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Astron. Astrophys. 325, 685-692 (1997)

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3. Data analysis and results

3.1. Luminosity determination

We calculated the bolometric luminosity of the observed sources by integrating the continuum emission between 12 µm and 1.3 mm using the IRAS fluxes from SNS and our submillimetre measurements. Emission outside the 12 µm - 1.3 mm range does not significantly contribute to the total luminosity (see also Reipurth et al. 1993) because, for Class I sources, the spectral energy distribution is sharply peaked around 100 µm (see Fig. 1 and Lada 1991).
For the IRAS fluxes we adopted the SNS values because they used one-dimensional co-adds and determined more accurate fluxes than those of the IRAS Point Source Catalog (1988; PSC). Moreover, the PSC gives upper limits for six of the sources. Chen et al. (1993b) also determine IR fluxes using co-added images and aperture photometry. They determine fluxes with very small uncertainties (4% at 100 µm for four sources of our sample). In spite of this, we have discarded these fluxes because we find that they do not fit the overall observed continuum. Luminosity errors given in Table 1 take into account all the errors in flux determination between 12 µm and 1.3 mm. The results of our calculations are given in column 4 of Table 1. Our luminosity determinations generally agree, to within the errors, with those of Chen et al. (1993b).

[FIGURE] Fig. 1. Observed continuum emission from 60 µm to 1.3 mm for 9 sources of our sample. The solid line represents the best-fit result. The continuum emission of two T Tauri stars is also shown for comparisons

3.2. The circumstellar dust

In order to obtain the mass, temperature and [FORMULA] of the circumstellar dust, we used the formula (Hildebrand 1983) which, in an optically thin case, expresses the observed flux, [FORMULA], as

[EQUATION]

where D is the cloud distance, [FORMULA] is the Planck function, M is the total mass, assuming a gas-to-dust ratio of 100, and [FORMULA] = [FORMULA] =0.1 cm2 /g is the dust opacity (Hildebrand 1983). In Sect. 3.4 we discuss the value of [FORMULA].

The optically thin regime can be assumed at submillimetre wavelengths because dust emission is optically thin up to column densities of [FORMULA]   [FORMULA]  cm-2 (Hildebrand 1983). We have checked that the estimated values for dust opacity verify this assumption.

We used a least-squares fit method to determine the best values of T, [FORMULA] and M to fit, with Eq.  1, the far IR and submillimetre data. The use of the far IR fluxes at 60 and 100 µm in this procedure is crucial, as it defines the emission peak and then greatly influences the obtained dust temperature. For all sources, started from the same initial set of parameters (T =20 K, [FORMULA] =1, M =0.1  [FORMULA]) we computed the flux [FORMULA] and the quantity

[EQUATION]

where, for each wavelength i, [FORMULA] is the flux value computed from Eq.  1, [FORMULA] and [FORMULA] 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 [FORMULA] was reached. Then, the same process was repeated sequencially for the two other parameters. When a minimum value of [FORMULA] 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 [FORMULA] 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 µm and the maximum (minimum) flux at 100 µm, both compatible with a 1 [FORMULA] error in these fluxes. Therefore, two limiting values are given for each parameter together with the best-fit one.

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 µm and 1.3 mm using a single temperature greybody. The only noticeable exception is for S11 whose fit is too low to agree with the observed 100 µm flux. We suspect that this flux is overestimated because, taking the mean value of log([FORMULA] / [FORMULA])= -0.4 derived for Class I sources in L1641 (Chen et al. 1993b), we find for S11 an "expected" 100 µm flux of 60 Jy that agrees with our fit.

The parameters ([FORMULA], M and T) obtained with these assumptions are presented in Table 2 together with the values of the optical depth at 60 and 100 µm .


[TABLE]

Table 2. Obtained parameters (isothermal and isodense approximation)


Columns 2 to 7 present the [FORMULA] 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 [FORMULA] 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 µm obtained using the relation between the optical depth and the dust opacity law

[EQUATION]

where N is the dust column density (the mass divided by the beam area at the cloud distance) and [FORMULA] is the dust opacity.

The temperatures are in the range 26 - 40 K as expected for YSOs, with a continuum emission that peaks around 100 µm (see also Reipurth et al. 1993; André & Montmerle 1994; Henning et al. 1994). This temperature can be interpreted as the mean dust temperature (Walker et al. 1990), mass-averaged (Henning 1983) over the envelope. The mass values are in the range 0.5 -  4  [FORMULA]. Hence we are dealing with cold objects of relatively low mass. We note that the masses are function of the value chosen for [FORMULA] ; a higher [FORMULA] would lead to a smaller mass, but [FORMULA] is not well known (see Sect. 3.4).

We find [FORMULA] in the range 1.1 - 1.9. This range represents the very extreme limits. The value of [FORMULA] 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 µm for all sources (see Table 2). Therefore opacity effects should not influence the parameter determinations.

3.3. Validity of the approach

In 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

[EQUATION]

where [FORMULA] and [FORMULA] represent, respectively, the dust temperature and density at the external radius ([FORMULA]) of the envelope.

In Eq.  4we fix the density and temperature exponents p and q to 1.5 and 0.3 respectively, as expected in case of free-fall collapse (Adams 1991). We also fix the exponent [FORMULA] of the extinction law to the values obtained using Eq.  1(see Table 2). This assumption should be correct as the main part of the far IR and submillimetre emission comes from the external shells where the temperature variation is weak. Therefore, possible variation of [FORMULA] due to temperature variation in the envelope should also be weak.

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 ([FORMULA]) 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 105 years, as expected from the star formation models. This age is in agreement with the upper limit of 3 106 years and the outflow dynamical time which is about 5 104 years for these objects (SNS). An external radius of 0.02 pc at a distance of 480 pc correspond to 10 [FORMULA] , nearly half of the JCMT beam. Stopping the integration at this radius implies that the residual emission detected in the beam is subtracted by the chopping technique.

The inner radius is computed by fitting the data of each source with Eq. 4and corresponds to the radius where the dust destruction temperature ([FORMULA] =1500 K) is reached.

The free parameters of our model are the external temperatures and densities. The results are given in Table 3.


[TABLE]

Table 3. Spherical model results


The values of [FORMULA] and [FORMULA] are, respectively, in the range 21 to 26 K and 105 to 106  cm-3, in good agreement with the values obtained for the L1641 cloud (15 to 30 K and 102 to 107  cm-3 ; Sakamoto et al. 1994).

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 [FORMULA] 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

[EQUATION]

which, for the adopted values of p and q (1.5 and 0.3 respectively), gives  [FORMULA]. This brings the external temperatures determined with the shell model in excellent agreement with the average temperatures obtained using Eq.  1.

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 of [FORMULA]

The value of [FORMULA], the normalisation factor of the dust opacity law at 250 µm, is poorly known, mainly due to uncertainties in grains structure (Colangeli et al. 1995; Ossenkopf & Henning 1994). Laboratory measurements and theoretical calculations agree better if a CDE (Continuous Distribution of Ellipsoids) model is used (Colangeli et al. 1995).

According to Reipurth et al. (1993), the value of [FORMULA] is in the range 3 10-3 to 2 10-2  cm2 /g. Using our [FORMULA] values in the range 1.1 - 1.9 and taking [FORMULA] =0.1 cm2 /g, we find [FORMULA] in the range 4.4 10-3 to 1.7 10-2 cm2 /g, in good agreement with the above values (see also Ossenkopf & Henning 1994).

To better constraint the value of [FORMULA], we estimated, in an independent way, a total mass for the L1641 sources: [FORMULA]  ([FORMULA]) = 0.025  [FORMULA] (see Walker et al. 1990), using [FORMULA] values given by SNS and Chen & Tokunaga (1994) and assuming the circumstellar material to be distributed in a sphere (R = [FORMULA] =10[FORMULA]). Results are presented in Table 4.


[TABLE]

Table 4. Independent mass estimate


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 [FORMULA] value. This method does not allow us to give a precise value for [FORMULA] 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.

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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