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Astron. Astrophys. 363, 675-691 (2000)

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9. Discussion and conclusion

Detailed evolutionary calculations of the visual binary [FORMULA] Centauri, including the pre main-sequence have been performed using the recent mass determinations of Pourbaix et al. (1999). Models have been constructed using the CEFF equation of state, OPAL opacities, NACRE thermonuclear reaction rates and microscopic diffusion. We have revisited the effective temperatures, surface gravities and metallicities, using published spectroscopic data and taken these quantities as observational constraints. We have determined the most reliable solution within the confidence domains of the observable constraints via a [FORMULA] minimization. Each solution is characterized by [FORMULA], where [FORMULA] is the age of the system, [FORMULA] the initial helium content, [FORMULA] the initial metallicity and [FORMULA] and [FORMULA] the convection parameters of each star model. We obtained calibrations using different convection theories and adapted values for the mixing-length parameter of each component.

With the basic Böhm-Vitense (1958) mixing-length theory of convection, we derived [FORMULA]. With a convective core overshoot of [FORMULA], we obtained [FORMULA]. With the Canuto & Mazitelli (1991 , 1992) convection theory, we get [FORMULA]. Using the mass values and the observational constraints of the "best" model of Guenther & Demarque (2000), with the basic mixing-length theory, we obtained [FORMULA].

We have also performed [FORMULA] minimizations forcing the use of a unique value for the convection parameters of models of [FORMULA] Cen A & B. We have not reported the results as they are not significantly different from the former ones.

The most striking fact in our results is the small values obtained for the ages which are noticeably smaller than previous studies, except those of Boesgaard & Hagen (1974) and Pourbaix et al. (1999). With respect to the recent models of Guenther & Demarque (2000) this is due mainly to the mass discrepancies resulting from the small differences in distances. The [FORMULA] increase between the masses of Pourbaix et al. (1999) we used and those used by Guenther & Demarque (2000) results from by the 1.5% disparity between the revised Hipparcos parallax (Söderhjelm 2000) used by Guenther & Demarque and the orbital parallax determination of Pourbaix et al. As revealed by the [FORMULA] minimization, in the case of the present calibration of [FORMULA] Cen, the determination of the age appears to be more sensitive to the mass differences than to the basic observational atmospheric constraints namely, effective temperature, gravity - or luminosity - and metallicity. According to our results, no really satisfactory criterion allows to discriminate among different sets of modeling parameters: they all generate models which fit the observational constraints based on H-R diagram analysis and metallicities. A naive argument, however, can plead in favor of a subsolar age: as the [FORMULA] Cen binary system is formed of metal enriched material, it formed from more processed interstellar matter than the Sun; this may indicate that [FORMULA] Cen is younger than the solar system.

For the models computed with the basic MLTBV theory and fitted on our observational constraints, the mixing-length parameters are noticeably smaller than the convection parameter [FORMULA] of the solar model calibrated with the same physics; the differences are larger than expected from the Ludwig et al. (1999) simulations. For the models fitted to the observational constraints of the "best" model of Guenther & Demarque (2000), the values derived for [FORMULA] bracket the calibrated solar value. For all models computed with MLTBV theory we note that the smaller the age, the smaller the mixing-length parameters.

For models computed with MLTCM convection theory, in accordance with Canuto & Mazitelli (1991 , 1992) we obtain mixing length parameters both close to unity and to the convection parameter [FORMULA] of the solar model calibrated with the same physics, [FORMULA]. This indicates that the convection theory of Canuto & Mazitelli may support the assumption of a universal convection parameter and it seems to provide better fits to observations, as already pointed out by helioseismology as aforesaid in Sect. 3.

For the [FORMULA] Cen A models fitted on our observational constraints, as in the solar model, microscopic diffusion alone is not efficient enough to account for the observed surface lithium depletion. This may indicate that an unknown physical process, at work beneath the outer convection zone, reinforces the microscopic diffusion and gravitational settling to transport the material down to the lithium burning zone. For [FORMULA] Cen B, the only available observation of the surface lithium abundance is an upper limit (Chmielewski et al. 1992): all [FORMULA] Cen B models predict values compatible with this observation except model [FORMULA]. The surface lithium depletion of models calibrated with the Guenther & Demarque (2000) observational constraints are close to the observations and, in that sense, they appear to be more satisfactory than models fitted on our observational constraints: in these models there is no need for additional physical process to transport the lithium to its burning zone. But this imply that the small lithium depletion found in the standard solar model and the too large efficiency of the microscopic diffusion in more massive stars result from physical processes which are not active in [FORMULA] Cen, though masses and rotation status are close to solar ones.

All our models with large masses have initial helium contents close to the solar value [FORMULA], while the models with low masses have a high initial helium content [FORMULA]. With a primordial helium abundance of [FORMULA] we get a galactic enrichment of [FORMULA], both for the Sun and low mass models and [FORMULA] for high mass models. The differences in [FORMULA] between our [FORMULA] Cen models and the Sun are compatible with the scattering found in the solar neighborhood (Pagel & Portinari 1998) and in other binary system calibrations (Fernandes et al. 1998).

We have computed the large and small frequency spacings of acoustic oscillations for all the models. The large separation [FORMULA] for the three models with large masses are within [FORMULA]Hz, despite their differences in age and physics. They are such that [FORMULA]Hz. The variations of [FORMULA] taking into account the uncertainties in the observable constraints, effective temperatures, gravities and metallicities, are of about [FORMULA]Hz. For the model [FORMULA] of small mass, the large separation, [FORMULA]Hz, is in agreement with the value of Guenther & Demarque (2000) and significantly lower for our other models. The differences in [FORMULA] are mainly explained by the differences in mass and radius.

For the three models with large masses the small separations [FORMULA] vary from 7.5 to [FORMULA]Hz for [FORMULA] Cen A and are of about [FORMULA]Hz for [FORMULA] Cen B. For each model the variations in [FORMULA] within the observable uncertainties are [FORMULA]Hz. For the model [FORMULA], [FORMULA] is much smaller and in agreement with the value of Guenther & Demarque. All these differences in [FORMULA] are mainly related to the differences in central hydrogen content, hence in the age.

These results show that the determination of [FORMULA] and [FORMULA] by seismological observations would help to discriminate between the models of [FORMULA] Cen A computed with different masses and to confirm or not the new determination of the masses by Pourbaix et al. This implies an improvement of the accuracy of the observables used to constrain the calibrated models of [FORMULA] Cen A & B. Concerning the comparison with the seismic observations, the large splitting estimated by Pottasch et al. (1992) and Edmonds & Cram (1995) favor Pourbaix et al. (1999). masses. The different possible estimations for the large and small spacings [FORMULA] and [FORMULA] by Kjeldsen et al. (1999a) do not allow to discriminate between the models. We note, however, that their estimation [FORMULA]Hz and [FORMULA]Hz is highly improbable.

We conclude that, even for [FORMULA] Cen, the best known binary system, the models are not strongly enough constrained by the available astrometric, photometric and spectroscopic data. In order to deeply test stellar physics additional information on the internal structure is needed. Up to now ground-based observations give tentative evidence for acoustic oscillations in [FORMULA]Cen A. In a few years from now, one can expect that asteroseismology from space e.g. COROT (Baglin et al. 1998), MONS (Kjeldsen et al. 1999b) and MOST (Mattews 1998) missions and from ground, e.g. Concordiastro (Fossat et al. 2000), will provide data accurate enough to improve our knowledge of stellar interiors.

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Online publication: December 11, 2000