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This section summarizes the results of a recently completed detailed numerical study designed to put a firmer bounds on the age and on the uncertainty in age of the oldest globular clusters in the Galaxy. The discussion consists of two parts. In the first part, 1000 Monte Carlo realizations of stellar isochrones, based on estimates of the range and distribution of input parameters in the stellar evolution code, were produced, which were then used to derive the ages of the 17 oldest globular clusters. The evolutionary tracks were computed using the Yale stellar evolution code, and were constructed using the same physical assumptions as in Guenther et al. (1992), and Chaboyer & Kim (1995), with a few relevant updates in the equation of state and the [alpha / Fe] element ratio discussed in the previous section. These updates are not trivial when it comes to age determinations, as they altogether account for a reduction of as much as 17% of the median maximum age over the Chaboyer et al. (1996a) age scale (shown in Figure 2). Then, incorporating estimates of the observational uncertainties in the measured color-magnitude diagrams and in the fitting procedure to the theoretical isochrones, a probability distribution for the mean age of the 17 oldest clusters was derived. The ages were derived using a semi-empirical DeltaV technique, in which the absolute calibration was based on the empirical absolute magnitude of the RR Lyrae variables Mv(RR) = 0.6 ± 0.08 (rather than a purely theoretical calibration) (Layden et al. 1994). The abundance of the alpha-elements was taken to be [alpha/Fe] = 0.55 ± 0.05 (Gaussian) ± 0.2 (top- hat) (King 1993; Nissen et al. 1994). Two totally independent color calibration tables of Green et al. (1987) and Kurucz (1992) were used.

The age distribution for the oldest globular clusters is plotted in Figure 9. We see that the median age of the distribution is 14.56 Gyr, with a one-sided 95% Confidence Limit lower bound occurring at an age of 12.07 Gyr.


Figure 9. Histogram showing the relative number of realizations of mean cluster ages drawn randomly from the Monte Carlo data set using Mv(RR) = 0.6 ± 0.08, chosen from a Gaussian distribution. The dashed curve is a Gaussian approximation to the actual distribution (from the work of Chaboyer et al. 1996b).

The reader is referred to the original paper for details in the input parameters and error estimates beyond what is discussed in Section 4 (Chaboyer et al. 1996b). The Monte Carlo analysis yields quantitative estimates of the main sources of uncertainties in globular cluster ages. These uncertainties are listed in Table 1. The Monte Carlo result is consistent with former discussions of the problem of globular cluster ages in identifying the distance estimate (in this discussion the adopted Mv(RR)), as the largest source of age uncertainty (e.g. Renzini 1991). Other major uncertainties include the uncertainty in the CNO abundance (or the [alpha/Fe] ratio), and the choice of mixing length in the convection zone. This last uncertainty, usually ignored in the past primarily because of its intractability, is, as explained in Section 4, one that needs to be explored more fully. This is because the Monte Carlo realizations, while varying the value of the mixing length parameter from realization to realization, assume that this parameter remains constant during evolution. Further work will need to explore the possibility of a distortion to the shape of evolutionary turnoff due to possible variations in effective mixing length as a star evolves. However, the Monte Carlo data also show that unless we find that convection behaves in a grossly unexpected way at low metallicities near the turnoff, the errors on the ages is likely to amount to only a few percent in either direction. We must conclude from this study that to the best of our knowledge, the mean age of the oldest globular clusters is 14.6 Gyr, with a 5% probability that it is below 12.1 Gyr or above 17.1 Gyr. And that the probability that globular clusters have the same age as the oldest disk stars (i.e. 10 Gyr), is vanishingly small.

Table 1. Globular cluster ages error budget 1
1Derived by varying each parameter over its full 2alpha range, keeping all other parameters fixed.

Error (±%)
Mv(RR)(distance) 16
[alpha/Fe] 7
Treatment of convection 5
Helium diffusion 4
1415N(p, y) O rate 3
Color transformation 3

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