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

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4. Synthetic calculations with dredge-up

Here we present the results of synthetic calculations carried out with different technical [FORMULA] relations including a composition dependence and the first subluminous pulses (see Figs. 3 and 4). The models are meant to be useful experiments, giving a first hint of how the quiescent luminosity of a TP-AGB star may behave when dredge-up events strongly alter the envelope composition.

[FIGURE] Fig. 3. Evolution of the pre-flash quiescent luminosity (squares) for the [FORMULA] model, assuming [FORMULA], the composition of the dredged-up material from Herwig et al. (1997), and the [FORMULA] relation from Boothroyd & Sackmann (1988a). The grid of dotted lines corresponds to [FORMULA] relations for various values of µ and [FORMULA] (both increasing with L). The filled squares mark a few selected values of the quiescent luminosity, each determined, at given core mass, by that [FORMULA] relation of the grid which is consistent with the current surface chemical composition.

[FIGURE] Fig. 4. Quiescent luminosity (i.e. at the pre-flash maximum) as a function of the core mass for an evolving [FORMULA] TP-AGB star (dotted lines). The luminosity evolution is derived adopting various [FORMULA] relations (i.e. Boothroyd & Sackmann 1988a, BS88; Tuchman & Truran 1998, TT98; Wagenhuber & Groenewegen 1998, WG98), assuming two different prescriptions for the chemical composition of the dredge-up material (BS88 or HBSE97; see the text for more details), and different choices for the efficiency parameter of dredge-up ([FORMULA]) as indicated nearby the corresponding curve. In each panel, the solid line refers to the [FORMULA] relation consistent with the envelope composition at the onset of the full-amplitude regime.

Calculations are carried out over a limited number of inter-pulse periods for a 3 [FORMULA] TP-AGB star with original solar composition (i.e. [FORMULA], [FORMULA]). The chemical composition of the envelope at the first thermal pulse is characterised by ([FORMULA]). The third dredge-up is assumed to occur once the full amplitude regime is attained, i.e. after the first subluminous pulses when the linear [FORMULA] relation is approached. We adopt various values of the dredge-up parameter ([FORMULA]) and two prescriptions for the composition of the dredged-up material. They are (in mass fraction):

Here we are not concerned to give a detailed description of dredge-up and its properties. For instance, we assume that dredge-up events take place at each thermal pulse in the full amplitude regime, whereas we expect that a significant increase of the envelope metallicity could, at a certain stage, even inhibit further occurrence of the process by decreasing the temperature at the base of the convective envelope during the post-flash luminosity maximum (see e.g. Boothroyd & Sackmann 1988c).

Since most available [FORMULA] formulae were obtained for relatively small ranges of metallicity, usually not super-solar, they may not give realistic results if the envelope metallicity increases to very high values. However, in this respect, the recent analysis developed by Tuchman & Truran (1998) is relevant. They have quantitatively investigated the composition influence upon the [FORMULA] relation, in order to better estimate the luminosity of classical novae, objects in which shell hydrogen burning is known to occur in extremely metal-rich material (e.g. [FORMULA]). At such high values of the metallicity, the corresponding [FORMULA] relation is shifted to significantly higher luminosities than predicted for solar composition.

Fig. 3 shows the locus traced by the [FORMULA] TP-AGB model experiencing dredge-up, adopting an efficiency [FORMULA] and the HBSE97 prescription for the chemical composition of the inter-shell. A grid of classical [FORMULA] relations (from Boothroyd & Sackmann 1988a) is also plotted for increasing values of the mean molecular weight, ranging from [FORMULA] to [FORMULA] in steps of about 0.0025. The envelope composition of the last calculated model ([FORMULA] pulse) is characterised by ([FORMULA]). For a core mass [FORMULA] the quiescent luminosity is [FORMULA], corresponding to an over-luminosity of about [FORMULA] with respect to the case in which the chemical composition were unchanged and equal to that of the first thermal pulse.

Other examples are presented in Fig. 4. All models show that the deviations from the [FORMULA] relation at constant metallicity are greater for increasing [FORMULA] and/or for higher abundances of carbon and oxygen in the inter-shell. We can also note that models with [FORMULA] also evolve to luminosities above the reference [FORMULA] relation, despite the effective decrease of the core mass.

Finally, we remark that our synthetic results shown in Fig. 4 reproduce the behavior of the luminosity as found by HSB98. For instance, the cases with [FORMULA] and [FORMULA] clearly resemble the sequences [FORMULA] and B[FORMULA] in their Fig. 2, respectively. However, it must be specified that in our calculations with extremely efficient dredge-up ([FORMULA]), a considerable over-luminosity above the [FORMULA] relation shows up after a much larger number of dredge-up episodes ([FORMULA]) if compared to the results by HSB98 ([FORMULA]). This difference can be partly ascribed to the fact that in our case the onset of third dredge-up occurs only when the full-amplitude regime is attained, whereas in HSB98 dredge-up takes place from the first thermal pulses on, when other effects (in addition to the chemical composition, see Sects. 2.1 and 3) are likely to play a role.

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© European Southern Observatory (ESO) 1999

Online publication: November 2, 1999