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Astron. Astrophys. 332, 204-214 (1998)

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4. Determination of a critical µ-gradient from the observed carbon isotopic ratios

Recently, different groups have simulated extra-mixing between the base of the convective envelope and the hydrogen burning shell in order to reproduce the CNO abundances in RGB stars. Denissenkov & Weiss (1995) modeled this deep mixing by adjusting both the mixing depth and rate in their diffusion procedure. Wasserburg et al. (1995) and Boothroyd & Sackman (1997) used an ad-hoc "conveyor-belt" circulation model, where the depth of the extra-mixed region is related to a parametrized temperature difference up to the bottom of the hydrogen-burning shell. On the other hand, other authors attempted to relate the extra-mixing with physical processes, among which rotation seems to be the most promising. Sweigart & Mengel (1979) suggested that meridional circulation on the RGB could lead to the low 12 C/13 C observed in field giants. Taking into account the interaction between meridional circulation and turbulence induced by rotation in stars, Charbonnel (1995) showed that rotation-induced mixing can account for the observed behavior of carbon isotopic ratios and for the Li abundances in Population II low mass giants.

In these different approaches, the common underlying question is the determination of the extension of the region where extra-mixing occurs, and more precisely the nature of the stabilizing factor. An investigation into the mechanism of extra mixing must include an understanding of why the mechanism does not operate at luminosities below that of the RGB bump, and what is the nature of the factor that stabilizes the star against extra mixing before it reaches this point. When suggesting that the extra-mixing on the RGB could be related to rotation, Sweigart & Mengel (1979) and Charbonnel (1994, 1995) underlined the inhibiting effect of molecular weight (or µ) barriers. Indeed, as argued by Mestel (1957), gradients of molecular weight tend to restrain the circulation and to stabilize the mixing processes related to rotation. We will show now how the observations gathered in this paper allow us to constrain, from simple considerations, the conditions in which a µ-barrier shields the central regions of a star from extra-mixing.

We have computed some standard stellar models for exploration of these questions. The evolutionary sequences were followed from the pre-main sequence up to the RGB tip with the Toulouse-Geneva code (see Charbonnel et al. 1992). The input microphysics is the same as used in Charbonnel et al. (1997). The relative ratios for the heavy elements correspond to the mixture by Grevesse & Noels (1993). We take the same isotopic ratios as Maeder (1983). Neither element segregation or rotation-induced mixing are included in the present computations.

The gradients of molecular weight ([FORMULA]) in a standard stellar model of [FORMULA] with [FORMULA], typical of the present observed sample, are presented in Figs. 4, 5 and 6 at three different evolutionary stages: at the end of the main sequence, after the completion of the first dredge-up, and shortly after the RGB bump. We also show the abundance profiles for H, 3 He, 13 C and 14 N, and the 12 C/13 C ratio.

While on the main sequence, nuclear burning leads to an increase of the mean molecular weight in the central regions of the star (see Fig. 4). During the first dredge-up phase, the convective envelope homogenizes the star down to very deep regions, and builds a very steep gradient of molecular weight at the point of its maximum penetration ([FORMULA] at [FORMULA] in Fig. 5). Further on, as the star keeps ascending the RGB, the hydrogen burning shell becomes thinner and moves outwards in the mass scale, while the convective envelope retreats. As discussed in Sect. 3, our observations provide strong evidence that the extra-mixing is inhibited until the star reaches the LFB. The shape of [FORMULA] just after this evolutionary point, i.e. after the encounter of the hydrogen burning shell with the previously mixed region, is shown in Fig. 6. Below the base of the convective envelope [FORMULA] is much lower at this time.

[FIGURE] Fig. 4. Molecular weight gradient (in log scale, bold dashed line, left axis) and composition profiles (right axis) in a [FORMULA], [Fe/H]=-0.45 stellar model, at the end of the main sequence. The mass fractions are multiplied by 100 for 3 He and 14 N, by 1000 for 13 C; the ratio 12 C/13 C is divided by 100. The thick cross corresponds to [FORMULA], the critical molecular weight gradient obtained with the prescription by Huppert & Spiegel (1977) discussed in Sect. 5

[FIGURE] Fig. 5. Same as Fig. 4, after the completion of the first dredge-up, in the H-burning shell region. The base of the convection zone is located at [FORMULA].

[FIGURE] Fig. 6. Same as Fig. 4, after the RGB bump, in the H-burning shell region. The base of the convection zone is located at [FORMULA].

Our observations provide a precise clue on the extension of the extra-mixed region down to the nuclear burning layers. The low 12 C/13 C ratios we see in our sample ([FORMULA] 7) can indeed be reached if the extra-mixing develops down to [FORMULA] in our [FORMULA] model (see Fig. 6). At this point, [FORMULA]. We now assign this value to ([FORMULA], that we define as the "observational" critical µ-gradient which appears to shield the central regions of the star from extra-mixing. In our [FORMULA] model, the temperature difference [FORMULA] between the bottom of the hydrogen burning shell and this point is equal to 0.26. This value for [FORMULA] is in agreement with the one Boothroyd & Sackman (1997) have to impose in their parametrized mixing model in order to match the observed 12 C/13 C ratios at solar metallicity.

Let us check now to what region ([FORMULA] corresponds, in terms of expected 12 C/13 C, at different evolutionary stages, and for stars of different metallicities. We see in Fig. 4 that while the star is on the main sequence, the real [FORMULA] becomes higher than ([FORMULA] in the very external part of the star, around [FORMULA] equal to 0.75. If we assume that such a µ-barrier can not be penetrated, then on the main sequence no extra-mixing is expected to occur in the stellar region of energy production. This explains the observations of the carbon isotopic ratio in [FORMULA] Ori, in the less luminous stars of SSP93's sample, and in the less evolved RGB stars of M67 (GB91), where we see a perfect agreement with standard theoretical dilution. This is also in agreement with helioseismological constraints. Indeed, solar structure computations bring important information on the critical µ-gradients that limit the mild mixing which is necessary to explain the lithium depletion in the Sun. Richard et al. (1996) and Richard & Vauclair (1997) showed that the best solar models (for what concerns helioseismological comparison and agreement with Li and Be observations) are those including both element segregation and rotation-induced mixing, where the mixing is cut-off when the µ-gradient becomes [FORMULA] to 1.5 - 4 [FORMULA]. It is highly satisfactory to get the same value for ([FORMULA] from two completely different observational constraints.

The gradients of molecular weight shortly after the RGB bump in a model of 0.9M [FORMULA] with [Fe/H] = -2 (typical of a globular cluster giant) and in a model of 1.25M [FORMULA] at solar metallicity (typical of a M67 giant star) are presented respectively in Figs. 7 and 8. If we assume that the extra-mixing reaches down to the same [FORMULA] whatever the star, then the 12 C/13 C is expected to decrease down to its equilibrium value for the globular cluster giant, and down to [FORMULA] 12 in the open cluster giant. These values, and the dependency with metallicity, are exactly as observed in the giants in open clusters by Gilroy (1989) and are in agreement with the other observations of field stars and globular cluster giants. They are independent of any modeling for the extra-mixing. It appears then that the value we derived empirically for [FORMULA] is "universal", in the sense that an unique value is sufficient to describe different constraints.

[FIGURE] Fig. 7. Same as Fig. 6 in the [FORMULA], [Fe/H]=-2 stellar model. The base of the convection zone is located at [FORMULA]. The mass fractions are multiplied by 100 for 3 He, and by 50000 and 5000 respectively for 13 C and 14 N; the ratio 12 C/13 C is divided by 100

[FIGURE] Fig. 8. Same as Fig. 6 in the [FORMULA], [Fe/H]=0 stellar model. The base of the convection zone is located at [FORMULA]. The mass fractions are multiplied by 100, 1000 and 50 respectively for 3 He, 13 C and 14 N; the ratio 12 C/13 C is divided by 100

As already shown by C94, if molecular weight barriers are the inhibiting factor, then no extra-mixing is expected to occur in stars which ignite helium in a non-degenerate core, i.e. in stars with masses higher than 1.7 to [FORMULA] (the exact value depends on the metallicity). Indeed, in these more massive stars, the hydrogen burning shell does not have the time to reach the regions that have been homogeneized during the first dredge-up, so that [FORMULA] below the convective envelope is always higher than [FORMULA]. Under these conditions, extra-mixing is inhibited in these stars. This is confirmed by the observational data in open clusters with turn-off masses higher than about [FORMULA] (Gilroy 1989).

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Online publication: March 10, 1998