2 From theory to observation

In order to cover with sufficient details the range in metallicity $Z=2\ 10^{-4} - 6\ 10^{-3}$ which appears adequate for galactic globulars, computations presented in Paper I have been supplemented with an additional set of stellar models with Z=0.0006. As in Paper I, evolutionary models have been computed by adopting an amount of original He Y=0.23 and with a mixing length parameter l = 1.6 Hp. Data for these new models are reported in the Appendix of this paper.

On the basis of these new results, in this paper we will rely on a grid of evolutionary models covering, at least, the range of cluster ages 8 -18 Gyr for the assumed metallicities Z= 0.0002, 0.0006, 0.001, 0.006. The computations have been performed both by including or neglecting element sedimentation. In addition, for Z=0.001 we computed two set of evolutionary tracks with element sedimentation, under the same assumption about the mixing length but with Y= 0.21 or 0.25, respectively. These models will allow to discuss the influence of original He on models where element diffusion is taken into account. According to the usual procedure, the evolutionary models have been used to derive cluster isochrones in the $\log L$, $\log T_{\rm e}$ plane.

To allow a comparison with observation, this theoretical scenario has to be implemented with suitable procedures transforming the theoretical quantities L (luminosity) and $T_{\rm e}$ (effective temperature) into reliable predictions about magnitudes in selected photometric bands. To look into this problem Fig. 1 shows our Z=0.0002 12 Gyr isochrone (with diffusion) as transformed into the CM diagram according to selected available choices about the color-temperature and magnitude luminosity transformations. One finds that the most recent evaluations given by Castelli et al. (1997a,b) are very similar to previous evaluations given by Buser & Kurucz (1978, 1992) all along the TO region as well as in the redder portion of the HB, whereas Kurucz (1992) gives slightly redder TO colors. In all cases one finds that the difference affects mainly the colors, whereas bolometric corrections appear remarkably similar. Thus V magnitudes appears independent of the choice of the models.

  

Figure 1: The run of the theoretical 12 Gyr isochrone in the V, B-V diagram for the labeled choices about the adopted model atmospheres

If not otherwise advised, in this paper we will rely on the model atmospheres by Castelli et al. (1997a,b). Here we note that preliminary computations (Castelli, private communication) for the case Z=0.0002 have already shown that the model atmospheres can be further improved. As a matter of fact, one finds that when both the new solar abundances and the enhancement of $\alpha$-elements is taken into account, model atmospheres for metal poor stars ([Fe/H] = -2.0) reach a better agreement with available empirical estimate by Alonso et al. (1996). Fortunately, the same Fig. 1 shows that such improvement affects only the slope of the MS at the lower luminosity, leaving in particular unaffected the predicted magnitude of candidate candle stars.

As is well known, MS models are also affected by an intrinsic uncertainty due to the assumptions about the value of the mixing length parameter. The effect of varying the adopted mixing length within the (reasonable) interval l= 1.3-2.0 Hp is shown in Fig. 2, where we report selected isochrones for the labeled values of ages and mixing length parameters. One finds that for ages ranging from 9 to 15 Gyr the MS color at M _v=6.0 decreases by about $\Delta(B{-}V)\sim\!0.02$ mag, whereas decreasing the mixing length from 2.0 down to 1.3 Hp (the local pressure scale height) gives again a decrease of this color of about $\Delta(B{-}V)\sim\!0.03$ mag, depending on the cluster age. Thus the color of the MS at M_v=6.0 is far from being a firm prediction of theory. The same figure shows that the magnitude of the MS at B-V=0.55 (i.e. around theoretical predictions for M_v=6) for a given cluster metallicity can move by about 0.28 mag ($\sim$ 0.18 mag for a variation from 1.3 to 2.0 Hp and $\sim$ 0.1 mag for a difference in age from 9 to 15 Gyr), preventing the use of theoretical MS models as precise distance calibrators.

When moving toward more advanced evolutionary phases, as already noticed by Chaboyer (1995), the calibration of TO magnitudes in terms of cluster ages does depend on the assumptions about the mixing length parameter. One finds that such a dependence increases when the cluster ages and/or the metallicity is increased, for the simple reason that in both cases TO stars become cooler and, thus, more affected by external convection. As a result, the most metal poor and younger TO (i.e., the hottest TO) are barely affected by the treatment of convection all over the explored range of mixing length parameters (1.3 $\div$ 2.0 Hp). As an example, for Z=0.0002 and for the age t= 9 Gyr, a change from l= 1.3 to 2.0 Hp moves the TO magnitude by only 0.02 mag. However, for the same assumed metallicity, at t=14 Gyr, the same variation of the mixing length parameter gives a difference in the TO magnitude of the order of 0.1 mag.

  

Figure 2: The 9, 12 and 15 Gyr isochrones as computed under the labeled assumptions about the mixing length parameter and translated in the CM diagram according to Castelli et al. (1997a,b)

According to the quoted evidences, we will proceed in the discussion of theoretical predictions with the warning that discussed magnitudes should be reliable, whereas colors are affected by the discussed uncertainties in both stellar and atmospheric models. Detailed data for the computed isochrones are available by anonymous ftp at astr18pi.difi.unipi.it (/pub/globular).

In the next section we discuss low mass stars during the major phase of H burning, namely from the initial main sequence to the subgiant phase, just after the exhaustion of central H. We selected this evolutionary phase according to the evidence that the corresponding stellar models rely on the "minimum" input physics as given by an equation of state (EOS) not too far from a perfect gas, by radiative opacities and, finally, by sufficiently well known H burning rates (see, e.g., Brocato et al. 1998) Correspondingly, the quoted evolutionary phases have to be regarded as the most solid predictions of theory. Section 4 will deal with a discussion of more advanced evolutionary phases, from the red giant branch to the AGB through the phase of central He burning, where more physics has to be added, like neutrino production by weak interactions or the physical behavior of electron-degenerate matter. In both cases we find that theoretical predictions reach a good agreement with observational constraints. Following such an agreement, we discuss relevant evolutionary features and, in particular, the estimate of cluster ages based on the HB luminosity level. A brief final discussion closes the paper.