribution:We employed the standard flared disk model with well-mixed gas and dust, which has been successfully used to explain the observed SEDs of a large sample of young stellar objects and BDs (e.g., Wolf et al. 2003; Sauter et al. 2009; Harvey et al. 2012a; Joergens et al. 2013; Liu et al. 2015). The structure of the dust density is assumed with a Gaussian vertical profile (1)\begin{equation} \rho_{\rm{dust}}=\rho_{0}\left(\frac{R_{*}}{\varpi}\right)^{\alpha}\exp\left[-\frac{1}{2}\left(\frac{z}{h(\varpi)}\right)^2\right], \label{dust_density} \end{equation}ρdust=ρ0R∗ϖαexp−12zh(ϖ)2,and the surface density is described as a power-law function (2)\begin{equation} \Sigma(\varpi)=\Sigma_{0}\left(\frac{R_{*}}{\varpi}\right)^p, \end{equation}Σ(ϖ)=Σ0R∗ϖp,where ϖ is the radial distance from the central star measured in the disk midplane, and h(ϖ) is the scale height of the disk. The disk extends from an inner radius Rin to an outer radius Rout. To the best of our knowledge, among our sample, there are five objects that have been identified as binary systems so far. They are 2M1207 (a~55 AU, Chauvin et al. 2004), J04221332+1934392 (a~7 AU, Todorov et al. 2014), J04414489+2301513 (a~15 AU, Todorov et al. 2014), USD161833 (a~134 AU, Bouy et al. 2006), and USD161939 (a ~ 26 AU, Bouy et al. 2006), where a refers to the separation within the system. The disks around individual components in binary systems are expected to have truncation radii of the order of (0.3 − 0.5)a (Papaloizou & Pringle 1977). We adopted 0.5 a as the disk outer radii for 2M1207, USD161833, and USD161939. For the close pairs (a ≲ 15 AU, J04221332+1934392 and J04414489+2301513), dynamical simulations of star-disk interactions suggest that individual disks are unlikely to survive (e.g., Artymowicz & Lubow 1994). Disk modeling is complicated in those close multiple systems. For simplicity, we assume that the emission is associated with circumbinary disks of 100 AU in size. For other objects, we fix Rout = 100 AU in the modeling because the choice of this parameter value makes essentially no difference to the synthetic SEDs in the simulated wavelength range (Harvey et al. 2012a). The scale height follows the power-law distribution(3)\begin{equation} h(\varpi) = H_{100}\left(\frac{\varpi}{100\,\rm{AU}}\right)^\beta,\\ \end{equation}h(ϖ)=H100ϖ100 AUβ,with the exponent β characterizing the degree of flaring and H100 representing the scale height at a distance of 100 AU from the central star. The indices α, p, and β are codependent through p = α − β. We fix p = 1, which is the typical value found for T Tauri disks in the sub-millimeter (e.g., Isella et al. 2009; Guilloteau et al. 2011), since only spatially resolved data can place constraints on this parameter (e.g., Ricci et al. 2013, 2014).               Dust properties:We assume the dust grains to be a homogeneous mixture of 75% amorphous silicate and 25% carbon with a mean density of ρgrain = 2.5 g cm-3 and the complex refractive indices given by Jäger et al. (1994, 1998), and Dorschner et al. (1995). Porous grains are not considered because the fluxes at wavelengths beyond ~ 2 μm are almost independent of the degree of grain porosity in low-mass disks, as shown by Kirchschlager & Wolf (2014). The grain size distribution is given by the standard power law dn(a) ∝ a-3.5da with minimum and maximum grain sizes amin = 0.1 μm and amax = 100 μm, respectively. The choice of the minimum value for the grain size, amin, ensures that its exact value has a negligible impact on the synthetic SEDs. Since there is no information about the maximum grain sizes of our target disks, as provided, e.g., by the (sub)millimeter spectral index, we adopt amax = 100 μm. The Herschel/PACS far-IR observations are sensitive to dust grains with this assumed sizes. Strong grain growth up to millimeter sizes as detected in some BD disks (e.g., Ricci et al. 2012, 2013, 2014; Broekhoven-Fiene et al. 2014) would remain undetected in our data and could affect the disk mass. Our prescription for the dust properties is identical to those used in Liu et al. (2015).
Citation: Todorov et al. 2014
Citation function: Uses
Citation: Todorov et al. 2014
Citation function: Uses