## 3. Data analysis and results## 3.1. Luminosity determinationWe calculated the bolometric luminosity of the observed sources by
integrating the continuum emission between 12
## 3.2. The circumstellar dustIn order to obtain the mass, temperature and of the circumstellar dust, we used the formula (Hildebrand 1983) which, in an optically thin case, expresses the observed flux, , as where D is the cloud distance, is the Planck
function, The optically thin regime can be assumed at submillimetre
wavelengths because dust emission is optically thin up to column
densities of
cm We used a least-squares fit method to determine the best values of
where, for each wavelength i, is the flux value computed from Eq. 1, and are the observed ones and their error (from Table 1). One of the three parameters was changed in step of 10% until a minimum value of was reached. Then, the same process was repeated sequencially for the two other parameters. When a minimum value of is reached with these three parameters, we decrease the step by a factor 1.25 and reiterate the process described here above. At each decrease of the step, the best-fit procedure started with the values that minimised at the end of the precedent cycle. Convergence was reached after a maximum of 20 iterations. We checked that, changing the initial values and the order of the parameters does not change the result. In order to evaluate the uncertainties associated with the
parameters, mainly due to the errors on IR fluxes, two different fits
have been obtained. The first (second) curve passes through the
minimum (maximum) flux at 60 Fig. 1 presents the observed submillimetre spectra together with the best-fit result (solid line) for nine of our ten sources. S31 was not considered because SNS did not obtained IR fluxes for this source. In Fig. 1 we also show the spectral energy distribution of two T Tauri stars (Class II sources; IR and submillimetre measurements are from the IRAS-PSC 1988 and Mannings & Emerson 1994, respectively). There is a clear difference between the SEDs of Class I and Class II objects, probably due to differences in the geometry of the emitting regions (probably disk-dominant in case of T Tauri stars, Mannings & Emerson 1994). Therefore, because the Hildebrand formalism assumes a spherical geometry, no fit is obtained for these two T Tauri stars. Fig. 1 shows that we can fairly well reproduce the continuum
emission observed for our sources between 60 The parameters (,
Columns 2 to 7 present the values, dust
temperature and the total circumstellar mass, along with limiting
values (obtained with the two extreme fits; see above). For each
source, the highest value is associated with the
highest mass and the lowest temperature. Columns 8 and 9 present,
respectively, the values of the optical depth at 60 and
100 where The temperatures are in the range 26 - 40 K as expected for
YSOs, with a continuum emission that peaks around
100 We find in the range 1.1 - 1.9. This range
represents the very extreme limits. The value of
should characterize the dust type that constitutes the envelope.
However, opacity effects can weaken or cancel the link between the
slope and the intrinsic dust properties (see Reipurth et al. 1993).
This point is crucial for young sources, as opacity effects could lead
to a "look-alike" flatter spectrum as observed for the Class 0
source HH24MMS (Ward-Thompson et al. 1995). Using Eq. 3, we
verify that the optical depth is below unity both at 60 and
100 ## 3.3. Validity of the approachIn order to see possible limitations arising from the adopted assumption of a unique dust temperature and density (as implied by the use of Eq. 1) we developed a spherical model assuming a power law distribution of temperature and density where and represent, respectively, the dust temperature and density at the external radius () of the envelope. In Eq. 4we fix the density and temperature exponents The continuum emission of each shell is computed taking into
account its position in the envelope and the relative extinction. We
verify that the envelope remains optically thin at the considered
wavelengths. The total mass is then calculated, integrating the
density over the total volume between the inner and outer radius of
the envelope, fixed as follows: the outer radius
() is fixed at 0.02 pc, which corresponds
to the free-fall radius for a mean sound speed of 0.2 km/s and an
age of 10 The inner radius is computed by fitting the data of each source with Eq. 4and corresponds to the radius where the dust destruction temperature ( =1500 K) is reached. The free parameters of our model are the external temperatures and densities. The results are given in Table 3.
The values of and
are, respectively, in the range 21 to 26 K and 10 The masses obtained by integrating the density profile over the envelope is generally about 20% higher than the ones obtained assuming a unifor density and isothermal envelope. This may be due to the contribution of the higher density central regions of the envelope. On the other hand, the external temperatures estimated with the shell model are on average 6 - 7 K lower than the mean temperatures estimated with the first method. This finding is reasonable because the mean temperature is an average for the envelope temperature. André & Montmerle (1994) expressed the averaged temperature of the envelope as which, for the adopted values of Therefore, we believe that the assumption of a homogeneous isothermal dust sphere adopted in Eq. 1is a viable approach for a reliable estimate of the dust mass and temperature; this result, valid for Class 0 sources (Ward-Thompson et al. 1995), is verified here for Class I sources. ## 3.4. The value ofThe value of , the normalisation factor of
the dust opacity law at 250 According to Reipurth et al. (1993), the value of
is in the range 3 10 To better constraint the value of , we
estimated, in an independent way, a total mass for the L1641 sources:
() = 0.025
(see Walker et al. 1990), using
values given by SNS and Chen & Tokunaga
(1994) and assuming the circumstellar material to be distributed in a
sphere (
Column 2 gives the visual extinction value. The third column gives the mass, assuming spherical geometry. Column 4 gives the mass ranges determined using Eq. 1. A general agreement is found between the two mass determinations, supporting the chosen value. This method does not allow us to give a precise value for but indicates that a high value (leading to relatively low masses; see Eq. 1) is needed to obtain an agreement between the two independent mass estimates. © European Southern Observatory (ESO) 1997 Online publication: April 28, 1998 |