4 Comparisons with other sets of models and with observations

We now compare our models with recent models computed by other groups and with various observations.

The strongest approximation in our models lies in the treatment of the atmosphere and of the surface boundary conditions, which are specified by the Eddington approximation. In Figs. 5 and 6 we compare our results with the models of D'Antona & Mazzitelli (1994) and Tout et al. (1996) which also rely on a simple treatment of the boundary conditions. For the mass range considered in our work we obtain predictions very similar to D'Antona & Mazzitelli.

Sophisticated model atmospheres for the computation of very low mass stars have been developed recently (Baraffe et al. 1995, 1998; Brett 1995; Allard et al. 1997). Their use becomes crucial for very low mass stars ( ${M_{\odot}}$)down to the brown dwarf limit. We refer to Chabrier & Baraffe (1997) and Baraffe et al. (1998) for a discussion of the physical basis of the differences. As can be seen in Fig. 5, the Eddington approximation we use results in higher $T_{\rm eff}$ (from 2 to 5$\%$ depending on the metallicity) compared to the Chabrier & Baraffe (1997) models at our low mass end; our models also have slightly smaller radius (see Fig. 6). In this mass range the predictions agree well with the observations of Popper (1980) and Leggett et al. (1996) and do not allow to disregard one model with respect to the other.

  

Figure 5: Theoretical HR diagrams at two values of the metallicity. Solid lines: our models (Y=0.28 for Z=0.02); dashed lines: Chabrier & Baraffe (1997); dashed-dotted lines: Tout et al. (1997); open squares: D'Antona & Mazzitelli (1994). The models are taken at 0.1 Gyr, except for the zams models of T97. The observational data are from Popper (1980; black points) and from Leggett et al. (1996; black triangles)

  

Figure 6: Dependence of the stellar radius on mass for solar metallicity models. See Fig. 5 for the references of the models and observations

  

Figure 7: Mass-luminosity relations at 0.1 and 10 Gyr (solid and dashed lines respectively) for the solar (with Y=0.28) and low metallicity models (lower and upper curves respectively). The observations are from Andersen (1991) and Henry & McCarthy (1993) (white and black circles respectively)

In Fig. 7 we show the predicted mass-luminosity relations for the V and K band for different metallicities and ages. The magnitudes were derived from the standard stellar library for evolutionary synthesis of Lejeune et al. (1998), which provides empirically calibrated colours for solar metallicity and a semi-empirical correction for non-solar metallicities. The comparison with the observations of Andersen (1991) and Henry & McCarthy (1993) shows a good agreement with our predictions for both the V and K band (Fig. 7). Again a similar agreement is obtained with the models of Brocato et al. (1998) and Baraffe et al. (1998).

From the comparisons in Figs. 5 to 8, we conclude (in agreement with Alexander et al. 1997) that for the mass range considered in this work an approximate treatment of the stellar atmosphere leads to a satisphactory agreement between theoretical predictions and observations. Independently ab initio stellar interior and atmosphere models allowing the detailed predictions of all observational properties are of fundamental importance for our understanding of very low mass stars which are out of the scope of the present grids.