## 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 *F*_{1} and
*F*_{2} to be the flux density at frequencies
*ν*_{1} and *ν*_{2} and
*β* can be
expressed as in Eq. (1): (A.1)We assume that
the variables *F*_{1} and *F*_{2} are
independent and that *σ*_{1} and *σ*_{2} are their
standard deviations; propagating the errors we obtain (A.2)Calculating the
partial derivative of Eq. (A.1) in both
the variables *F*_{1} and *F*_{2}, 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*