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Astron. Astrophys. 349, 243-252 (1999)

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4. Dynamical models

4.1. Modelling method

To obtain the structure of the stellar atmosphere and circumstellar envelope as a function of time we solve the coupled system of radiation hydrodynamics and time-dependent dust formation (cf. Höfner et al. 1995, Höfner & Dorfi 1997 and references therein). The gas dynamics is described by the equations of continuity, motion and energy, and the radiation field by the grey moment equations of the radiative transfer equation (including a variable Eddington factor). In contrast to the models presented in Höfner & Dorfi (1997) we use a Planck mean gas absorption coefficient based on detailed molecular data as described in Höfner et al. (1998). Dust formation is treated by the so-called moment method (Gail & Sedlmayr 1988; Gauger et al. 1990). We consider the formation of amorphous carbon in circumstellar envelopes of C-rich AGB stars.

The dynamical calculations start with an initial model which represents the full hydrostatic limit case of the grey radiation hydrodynamics equations. It is determined by the following parameters: luminosity [FORMULA], mass [FORMULA], effective temperature [FORMULA] and the elemental abundances. We assume all elemental abundances to be solar except the one of carbon which is specified by an additional parameter, i.e. the carbon-to-oxygen ratio [FORMULA]. The stellar pulsation is simulated by a variable boundary (piston) which is located beneath the stellar photosphere and is moving sinusoidally with a velocity amplitude [FORMULA] and a period P. Since the radiative flux is kept constant at the inner boundary throughout the cycle the luminosity there varies according to [FORMULA].

4.2. Dust opacities

The self-consistent modelling of circumstellar dust shells requires the knowledge of the extinction efficiency [FORMULA] of the grains, or rather of the quantity [FORMULA], which is independent of a in the small particle limit which is applicable in this context. For the models of long period variables presented in Höfner & Dorfi (1997) and Höfner et al. (1998) a fit formula for the Rosseland mean of [FORMULA] derived from the optical constants of Maron (1990) was used.

One important point of this paper is to investigate the direct influence of [FORMULA] on the structure and wind properties of the dynamical models. Therefore we have computed Rosseland and Planck mean values of [FORMULA] (see Fig. 2) based on various optical constants derived from laboratory experiments (see Sect. 3 for details about the samples).

[FIGURE] Fig. 2. The Planck mean and the Rossland mean calculated from the optical constants of amorphous carbon as derived from various experiments. See Table 1 for details. The Preibisch data would fall right on top of the Rouleau data.

For the dynamical calculations presented here we have selected the following data sets (see Table 1 for a detailed specification): Jäger400 and Jäger1000 (representing the extreme cases), Rouleau (closest to the Maron data used in earlier models but extending to wavelengths below [FORMULA]m) and Zubko . The data of Preibisch are almost identical to the data of Rouleau .

Fig. 2 demonstrates that for a given data set the difference between Planck and Rosseland means is relatively small. This is due to the fact that amorphous carbon grains have a continuous opacity with a smooth wavelength dependence 1.

4.3. Wind properties

All models discussed here are calculated with the same set of stellar parameters, i.e. [FORMULA], [FORMULA], [FORMULA] K, [FORMULA], [FORMULA] d, [FORMULA] km s-1, corresponding to model P13C14U4 in Höfner et al. (1998). The only difference between individual models is the adopted mean dust opacity. Most models have been calculated with Rosseland mean dust opacities to allow us a direct comparison with earlier models based on Rosseland means derived from the Maron data 2.

Wind properties like the mass loss rate [FORMULA] or the time-averaged outflow velocity [FORMULA] and degree of condensation [FORMULA] are direct results of the dynamical calculations. The Rosseland mean models in Table 2 (first group) are listed in order of decreasing dust extinction efficiency. Both [FORMULA] and [FORMULA] change significantly with the dust data used. [FORMULA] increases with decreasing dust extinction efficiency while [FORMULA] decreases, reflecting a lower optical depth of the circumstellar dust shell (see also Sect. 5). The mass loss rates seem to show a weak overall trend but it is doubtful whether the differences between "neighbouring" models in Table 2 are significant. Since the mass loss rate varies strongly with time the average values given in the table are more uncertain than the ones for the velocity and the degree of condensation which both do not show large variations with time.


Table 2. Comparison of modelling results for different dust opacity data: mass loss rate [FORMULA] (in [FORMULA]), mean velocity at the outer boundary [FORMULA] (in km s-1), mean degree of condensation at the outer boundary [FORMULA]; model parameters: [FORMULA], [FORMULA], [FORMULA] K, [FORMULA], [FORMULA] d, [FORMULA] km s-1; `R' denotes a Rosseland mean dust absorption coefficient, `P' a Planck mean; for details see text.
*) model P13C14U4 of Höfner et al. (1998)

The behaviour of the wind properties can be explained in the following way: The stellar parameters of the models presented here were chosen in such a way that the models fall into a domain where dust formation is efficient and the outflow can be easily driven by radiation pressure on dust (luminous, cool star, relatively high C/O ratio). In this case, the mass loss rate is essentially determined by the density in the dust formation zone which mainly depends on stellar and pulsation parameters (see e.g. Höfner & Dorfi 1997). Therefore it is not surprising that the mass loss rates of the different models are quite similar.

On the other hand, in a self-consistent model, the degree of condensation (dust-to-gas ratio) depends both on the thermodynamical conditions in the region where the dust is formed and on the specific grain opacity. The higher the mass absorption coefficient the faster the material is pushed out of the zone where efficient dust formation and grain growth is possible. Therefore the degree of condensation decreases with increasing dust absorption coefficients (i.e. higher radiative pressure) as grain growth is slowed down by dilution of the gas. Note that even in the model with the lowest dust absorption coefficient (DJ4R) the condensation of "free" carbon (i.e. all carbon not locked in CO) is far from complete ([FORMULA]).

For the two extreme cases (Jäger1000 and Jäger400 ) we have also calculated models with Planck mean dust opacities. As shown in Table 2 the wind properties of the corresponding Planck and Rosseland models (DJ1P/DJ1R and DJ4P/DJ4R) are very similar (if the differences are significant at all, see above). The two Planck mean models fit nicely into the dust extinction efficiency sequence discussed before for the Rosseland mean models.

As demonstrated in many earlier papers (e.g. Winters et al. 1994; Höfner & Dorfi 1997) the dust formation in dynamical models is not necessarily periodic with the pulsation period P. In general the models are multi- or non-periodic in the sense that the dust formation cycle is a more or less well defined multiple of P. In the models discussed here, a new dust shell is formed about every 5-6 pulsation periods.

In contrast to the hotter inner regions, the atmospheric structure below about 1500 K does not repeat after each pulsation cycle but is more or less periodic on the dust formation time scales which span several pulsation cycles. While it is easy to compare the time-averaged wind properties of the models it is much more problematic to find comparable "snapshots" in different model sequences for the discussion of observable properties presented in the next section. Therefore in all cases involving detailed radiative transfer on top of given model structures we have decided to show statistical comparisons including several maximum and minimum phase models of each sequence.

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© European Southern Observatory (ESO) 1999

Online publication: August 25, 1999