Problems in the interpretation of the observed flux distribution of dust-obscured stars

Absorption and/or emission features observed at 11mm in low resolution infrared spectra of late-type stars are the signatures of dust-obscured evolved objects. In particular, Amorphous Carbon (AC) and Silicon Carbide (SiC) are believed to be the main components of the circumstellar matter around C-rich stars, while dust shells composed of Silicate surround O-rich stars. A modelling technique focussed on the analysis of the whole spectral flux distribution, including broadband photometry from the optical to the sub-mm and infrared spectra (7—23 mm), represents a natural tool to tackle this problem. Several authors have developed numerical codes to

Figure 1. Top: a model showing an excellent fit to the mid-infrared spectra of UU Aur. Bottom: an enlarged view of the above model which shows a poor fit to the overall photometric distribution

Figure 2. A contrary example to Fig.1 where a model shows a poor fit to the mid-infrared spectra of UU Aur (top), but a good fit to the overall photometric distribution (bottom)

solve the transfer of radiation through a medium composed of dust grains of different sizes and different chemical compositions: then by comparing theoretical models with observational data, one obtains information on the nature of the envelopes around the objects. However, as in the modelling of any astronomical object, there are limitations in the technique; below we outline a few of these.

Broadband spectral modelling

The importance of reproducing the whole spectral distribution, including broadband photometry and infrared spectra is emphasized by the examples given in Fig. 1 and Fig. 2. In Fig. 1 (top) we show the low resolution spectra from IRAS of the C-rich star UU Aurigae (empty squares), which has been well reproduced by a model which pictures the dust shell as composed of a mixture of SiC and AC (continuous line). Although, this model produces an excellent fit to the mid-infrared spectra, it can not account for the observed broadband photometric distribution without imposing additional constraints (Fig. 1, bottom).

A contrary example is given on Fig. 2: with a model which fits the gross properties of the whole flux distribution but is unable to reproduce the feature around 11.3 mm.

Figure 3. Models for UU Aur with two dust components. Continuous line: AC and SiC. Dashed line: silicate and SiC.

Actually, even a multi-spectral analysis does not lead to a unique model fitting the observational data. For example, using a modelling technique which describes a multi-component dust shell without assuming a drastic number of approximations, one typically deals with about 10 parameters: e.g. stellar parameters such temperature and radius; geometrical extension of the dust shells; sizes, chemical composition and numerical density of dust grains.

Moreover, extinction and scattering coefficients of a given dust component are not always very well known, and sometime remarkable discrepancies are found in the numerical values suggested by different authors. With such a large number of free parameters, there is a suspicion in some circles that the above modelling technique is able to reproduce any kind of observed flux distribution. In true second-hand car salesman speak, this of course is not the case. Below, we discuss the modelling of UU Aur, outlining some of the problems.

The irritating case of UU Aur

UU Aur is a star which clearly shows a far infrared excess. A 'traditional' modelling technique picturing the star as surrounded by a shell composed of a mixture of AC and SiC, with a density distribution as r-2 (r being the distance from the star) does not account for the observed infrared excess. The density law in r-2 implies a mass loss rate constant with the time, while the observed infrared excess at the long wavelengths needs a different slope to the density distribution. In order to modify the density distribution, a simple approach is to assume that the condensation of one of the dust components arises at a very large distance from the star.

Figure 3 shows the best-fit obtained assuming that SiC condense at a distance of few stellar radii, and AC condense at the distance of a few hundred stellar radii. By contrast, assuming that SiC also condenses farther away from the star, the gross properties of the flux distribution are still well reproduced, but it is not possible to fit the feature seen in the mid-infrared spectra.

Actually, in order to fit the data of UU Aur, AC is not strictly required, provided that an efficiency law in l- 1 at long wavelengths is assumed. The dashed line in Fig. 3 shows the best fit obtained by substituting the amorphous carbon with silicate, and assuming silicate condenses at a few hundred stellar radii.

All the above models fit poorly the near infrared photometry. A possible explanation would be to invoke variability: in the AGB stars, variations of 2 magnitudes during a period of a few months are easily found in the optical and in the near infrared photometric measurements, furthermore there is evidence of noticeable variations of flux in the sub-mm. Thus a multi-spectral coverage of measurements taken at the same time appears to be the only option in order to analyse the flux distribution of AGB stars. Therefore, the lack of co-eval data for UU Aur could explain the poorness of the fit. Another explanation is that we have modelled the flux emerging from the star as a blackbody, whereas C-star photospheric spectra are heavily line blanketed.

Figure 4. Models for UU Aur with three dust components: AC, SiC and silicate. Continuous line: both silicate and AC are present in the outer parts of the shell. Dashed line: a 'detached' shell composed of silicate only is considered.

However, if one strongly wishes to see a good fit from the optical to the sub-mm, we may consider a three- component model, including AC, SiC and silicate (the silicate condensing farther away from the star): see Fig. 4, continuous line.

The idea that detached shells composed by silicate surround an inner shell of AC and SiC is an attractive one as it leads one to speculate that O-rich stars may represent an earlier stage in the evolution of AGB stars, which then evolve as C-rich stars. The dashed line in Fig. 4 shows the results in testing this hypothesis: the circumstellar envelope is composed of an inner shell of AC and SiC, then the density of the shell is dropped, and a new shell composed of silicate is considered. This situation is different from the previous one, where silicate, AC and SiC coexist in the outer part of the shell.

We can even employ a more sophisticated model via taking into account a time-dependent mass loss rate. For example, the continuous line in Fig. 5 shows such a beast where the best-fit is obtained assuming a dust- shell composed of amorphous carbon and SiC as originated by a mass loss process which has been interrupted for a few hundred years in the recent past.

Figure 5. A time-dependent mass loss rate model for UU Aur, including AC and SiC.

Conclusions and what of the future?

As shown in the different plots it is rather difficult to discriminate between the various proposed models. From this analysis - which has been intentionally made only on the observed spectral distribution - what conclusions can be derived?

First of all, the results from the analysis of the flux distribution of C-rich (and O-rich) stars, should be regarded with some caution. It is worth to point out this concept with a numerical example, e.g. the model shown in Fig. 2 (continuous line) has been obtained assuming a two component model with AC and SiC and a constant mass loss rate, has a ratio SiC/AC = 0.01. On-the-other-hand, the model shown in Fig. 5, obtained assuming a time-dependent mass loss rate, has a ratio SiC/AC = 1.

Figure 6. The normalized surface brightness in the millimeter regions for the UU Aur models involving a two dust component model with AC and SiC (continuous line), a three dust component model with AC, SiC and silicate (dotted line), and a time-dependent mass loss rate model including only AC and SiC (dashed line).

Indeed, rather than provide parameters of the best-fits, it seems more convenient to try an answer the following questions: which features of the models must be excluded and which features cannot be ruled out?

In the case of UU Aur, the spectral analysis shows that:

i) SiC is required as a dust-shell component, and it must condense near the star (within a few stellar radii), otherwise the feature around 11.3 mm cannot be explained.

ii) A density law in r-2 is not sufficient to account for the observed flux distribution at the long wavelengths.

iii) Provided that the flux observed at the long wavelengths can be explained as due to an episode of change in the mass loss rate, such episode could have occurred in a time-scale of a few hundred years.

This is not much... however, all is not lost. One possible way of distinguishing between the various models outlined in Figs. 3, 4 & 5 maybe via imaging in the millimeter region. In Fig. 6, we reproduce the normalized surface brightness plots as a function of angular resolution for three models. As is quite noticeable, these models have different surface brightness and provided one can use instruments with sufficient angular resolution, the millimeter region offers a potentially valuable diagnostic tool. But what of current instrumentation. With SCUBA on the JCMT, the best angular resolution is 6 arc sec at 450 mm. Using the JCMT and the CSO in an interferometer mode will achieve 1.4 arc sec at 1 mm or 0.6 arc sec at 450 mm, allowing at least to discriminate if the outer part of the dust shell is composed of silicate or AC grains.

S. Bagnulo & J.G. Doyle, Armagh Observatory,

C.J. Skinner, Space Telescope Science Institute