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

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3. The dependence on the core radius

As remarked above, most of the luminosity behavior of the HSB98 tracks can be understood by means of already known effects taking place during the TP-AGB evolution. HBS98, however, explicitly mention a violation of the classical [FORMULA] relation caused by dredge-up, and provide an explanation for the unusual behavior of their tracks based on the stellar core radius.

The authors find that the core radii, [FORMULA], of their TP-AGB models follow quite different paths in the [FORMULA] plane, depending on whether the models experience dredge-up or not. Then, considering the apparent lack of a unique [FORMULA] relationship and using the homology relation [FORMULA], HSB98 conclude that the luminosity is not a function of [FORMULA] alone, but also of [FORMULA]. As a consequence, the [FORMULA] relation should rather be seen as a [FORMULA] relation.

Moreover, HSB98 claim that the over-luminosity above the [FORMULA] relation of their evolutionary sequences with [FORMULA] can be explained assuming


a fact we cannot confirm. In Fig. 2, we plot the luminosity evolution of the HSB98 [FORMULA] sequences, as derived both from their L and [FORMULA] values (from their Figs. 2 and 3; filled symbols). We then use the [FORMULA] and [FORMULA] values in order to obtain the equivalent luminosity from Eq. (1) above. This procedure however requires that we fix a value to the constant in this equation. The first point of the [FORMULA] sequence is used for this purpose, so that we obtain the relation [FORMULA] (where all quantities are in solar units). The open symbols in Fig. 2 then show the luminosities as obtained from this latter relation.

[FIGURE] Fig. 2. Evolution of the pre-flash quiescent luminosity (squares) for the HSB98 [FORMULA] models (filled dots). The open dots represent the luminosity predicted by means of Eq. (1).

In this way, we have obtained the equivalent of Fig. 4 in HSB98. We find that the luminosities as derived from Eq. (1) are far from reproducing those given by the complete evolutionary sequences, although the general behaviour is similar. In contrast, HSB98 obtain quite a good match between the two sets of curves. Examining their Fig. 4, and comparing it with Fig. 2, we conclude that HBS98 adopt two different scales in their plot (i.e. for the true luminosity and for the luminosity as obtained from Eq. (1)), which are not related by a single multiplicative constant. Thus, their explanation of the luminosity evolution in terms of Eq. (1) is misleading, since obviously, the "constant" they adopt varies from model to model along the evolutionary sequences.

The possible dependence of the luminosity on core radius deserves the following remarks. This dependence would reflect the release of gravitational energy by the contracting core. During the TP-AGB evolution, the core contracts more rapidly during the first thermal pulses, until an almost constant and very low contraction rate is established in the full-amplitude regime (Herwig 1998). Another concurring effect comes from the decrease of the ratio [FORMULA] between the gas and the total pressure at increasing luminosities. The homology relation actually predicts [FORMULA], where the exponents [FORMULA] and [FORMULA] are given in Eq. 3 of HSB98. Since [FORMULA], the radius dependence in Eq. (1) vanishes as we increase L and hence [FORMULA].

For both reasons, an [FORMULA] relation independent of [FORMULA] should hold after a certain time. Unfortunately, the calculations by HSB98 have been stopped at the most important point, i.e. where the evolution of the core radius as a function of the core mass for the models with efficient dredge-up joins the standard [FORMULA] relation described by the models without dredge-up after the initial pulses (see their Fig. 3). If, from this point on, both relations follow the same path, then the entire effect presented by HSB98 is indeed related to peculiar behaviour of the first pulses, before the settling of the full-amplitude regime. In this case, there would be no real violation of the [FORMULA] relation.

Otherwise, if there is a different dependence of the core radius upon the core mass, the [FORMULA] relation might be at least partly modified. If this were the case, it would be quite important to single out the physical effect produced by the convective dredge-up, which occurs at a thermal pulse, on the core radius during the subsequent quiescent evolution. In other words, why should the evolution of the core radius turn out different between models with and without dredge-up? This point is not clear in the HSB98 analysis.

To this respect, an interesting point is discussed by Tuchman et al. (1983), in their analytical demonstration of the [FORMULA] relation from first principles. In brief, the authors shows that core radius of an AGB star is larger than the radius of an ideal zero-temperature white dwarf (for which a unique linear [FORMULA] relation exists) by a multiplicative factor, [FORMULA], which depends both on the core mass [FORMULA] and on the temperature [FORMULA] at the top of the H-burning shell (i.e. bottom of the overlying inert radiative buffer; see their Eqs. 1.18 and 1.19). This factor [FORMULA], being typically [FORMULA] for relevant burning shells, decreases with [FORMULA] and increases with [FORMULA]. Hence, in order to get a greater shrinkage of the core in the AGB models with efficient dredge-up, while the core mass is kept constant, a lower temperature [FORMULA] should be attained during the quiescent regime. This, in fact, would result in a smaller [FORMULA], and hence in a smaller [FORMULA]. The final result would be a certain excess of luminosity with respect to the reference [FORMULA] relation. Unfortunately, no information about [FORMULA] is given in HSB98, but it would be worth investigating this point with the aid of full AGB calculations.

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Online publication: November 2, 1999