5 The role of the original He content

All the theoretical computations referred in the previous sections assume an amount of original He content equal to Y=0.23, which is, at the present, the most popular and widely adopted estimate for the original He abundance in metal poor, Population II stars. However, in Paper I we already discussed the large uncertainty in the evolutionary determination of this parameter. According to such an evidence, we will investigate the role played by the assumption about Y on the present theoretical scenario. To this purpose we reinvestigated the evolutionary behavior of the Z=0.001 models (with element diffusion) but under the two alternative assumptions Y=0.21 or Y=0.25.

  

Figure 10: Present isochrones in the CM diagram for Z=0.001, and the labeled values of age and original He abundances

Figure 10 compares selected isochrones for Z=0.001 and for the three adopted values of helium abundance. As a result one finds the following relations:

unevolved MS (at B-V=0.6):
$\Delta M_{ v}/\Delta$$Y\sim$2.5 (slowly depending on (B-V)

TO (at t=10 Gyr):
$\Delta M_{ v}= 0.019$ ($Y=0.21\div 0.23$)
$\Delta M_{v}=0.034$ ($Y=0.23\div 0.25$)

HB (at $\log T_{\rm e}=3.85$):
$\Delta M_{ v}= -0.084$ ($Y=0.21\div 0.23)$
$\Delta M_{ v}= -0.068$ ($Y=0.23\div 0.25).$

As a whole, present results (where element sedimentation is taken into account) appear in reasonable concordance with canonical evaluations given by Renzini (1991) for the TO magnitudes and by Buzzoni et al. (1983) or, more recently, by Bono et al. (1995) for HB luminosities. One finds that a variation of Y within the assumed limits (Y=0.23 $\pm$ 0.02) gives a maximum variation of 0.034 mag in the TO magnitude and of 0.10 mag in the difference of magnitude between TO and HB. According to the calibration given in previous sections this uncertainty implies an error of about 0.2 Gyr for ages from the TO magnitude, and an error of about 1 Gyr for ages from the "vertical" method, i.e., from the difference in magnitude between TO and HB.

From the previous sections one also derives $\Delta M_{ v}$(TO)$\sim$ 0.37 $\Delta$[Fe/H] and $\Delta M_{ v}=0.18$ $\Delta$[Fe/H]. Assuming an uncertainty of $\pm$ 0.2 dex on current evaluations of cluster metallicity, we obtain that such an error drives, by itself, a variation of, about, 0.08 mag in the predicted TO, and of about 0.04 mag on the difference in magnitude between TO and HB. As a result, one finds that with the assumed uncertainty in both Helium and metal content, even with perfect photometry, the age of a cluster cannot be determined better than $\pm$ about 0.8 Gyr from the TO magnitude or better than, about, 1.4 Gyr from the vertical method.

We remind that in all cases we assumed scaled solar composition as given by Grevesse & Noels (1993). However, for metallicity not too much high, the effect of the $\alpha$-enhancement on the evolutionary tracks and isochrones can be simulated by using a scaled solar mixture of total metallicity equal to the actual one (see Salaris et al. 1993). Detailed calculations recently performed by Salaris & Weiss (1998b) show that the $\alpha$-enhancement does not influence at all the $\Delta V$(TO-HB) parameter for metallicities up to $Z\approx 0.004$, while for Z=0.01 the variation of $\Delta V$(TO-HB) is about 0.1 mag. Thus the influence of the $\alpha$-enhancement on $\Delta V$(TO-HB) is expected to be negligible up to our highest adopted metallicity (Z=0.006). In the case of $\alpha$-enhancement we only expect, for Z=0.006, a shift of the color of the RG branch, well within the known uncertainties due to the uncertainty on the efficiency of superadiabatic convection and on the adopted models atmospheres.

As already discussed by Caputo et al. (1983) one finally finds that MS and HB have opposite behavior as far as the amount of original He is concerned. This implies that the difference in magnitude between the HB and the MS at a given color is in principle an indicator of the original He content independent on the cluster age and not affected, as the well known parameter R is, by the still large inaccuracy on the rate of the ¹²C$-\alpha$ reactions. This indicator is becoming more and more relevant vis-a-vis the increased capability of precise photometry of faint MS stars in galactic globulars. Combining the previous relations one finds $\Delta M_{ v}$(HB-MS)$\sim 6.5\Delta Y$. Thus $\Delta M_{ v}$(HB-MS) appears rather sensitive to the amount of original He, increasing by 0.07 mag when Y is increased by only 0.01. However one should remember that significant theoretical uncertainties, due to the uncertainty in the efficiency of superadiabatic convection and to the choice of color transformations, together with observational uncertainties, due to the evaluation of the cluster reddening and metallicity, make de facto this quantity a difficult parameter to be used as helium calibrator.