## 4. An empirical calibration of vs [Fe/H]## 4.1. Problematic aspects of the "horizontal" methodThe main advantage of the age parameter (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 . 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
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 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 and [Fe/H] =-1.91 , Zinn 1985), then the coincidence of the parameters would actually imply an age difference of 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 on metallicity, it is
almost impossible to build up a self-consistent and reliable
## 4.2. The new calibrationIn 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 -method (and that
passed the -test; see sec. 3.2.1), to
calibrate the actual dependence of on
metallicity. If reliable, such a "direct" calibration (note that the
ages are now assumed to be In Fig. 11 it is reported a plot of the individual
parameters 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.
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 due to the observational uncertainties in the metallicity [Fe/H] : a typical error in [Fe/H] of 0.2 dex reflects into an indetermination of about 0.008 mag in and, in turn, about 1 Gyr in the relative age. ## 4.3. Comparison with the theoretical models: there is a discrepancySupposing that the Fig. 12 displays the "constant-age" loci as derived from the
models (
In fact, there is no combination of models and/or transformations
able to yield 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
values do not
In our view there are essentially only two possible answers: - 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. - 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 ( 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 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 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 vs. [Fe/H] and 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 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: - 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.
- One can hardly assume that any of the relative-age scales available in the literature so far can actually yield ages to better than 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 exampleAs noticed at the end of Sect. 3.2, from the analysis of
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 However, it is worth checking whether the two indicators,
and , 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
( ) 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 4 Gyr younger than the average using , and nearly coeval to the average using . 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 , where indicates the algebrical residual between the observed parameter 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 In Fig. 16 the of all the
clusters in Table 1 have been reported on a "differential-age"
grid constructed with the
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 However, combining Eqs. (1) and (2) we obtain the final calibration: This Eq. (3) yields 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 , to
select a subsample of coeval clusters, and to derive the relative ages
of 6 well-known young clusters. Therefore, ## 4.5. A consistency checkIt 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 are plotted against the relative ages obtained using the horizontal parameter . 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 1.5 Gyr for the vast majority of the clusters, and most of the younger clusters are confirmed to be so by both methods.
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
relationship described via Eq. (2). For
instance, the differential age loci plotted as 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] ) and other estimates (for instance from CMD morphology, [Fe/H] ) 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 ( 1 Gyr). As already argued by VBS, who carefully compared the CMDs of these two clusters using the 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 0.015 mag difference in . 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 determinationsConfirming 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 As described at Sects. 2.4 and 4.2, the zero-point of our
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
(
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 Second, the relative ages derived for the blue-HB clusters by CDS96 using the 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 method for this kind of clusters. ## 4.7. Distributions of the adopted relative ages with metallicity and Galactocentric distanceAlthough further significant improvements are surely urged, the
combined 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.
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: - 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).
- 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] , which as well known presents the most evident signature of the so-called second parameter effect.
- the oldest objects in our sample are not the most metal poor but, rather, they are found at the same metallicity ([Fe/H] -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 derived from Eq. (3).
- 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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