As a preliminary point, let us recall that in Paper I HB models have been computed assuming initial He core masses from a 0.8 $M_{\odot}$ progenitor. Strictly speaking, according to data given in the same paper, this implies an age of the order of 11 Gyr for the most metal poor cluster, increasing up to 16 Gyr for Z=0.006. In Paper I we already discussed this issue, reaching the conclusion that the expected variation of HB luminosity can be neglected over a reasonable range of ages. Now we add that detailed numerical experiments show that a variation of the cluster ages of $\pm$ 5 Gyr gives a variation in the predicted HB magnitude of $\pm$ 0.02 mag. According to the calibration given in Fig. 5 a similar error in the cluster distance modulus would produce an error in the cluster age not larger than 0.3 Gyr which, however, could be easily taken into account whenever a better precision would be required.

**Table 2:** Selected evolutionary quantities in the observational plane for ZAHB models with Y=0.23 and element diffusion. Masses are in solar units

Table 2 gives details for HB structures in the observational plane at the various metallicities. Selected evolutionary quantities for the new HB models with Z=0.0006 are given in the Appendix of this paper. A comparison of theoretical predictions for HB with Hipparcos data has been already presented in Paper I. Figure 8 compares the magnitude of the ZAHB (at $\log T_{\rm e}=3.85$ ) with previous evaluations on the matter. One recognizes the not negligible increase in the predicted HB luminosity induced by the improved physics as well as the fair agreement, toward the lower metallicities, with the recent computations by Caloi et al. (1997). As already predicted (Castellani et al. 1991) one finds that the dependence of the ZAHB luminosity on the metallicity increases with metallicity. The best fit of data in Fig. 8 gives:

which reproduces the theoretical predictions for ZAHB with diffusion within less than 0.01 mag all over the assumed range of metallicities. If diffusion is not taken into account the above magnitude has to be decreased by about 0.04 mag.

$\begin{figure} \includegraphics []{h0955f8.eps}\end{figure}$

Figure 8: Visual magnitude of the ZAHB at $\log T_{\rm e}=3.85$ as a function of the metallicity for present models with (solid line) and without diffusion (dashed line) compared with values for the same quantity available in the literature. CCP91 indicates models by Castellani et al. (1991)

For metallicities lower than Z=0.001 the above relation can be approximated by a linear relation, as usually adopted in the literature. We find:

These results can be usefully compared with the fairly large amount of observational relations presented in the literature:

It appears that present predictions show a dependence on the metallicity in reasonable agreement with the evaluation by Walker (1992) and the more recent evaluation given by Gratton et al. (1998b) based on Hipparcos results, and definitely smaller than required by Sandage in his scenario for explaining the Oosterhoff dichotomy. However, before comparing the zero point of the magnitudes one has to recall that our previous relations refer to the ZAHB luminosity level, whereas observational data refer to the mean luminosity of the HB at the color of the RR Lyrae Gap. The connection between the two luminosity level has been already discussed in several paper (see Caputo et al. 1987; Carney et al. 1992; Cassisi & Salaris 1997). One can safely assume $\Delta M_{ v}=0.08$ mag as a suitable estimate of the difference in magnitude between RR and ZAHB. Thus our zero points will become M_v=0.69 (diffusion) or M_v=0.66 (no diffusion), respectively. One finds that we are predicting HB with the same dependence on metal content and only 0.05 mag brighter with respect to the recent estimates by Gratton et al. (1997).

One can finally connect theoretical results concerning HB stars with previous predictions about the TO magnitudes to give a theoretical calibration of the difference in magnitude between HB and TO, $\Delta V$ (TO-HB), often used as age indicator for galactic globulars. This is shown in Fig. 9, where we report $\Delta V$ (TO-HB) as a function of the age for the four selected assumptions about stellar metallicity.

$\begin{figure} \includegraphics []{h0955f9.eps}\end{figure}$

Figure 9: The difference in magnitude between HB and TO as a function of the age for present models with (solid line) and without (dashed line) element diffusion for the labeled values of metallicities

4 Horizontal branches