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

1. Introduction

The third dredge-up (3DUP) phenomenon is the process by which ashes from helium burning nucleosynthesis are transported from the interior to the surface of low- and intermediate-mass stars () during the asymptotic giant branch (AGB) phase (see, e.g., Mowlavi 1998a for a general introduction to the structural and chemical evolution of AGB stars). It contrasts with the first and second dredge-ups which bring to the surface only ashes from hydrogen burning. The signature of those 3DUP events is observable in the specific chemical composition characterizing many AGB stars as compared to those of red giant stars (see, for example, Mowlavi 1998b). In particular, the high C/O ratios observed in MS, S, SC (C/O1 in number) and C (C/O1) stars are explained as resulting from an increase in the carbon abundance due to the 3DUP (solar C/O0.42). Similarly, the overabundances (as compared to those in red giants) of fluorine and s-process elements (Jorissen et al. 1992) attest to the operation of dredge-up in AGB stars.

Stellar models show that thermal instabilities (called pulses) develop periodically in the He-burning shell of AGB stars (Schwarzschild and Härm 1965), leading to structural readjustments which provide an adequate stage for the operation of the 3DUP (Iben 1975, Sugimoto and Nomoto 1975). Unfortunately, model predictions available in the literature do not agree on the characteristics of the 3DUP, such as the pulse number at which it begins to operate along the AGB or the extent of envelope penetration in the C-rich layers. The problem is of particular relevance when confronting chemical abundances observed at the surface of AGB stars to model predictions. For example, the famous `carbon star mystery' put forward by Iben (1981) about the luminosity function of C stars in the Magellanic Clouds revealed, among other things, the difficulty of models to explain the low luminosity C stars (see Mowlavi 1998c for the present state on that question). The few - but isolated - successes of AGB models in obtaining low-luminosity carbon star models in the late 80's suggested that the solution to the carbon star mystery resides in the use of new radiative opacities and adequate mixing length parameters (Sackmann & Boothroyd 1991). Yet, many recent AGB calculations do not confirm those expectations (Vassiliadis & Wood 1993, Wagenhuber & Weiss 1994, Blöcker 1995, Forestini & Charbonnel 1997), while some others do succeed in obtaining dredge-up (Frost & Lattanzio 1996, Straniero et al. 1997, Herwig et al. 1997, this work).

The disagreement between AGB model predictions on the issue of dredge-up prevents us to provide reliable and consistent chemical yields of low- and intermediate-mass stars. Why do some models obtain dredge-up while others fail to do so? Herwig et al. (1997) succeed in obtaining dredge-up from the pulse on in a Pop. I star model by using an overshooting algorithm below the convective envelope. Some sort of extra-mixing beyond the convection borders defined by the Schwarzschild criterion was already used by Boothroyd & Sackmann (1988a) and Lattanzio (1986), and its inclusion is found to improve the dredge-up efficiencies predicted by the models of Frost & Lattanzio (1996). These last authors further note the critical dependency of dredge-up on the numerical treatment of convection in massive AGB models. In contrast, Straniero et al. (1997) claim to obtain dredge-up consistently without invoking any extra-mixing. A similar claim is also reported by Lattanzio (1989). According to Straniero et al. (1997), the solution resides rather in the time and mass resolution of the models. Clearly, several questions must be clarified. How influential is the numerical accuracy of AGB models in obtaining dredge-up? What is the role of extra-mixing below the envelope? Is extra-mixing necessary for obtaining 3DUP in current AGB models using the Schwarzschild criterion? If yes, how sensitive are the dredge-up characteristics to the extra-mixing parameters, about which little is known today? And, finally, what are the characteristics governing the 3DUPs?

The aim of this paper is to shed some light on these questions. They are, for a great part, related to the delicate question of convection and the determination of their boundaries in AGB models. The use of a local theory of convection, such as that of the mixing length theory (MLT), is certainly one of the greatest shortcomings still affecting stellar model calculations in general, and AGB models in particular. New non-local formulations are being developed (e.g. Canuto & Dubovikov 1998) and may, hopefully, be included in future calculations. For the present time, however, evolutionary AGB models are still performed using the MLT prescription - mainly due to computer time requirements -, and we also adopt it in this study. During the third dredge-up, however, the use of such a local prescription leads to the development of an (unphysical) discontinuity in the abundance profiles when the H-rich envelope penetrates into the H-depleted layers. The determination of the lower boundary of the convective envelope then becomes problematic, and requires a careful analysis of its stability against mixing across the discontinuity (i.e. extra-mixing). That problematic is set forth in Sect. 2. Several definitions involving the use of the Schwarzschild criterion and of extra-mixing are also presented in that section. Sects. 3 to 6 then analyze the 3DUP predictions from AGB model calculations of a star of solar metallicity, computed both with and without extra-mixing. Sect. 3 describes the main characteristics of the code used to compute the models. A first set of calculations performed without any extra-mixing is analyzed in Sect. 4. It is shown that dredge-up does not occur in those conditions. In contrast, Sect. 5 shows that models calculated with extra-mixing lead naturally to the occurrence of the 3DUP phenomenon. The dredge-up laws derived from those calculations are presented in Sect. 6. Finally, some implications of those laws on stellar evolution along the AGB are analyzed in Sect. 7. Conclusions are drawn in Sect. 8.

© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999
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