2. "Best" canonical models

The evolutionary behavior of our "best" models, as defined by step 8 in the previous section, has been investigated for selected choices on the assumed star metallicity and adopting everywhere an original amount of He given by Y=0.23 as a suitable value for population II stars. In all cases we assumed solar scaled composition as given by Grevesse & Noels (1993). However, alpha-enhanced distributions can be taken into account bearing in mind the scaling law discussed by Salaris et al. (1993). It is worth noting that the validity of such a relation has been recently questioned by VandenBerg & Irwin (1997), but for a metal-rich regime and for large -enhancement factors ([/Fe]> 0.3; see also Weiss et al. 1995 for the same topic), i.e., for values beyond the range suitable for globular cluster stars. Table 2 (click here) gives selected data of the models at the track turn-off (TO) for the various choices for the stellar mass and for metallicities Z=0.0001, 0.0002, 0.001 and 0.006. Left to right one finds: the metallicity (Z), the mass of the model (M), the age (), the luminosity (Log) and the effective temperature (Log) at the track Turn Off (TO).

 

Z M

(Gyr) (K)

0 . 0001 0 . 8 11 . 6 0 . 410 3 . 826

0 . 0002 0 . 6 33 . 3 -0 . 021 3 . 773

0 . 0002 0 . 7 18 . 6 0 . 188 3 . 797

0 . 0002 0 . 8 11 . 2 0 . 378 3 . 824

0 . 0002 0 . 9 7 . 4 0 . 572 3 . 859

0 . 0002 1 . 0 5 . 0 0 . 723 3 . 899

0 . 001 0 . 6 35 . 6 -0 . 178 3 . 755

0 . 001 0 . 7 20 . 0 0 . 060 3 . 777

0 . 001 0 . 8 11 . 7 0 . 231 3 . 799

0 . 001 0 . 9 7 . 4 0 . 393 3 . 822

0 . 001 1 . 0 5 . 0 0 . 577 3 . 852

0 . 006 0 . 65 38 . 3 -0 . 250 3 . 729

0 . 006 0 . 7 29 . 1 -0 . 140 3 . 740

0 . 006 0 . 8 16 . 8 0 . 017 3 . 759

0 . 006 0 . 9 9 . 5 0 . 134 3 . 777

0 . 006 1 . 0 5 . 5 0 . 263 3 . 796

. . . . .

Table 2: Evolutionary properties at the TO for step 8 "best" models. The age at the track TO is in Gyr

 

 

(Gyr) (K)

0.0002 8 0.595 3.853 0.877

0.0002 9 0.547 3.843 0.859

0.0002 10 0.500 3.835 0.834

0.0002 11 0.457 3.827 0.812

0.0002 12 0.393 3.821 0.790

0.0002 13 0.357 3.817 0.773

0.0002 14 0.329 3.813 0.758

0.0002 15 0.303 3.809 0.744

0.0002 16 0.278 3.805 0.731

0.0002 17 0.255 3.802 0.719

0.0002 18 0.234 3.799 0.709

0.001 8 0.455 3.820 0.897

0.001 9 0.411 3.814 0.869

0.001 10 0.373 3.809 0.845

0.001 11 0.339 3.804 0.824

0.001 12 0.309 3.799 0.806

0.001 13 0.256 3.796 0.786

0.001 14 0.231 3.793 0.772

0.001 15 0.208 3.790 0.758

0.001 16 0.187 3.787 0.746

0.001 17 0.167 3.784 0.734

0.001 18 0.148 3.782 0.723

0.006 9.5 0.281 3.781 0.932

0.006 10 0.262 3.779 0.921

0.006 11 0.229 3.775 0.900

0.006 12 0.203 3.772 0.881

0.006 13 0.167 3.769 0.862

0.006 14 0.146 3.767 0.847

0.006 15 0.114 3.764 0.831

0.006 16 0.096 3.762 0.819

0.006 17 0.079 3.760 0.807

0.006 18 0.059 3.757 0.796

Table 3: Selected TO quantities for H burning isochrones for the labeled values of metallicity and age

 

On the basis of these evolutionary tracks H burning isochrones have been computed for the quoted assumed metallicities and covering the range of ages suitable for galactic globular cluster stars. Table 3 (click here) gives detailed information on the isochrone TO luminosity and effective temperature. Left to right one finds: the metallicity (Z), the age (), the luminosity (Log), the effective temperature (Log) and the mass of the model () at the isochrone Turn Off (TO). As expected, data for the case Z=0.0002 overlap similar computations presented in Paper I, since passing from step 4 to step 8 affects only the advanced evolution of RG and HB structures. Thus present computations may be regarded as an extension to larger metallicities of the quoted computations.

We agree with the comment of our unknown referee about the risk of using TO luminosity as a parameter to derive cluster ages. From an observational point of view it appears quite difficult to define this parameter with high accuracy (see, e.g., Richer et al. 1988); the average uncertainty on the TO magnitude can be estimated of the order of  mag, which leads to an uncertainty on the derived age of the order of 1.5 Gyr (see Chaboyer et al. 1996a for a discussion on this point). Moreover, from a theoretical point of view, the TO being defined as the bluest point on the isochrone, the determination of the TO magnitude M_v(TO) is related to the isochrone color (Chaboyer 1995; Chaboyer et al. 1996b), thus depending on the stellar effective temperature, i.e., on the stellar radius. This radius can be affected by significant uncertainties, depending on the theoretical treatment of convection in superadiabatic layers (MDC). For such reasons, the use of other age indicators (see, e.g., Chaboyer et al. 1996b) has been suggested in several investigations. However, since such a parameter is still widely in use, let us discuss in some detail our results, to allow a comparison with previous results appeared in the literature.

The best fit of the data for the dependence of the TO luminosity on the cluster ages gives the analytical relations:
Log
Log
Log

where t₉ is the cluster age in Gyr. These relations reproduce the computational results in the range 8 to 18 Gyr with a maximum error of a few times 10⁸ years. As a result, one finds that, for each given age, the TO luminosities are predicted to decrease with metallicity with a slope in fair agreement with a large body of previous predictions but with lower predicted luminosities for each given age. This is shown in Fig. 1 (click here), which compares the dependence of present TO luminosities on the assumed metallicity for a given age (t=12 Gyr) with similar results already appeared in the literature. In the figure (as well as in some other following figures) theoretical expectations for O-enhanced mixtures (Bergbusch & VandenBerg 1992; VandenBerg 1992; Dorman et al. 1993) are reported in terms of the total fraction of heavy elements.

  
Figure 1: Behavior of the TO luminosity on the assumed metallicity for a given age (t=12 Gyr). Results for present "best" canonical models are compared with similar results available in the literature. For the MDC 1995 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

Figure 1 (click here) shows that present results represent in all cases a lower boundary for current evaluations of TO luminosities, thus decreasing current age estimates for each given TO luminosity. More in detail, present results predict TO luminosities systematically lower by about with respect to previous computations, with the exception of Mazzitelli et al. (1995: MDC) who predict again larger luminosities, but with a difference which decreases at the larger metallicities. The equations given above imply that the quoted decrease gives a decrease by about 10% in previous estimates of globular cluster ages. We will discuss this point further in the final section.

Apart from the problem of TO luminosities, H burning models deserve further attention as progenitors of He burning models, determining the structural parameters which will constrain the evolutionary behavior and, in particular, the luminosity of HB stars.

 

0 . 0001 0 . 515 0 . 238 13 . 22 3 . 322 2 . 245

0 . 0002 0 . 511 0 . 239 13 . 22 3 . 349 2 . 168

0 . 001 0 . 503 0 . 242 14 . 52 3 . 396 1 . 943

0 . 006 0 . 494 0 . 246 20 . 89 3 . 442 1 . 535

. . . . . .

Table 4: Selected evolutionary quantities for Red Giant models without diffusion. The age at the He flash is in Gyr

 

Computational results concerning those parameters are reported in Table 4 (click here) for the labeled assumptions about the cluster metallicity. Left to right one finds: the metallicity (Z), the mass () of the He core at the He flash, the surface helium abundance () after the first dredge-up, the age () and the luminosity (Log) at the He flash and the mean value between the minimum and the maximum in luminosity (Log) during the RGB "bump". Data in Table 4 (click here) will allow the approach of He burning phases adopting self-consistent evolutionary values for the two parameters characterizing a ZAHB structure, namely the mass of the He core and the He abundance in the stellar envelope. Since both values depend only marginally on the assumptions made about the cluster ages, Table 4 (click here) reports the values corresponding to a evolving Red Giant which can be safely assumed as representative of theoretical expectations in a sufficiently large range of ages. Here let us notice that the discussed increase (see Table 1 (click here)) of the predicted luminosity of the RG tip would affect the current estimate of the Hubble constant H₀ when using such a feature as a distance indicator. As a matter of the fact, one easily finds that the quoted increase by 0.2 mag in the top RG luminosity implies an increase by about 10% in the distance and, in turn, a decrease by the same amount of the H₀ estimate.

  
Figure 2: He core masses at the He flash as a function of metallicity for present models (canonical and with element diffusion) as compared with similar data already appeared in the literature

The amount of extra He brought to the stellar surface by the first dredge up appears in good agreement with similar evaluations already given in the literature (see, e.g., Castellani & Degl'Innocenti 1995 and references therein). Figure 2 (click here) compares present masses of the He cores in the flashing Red Giants with previous results. Again one finds that all current evaluations but MDC have a rather similar dependence on the assumed metallicity. However, one finds that our "best" models in all cases predict values larger than previous predictions; this acts in the sense of increasing the expected luminosity of ZAHB structures.

 

M LogL

(K) (Myr) (Myr)

0 . 53 1 . 357 4 . 410 93 . 0 -

0 . 55 1 . 395 4 . 320 88 . 2 113 . 6

0 . 60 1 . 471 4 . 217 82 . 6 105 . 3

0 . 65 1 . 582 4 . 108 78 . 5 100 . 6

0 . 70 1 . 662 4 . 010 75 . 0 96 . 9

0 . 75 1 . 727 3 . 926 74 . 4 94 . 7

0 . 77 1 . 744 3 . 900 73 . 9 -

0 . 80 1 . 769 3 . 860 - 91 . 3

M LogL

(K) (Myr) (Myr)

0 . 53 1 . 328 4 . 331 98 . 8 -

0 . 55 1 . 373 4 . 257 94 . 2 119 . 9

0 . 60 1 . 562 4 . 084 86 . 6 111 . 2

0 . 65 1 . 688 3 . 855 84 . 0 106 . 8

0 . 70 1 . 747 3 . 739 82 . 5 104 . 3

0 . 75 1 . 777 3 . 726 81 . 6 101 . 9

0 . 80 1 . 794 3 . 722 - -

. . . . .

Table 5: Selected evolutionary quantities for ZAHB models without diffusion. The time spent during the central helium burning phase (in Myr) is compared with results from CCP

 

By adopting and values from H burning models we are now in the position of predicting the evolutionary behavior of He burning Horizontal Branch (HB) structures. Table 5 (click here) gives detailed informations on the HR diagram location of Zero Age Horizontal Branch (ZAHB) together with a comparison between present and CCP He burning lifetimes. Left to right one finds: the mass (M), the luminosity and the effective temperature of the zero-age horizontal-branch, ZAHB, model (following CCP we assume as ZAHB structures the models already evolved by 1 Myr), the time () spent during the central He burning (until the disappearance of the convective core) and the same quantity () but for the CCP models.

  
Figure 3: The ZAHB luminosity at , as a function of metallicity for present models, compared with previous results, as labeled (BCFN94 = Bertelli et al. 1994)

As expected on the basis of the exploratory computations given in the first part of this paper, one finds that the expected luminosity of ZAHB models is substantially increased whereas He burning lifetimes in all cases decrease by more than 20%. Figure 3 (click here) presents predictions about the luminosity of the ZAHB model in the RR Lyrae instability strip together with previous results. One finds that "old" computations, as given by CCP or Lee & Demarque (1990) agree in predicting lower luminosities, by about Log0.05. This implies that, when using ZAHB models as "standard candles" to constrain the cluster distance modulus (DM), "old" computations would produce smaller DM, thus lower luminosities of the observed TO and, finally, greater ages. The same figure shows that all the most recent computations agree in predicting more luminous ZAHBs. In particular one finds that at the lowest metallicity, we predict luminosities in close agreement with MDC, notwithstanding the (small) difference in the He core masses. Note that the difference at the larger metallicities can be understood in terms of the different slope of the -metallicity relation already shown in Fig. 2 (click here).

  
Figure 4: Central He-burning lifetimes as a function of the ZAHB effective temperature for present models (solid line) compared with similar data in CCP (dashed line) and with the predictions by Buzzoni et al. (1983) for HB models with (stars). Metallicities as labeled

Figure 4 (click here) finally compares present He-burning lifetimes with the ones given in CCP and with the value originally predicted by Buzzoni et al. (1983) for the two assumed metallicities. The emerging scenario concerning current evaluation of the amount of original He in globular cluster stars will be discussed in Sect. 5.