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2.1. Use of color-magnitude diagrams

Dating globular requires on the observational side a color-magnitude diagram (CMD) corrected for interstellar reddening, a knowledge of the chemical composition, and a distance modulus for the cluster. On the theoretical side, isochrones are derived from stellar evolutionary tracks for the appropriate chemical mixture. The evolutionary tracks themselves are constructed using a stellar evolution code, which calculates as a function of time the evolution of a stellar model with a given mass and chemical composition.

The theoretical isochrones, which are calculated in the [log Teff - logL] plane, must be converted to the observed CMD with an appropriate color conversion (or color calibration) table. This table translates from the [log Teff - log g] plane (the plane of stellar atmosphere parameters) for the cluster composition to the color system used for the observations (log g is the surface gravity). This is possible because each point along the theoretical isochrone corresponds to a unique stellar mass.

The traditional way of dating a globular cluster is to match the luminosity of the turnoff point on the cluster CMD (definest as the bluest or hottest point near the turnoff), to a theoretical isochrone derived for the appropriate cluster chemical composition. This simple approach has been justified until now in view of the large uncertainties in the data, both observational and theoretical. It is still the basis of the ages presented in this paper, and used in the Monte Carlo simulation described in Section 5. However, the extraordinary quality of the new HST data, and the equally remarkable strides in stellar structure theory in the last few years, will justify increasingly sophisticated approaches in the fitting procedure (for a step in this direction, see e.g. Chaboyer et al. 1996c; Rubenstein & Bailyn 1997).

2.2. The DeltaV and Delta(B - V) techniques

The DeltaV technique (Iben and Renzini 1983) makes use of the quantity DeltaV which is the difference in magnitude between the main sequence turnoff and the HB luminosity at the same color. It has the advantage of being insensitive to interstellar reddening. And in its purest form, i.e. when it relies on theoretical models of the HB to set the HB luminosity, it provides directly a completely theoretical calibration for the globular cluster ages, given a chemical composition, which is independent of distance. For these reasons and because of its convenience, the DeltaV technique, or a modified version of it, has often been favored in recent years for age determinations. The motivation for modifying the DeltaV calibration is the introduction of an empirical or semi-empirical calibration for the horizontal branch luminosity, rather than using the theoretical HB luminosity calibration. This is the form in which the DeltaV technique will be used in Section 5. The method is illustrated in Figure 3.


Figure 3. Definitions of DeltaV and Delta(B - V) in the absolute magnitude-intrinsic color plane. The horizontal branch stays constant in luminosity with time (note the shifted scale for the schematic HB), whereas the turnoff becomes fainter and redder. We see that DeltaV increases and Delta(B - V) decreases with age.

The Delta(B - V) technique (Sarajedini and Demarque 1990), also illustrated in Figure 3, and a similar technique developed by VandenBerg et al. (1990), make use of the color extent of the subgiant branch. It is most reliable for comparing cmd's of star clusters of the same composition but different ages. Because it is more sensitive to the radius evolution than the DeltaV index, the Delta(B - V) index contains complementary information, which will become useful as our ability to calculate stellar radii improves.

2.3. Luminosity functions

Although it is an old and fundamental test (Sandage 1953, 1957), it has in the past been not been possible to use luminosity functions for precision work. But there has recently been an increasing interest, primarily because luminosity functions can now be measured more reliably and completely to faint magnitudes. In principle the luminosity function alone could be used to derive a cluster age estimate, but in practice the best use of the luminosity function is in conjunction with the CMD. Used in this way, the luminosity function provides a stringent test of stellar evolutionary rates, both in luminosity and in color.

Of special interest is the comparison of turnoff and giant branch luminosity functions. Bolte (1994) has measured the luminosity function of M30 and found a discrepancy between the rates of evolution predicted by theory and the cluster luminosity function, but the problem is complicated by dynamical effects. On the other hand, the globular cluster M5, which provides a more straightforward test of stellar evolution since it is freer of dynamical complications, has been shown to have a luminosity function in agreement with standard stellar models. In all cases, special care must be taken with completeness corrections at the faint end of the luminosity function, as shown in the recent work of Bromm et al. (1996) and Sandquist et al. (1996).

Another related test of stellar evolution have been proposed by Jimenez & Padoan (1996), which is based on giant branch luminosity functions and estimated HB masses. While such a method is in principle incapable of precise absolute age determinations, its value lies in that it offers a different test of stellar evolutionary rates. Future work involving the envelope entropy function in globular clusters, will provide additional tests of the reliability of stellar models for age determinations (see Section 6).

2.4. Dynamical information

HST observations reveal the presence of large numbers of binaries in globular clusters (Rubenstein & Bailyn 1996; 1997). Studies of spectroscopic and eclipsing binary systems in star clusters hold great promise for improving cluster distance determinations (the major source of error in cluster dating), and the mass-luminosity relation near the main sequence turnoff (cf. the paper by Paczynski at this Symposium).

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