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Astron. Astrophys. 355, 966-978 (2000)

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5. Calibration of the indices. Introduction

In order to obtain analytic relations between the indices and the actual metallicity, our photometric parameters were compared both with the ZW and the CG values. A summary of the resulting equations is given in Table 6. For each index (first column) both linear and quadratic fits were tried, of the form: [FORMULA] and [FORMULA]. The coefficients of the calibrating relation are given in the columns labelled [FORMULA], [FORMULA], and [FORMULA]; in Column 7, the rms of the residuals is also given. In the case of the [FORMULA] and [FORMULA] indices, neither the linear nor the quadratic fits give satisfactory results, when the CG scale is considered. Instead, a good fit is obtained if a change of variables is performed, setting [FORMULA], and linearly interpolating in the index (i.e. setting [FORMULA]). The Column 8 of Table 6 identifies the kind of fitting function that is used for each parameter/metallicity combination: the symbols "1", "2" and "z" refer to the linear, quadratic, and linear in z fits, respectively. Relations on both the CG and ZW metallicity scales are given, and Column 3 flags the [Fe/H] scale that is used.


Table 6. Coefficients of the calibrating relations for the indices (see text for the definition of the equations). NGC 6656 was excluded from the fits

In order to measure the [FORMULA] and [FORMULA] indices (cf. Sect. 3) a distance scale must be adopted. The most straightforward way is to use the observed [FORMULA] (cf. Table 1) coupled with a suitable law for the HB absolute magnitude.

It has become customary to parameterize this magnitude as [FORMULA], although there is no consensus on the value of the two parameters a and b. The current calibrations of these two metallicity indices were obtained by Da Costa & Armandroff (1990) and Lee et al. (1993), and they are based on the Lee et al. (1990; LDZ) theoretical luminosities of the HB. LDZ gave a relation [FORMULA] valid for [FORMULA].

As discussed in Sect. 4, since many current determinations of Population II distances within the Local Group are based on the Lee et al. (1990) distance scale, and for the purpose of comparison with previous studies, we provide a calibration using the latter HB luminosity-metallicity relation. However, in the last ten years revisions of this relation have been discussed by many authors, so we also calibrated the two indices using [FORMULA] (Carretta et al. 1999), which is one of the most recent HB-based distance scales.

We must stress that metallicities on the ZW scale must be used in the [FORMULA] vs. [Fe/H] relation. Indeed, CG showed that their scale is not linearly correlated to that of ZW, so not even the [FORMULA] vs. [Fe/H] relation will be linear: if one wishes to use the new scale, then the absolute magnitude of the HB must be re-calibrated in a more complicated way.

The best calibrating relations are shown in Figs. 8 to 11. In the following sections, for each index a few remarks on the accuracy of the calibrations and comparisons with past studies are given.

[FIGURE] Fig. 8. Calibration of the index S (cf. Fig. 1, left panel) as a function of [FORMULA] on the Zinn & West (1984) scale (top panel) and on the Carretta & Gratton (1997) scale (bottom panel). Linear (top panel) and parabolic (bottom panel) fits of the data are also represented. The cluster marked with open circle was excluded from the fit (see text for details). Starting from this figure (to Fig. 11), the open squares connected by a solid line represent the mono-parametric approximation (see Sect. 4)

[FIGURE] Fig. 9. Calibration of the index [FORMULA]. The solid lines represent the equations described in the text, while the dashed curve represents the the DA90 calibration.

[FIGURE] Fig. 10. Calibration of the index [FORMULA] The solid lines represent the equations described in the text, while the dashed curve represents the the Lee et al. (1993) calibration. The Caldwell et al. 1998relation is also shown with plus symbols

[FIGURE] Fig. 11. Calibration of the index [FORMULA] . The solid lines represent linear (top panel) and quadratic (bottom panel) fits to the data

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Online publication: April 3, 2000