Online material

Appendix A: Error estimate of the approximate dust opacity spectral index

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)

Appendix B: Estimate of the protostars’ parameters

In order to model the mm emission coming from the disks, the bolometric luminosity L_bol and the effective temperature T_eff 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

(Evans et al. 2009) and are shown in Table 3. On the other hand, in order to obtain T_eff some assumptions are needed.

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 T_eff given L_bol (Evans et al. 2009) as inputs.

We expect Class I YSOs to have two important contributions from the central protostar: the photospheric stellar luminosity, L_phot (the one just considered), and the accretion luminosity, L_acc. At this stage of evolution L_acc can be comparable with L_phot, while using this simple procedure we are assuming L_acc ≪ L_phot, therefore negligible.

This could in principle lead to some differences in the results, in particular because L_acc 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 T_eff 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 L_acc 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 M_s given by Jørgensen et al. (2009) and Lommen et al. (2008), presented in Table 1.

From L_bol and T_eff we obtain R_eff using the standard photosphere relation as follows: (B.1)Then, to deduce M_eff, 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