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

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2. Violations of the classical [FORMULA] relation

A violation of the classical [FORMULA] relation implies that, for some reason, the configuration defined in Tuchman et al. (1983) is altered. For instance, the occurrence of hot-bottom burning (or envelope burning) in the most massive TP-AGB stars causes the inert radiative buffer to disappear, due to the deep penetration of the convective envelope into the H-burning shell. Another example refers to the first inter-pulse periods, when the luminosity of a TP-AGB star is found to be lower than predicted by the classical [FORMULA] relation for the same [FORMULA]. In these initial stages the condition [FORMULA] is not actually fulfilled, as the gravitational contraction of the core and the He-burning shell provide non-negligible contributions to the surface luminosity.

In the context of the recent results by HSB98, the first natural question is: does the third dredge-up in low mass TP-AGB stars lead to a real violation of the classical [FORMULA] relation? In other words, is any of the conditions listed above not fulfilled?

The answer is: no. In fact, the degenerate core and the H-burning shell still exist in the quiescent regime after the dredge-up has occurred, as does the radiative inert buffer, since HSB98 are only considering stars which do not experience hot-bottom burning.

The second question is: if the basic conditions for the existence of the classical [FORMULA] relation are still fulfilled, what causes the deviation from the linear [FORMULA] relation?

2.1. The first pulses

In order to answer the latter question, let us consider the previously known deviations from the classical [FORMULA] relation.

The first effect to consider is the initial luminosity evolution of TP-AGB stars. In complete calculations of AGB stars the first thermal pulses still take place during a phase of fast core contraction, at luminosities lower than given by the classical, linear [FORMULA] relation. During a few thermal pulses, the luminosity gradually approaches this relation, up to the so-called full-amplitude regime . During these first pulses unique relations between [FORMULA] and L are not expected to exist.

The sequences of models shown by HSB98 in their Figs. 2 and 3 refer to a relatively small number of pulse cycles, most of which have not yet attained the full-amplitude regime. More specifically, the two evolutionary sequences they computed with overshooting, for [FORMULA] (labelled [FORMULA] in HSB98) and [FORMULA] ([FORMULA]) models, present only 14 and 12 thermal pulses, respectively. In each of these sequences, at least 6 of the pulse cycles are clearly in the sub-luminous phase which characterizes the first pulses. This can be seen in Fig. 1.

[FIGURE] Fig. 1. Evolution of the pre-flash quiescent luminosity for the HSB98 models. The symbols refer to the inter-pulse periods, and are taken from Fig. 2 of HSB98. The continuous line represents a linear [FORMULA] relation which is chosen here to describe the full-amplitude regime of the [FORMULA] and [FORMULA] sequences. The dotted line shows the [FORMULA] relation of Blöcker (1993).

HSB98 compare their sequences with the [FORMULA] relation from Blöcker (1993). The latter is shown as a dotted line in Fig. 1. This relation clearly predicts too faint luminosities if compared to the most luminous points in the sequences of models with no or little dredge-up, [FORMULA] and [FORMULA]. This inappropriateness of the Blöcker [FORMULA] relation to describe the present HSB98 models probably derive from the different input physics used in both sets of models.

We therefore prefer to define another linear [FORMULA] relation, more appropriate to describe the asymptotic behaviour of the HSB98 models without dredge-up. This is shown as the solid line in Fig. 1. This has been chosen to be the one which reasonably fits the 16 last inter-pulse periods (out of 19) in the [FORMULA] sequence, and the last 3 or 4 (out of 23) in the [FORMULA] one. Both evolutionary sequences asymptotically approach this linear relation.

In all the HSB98 evolutionary tracks, the first thermal pulses have luminosities which are lower than predicted by this asymptotic linear [FORMULA] relation we adopt (solid line). In the sequences with efficient dredge-up, only the very last quiescent models are more luminous than predicted by this relation for the same core masses. Specifically, only the last 3 inter-pulse periods of the [FORMULA] track, and the last 5 of the [FORMULA] one are above the [FORMULA] relation. The remaining points are all steadily increasing in luminosity, which is just the behavior expected for the first thermal pulses.

The luminosity increase in the initial phase of the TP-AGB evolution is partly due to the release of gravitational energy by the contracting core, and clearly constitutes a violation of the assumptions for the validity of the classical [FORMULA] relation. However, this effect is well known and already taken into account in the technical [FORMULA] relations in synthetic models (e.g. Groenewegen & de Jong 1993; Marigo et al. 1996, 1998, 1999; Marigo 1998; Wagenhuber & Groenewegen 1998). We note, however, that the behavior of [FORMULA] for the two sequences with efficient dredge-up clearly deviates from those of the sub-luminous pulses without dredge-up, such that the presence of an additional effect resulting from the dredge-up is likely.

2.2. The composition dependence in the [FORMULA] relation

Another important point is related to the change of the surface chemical composition produced by the third dredge-up, and thus to the composition dependence of the [FORMULA] relation.

Indeed, the fact that changes in the chemical composition of the envelope may affect - but do not violate - the classical [FORMULA] relation had already been pointed out long ago from theoretical arguments (e.g. Refsdal & Weigert 1970; Kippenhahn 1981; Tuchman et al. 1983). As clearly derived from Tuchman et al. (1983; see their Eqs. (1.17) and (1.29)) the [FORMULA] relation contains a non-negligible dependence on the composition of the envelope, essentially expressed by three parameters:

  • a factor ([FORMULA]) from the electron scattering opacity,

  • a factor ([FORMULA]) from the mean molecular weight ([FORMULA] for a fully ionized gas),

  • a factor ([FORMULA]) from the hydrogen burning rate.

Since then, various [FORMULA] relations, both classical linear and technical ones which include a composition dependence, have been presented by different authors (e.g. Lattanzio 1986; Boothroyd & Sackmann 1988a; Wagenhuber & Groenewegen 1998; Tuchman & Truran 1998). From these studies it turns out that at any given core mass, the quiescent luminosity of a TP-AGB star increases with increasing metallicity Z, helium content Y, (both leading to a higher mean molecular weight µ), and CNO abundances [FORMULA]. For instance, based on calculations of full AGB models, Boothroyd & Sackmann (1988a) carefully analyzed the composition dependence, deriving a proportionality factor [FORMULA] in their fitting formula of the [FORMULA] relation. They found that at given core mass, stars of solar composition ([FORMULA]) are [FORMULA] more luminous than metal poor stars ([FORMULA]).

It follows that the occurrence of recurrent dredge-up episodes in TP-AGB stars is expected to alter (not to break) even the classical [FORMULA] relation, as a consequence of the increase of the mean molecular weight in the envelope.

Of course, this effect has already been included in several synthetic calculations (e.g. Groenewegen & de Jong 1993; Marigo et al. 1996, 1998; Marigo 1998), where [FORMULA] relations with a metallicity dependence have been adopted. Marigo (1998) has already pointed out that a deviation from the [FORMULA] relation, corresponding to constant metallicity, can be caused by changes in the envelope composition.

We have estimated, from the data presented in Herwig et al. (1997; hereinafter HBSE97), Herwig (1998) and HSB98, the total change in the envelope composition due to the dredge-up events. For the [FORMULA] [FORMULA] track, the mean molecular weight µ is estimated to increase from 0.6314 to 0.6394 during the TP-AGB evolution, whereas for the [FORMULA] [FORMULA] one, it increases from 0.6304 to 0.6376. This implies that in both cases µ increases by 1.3% in total. Assuming [FORMULA] (following Boothroyd & Sackmann 1988a), this change in the envelope chemical composition would imply a change of 4% in the luminosity predicted by the linear [FORMULA] relation for constant metallicity. This already accounts for one-third of the luminosity increase above the [FORMULA] relation drawn in Fig. 1.

2.3. The presence of hot-bottom burning

HSB98 claim that their evolutionary sequences do not present hot-bottom burning, since their core masses are "lower than those associated to hot-bottom burning" (HBB). The highest core mass in their tracks is [FORMULA], whereas they consider HBB to be present only at higher core masses.

However, the knowledge of the core mass is not enough to diagnose the possible occurrence of HBB. Several authors (Boothroyd & Sackmann 1992; Vassiliadis & Wood 1993; D'Antona & Mazzitelli 1996; Marigo 1998) find that the presence of HBB, and its associated "over-luminosity", are sensitive to other stellar parameters as well, as e.g. the envelope mass, metallicity, mixing-length parameter, and to the details of the convection theory.

The latter results have been obtained by means of stellar models that adopt canonical convection theories. It would be interesting to quantify whether the diffusive overshooting scheme applied by HSB98 to all convective boundaries, may also produce conditions favorable to HBB, i.e. higher temperatures at the bottom of the convective envelope at lower core masses. If this were the case the over-luminosity of the tracks may be partially ascribed to the occurrence of (a possibly mild) HBB. In this respect, however, no conclusion can be drawn without additional information about the HBS98 tracks.

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Online publication: November 2, 1999