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Astron. Astrophys. 360, 952-968 (2000)

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6. Surface properties

In the previous sections we have described the different mechanisms by which overshoot influences the model properties. We will now focus on the surface properties of the models with overshoot.

Stellar parameters. The comparison of the core mass-luminosity relation of models with and without efficient dredge-up shows clear differences Herwig et al. 1998. While models without dredge-up follow a linear relation when the asymptotic regime has been reached, models with very efficient dredge-up ([FORMULA]) continue to increase in luminosity even if the core mass no longer increases. Here, the continuing radius decrease leads - according to simple homology relations Refsdal & Weigert 1970 - to an increase in luminosities. This effect is most efficient over the earlier thermal pulses where the relative radius decrease per TP cycle is larger than after many thermal pulses when the core asymptotically resembles a white dwarf. The radius effect is responsible for the sub-luminous phase (compared to the luminosities expected from the core mass - luminosity relation) of the first few TPs, which is well known from any TP-AGB model sequence. However, the luminosity evolution continues to be strongly coupled to the core radius evolution. This becomes apparent only if the core mass is prevented from growing continuously in accordance with the radius shrinkage, as in models with efficient dredge-up. In addition, as Marigo et al. Marigo et al. (1999) have pointed out, possibly up to one third of the luminosity increase observed by Herwig et al. Herwig et al. (1998) can be ascribed to the well known effect of the molecular weight increase in the envelope as a result of dredge-up of processed material.

The rather short-term variation of the stellar parameters during and between the thermal pulses are affected as well. In general the surface luminosity reacts to the TP in the deep interior by a sudden ([FORMULA]) and drastic luminosity decline of about [FORMULA] of the pre-TP luminosity. It is followed (again on the thermal time scale) by an immediate luminosity jump which forms the more or less pronounced TP surface luminosity peaks (see Fig. 15 at [FORMULA]). The model sequence without overshoot shows only small TP-luminosity peaks after the first TPs with peak luminosities lower than the quiescent interpulse luminosity. Only after the third TP does the peak luminosity exceed the interpulse luminosity and over the following TPs the ratio of peak and interpulse luminosity grows. The model sequence with overshoot shows very large peaks already after the first TP with a ratio of [FORMULA] between the peak and the interpulse luminosity which is gradually decreasing towards later TPs. At the fifth TP (Fig. 15) the surface luminosity peak is similar for the two sequences. The actual local He-luminosity peak during a TP is followed by a much smaller secondary peak. As this secondary maximum decays in the intershell the surface luminosity does also decline to the second surface luminosity minimum at [FORMULA] in Fig. 15. Another difference caused by overshoot is a faster recovery of the H-burning shell after the TP (top panel Fig. 15). This is also reflected by the evolution of the surface parameters. For the model sequence with overshoot it takes only [FORMULA] from the TP until a [FORMULA] level of the previous interpulse luminosity at the surface has been regained while the sequence without overshoot requires [FORMULA]. This effect is even more pronounced at later TPs, however it is then intermingled with the effect of different core mass evolution due to the different efficiency of the dredge-up.

[FIGURE] Fig. 15. Luminosity of H-burning (top panel), effective temperature (middle panel) and surface luminosity (bottom panel) for one complete pulse cycle of the sequence with (solid line) and without (dashed line) overshoot respectively. Time has been set to zero at the fifth TP.

Chemical evolution. The chemical evolution of AGB models with overshoot is different compared to models without overshoot due to four main effects:

  1. The formation of a [FORMULA] pocket in the upper part of the intershell region (Sect. 4.2). The consequences for the n-capture nucleosynthesis will be investigated elsewhere.

  2. Substantially larger dredge-up, which leads to more efficient transport of processed material to the surface.

  3. Higher temperature in He-flash convection zone during the TP (Fig. 10).

  4. Depletion of helium and larger [FORMULA] and [FORMULA] abundance in the intershell due to deeper penetration of the bottom of the He-flash convection zone into the C/O-core.

These effects are reflected by the chemical evolution at the stellar surface (Tables 1 and 2).

C/O and carbon isotopic ratio. Recurring and efficient dredge-up transports primary [FORMULA] into the envelope after each TP. [FORMULA] is not produced or destroyed in the stars of the mass range discussed here. This leads to a steady increase of the [FORMULA] ratio at the surface. At the same time also the C/O ratio increases from its initial value of [FORMULA] and exceeds unity after a number of TPs. Note that in calculations with overshoot [FORMULA] is also mixed into the envelope at each dredge-up event. Because of this additional [FORMULA] the C/O ratio grows more slowly with overshoot than without overshoot if [FORMULA] where the same. However, this effect is more than compensated for by two other effects: the dredge-up itself is larger and the mass fraction of [FORMULA] and [FORMULA] of the dredged-up intershell material is larger. This leads to a larger increase of the C/O ratio for each TP. For example, as can be seen from Table 2 the 4[FORMULA] sequence becomes C-rich at the tenth TP contrary to the results of Forestini & Charbonnel Forestini & Charbonnel (1997) and Marigo et al. Marigo et al. (1996) who both find that a 4[FORMULA] AGB star of solar metallicity does not become a C-star at all. Note that these authors assume dredge-up to operate and that HBB cannot prevent a 4[FORMULA] star from getting C-rich.

Fig. 16 compares observational data of the chemical composition of AGB stars with the model data of the sequences with overshoot. The filled symbols in the lower left corner correspond to the first TPs. The observational data shows properties of M- S- and C-giant 4 stars with different masses (most of them smaller than those of the model sequences) which presumably also have experienced a different number of TPs. 5 However, apart from giants with [FORMULA] and very low [FORMULA] ratio (J-giants), a correlation between the two ratios is apparent. With respect to the model data this correlation must be interpreted as a time evolution. The model sequences do generally follow the observed trend. However, the theoretical relation shows an offset to the observations and also a somewhat smaller slope. In addition the increment of the ratio from TP to TP is rather large and depending on mass loss several more TP are likely for these model sequences, leading to very large [FORMULA] ratios. A somewhat smaller overshoot efficiency would result in a lower dredge-up efficiency and a smaller amount of dredged-up oxygen. While the first effect would decrease the steps at which the model sequence passes along the diagram, the latter would increase the slope of the theoretical relation somewhat. As another detail, the initial abundance ratios have a considerable influence on the theoretical prediction. Most of the displayed stars are probably of lower mass and therefore their initial [FORMULA] ratio before the first TP was closer to [FORMULA] Gilroy 1989 than around 20. This initial [FORMULA] ratio determines the the slope of the [FORMULA] - C/O relation as well, with a smaller initial ratio leading to a larger slope. Thus, models with initial masses around or below 2[FORMULA] should show a larger slope due to this effect and should show a better agreement with the observations.

[FIGURE] Fig. 16. Reproduction of [FORMULA] versus [FORMULA] plot in Smith & Lambert (1990) (open symbols) together with the surface ratios of the [FORMULA] (filled circles) and [FORMULA] (filled pentagons) model sequence. See text for details.

Oxygen isotopic ratios.

Observationally the oxygen isotopic ratios [FORMULA] and [FORMULA] of AGB stars extend to much larger values than predicted by evolutionary models. Theoretical models without overshoot cannot account for increasing [FORMULA] and [FORMULA] ratios because dredge-up is only achieved with difficulty for the relevant masses. Even if this problem is circumvented by assuming dredge-up to be present then the dredged-up material leads only to a rather inefficient dilution of [FORMULA] and [FORMULA] in the envelope because the dredged-up material is depleted in these isotopes compared to the envelope value. The [FORMULA] abundance in the intershell of models without overshoot is similar to the envelope abundance and no modification can be expected there.

Harris et al. Harris et al. (1985); Harris et al. (1987) determined for [FORMULA] and [FORMULA] values between 600 and 3000. It is also interesting to note that the two ratios are correlated. Moreover, both ratios seem to be correlated with the neutron exposure associated with the observed s-process abundances and also with the carbon abundance. Both, an increasing carbon abundance and an increasing neutron exposure are indicating more efficient or more frequent dredge-up because s-process elements and carbon are not made in or at the bottom of the envelope. It is therefore likely that the enormous oxygen isotopic ratios are related to the TDUP.

Forestini & Charbonnel Forestini & Charbonnel (1997) have assumed dredge-up and predict from an extrapolation of their models without overshoot an increase of the [FORMULA] ratio of [FORMULA] ([FORMULA]) from initially [FORMULA] for the entire AGB evolution of their solar 3[FORMULA] (4[FORMULA]) case and almost the same values for the [FORMULA] ratio starting from initially [FORMULA]. The evolutionary effect in the overshoot model is significantly larger because the abundance of [FORMULA] in the dredged-up material is five to ten times larger if overshoot is considered compared to models without overshoot. Over the first dozen TPs available in Table 1 and 2, the oxygen isotopic ratios increase by about [FORMULA] ([FORMULA]) for the 3[FORMULA] (4[FORMULA]) case. Similar to what has been said previously a comparison of absolute numbers observed to the model predictions is misleading at this stage. The initial model [FORMULA] ratios are probably too small, possibly due to a wrong [FORMULA] reaction rate. Also, most of the observed data will belong to less massive stars for which comparable model data do not yet exist. Clearly, this issue needs a more detailed investigation. However, it seems that overshoot can contribute to the solution of this problem.

Magnesium isotopic ratios. The evolution of the magnesium isotopic ratios [FORMULA] and [FORMULA] are not only tracers of the TDUP but also of the temperature in the He-burning shell during the thermal pulses. While [FORMULA] is not produced or destructed in AGB stars the other Mg isotopes are produced during the TP by [FORMULA] captures of [FORMULA]. The latter is abundant in the intershell as a result of two [FORMULA] captures on [FORMULA]. Since the temperature at the bottom of the He-flash convection zone is larger if overshoot is present (Fig. 10) the reactions [FORMULA] and [FORMULA] are more efficient. The 3[FORMULA] (4[FORMULA]) sequence accordingly shows a decrease of [FORMULA] from 8.0 (8.1) at the first TP to 4.9 (3.9) at the last TP computed. The [FORMULA] ratio is 7.1 (7.0) for the 3[FORMULA] (4[FORMULA]) sequence at the first TP and decreases to 5.8 (5.3) at the last computed TP (see previous tables). Again, this decrease for the first dozen TP is stronger than that predicted by Forestini & Charbonnel Forestini & Charbonnel (1997) for the entire evolution.

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© European Southern Observatory (ESO) 2000

Online publication: August 23, 2000