6 Age estimates

For reasons discussed by us in detail elsewhere (see, for instance, Alcaino et al. 1998), we continue our homogeneous approach to the determination of age of globular clusters, awaiting the results of the current polemic on the controversy of globular cluster ages and the Hubble constant, in particular, with relation to the recent HIPPARCOS results. This controversy has not been completely solved until now. Chaboyer et al. (1998) argue that, from several methods including HIPPARCOS results, a consistent revision of RR Lyrae absolute magnitudes follows leading to mean age of $11.5\pm 1.3$ Gyr for oldest globular clusters. However, Layden (1998) claims that no significant revision of the distance scale for RR Lyraes follows from HIPPARCOS data, and Frolov & Samus (1998) obtain the distance scale for globular clusters, quite close to the traditional one, from infrared absolute magnitudes of RR Lyrae variables; these results tend to support "old'' globular clusters. Fernley et al. (1998), on the base of HIPPARCOS parallaxes and proper motions, derive statistical parallaxes of RR Lyraes and obtain, for the mean age of Galactic globular clusters, the value $17.4\pm3.0$ Gyr.

In age determinations, we use the approach similar to that used by us for M 79 (Kravtsov et al. 1997), NGC 6397 (Alcaino et al. 1997), and M 30 (Alcaino et al. 1998): we do not fix any values of reddening and distance modulus beforehand, but, vary these parameters in order to determine which isochrone best fits the observations, especially in the region of the lower giant branch and the turnoff point. The optimal position of the isochrone is determined by the best reproduction of the shape of the subgiant branch and the slope of the main sequence as well as of the color and luminosity of the turnoff. We do not attribute much physical significance to the resulting reddening values and we are going to study the reasons for these values being different from those derived by more direct methods in our future research.

Figures 10a,b,c show the $V,\, B-V$ diagram with ridge sequences of NGC 6723 (cf. Sect. 5.1 and Table 2), with superimposed oxygen-enhanced isochrones (Bergbusch & VandenBerg 1992) for three [Fe/H] values bracketing the metallicity range discussed for NGC 6723. We used the following approach to fit the ridge sequences. The HB absolute magnitude, $M_V({\rm HB})$, is given as a function of metallicity by the relation from Harris (1996):

$$M_V({\rm HB}) = 0.2\,{\rm [Fe/H]} + 1.0.$$ (1)

The apparent distance modulus is defined as $V - M_V = V_{\rm HB} - M_V({\rm HB})$, where for $V_{\rm HB}$ we adopted, in accordance with Sect. 5.1, the value $15\hbox{$.\!\!^{\rm m}$}48$. The color excess E_B-V optimal for the given chemical abundance was found from the best coincidence between the ridge sequences and the set of isochrones.

We estimated the age from the behavior of the ridge sequences in the region of the main-sequence turnoff and the subgiant branch, paying close attention to the quality of the fit of the observed sequences with the isochrones in this region for each adopted abundance.

In Fig. 10a, we present the isochrones for $\rm [Fe/H] =-0.78$. For the chosen parameters, the agreement with the ridge sequences is excellent at the main sequence, but the observed slope of the subgiant branch is somewhat steeper than the theoretical one, and we are unable to find a unique age value for the cluster. The position of the giant branch disagrees with the cluster age that follows from the turnoff position. The resulting color excess is somewhat lower than that determined in Sect. 5.1, and the distance modulus is lower than both our value (Sect. 5.1) and the value cataloged by Harris (1996). If, however, the metallicity of NGC 6723 is really close to $\rm [Fe/H] =-0.78$, then its turnoff age, in the Bergbusch & VandenBerg scale, is 16 - 17 Gyr.

  

Figure 10: The $(V,\, B-V)$ diagram with ridge sequences of NGC 6723 (Table2), with superimposed oxygen enhanced isochrones of Bergbusch & VandenBerg, for the indicated adopted parameters. a) $\rm [Fe/H] =-0.78$; b) $\rm [Fe/H] =-1.03$; c) $\rm [Fe/H] =-1.26$.The $M_V({\rm HB})$ values indicated in each panel are from Eq. (1), for the given metallicity value; for panel c) this value does not agree with the adopted distance modulus, as discussed in the text. The isochrones span the age range from 5 to 18 Gyr (panel a) and from 8 to 18 Gyr (panels b and c), the increment in age between subsequent isochrones is 1 Gyr

Figure 10b shows isochrones for $\rm [Fe/H] =-1.03$. Like in the previous case, the representation of the ridge main sequence for the adopted parameters is excellent. However, the observed slope of the subgiant branch now is close to the theoretical one, and we can also use this branch for age estimates. Agreement with theory is also found for the observed distance between the turnoff and the base of the giant branch. The observed giant branch runs along the red edge of the theoretical RGBs. For $V \approx 15$ and $V\approx 13$, it is even below this edge. Note however that our red giant ridge sequence is not quite reliable for V < 15 because of the low number of measured red giants. Still, as a whole, Fig. 10b demonstrates a much better agreement between theory and observations than Fig. 10a and Fig. 10c (described below). The color excess shows good agreement with our determination (Sect. 5.1). The distance modulus also shows better agreement, both with our determination and with the value from Harris (1996), than in the cases of Figs. 10a and 10c. If the metallicity of NGC 6723 is close to $\rm [Fe/H] =-1.03$, then its age in the Bergbusch & VandenBerg scale, from the turnoff and the subgiant branch, is 15 - 16 Gyr.

Figure 10c presents Bergbusch & VandenBerg isochrones for $\rm [Fe/H] =-1.26$, the metallicity value closest to the modern estimates for NGC 6723. For the adopted parameters, the agreement with the ridge sequences for the main sequence is the worst of the three cases. The observed slope of the subgiant branch agrees with the theoretical one. The observed lower giant branch tends towards the blue edge of the theoretical RGBs, i.e. towards lowest ages in the Bergbusch & VandenBerg set of isochrones, and its position does not agree with turnoff ages. The color excess is significantly higher than our determination (Sect. 5.1), and the distance modulus exceeds both our value and the value quoted by Harris (1996). The agreement between the theoretical and the observed diagrams is achieved at $V - M_V = V_{\rm HB} - M_V({\rm HB}) + 0.2$. If, however, the metallicity of NGC 6723 is really close to $\rm [Fe/H] =-1.26$, its age in the Bergbusch & VandenBerg scale, from the turnoff and the subgiant branch, would be 13 - 14 Gyr, the minimum value among the discussed cases.

We conclude that, from the point of view of the agreement between the theoretical models and the observed sequences, the best metallicity value is close to $\rm [Fe/H] =-1.03$. It leads to the cluster's photometric parameters (reddening and distance modulus) being in satisfactory agreement with independent estimates and to the age of 15 - 16 Gyr in the scale of Bergbusch & VandenBerg.