3 H burning phases

As a first test of theoretical results, Figs. 3a to c shows a comparison between isochrones (with sedimentation) and Hipparcos data (Gratton et al. 1997; Chaboyer et al. 1998) for field subdwarfs in selected intervals of [Fe/H] values. Each panel compares isochrones at t=10 and 12 Gyr and for a given [Fe/H] value with subdwarfs with [Fe/H] estimates within $\pm$0.1 dex. Triangles in each panel show the estimated shift in the isochrone if the suggested enhancement in $\alpha$-elements [$\alpha$/Fe] $\simeq$ 0.3 (Carretta & Gratton 1997; Gratton et al. 1998a) is accounted as an increase in the total value of Z. For the sake of comparison, each panel shows also the location of the MS loci but for different metallicities.

Inspection of the three panels shows that $\alpha$-elements play a minor role, since the corresponding shift of the isochrone lies within the observational errors. Bearing in mind that sedimentation does not affect the location of MS stars, one finds that the agreement between theory and observation appears as fine as it can be expected according to the standard errors in absolute magnitudes and the spread in metallicity. One finds that the popular Reid's statement "current models overestimate the change in luminosity with decreasing abundance for extreme metal poor subdwarf" is hardly supported. In fact, the three panels of Fig. 3 suggest that no contradiction appears between extant theory and Hipparcos observations.

  

Figure 3: Comparison of present isochrones with Hipparcos subdwarf absolute magnitudes (Gratton et al. 1997). Each panel shows the run of the 10 and 12 Gyr isochrones (full lines) for the given value of [Fe/H]. Triangles (and the dashed line) show the shift of the 10 Gyr isochrones caused by an enhancement [$\alpha$/Fe] $\simeq$ 0.3. As a comparison, dotted lines show the run of 10 Gyr isochrones with different [Fe/H], as labeled. Filled squares and open squares indicate single stars and detected or suspected binary stars, respectively

Figure 4 shows the comparison between present predictions of B-V colors at M _v= 6.0 and Hipparcos observational data as presented by Gratton et al. (1997). One finds that theoretical data appear in satisfactory agreement with observations, whose error appears of the same order of magnitude of the theoretical uncertainty discussed in the previous section. The best fit to the theoretical data gives:

(B-V) = 0.890 + 0.256 [Fe/H] + 0.048 [Fe/H]²

(at M _v=6.0)

which practically overlaps the relation given by Gratton et al. (1997a) except for a difference in the zero point by $\Delta(B-V$) = 0.014. As an alternative approach to the effect of metallicity one finds for the MS magnitude at (B - V) = 0.6:

$M_{v}= 3.889 - 2.160[{\rm Fe/H}] - 0.509[{\rm Fe/H}]^2$

$({\rm at}\ (B-V) = 0.6).$

One can use either of these two relations for correcting the metallicity effects in data in Fig. 3, by shifting the observational data to the [Fe/H] value of the computed isochrone. However, one finds that such a procedure does not improve the fitting given in Fig. 3, as an evidence that parallax errors (i.e., errors in absolute magnitudes) overcome the effect of metallicity in the chosen dwarf samples.

  

Figure 4: Present predictions of B-V colors at M v=6.0 for the various metallicities compared with observational data given by Gratton et al. (1997a) on the basis of Hipparcos observations

   Table 1: Coefficients of the linear relation: $M_{ v}^{\rm TO}$ = aLog(age) + b which connects the visual magnitude at the isochrone TO ($M_{ v}^{\rm TO}$) with the TO age for present models with element diffusion and Y=0.23. Age is in Gyr, metallicity as indicated

When moving from the MS to more advanced phases, Fig. 5 gives TO V-magnitudes as a function of age for the various adopted metallicities. One finds that for each given metallicity the TO magnitudes can be arranged as a linear function of the logarithm of the age, with a maximum error (for Z=0.001) not exceeding 0.4 Gyr. Table 1 gives the coefficients of the linear relations connecting the TO visual magnitudes with cluster ages (in Gyr) for isochrones with element diffusion taken into account. Figure 6 shows the comparison between current predictions for the TO magnitudes at t=10 Gyr and for the four adopted metallicities with similar results in the literature. As already predicted in Paper I, one finds that present evaluations tend to decrease the cluster age for any given TO magnitude, the differences in Fig. 6 being mainly the consequence of the difference already discussed in Paper I concerning theoretical luminosities. In passing we note that at the larger metallicities present magnitudes do not overlap Mazzitelli et al. (1995) predictions, as occurred for luminosities. This is mainly due to a small difference in adopted visual magnitude of the Sun: MDC adopted $M_{ v\odot}=4.79$ mag against our preferred canonical value $M_{ v\odot}=$ 4.82 mag.

  

Figure 5: Visual magnitude of the TO isochrones as a function of the logarithm of the age, for the labeled metallicities for models with (solid line) and without (dashed line) element diffusion

  

Figure 6: TO visual magnitudes for t=10 Gyr as a function of the metallicity for present models as compared with similar results available in the current literature. For DCM 1997 models CM indicates the adoption by the authors of the Canuto & Mazzitelli (1991) treatment of overadiabatic convection while MLT indicates the adoption of the usual mixing length theory

As already known, the TO magnitude, which is defined as the magnitude of the bluest point of the isochrone, is a difficult observational parameter in particular in the most metal poor clusters. Moreover, small variations in the shape of the theoretical isochrone can produce relevant variation in the magnitude of the nominal TO. This explains the small departures from linearity disclosed by data plotted in Fig. 5. In a recent paper Chaboyer et al. (1996) suggested a different calibrator of the globular cluster age, as given by the visual magnitude of the so called BTO point, that is the point brighter than the TO and redder by $\Delta(B {-} V)=0.05$.The main advantage of M _v(BTO) is that this point is not in the vertical turn-off region and thus it is easier found in the observed color magnitude diagram. Figure 7 shows that the visual magnitude of theoretical BTO appears much more regularly depending on the cluster age than M _v(TO) does, further supporting the use of such a parameter.

  

Figure 7: Visual magnitude at the isochrone BTO for present models with diffusion as a function of the logarithm of the age, for the labeled metallicities