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Astron. Astrophys. 319, 648-654 (1997)

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4. Mass loss

In contrast to the preceeding section we now are interested in time-averaged quantities characterizing the mass loss, i.e. the mass loss rates and the outflow velocities of the models. We investigate their dependence on various physical parameters and compare the modelling results to observations.

4.1. Parameter dependence

According to their influence on the models, the physical parameters can be split into the following groups: [FORMULA], [FORMULA] and [FORMULA] determine the spatial structure of the hydrostatic initial model and, consequently, (to a certain extent) the background structure of the inner parts of the dynamical model. P and [FORMULA] characterize the stellar pulsation (period and amplitude) and thus the levitation of the atmosphere. Together these parameters largely determine the conditions in the dust condensation zone while [FORMULA] controls the efficiency of grain formation and growth for given temperature and density.

Models R7, R7C20 and R7C18 show the effects of different carbon abundances while all other parameters are held constant. Comparing the differences of [FORMULA] and [FORMULA] between the models R7 and R7C20 as well as R7C20 and R7C18, the larger change in the abundance of condensible carbon [FORMULA] in the first pair causes only a relatively small effect compared to the second pair. This nicely demonstrates the non-linearities of the dust condensation and wind mechanism. As long as the models are in a parameter domain where both the dust condensation and radiative driving work efficiently, [FORMULA] depends only slightly on [FORMULA]. However, as the abundance of condensible carbon is reduced it becomes more and more difficult to drive a stellar wind and [FORMULA] starts to decline significantly with decreasing [FORMULA]. The same effect is seen in models R10, R10C18, R10C16 and R10C15, another series of models with different values of [FORMULA] but otherwise identical parameters. Fig. 2 shows models R10C16 and R10C15 which differ less than 20 percent in [FORMULA] but exhibit a completely different spatial structure. R10C16 shows a pronounced shell-like structure associated with strong shock waves and complete condensation occuring within the dust layers. R10C15 exhibits only small variations in the various quantities plotted, the maximum degree of condensation is far from complete and both the velocity and the mass loss rate are significantly lower than in model R10C16.

[FIGURE] Fig. 2. Radial structure of models R10C16 (dashed) and R10C15 (full line): velocity a, density b, gas temperature c and degree of condensation d.

A moderate change of the pulsation period seems to have little effect on the mass loss rate and the velocity as inferred from models R5 and R5P or models R7C20 and R7C20P, respectively. The piston velocity amplitude on the other hand may have considerable influence on the mass loss as demonstrated by models R7 and R7U as well as models R13 and R13U. The mass loss rate increases with [FORMULA] since (for given stellar parameters) the shock waves caused by the pulsation determine the density in the dust condensation and wind acceleration region and thus [FORMULA].

Models R5 and R7, R7C20 and R10 as well as R10C16 and R13C16 show how a combined change of [FORMULA] and [FORMULA] due to an evolution along the AGB (with [FORMULA], [FORMULA] and [FORMULA] held constant and [FORMULA] changing accordingly) affects the mass loss. Since both increasing [FORMULA] and decreasing [FORMULA] lead to more extended atmospheres, the mass loss rate increases strongly. As the mass loss at the same time reduces [FORMULA] the effect will even be larger. A comparison of models R7 and R7M or models R13U and R13MU demonstrates how sensitively the mass loss depends on [FORMULA].

For stellar winds which are driven by radiation pressure on dust a close correlation should exist between the outflow velocity and the strength of radiation pressure relative to gravitation. In Fig. 3 we have plotted [FORMULA] as a function of the quantity


which is proportional to the ratio of the radiation pressure term and the gravitation term in the equation of motion. The dust opacity [FORMULA] is proportional to the dust-to-gas mass ratio [FORMULA] which is related to the degree of condensation [FORMULA] by


where [FORMULA], [FORMULA] and [FORMULA] are the atomic masses of carbon, hydrogen and helium and [FORMULA] is the helium abundance. ([FORMULA], [FORMULA], [FORMULA] ; [FORMULA]). We find a good correlation of [FORMULA] and [FORMULA] except for the models of series R5 which lie distinctly above the bulk of the models. We interpret this as a consequence of a larger relative importance of the momentum input by the piston in the models with lower luminosities.

[FIGURE] Fig. 3. Outflow velocity as a function of a quantity characterizing the strength of radiation pressure relative to gravitation; symbols for models of series R: [FORMULA], [FORMULA], [FORMULA], [FORMULA] ; models of series P are represented by triangles.

Note that in our sample of models no apparent correlation exists between the outflow velocity and the mass loss rate. Depending on the stellar parameters and pulsational properties a wide range of combinations seems possible.

4.2. Comparison with observations

In Fig. 4 we have plotted the mass loss rate versus the pulsation period to compare the modelling results with mean [FORMULA] -P relations deduced from observations of Miras. We find good agreement with the results of Groenewegen (1995, for carbon Miras) and Whitelock (1990, O-rich Miras in the Galactic Bulge), especially when considering the fact that the scatter in each of the observed relations is about 1 dex for a given period and that a certain overlap exists between C- and O-rich objects. Note that the range of possible mass loss rates for a given period can be can be accounted for by differences of the pulsation amplitude (e.g. models R13 and R13U, [FORMULA] d) or the abundance of condensible material (non-linearities in the dust formation and radiative acceleration process; e.g. model series R10, [FORMULA] d).

[FIGURE] Fig. 4. Comparison of modelling results with mean [FORMULA] -P relations deduced from observations: Groenewegen (1995, for carbon Miras; full line), Whitelock (1990, O-rich Miras in the Galactic Bulge; dashed line). Note that the scatter in the observed relations is about 1 dex for a given period.

The periods of model series R have been chosen according to the rather well established P -L relation for O-rich Mira variables (e.g. Feast et al. 1989, Whitelock 1993). In general, the periods of C-rich Miras are expected to be somewhat longer at a given luminosity than for their O-rich counterparts. However, as demonstrated in the preceeding section the mass loss rate does not depend critically on P. Using larger values of P for a given model would then basically result in shifting the points plotted in Fig. 4 to the right (higher periods) as inferred from models R5/R5P and R7C20/R7C20P. Recently, Groenewegen & Whitelock (1996) have derived a revised P -L relation ([FORMULA]) for carbon Miras in the Galaxy. Interpolating for the corresponding periods between models R5 and R5P ([FORMULA] 334 d) or R7C20 and R7C20P ([FORMULA] 463 d) leads to points that coincide with the mean [FORMULA] -P relation for carbon Miras of Groenewegen (1995).

In Fig. 5 we compare our results to the well-known mass loss law of Reimers (1975)


([FORMULA], [FORMULA] and [FORMULA] in solar units) which - directly or with modifications - has been widely used in calculations of AGB evolution though it was originally found for RGB stars. While at [FORMULA] the models lie within the range defined by the parameter [FORMULA] the mass loss in our models increases much steeper with luminosity so that at [FORMULA] there is practically no overlap with the range given by Reimers' law. Recent evolution calculations by Blöcker (1995) use different mass loss laws for the RGB, AGB and post-AGB phases. During the AGB phase mass loss is described by a modified version of Reimers' law based on the dynamical models of Bowen (1988). An additional factor depending on [FORMULA] leads to a steeper increase of mass loss with luminosity (about one order of magnitude for the range plotted in Fig. 5) which is in much better agreement with the results presented here.

[FIGURE] Fig. 5. Comparison of modelling results (only models of series R with [FORMULA] are plotted) with the mass loss law of Reimers (1975) for [FORMULA] and [FORMULA] and corresponding stellar parameters (dotted lines).
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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998