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Astron. Astrophys. 360, 213-226 (2000)

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2. Method and assumptions

2.1. Model approach and assumptions

In this study we have chosen to model the CSM of the programme stars using a simple optically thin dust model. In view of the above discussion it is clear that this is a simplified approach. Still, we have adopted this approach for the following reasons:

(i) A main focus of this paper is to identify and model the solid-state features present in the rich ISO spectra of AB Aur and HD 163296. For this, we employ a library of laboratory measurements of the optical constants of approximately fifty different grain materials of interest. From this collection we selected eight species, listed in Table 2, which are most likely present in Herbig Ae/Be systems. These species are discussed in the next subsection. We attempt to constrain composition, abundances, size distributions, shape properties and mass over temperature distribution of the dust grains present. This goal requires an extended parameter study, which at the present day cannot be done using a consistent multi-dimensional dust transfer model.


[TABLE]

Table 2. Sources of the optical constants.
Notes:
Abbreviations used to designate the solid state: A = Amorphous; C = Crystalline; G = Grafitic; M = Metallic.
References:
(1) Dorschner et al. (1995); (2) Henning et al. (1995); (3) Laor & Drain (1993); (4) Preibisch et al. (1993); (5) Warren (1984); (6) Scott & Duley (1996); (7) Servoin & Piriou (1973); (8) Henning & Stognienko (1996).


(ii) The above discussion on the geometry of the CSM around AB Aur and HD 163296 shows that part of the emission originates from an extended optically thin medium, e.g. the broad emission complex from [FORMULA] 14 to 28 µm and the 9.7 µm silicate emission, and which suggests that the optically thin model has at least partial validity.

Still, the next step should be to model these two nearby Herbig Ae systems using a 2D-model. Any constraints on the chemical properties derived in this exploratory study will then contribute significantly to the feasibility of such a complex follow-up investigation. Below, we will discuss model assumptions and adopted optical properties of the dust constituents.

In order to model IR spectra we have developed the radiative transfer code MODUST. This code treats the transfer of radiation through spherical density distributions of dust grains. The (multiple) dust shells are irradiated by a central source, either a black body or a Kurucz model. Any prescription for the dust distribution in the shell may be adopted. In the present analysis, we use this model in the optically thin mode so as to be able to perform an extensive parameter study. In the models used to fit the spectra of AB Aur and HD 163296 we assumed a power law shape for the density [FORMULA] as well as for the grain size distribution [FORMULA], i.e.

[EQUATION]

The shell is confined between an inner and outer edge. [FORMULA] is the density at the inner edge [FORMULA] of the dust shell. Grain sizes range between a minimum [FORMULA] and maximum size. The constant A in the equation for the grain size distribution is a normalization constant depending on the bulk density and size distribution of the grains. A power-law gives a good description of a grain size distribution. Observations of interstellar extinction show a size distribution with [FORMULA] (Mathis et al., 1977). Theoretical work (Biermann & Hartwit 1980) predicts a power-law distribution whenever there is shattering and coagulation of grains through grain-grain collisions. Dust condensation models (e.g Dominik et al. 1989) also predict a power law distribution to be applicable over a large range of grain sizes. We use a multi-component mixture of grain species where the grains are homogeneous in composition and in bulk density. Table 2 lists the grain species of interest we used for our modelling (for a discussion see below). We used the optical constants from laboratory measurements to calculate the extinction properties; for a discussion we refer to the references given in Table 2. At present our model incorporates spherical dust grains, for which we use Mie calculations, and a continuous distribution of ellipsoidal dust grains, for which we use CDE calculations to determine the absorption and scattering coefficients (see Bohren & Huffman 1983 for a full review on these methods). Table 2 also lists the wavelength ranges in which laboratory measurements were available. Outside these ranges we made the following extrapolations. At short wavelengths, i.e. in the UV or optical, the dielectric function [FORMULA] can be extrapolated by,

[EQUATION]

With [FORMULA] the frequency, [FORMULA] the plasma frequency, and [FORMULA] a damping factor of the electromagnetic wave. Both insulators and metals show the same behaviour at these wavelengths. These extrapolations are reasonable for wavelengths shorter than [FORMULA]m. At long wavelengths (sub-millimetre and millimetre) there is a marked difference in optical properties between grains of different solid-state structure, composition and shape. In this regime, we therefore extrapolate the extinction coefficient,

[EQUATION]

with [FORMULA] equal to 1 for amorphous and CDE dust grains and [FORMULA] for spherical metallic and crystalline grains.

The temperature distribution of the dust follows from the equation of radiative equilibrium. The adopted optically thin limit implies that the dust particles absorb the full radiation from the direction of the stellar disk only. One can easily show that if the absorption coefficient of the dust would behave as a power law, [FORMULA], then the resulting temperature structure at sufficient distance from the star is also given by a power law, with slope [FORMULA]. Although in reality [FORMULA] contains resonances, the overal run of temperature is usually well represented by a power law. Therefore, in the remainder of this paper, we will characterize the temperature distribution by providing the values at the minimum and maximum radius only.

2.2. Adopted chemical composition of the dust

From the SWS and LWS spectra one can derive information about the mass over temperature distribution of the dust, and by looking at spectral features, about the chemical composition of the dust. However, if only spectral information is available the geometry and properties of the circumstellar material can not be uniquely determined (e.g Bouwman et al. 1999). To determine such properties as density and grain size distribution one has to constrain the models either by supplementary observations, such as imaging, or by using theoretical arguments, e.g concerning dust destruction temperatures, crystallisation time-scales or grain removal by radiation pressure.

One of the main difficulties with the determination of the chemical composition of the circumstellar dust by comparison with laboratory measurements, is that solid-state resonance bands are rather broad. With different grain species contributing at the same wavelengths, this will result in a more or less continuous spectra without a clear spectral signature that can be uniquely attributed to a single grain species. Exceptions to this are for instance the crystalline silicates such as forsterite (Mg2SiO4) and enstatite (MgSiO3) which have strong and narrow resonances in the IR, and the vibrational modes in disordered (amorphous) silicate grains around [FORMULA]m. To determine which dust species contribute at parts of the spectra where no clear spectral feature is present, additional constraints need to be imposed (e.g. abundance constraints, condensation temperature of the individual dust species). Taking the results of dust nucleation models, one can estimate which dust species are present.

We assume that all the Mg, Fe, and Si is incorporated into dust grains. Evidence for this comes from abundance studies of the ISM, which show a correlation between heavy element depletion and density in the ISM (O'Donnel & Mathis 1997). This points to a very efficient mechanism for accretion of these species in dust grains. Studies of these elements in the gas phase also show abundances less than solar. This could be a result of either very hardy dust grains or could indicate that the elements in the ISM have abundances less than solar. The observed near solar gas-phase abundance of S points to a solar composition of the ISM (Fitzpatrick & Spitzer 1997), suggesting Mg, Fe and Si are in grains which are very difficult to destroy. If so, the ratio of iron plus magnesium to silicon atoms in the dust (3.4:1) is greater than the maximum ratio that can be accounted for by silicate grains alone (2:1). This result seems to imply that a substantial fraction of the iron or magnesium is in a grain population other than silicates (Fitzpatrick 1997). Theoretical work on dust condensation (Gail & Sedlmayer 1999) shows that metallic iron could be one of the primary condensates in outflows around M-type giants. Most of the Mg will be incorporated into silicate. Iron can enter the silicate at a lower temperature forming iron rich amorphous silicates (Gail 1998). In the ISM, metallic iron can be oxidized to FeO at temperatures below 400 K on a time scales [FORMULA] (Jones 1990). Taking a typical density of [FORMULA] [FORMULA] in diffuse clouds, small iron grains (0.01-[FORMULA]m) will be fully oxidized in [FORMULA]-[FORMULA] year. In view of typical lifetimes of diffuse clouds ([FORMULA]-[FORMULA] yr), one may expect that all small Fe grains are oxidized. This also seems to be confirmed by mass spectrometer data on Halley and chemical analysis of interplanetary dust particles (IDPs) (Bradley et al. 1992; Schulze et al. 1997). These studies show that most Mg is confined to silicates but that Fe is also incorporated in different materials such as metals, oxides and sulphides.

The nature of the carbonaceous dust is not clear. The only direct observational evidence for the presence of carbon around the Herbig Ae stars comes from the observed PAH features around AB Aur and from the presence of carbon ions in their winds and in in-falling gas. The carbonaceous dust in the ISM is modelled with a variety of materials (for a overview see Henning 1997). Most models include graphite and a form of disordered carbon. Since one is looking at (reprocessed) interstellar dust one can expect these materials also to be present around young stellar objects (YSOs).

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Online publication: July 27, 2000
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