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Astron. Astrophys. 344, 617-631 (1999)

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8. Conclusions

The analysis presented in this paper addresses several questions regarding the occurrence of 3DUP in AGB star models, the laws governing that phenomenon, and some of its implications on the structural and chemical evolution of those stars. The analysis is based on model calculations of a [FORMULA] star using the Schwarzschild criterion and performed either with or without extra-mixing.

Modeling dredge-up in AGB stars. The use of a local criterion to delimit convection borders, such as that of Schwarzschild, and without any extra-mixing is shown to lead to an unphysical situation which prevents the deepening of the convective envelope into the H-depleted regions, and which thus also prevents the occurrence of the 3DUP (Sect. 2). The Schwarzschild layer is undefined in those models due to the development of a discontinuity in the hydrogen abundance profile at the bottom of the convective envelope. Model calculations using no extra-mixing confirm the failure to obtain dredge-up in those conditions (Sect. 4). That conclusion is found to be independent of the mixing length parameter, stellar mass, or numerical space and time resolution of the models. Models using the Schwarzschild criterion with no extra-mixing are thus inadequate to describe the 3DUP phenomenon.

Models of the same [FORMULA] star but using extra-mixing, on the other hand, lead to efficient dredge-ups from the 11th pulse on (Sect. 5). This results directly from the unstable character of the lower boundary of the envelope against extra-mixing (Sect. 2.2). The calculations further reveal that the dredge-up characteristics are insensitive to the extra-mixing parameters (such as the extra-mixing extent and efficiency). This important conclusion is most welcomed since nothing is known yet about the characteristics of the extra-mixing expected to occur in AGB stars. It enables, in particular, to study the dredge-up characteristics with some confidence and without worrying about the extra-mixing parameters.

Although the dredge-up predictions are rather insensitive to the extra-mixing parameters, a proper handle of the extra-mixing procedure is essential in order to obtain reproducible predictions. In particular, a purely numerical extra-mixing can lead to model-dependent predictions (Sect. 5.3; see also Mowlavi 1999a). The use of instantaneous mixing in the extra-mixing region can also lead to convergence difficulties. The calculations presented in this paper use a diffusive overshooting with a decreasing bubble velocity field in the extra-mixing region. It leads to smooth chemical abundance profiles and to a proper location of the Schwarzschild layer.

The dredge-up process. The insensitivity of the dredge-up predictions on the extra-mixing parameters results from the fact that the dredge-up rate is limited by the time scale of the thermal readjustment of the envelope (Sect. 6.1). As dredge-up proceeds, H-depleted matter is lifted from the core into the envelope and its thermal state must relax to that of the envelope (see also Mowlavi 1999b). The dredge-up rate is estimated to [FORMULA]/yr from model calculations (Sect. 5). This is also what is expected from an analytical estimation using simplified arguments of the physics involved in the dredge-up process (Sect. 6.1).

Dredge-up laws. The model calculations performed with extra-mixing on selected afterpulses of the standard [FORMULA] star predict linear relations both between the dredge-up rate and core mass (Eq. 9) and between dredge-up efficiency and core mass (Eq. 10). In those calculations where the feedback of dredge-up on the AGB evolution is not taken into account, the dredge-up rate and efficiency further depend linearly on the stellar luminosity.

When the feedback of dredge-up on the AGB evolution is taken into account, the analysis presented in Sect. 6 suggests that the dredge-up rate keeps its linear dependence on the core mass. This conclusion is valid if the [FORMULA] relation is of the form of Eq. 14, [FORMULA] being the radius of the H-depleted core. In that case, we expect the dredge-up efficiency to level off at [FORMULA], at which point the core mass does not increase anymore from one pulse to the next. The asymptotic core mass of the [FORMULA] star model presented in this paper is [FORMULA] (Sect. 7.1).

Consequences on the AGB evolution. The synthetic calculations performed in Sect. 7 confirm the asymptotic evolution of the dredge-up efficiency towards unity and of the core mass towards [FORMULA]. This helps to constraint the initial-final mass relation of white dwarfs, [FORMULA] giving an upper limit for the mass f the white dwarf remnant.

The surface luminosity is not as much affected by the dredge-ups as is [FORMULA]. This results from the adopted [FORMULA] relation combined with the continuous decrease of [FORMULA]. The non-validity of the [FORMULA] linear relation, assumed in Eq. 18, is made evident in Fig. 12. The increase in L with time predicted by the synthetic calculations is only slightly less steep than the [FORMULA] increase predicted in the standard calculations without dredge-up (Fig. 11).

Mass loss can further constrain the initial-final mass relation of white dwarfs, because it depends on the luminosity through the stellar radius and effective temperature (Bowen & Willson 1991), and because the luminosity increases with time despite the leveling of [FORMULA]. We thus expect mass loss [or superwind(s)] to peal off the AGB's envelope at a lower core mass than in models not experiencing dredge-up. For example, if a superwind is assumed to eject all the envelope when the star reaches [FORMULA], the [FORMULA] star models with dredge-up predict the formation of a white dwarf remnant of [FORMULA]. This has to be compared with a predicted remnant of [FORMULA] in the absence of dredge-up. The synthetic calculations presented in Sect. 7 are, however, preliminary and based on simplified assumptions. Future investigations along these lines should refine the quantitative predictions given in this paper.

Finally, a [FORMULA] carbon star is predicted to form after about twenty pulses experiencing dredge-up (i.e. at about the 25th-30th pulse on the AGB), at [FORMULA]. Taking into account the luminosity dip experienced during about 20% of the following interpulse time, such a C star could be observed at luminosities as low as [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999