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

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3. The puzzle of the mixing-length parameter

The calibration of a solar model provides the value of the solar mixing-length parameter [FORMULA] which in turn is currently used to model other stars with the same input physics. The value of [FORMULA] changed in time because of successive updates of the input physics, in particular the low-temperatures opacities and model atmospheres. Apart from the difficulty of evaluating [FORMULA], the question arises whether stars of different masses, initial chemical composition and evolutionary status can be modeled with a unique [FORMULA]. Because models are also used to calculate isochrones, it is important to try to clarify the situation: if [FORMULA] is proved to vary significantly with mass, metallicity and with evolution, the isochrones might have quite different shapes which could modify the age estimates. On the other hand, models of various masses, all computed with [FORMULA], can reproduce quite well the slope of the main-sequence of the Hyades cluster (Perryman et al. 1998) and of field stars (Lebreton et al. 1999) observed by Hipparcos, indicating that [FORMULA] does not vary much for masses close to the solar mass.

For binaries with well-known properties as [FORMULA] Cen, the question of the universality of the mixing-length parameter has been examined many times. Table 1 shows that among the attempts of calibrating [FORMULA] Cen A & B in luminosities and radii using models based on the MLTBV, two yield values of [FORMULA] different for each component and different from [FORMULA] (Lydon et al. 1993; Pourbaix et al. 1999) while other calibrations suggest similar values for the two stars (Edmonds et al. 1992; Fernandes & Neuforge 1995). Also, Fernandes et al. (1998) calibrated three other binary systems and the Sun with the same input physics and concluded that [FORMULA] is almost constant for [FORMULA] in the range [FORMULA] dex and masses in the range [FORMULA] while Morel et al. (2000b) found a small difference of [FORMULA] ([FORMULA]) in the two components of the [FORMULA] Peg system.

As pointed out by many authors (e.g. Lydon et al. 1993; Andersen 1991) accurate masses and effective temperatures are required to draw firm conclusions on whether the MLTBV parameters are different or not. This is also illustrated by the large error bars on [FORMULA] ([FORMULA]) quoted by Pourbaix et al. (1999) which include all possible observable sources of errors including the errors in the masses. For a detailed estimate of the error budget see e.g. Guenther & Demarque (2000).

Some 2-D and 3-D numerical simulations of convection have been performed and translated into effective mixing length parameter (Abbett et al. 1997; Ludwig et al. 1999; Freytag et al. 1999). They suggest that, for stars with effective temperature, gravity and metallicity close to solar the mixing-length parameter remains almost constant. Similar results are obtained by Lydon et al. (1993) who compared models of [FORMULA] Cen A & B based on the results of numerical simulations of convection and found that a single value of [FORMULA] may be applicable to main sequence solar type stars. In the MLTCM convection theory, Canuto & Mazitelli (1991 , 1992) used two modern theories of turbulence to establish a formula for the turbulent convective flux which replaces the MLTBV expression (because the full spectrum of convective eddies is taken into account, the convective efficiency is magnified by about one order of magnitude). The MLTCM theory also makes use of a mixing length which is equal, either to the distance towards the outermost limit of the convection zone, or to the fraction ([FORMULA]) of the pressure scale height. The solar calibration with the MLTCM treatment yields an [FORMULA] value of the order of unity. It is to be noted that for the Sun, models calibrated with MLTCM have frequencies which are closer to the observations than those calibrated with MLTBV (Christensen-Dalsgaard et al. 1996).

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© European Southern Observatory (ESO) 2000

Online publication: December 11, 2000