Before closing the paper let us use the presented theoretical scenario to fit the $\rm C-M$ diagram of selected globular clusters. For a first test we chose $\rm C-M$ diagram presented by Walker (1994) for the metal poor cluster M 68. A preliminary discussion on that matter has been recently presented by Brocato et al. (1997) on the basis of Paper I evolutionary results and using Kurucz (1992) transformations. Without repeating the discussion given in that paper we present in Fig. 11 the best fitting of the observational diagram as obtained on the basis of our Z=0.0002 isochrones with element diffusion, adopting the very last version of Kurucz's model atmosphere (Castelli 1997a,b), where the new solar abundances and the enhancement of $\alpha$ -elements have been taken into account.

$\begin{figure} \includegraphics []{h0955f11.eps}\end{figure}$

Figure 11: Isochrones for ages between 10 and 12 Gyr and ZAHB compared to the CMD of M 68 (data from Walker 1994). Composition, distance modulus and reddening used for the fit as labeled. The adopted mixing length is 2.0 Hp

$\begin{figure} \includegraphics []{h0955f12.eps}\end{figure}$

Figure 12: Isochrones for age between 9 and 11 Gyr and ZAHB compared to the CMD of M 5 (data from Sandquist et al. 1996). Composition, distance modulus and reddening as labeled. The adopted mixing length is 2.3 Hp

As a test of theory at larger metallicity, Fig. 12 present the best fitting of the $\rm C-M$ diagram presented by Sandquist et al. (1996) for the intermediate metallicity cluster M 5. Now the best fitting gives an ages of about 10 Gyr and a distance modulus DM = 14.54. On the basis of Hipparcos parallaxes Gratton et al. (1997) give DM = 14.60 $\pm$ 0.07 with a (mean) age of 10.5 Gyr. On the same ground, Chaboyer et al. (1998) gives DM = 14.51 $\pm$ 0.09 with an age of 8.9 $\pm$ 1.1 Gyr.

The presented evolutionary models appears in both cases to give excellent agreement with independent evaluations of the cluster distance moduli based on Hipparcos data. This strongly increases the confidence in HB stars as (theoretical) standard candles and, in the same time, in the reliability of the derived cluster ages. One may notice that both clusters we are dealing with have been recently fitted with less updated physics, corresponding to step 4 in Paper I. In that case it was derived t= 12.2 Gyr for M 68 (Salaris et al. 1997) and, with slight different assumptions about the cluster chemical composition (Y=0.235, Z=0.0015), t= 10.9 Gyr for M 5 (Salaris & Weiss 1998a). Comparison with present results casts new light on the further rejuvenation of cluster ages induced by both the subsequent updating of the physics and the introduction of the element diffusion.

One can safely assume a conservative error of $\pm$ 0.1 mag in our estimates of the difference in magnitude between TO and HB, due to the arbitrarity introduced when the theoretical HB and isochrones are fitted to an observational CM diagram. As an example, we note that Brocato et al. (1997) using the theoretical models of Paper I, find for M 68 a distance modulus (DM=15 $\pm$ 0.25) slightly different from the present result, due to the slightly different way in which the theoretical HB is fitted to the observational one. This variation in the distance modulus together with the adopted color transformations (which influence the look of the fit) yields a difference of $\sim$ 1 Gyr in the estimated age. With the above quoted assumptions about the uncertainty in the chemical composition we conclude that the adopted theoretical scenario gives for our clusters:

where the first error is due to the uncertainty in the chemical composition, while the second represents the uncertainty in the fit. In passing, we note that the close similarity between the C - M diagrams of M 68 and M 92 (as, e.g., discussed in Brocato et al. 1997 and Salaris et al. 1997) drives to the conclusion that both clusters should have quite similar ages. According to the above discussion we suggest for these very metal poor globulars an age of the order of 11 Gyr against the 14 Gyr recently derived by Pont et al. (1998). One may notice that the above age evaluations could suggest a possible correlation between cluster ages and metallicity, the more metallic cluster being also the younger one. The evidence from the figure appears clear enough, however the source of possible errors in the photometry and in the fitting do not allow firm conclusions about a problem which deserves much more accurate investigations.

As a final remark, let us here remind once more that the above age estimates rely on the theoretically predicted HB luminosity. We have already quoted the good agreement of such a prediction with cluster distance moduli as derived by the fitting of Hipparcos subdwarf magnitudes. However one has also to remind that several estimates of RR Lyrae luminosities based on Hipparcos data give sensitively fainter magnitudes. As shown in Fig.13, the issue is far from being clearly settled. Here we can only say that if such faint magnitudes will be eventually confirmed, present theory is obviously overestimating the He cores at the end of RG evolution, likely as a consequence of a corresponding overestimate of the efficiency of cooling along the RG phase. In this case, data in Table 1 indicate that cluster ages should be increased by $\Delta\log t\, \sim\, 0.4 \delta M_{ v}$ , where $\delta M_{ v}$ represents the difference between the actual and the predicted magnitudes of RR Lyrae stars.

$\begin{figure} \includegraphics []{h0955f13.eps}\end{figure}$

Figure 13: Theoretical predictions concerning the HB magnitudes from the present paper (full line: diffusion) or from CCP (dashed line) compared with recent observational estimates as derived either from the subdwarf fitting (open symbols) or from RR Lyrae parallaxes (filled symbols)

It is a pleasure to thank Giuseppe Bono for a critical reading of the manuscript and for valuable suggestions. One of the authors, S.C., acknowledges the grant from C.N.A.A.

6 Discussion and final remarks