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Astron. Astrophys. 360, 952-968 (2000)

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2. Some properties of AGB stars, stellar models, and overshoot

Carbon stars and the third dredge-up. AGB stars show a large range of surface C/O ratios from about the solar value [FORMULA] up to well above unity. In particular many low-mass AGB stars show up as carbon stars with [FORMULA] Smith et al. 1987; Frogel et al. 1990. Half of all planetary nebulae are carbon rich as well Zuckerman & Aller 1986. Already the early stellar evolution calculations Iben 1975; Iben 1977; Iben & Truran 1978; Sackmann 1980 pointed towards the TDUP to provide a link between the intershell region where carbon is present and the bottom of the envelope convection. 1 However, these models have not shown sufficient dredge-up for AGB models with low (core) masses. Lattanzio Lattanzio (1989) used a method to determine the convective boundary which ensures that the ratio of radiative and adiabatic temperature gradient approaches unity smoothly at the convective boundary. This method favors the occurrence of the TDUP even for low mass stars of solar metallicities. Hollowell & Iben Hollowell & Iben (1988) found for low mass and low metallicity that the dredge-up of processed material occurs if some additional mixing is assumed. Wood Wood (1981) and Boothroyd & Sackmann Boothroyd & Sackmann (1988b) showed that lower metallicity and increased mixing length parameter enhances TDUP. Straniero et al. Straniero et al. (1997) find the TDUP for low-mass AGB stars and relate its occurrence to increased numerical resolution. Calculations which do not find dredge-up for low core masses are the rule rather than the exception Vassiliadis & Wood 1993; Blöcker 1995; Forestini & Charbonnel 1997; Wagenhuber & Groenewegen 1998; Langer et al. 1999. Herwig et al. Herwig et al. (1997) presented first results of a 3[FORMULA] stellar model sequence which had been computed with convective overshoot, treated time-dependently at all convective boundaries. With this approach TDUP was very efficient for low core masses and solar metallicity.

Intershell abundances. The abundance distribution in the intershell is of great importance because in this region the major nuclear burning and mixing processes associated with the He-flash and the TDUP take place. The intershell abundances are determined by nuclear burning during the quiescent interpulse phase (mainly hydrogen burning), the nuclear processing during the TP, the convective mixing in the pulse-driven convective zone, and third dredge-up. The intershell abundance is significantly affected if overshoot is applied to the PDCZ because additional material is mixed into the intershell from the C/O core (Sect. 5.2).

It affects the local nuclear production, the surface enrichment and also the structure of the star. It is also of relevance for the interpretation of surface abundances of hydrogen-deficient post-AGB stars (see e.g. Schönberner 1996; Iben et al. 1996 or Werner et al. 1999).

Neither Schönberner Schönberner (1979) nor Boothroyd & Sackmann Boothroyd & Sackmann (1988a) considered overshoot and both found that the abundance distribution in the intershell evolves with each TP. Starting with an almost pure He composition from previous hydrogen burning at the first TP, the abundances approach typical values of (He/C/O)=(0.76/0.22/0.02) (mass fractions) after a few TPs.

[FORMULA] production. The observed correlation of s-process elements and carbon in low-mass stars Smith & Lambert 1990 points to low-mass stars as a likely site for n-capture nucleosynthesis Gallino et al. 1997; Wallerstein et al. 1997. Gallino et al. Gallino et al. (1998) have demonstrated that, if hydrogen ingestion from the envelope into the intershell region at the end of the TDUP is assumed then the subsequent formation of [FORMULA] via the nuclear reaction [FORMULA] does indeed predict a neutron exposure which leads to a s-process enhancement in compliance with the solar main component of heavy elements. However, the physical mechanism of H-ingestion is unclear. Depth-dependent overshoot naturally predicts the formation of a [FORMULA] pocket as needed for the s-process nucleosynthesis.

Mixing and overshoot. Mixing of elements in stars is attributed to a number of processes Pinsonneault 1997 of which convection is the most effective. Stellar evolution models commonly employ the mixing-length theory (MLT) Böhm-Vitense 1958 or some descendent thereof. The boundary of convective instability is determined by the local Schwarzschild condition [FORMULA] where [FORMULA] and [FORMULA] are the adiabatic and radiative gradient Kippenhahn & Weigert 1990. However, neighboring layers are related by inertia, momentum transfer and the equation of continuity and therefore convective elements might overshoot beyond the boundary of convection (e.g. Shaviv & Salpeter 1973; Maeder 1975; Roxburgh 1978; Bressan et al. 1981; Langer 1986). No commonly accepted quantitative theoretical description of convective overshoot currently exists Renzini 1987.

Comparison of stellar models with observational findings indicates that convective overshoot occurs in real stars Mermilliod & Maeder 1986; Maeder & Meynet 1989; Andersen et al. 1990; Stothers 1991; Napiwotzki et al. 1991; Alongi et al. 1991; Alongi et al. 1993; Schröder et al. 1997; Kozhurina-Platais et al. 1997. Overshooting in stellar evolution calculations has been simulated by an extension of the instantaneous mixing beyond the convective boundary (instantaneous overshoot). The widely used set of models by Schaller et al. Schaller et al. (1992) has been calculated with an overshoot distance of one fifth pressure scale height ([FORMULA]) which was calibrated in order to fit the observed terminal age main sequence of 65 stellar clusters and associations. Improving on the instantaneous treatment, Deng et al. Deng et al. (1996a); Deng et al. (1996b) and Salasnich et al. Salasnich et al. (1999)) have explored the effects of turbulent diffusion in the overshoot region of massive stars.

Due to the insights of two- as well as three-dimensional hydrodynamical simulations convection is nowadays pictured in terms of downdrafts and up-flows rather than as a hierarchy of eddies Stein & Nordlund 1998. Two-dimensional simulations showed that prominent downward-directed plumes can overshoot a substantial distance into the stable region Hurlburt et al. 1986; Hurlburt et al. 1994; Freytag et al. 1996. These models show emerging and vanishing patterns of curls, fast narrow downdrafts and broad up-flow regions.The turbulent velocity field decays exponentially beyond the convective boundary (see also Xiong 1985; Asida & Arnett in prep.). These results have been applied via a time-dependent treatment of convective mixing to low- and intermediate mass stellar models by Herwig et al. Herwig et al. (1997); Herwig et al. (2000) (low mass TP-AGB), Ventura et al. Ventura et al. (1998) (main sequence) and Mazzitelli et al. Mazzitelli et al. (1999) and Blöcker et al. Blöcker et al. (2000) (massive AGB stars, HBB).

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Online publication: August 23, 2000