6 Concluding remarks

The stellar evolutionary tracks described here constitute a very extensive and homogeneous grid. They are therefore suited to the purposes of evolutionary population synthesis, either of simple (star clusters) or complex (galaxies) stellar populations. They can be used to describe populations older than $6\ 10^7$ yr, for a wide range of metallicities. This range does not include, however, that of very low-metallicities found in some globular clusters, and the tail of very high metallicities which may be found in the most massive elliptical galaxies and bulges. We intend to extend the data-base in order to cover these intervals in forthcoming papers.

One of the main characteristic of this data-base is that stellar tracks are presented at very small mass intervals. The typical mass resolution for low-mass stars is of 0.1 $M_{\odot }$. This is reduced to 0.05 $M_{\odot }$ in the interval of very-low masses (  $M_{\odot }$), and occasionally in the vicinity of the limit mass $M\mbox{$_{\rm Hef}$ }$ between low- and intermediate mass stars. The mass separation between tracks increases in the range of intermediate-mass stars, but anyway we provide enough tracks to allow a very detailed mapping of the HR diagram. As a result of this good mass resolution, the derived theoretical isochrones are also very detailed.

Due to their characteristics, the data-base is particularly suited to the construction of synthetic colour-magnitude diagrams (CMD). The latter are important tools for the correct interpretation of the high-quality photometric data which is becoming available for Local Group galaxies. Among these data, we mention the HIPPARCOS results for the solar vicinity (Perryman et al. 1995), the extensive photometric data-bases derived from the search of micro-lensing events towards the Magellanic Clouds and the Galactic Bulge, and deep HST images of particular galaxy fields.

As examples of the level of detail provided by the present tracks and isochrones, we mention the results obtained in the works by Girardi et al. (1998) and Girardi (1999). They have simulated synthetic CMDs for different model galaxies, finding substructures in the red clump region of the CMD. In particular, a faint secondary clump happens to be present in such models, as a result of the fine mass resolution adopted in the calculation of the evolutionary tracks with mass $M\sim2$ $M_{\odot }$. Counterparts to these clump substructures have been noticed in the CMDs derived from HIPPARCOS data, and those of some Magellanic Cloud fields observed to date (see Girardi 1999).

In addition, the present tracks behave in a very regular way for different metallicities, so that the construction of isochrones for any intermediate value of metallicity (in the interval $0.0004\le Z\le0.03$) is possible. These isochrones for arbitrary values of Z are available upon request, but are not included in the present electronic data-base.

An open question is whether the present models can be complemented with those of Bertelli et al. (1994) and Girardi et al. (1996a), for metallicities lower than Z=0.0004 and higher than Z=0.03, or masses higher than 7 $M_{\odot }$. The answer is not simple. In fact there are several differences between both sets of models. First of all, for metallicities higher than Z=0.008, the adopted Y(Z) relations are different. It generates systematic, although small, differences in the model luminosities and lifetimes for the sets of highest metallicities. Whether they are significant, however, is something that depends on the level of detail one is interested to look at.

The Bertelli et al. (1994) data-base does not contain Z=0.001 models with OPAL opacities, which are included in the present one. With respect to lower metallicities, the Girardi et al. (1996a) models with Z=0.0001, are probably a valid complement to the present ones, but present a lower mass resolution than now adopted, and a different prescription for overshooting in the mass range $1.0<(M/\mbox{$M_{\odot}$ })<1.5$.

On the other hand, a preliminary comparison between the present and previous Padova evolutionary models, in the range from about 5 to 7 $M_{\odot }$, reveals that they have almost identical tracks in the HRD and lifetimes. Therefore, they can be used to complement the present models for masses higher than 7 $M_{\odot }$, or equivalently for ages younger than 10⁸ yr.

Acknowledgements

We thank D. Mihalas and R. Wehrse for kindly providing us with the "MHD'' code for the equation of state, and E. Bica, P. Marigo, B. Salasnich and A. Weiss for the many useful discussions. L. Girardi acknowledges the many people who, in the last years, either lent him computers and CPU time in order to make preliminary calculations to the present tracks (especially M.V.F. Copetti, A.A. Schmidt, S.O. Kepler), or helped in the solution of daily computer problems (especially A. Weiss and B. Salasnich). The work by L. Girardi has been initially funded by the CNPq Brazilian agency, and later by the German Alexander von Humboldt-Stiftung. This work was completed with funding by the Italian MURST.