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

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2. A new observational age-parameter

2.1. Definitions

To overcome the first of the two problems listed above, we suggest a new method, hereafter called vertical method, based on the measure and calibration of a new CMD observable (see Fig.  1): [FORMULA]  , the distance in magnitude between the HB at the color of the TO (like in the [FORMULA]  -method) and an arbitrary point on the upper main sequence, near the TO but where the main sequence has a non-vertical slope. For the precise location of this arbitrary point, we choose for simplicity to follow VBS and use 5, the well-defined point on the main sequence which is 0.05 mag redder than the TO.

A similar approach to the same problem has been adopted by Chaboyer et al.  (1996). They defined the observable [FORMULA] as a point which is brighter than the TO and 0.05 mag redder (i.e., at the base of the Sub Giant Branch). On the other hand, 5 is a point 0.05 mag redder than the TO but dimmer than the TO itself (see Fig.  1). The philosophy at the base of both approaches is the same, but we prefer to use an observable which is defined in a portion of the Color-Magnitude Diagram populated by less evolved Main Sequence stars and, thus, presumably less sensitive to uncertainties in several parameters, like the mixing length, etc.

[FIGURE] Fig. 1. Definition of the new CMD observable proposed as age-parameter (see Sect. 2.1)

The age parameter [FORMULA]   defined here presents two main advantages:

  1. It shares with [FORMULA]   the firm theoretical background and the independence of distance and reddening.
  2. It offers an intrinsic higher accuracy in its practical measure from the observed CMDs.

On the other hand, we should recall that:

  1. This new parameter cannot be applied to clusters having only blue HB stars, and the limitation discussed for the [FORMULA]  -method at point (2) above is still present. This does not allow us to apply the [FORMULA]  -method to the whole set of well observed clusters, but we present at Sect. 3 a procedure to partially overcome the problem.
  2. Since the measure of the observable for the MS is not exactly the TO but rather a point shifted in color along the MS, actually one contaminates the observable [FORMULA], which in the models depends only on luminosities (and, in turn, on "safe" nuclear burning), with a "horizontal component" (due to the shift in color by 0.05 mag), which in the models depends on more uncertain quantities, like the mixing length, the color transformations, etc.

2.2. Dependence on the MS morphology

Since the reference point on the main sequence -5- has been selected using the TO color, we must study its dependence on the adopted theoretical isochrones. Together with the discussion presented in the following paragraph on the HB, this analysis is necessary to achieve a proper calibration in terms of age of the adopted observable as well as of any other "vertical" observable (like [FORMULA]  ) used so far.

2.2.1. Properties of 5 with varying the adopted models

Although it is commonly accepted that the most widely-used theoretical isochrones for Pop II stars produce substantially the same TO luminosity, it is nevertheless well known that they actually differ when compared in detail, especially after applying transformations into the observational plane.

Leaving out a complete comparison which is beyond the present purposes, in order to understand how these subtle differences may play a rôle in the estimate of relative ages, we report in Fig.  2 the comparison of three of the latest theoretical isochrone sets kindly made available to us in machine-readable form by the authors.

[FIGURE] Fig. 2. Three recent sets of isochrones (Straniero and Chieffi 1996-SC96, D'Antona et al.  1997-DCM97, and VandenBerg 1996-VDB96) are compared for two values of age and metallicity. The transformations to the observational plane by Kurucz (1993) are used in all the cases. The relevant features of the CMD show very similar luminosities while the colors differ noticeably

In particular, we have compared the latest models computed by VandenBerg (1996-VdB96) and by Straniero and Chieffi (1996-SC96), which are based on the recent input physics and opacities and on the traditional mixing length approach to deal with convection, and those presented by D'Antona, Caloi & Mazzitelli (1997-DCM97) who have used the Canuto and Mazzitelli (1991, 1992) treatment for the convective layers.

For euristic purposes we adopted the same set of the Kurucz (1993-K93) transformations from the theoretical plane into the observational one, in order to separate the effects of the models from those of the transformations. Fig.  2 reveals that the models are substantially coincident in the TO region for the two metallicities and ages considered, while they are progressively more discrepant going towards redder colors, both along the MS and the base of the RGB. Note that the discrepancy in color for the faint MS ([FORMULA]) is of the order of 0.04 mag in the worst case, while the base of the giant branches differs by about 0.1 mag for the two extreme cases (DCM97 and SC96).

In Fig.  3 (panels a, b) we compare, from the theoretical point of view, the behaviour of 5 and [FORMULA] as a function of metallicity and age with varying the isochrone-sets, but adopting the same color-transformations.

[FIGURE] Fig. 3. The luminosity of the MS-TO point (panel a) and of the MS-point 0.05 mag redder than [FORMULA] (panel b) as derived from the three isochrone-sets, are compared as a function of metallicity. The apparent better agreement for [FORMULA] is mainly due to a zero-point difference in 5

What matters in Fig.  3 is the dependence of the two observables, 5 and [FORMULA], on the adopted model. Inspection of the plots reveals that while [FORMULA] is substantially stable under this aspect, different models produces differences in 5 that can reach 0.2 mag for the extreme cases. In other words, the calibration of 5 in terms of age presents an indetermination of about 2 Gyr due to disagreements of the theoretical models. This drawback however, disappointing as it may be, is only a piece of a more general picture of indeterminacies which affect the calibration of many observables, including [FORMULA], as one passes from the theoretical to the observational CMD (see Sect. 2.2.2)

2.2.2. Properties of 5 with varying the adopted transformations

Turning to the comparison of the same isochrone-set (SC96) but using different transformations, we examined three sets of transformations from the theoretical to the observational plane: K93, Buser and Kurucz (1992-BK92) and VandenBerg (1992-VDB92). The normalization has been achieved, imposing that [FORMULA] =4.82 for all the three trasformations.

The effect of adopting different theoretical-to-observational plane transformations is displayed in fig. 4. The isochrones by SC96 transformed following K93, BK92 and VDB92 are plotted for two ages and two metallicities. Sizeable variations of the colors, the color-differencies, and also of the luminosities are clearly evident.

Fig.  4 shows that this choice actually matters. In fact, the effect of changing color-transformations is almost more important than the use of different original isochrones. In particular, all the panels of Fig.  4 show a clear difference in the color zero-point, but also a more evident discrepancy in the color-differential quantities ([FORMULA]  , for example).

[FIGURE] Fig. 4. The effect of adopting different theoretical-to-observational plane transformations. The isochrones by SC96 transformed following K93, BK92 and VDB92 are plotted for two ages and two metallicities. Sizeable variations of the colors, the color-differencies, and also of the luminosities are clearly evident

Moreover, also in the V-magnitudes scale, the three transformations give different values for [FORMULA] and 5. This disturbing evidence is further confirmed by the plots shown in Fig.  5, where the values obtained for 5 and [FORMULA] from the SC96 models are compared with varying metallicity, after applying the three different transformations, i.e. K93, BK92 and VDB92.

[FIGURE] Fig. 5. The TO (panel a) and the 5 (panel b) luminosities derived from the SC96 isochrones, converted into the observational plane using three different temperatute-color and luminosity-magnitude transformations compared as a function of metallicity, [Fe/H] 

In particular, it is disappointing to note that with changing transformations, the trend of both [FORMULA] and 5 tends to diverge at the metal poor end, which, as well known, has the major cosmological impact. Furthermore, it is evident from Fig.  5 that the effect of the adoption of different trasformations is slightly greater on 5 than on [FORMULA]. As said in Sect. 1, the adoption of an observable different from the "pure" [FORMULA] can have significant advantages from an observational point of view, but it presents some drawbacks in terms of theoretical calibration.

2.3. Dependence on the HB properties

Since the calibration of [FORMULA]   and [FORMULA]   rests on the theoretical models, the main point is to use sets of models as homogeneous as possible to avoid spurious differential effects introduced by variations in the input physics and in the treatments adopted in the computations. This means that it is better to adopt MS and HB models computed by the same authors, if available, and to apply the same set of color transformations.

However, it is also well known that for the specific case of the HB, there is still room for a residual discrepancy between the models and the observational data concerning the precise dependence of the absolute magnitude of the HB, - [FORMULA] -, on metallicity, -[Fe/H] . In particular, if we assume a linear relationship between [FORMULA] and [Fe/H]  (see for references Chaboyer, Demarque & Sarajedini 1996), the slope can vary from about 0.15 (the standard theoretical models) up to 0.35 (Sandage 1982, 1993). Moreover, the zero-point is uncertain at the 0.2 mag level, but since we are mostly interested in the relative ages, this item is not so important here.

In order to show the size of the effects due to significantly different choices, we have reported in Fig.  6 (panel a) the [FORMULA] vs.  [Fe/H]  relationship recently obtained by SC96, by DCM97 and by VdB96 transformed to the observational plane adopting K93 as already done for the corresponding MS. As can be seen, the three theoretical loci are quite different. The mean slope goes from 0.18 to 0.26 for SC96 and DCM97, respectively, and covers almost the whole range of the observational estimates quoted above.

[FIGURE] Fig. 6. The luminosity of the HB at the RR-Lyrae color as a function of metallicity: a SC96, DCM97 and VDB96 models, b SC96 models (transformed with K93, BK92 and VDB92). The slope of the [FORMULA] vs [Fe/H]  function varies with the models, the zero-point changes also with the adopted transformations

On the other hand, also the use of different color transformations makes a difference, as noted for the MS. As can be seen from Fig.  6b, where the ZAHB tracks computed by SC96 have been transformed using the quoted relationships, the differences in luminosities are quite sizeable. This reinforces the note that the use of different models and even of different transformations from the theoretical to the observational plane strongly affects the calibration of differential age observables like [FORMULA]   and [FORMULA]  .

In synthesis, we can conclude that in the definition of the "constant-age loci" in the plane [FORMULA]   or [FORMULA]   vs [Fe/H] , the choice of the dependence of the HB luminosity [FORMULA] on the metal-content [Fe/H]  plays a crucial rôle.

The effect of choosing one of the very different slopes shown in Fig.  6a seems to induce insurmountable difficulties in the use of such a calibration (which is indeed present in any age-calibration somehow resting on the HBs). However, there are two important items to consider:

  1. The recent determinations of the slope [FORMULA] [FORMULA] / [FORMULA] [Fe/H]  from various methods (RR-Lyrae pulsational properties, Baade-Wesselink studies of individuul RR-Lyrae stars, HB studies of the M31 globular clusters with HST, etc.) seem to converge towards a mean value of about 0.20 or smaller (see for references Chaboyer, Demarque & Sarajedini 1996), which is very close to the mean slope of the standard theoretical HB models.
  2. Since we aim here at studying just the relative ages, we propose a differential use of the [FORMULA]   parameter (which is, by the way, already a differential quantity, so as to yield a double differential method). This choice eliminates the problem of the inaccurate knowledge of the zero-point of the [FORMULA] vs [Fe/H]  relationship.

2.4. The [FORMULA]   vs.  age calibrations

Once that a self-consistent set of isochrones has been adopted, the choice of the [FORMULA] vs [Fe/H]  relationships allows us to draw plots of the resulting isochrones in the [FORMULA]   vs [Fe/H]  plane.

Since there are various possible calibrations depending on the actual choices, we have reported the various constant-age loci in different panels of Fig.  7, 8, and 9. On the same plots we have also reported the observed values for a sample of Galactic globular clusters for which accurate photometry is available.

[FIGURE] Fig. 7. [FORMULA]   for 24 clusters superimposed to a theoretical calibration grid derived from a SC96, b DCM97 and c VDB96. The transformations are those by K93. The clusters have been selected according to the requirement of having a well-populated horizontal portion of the HB. The bulk of the clusters turn out to be confined in a quite narrow age range, with only a few clusters clearly younger than the average. Note the 14 clusters (marked as full dots) located within a 2 Gyr strip in all the calibrations

[FIGURE] Fig. 8. The effect of different color-transformations on the theoretical calibrations of [FORMULA]   derived from SC96. The position and the shape of the calibration-grid show some variations, but it is always possible to select the same sample of 14 clusters, included within a 2 Gyr strip

[FIGURE] Fig. 9. The observed [FORMULA]   of the 14 clusters selected in Fig. 7 and Fig.  8 superimposed to the calibration of [FORMULA]   derived from the SC96 and DCM97 models. Though there is some dispersion (possibly due to observational difficulties in deriving [FORMULA]  ), yet 11 out of the 14 clusters selected with the procedure described at Sect. 3.2 (and Fig.  7, 8) can still be considered "coeval" within the uncertainties

From the analysis of the theoretical calibrations, one can draw some general considerations:

  1. although the calibrations clearly change with varying ingredients, it is evident that the overall behaviour is substantially the same. The variation of the observable [FORMULA]   with the age is quite large (i.e. the sensitivity to age variations is high), and the dependence on metallicity is small. This ensures that errors in metallicity (still as large as 0.2 dex) have negligible impact on the estimated ages
  2. by interpolating in any plot in Fig.  7, 8, and 9, it is possible to estimate age differences -9.- for any cluster with known [FORMULA]   (or [FORMULA]  ) and [Fe/H]  relative to an arbitrary isochrone taken as the reference zero-point
  3. since this parameter is differential, the problematic effects induced by the use of different transformations are reduced. In fact, as we see in Fig.  8, a compensating mechanism is at work which in practice makes almost neglegible the impact of this specific item. This represents of course one of the major advantages of differential methods
  4. while the adoption of different slopes in the [FORMULA] vs [Fe/H]  calibration (or in any sub-interval of the whole metallicity range) modifies the shapes of the isochrones, a simple shift in the zero-point of the same relationships would correspondly shift the whole pattern, without affecting the relative ages

In conclusion, despite several differences do exist between different calibrations and transformations, it is possible to select a region of the [FORMULA]   (or [FORMULA]  ) vs [Fe/H]  plane where to locate clusters coeval within a given age interval, independent of the chosen calibration. In particular, in the following sections, we will show that it is actually possible to extract a sub-set of clusters which can be considered to be substantially coeval (i.e. within a conservative estimate of the errors) independently of the adopted calibrations. This sample of "coeval" clusters will allow us to make some interesting checks on the internal consistency of the models themselves.

2.5. Additional useful remarks on the 5 parameter

To have a better insight on the properties of this new parameter, we report the results of several specific tests we originally carried out using the former isochrones computed by Straniero & Chieffi (1991 -SC91) and the ZAHB models of Castellani, Chieffi & Pulone (1991). These theoretical models have the advantage to have adopted homogeneous input physics with no "a priori" variations of the helium content Y and of the mixing-length parameter [FORMULA], nor of the alpha-elements (O, Ca, Si, etc.) as a function of [Fe/H] . They represent thus a "classic, standard" reference grid.

Within this framework, one may wonder whether the present uncertainties in the treatment of convection and in the assumed constancy of the mixing-length parameter for stars of different metal abundance may undermine the quasi-independence of [FORMULA]   from the metallicity. To check this point we computed specific models using the same code as SC91 and obtained that for Z=0.001 and t=16 Gyr, [FORMULA]   varies from 4.960, to 4.980 and to 5.013 mag, with [FORMULA] varying from 2.0, to 1.6 and 1.0, respectively. In turn, having fixed all the other quantities, the largest expected difference in age due to mixing-length variations is of only 0.5 Gyr.

As regards the effect on [FORMULA]   of possible enhancements of the [FORMULA] -elements, from the models of Salaris, Chieffi & Straniero (1993), one gets a maximum variation of [FORMULA] -0.7 Gyr with an overabundance of +0.3 dex in the [FORMULA] -elements at the low metallicity tail ([Fe/H]  [FORMULA] -2.3), and the difference decreases progressively with increasing metallicity.

Variations of the primordial helium content have an important effect on the [FORMULA]   parameter as well as on [FORMULA]  . In fact, changing Y one gets sizeable differences in [FORMULA] but small variations of 5, and therefore [FORMULA]   varies significantly. By theoretical isochrones computed at different Y, we have obtained [FORMULA]   / [FORMULA] Y [FORMULA] 2. Hence, [FORMULA] Y [FORMULA] mimics a difference in age of about 1 Gyr. However, since the available estimates of Y in GGCs indicate a constant primordial helium for all the GGCs within [FORMULA] Y [FORMULA] (Buzzoni et al.  1983), then the dependence of [FORMULA]   on Y should be actually negligible.

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998