3. Dynamics and periodicity
In general, models which include a time-dependent description of the dust component do not show a periodic temporal behaviour of the circumstellar envelope in response to a periodic driving at the inner boundary. This is due to the facts that (i) the formation and growth of dust grains is governed by its own time-scales which need not be associated with those of the pulsation (piston) and (ii) that - due to its high opacity - the dust tends to dominate the dynamics and thermodynamics of the circumstellar envelope.
However, most of the models presented in the literature (e.g. Fleischer et al. 1992, Winters et al. 1994, Höfner et al. 1996) are single- or multi-periodic, i.e. the dust formation and dynamics of the circumstellar envelope repeat on a timescale that is an integer multiple of the piston period P. This gives a somewhat distorted view of the situation because usually careful fine-tuning of parameters is necessary to produce such well-behaved models.
In this paper we present two groups of models (cf. Table 1): The parameters of series P have been chosen to produce certain periodic models which are regarded as prototypes for the following discussion. The purpose of model series R is to demonstrate the dependence of outflow characteristics (mass loss rate, terminal velocity) on various parameters. For the investigation of mass loss (cf. Sect. 4) the occurence of periodicity in some models will only be regarded as a by-product but in this section we want to demonstrate that different kinds of (multi-) periodicity can give valuable hints on the nature of the corresponding model. Since the dust formation process influences not only the dynamics but also the near IR light curves (e.g. Winters et al. 1994) detailed observations could possibly help to constrain model parameters.
Table 1. Model parameters (, , , , P, ) and results: mass loss rate , mean velocity at the outer boundary , mean degree of condensation at the outer boundary and the corresponding dust-to-gas mass ratio (see text); and are periods (in units of the piston period P) characterizing the formation of new dust layers and the dynamics of the circumstellar envelope, respectively (a 'q' behind a number indicates a quasi-periodic behaviour). is the stellar radius calculated from and . The last column gives the plot symbols used in Figs. 4 and 5.
3.1. Prototype models
Model P1 is a typical single-periodic model. A new dust layer is formed each piston cycle triggered by the enhanced density behind the shock waves caused by the pulsation. The dynamics of the atmosphere and the inner parts of the circumstellar envelope is depicted in Fig. 1 (top) showing the positions of selected mass shells (test particles) as a function of time. Below about we see the dust-free atmosphere which is periodically passed by strong shocks (marked by the sharp bends in the lines). Between 2 and the formation of dust layers and their subsequent acceleration due to radiation pressure (indicated by the steepening of the lines) takes place. Above, we see the innermost parts of the circumstellar dust shell and stellar wind zone (note that the distribution of the lines has no physical meaning in itself but results from the fact that the selected mass shells correspond to actual grid points in one model of the time sequence which are concentrated at shock fronts and other important spatial features).
The time-scales of the grain formation and growth depend strongly on the densities of the relevant chemical species. Therefore, if we reduce the condensible material available (by decreasing ) the dust formation becomes slower and the periodicity vanishes. In model P2 (which differs from model P1 by a lower value of ) we encounter a simple form of multiperiodicity: A dust layer is formed every second piston period (Fig. 1, center).
A different kind of multiperiodicity is observed in model P4 (model V1 in Höfner et al. 1996). A dust layer is formed every piston period but alternately this process is triggered by a passing shock (as in the single-periodic model P1) or occurs spontaneously (i.e. without a density enhancement caused by a shock) in the wake of the preceeding dust shell, above the next dust-free shock wave created by the pulsation (Fig. 1, bottom). In Höfner et al. (1996) we have interpreted this behaviour as a consequence of a higher background density of the atmosphere compared to a single-periodic model. The mass loss rate (which can be used as an indicator of the density in the acceleration region of the outflow) is more than a factor of two higher than in the single-periodic model P1. Actually, the effective temperature of model P4 is 100 K cooler than in model P1 which - for a given luminosity - leads to a more extended stellar atmosphere.
A similar effect can be achieved by changing the luminosity while keeping the effective temperature constant. The luminosity of model P3 is about 15 percent higher than in model P1 and the carbon abundance is the same as in the cooler model P4. Note, that the stellar radii of models P3 and P4 are almost identical. P3 and P4 show a similar multiperiodic behaviour and the mass loss rates are comparable.
3.2. A random sample
If we regard model series R (cf. Table 1) we find that only 2 of the 17 models show a well-defined periodicity of the circumstellar dust shell. Model R5P is single-periodic and model R10C18 shows a multiperiodic behaviour similar to models P3/P4. About half of the remaining models exhibit a quasi-periodicity (indicated by a 'q' behind the corresponding period) and the rest shows irregular temporal variations.
The relation between models R7 and R7C20 is analogous to P1 and P2 in that they differ only by the carbon abundance. In the model with the higher value of (R7) a dust shell forms every piston cycle while in R7C20 the formation of a new dust layer occurs every second piston period. Model R7M has the same parameters as model R7 except for the mass which is lower by 10 percent (leading to a more extended structure of the atmosphere) and exhibits a behaviour similar to models P3/P4, i.e. a formation of dust layers once per piston period which alternately is triggered by a shock or happens spontaneously. The mass loss rate of R7M is twice as high as in model R7 supporting our argument about the mean density and this type of multiperiodicity. Models R5 and R5P which differ only by the pulsation period demonstrate the influence of time-scales. In the model with the shorter period (R5) a dust layer forms about every second period while the longer period of model R5P allows for the occurence of dust formation once each piston cycle. Note in this context that the mass loss rates and outflow velocities of the two models are almost identical.
3.3. Influence on light curves
Observable properties of LPVs like light curves are influenced both by the pulsation of the star and the dust formation in the circumstellar envelope. The pulsation (which is believed to be due to a -mechanism operating in the H and first He ionisation zones) is associated with changes of the luminosity, i.e. bolometric variations. In contrast, the formation of dust layers basically causes a spectral redistribution of the stellar radiation. The grains absorb effectively at short wavelengths and reemit the energy at longer wavelengths. Thus, in principle, it should be possible to disentangle the variability caused by stellar pulsation from the effects of dust formation and dynamics in the atmosphere by a simultaneous photometric monitoring of LPVs at different wavelengths. A possible (multi-) periodicity in combination with known outflow characteristics (velocity, mass loss rate) could then help to restrict model parameters like e.g. the abundance of the condensible material.
Winters et al. (1994) have calculated light curves of C-stars based on dynamical models of the circumstellar dust shell which are - regarding the physical input - largely comparable to the models presented here (cf. Höfner et al. 1996 for a discussion). They demonstrate that the formation of dust layers decisively influences the shape of the light curves and that the dynamics of the circumstellar dust shell (e.g. multiperiodicity) may be reflected in the long-term behaviour, i.e. variations of the light curve over several pulsation periods which are superimposed on the variation caused by the pulsation. Note that such features are actually found in observed light curves of LPVs (e.g. Le Bertre 1992). Preliminary calculations of near-IR light curves based on our own models seem to support the results of Winters et al. (1994). In the case of the multiperiodic model P4 we find - depending on the wavelength and thus on the spatial region where the radiation comes from - a periodicity of the light curves on a time-scale of either 1 or 2 .
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998