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Astron. Astrophys. 324, 617-623 (1997)

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2. Modelling method

2.1. Dynamical models

The results presented in this paper are based on self-consistent time-dependent models of the atmosphere and circumstellar envelope of LPVs. The models are obtained by solving the system of grey radiation hydrodynamics (describing the energy and momentum balance of gas and radiation) together with a detailed treatment of dust condensation. Considering a carbon-rich environment we assume the formation of amorphous carbon grains (Gail & Sedlmayr 1988, Gauger et al. 1990). The effects of stellar pulsation are included in the calculations by applying a piston accompanied by a variable luminosity at the inner boundary which is located below the stellar photosphere. A thorough description of the dynamical models actually used in this paper (cf. Table 1, model series R and model P5) together with a discussion of the corresponding parameters is given in Höfner & Dorfi (1997). In addition, we have included a few stationary wind models which have been taken from earlier publications (VS: Höfner et al. 1996, L12: Dorfi & Höfner 1996, A23 and B19: Höfner et al. 1995).


Table 1. Parameters of dynamical models ([FORMULA], [FORMULA], [FORMULA], [FORMULA], P, [FORMULA]) and resulting outflow characteristics (mass loss rate [FORMULA], velocity at the outer boundary [FORMULA], the dust-to-gas mass ratio [FORMULA], the mean radius [FORMULA] where the optical depth becomes [FORMULA] at [FORMULA] m, the mean optical depth of the dust envelope [FORMULA] at [FORMULA] m and the mean flux-weighted optical depth [FORMULA]). The first group of models consists of LPV envelopes presented in Höfner & Dorfi (1997), the second group contains stationary wind models taken from various other papers. See text for details.

The models are determined by the following set of parameters: stellar mass [FORMULA], luminosity [FORMULA] and effective temperature [FORMULA] of the hydrostatic initial model, carbon-to-oxygen abundance ratio [FORMULA] as well as the piston parameters period P and velocity amplitude [FORMULA]. The dynamical calculations yield the spatial structure of the circumstellar envelope (density, velocity, temperature, degree of condensation, mean grain size, etc.) as a function of time as well as average quantities characterizing the outflow like the mass loss rate [FORMULA], outflow velocity [FORMULA] and the dust-to-gas mass ratio [FORMULA].

In the context of IR properties a point of great significance is the spatial extension of the models. All relevant physical phenomena which determine the dynamics of the outflow are concentrated within a spatial region close to the star and are fully covered by the self-consistent calculations. However, to obtain realistic spectral energy distributions it is necessary to consider the entire circumstellar envelope because the extended circumstellar dust shell around the star contributes significantly to the observed IR fluxes causing an infrared excess. As the wind acceleration and the temporal variations affecting the dust-driven wind are essentially restricted to a region within a few photospheric radii we can assume a constant velocity outside our computational outer boundary, typically located at about [FORMULA]. At this point the variability of the outflow velocity is already small compared to the average value (less than 10%). Since it is quite evident that the far-IR emission comes from a spatially very large region, we characterize the outer envelope by mean values of the mass loss rate and outflow velocity averaged over many pulsational cycles. As the deviations of the density and dust-to-gas ratio from the time-average values occur on spatial scales much smaller than the characteristic dimensions for temperature changes it is sufficient to take temporal mean values for the purpose of radiative transfer calculations. From the hydrodynamical calculations we obtain these time-averages of mass loss, degree of condensation and outflow velocity and use them to determine the density, dust-to-gas ratio and velocity of the expanding circumstellar envelope up to about [FORMULA]   [FORMULA]. At such large radii the temperature can be accurately determined by the Lucy approximation (Lucy 1971, 1976).

2.2. Frequency-dependent radiative transfer

The dynamical models are calculated with a grey approximation of radiative transfer where the corresponding Rosseland mean of the frequency-dependent dust opacities is used. To obtain spectral energy distributions we solve independently for each frequency the time-independent radiative transfer equation along parallel rays (Yorke 1980). The gas opacity is taken from the dynamical models and the dust opacity is calculated from the optical properties given by Maron (1990) for amorphous carbon and by Pégourié (1988) for SiC adopting the small particle limit of the Mie theory (grain sizes are typically between [FORMULA] m and [FORMULA] m).

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998