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Astron. Astrophys. 336, 565-586 (1998)

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4. Radiative transfer calculations

In this chapter, we consider four 1.3 mm continuum sources for a more detailed discussion of their spectral energy distribution (SEDs). Two sources (AB Aur (Lada class II), V 1331 Cyg (class I)) belong to the group of point-like objects, the other two sources (VY Mon (class I), LkH[FORMULA] 234 (class II)) to the group of objects with a core/envelope structure. We consider the point-like objects as very good disk candidates. However, we should not forget that the appearance of an object as an unresolved or core/envelope source depends not only on the actual physical structure, but also on the distance and the size of the telescope beam. The supposition of the presence of disks associated with unresolved sources was recently confirmed in the case of the two Herbig Ae stars AB Aur and HD 163296 by millimetre interferometry (Mannings & Sargent 1997). The question is if all of our 6 point-like objects have disks or not. Only with high-resolution interferometric observations of these objects, we will be able to answer this question. However, we can apply radiative transfer calculations to test if spherically symmetric models are sufficient to model the broad-band SEDs of these stars. Already here we should stress that a successful fit with such models would not imply the lack of a small-scale disk (see Henning et al. 1994).

For the modelling of both, the point-like and the core/envelope sources, we applied a spherically symmetric radiative transfer code developed by Egan et al. (CSDUST3, 1988). This program includes scattering and the properties of different dust populations. The motivation for using the spherically symmetric model comes from the successful application of such a code for a number of HAEBE stars (HD 100546, CoD-42o11721) by Henning et al. (1994). Only in the case of HD 163296 a disk model was necessary to get a good fit. Our millimetre maps show that this star has a distinct point-like millimetre emission, whereas HD 100546 and CoD-42o11721 show extended millimetre emission.

In the model, the star is located in the centre of a cavity free of any absorbing material surrounded by a spherically symmetric dust shell extending from the inner radius [FORMULA] to an outer radius [FORMULA]. The density distribution can be given by two power laws ([FORMULA][FORMULA]) for the inner and outer parts of the shell, separated by a spherical surface at the radius [FORMULA]. The dust temperatures range between the lowest values adopted by the silicate grains at the upper size limit and the highest values adopted by the carbon grains at the lower size limit.

We performed radiative transfer calculations with a variety of dust compositions (see Henning et al. 1994), including silicates, graphite, amorphous carbon, and ice mantles of different thickness. Testing most of these dust models, we found that the "fluffy grain model" provides the best fits to the SEDs. This model consists of fluffy particles with a carbon-to-silicate volume ratio of 0.89:1 and a vacuum volume fraction of 70%. The lower and upper limits of the size distribution ([FORMULA] a-3.7) are 15 and 240 nm, respectively. The success of this grain model comes mainly from the higher values of the opacity in the FIR/submm region compared to a "compact" grain model. The total mass absorption coefficient at 1.3 mm is 4.5 cm2g-1 (per gram dust) which is a factor of 4.5 higher than used in Sect. 3.3 for estimating the dust masses of the cores.

In Fig. 3a and b and 4a and b, we present the calculated SEDs together with the observations for the point-like and core/envelope sources, respectively. Table 6 contains the model parameters (stellar temperature and luminosity, radii [FORMULA], [FORMULA] and [FORMULA], hydrogen number densities and exponents of density distributions) and the calculated values for particle temperatures, gas masses, and visual extinctions together with the distances of the objects.

[FIGURE] Fig. 3a and b. Radiative transfer models of point-like sources: a AB Aur, b V 1331 Cyg. The observations are marked by triangles. Calculated flux densities/beam are given by asterisks. The solid lines refer to the total flux densities.

[FIGURE] Fig. 4a and b. Radiative transfer models of core/envelope sources: a VY Mon, b LkH[FORMULA] 234 (the IRAS data points are marked because of the discrepancy between the IRAS and stellar positions). The observations are shown by triangles. Calculated flux densities/beam are given by asterisks. The solid lines refer to the total flux densities.


[TABLE]

Table 6. Model parameters of the selected HAEBE and FU Ori stars.


For the construction of the SEDs, we used the total 1.3 mm continuum flux densities derived from the maps. In the case of the core/envelope sources, we also give the values for the cores (see Fig. 4a and b).

The optical and near/mid-infrared parts of the SEDs of AB Aur come from data given by Hillenbrand et al. (1992) and Berrilli et al. (1992). In the case of AB Aur, the mid-infrared region of the SED was complemented by a low-resolution IRAS spectrum where we only show 20 rebinned data points instead of the original 82 values. Between 0.35 mm and 1.3 mm various observations were taken from Mannings (1994).

The spectral energy distribution of V 1331 Cyg was constructed from data given by Chavarría-K. (opt./IR, 1981), IRAS data, and submm/mm measurements performed by Weintraub et al. (1991).

In the case of VY Mon all flux densities used for the construction of the SED are taken from Casey & Harper (1990). This paper includes KAO, IRTF, and IRAS flux densities.

The construction of the SED of LkH[FORMULA] 234 was more complicated. Whereas data with comparably high spatial resolution are available for the H and K bands (Weintraub et al. 1994, 5.6" beam size) and for different narrow-band filters between 8.3 and 17 µm (Cabrit et al. 1997, 2.5" beam size), and additional optical and near/mid-infrared data can be taken from Hillenbrand et al. (1992), problems appeared at far-infrared wavelengths. According to Fig. 2, the distance between the stellar position and the centre of the IRAS error ellipse is approximately 35". Despite the assignment of the IRAS source to the HAEBE star often found in the literature, the IRAS data points have to be considered with caution. In Fig. 4a and b we plotted the IRAS data points because of their good agreement with the shape of the calculated SED and the 53 µm KAO observation of Berrilli et al. (1992). In addition, a 1.3 mm observation of Altenhoff et al. (1994) is available.

In the case of all four sources, we obtained good fits to the SEDs using the spherically symmetric model. This is amazing especially in the case of AB Aur where Mannings & Sargent (1997) recently detected a disk-like structure. This clearly shows that one should be extremely cautious with any statement that a good fit by spherically symmetric models is a proof against the presence of disks. Here, it is interesting to note that Mannings & Sargent (1997) also detected a disk around HD 163296 for which Henning et al. (1994) could only obtain a fit of the SED with a disk model.

We tried to use model parameters close to the observations. The stellar temperatures and luminosities as well as the visual extinction values are in agreement with the values generally adopted for the objects. The outer radii obtained from the model fits are very similar to the source sizes determined from the maps. Taking the difference between the absorption coefficients used in the radiative transfer models and the [FORMULA] values assumed for the mass estimates given in Table 5 into account, the masses obtained from radiative transfer modelling agree rather well with these values.

We also tried to model the objects Elias 1 (point-like) and MWC 297 (core/envelope). In the case of Elias 1 a spherically symmetric model can be fitted to the data. For MWC 297 a satisfactory fit could not be easily reached. Another object (IRAS 12496-7650) was modelled by Henning et al. (1993). In this case they suggest a disk/envelope model because neither the disk nor the envelope model could fit the spectral energy distribution. Recently Turner et al. (1997) modelled the spectral energy distribution of V 1515 Cyg using models of outbursting accretion disks very successfully. V 1515 Cyg is a source which could not be detected in our mapping observations.

Summarizing, we can say that we could not find a clear tendency within the modeling of our 1.3 mm sources representative for the different groups (core vs. core/envelope sources). Despite the clear evidence for the presence of a disk around AB Aur, we could fit the SED by a spherically symmetric model. Therefore, high-resolution observations are indispensable to find out what the small-scale circumstellar structure of the dust around our target stars is.

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© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998
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