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Astron. Astrophys. 322, 147-154 (1997)

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4. Comparing different stellar evolutionary models

During the last decades there have been several attempts to reproduce the evolution of stars, motivated, in part, by the increasing power of computers and the continuous improvement of the input physics. In Table 1 some of the most recent models are listed. Differences between them are due to the use of different physical ingredients in their calculation, such as the free parameter [FORMULA] of the mixing length theory for the convection treatment, the free parameter [FORMULA] which allows us to consider the effect of overshooting as a mechanism to extend the convective shells, the nuclear reactions rates, the mean opacities, or the mass loss rates. Since the ages and masses obtained with our algorithm will depend on the SEM we use, a classical test has been developed to decide which of these recent models is more realistic. Comparison only consider solar composition models.


Table 1. Main characteristics of some of the most recent stellar evolutionary models. LAOL= Los Alamos Opacity Library (Huebner et al. 1977); OPAL= Opacity Library by Rogers & Iglesias (1992). [FORMULA] The core overshooting parameter is [FORMULA] = 0.25 in the mass range 1.0 [FORMULA] M [FORMULA] 1.5, and [FORMULA] = 0.50 above it; the envelope overshoot is [FORMULA] = 0.7 all over the mass range

The most accurate information on fundamental parameters of stars (mass and radii) is obtained from the analysis of the light and radial-velocity curves of detached, double-lined eclipsing binary systems. Based on Andersen's (1991) and Popper's (1980) compilations of these kinds of binary stars, we constructed a sample of 61 stars with masses between 1 and 3  [FORMULA], covering in this way the Main Sequence A-type range. According to Andersen (1991) the mean precision in R and M is 1.5 % and 1.4 % respectively. The log  [FORMULA] for these stars, coming from spectroscopic and photometric measurements from several authors, is more poorly determined (0.013 dex).

From R and M a very accurate [FORMULA] can be computed (mean error of 0.015 dex). Similarly to the appendix, log M allows to linearly interpolate an evolutionary track among a set of SEM, whereas [FORMULA] is used to interpolate the log age ([FORMULA]) and effective temperature ([FORMULA] [FORMULA]) between two EEP inside this evolutionary track for every component of each binary system. These two new values are very precise too; they will constitute the key to our SEM test.

The main evaluation of the models of Table 1 is based on the hypothesis of "same age" for the two components of every binary system. We define [FORMULA] as follows:


where [FORMULA] is the number of binary systems in our sample, [FORMULA] are weights given by


and [FORMULA] and [FORMULA] are the errors propagated directly from the individual errors in R and M, as in Sect. 3. These weights prevent the very uncertain ages -such as those of stars near the ZAMS- from biasing the result.

In addition, the comparison between [FORMULA] and the observed value [FORMULA] gives us a secondary test. Let us define [FORMULA] as:


where [FORMULA] is the number of stars. The use of weights in this case is not recommended, since in practice they depend on the errors in the determination of the observational [FORMULA], which are quite arbitrarily estimated.

The smaller [FORMULA] and [FORMULA] [FORMULA] are, the closer the agreement between the SEM and these accurate observations. In fact, the first quantity is more reliable since it has been calculated solely from fundamental stellar parameters.

After rejecting ten stars located below the ZAMS we calculated [FORMULA] and [FORMULA] [FORMULA] for three subsamples to evaluate the SEM fit in different HR-diagram regions:

i The whole sample (51 stars): it will show us the general fit in the 1 to 3  [FORMULA] range;

ii Stars more massive than 1.5 [FORMULA] (42 stars), excluding in this way the stars located in the conflictive Main Sequence transition between radiative and convective cores;

iii Six moderately evolved binary systems, already analyzed by Andersen et al. (1990). This subsample will allow us to evaluate the importance of the overshooting effect, since the stars in it may be classified as Main Sequence or Subgiant branch stars as a function of the adopted [FORMULA] value.

Fig. 4 shows the [FORMULA] for different SEM. It is obvious from this figure that those models that do not take into account the overshooting effect on the convective layers (VAN and CCS) assign quite different ages to the components of the binary systems, especially for the most evolved ones [sample iii ]. This is related to the fact that the overshooting effect tends to widen the HR-diagram. Results obtained with samples i and ii are almost equal, which indicates that the fit of the observations does not depend on the mass range. These results suggest that the behavior of all models with [FORMULA] (MM, SSMM, CG and BFBC) in this range of masses is, for the regions tested, very similar.

[FIGURE] Fig. 4. [FORMULA] as a function of different SEM. Solid line: the whole sample of eclipsing binary systems; dashed line: stars more massive than 1.5  [FORMULA] ; long-dashed line: moderately evolved binary systems (see text)

The general [FORMULA] fit does not change markedly for samples i and ii when different SEM are considered, so it is independent of the mass (Fig. 5). For evolved stars (sample iii) the fit is clearly improved when the overshooting effect is included. The better fit obtained for moderately evolved stars (sample iii) is a consequence of their position in the HR diagram, far from the regions where errors are large.

[FIGURE] Fig. 5. [FORMULA] [FORMULA] for different SEM. Line style like in Fig. 4

From left to right, the general trend in Figs. 4 and 5 is to improve the match between observational information and the set of SEM, which almost implies a chronological improvement. An important contribution to this improvement is the application of the overshooting effect to better define the limits of the convective regions of stars. In particular, the best fits are obtained for SSMM and BFBC models. Some new studies on dynamo action in stratified convection with overshooting (Nordlund et al., 1992) and rotational effect on convection (Pulkkinen et al., 1993) will introduce important advances in the description of stellar evolution in the near future.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998