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Astron. Astrophys. 333, 926-941 (1998)

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3. The deep mixing scenario

By the end of the 1970s the fact of star-to-star abundance variations in GCRGs was well established for C and to a lesser extent for N (Bell et al. 1979, Dickens et al. 1979, Norris & Cottrell 1979, Da Costa & Cottrell 1980), while little quantitative information was available for O. In 1979 Sweigart & Mengel found that in low metallicity red giant models the radiative layer where C was transformed into N, and for very low metallicity even the layer where O was transformed into N, were rather well separated from the main part of the HBS where H was transformed into He. This meant that rotationally-driven meridional circulation currents, if present beneath the BCE, could freely penetrate close enough to the HBS and transport outwards material with depleted C and O and enhanced N abundances; usually, a large mean molecular weight gradient forms a barrier which cannot be penetrated by meridional circulation. Further developments of Sweigart & Mengel's idea have been commonly referred to as "the deep mixing (or evolutionary) scenario". It should be noted that in this model the nature of the mixing mechanism is usually not specified. Exceptions are the pioneering work of Sweigart & Mengel (1979) itself and that of Smith & Tout (1992) where meridional circulation in its simplest classical treatment was shown capable of providing the required rate of mixing, and the recent paper of Charbonnel (1995) who considered the more complicated mixing algorithm elaborated by Zahn (1992) which takes into account the interaction between meridional circulation and turbulent diffusion. Other works concentrate on nucleosynthesis aspects of the problem and try to answer the question of whether any postulated mixing can explain the whole spectrum of abundance variations (and correlations) seen in globular clusters (as does the present work).

In the 1980s, following the first reports by Cottrell & Da Costa (1981) and Norris et al. (1981) that in NGC 6752 the N enhancements were accompanied by overabundances of Na and Al, evidence accumulated that this was a common feature of many clusters. Moreover, Paltoglou & Norris (1989) found that in [FORMULA]  Cen there is an anticorrelation between Na and O, which anticipated the discovery of the tight global anticorrelation of [O/Fe] versus [Na/Fe] by Kraft et al. (1993). Because it was then absolutely unclear how Na and Al, with rather large nuclear charges (and therefore rather strong Coulomb barriers against charged particle nuclear reactions), could be produced during hydrogen burning in low mass stars, the deep mixing scenario had begun to lose its supporters.

Its status was rehabilitated by low energy resonances in the reactions 22 Ne(p, [FORMULA] Na, 25 Mg(p, [FORMULA] Al and 26 Mg(p, [FORMULA] Al. First, Denissenkov & Denissenkova (1990) demonstrated that the temperature in the O depleted layer is high enough for the reaction 22 Ne(p, [FORMULA] Na to proceed (due to a resonance!) even faster than the reaction 16 O(p, [FORMULA] F responsible for the O depletion. Second, Langer et al. (1993) found that 27 Al can also be synthesized (mainly at the expense of 25 Mg) beneath the O layer in low metallicity red giants. These findings are illustrated in Fig. 1, where abundance profiles for a number of nuclides participating in the CNO-, NeNa- and MgAl-cycles are plotted in the radiative layers adjacent to the HBS in a [FORMULA] model star having [FORMULA] and [FORMULA]. The vertical segments on the abscissa show the locations of layers where (from right to left) 1, 5 and 10 percent of H have been consumed. We will not consider mixing penetrating too deeply into the HBS because: (i) on theoretical grounds the molecular weight gradient is expected to stabilize processes inducing both meridional circulation and turbulent diffusion (e.g. Kippenhahn 1974; Talon & Zahn 1997); (ii) observed Na abundances constrain the mixing depth (and rate) too (see below); and (iii) from the computational point of view, deep mixing is not allowed to bring too much fresh hydrogen fuel into the HBS because otherwise one would then have to take into account feedback of the mixing on the internal structure and evolution of red giants.

[FIGURE] Fig. 1. Abundance profiles for a number of nuclides participating in the CNO-, NeNa- and MgAl-cycles in radiative layers adjacent to the HBS in a [FORMULA] model star having [FORMULA] and [FORMULA] approximately matching metallicities of the globular clusters M 13 and [FORMULA]  Cen. The mass coordinate [FORMULA] is measured from the HBS in units of the mass separating the HBS and BCE. The vertical segments on the abscissa show locations of layers where (from right to left) 1, 5 and 10 percent of H were consumed

Nucleosynthesis in GCRGs and subsequent mixing have also been discussed recently by Cavallo et al. (1996, 1997) and Langer et al. (1997). The Cavallo et al. papers use detailed stellar interior models to give the physical conditions for the nuclear reactions while that of Langer et al. (1997) does not. They also give useful results for our work and supplement it since they consider effects of different overall metal abundances on the efficiency of deep mixing and also show how nucleosynthesis and mixing in GCRGs self-consistently explain the anomalous abundances of C (as well as the 12 C/13 C ratios), N, O and Na.

3.1. The initial chemical composition

To calculate the abundance profiles in Fig. 1 we adopted the following initial chemical composition for elements heavier than He: first, all the relevant solar abundances (Anders & Grevesse 1989) were multiplied by a factor [FORMULA] ; then the scaled abundances of [FORMULA] -elements (16 O, 20 Ne, 24 Mg and 28 Si) were increased by a factor 2.512 to agree with the average value [FORMULA] /Fe [FORMULA] inferred from field Population II dwarfs (Wheeler et al. 1989); next, abundances of Na and Al were reduced to [Na/Fe] = [Al/Fe] = -0.4 to take into account the even/odd effect (ibid.); and finally, to comply with the global anticorrelation of [O/Fe] versus [Na/Fe] the 22 Ne abundance was also decreased to ensure that [22 Ne/Na] = 0 (Paper I ).

We did not introduce in the initial chemical composition any corrections resulting from the first dredge-up because they were negligible (Sect. 1) compared with the surface abundance changes produced by the following deep mixing. For completeness we note that in our calculations we adopt isotopic ratios 24 Mg/25 Mg/26 Mg = 90/4.5/5.0, except as otherwise stated.

3.2. The CNO and Na abundances

Three further details evident in Fig. 1 are important for the following discussion, and while well-known, are worthy of note. (i) As one approaches the HBS the Al abundance increases to a considerably smaller degree than that of Na, with the latter experiencing two successive rises, the first of which appears at the expense of 22 Ne while the second is due to consumption of the much more abundant 20 Ne. (ii) The 24 Mg abundance shows no changes at all. It should be emphasized that if one took a pre-core-helium-flash model star one would still find very little 24 Mg depletion even well inside the HBS. (iii) 12 C first decreases with depth and then, after the CN-cycle comes to equilibrium, goes up because in CN-cycle equilibrium the abundance of 12 C (and 13 C) follow that of N. The latter begins to increase further with depth at the expense of O. Thus the surface abundances of 12 C and 13 C are expected to behave qualitatively differently from that of, say, O with changing mixing depth (and rate). For example, in a star with deeper mixing ([FORMULA] [FORMULA]) more C can survive at the surface than in a star with shallow mixing ([FORMULA] [FORMULA]).

We have found that deep mixing with the parameters of depth and rate [FORMULA] = 0.05, [FORMULA] = [FORMULA] cm2 s-1 reproduces quite well all of the observed correlations (or, more precisely, abundance trends) except that of O vs. Al seen in [FORMULA]  Cen, as is shown by the solid lines in Fig. 2. The observational data in the figure have been taken from ND95. We have corrected their N abundances (the average shift applied to [N/Fe] was +0.5 dex) to agree better with the data of Brown & Wallerstein (1993) and Norris & Da Costa (in preparation).

[FIGURE] Fig. 2. The abundance trends seen in [FORMULA]  Cen giants (symbols; from ND95) are compared with the results of our deep mixing calculations for two sets of mixing depth and rate ([FORMULA] ; [FORMULA], cm2 s-1): (0.05; 5 108) - solid, dotted and short-dashed lines, (0.06; 2.5 109) - long-dashed and dot-short-dashed lines. The former pertains to [FORMULA]  Cen while the latter corresponds to the best fit to the anticorrelation of [O/Fe] versus [Na/Fe] in M 13 (Fig. 3). In panel b the dotted line was calculated with an initial abundance [25 Mg/Fe] = 1.2, whereas both the short-dashed and dot-short-dashed lines were determined with [25 Mg/Fe] = 1.1 and the 26 [FORMULA] (p, [FORMULA] Si reaction rate increased to 103 times the value given by CF88 (see discussion in Sect. 3.4). Open and filled symbols refer to CO-strong and CO-weak stars, and x-es denote stars with unidentified CO status, following ND95. In panel d the N abundances of ND95 have been shifted by +0.5 dex (for them to agree better with the data of Brown & Wallerstein (1993) and Norris & Da Costa (in preparation)). The large crosses indicate observational error bars

We started our deep mixing calculations with the initial chemical composition described in Sect. 3.1. We do not assume that all low-mass stars in [FORMULA]  Cen (and in any other cluster) had exactly the same abundances before they became red giants because those could be changed by accreting nuclear processed material from a more massive and consequently more evolved companion star in a binary system, for example. Such differences in [C/Fe] among [FORMULA]  Cen giants (see the group of CO-strong stars in Fig. 2) are actually observed. Nevertheless for the present calculations we assume that any range of initial composition is small.

To explain the anticorrelation of O vs. Na in M 13, which is a very good representative of the global anticorrelation of [O/Fe] versus [Na/Fe] in "normal" clusters, we needed to assume a somewhat less deep but faster mixing than in the case of [FORMULA]  Cen: [FORMULA] = 0.06, [FORMULA] = [FORMULA] cm2 s-1. The reason for these required different mixing rates will not be understood until the deep mixing mechanism itself is understood. This model is shown by the long-dashed lines in Fig. 3 where the observed abundances of Na and Al have been taken from Kraft et al. (1997), and corrected by +0.05 dex and -0.25 dex, respectively, to compensate for differences between the gf values adopted in the two works 2. Note that here, too, there is no agreement between the observed and model results for [Al/Fe] versus [O/Fe]. It is worth commenting that even with the mixing penetrating as deep as [FORMULA] [FORMULA] 0.05 - 0.06 (cf. Fig. 1), the diffusive mixing rate is such that the amount of additional hydrogen processed in the HBS is relatively small. In the two cases reported here, the final surface H abundances are decreased by only 9.9 and 7.5%, respectively, and Sweigart (1997) has shown that under these circumstances the giant branch evolution is essentially unaltered.

[FIGURE] Fig. 3. The anticorrelations of [O/Fe] versus [Na/Fe] and [Al/Fe] and the correlation of [O/Fe] versus [Mg/Fe] seen in M 13 giants (symbols) compared with the results of our deep mixing calculations for [FORMULA] = 0.06 and [FORMULA] = 2.5 109 cm2 s-1. Observational data are taken from Kraft et al. (1997) with corrections of +0.05 dex and -0.25 dex applied by us to their [Na/Fe] and [Al/Fe] values, respectively, to compensate for differences between their adopted gf values and those of ND95. The dashed lines were computed with standard input physics. The dot-long-dashed line in panel a was calculated with the new NeNa-cycle reaction rates from El Eid & Champagne (1995), while the dot-short-dashed lines (panels b and c) were calculated with the initial abundances [24 Mg/Fe] = 0 (as opposed to the value +0.4 while we normally adopt), [25 Mg/Fe] = 1.1 and the 26 [FORMULA] (p, [FORMULA] Si reaction rate increased to 103 times the CF88 value. The large crosses indicate observational error bars

For comparison, the deep mixing solution giving the best fit for the global anticorrelation is also plotted in Fig. 2 (long-dashed lines). From Fig. 2a we infer that [FORMULA]  Cen may contain red giants exhibiting in their atmospheres Na produced not only at the expense of 22 Ne but also from 20 Ne. That is to say, in some [FORMULA]  Cen stars deep mixing may penetrate even to the second rise of Na seen in Fig. 1. To our knowledge [FORMULA]  Cen is the only example of globular clusters observed to date where some red giants exhibit in their atmospheres Na produced from both 22 Ne and 20 Ne, as indicated by our model calculations.

Closer comparison of the results for [FORMULA]  Cen and M 13 shows that the apparent difference between the two clusters is driven by the more metal-rich objects in [FORMULA]  Cen, and we shall return to this point in Sect. 3.5. It suffices here to note that our conclusion on the need for deeper mixing in the [FORMULA]  Cen stars with very strong Na (which also have larger [Fe/H]) will stand, because in more metal-rich objects the regions in which Na is synthesized are closer to the HBS and less accessible for mixing than in the more metal-poor ones (as confirmed by Cavallo et al. 1997).

A detailed analysis of structural and evolutionary differences in the parameters important for the deep mixing scenario between models of red giants having different metallicities has been carried out recently by Cavallo et al. (1997). Their main conclusions concerning the metallicity-dependent effects can be summarized as follows: (i) The HBS burns outward in mass quicker with increasing metallicity which allows less time for the nuclear processing of material above the HBS; (ii) in the more metal-rich models layers with temperature high enough to alter the envelope abundances lie closer to the HBS, the temperature in these models being lower than in the metal-deficient models. These points have led Cavallo et al. (1997) to the conclusion that "the low-metallicity (evolutionary) sequences should experience greater variations in their surface abundances in more elements than the high-metallicity sequences, assuming mixing does occur". In contradiction to these theoretical expectations the metal-rich giants in [FORMULA]  Cen have larger Na abundances than their metal-poorer counterparts (ND95, for more details see our Sect. 3.5). However, at present this cannot be considered as a serious disagreement between theory and observations because the metal-rich giants in question were selected by ND95 in an extremely biased manner.

A further unbiased analysis of abundances in [FORMULA]  Cen giants which are known to show a large range of metallicity is necessary to find observational evidence of the metallicity-dependent effects in deep mixing. Unfortunately, a comparison of abundances in red giants selected from different globular clusters may not be conclusive in this regard because the efficiency of mixing may vary from cluster to cluster.

In ND95 (Sect 5.3.1) the presence of a "floor" to the carbon abundance distribution at [C/Fe] [FORMULA] was suspected. This "floor" finds a natural explanation in the present simulations: sufficiently deep mixing which touches the C rise seen in Fig. 1 (below [FORMULA] = 0.15) (see comment (iii) at the beginning of this section) cannot produce very large surface carbon depletions (compare the solid and the long-dashed lines in Fig. 2c).

The deep mixing calculations for [FORMULA]  Cen giants show that during the last 50% of time spent by the model star between the onset of mixing and the core helium flash the surface 12 C/13 C ratio declines gradually from 9 to 6, which is in good agreement with values 4-6 reported by Brown & Wallerstein (1989) for objects near the tip of the giant branch.

The dot-long-dashed line in Fig. 3a presents results of our deep mixing calculations performed with the new NeNa-cycle reaction rates (El Eid & Champagne 1995). Its form differs noticeably from that of the long-dashed line calculated with the CF88 reaction rates, especially in the range [FORMULA] [O/Fe] [FORMULA]. This difference is entirely due to the considerably higher new rate of the reaction 22 Ne(p, [FORMULA])23 Na which causes a shift of the first rise of the Na abundance far outwards from the O depleted layer (Fig. 4). As a result, deep mixing first quickly produces a large Na enrichment at the stellar surface and only then does the surface O abundance begin to decline. Unfortunately, observational errors in the spectroscopic abundance analysis do not allow us to make a definitive choice between the old and new reaction rates from inspection of Fig. 3a.

[FIGURE] Fig. 4. Na and Al abundance distributions close to the HBS, calculated under different assumptions: standard assumptions (see text) - solid lines; new NeNa-cycle reaction rates from El Eid & Champagne (1995) - dot-long-dashed line; initial abundance [25 Mg/Fe] = 1.2 - dotted line; [25 Mg/Fe] = 1.1 and the 26 [FORMULA] (p, [FORMULA] Si reaction rate from CF88 multiplied by the factor 103 - short-dashed line

To summarize, the deep mixing scenario can explain evolutionary changes in the abundances of C (and 12 C/13 C ratios), N, O and Na. It fails, however, to interpret the anticorrelation of [O/Fe] vs. [Al/Fe] (Fig. 2b, solid line and Fig. 3c, long-dashed line) and the correlation of [O/Fe] vs. [Mg/Fe] (Fig. 3b, long-dashed line).

3.3. When does the deep mixing start?

In the above considerations of both M 13 and [FORMULA]  Cen we started our deep mixing calculations with a model star in which the HBS had just crossed the H-He discontinuity left behind by the BCE at the end of the first dredge-up. This and several neighbouring models are characterized by a small drop in luminosity caused by the adjustment of the HBS to an increased fuel supply when it encounters the H-He discontinuity. Stellar evolution slows down near this point and as a result a (subgiant) bump in the globular-cluster luminosity function appears at the corresponding visual magnitude. It was Sweigart & Mengel (1979) who proposed that deep mixing started only with that model because in preceding models there was a molecular weight gradient between the HBS and the BCE built up during the MS hydrogen burning which did not permit mixing (meridional circulation) to operate. Observations of anomalous evolutionary changes of 12 C/13 C and of the 3 He and 7 Li abundances in evolved halo stars seem to support this idea (Charbonnel 1995). It is, however, clearly at odds with the observations in clusters such as M92 which show depletions of carbon well below the level postulated by Sweigart & Mengel (Langer et al. 1986). Moreover, recent spectroscopic analysis of red giants in M 13 (Kraft et al. 1997) has revealed stars with enhanced Na and Al and depleted O (and even Mg!) at luminosities low enough to draw the conclusion that at least in the giants of M 13 deep mixing (if it is responsible for the observed abundance variations) begins earlier than Sweigart & Mengel thought. Our comment on this point is as follows. Inspection of the chemical structure of our models shows that the molecular weight gradient built up during the MS hydrogen burning is by far less steep than the gradient at the point of deepest penetration of the postulated deep mixing which has to be overcome in order to dredge up freshly synthesized Al and even Na (if the reaction rates of CF88 are used). We therefore infer that in every globular cluster having red giants with enhanced Al (and Na) and depleted O, even though we cannot identify the process responsible, deep mixing began to operate well before the point suggested by Sweigart & Mengel.

A potential serious problem for deep mixing could be overcoming the H-He discontinuity where, in canonical evolution, the mean molecular weight experiences a jump within a very narrow mass interval as a result of the deepest penetration of the convective envelope. Since mixing signatures, such as C-depletions, are seen at luminosities well below that at which the HBS reaches this composition discontinuity (e.g. M92, Langer et al. 1986), we must postulate that the mixing processes, perhaps acting in concert with variable convective overshoot, prevent this discontinuity from arising (or smooth it out). On the other hand, the luminosity function bump that is predicted when the HBS burns through this discontinuity, is observed in many globular clusters (e.g. Alongi et al. 1991). Consequently, in these clusters, not all of the evolving red giants can have had the discontinuity smoothed away.

3.4. The Mg and Al abundances

As may be seen in Fig. 1, aluminium cannot be produced at the expense of 24 Mg in standard evolutionary calculations (see also comment (ii) on Fig. 1 at the beginning of Sect. 3.2). On the other hand, 25 Mg does make Al (Fig. 1) but in amounts which are too small to explain the observations in [FORMULA]  Cen (see Fig. 2b, solid line) and M 13 (Fig. 3c, long-dashed line). Until recently one could speculate that in (some) GCRGs the initial 25 Mg abundance might be anomalously large as compared to its scaled solar value (e.g. Langer & Hoffman 1995; Paper I ). In the solar chemical composition 24 Mg is the most abundant magnesium isotope: 24 Mg/25 Mg/26 Mg = 79/10/11 (Anders & Grevesse 1989). If one assumes that [25 Mg/Fe] [FORMULA], then one can produce the observed Al enhancements of about +1 dex (see below). Due, however, to an increased contribution of 25 Mg to the sum Mg = 24 Mg+25 Mg+26 Mg and the evolutionary transformation of this 25 Mg into Al one would then expect the total magnesium abundance [Mg/Fe] to decline with increasing [Al/Fe]. Recently such an observational trend has been found in M 13 giants (Fig. 3b, symbols) by Shetrone (1996a) and Kraft et al. (1997). On the other hand, analysis of isotopic magnesium composition in a sample of 6 stars in M 13 (S96; Shetrone 1997) unexpectedly demonstrated that stars with extremely large [Al/Fe] possessed noticeably reduced 24 Mg and not 25 Mg ([FORMULA] [24 Mg/Fe] [FORMULA] for 5 stars with the largest [Al/Fe]). It should be noted that the same analysis revealed (for the first time in any star ever observed!) anomalous magnesium isotopic ratios with an unusually increased contribution of the sum 25 Mg+26 Mg ([25 Mg+26 Mg/Fe] up to +0.21, average fractions of the magnesium isotopes [FORMULA] Mg [FORMULA] Mg [FORMULA] Mg [FORMULA] = 56/22/22). Unfortunately, S96's analysis was not able to separate the 25 Mg and 26 Mg isotopes and gave only their summed abundance, and consequently, in deriving the magnesium isotopic ratios 25 Mg and 26 Mg were assumed to have identical abundances.

A straightforward interpretation of the anticorrelation of [Mg/Fe] versus [Al/Fe] in the M 13 giants, which guarantees an explanation of the O vs. Al anticorrelations in the clusters M 13 and [FORMULA]  Cen and which also takes into account the results of S96's isotopic analysis is to suppose that there is a strong but still undetected low energy resonance in the reaction 24 Mg(p, [FORMULA] Al. Unfortunately, at present we cannot estimate the strength of the resonance because we do not know its energy which also strongly influences the reaction rate. However, we note that the resonance has to provide this reaction with a rate comparable with that of the 25 Mg(p, [FORMULA] Al one in order to ensure that the deep mixing reaches the depth where 24 Mg is depleted (see the 25 Mg profile in Fig. 1). In this case aluminium would be a product of the chain of reactions 24 Mg(p, [FORMULA] Al [FORMULA] Mg(p, [FORMULA] [FORMULA] (p, [FORMULA] Si [FORMULA] Al and the beta-decay 26 [FORMULA] Mg together with the channel 25 Mg(p, [FORMULA] [FORMULA] Mg would take care of a large final abundance of the sum 25 Mg+26 Mg then dominated by 26 Mg. "Unfortunately", nuclear physicists seem to have little (if any) doubt concerning the current 24 Mg(p, [FORMULA] Al reaction rate (Arnould et al. 1995; Zaidins & Langer 1997). Nevertheless, it would be interesting to know whether they can guarantee that such a low energy resonance does not exist. Of course, any further experimental studies of the MgAl-cycle reaction rates would be well worthwhile.

It should be noted that uncertainties in the NeNa cycle reactions rates also affect the overall Mg abundances because of leakage from the NeNa cycle into 24 Mg, the rates advocated by El Eid & Champagne (1995) giving more efficient leakage than the rates of CF88. However, as a result of using the El Eid & Champagne NeNa cycle rates it becomes even more difficult to reduce the 24 Mg abundance in the HBS (Cavallo et al. 1997) which in turn makes it more difficult to explain the MgAl anticorrelation in the deep mixing scenario.

An alternative is that in GCRGs with enhanced Na and Al and depleted O and Mg abundances we actually observe products of hydrogen burning which has occurred at much higher temperatures (say, at [FORMULA]) than those reached in the HBS in the standard stellar models ([FORMULA]). We will explore this possibility in the next section.

In the event that the above two suggestions cannot be realized, a third and final possibility is to postulate that the extremely large Al enhancements in GCRGs are a signature of an unusually overabundant initial 25 Mg. But now, to comply with the results of S96's magnesium isotopic analysis, we have also to explain how stars with especially large initial 25 Mg abundances acquired a deficit in 24 Mg. In Sect. 4 we will consider a primordial source which is able (in principle) to produce such an abundance mixture and until then we merely assume that low mass stars in M 13 and [FORMULA]  Cen had initially increased 25 Mg.

The dotted line in Fig. 2b (see also Fig. 4) shows how [Al/Fe] evolves with [O/Fe] in [FORMULA]  Cen giants in the case of an initial abundance [25 Mg/Fe] = 1.2. Here we used the same values of [FORMULA] and [FORMULA] as earlier because they had given good fits to the other three abundance trends seen in [FORMULA]  Cen. Unfortunately, the new calculations disagree with observations both quantitatively (there is not enough Al produced) and qualitatively (the dotted line has a slope different from that hinted at by the observations). The quantitative disagreement is caused by the fact that with CF88 reaction rates the channel 25 Mg(p, [FORMULA] Al [FORMULA] Mg dominates over the 27 Al producing channel (see above), and as a result, a large fraction of 25 Mg is wasted to synthesize 26 Mg instead of 27 Al. The wrong slope of the theoretical dependence of [Al/Fe] on [O/Fe] comes from the necessity of waiting until a large enough abundance of 26 [FORMULA] is built up for the chain of reactions 26 [FORMULA] (p, [FORMULA] Si([FORMULA] Al to begin competing with the beta-decay 26 [FORMULA] Mg. These undesirable effects can be diminished by choosing a faster 26 [FORMULA] (p, [FORMULA] Si reaction rate. Indeed, according to Arnould et al. (1995), for the range of temperatures found in the HBS in GCRGs ([FORMULA]) the 26 [FORMULA] (p, [FORMULA] Si reaction rate may be underestimated by a large factor of [FORMULA] in CF88. If this acceleration factor is applied, the 27 Al producing channel gets so wide that now one can even use a somewhat smaller value of the initial 25 Mg abundance. In Fig. 2b (see also Fig. 4) the short-dashed line was calculated with [25 Mg/Fe] = 1.1 and the 26 [FORMULA] (p, [FORMULA] Si reaction rate equal to [FORMULA] times its CF88 value. The dot-short-dashed lines in Figs. 2b and 3c were calculated under the same assumptions but for the mixing depth and rate required by the O vs. Na anticorrelation in M 13. Now theory matches the observations! The initial 24 Mg abundance has no influence on the results of these calculations. In Fig. 3b the dot-short-dashed line is obtained assuming the initial abundance [24 Mg/Fe] = 0 (instead of +0.4) which, as expected, produces a decline of the total magnesium abundance [Mg/Fe] with increasing [Al/Fe] due to consumption of the now abundant 25 Mg. It seems, however, that observations demand even a larger (initial) deficit in 24 Mg.

The adoption of [24 Mg/Fe] = 0, [25 Mg/Fe] = 1.1 and [26 Mg/Fe] = 0 results in [25 Mg+26 Mg/Fe] = 0.81, [Mg/ Fe] = 0.33 and 24 Mg/25 Mg/26 Mg = 37/58/5, i.e. a 25 Mg dominated mixture of the magnesium isotopes; and for the deep mixing presumably operating in M 13 giants ([FORMULA] = 0.06, [FORMULA] = 2.5 [FORMULA] cm2 s-1) just before the core helium flash we find [24 Mg/Fe] = 0.004, [25 Mg/Fe] = -0.86, [26 Mg/Fe] = 0.61 and, consequently, [25 Mg+26 Mg/Fe] = 0.35, [Mg/Fe] = 0.10 and 24 Mg/25 Mg/26 Mg = 64/1/35. Since, however, the observations cannot separate 25 Mg and 26 Mg this final mixture is not distinguished from the one 24 Mg/25 Mg/26 Mg = 64/18/18 which has the ratios close to the average ones reported by S96.

Shetrone interpreted his results of the magnesium isotopic analysis in terms of the 25 Mg and 26 Mg abundances remaining unchanged during deep mixing. But as we have seen there are two alternative interpretations: (i) initially the magnesium isotope mixture consisted of almost pure 24 Mg (as, for instance, in the M 13 giant L598 of S96's sample which has 24 Mg/25 Mg/26 Mg = 94/3/3, and which shows little evidence of mixing); if, in addition, there is a strong low energy resonance in the reaction 24 Mg(p, [FORMULA] Al, then after mixing we will have 24 Mg depleted and both the sum 25 Mg+26 Mg and 27 Al, enhanced; (ii) initially 24 Mg was depleted and 25 Mg substantially increased (primordially!); during deep mixing the 24 Mg abundance remains constant and that of 27 Al increases at the expense of 25 Mg, the sum 25 Mg+26 Mg being dominated by 26 Mg which is also produced from 25 Mg. In both cases one obtains approximately what Shetrone has observed.

3.5. Metallicity dependent effects in [FORMULA]  Cen

We noted in Sect. 3.2 that the more metal-rich stars in [FORMULA]  Cen were in large part responsible for the apparent differences between it and M 13. In Fig. 5 we divide the [FORMULA]  Cen sample into two abundance groups: on the left are objects having [Fe/H] [FORMULA] -1.3, and in the middle those with [Fe/H] [FORMULA] -1.3. On the right, for comparison purposes, we also show data for the metal-richer globular clusters M5 ([Fe/H] = -1.2) and M71 ([Fe/H] = -0.8) from Shetrone (1996a). To focus further discussion we also superimpose on the diagrams the model fits for Na in M 13 (the dashed line from Fig. 3a).

[FIGURE] Fig. 5. The dependence of [Na/Fe], [Mg/Fe] and [Al/Fe] on [Fe/H] for (left) [FORMULA]  Cen giants with [Fe/H] [FORMULA] -1.3, (middle) [FORMULA]  Cen stars with [Fe/H] [FORMULA] -1.3, and (right) M5 and M71 giants. For [FORMULA]  Cen the symbols are as defined in Fig. 2, while for M5 (open triangles) and M71 (filled triangles) the data have been taken from Shetrone (1996a) corrected by +0.05 and -0.25 for Na and Al as in Fig. 3. The dashed lines are the deep mixing simulation for Na in M 13 shown in Fig. 3a

Inspection of the figure reveals several interesting points. First, concerning Na, one sees that the metal-poorer objects in [FORMULA]  Cen have an [Na/Fe] versus [O/Fe] correlation similar to that seen in M 13. In contrast, however, the metal-rich [FORMULA]  Cen stars have [Na/Fe] values which are larger by [FORMULA] 0.4 dex than in M 13. Such large Na enhancements do not exist in the M5 and M71 sample, which has comparable [Fe/H].

Second, concerning Mg, we see among the metal-poor [FORMULA]  Cen group the same [Mg/Fe] versus [O/Fe] correlation as discussed by Shetrone (1996a) and Kraft et al. (1997) for M13. The spread in [Mg/Fe] ([FORMULA] 0.5-0.6), appears to be slightly larger that observed for M 13 in Fig. 3 ([FORMULA] 0.3), suggestive of more extreme mixing in [FORMULA]  Cen. This is, however, very different from the behaviour of [Mg/Fe] versus [O/Fe] in the metal-rich [FORMULA]  Cen stars, where one sees little evidence for a spread in [Mg/Fe].

Finally, both groups of [FORMULA]  Cen stars appear to exhibit the same [Al/Fe] versus [O/Fe] behavior. A concomitant, important, point is that there is little if any dependence of [Mg/Fe] on [Al/Fe] in the metal-rich [FORMULA]  Cen giants, in stark contrast to what is found in the metal-poor [FORMULA]  Cen and M 13 giants!

One may summarize the above results by stating that the mixed metal-rich giants in [FORMULA]  Cen, in comparison with their metal-poorer counterparts, have similar Al enhancements, larger Na enhancements, and no accompanying Mg depletion. The metal-rich [FORMULA]  Cen stars clearly offer an important constraint on our understanding of the manner in which they have been enriched. For those who would take up the challenge we provide the following information and caveats. The mixed objects in question are ROA 150, 162, 231, 248, 357, 371 and 480, and were included in the work of ND95 in an extremely biased manner as examples of objects having the most extreme abundance peculiarites in the cluster. All have enhancements of the s-process elements, which ND95 attribute to primordial enhancement from low mass AGB stars. They have no counterpart in the more metal-rich clusters, such as M5 and M71.

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© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998