FIG. 1. The CMD of M5 and three different isochrones. Notice that a vertical shift in any of the nonfitting isochrones (10 and 14 Gyr) will make them fit the observed CMD perfectly well. This is equivalent to a change in the distance modulus.

inferred by the subdwarf method to GCs and therefore reduced the ages of GCs (2).

To circumvent the need for the distance determinations in computing the age, Iben and Renzini (3) proposed an alternative method for deriving ages using the MSTO, the so-called ∆V method. The method exploits the fact that the luminosity of the MSTO, and not only its Teff, changes with mass (age), and also that the luminosity of the HB does not change because the core mass of the He nucleus is the same independently of the total mass of the stars (provided we are in the low mass range), and the luminosity in the HB is provided by the He core burning. Because the method is based on a relative measure (the distance between the HB and the MSTO), it is distance independent. Of course, the method needs the knowledge of at least one GC distance to be zero-calibrated. Unfortunately, the method has a serious disadvantage: the need to know accurately the location of the MSTO point. This turns out to be fatal for the method because it has associated an error of 3 Gyr in the age determination (see Fig. 1 around the MSTO).

Furthermore, all the above methods are affected by three main diseases: the calibration color-Teff, the calibration of the mixing-length parameter (α), and the need to fit morphological features in the CMD (i.e., the MSTO). See Table 2 for a detailed review of all errors involved in the different methods.

The most common ages obtained for the oldest GCs by using the MSTO method are in the 14–18 Gyr range. Nevertheless, an error bar of 3.5 Gyr is associated with all age determinations using the MSTO methods described above.

The Horizontal Branch Morphology Method

We obtained UBVRIJHK photometry of the GCs M22, M107, M72, M5, and M68 at La Silla (Chile) by using the Danish 1.5-m telescope and the ESO 2.2-m telescope. We added to our sample four other GCs: M92, M3, M55, and 47 Tuc.

The spread of stars along the HB is mainly caused by previous mass loss that varies stochastically from one star to another (4). The range of colors where zero-age HB stars are found is a function of metallicity (the “first parameter”) and the range of ZAHB masses. More precisely, the ZAHB color at a given metallicity depends on both the star’s total mass and the ratio of core mass to total mass, but the core mass essentially is fixed by the physics of the helium flash and is quite insensitive to the mass and metallicity. For a given average mass loss, the average final mass is thus a decreasing function of age, which is therefore a popular candidate for the “second parameter” (5), although other candidates such as CNO abundance also have been suggested. A strong case for age as the chief (though perhaps not necessarily the only) second parameter has been made by Lee, Demarque. and Zinn (6), who find a tendency for the clusters to be younger in the outer Galactic halo. Jorgensen and Thejll (7), by using analytical fits to a variety of RGB models and following evolution along the RGB with mass loss treated by Reimers’ formula (8), showed that, for clusters with narrow RGBs (the majority), star-to-star variations in initial mass, metallicity, or mixing-length parameter can be ruled out as a source of the spread along the HB. This leaves as likely alternatives only either variations in the Reimers’ efficiency parameter η (or some equivalent) or a delayed helium flash caused by differential internal rotation. The later alternative would lead to a fuzzy distribution of stars at the RGB tip.

With our data we can analyze these propositions. Assume there was a variation in the total mass at the flash caused by mass loss. The effect on the luminosity at the helium core flash is small ≈0.01 mag, but the effect on the temperature is quite significant ≈110 K. On the other hand, a delayed helium core flash would not produce any effect on the effective temperature but would make stars appear above the theoretical helium core flash in a bin of 0.3 mag. Considering that the evolution time in this very last bin would be the same as in the last bin before the theoretical helium core flash, we would expect the same number of stars in these two bins of the diagram. So, for a typical GC we would expect 3–4 stars. Variations in the mass loss certainly will produce variations in the morphology at the red giant branch tip.

Following this strategy we looked at the previous set of observations and counted the number of stars that were expected in every bin of luminosity. By using the set of three clusters where it was possible to distinguish the RGB from the asymptotic giant branch (M72, M68, and M5), we had a relatively good statistical sample to test the theory of a delayed helium core flash. We counted the RGB stars and compared them with the theoretical predictions. To calculate the number of stars expected in every bin of luminosity we used stellar evolutionary tracks to compute the time spent there and then used the fuel consumption theorem (9) to compute the number of stars—the integrated luminosity of the cluster was properly scaled to the area covered by the charged coupled device. We have concluded from the set of observations that there is no GC where there appears to be an extra number of stars populating the RGB beyond the helium core flash (see figure 14 in ref. 1). This argument rules out, to a level of 0.01 M, variations of the core mass at the flash as the cause of HB color variations.

A method that is independent of the distance modulus can be developed, by using the fact that the spread of stars along the HB is mainly because of previous mass loss that varies stochastically from one star to another. It is therefore meaningful to proceed to an analysis of both the right giant branch tip and the HB and to link them together to deduce general properties from morphological arguments.

The procedure that we use to analyze the morphology of the RGB and the HB together and constrain the mass of the stars at the RGB is as follows:

  • Because the vertical position of the RGB depends only on metallicity and α, once the metallicity is known α is the only free parameter. Therefore, we can find a fit for the best value of α, using the vertical position of the RGB.

  • The reddest point of the HB corresponds to zero mass loss and therefore to the most massive stars that are alive in the GC and therefore the oldest.

  • By using HB theoretical models we can determine the mass of the reddest point of the HB. So it is possible to compute stellar tracks for a certain input mass and iterate until the track at the zero age horizontal branch matches the reddest point of the observed HB.

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