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

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5. Concluding remarks

In this paper we claim that the results from HSB98 may partially be understood by means of the already known deviations from the classical linear [FORMULA] relation.

An important effect certainly present in their evolutionary calculations is the increase in luminosity associated with the initial core contraction that occurs during the first thermal pulse cycles of any TP-AGB star. This phase of rapid luminosity evolution represents a substantial fraction of the tracks presented by HSB98. In order to determine if dredge-up really leads to a violation of the classical [FORMULA] relation, which is expected to hold for the later evolution of AGB stars, the HSB98 evolutionary sequences should be extended in order to include a much larger number of thermal pulses.

In fact, the most evident effects of the efficient dredge-up in HSB98 evolutionary sequences are:

  1. The small or negative changes in the core mass from pulse to pulse, which cause the tracks to evolve almost vertically in the [FORMULA] diagram, instead of along a line of increasing core mass and luminosity.

  2. The changes in the surface chemical composition which make their quiescent luminosity deviate from that predicted by an [FORMULA] relation obtained for a constant value of metallicity.

None of these effects, however, implies a violation of the classical [FORMULA] relation. The structural conditions for the existence of a [FORMULA] relation are expected to hold only after the tracks enter in the full-amplitude regime, as remarked above.

In this regard, we remark that the evolutionary tracks should be compared with the [FORMULA] relation obtained from the current chemical composition of the envelope, and not with those obtained from tracks of constant metallicity. Also, the possible presence of hot-bottom burning should be completely ruled out before we can tell about deviations from the [FORMULA] relation. It would be of particular interest, for instance, to investigate the evolution of low-mass stars ([FORMULA]) computed with a similar algorithm for convection as in HSB98.

It turns out that the correct interpretation of HSB98 results requires the analysis of additional quantities along their evolutionary tracks, other than the core mass, luminosity, and core radius. These quantities are: the fraction of the stellar luminosity provided by the release of gravitational energy (necessary to identify if the full-amplitude regime has been reached), the surface chemical composition of the models (necessary to better quantify the deviation from the initial [FORMULA] relation due to composition changes); the luminosity provided by nuclear burning in the convective envelope (necessary to rule out the presence of hot-bottom burning); and the temperature [FORMULA] at the top of the H-burning shell (useful to investigate its effect on the factor [FORMULA], defined in Sect. 3, and hence on the core radius [FORMULA]). Unfortunately, this information is not provided by HSB98.

We stress once more that synthetic TP-AGB models have already been adopting technical non-linear [FORMULA] relations, i.e. including significant deviations from linearity due to the sub-luminous first thermal pulses and changes in the surface chemical composition produced by dredge-up (e.g. Groenewegen & de Jong 1993; Marigo et al. 1996; Marigo 1998). Moreover, the real breakdown of the [FORMULA] relation caused by hot-bottom burning in the most massive AGB stars have been accurately taken into account (Marigo et al. 1998; Marigo 1998; Wagenhuber & Groenewegen 1998) in these models. Finally, we recall that the [FORMULA] relation applies only to the quiescent inter-pulse periods, but not to the luminosity variations driven by thermal pulses. Even the effect of the post-flash low-luminosity dip is usually included in synthetic TP-AGB calculations.

Therefore, synthetic AGB evolution calculations already include all known effects affecting the [FORMULA]-relation and do not rely on the assumption that the classical, linear [FORMULA] relation is valid. A corresponding comment in HSB98 turns out to be inappropriate. As such, any new effect, as possibly indicated by the HSB98 calculations can easily be incorporated after sufficient data from full calculations are available.

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© European Southern Observatory (ESO) 1999

Online publication: November 2, 1999
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