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Astron. Astrophys. 351, 161-167 (1999)

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1. Introduction

The [FORMULA] relation for TP-AGB stars, first discovered by Paczynski (1970a), has since then been employed in many studies involving this evolutionary phase. It is a basic ingredient in synthetic calculations of TP-AGB evolution, and an important tool for the interpretation of observational data for AGB stars.

What the [FORMULA] relation means in the classical sense is simply that the quiescent luminosity of a TP-AGB star in the full-amplitude regime is mainly controlled by its core mass, without any dependence on the mass of its outer envelope.

Various theoretical analyses have been performed in the past to explain the existence of the [FORMULA] relation from first principles, using either homology relations (Refsdal & Weigert 1970; Havazelet & Barkat 1979; Kippenhahn 1981), or the equations of stellar structure under specific physical conditions (see Eggleton 1967; Paczynski 1970b; Tuchman et al. 1983; Jeffery 1988).

A transparent discussion of the validity of the [FORMULA] relation was presented by Tuchman et al. (1983), to whom the reader should refer. There it is shown that an [FORMULA] relation necessarily holds when the star consists of:

  • a degenerate core of mass [FORMULA] surrounded by

  • a narrow radiative burning shell (or double shell) source providing most of the luminosity ([FORMULA]), beyond which there must exist

  • a thin (with a mass [FORMULA]) and inert (the luminosity is constant) transition region in radiative equilibrium , extending up to the base of the convective envelope.

Then, because of the extreme steepness of the structural gradients across the radiative inert zone, it follows that the thermal evolution of the core is decoupled from that of the envelope. The relationship between the core mass and the luminosity, defined on the ground of this physical picture, is of linear nature, as confirmed by numerical results (e.g. Paczynski 1970a; Iben 1977; Wood & Zarro 1981; Boothroyd & Sackmann 1988a). Hereinafter, such linear relation will be referred to as the classical [FORMULA] relation .

However, such a simple [FORMULA] relation does not hold for all AGB stars. Blöcker & Schönberner (1991) have shown that the [FORMULA] relation can indeed break down in the most massive AGB stars ([FORMULA] depending on the metallicity) experiencing envelope burning (or hot-bottom burning). In this context, substantial efforts have been made in order to accurately include this effect in synthetic TP-AGB calculations (Marigo et al. 1998; Marigo 1998; Wagenhuber & Groenewegen 1998). It must be emphasized that [FORMULA] relations in synthetic calculations are always technically motivated relations intended to fit results of full stellar evolution calculations. By no means they are just the classical, physically motivated linear relations mentioned above. In the following, such relations will be referred to as technical [FORMULA] relations .

Very recently, Herwig et al. (1998, hereinafter HSB98) have claimed that the classical [FORMULA] relation may also be violated in low-mass AGB stars, as a consequence of efficient third dredge-up. More specifically, they present evolutionary sequences with a dredge-up efficiency close to [FORMULA] or even higher 1, i.e. characterized by an almost constant or slightly decreasing core mass [FORMULA]. Despite of this fact, these sequences are found to evolve at increasing luminosity. This behavior is in apparent contradiction with the trend expected from the classical [FORMULA] relation, predicting lower luminosities at lower core masses.

In the present study we address the question whether these results present a further deviation from the classical [FORMULA] relation, as claimed by HSB98, or if they can be explained, at least partly, by the already known (and understood) deviations.

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© European Southern Observatory (ESO) 1999

Online publication: November 2, 1999