In the optically thin limit and the Rayleigh-Jeans regime, the dust opacity spectral index β can be approximated using the flux density at two wavelengths. In this appendix, we discuss the error propagation from the observational uncertainty to the deduced β value as done by Chiang et al. (2012). We take F1 and F2 to be the flux density at frequencies ν1 and ν2 and β can be expressed as in Eq. (1): (A.1)We assume that the variables F1 and F2 are independent and that σ1 and σ2 are their standard deviations; propagating the errors we obtain (A.2)Calculating the partial derivative of Eq. (A.1) in both the variables F1 and F2, we obtain Combining these two results with Eq. (A.2) we obtain the uncertainty on β: (A.3)
In order to model the mm emission coming from the disks, the bolometric luminosity Lbol and the effective temperature Teff of the central objects are needed. Once these two parameters are known, then one can obtain the black-body emissions related to the central protostars which are used as the input sources of energy for the computation of the disks dust temperature.
The bolometric luminosities were obtained with Spitzer photometry from the “Cores to Disk” legacy program
We assume that our two Class I YSOs lie along the stellar birthline, namely the locus in the H-R diagram along which young stars first appear as optically visible objects (Stahler 1983). We adopt the birthline constructed by Palla & Stahler (1990) to obtain Teff given Lbol (Evans et al. 2009) as inputs.
We expect Class I YSOs to have two important contributions from the central protostar: the photospheric stellar luminosity, Lphot (the one just considered), and the accretion luminosity, Lacc. At this stage of evolution Lacc can be comparable with Lphot, while using this simple procedure we are assuming Lacc ≪ Lphot, therefore negligible.
This could in principle lead to some differences in the results, in particular because Lacc tends to enhance the temperature and therefore to change the stellar spectrum. Since we do not have any kind of observation to constrain how much the accretion contributes to the total stellar luminosity, we only assume a photospheric contribution. This could lead to underestimating the temperature in the internal regions of the disk-envelope structure, and thus to overestimating the total dust mass. However, we do not expect this effect to be crucial: Natta et al. (2000) show that the average temperatures of disks around stars with very different Teff change only by a factor of two. This means that our errors on the estimation of the dust mass given by the lack of informations on Lacc is only of a factor of two, which is comparable to the uncertainties we have on the dust opacities.
We can anyway check whether our results are reasonable or not, deducing the effective masses of the central objects and comparing it with the masses Ms given by Jørgensen et al. (2009) and Lommen et al. (2008), presented in Table 1.
From Lbol and Teff we obtain Reff using the standard photosphere relation as follows: (B.1)Then, to deduce Meff, we use the radius vs. mass relation obtained by Palla & Stahler (1990) for a spherical protostar accreting at a rate of 10-5 M⊙ yr-1.
All the effective parameters obtained with the birthline method are summarized in Table 3 together with the measured protostars masses already present in the literature. The effective mass of Elias 29 is compatible with the measured ones.
© ESO, 2014