Da Costa & Armandroff (1990) presented in tabular form the fiducial GGC branches of 6 globulars, covering the metallicity range . The RGBs were corrected to the absolute plane using the apparent V magnitude of the HB, and adopting the Lee et al. (1990) theoretical HB luminosity. Since the DA90 study, these branches have been widely used for stellar population studies in the Local Group. Based on these RGBs, in particular, a method to determine both the distance and mean metallicity of an old stellar population was presented by Lee et al. (1993).
Both DA90 and Lee et al. (1993) provided a relation between the metallicity [Fe/H] and the color of the RGB at a fixed absolute I magnitude ( and -3.5, respectively), and recently a new relation for has also been obtained by Caldwell et al. (1998). Once the distance of the population is known (e.g. via the luminosity of the RGB tip), then an estimate of its mean metallicity can be obtained using one of the calibrations. It is assumed that the age of the population is comparable to that of the GGCs, and that the age spread is negligible compared to the metallicity spread (RSPA99).
In such case, one expects that any RGB star's position in the absolute CMD is determined just by its metallicity, and that a better statistical determination of the population's metal content would be obtained by converting the color of each star into a [Fe/H] value. With this idea in mind, in the following sections we will show that this is indeed possible, at least for the bright/most sensitive part of the giant branch. We found that a relatively simple continuous function can be defined in the space, and that this function can be used to transform the RGB from the plane to the plane.
In order to obtain this function, we first selected a subsample of clusters with suitable characteristics, so that a reference RGB grid can be constructed. The fiducial branches for each cluster were then determined in an objective way, and they were corrected to the absolute plane. In this plane, the analytic function was fitted to the RGB grid. These operations are described in the following sections.
4.1. Selection of clusters
The clusters that were used for the definition of the fiducial RGBs are listed in Table 3, in order of increasing metallicity. The table reports the cluster name, and some of the parameters listed in Table 1 are repeated here for ease of use. The values of the reddening were in some cases changed by a few thousandth magnitudes (i.e. well within the typical uncertainties on ), to obtain a sequence of fiducial lines that move from bluer to redder colors as [Fe/H] increases, and again the corresponding values were obtained assuming that (Dean et al. 1978). Indeed, due to the homogeneity of our sample, we expect that if a monotonic color/metallicity sequence is not obtained, then only the uncertainties on the extinction values must be taken into account.
In order to single out these clusters from the total sample, some key characteristics were taken into account. In particular, we considered clusters whose RGBs are all well-defined by a statistically significant number of stars; they have low reddening values (); and they cover a metallicity range that includes most of our GGCs ( on the ZW scale).
The DA90 fiducial clusters were NGC 104, NGC 1851, NGC 6752, NGC 6397, NGC 7078 and NGC 7089 (M2). NGC 104 is the only cluster in common with the previous study, and M2 is not present in our dataset. The other objects have been excluded from our fiducial sample since they have too large reddening values ( for NGC 6397 and NGC 7078), or their RGBs are too scarcely populated in our CMDs (NGC 1851 and NGC 6752). Nevertheless, the calibrations that we obtain for the and are in fairly good agreement with those obtained by DA90 (for the small discrepancies at the high metallicity end, cf. Sects. 6.2 and 6.3), and in particular with the recent Caldwell et al. (1998) calibration for the index.
The ridge lines of our fiducial RGBs were defined according to the following procedure. The RGB region was selected from the calibrated photometry, by excluding both HB and AGB stars. All stars bluer than the color of the RR Lyr gap were removed; AGB stars were also removed by tracing a reference straight line in the CMD, and by excluding all stars blue-side of this line. This operation was carried out in the plane, where the RGB curvature is less pronounced, and a straight line turns out to be adequate.
The fiducial loci were then extracted from the selected RGB samples. The and I vectors were sorted in magnitude, and bins were created containing a given number of stars. Within each bin, the median color of the stars and the mean magnitude were used as estimators of the bin central color and brightness. The number of stars within the bins was exponentially increased going from brighter to fainter magnitudes. In this way, (a) one can use a small number of stars for the upper RGB, so that the color of the bin is not affected by the RGB slope, and (b) it is possible to take advantage of the better statistics of the RGB base. Finally, the brightest two stars of the RGB were not binned, and were left as representatives of the top branch. After some experimenting, we found that a good RGB sampling can be obtained by taking for each bin a number of stars which is proportional to , where i is an integer number. The resulting fiducial vectors were smoothed using an average filter with a box size of 3.
The RGB regions of the 6 clusters are shown in Fig. 5, together with the fiducial lines: it can be seen that in all cases the AGBs are easily disentangled from the RGBs. The values of the fiducial points corresponding to the solid lines in Fig. 5, are listed in Table 4.
The fiducial branches defined in Sect. 4.2 were fitted with a parametrized family of hyperbolae. First, the RGBs were moved into the absolute plane. The distance modulus was computed from the apparent magnitude of the HB (cf. Table 3) and by assuming the common law ; in order to compare our results with those of DA90, and were used, but we also obtained the same fits using more recent values as in Carretta et al. (1999), i.e. and . The RGB was modeled with an hyperbola as in Rosenberg et al. (1999a), but in this case the coefficients were taken as second order polynomials in [Fe/H]. In other words, we parametrized the whole family of RGBs in the following way:
The list of the parameters of the fits in magnitude is reported in Table 5, together with the rms of the residuals around the fitting curves. The table shows that the parameter d does not depend on the choice of the distance scale, as expected. Even the other coefficients are little dependent on the distance scale, apart from . It is affected by the zero-point of the HB luminosity-metallicity relation, and indeed there is the expected mag difference going from the LDZ to the C99 distance scale.
Table 5. The coefficients that define the functions used to interpolate our RGBs (see text); the top header line identifies the two distance scales used, while the two metallicities are identified in the second line of the header
One could question the choice of a constant d, but after some training on the theoretical isochrones, we found that even allowing for a varying parameter, its value indeed scattered very little around some mean value. This empirical result is a good one, in the sense that it allows to apply a robust linear least-square fitting method for any choice of d, and then to search for the best value of this constant by a simple rms minimization. We chose to fit the function, and not the function, since the latter one would be double-valued for the brightest part of the metal rich clusters' RGBs. This choice implies that our fits are not well-constrained for the vertical part of the giant branch, i.e. for magnitudes fainter than . However, we show in the next section that our analytic function is good enough for the intended purpose, i.e. to obtain the [Fe/H] of the RGB stars in far Local Group populations, and thus to analyze how they are distributed in metallicity.
Our synthetic RGB families are plotted in Figs. 6 and 7, for the LDZ distance scale. In the former figure, the ZW metallicity scale is used, while the CG scale is used in the latter one. The figures show that the chosen functional form represents a very good approximation to the true metallicity "distribution" of the RG branches. The rms values are smaller than the typical uncertainties in the distance moduli within the Local Group. We further stress the excellent consistency of the empirical fiducial branches for clusters of similar metallicity. We have two pairs of clusters whose metallicities differ by at most 0.03 dex (depending on the scale): NGC 288 and NGC 5904 on the one side, and NGC 5272 and NGC 6205 on the other side. The figures show that the fiducial line of NGC 288 is similar to that of NGC 5904, and the NGC 5272 fiducial resembles that of NGC 6205, further demonstrating both the homogeneity of our photometry and the reliability of the procedure that is used in defining the cluster ridge lines.
If the coefficients of the hyperbolae are taken as third order polynomials, the resulting fits are apparently better (the rms is mag); however, the trends of the metallicity indices show an unphysical behavior, which is a sign that further clusters, having metallicities not covered by the present set, would be needed in order to robustly constrain the analytic function.
In the following section, the indices are calibrated in terms of metallicity, so that in Sect. 7 they will be used to check the reliability of our generalized fits.
© European Southern Observatory (ESO) 2000
Online publication: April 3, 2000