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

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5. The role of overshoot at the boundaries of the He-flash convection zone

During the He-flash the intershell region becomes convectively unstable due to the huge energy generation of the He-burning shell. Overshoot at the bottom of this He-flash convection zone leads to a deeper mixing from the region below the He-shell (the C/O core) into the intershell. This process could be called the fourth dredge-up Iben 1999 or intershell dredge-up. Both the abundance in the intershell and also the structural parameters in the convection zone are affected by overshoot at the base of the He-flash convection zone. In particular, the change of the intershell abundance and the larger energy generation during the TP leads to very efficient TDUP Herwig et al. 1999b. If these overshoot AGB models are used as starting models for the post-AGB evolution the long-standing discrepancy between observed surface abundances of H-deficient post-AGB stars and stellar models of this phase can be resolved Herwig et al. 1999a.

5.1. Intershell structure during the flash

During the onset of the He-flash, when helium-burning increases rapidly, the layers just above the position of maximum energy generation and temperature become convectively unstable and the pulse-driven convection zone develops (Fig. 8). In this situation overshoot at the bottom of the PDCZ supplies additional fresh helium into the underlying nuclear burning region and supports the nuclear runaway. The situation is somewhat similar to hot-bottom burning were the hydrogen burning shell obtains additional nuclear fuel from the deep envelope convection and accordingly increases its energy production. Similarly the extra mixing below the PDCZ leads to a more violent He-flash with larger peak He-burning luminosities (Fig. 9) and larger mass [FORMULA] of the convectively unstable region (Fig. 7d). The stronger He-flashes are responsible for the structural and abundance consequences discussed below. The larger extent of the He-flash convection zone has immediate consequences, e.g. for the s-process nucleosynthesis: Heavy elements produced during the interpulse phase under radiative conditions are diluted in the convective intershell during the pulse before they are dredged-up into the envelope. For a larger He-flash convection zone the dilution effect will be larger.

[FIGURE] Fig. 7a-d. Evolution of the position and size in mass coordinates of several quantities at each TP as a function of time. Full symbols show the 3[FORMULA] sequence computed with overshoot at all convective boundaries, open symbols represent the comparison sequence without any overshoot and started with the same initial model before the first TP as the 3[FORMULA] sequence with overshoot. Panel a : H-free core at the TP, the increase by H-burning during the interpulse period is counterbalanced by the dredge-up after the TP for the overshoot sequence; b : difference of core mass at TP and smallest mass coordinate of convective envelope bottom achieved after the TP, negative values indicate DUP; c : difference of core mass at TP and the largest mass coordinate of the PDCZ top achieved during the TP, d : difference of top and bottom mass coordinate of PDCZ.

[FIGURE] Fig. 8. Onset of He-flash convective instability in the intershell region at the first TP of 3[FORMULA] sequence with overshoot. The two vertical lines show the upper (right) and lower (left) boundary of convection.

[FIGURE] Fig. 9. The maximum helium-burning luminosities during the He-flash for the 3[FORMULA] model sequence with (filled triangles) and without (open triangles) overshoot. Each symbol corresponds to one TP. The generally larger peak flash luminosities of the overshoot TPs are due to the extra-mixing below the bottom of the He-flash convection zone which leads to higher temperatures in the He-shell during the flash (Fig. 10). The relation runs vertically at [FORMULA] for the OV case because efficient dredge-up prevents further core mass growth ([FORMULA]).

Apparently there is no noticeable effect of overshoot at the top of the PDCZ. For the metallicity and overshoot efficiency considered here the minimum mass layer remaining between the H-rich envelope and the top of the PDCZ at its largest extension is independent on overshoot as can be seen from Fig. 7c.

The sequence without overshoot does not show any TDUP, and accordingly the whole intershell with the He-burning, and the H-burning shell at each boundary respectively is shifted outward by mass. For the 3[FORMULA] sequence with overshoot dredge-up sets in at the third TP and the effective core mass growth ([FORMULA]) is slowed down (Fig. 7a,b). At the last computed TP of the 3[FORMULA] sequence the intershell no longer evolves outward with mass. During this stationary shell burning the nuclear fuel for both the hydrogen and helium shell is transported downward by dredge-up. Over the pulse cycle the dredged-down hydrogen is first converted into helium which is then further mixed down by the He-flash convection zone. Finally the burning products are exchanged by dredge-up with fresh envelope material. In this situation dredge-up is not only important to mix processed material up but also to mix fresh material down and feed the stationary shells.

For the 4[FORMULA] sequence the core mass actually decreases from pulse to pulse because [FORMULA] (Table 2). In that case the nuclear burning shells are shifted inward with respect to mass. Obviously this situation can only last because fresh material for the nucleosynthesis is transported downward by dredge-up.

At the fifth TP (3[FORMULA]) the core masses of the sequence with and without overshoot have not yet diverged too much due to the dredge-up difference between the two cases. However, the differences due to overshoot are already well established. Not only is the mass-wise extent of the PDCZ larger with overshoot (Fig. 7d) but in addition the convective instability is longer (Fig. 10a). The He-flash convection zone develops and disappears faster with overshoot. The overall duration of convective instability in the intershell is about [FORMULA] with overshoot and more than [FORMULA] for the case without overshoot.

[FIGURE] Fig. 10a-c. Panel a : The mass coordinates of the PDCZ boundary during the fifth TP of the 3.0[FORMULA] with and without overshoot. The time has been set to zero at the moment where the He-burning luminosity has reached the maximum. Panel b : Temperature at the boundary of the He-flash convection zone of the same TP. Panel c : Geometric position of the PDCZ boundary (CB) and of the maximum energy release by helium burning.

With overshoot the temperature at the bottom of the He-flash convection zone is larger while at the top of the convectively unstable zone the temperature is smaller (Fig. 10b). Although the overall duration of the convective instability is larger without overshoot it is important to note that the high-T phase is much longer with overshoot. The duration of the high-T phase and the temperature reached during this phase is important for the analysis of the nucleosynthesis during the He-flash, in particular of the [FORMULA] neutron source ([FORMULA]). Without overshoot the maximum temperature reached at the bottom of the He-flash convection zone during the fifth TP is [FORMULA]. With overshoot the temperature at the bottom of the He-flash convection zone exceeds [FORMULA] for [FORMULA] while [FORMULA] for [FORMULA]. Over the following TPs the temperature in the He-flash convection will increase steadily. However, the general trend described above will remain preserved. Therefore, the exact mechanism of the s-process and also other aspects of the nucleosynthesis will be affected not only by the different intershell abundances due to overshoot but also because of different temperatures and time scales in the intershell during the He-flash.

Fig. 10c displays the geometric evolution of the He-flash convection zone together with the position of the maximum energy generation by helium burning. The fact that the temperature at the top of the He-flash convection zone is lower with overshoot is closely related to the greater expansion of the intershell in this case. With overshoot the largest geometric extent of the He-flash convection zone is about 1.4 times the distance found without overshoot.

5.2. The abundances in the intershell region

Models with overshoot have larger intershell mass fractions of carbon and oxygen at the expense of helium. In Fig. 11 the variation of the intershell abundances from pulse to pulse is shown. These models with overshoot show qualitatively the same dependence on the pulse number like models without overshoot Schönberner 1979; Boothroyd & Sackmann 1988b but the quantitative abundances are very different. During the first TPs the [FORMULA] and [FORMULA] abundance increases strongly at the expense of [FORMULA]. After about six TPs the [FORMULA] abundance in the intershell reaches a minimum while [FORMULA] and [FORMULA] go through a maximum. After about five to ten additional TPs all abundances are leveling at values which are similar to those found after the second or third TP. Quantitatively, overshoot with the efficiency [FORMULA] increases the amount of carbon by about a factor of two and the amount of oxygen by a factor of 10 to 20.

[FIGURE] Fig. 11. Time evolution of abundances (mass fraction) in the upper part of the intershell shortly after each TP of the [FORMULA] and [FORMULA] model sequence with overshoot (compare with Boothroyd & Sackmann Boothroyd & Sackmann (1988a), Fig. 9). Each mark indicates one thermal pulse starting at the first TP. The zero point for the [FORMULA] sequence has been set at the third TP where the TDUP stars.

Test calculations in order to study the dependence of the intershell abundance on the efficiency of overshoot reveal that there is a relation between the overhoot parameter and the intershell abundance: larger overshoot leads to larger carbon and oxygen abundances and consequently smaller helium abundance. We have followed the evolution from the same starting model before the first TP over three TPs with different values of f. In Fig. 12 we display the abundance in the upper part of the intershell after the third TP. For [FORMULA] the abundances are the same as in Schönberner Schönberner (1979). With larger and larger overshoot efficiency, the amount of oxygen increases almost linearly.

[FIGURE] Fig. 12. Intershell abundances for different overshoot efficiencies after the third TP. Test model sequences have been calculated from the same starting model before the first TP with different values of f. The abundances for the sequence with [FORMULA] are computed without overshoot.

This correlation leads to a quite stringent constraint for the possible range for the overshoot efficiency f at the bottom of He-flash convection zone. We compare our intershell abundances with the surface abundances of the [WC]-type central stars of planetary nebulae Koesterke & Hamann 1997; Hamann 1997; De Marco et al. 1998 and the PG 1159 stars Dreizler & Heber 1998; Werner et al. 1999. The spectroscopic abundance analysis shows that these stars are very carbon rich and also oxygen rich. Typically, one finds (He,C,O)=(0.50,0.33,0.17) (mass fractions) for PG 1159 stars. Post-AGB stars become hydrogen-deficient because the intershell material appears at the surface in the aftermath of a very late TP Herwig et al. 1999a. Thus, the observed surface abundances of H-deficient post-AGB stars must be interpreted as the intershell abundances of the progenitor AGB stars Herwig & Blöcker 2000. Models without overshoot never show more than [FORMULA] of oxygen in the intershell and are thus unable to reproduce the large mass fraction of oxygen observed. This is a strong indication of the presence of at least some overshoot at the bottom of the He-flash convection zone. However, practically no H-deficient post-AGB stars are known to show more than about [FORMULA] of oxygen 2.

The oxygen abundances shown in Fig. 12 are an estimate of the surface oxygen abundance of a H-deficient post-AGB model evolved from the respective AGB model because the oxygen abundances after the third TP is already similar to the later TPs (Fig. 11). Any overshoot parameter much larger than [FORMULA] would lead to H-deficient post-AGB models with too large an oxygen abundance. Therefore, observations - together with the theoretical understanding of the evolutionary origin of H-deficient post-AGB stars - do constrain the overshoot efficiency at the bottom of the He-flash convection zone to a narrow range of [FORMULA].

5.3. The third dredge-up and its dependence on overshoot at the bottom of the He-flash convection zone

One of the most important properties of TP-AGB models with overhoot is the efficient TDUP, which solves the problem of previous models to account for the typically low luminosities of carbon stars. In Table 1 and 2 the amount of dredged-up intershell mass and the related values for the dredge-up efficiency [FORMULA] 3 are given for the 3[FORMULA] and 4[FORMULA] sequence with overshoot, and these results will be further discussed in Sect. 6.

Here, we consider the dependence of [FORMULA] on the efficiency of overshoot. In particular it is important to clarify the different roles of overshoot below the bottom of the He-flash convection zone and below the bottom of the envelope convection zone (see also Sect. 4.1). For this purpose we have made a numerical experiment which involved the computation of one TP cycle with different values for the overshoot efficiency parameter f. The computations were started immediately after the He-flash convection zone of the eighth TP had disappeared. Thus, the dredge-up episode following this TP has been computed with different f values at the envelope convection zone only, while during the preceding TP f was the same for all convective boundaries. The resulting values for [FORMULA] are shown in Fig. 13 as open symbols and demonstrate the dependence of the third dredge-up on the overshoot efficiency below the convective envelope. Within the considered range of f-values practically no dependence of [FORMULA] on the overshoot efficiency exists. Only without any overshoot ([FORMULA]) does dredge-up not occur for this TP. Even with f as small as 0.004 dredge-up is found with practically the same efficiency than with larger f (see Sect. 4.1). Note that the value of [FORMULA] displayed for the open symbols in Fig. 13 is just the value found for the eighth TP of the 3[FORMULA] overshoot sequence (Table 1). If another TP had been chosen for this numerical experiment [FORMULA] would have had the correspondingly larger or smaller value. Fig. 13 does not mean that models with overshoot applied only to the envelope convection always have [FORMULA].

[FIGURE] Fig. 13. Variation of the dredge-up efficiency [FORMULA] with the overshoot efficiency. The open symbols represent cases where the f value has been changed just after the pulse-driven convective zone has disappeared after the eighth TP but before the bottom of the envelope convection has come close to the mass coordinate of the H-free core. Thus, in this case a different overshoot parameter is only effective at the bottom of the envelope during the dredge-up phase. The full symbols show the dredge-up efficiency after the following ninth TP which has been computed with the respective f-value also for the He-flash convection zone. In this case a different overshoot parameter has been effective at the bottom of the He-flash convection zone.

The test model sequences have then been evolved beyond the dredge-up episode after the ninth TP. The respective f-values have now been applied to the convective boundaries of the He-flash convection zone as well. The resulting [FORMULA]-values are represented by filled symbols in Fig. 13 and show the dependence of the dredge-up on the overshoot efficiency at the bottom of the He-flash convection zone. The full symbol at [FORMULA] shows [FORMULA], the value given in Table 1 for the ninth TP. f and [FORMULA] are correlated: larger overshoot leads to larger dredge-up. The relation defined by the full symbols is specific to the chosen TP of this sequence. However, the trend is generally valid. A combination of effects is responsible for this correlation. With intershell overshoot the peak He-burning luminosities are substantially larger (Fig. 9). At the 14th TP of the overshoot sequence ([FORMULA]) the He-flash peak luminosity is [FORMULA]. At the 14th TP of the sequence without overshoot ([FORMULA]) the peak luminosity is [FORMULA]. Boothroyd & Sackmann Boothroyd & Sackmann (1988a) found for a 3[FORMULA], Z=0.02 model sequence [FORMULA] after about 20 TPs at a core mass [FORMULA]. These models were computed without overshoot during the pre-AGB evolution and therefore have a lower core mass at the first TP [FORMULA].

Greater He-flash strength leads to a greater expansion and cooling of the upper layers of the intershell (Fig. 10) and favors the occurrence of the TDUP ([FORMULA]). In Fig. 14 one can see that the modification of the intershell abundance during only one TP is considerable. The smaller He abundance with larger f leads to a larger opacity [FORMULA] which favors the dredge-up also because [FORMULA].

[FIGURE] Fig. 14. The intershell abundances after the ninth TP as a function of f. While for Fig. 12 the different overshoot values have been applied over all previous TPs here we have computed only the ninth TP with different overshoot values. The starting model has been a 3[FORMULA], [FORMULA] model before the ninth TP. This figure demonstrates how the abundances are modified due to overshoot during only one TP.

Thus the overshoot at the base of the He-flash convection zone strongly affects TDUP. In the previous section we have shown that f is constrained by the observational properties of H-deficient post-AGB stars. Therefore we conclude that the upper limit of the efficiency of TDUP can be constrained from observational properties of post-AGB stars.

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Online publication: August 23, 2000
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