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Astron. Astrophys. 357, 669-676 (2000)

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4. Comparison with AGB model predictions

We used TP-AGB models obtained with the FRANEC evolutionary code for solar metallicity and initial masses 1.5, 3 and 5 [FORMULA] , adopting the Reimers' parameterization (Reimers 1975) for mass loss ([FORMULA] [FORMULA] 0.7, 1.5 and 10, respectively). In the 1.5 and 5 [FORMULA] models, mass loss was actually accounted for in a post-process calculation, after computing the series of thermal pulses from a model with constant mass (Gallino et al. 1999). TDU was found to start at the 11th (1.5 [FORMULA] ), 8th (3[FORMULA] ) and 4th (5 [FORMULA] ) thermal pulse, and the parameter [FORMULA] was on average 0.24, 0.26 and 0.38 in the three cases. The model details are described elsewhere (Straniero et al. 1997; Gallino et al. 1998; Vaglio et al. 1998; Straniero et al. 2000).

In the most massive stellar model discussed here (the 5 [FORMULA] star), the temperature at the base of the convective envelope during the interpulse never exceeds 50 [FORMULA] K so that a negligible Hot Bottom Burning (HBB, i.e. burning at the inner edge of the convective envelope) occurs, without affecting the CNO nuclei. With FRANEC, efficient HBB is currently found in models of 6 and 7 [FORMULA] and solar metallicity. The minimum mass for its activation decreases with decreasing metallicity (Lattanzio & Forestini 1999; Straniero et al. 2000).

As recalled in the introduction, in TP-AGB stars two neutron bursts are released in different conditions, by the 13C and by the 22Ne neutron source. The effectiveness of the 13C neutron source requires the penetration of a limited amount of protons from the envelope into the He intershell zone at every TDU. This allows the formation of a tiny 13C pocket in the top layers of the He intershell at H-shell burning reignition. Subsequently, the 13C nuclei are fully consumed by [FORMULA]-captures in radiative conditions already in the interpulse phase (Straniero et al. 1997). Though some successful models of the 13C -pocket formation have been presented, the details are still a matter of debate (see for instance Herwig et al. 1997; Langer et al. 1999) and the amount of 13C burnt per nucleosynthesis episode must be assumed as a free parameter of the model. However, this remarkable source of uncertainty does not affect much the resulting abundances of the nuclei discussed here, which mostly depend on the activation of the 22Ne source, hence on the maximum temperature at the base of the TP convective zone, which in turn depends on the stellar mass. In order to illustrate this, in Fig. 3 we show the temporal evolution of the stellar structure during an interpulse - pulse cycle from our 1.5 [FORMULA] model, while in Table 3 we present the enhancement factors with respect to solar of relevant isotopes across the same cycle. This corresponds to the interval between the 14th and the 15th TDU episode.

[FIGURE] Fig. 3. Structure vs time of a AGB star model of 1.5 [FORMULA] evolving trough the interpulse - pulse cycle illustrated in Table 3. The zero of the temporal scale in abscissa corresponds to [FORMULA] = 1.28[FORMULA]108 yr after the start of core He burning. [FORMULA] is the mass coordinate in [FORMULA] . The upper line (labelled [FORMULA]) represent the bottom of the convective envelope, the middle line (labelled [FORMULA]) indicates the H/He discontinuity, and the bottom line (labelled [FORMULA]) indicates the He/C-O core discontinuity. The zone comprised between the two last lines is the He intershell. The insert is an enlarged view of the 25th convective pulse, which precedes the 15th TDU episode. The shaded region corresponds to the layers of the 13C pocket. For sake of clarity, this region was shifted somewhat downwards from the H/He discontinuity left by the TDU episode. There, a certain amount of protons are assumed to penetrate in the radiative He intershell (the proton pocket, corresponding to the area below TDU, not shaded). Immediately before H reignition protons are captured by the abundant 12C , giving rise to the formation of a 13C pocket. Before the next TP, all 13C nuclei are consumed so that the 13C pocket changes into an s pocket, before being ingested by the next growing convective instability. This structural scheme can be considered as representative of all AGB models with TDU, apart from numerical details. The bracketed numbers illustrate where and when the abundances reported in the corresponding column numbers of Table 3 were calculated.


Table 3. Enhancement factors with respect to initial abundances at the 25th pulse of the 1.5 [FORMULA] model, Z = 0.02.

The phases shown include the end of the previous thermal pulse (24th, whose relevant abundances are listed in Column 2 of Table 3), the s processing through radiative burning inside the tiny 13C pocket during the interpulse (Column 3) followed by dilution of the highly s-enriched pocket by a factor 20 in mass when the 25th convective TP spreads over the whole He intershell. Here the pocket is mixed with material composed at 50% of H-burning ashes and 50% of s-processed material (in the lower part of the He intershell) from the previous TPs. The composition after this dilution is shown in Column 4. Further s-processing occurs through partial 22Ne burning in the TP. This occurs when the convective instability reaches its maximum extension, and lasts for about 6 yr. The final composition in the He intershell after the 25th TP is listed in Column 5.

From a comparison between the various columns one can understand which phases are dominant in the production of the various isotopes. Species lighter than Fe are controlled by 22Ne burning, hence by the pulse temperature. In particular, for 26Mg and 37Cl the 22Ne burning episode increases the previous abundance in the He intershell by a factor [FORMULA] 1.5, and 25Mg by a factor 2.7 (see Columns 4 and 5). The last column of the table gives the enhancement factor in the envelope after the 15th TDU episode, when C/O equals 1.3. Owing to dilution by TDU episodes with the original envelope, the surface enrichment with respect to solar at this stage is rather low, amounting to 11% and to 5%, respectively, in the case of 26Mg and 26Mg, and to 35% in the case of 37Cl. Conversely, the most abundant isotopes 24Mg and 35Cl in the envelope are left about unchanged.

In the last column we also show (in brackets) the envelope enhancement factors resulting by the computation of a test case in which the 22Ne([FORMULA],n)25Mg reaction was switched off. By comparing the two numbers in Column 6 for each isotope one can see how much of the production factor in the envelope has to be ascribed to the 22Ne source.

In the last row of Table 3 we have also included the heavy s-only isotope 150Sm, in order to better state the difference between the heavy neutron-rich species and the lighter nuclei studied here. This last row shows that the abundance of 150Sm is almost the same for the two cases of Column 6, since its production is dominated by the 13C neutron source.

The observational results for Mg and Cl isotope ratios in the circumstellar envelope are compared in Fig. 4 with envelope predictions from the mentioned AGB models of two different initial mass and for different choices of the 13C amount in the pocket (for silicon and sulfur, the measured and predicted ratios are compatible but will not be discussed here since the present observational errorbars are too large to provide any constrain on the models). Cases labelled ST correspond to the models defined as standard in Gallino et al. (1998) for stars up to 3 [FORMULA] , and in Vaglio et al. (1998) for intermediate mass AGB models. The other two are obtained by scaling the 13C amount downward by a factor of three (d3), or upward by a factor of two (u2) (see Busso et al. 1999; Lugaro et al. 1999). In Fig. 4 each dot corresponds to a TDU episode; open symbols refer to the late TP-AGB phases where C/O [FORMULA] 1.

[FIGURE] Fig. 4. a  Computed isotopic ratios of 26Mg/24Mg vs 25Mg/24Mg during the TDU phases in the envelope of AGB stars of solar metallicity, initial masses 1.5, 3, and 5 [FORMULA] and for different choices of the 13C amount (d3, ST, u2, see Busso et al. 1999) as compared with measured isotopic ratios (full symbol). Large open symbols are for C-rich envelope, dots are for O-rich conditions. b  The same as panel a for the ratio 37Cl/35Cl plotted versus 12C/13C.

The mass of the envelope is progressively eroded by stellar winds and in a minor way by the growing of the H-burning shell. The TDU mechanism ceases to operate when the envelope mass approaches 0.5 [FORMULA] . This corresponds to the last representative point shown in Fig. 4. There the photospheric ratios C/O and 12C/13C reach 1.4 and 63, respectively, for the 1.5 [FORMULA] model, 1.1 and 103 for the 3 [FORMULA] model, 1.3 and 122 for the 5 [FORMULA] model. From then on the envelope composition does not change anymore, while the star will eventually encounter a phase of superwind (e.g., Iben & Renzini 1983) blowing off the remaining envelope. The cause of this phenomenon may be identified in a dynamical instability driven by radiation pressure right at the bottom of the envelope when the luminosity exceeds a critical value (Sweigart 1998; Straniero et al. 2000).

The results shown in Fig. 4 make clear that model predictions are almost independent of the amount of 13C consumed in the pocket (see Lugaro et al. 1999for a detailed discussion of the behaviour of Si isotopes). From Fig. 4 it appears that the 1.5 and 3 [FORMULA] models are both compatible with the observed Cl and Mg isotopic ratios, whereas the 5 [FORMULA] model predicts too high values for the 37Cl/35Cl and [FORMULA]Mg/24Mg ratios with respect to the observations.

Let us discuss the above result in the more general context of AGB modelling. The final Cl and Mg abundances in the envelope might in principle depend of the choices for TDU and mass loss as well as on the efficiency of the 22Ne neutron source during the TPs. As it is well known the lack of a reliable theory for stellar convection and the difficulty of evaluating a suitable mass loss rate may substantially affect our comprehension of AGB evolution. The models here adopted have been computed by using the Schwarzschild criterion for convection, without allowing for any extramixing. This likely provides a minimal efficiency for TDU. We have also used the Reimers' formula to account for the mass loss rate. Other stellar evolutionary models make use of various diffusive or overshoot prescriptions for TDU (e.g., Frost & Lattanzio 1996; Herwig et al. 1997), and of different mass loss criteria.

We can have hints on the effects that a more efficient TDU in the first TPs may have on the predicted isotopic ratios by considering how these last vary from pulse to pulse. For the 1.5 [FORMULA] model the abundance by mass of 37Cl in the He intershell varies from the 15.th TP (5.th with TDU) to the 25.th TP by a factor 1.5. This increase is related to the slight progressive increase of the peak temperature at the bottom of the TPs, from 2.78 [FORMULA] 108 K at the 15.th TP to 2.99 [FORMULA] 108 K at the 25.th TP. Other factors are involved, such as the decrease with pulse number of the mass of the He intershell and of the overlap factor among adjacent pulses (e.g., Gallino et al. 1998). All these structural chacteristics are quite general in low mass AGB stars, being governed by the relatively low value of [FORMULA] at the first TP with TDU (e.g., Gallino et al. 1998). In models with a more efficient TDU since the first TPs, and a lower number of pulses on the AGB so that about the same final carbon enrichment in the envelope would result, the 37Cl abundance in the envelope might achieve a slightly lower surface enrichment than we do. However, because of dilution with the envelope the effect is not large and our general conclusions would not change. In the case of the 5 [FORMULA] model, the value of [FORMULA] at the first TDU is already 0.88 [FORMULA] , and the various structural and physical characteristics in the He intershell, among which the temperature history in the TPs, remain almost the same from pulse to pulse (Straniero et al. 2000). Thus the abundances of Cl isotopes in the He intershell reach asymptotic values very quickly, so that any choice for TDU and mass loss would lead to a too high 37Cl/35Cl ratio at the stellar surface to be compatible with observations. Similar considerations hold for the Mg isotopes.

We therefore argue that, as for the constraints derived from Cl and Mg isotopes, a higher mass estimate for CW Leo is possible only from models where the temperature in the pulses, the temporal evolution of the mass of the He intershell and overlap between adjacent TPs, or the rate of the 22Ne([FORMULA],n)25Mg reaction (i. e. the fundamental structural parameters in the TP-AGB phase), are substantially different than in our model. Whether these differences are possible is something that deserves scrutiny in another context. In any case, the precision of the new Cl measurement allows us to reduce the ambiguity on the initial stellar mass, showing that this last has to be low enough to keep at low efficiency the 22Ne neutron source. With the models presently adopted this actually requires a mass below 3 [FORMULA] .

A further comment concerns the luminosity. Our 1.5 [FORMULA] model reaches a maximum luminosity at the tip of the TP-AGB phase of 1.15 [FORMULA] 104 [FORMULA], which fits in the range deduced by observations. On the contrary, our 5 [FORMULA] is too luminous (4.15 [FORMULA] 104 [FORMULA]). This is another argument in favour of a low initial mass.

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Online publication: June 5, 2000