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

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4. An empirical calibration of [FORMULA]   vs [Fe/H] 

4.1. Problematic aspects of the "horizontal" method

The main advantage of the age parameter [FORMULA]   (VBS, SD, SC91) is that it can be applied to any cluster having a good photometry in the TO region, irrespective of its HB morphology.

Nevertheless, this method presents some disadvantages, the most important being a large dependence on metallicity of the colors of the two reference points (located at the main sequence TO and on the RGB) and, in turn, of their difference [FORMULA]  .

This intrinsic weakness of the method has been widely recognized, and even its proposers (VBS, SD) recommended not only to apply this procedure to study just relative ages, but also to restrict the direct comparisons to clusters having the same metallicity (within the current measuring errors [FORMULA] dex in [Fe/H] ). In particular, VBS grouped the GGCs into "metallicity boxes", but often the CMDs are hardly comparable and the claimed age differences could also be interpreted as due to a difference in metallicity.

[FIGURE] Fig. 10. Comparison of M92 fiducial ridge lines for the main sequence (Stetson & Harris 1988) and horizontal branch (Buonanno et al.  1985) with the corresponding ones of NGC 6397 (Buonanno et al.  1989, Alcaino et al.  1987). The latter were registered to the former by adding to the observed magnitudes and colors [FORMULA] mag and [FORMULA] mag. The abscissa and ordinate zero-points correspond to the M92 turnoff color [FORMULA] =0.388 and 5 =19.64. Even if the overall coincidence of the two fiducial main sequences seems to indicate a similar age, it appears a clear vertical displacement between the HBs. This can be interpreted as due to age differences (see Sect. 4.1)

An example of this problem is shown in Fig.  10, where we report the fiducial sequences of M92 (Stetson & Harris 1988) compared with those of NGC 6397 (Buonanno, Corsi & Fusi Pecci 1989). The ridge lines of the two clusters have been registered following the procedure of VBS (in their Fig. 4 panel c). In particular, NGC 6397 has been shifted by +2.17 mag in V and -0.168 mag in B-V. As can easily be seen from Fig.  10, while the MS, the TO and the SGB of M92 and NGC 6397 are fully overlapping, the two HBs show a clear displacement. Hence, the conclusion drawn for instance by VBS that the two clusters are coeval within 0.5 Gyr based on the identity of the parameter [FORMULA]   is hardly supported at this level of accuracy by the comparison of the whole CMDs.

In our view, an alternative interpretation of this inconsistency is that, since M92 and NGC 6397 are different enough in metallicity ([Fe/H] =-2.24 [FORMULA] and [Fe/H] =-1.91 [FORMULA], Zinn 1985), then the coincidence of the [FORMULA]   parameters would actually imply an age difference of [FORMULA] Gyr, as indicated also by the HB luminosity displacement. Such a conclusion is moreover strengthened by its independence of the adopted theoretical models.

It is therefore evident that without a procedure able to calibrate the dependence of [FORMULA]   on metallicity, it is almost impossible to build up a self-consistent and reliable relative age scale.

4.2. The new calibration

In order to overcome the above limitations, we suggest to use the sub-sample of clusters we found to be "coeval" based on the metallicity independent [FORMULA]  -method (and that passed the [FORMULA]   -test; see sec. 3.2.1), to calibrate the actual dependence of [FORMULA]   on metallicity. If reliable, such a "direct" calibration (note that the ages are now assumed to be known independently) will allow (a) to know experimentally the dependence of the parameter [FORMULA]   on metallicity, (b) to determine age differences for clusters of any metallicity using the same calibrated scale, once the metallicity is ascertained. Moreover, such a procedure links in a self-consistent approach the two traditional "vertical" and "horizontal" methods for GGC dating ([FORMULA]   and [FORMULA]  ). The main step in this procedure is the selection of the "known calibrating" clusters discussed in Sect. 3.2 above.

In Fig.  11 it is reported a plot of the individual [FORMULA]   parameters vs [Fe/H]  for the 11 calibrating clusters. A linear best fit to the data of the adopted calibrating clusters (the full line in Fig.  11) yields:


Although the number of calibrating objects is small and the coverage of the metallicity range still poor, it is reassuring that the quality of the fit is good enough to define a clear-cut correlation, whose quality will be easily improved by further data of clusters with independent relative age estimates, or by the availability of more accurate measures for the same objects.

[FIGURE] Fig. 11. The observed horizontal parameter [FORMULA]   as a function of [Fe/H]  for the 14 clusters selected as "coeval" on the basis of [FORMULA]   (see Sect.3.2). A linear best fit is also reported (full line). The cross at the bottom-right shows the typical error for the two observables

The knowledge of the value of the slope estimated in Eq. (1) immediately allows one to estimate the errors involved in the determination of relative ages with [FORMULA]   due to the observational uncertainties in the metallicity [Fe/H] : a typical error in [Fe/H]  of [FORMULA] 0.2 dex reflects into an indetermination of about [FORMULA] 0.008 mag in [FORMULA]   and, in turn, about [FORMULA] 1 Gyr in the relative age.

4.3. Comparison with the theoretical models: there is a discrepancy

Supposing that the relative ages adopted as known calibrators for the coeval sub-sample are correct, it is useful to compare the empirical isochrone defined by the coeval sub-sample in the [FORMULA]   vs.  [Fe/H]  plane with the corresponding theoretical isochrones.

Fig.  12 displays the "constant-age" loci as derived from the models (dashed lines) computed by SC96 (panel a), VDB96 (panel b) and DCM97 (panel c), all based on K93 transformations. This figure shows how the isochrones are generally different either in the zero-point (absolute age), either in the dependence on metal content. The discrepancy is even more evident looking at Fig.  13, 14 and 15, where the calibrations obtained from the models of SC96, DCM97 and VDB96 transformed following K93, BK92 and VDB92 are compared to the empirical isochrone and the GGC data. As can be seen, the transformations have a very strong impact on the comparison. In particular, the 11 calibrating clusters could be "coeval" or even different in age (up to 5 Gyr) depending on the adopted set of models+transformations.

[FIGURE] Fig. 12. The "empirical isochrone" in the plane [FORMULA]  , [Fe/H]  of Fig.  11 compared to the theoretical calibration grids derived from three different isochrone-sets, all transformed to V, B-V using K93. As can be seen, the overall agreement between the empirical and the theoretical calibrations of [FORMULA]   is poor in any case

[FIGURE] Fig. 13. The effect of varying the temperature-color and luminosity-magnitude transformations on the theoretical calibration of [FORMULA]  . The SC96 models transformed following K93, BK92 and VDB96 are used to build the calibration grids reported in the three panels dashed lines. As expected both position and shape of the theoretical isochrones are strongly affected by the choice of the transformation. Yet, in no case there is a satisfactory agreement with the "empirical isochrone"

[FIGURE] Fig. 14. The DCM97 models are used to derive the calibration grids reported as dashed lines in the three panels for different temperature-color transformations. As in Fig.  12 and Fig.  13, the agreement between the theoretical and the empirical isochrones is poor, independent of the adopted choice

[FIGURE] Fig. 15. The case of the [FORMULA]   calibration using the VDB96 models converted into the observational plane with different trasformations is presented in the three panels. The agreement with the empirical calibration is still not satisfactory as in the previous plots

In fact, there is no combination of models and/or transformations able to yield [FORMULA]   values truly consistent with the previous conclusion that the 11 calibrating clusters are coeval within the quoted uncertainties. One could also note (see Fig. . 12 - 15) that in many cases the observed [FORMULA]   values do not match the theoretical grids, i.e. observations and theoretical predictions strongly disagree also in an absolute sense.

Why do the empirical and theoretical calibrations behave so differently?

In our view there are essentially only two possible answers:

  1. The clusters we have adopted to be "bona-fide" age calibrators (based on the vertical, differential methods) are actually non-coeval (i.e. they differ in age by more than 2 Gyr), and, consequently, the linear locus traced in Fig.  11 - 15 is not an empirical isochrone. If this is the case, one has to admit that the measurement errors of the vertical age parameters are, on average, greater than assumed, or that the calibrations of the same parameters are in error by a similar amount.
  2. If the adopted calibrating clusters are actually coeval within 2 Gyrs, then the theoretical models are somehow wrong, and in particular the scaling of the tracks in the observational plane with varying metallicity would be called into question.

A truly safe choice between the two alternatives can hardly be done at this stage, as the uncertainties affecting the estimates of the relative ages using the vertical methods are still large and, moreover, their calibrations depend on the the theoretical models. However, it is important to note that, while the vertical method rests only on model luminosities (i.e. on nuclear burning), the horizontal parameters obtained from the theoretical tracks depend on various uncertain quantities, like the mixing length, the color-temperature transformations, etc. and it is conceivable that errors (varying in size with varying metallicity) can still affect the final horizontal scaling of the tracks in the observational plane.

On the other hand, the existence of such a difficulty to compare horizontal parameters measured from the CMD of clusters having largely different [Fe/H]  was clearly pointed out also by the proposers of the "horizontal" method. For instance, VSB note that "it seems unlikely that the predicted variation of [FORMULA]   with metallicity can be quantitatively correct. Consequently, we recommend that our procedure be applied only to clusters that have the same [m/H] values to within current measuring errors, namely [FORMULA] dex".

In conclusion, unless the ages estimated from the vertical method for the 11 calibrating clusters are grossly in error, it seems evident that the comparison of the observational data in the planes [FORMULA]   vs.  [Fe/H]  and [FORMULA]   vs.  [Fe/H]  with the theoretical isochrones leads to a disagreement, worth of further analysis.

This fact carries further support to our approach: given the many problems and uncertainties associated with the theoretical modeling of the [FORMULA]   vs metallicity relation, an empirical calibration based on the most robust "vertical methods" can ultimately prove to be very fruitful, in perspective, to derive a reliable relative-age scale encompassing the whole Galactic globular cluster system. At the same time this procedure can be useful in constraining the models themselves.

In order to show how such a calibration can be obtained and used to derive an age scale, we complete the proposed pipeline in the following sections. Before proceeding, we note that, though based on only 11 clusters and affected by several possible uncertainties (discussed above), our method will yield a scale for the relative ages very similar to those recently obtained by using the "vertical-method" (CDS96) or the "horizontal-method" (Richer et al.1996) (see Sects. 4.6 and 4.7). In our view, such an agreement shows that:

  1. The present method already gives performances not worse than the "traditional" ones, and the semi-empirical calibration on which is based can be improved also without having a corresponding significant improvement in the detailed models.
  2. One can hardly assume that any of the relative-age scales available in the literature so far can actually yield ages to better than [FORMULA] Gyr allover the full metallicity range. In fact, only age differences between couples of clusters sharing the same metal content can be obtained at a higher precision level (1 Gyr), but they cannot then been grouped together to yield a homogeneous scale.

4.4. A possible extension of the calibration: just an example

As noticed at the end of Sect. 3.2, from the analysis of [FORMULA]   only a few clusters are significantly younger than the bulk, namely Pal 5, Pal 12, Ru 106, Terzan 7, Arp 2, NGC 1851 and IC 4499. This sample of young clusters, though well distributed in metallicity, is admittely too scanty to perform an accurate empirical calibration of the [FORMULA]   parameter in terms of age.

However, it is worth checking whether the two indicators, [FORMULA]   and [FORMULA]  , give consistent indications on the relative age of the objects considered in Table 1 and not included in the group of the 11 coeval calibrating clusters. To this aim, we use the "young" clusters in order to derive an "empirical guess" for the dependence of [FORMULA] ([FORMULA]  ) vs [FORMULA] age in the [FORMULA]   - [Fe/H] plane.

Before proceeding, we have also to note that the sample of young calibrating clusters cannot include IC 4499 as the available CMD (Ferraro et al. 1994) presents an intrinsic inconsistency between the two age parameters. In fact, the cluster turns out to be [FORMULA] 4 Gyr younger than the average using [FORMULA]  , and nearly coeval to the average using [FORMULA]  . No simple explanation seems to be viable at this stage, further observations are urged.

Adopting for the 6 remaining young clusters the differential age obtained through the vertical parameter, we computed for each cluster the ratio [FORMULA], where [FORMULA] indicates the algebrical residual between the observed parameter [FORMULA]   and the "coeval expected" one computed by inserting the relevant [Fe/H] in Eq. (1). By averaging over the six considered clusters we derive the slope:


The numerical value of the empirical slope in Eq. 2 is very close to the figures derived by VBS and SC91 from their models. However, we have now obtained a direct, observational constraint which allows one to adopt a given set of models+transformations. So doing, [FORMULA]   can be fruitfully used as a differential age parameter (provided that a proper calibration of its dependence on metallicity has been obtained!). 2

In Fig.  16 the [FORMULA]   of all the clusters in Table 1 have been reported on a "differential-age" grid constructed with the empirical "0-age" isochrone of Fig.  11 (full line) and the [FORMULA] Gyr isochrones obtained from the eq (2) (dashed lines). The bulk of the clusters sample is confined in the strip [FORMULA] Gyr, with only a few clusters turning out to be younger than the average. It is noticeable the existence of a spread out of the data, perhaps greater than in Fig.  7, where the [FORMULA]   parameter is adopted. This dispersion can be ascribed to two main reasons: the errors in the measure of the two adopted observables (note that 1- [FORMULA] errors are always reported in any figure), or the existence of a "true" difference in the relative ages (note also that the sample of clusters in Fig.  16 is about 30% richer than in Fig.  7).

[FIGURE] Fig. 16. The observed [FORMULA]   for all the clusters in Table 1 as a function of [Fe/H] . The full line (labeled "0") is the best fit to the coeval clusters full dots and the dotted lines are the relative-age isochrones, empirically calibrated with the [FORMULA]   of the young clusters. This represent the empirical calibration for the relative ages we adopt. Using this locus for each cluster with known [FORMULA]   and [Fe/H] , it is possible to derive a differential age with respect to the adopted zero-age reference line

Given the global uncertainties, it is difficult to choose for each individual cluster which might be the dominating factor that causes the observed deviation from the empirical isochrone.

However, combining Eqs. (1) and (2) we obtain the final calibration:


This Eq. (3) yields relative ages for all the clusters with known metallicity [Fe/H]  and color-difference [FORMULA]  , as calibrated using the whole procedure described so far. The relative ages determined for all the clusters in Table 1 using Eq. (3) are reported in column 4 of Table 2. Given the actual uncertainties in the whole procedure, we estimated that the relative ages so obtained can be calculated with an accuracy not better than [FORMULA] 2 Gyr.

We want to stress here that, in deriving Eq. (3), the theoretical models have been used to define the properties of the luminosity-difference parameter [FORMULA]  , to select a subsample of coeval clusters, and to derive the relative ages of 6 well-known young clusters. Therefore, the whole procedure leading to Eq. (3) rests on the reliability of the TO luminosities as predicted by the models. The uncertainties in the temperatures and colors typical of the isochrones do not affect the above conclusions and the reliability of the coefficients in the equations above can be improved by increasing the accuracy in the determination of the three involved observables: [FORMULA]  , [FORMULA]   and [Fe/H] .

4.5. A consistency check

It is possible to carry out consistency checks of the above procedure by simply comparing the relative ages obtained using both considered CMD age-parameters.

In Fig.  17, the relative ages obtained through the vertical parameter [FORMULA]   are plotted against the relative ages obtained using the horizontal parameter [FORMULA]  . Of course, the clusters with just blue HB, for which we estimated the age only via the color-difference, are not reported. The relative ages, evaluated using the two methods are in agreement within [FORMULA] 1.5 Gyr for the vast majority of the clusters, and most of the younger clusters are confirmed to be so by both methods.

[FIGURE] Fig. 17. The relative ages obtained using two different CMD observables (the magnitude difference [FORMULA]   and the color difference [FORMULA]  ) are consistent within [FORMULA] 1.5 Gyr. The RMS deviation of the 24 points from the 1-to-1 line reported in the figure is 1.1 Gyr. The three exception IC 4499, Palomar 12 and Terzan 7 are discussed in the text (Sect. 4.5)

There are only three clusters for which the two age determinations differ by (slightly) more than 2 Gyr: the first is IC 4499 already discussed above, the remainig two are the young metal-poor clusters Palomar 12 and Terzan 7.

The discrepancy resulting for Palomar 12 and Terzan 7 may be originated by the poorly-sampled linear approximation of the 9.vs [FORMULA] relationship described via Eq. (2). For instance, the differential age loci plotted as dashed-lines in Fig.  16 could differ from straight lines, parallel to the zero-age locus. Alternatively, the observed discrepancy could be due to an erroneous estimate of [Fe/H]  or to a peculiar distribution in the abundance ratios of the heavy elements, affecting the "horizontal" age determination.

Concerning this last possibility, as already noted by Armandroff and Da Costa (1991) and Buonanno et al. (1995b), there is a significant disagreement between the metal abundance derived from the spectroscopic determinations using the Ca-triplet ([Fe/H]  [FORMULA]) and other estimates (for instance from CMD morphology, [Fe/H]  [FORMULA]) for the young metal-rich clusters Pal 12 and Terzan 7. If their metallicity is actually very high, the age discrepancy noticed in Fig.  17 could partially disappear.

Another result emerging from the data in Table 2 and Fig.  17 concerns the classic "second-parameter couple" NGC 288 and NGC 362. In fact, based on the present results, the ages of these two objects commonly presented as the typical example of the existence of a significant age spread among Galactic globulars (Bolte 1989, Green and Norris 1990, but see Catelan and de Freitas Pacheco 1994), would be quite similar ([FORMULA] 1 Gyr). As already argued by VBS, who carefully compared the CMDs of these two clusters using the [FORMULA]   method, a difference in [Fe/H]  slightly larger than adopted ([Fe/H] =-1.27 for NGC 362 and [Fe/H] =-1.40 for NGC 288, Zinn 1985) could be (at least partially) responsible for the [FORMULA] 0.015 mag difference in [FORMULA]  .

The relative ages we finally adopted are listed in column 5 of Table 2. In particular, we decided to adopt the age resulting from Eq. (3) for the blue-HB clusters and the average of the "vertical" 9. (column 2 of Table 2) and the "horizontal" 9.(column 3 of Table 2) for the red-HB clusters. The errors adopted in column 5 of Table 3 are the combination of the errors in the two averaged estimates.

4.6. Adopted ages and comparison with other recent age determinations

Confirming the continuous interest devoted to the determination of both the absolute age and the formation time-scale of the Milky Way, there have been a number of recent papers specifically analysing the current status of the issue (see Stetson, VandenBerg and Bolte 1996, for review and references). Therefore, it may useful to briefly compare our results with other age scale determinations.

To this aim, we selected the very recent lists presented by Chaboyer, Demarque & Sarajedini (1996; hereafter CDS96) and Richer et al.  (1996), mainly because the first is based on the "vertical" parameter, while the second is based on the "horizontal" parameter.

Since we are interested in comparing only the relative ages, a normalization of the different age determinations is necessary before carrying out the comparisons.

As described at Sects. 2.4 and 4.2, the zero-point of our relative [FORMULA] t values reported in Table 2 has been fixed by using the 11 "coeval" clusters defined in Sect. 3.2. So as to allow a direct comparison, we have thus averaged the age determinations listed for the same 11 clusters by CDS96 and by Richer et al.  (1996), and have then computed the relative age for each cluster as the difference between the individual age and the average value adopted for each catalog, respectively.

The results of the comparisons are shown in Fig.  18 (panel a and b) for the lists of CDS96 and Richer et al.  (1996), respectively. The different symbols represent the blue HB clusters (full dots) and the red HB clusters (open dots).

[FIGURE] Fig. 18. Our adopted relative ages listed in Table 2 column 5 are compared to those deduced from [FORMULA]   by CDS96 (panel a) and from [FORMULA]   by Richer et al.  (1996) (panel b). The full dots represent the clusters with blue HB ((B-R)/(B+V+R) [FORMULA] +0.72)

Considering the different assumptions and methods used in the three compared studies and the uncertainties and difficulties involved in the whole age-determination, the overall agreement of the various relative ages is fairly good. In particular, one can add a few considerations.

First, the existence of a few young globular clusters is definitely confirmed by all the authors, and the dispersion of the bulk of the considered objects around the "equal-age" locus is comparable (with just a few exceptions) to the uncertainties in the individual age determinations.

Second, the relative ages derived for the blue-HB clusters by CDS96 using the [FORMULA]   parameter are older than obtained from both our procedure and by Richer et al.  (1996). In our view, this probably reflects the difficulty to estimate the HB luminosity when the HB parts adjacent to the instability strip are not populated. This confirms our choice of avoiding the use of the [FORMULA]   method for this kind of clusters.

4.7. Distributions of the adopted relative ages with metallicity and Galactocentric distance

Although further significant improvements are surely urged, the combined semi-empirical calibration of [FORMULA]   (our proposed new observable) and [FORMULA]   vs.  [Fe/H]  and [FORMULA] here presented can allow us to estimate via a self-consistent global procedure age differences among globular clusters having any metallicity and HB morphology.

Finally, one can briefly look at the distributions of our estimated relative ages as a function of metallicity (Fig.  19) and Galactocentric distance (Fig.  20) of the considered clusters.

[FIGURE] Fig. 19. Our relative ages listed in Table 2 column 5 are plotted vs.  [Fe/H] . The red-HB clusters are reported as open dots, the blue-HB as full dots. Leaving out Palomar 12 and Terzan 7 (see Sect. 4.5), any dependece of the age on metallicity seems hardly detectable on the basis of this sample of clusters

[FIGURE] Fig. 20. Our relative ages listed in Table 2 column 5 plotted vs.  the Galacticocentric distance. Open dots represent the red-HB clusters and full dots the blue-HB ones. A slight trend with ages decreasing at increasing distance seems to be present (at least for the red-HB subsample), but the main feature is the age dispersion resulting in the intermediate far halo (see Sect. 4.7)

Since, in order to avoid any possible bias, we have used a small sample of well-studied clusters, it is impossible to reach any conclusions based on our sample.

However, some preliminary indications can be noted:

  1. leaving out the two young metal-rich globulars, Pal 12 and Terzan 7, there is no compelling evidence for any clear-cut age-metallicity relationship among the clusters included in our sample (see Fig.  19).
  2. in Fig.  19, there is some spread partially compatible with the associated uncertainties, but could also be intrinsic. In particular, the largest spread in age is found for intermediate metal-poor clusters, [Fe/H]  [FORMULA], which as well known presents the most evident signature of the so-called second parameter effect.
  3. the oldest objects in our sample are not the most metal poor but, rather, they are found at the same metallicity ([Fe/H] [FORMULA] -1.8) where a large spread in age is observed. This result is not really new, as it was nevertheless present in Fig. 7 of VBS which shows that the TO regions of NGC 2298 and of M92 can be perfectly superimposed. Considering that these two clusters differ by 0.4 dex in [Fe/H]  (see VBS), the age difference of about 2.5 Gyr we found, would simply be a consequence of the slope [FORMULA] derived from Eq. (3).
  4. a weak trend with older clusters populating the inner halo and old and younger globulars present in the outer regions of the Galaxy seems to be in our data. Apparently, this evidence seems to be quite clear for the sub-sample of red-HB clusters, and uncertain or inexistent for the blue-HB objects. However, especially for this analysis our available sample is insufficient, and the the main feature emerging from Fig.  20 is the dispersion of the ages in the intermediate outer halo, already noticed by several authors (see Stetson et al.  1996 for references).
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Online publication: April 20, 1998