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

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4. Model results

In order to fit the IR spectra of the sources in our sample, for each set of dust opacities we computed a grid of models parameterized on the 10 [FORMULA] m optical depth [FORMULA], which is proportional to the dust mass loss rate of the envelope:


where [FORMULA] is the density of the dust grains (we assumed 3.0 g/cm3 for OS92 silicates, 1.85 g/cm3 for AC1 amorphous carbon and 2.5 g/cm3 for Pégourié 1988 SiC), [FORMULA] is the opacity at 10 µm, and [FORMULA] is the outflow velocity of the dust. The models were computed assuming d = 1 kpc, [FORMULA] = [FORMULA] cm, [FORMULA] = 2500 K, [FORMULA] = 2000 [FORMULA], [FORMULA] = 4 [FORMULA] if [FORMULA] and [FORMULA] = 5 [FORMULA] if [FORMULA].

Fig. 2 shows model spectra (in the wavelength interval 1-100 µm) of O-rich envelopes for increasing [FORMULA], from -3 to 1; in Fig. 3 the sequence of C-rich models with [FORMULA] from -3 to 0 is plotted. In both cases greater optical depths cause larger infrared excesses, and the dust features (silicates 9.8 µm and SiC 11.3 µm) change from emission to absorption. We stopped the C-rich sequence at [FORMULA] because this can be taken as a limit for this class of sources, considering that SiC features in absorption are rare in the LRS database (see Omont et al. 1993).

[FIGURE] Fig. 2. Model spectra for O-rich circumstellar envelopes; the curves have [FORMULA] = 0.001, 0.003, 0.01, 0.03, 0.1, 0.3, 1.0, 3.0 and 10.0; the IR excess increases with increasing optical depth.
[FIGURE] Fig. 3. Same as Fig. 2, for C-rich circumstellar envelopes having [FORMULA] = 0.001, 0.003, 0.01, 0.03, 0.1, 0.3, 1.0 and 3.0.

The van der Venn & Habing (1988) IRAS color-color diagram for the computed model sequences is shown in Fig. 4. The two curves are associated to increasing optical depth, and thus to increasing mass loss; note that they cross the appropriate regions of the diagram for the two classes of sources. The curves should not be considered as evolutionary tracks, because there is indication that mass loss rates do not increase monotonically during AGB evolution, rather they can have sudden variations in connection with thermal pulses (Vassiliadis & Wood 1993; see also Paper I).

[FIGURE] Fig. 4. IRAS [25]-[60] vs. [12]-[25] color-color diagram for O-rich (squares) and C-rich (triangles) model in Fig. 2 and 4 respectively; increasing optical depth results in increasing IR excess (from left to right).

4.1. Spectral distribution

By means of our model we finally fitted the spectral distribution of the sources listed in Table 1 and we obtained some relevant physical parameters for the star ([FORMULA], [FORMULA]) and the envelope ([FORMULA], [FORMULA], [FORMULA]).

Moreover, we derived the dust mass loss [FORMULA] (using Eq. 1, with [FORMULA] from Loup et al. 1993) and the bolometric luminosity [FORMULA].

From Loup et al. (1993) we also obtained values (based on radio OH and HCN observations and IRAS data) for the total mass loss rate [FORMULA] and the total emitted fluxes [FORMULA] ; we thus estimated the dust to gas ratio µ (in mass fraction) and the distance d for each source. Our results are summarized in Table 2.


Table 2. Model estimates for sources in Table 1. Envelope optical depth [FORMULA], dust mass-loss [FORMULA], dust to gas ratio µ, total luminosity [FORMULA] and distance d for each source are determined by model fitting parameters, while total mass loss [FORMULA], total fluxes [FORMULA] and gas outflow velocities [FORMULA] are derived by IR-radio-millimetric observations from Loup et al. 1993.

Note that the dust to gas ratios are generally low ([FORMULA]), with the exception of WX Ser. We would like however to stress that estimates of µ are affected by uncertainties in the grain average radius a ; Eq. 2 shows that [FORMULA] is proportional to a (assuming a correct fitting of [FORMULA]), which is not, in this case, simply a normalization factor. The distances we obtained are all in agreement with estimates by other authors (see e.g. Loup et al. 1993), with the exception of U Her, whose distance is clearly underestimated by our model, as explained below.

Fig. 5 shows the model spectra for a few sources reported in Table 1, superimposed to the IRAS LRS. All sources except those O-rich with less prominent silicate features (LML90's "Sil+", as in the case of U Her and "S" subclasses) can be satisfactorily fitted by the available opacities. In the case of the S-star W And ("S" subclass), to fit the LRS spectra was necessary to use a mixture of silicate and carbonaceous dust in the proportion 1:4, being impossible to model the source using silicates dust only. The models for the sources with weak silicate features in subclass "Sil+" were obtained by fitting the continuum only; this allows us to produce reasonable values for the physical parameters listed in Table 2, but is not able to reproduce the correct profile of the dust feature. In our opinion, this is probably related to an inadequateness of the available silicate dust opacities for sources of LRS subclass "Sil+" and "S", but we discuss other possibilities in Sect. 5.

[FIGURE] Fig. 5. Model spectra for 6 sources listed in Table 1. IRAS LRS data (triangles) are superimposed.

4.2. Temperature profiles

In spherical symmetry, the thermal structure of the model is expressed by a radial law [FORMULA] giving the self-consistent dust radiative equilibrium temperature for the dust grains. The temperature profiles thus obtained depend on the dust chemical composition and on the optical depth. For large envelope radii they can be approximated by a power law [FORMULA], where [FORMULA] is the asymptotic value for the logarithmic derivative of the temperature profile [FORMULA]:


The inferred [FORMULA] values are restricted to a limited range (see Table 3); however those for O-rich sources are systematically lower than those for C-rich ones (0.35 vs 0.40), while the case of the S-star modelled with a mixed dust composition is intermediate. This numerical result is in good agreement with the analytic formula first derived by Sopka et al. (1988) for spherically symmetric dust shells having power-law opacity [FORMULA] and density [FORMULA]:


For Oss1 silicates [FORMULA] in the mid- and far-IR wavelength range, while for AC1 Rouleau and Martin amorphous carbon is [FORMULA] in the same spectral range; Eq. 4 would give [FORMULA] for silicates and [FORMULA] for amorphous carbon, similar to our results, appropriate for the outer parts of the envelopes, where the mid- and far-IR thermal radiation is emitted.


Table 3. Asymptotic values of the thermal structure power-law exponent for the modelled sources, ordered for chemical type and increasing optical depth [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998