Astron. Astrophys. 333, 505-523 (1998)

3. The data

3.1. The cluster sample

For a practical application of this method we have selected from the literature the GGCs for which high precision CMDs exist for both the TO and the HB regions. In particular, we have restricted the sample to clusters having a well populated HB at the TO color to avoid any contamination due to uncertain extrapolation of the actual HB luminosity level. Such an assumption strongly limitates the extension of the sample, but it is useful to make the whole procedure as straightforward as possible.

As horizontal age-parameter we will use here   the difference between: 1) the color of the RGB at a luminosity level 2.5 mag brighter than the MS point which is 0.05 mag redder than the TO; 2) the color of the TO point. Such a parameter can be read on the CMD and can be measured for each cluster, independently of the HB morphology. It is a function of the cluster age, but also of the cluster metallicity, as discussed by VBS which used exclusively it in differential form and in narrow metallicity boxes.

Concerning the estimates of the various quantities deduced from the CMDs and involved in the definition of the age-parameters   and   with their associated errors, we are in presence of different situations:

1. Clusters for which the photometric data of the CMD are available to us in machine-readable form. These clusters have been marked with the label CMD in the "source" column of Table 1.

   For them, a statistical analysis of the distribution of the stars in the various CMD sequences allows a quantitative determination of the observational uncertainties associated to the measured quantities (,5, and ). The associated errors have been calculated following the procedure described by VBS (see their Sect. III and Table III) as we already did in previous papers (Buonanno et al.  1993 and Buonanno et al.  1995). Shortly, the errors have been estimated from the scatter of the stars with respect to the parabolic arc best fitting the considered CMD sequences.

2. Clusters for which the photometric quantities have been read on the published mean-ridge-lines. These clusters are identified by a label MRL in column 7 of Table 1.

   For these clusters, it is impossible to give a quantitative measure of the errors, so we are forced to adopt just educated estimates. To this aim, we based our estimates mainly on the similarity of the CMDs to clusters for which a machine-readable data-base is available.

The errors in   have been obtained by combining the estimated errors in and . The formal errors in   for the clusters with CMDs in machine-readable form turned out to be always less than 0.01 mag. Thus we adopted an error of 0.01 mag for all the   entries in Table 1.

Table 1. Observables of 36 selected clusters

Before proceeding, some further notes must be added for the metal rich GGCs ([Fe/H]  ): it is well known that for the most metal rich GGCs the HB collapses to a red clump of stars which is slightly brighter (by a quantity   mag) than the average luminosity of the RR Lyrae variables located at the center of the instability strip and used to measure   (see Sarajedini et al.  1995, Ajhar et al.  1996, Catelan and de Freitas Pacheco 1996, for discussion and references). Following the above authors, we have considered a correction   mag to the values of the observed for the clusters with [Fe/H]  , as a compromise.

Most of the adopted metallicities come from the compilation by Zinn & West (1984), superseded and supplemented by Armandroff & Zinn (1988). For the young clusters we adopted the metallicities derived from the CMDs; therefore in Table 1 are reported: for Ruprecht 106 the estimate of Buonanno et al.  (1993), for Arp 2 that of Buonanno et al.  (1995b), for Terzan 7 the estimate of Buonanno et al.  (1995a), and for IC 4499 that of Ferraro et al.  (1995). The helium abundance Y has been assumed as constant at Y for all the clusters, according to Buzzoni et al.  (1983).

A final caveat:

• We are aware that the true errors in the measured observables can be significantly larger, at least for some clusters. However, since our basic aim here is to present a procedure which can be improved with progressively improving the quality of the data-base, this would not undermine the overall approach.
• The same consideration can be applied to the adopted metallicity scale. In fact, it is well known that the scale of Zinn and West (1984) was at the state of the art at that time and deserves revisiting as for instance stated by Carretta and Gratton (1997). However, as noted by Rutledge et al.  (1997), it is still unclear which scale could approximate the true [Fe/H] scale more closely. Therefore, we will use the scale of Zinn and West (1984) allover the paper, and the possible impact of a different adoption has to be evaluated subsequently.

A summary of the adopted observables is listed in Table 1 which contains:

1. the cluster identification;
2. the adopted metallicity;
3. the apparent V magnitude of the HB - - at the TO color;
4. the reference point 5 ;
5. the magnitude difference   ;
6. the color difference   between thr RGB and the TO;
7. the source type of the data (machine-readable or mean-ridge-line);
8. the reference(s) to the adopted CMD(s).

3.2. The sub-sample of clusters "coeval" within the uncertainties

The observed   values for the 24 clusters with appropriate HB reported in Table 1 have been plotted in Fig.  7 a, b, c superimposed on the calibration of   in terms of the SC96 and DCM97 and VDB96 models. These values result restricted in a quite narrow range for the bulk of the clusters, independently of metallicity, with only 4-5 clusters showing clearly less-than-average values.

We intend to use the clusters in Table 1 and Fig.  7 in order to select a subsample of clusters to be considered "coeval" within the uncertainties, and then to derive differential ages for all the clusters in the sample.

As a selection criterion we consider as coeval all the clusters which have the observable   contained within the strip defined by the 14 and 16 Gyr isochrones in Fig.  7a. In particular, considering the 1 error in   reported in Table 1 and Fig.  7, we selected the clusters which have ages included between 14-16 Gyr with probability .

In other words, by adopting as "reference clock" the calibration of   obtained by using the SC96 models for both MS and HB with the K93 transformations, we pick up the clusters in Fig.  7a which have a mean (absolute) age of 15 Gyr with a "clock read-out-noise" of 1 Gyr. Note that the quantity 1 Gyr is not the error in the estimate of the (absolute) age, but only the arbitrary confidence limit we adopted for the selection of "bona fide" coeval clusters.

Following the above criterion, we selected as "coeval" the following clusters: NGC 104 (47 Tuc), NGC 362, NGC 1261, NGC 4147, NGC 4590 (M68), NGC 5053, NGC 5272 (M3), NGC 5904 (M5), NGC 6121, NGC 6352, NGC 6362, NGC 6584, NGC 6838 and NGC 7078 (M15).

Now the question is: is this sub-sample of claimed coeval clusters robust enough to survive as "truly" coeval with changing the reference clock (i.e. the theoretical models and/or the transformations)?

A quick look at Fig.  7 panel b,c gives already a first answer: the use of the isochrones computed by DCM97 and VDB96 would lead to the selection of the same sub-sample of "coeval" clusters, the only difference being a systematic shift in the absolute age.

The three panels shown in Fig.  8 allow also to check the degree of compatibility of our selected sub-sample of 14 "coeval" clusters (based on SC96 models and the K93 transformations) with alternative calibrations, where the transformations of VDB92 and BK92 are instead used. As already noted, both shape and position of the isochrones change with changing the color-transformations, but we can still claim that the 14 selected clusters can be considered coeval within the uncertainties, though the mean absolute age (the reference zero-point) is shifted.

From the above analysis, discussion and checks reported in Sect. 2.2-2.5, we are confident that the sub-sample of 14 clusters we have chosen can safely considered to be "coeval" (as measured by the   age-indicator) within the intrinsic "read-out-noise" of Gyr, independently of the assumed theoretical clock.

3.2.1. A further check

We have also verified how these 14 clusters behave, using the classic vertical parameter  . The   values (estimated from the CMDs in Table 1) for these clusters are for NGC104, NGC 362, NGC 1261, NGC 4147. NGC 4590, NGC 5053, NGC 5272, NGC 5904, NGC 6121, NGC 6352, NGC 6362, NGC 6584, NGC 6838 and NCG 7078, respectively.

Fig.  9 shows the calibration of   obtained using the SC96 (panel a), DCM97 (panel b), and VDB96 (panel c) models, with the observed values of   for the 14 clusters selected above superimposed to the grids.

It is readily evident that not all the selected "coeval" clusters pass the   - test. In fact only 11 out of 14 clusters still lie in the same "two-Gyr-strip", independently of the assumed model. The other 3 (NGC 104, NGC 4147, and NGC 7078) shows a spread in age as large as 4-5 Gyr as seen by the   observable.

While it is expected that data-points are more spread out in the plane of Fig.  9 than in the   vs plane because of the greater measurement errors associated with , such a large difference can hardly be accounted for given the adopted error bars. In our view there are only two possible explanations: a) there are systematic errors in the   measures of the three "anomalous" clusters, or b) the errors have been largely underestimated. It is very difficult to discriminate between these different possibilities: both cases a and b cannot be easily excluded since the sources of the photometry are necessarily very hetherogeneous and a really exaustive check is impossible.

However, since the main purpose of this paper is to define a procedure to establish a general scale for the relative-age of galactic globulars and to explore all the possible problems connected with the definition of such a scale in general, independent of the adopted method, it seems useful to proceed for a further step in our route using just the 11 clusters which survived the   -test, being at the same time aware of the uncertainties emerged till now.

3.2.2. The basis for a temptative relative-age scale

We can select above a sample of clusters of different metallicity, which are coeval within Gyr. The relative ages finally adopted for the GGCs listed in Table 1 (and having a measured  ) are reported in Table 2, column 3. They have been determined by using as reference the isochrone grid presented in Fig.  7 panel a and by computing (at the [Fe/H] value adopted for the cluster) the residual with respect to the 15 Gyr isochrone, taken as "0-age" reference line. The errors reported in Table 2 have been determined transforming the   error bars into using the procedure described above, with no allowance for errors in [Fe/H] .

Table 2. Relative ages

Since we now "know" that these clusters turn out to be coeval based on the vertical method, we can use them as reference grid to calibrate the horizontal method, overcoming so (at least partially) some of its major drawbacks.

© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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