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Astron. Astrophys. 322, 924-932 (1997)

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3. Our model

Radiative transfer in circumstellar dust shells has been previously discussed by many authors (Rowan-Robinson 1980; Griffin 1990; Justtanont & Tielens 1992, Groenewegen, 1993 and, more recently, Hashimoto 1995; a detailed discussion of these models is given in Habing, 1996). A fully self-consistent modelling of envelopes around evolved late-type stars requires coupling of radiative transfer with the time-dependent hydrodynamic equations of motion for the two interacting fluids (gas and dust) which constitute the stellar wind. Such an approach has been attempted by IE95: their treatment, in the hypothesis of a steady-state outflow due to constant dust-driven stellar wind, can successfully account for the observed mass loss rates and is able to reproduce the IRAS colors and visibility functions (see Ivezi and Elitzur 1996a, b) for a large sample of AGB objects in all LRS classes. More recently, models taking into account time dependent hydrodynamics have been developed to study the effect of sudden variations in the stellar parameters in connection with thermal pulses (Schönberner et al. 1996). This analysis is necessary in order to follow the latest stages of AGB evolution, which lead to the formation of a planetary nebula; however, time variations of the mass loss rate, on a thermal pulse time scale, have negligible effects on the IR properties of at least 95% of late-type stars associated with IRAS sources (IE95).

Further, two fluid (dust and gas) models for stationary dust-driven winds treating dust formation (with classical nucleation teory) and equilibrium chemical reactions have been developed to determine the gas molecular composition, dust to gas mass ratio and total mass loss in a self consistent way (Krüger et al. 1994)

In our model we totally neglect the dynamics of the envelope and the details of dust formation (the mass loss rate is a free parameter that is determined by fitting the data); rather, we concentrate our efforts on obtaining an accurate fit of the observed sources, paying special attention to the mid-IR interval, where the TIRCAM spectral bands are located (see Paper I). The model parameters are fixed by fitting the IRAS and TIRCAM data. For any dust mixture, the spectrum (and in particular the shape of the features) is used to estimate the optical depth [FORMULA] of the envelope, proportional to the dust opacity and the mass loss rate. Near- and Mid-IR photometric data allow us to determine the stellar parameters [FORMULA], [FORMULA] (the central star is assumed to radiate a black body spectrum at temperature [FORMULA]), and the inner CSE radius [FORMULA] ; the outer radius [FORMULA] is derived by fitting the IRAS far-IR photometry at 60 and 100 µm. The source distance enters in the model only as a scale factor for the flux densities, and is determined by normalizing the spectra with the observed photometry. We first assumed the value estimated by Loup et al. (1993); then we computed our own value of [FORMULA], where [FORMULA] is the total luminosity and [FORMULA] the total flux (in [FORMULA], assuming a reference distance of 1 kpc) obtained by Loup et al. (1993) using IRAS fluxes.

The code computes iteratively a self-consistent thermal structure of the envelope; the emergent spectra is calculated taking into account the effect of non-isotropic scattering, absorption and thermal reemission by grains. The computation is performed on the hypotheses of: (i) spherical symmetry of the dust shell, with an [FORMULA] density distribution, consistent with a steady outflow at constant velocity, (ii) balance between absorption and emission by the dust grains, and (iii) LTE dust radiation at the local temperature [FORMULA].

The dust grain composition is simulated adopting different sets of dust opacities [FORMULA] ; the dust grain size is usually assumed to satisfy a size distribution [FORMULA] with 0.01 µm [FORMULA] 0.25 µm (Mathis, Rumpl and Nordsieck 1977, hereafter MRN; see e.g. Jura, 1996 for more recent data). The main effect of considering a grain size distribution is a broadening of the 9.8 µm silicate feature (Simpson 1991), due to the non linear dependance of the opacity on a ; this is however important only for [FORMULA] and in the case of AGB CSE grains can be neglected. Hence we simply adopted for the grain size the average MRN value a = 0.1 µm.

3.1. Dust opacities

Following a Chan & Kwok (1990) scaling theorem, the solution of the radiative transfer equation in spherical symmetry is scaled on the flux-averaged optical depth [FORMULA], where:

[EQUATION]

In Eq. 1, [FORMULA] depends on frequency only through the opacity [FORMULA], which is thus the main parameter for modelling the spectra; for this reason, a great part of our preliminary work consisted in testing the various opacity profiles available in the literature. We now present the results of our analysis, separately for O-rich and C-rich dust.

It is well known that O-rich dust contains essentially silicate grains: they can be present as piroxens (Mg,Fe)SiO3, or as olivinæ ((Mg,Fe)2 SiO4). Actually, the olivina opacity profile of Krätschmer & Huffmann (1979) is suitable for supergiants, but not for Miras like WX Serpentis (see Griffin 1993). In the case of circumstellar envelopes the silicate dust probably consist of piroxens only, with various types of impurities making them "dirty silicates". We considered five opacity profiles for astronomical silicates.

The first is the one by Jones & Merril (1976); they introduced the concept of "dirty silicates", though many details of their opacity profile were subsequently modified.

The second is by Draine & Lee (1984); despite its wide use and accuracy, this is not well suited for our needs, since it is based on observations of the interstellar medium, and not of the circumstellar one (see also Ossenkopf et al. 1992, hereafter OS92, and Griffin 1993).

The third is by Volk & Kwok (1988): it is based on the observed spectra of 467 AGB stars with good quality photometric data in all four IRAS bands; the spectrum obtained from this opacity is similar to the observed one, but the 9.8 µm feature is slightly too much asymmetric and its peak is shifted towards redder wavelengths.

The fourth profile is the one by OS92: it is based on the previous but it is chemically and physically consistent, in the sense that computes the complex dielectric function for the dust using the Kramers-Krönig relations along with the Mie theory. Taking into account the possibility of dust annealing in the CSE, as illustrated by Stencel et al. (1990) and Nuth and Hecht (1990), the authors give the opacity in two versions, one for "warm" silicates (hereafter Oss1) and the other for "cold" ones (hereafter Oss2), depending on the physical conditions at their condensation.

The last and more recent is by David & Pégourié (1995), and is computed using Mie theory for a set of more than 300 IRAS sources with known LRS.

We obtained the best results when fitting our AGB O-rich sources with OS92 Oss1 and Oss2 opacities.

In the case of C-rich envelopes, the continuum in the spectra is usually ascribed to the presence of graphite or amourphous carbon, while the characteristic band at 11.3 µm is commonly attributed to SiC, although the presence of PAH has been proposed as well. Graphite is now commonly ruled out, because of the absence in the observed spectra of its typical narrow band at 11.52 µm (Draine 1984); moreover, the whole shape of the spectra obtained by using graphite (the most used opacities are by Draine & Lee 1984) does not fit the observed ones, as we tested directly. Therefore the main constituent of CSE carbonaceous dust seems to be amourphous carbon: due to the variability in its physical and optical properties, a comparison with observational data is fundamental. An accurate enough opacity profile, observationally tested for amorphous carbon is the one by Martin & Rogers (1987). More recently Rouleau & Martin (1991) presented a detailed study giving accurate sets of optical constants derived by Bussoletti et al. (1987): the set named AC1, in particular, is similar to the Martin & Rogers (1987) opacity, although more detailed; this is the opacity we used for our simulations.

Finally, in order to obtain spectra with the characteristic 11.3 µm band, we admitted the presence of SiC mixed with the amorphous carbon: we choose the opacity set for [FORMULA] -SiC (the exagonal rhomboedric cristalline form of SiC) from Pégourié (1988), which at the moment is the only accurate profile available in the literature; to obtain the optical properties for the mixture, we simply made a weighted average of the two species: we never found necessary to take a SiC percentage greater than 10%. This strongly depends on the opacity chosen for amorphous carbon; IE95, for example, used a more absorptive kind of amorphous carbon, together with Pégourié (1988) data for SiC, and found a SiC percentage up to 30%.

In Fig. 1 we plot the final opacity profiles for silicates, amorphous carbon and SiC used in our simulations.

[FIGURE] Fig. 1. Dust opacities used in our computations: Ossenkopf 1 "dirty silicates" profile by OS92 (dot-dash-dot line), amourphous carbon AC1 by Rouleau & Martin 1991 (dashed line) and Pégourié (1988) SiC (dotted line).

In our simulations we have not introduced Polycyclic Aromatic Hydrocarbons (PAH), because they are mainly excited by UV radiation, and their contribution to the dust heating and the final spectra is low for AGB stars, at least if a hot star companion is not present (Buss et al. 1991).

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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