Thus, for every combination of age and distance modulus, one can calculate the residual R(t, m-M)= [n_{2,th}(t)-n_{2,obs}(t,m-M)]^{2}+[n_{3,th}(t)-n_{3,obs}(t,m-M)]^{2}.
From a contour plot of R(t,m-M), the age and distance modulus may be determined from the minimum, and the contour lines measure the uncertainty.
The procedure to obtain the LF from evolutionary stellar tracks is illustrated in ref. 17. A power law stellar mass function is assumed here, as it was in that work.
We have shown that a careful binning of the stellar LF allows a very precise determination of age and distance of GCs at the same time.
If stellar counts with 5% 1s uncertainties in 1-mag wide bins are available, the age can be determined with an uncertainty of 0.5 Gyr and the distance modulus with an uncertainty of 0.06 mag.
This LF method is therefore an excellent clock for relative ages of GCs and also a very good distance indicator. In other words, its application will provide very strong constraints for the theory of the formation of the galaxy.
In Figs. 3 and 4 we show the result of applying the LF method to the galactic GCs M55 and M5. The plots show contour plots for the error in the determination of the distance modulus and age of M55 (M5) simultaneously. The contour plots correspond to different values for the uncertainty in the number of stars in the LF. As stated above, if stellar counts are within an uncertainty of 5%, then the age is determined with an uncer-
Table 2. Errors associated with different methods used to compute the age of the oldest GCs
Uncertainties |
MSTO |
HB |
LF |
Distance modulus |
25% |
0% |
3% |
Mixing length |
10% |
5% |
0% |
Color-T_{eff} |
5% |
5% |
0% |
Heavy elements diffusion |
7% |
2% |
7% |
a-elements |
10% |
5% |
10% |
Reddening |
5% |
10% |
0% |
tainty of 0.5 Gyr and the distance modulus with an uncertainty of 0.06 mag.
The age obtained for M55 (12 Gyr) confirms the conclusion of the HB morphology method that GCs are not older than 14 Gyr.
Table 2 lists the uncertainties involved in each of the three methods described above to determine GC ages. As already discussed the MSTO method is largely affected by the uncertainty in distance, but uncertainties in the mixing length, diffusion of heavy elements, and in the color-T_{eff} relation are important as well.
The HB method uses the fact that mass loss along the RGB is the chief of the HB morphology. This may seem to introduce an additional uncertainty in the method because mass loss in low-mass stars is unknown. In fact, the only stars used to determine the mass at the RGB are the reddest ones that did not suffer any mass loss. The evolution with mass loss along the RGB was done by using the method developed in Jimenez et al. (1) that describes the mass loss efficiency parameter with a realistic distribution function and minimizes its model dependence. Furthermore, the HB method is insensitive to changes in CNO abundances. The reason for this is that if CNO is enhanced with respect to iron the HB becomes redder, leading to a smaller mass for the reddest point of the HB, but because the stellar clock also goes faster, both effects compensate. The HB morphology method is weakly sensitive to diffusion by heavy elements (J.Mac-Donald, personal communication).
The LF method needs to know the metallicity of the GC. Apart from this, the LF method is the one with the smallest errors among the three methods described here. The biggest advantage of the LF method is that it is insensitive to the mixing-length, reddenning, and color-T_{eff} transformation. Because the LF method is based on counting stars in several bins, it is independent of fitting to morphological features in the observed CMD of the GC. Therefore the LF method is a superb technique to determine relative ages of GCs.
The main conclusions of this paper are:
Alternative methods to the traditional MSTO isochrone fitting are important to get a more accurate age (and distance) of the system of galactic GC. The HB morphology and LF methods are more accurate than the traditional MSTO method and overcome many of the uncertainties of the latter.
The three methods presented in this paper agree on an age of about of 13 Gyr for the oldest GCs. The minimum possible age is 10.5 Gyr and the maximum 16 Gyr, with 99% confidence.
More work needs to be done by using alternative methods to determine more accurately the age of GCs.
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