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

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6. Calibration of the indices. Discussion

6.1. S

On the CG scale, the second-order fit has a residual rms of 0.12 dex in [Fe/H]. On the CG scale, the linear fit is obtained with a rms of 0.12 dex. This index can therefore be calibrated on both scales, with a comparable level of accuracy. A parabolic fit does not improve the relation on the CG scale, since the coefficient of the quadratic term is very small (-0.004) and the rms is the same. These relations are shown in Fig. 8 as solid lines, where the upper panel is for the CG scale, and the lower panel for the CG scale (this layout is reproduced in all the following figures).

The cluster NGC 6656 (M22) was excluded from the fits, and is plotted as an open circle in Fig. 8. It is well-known that M22 is a cluster that shows a metallicity spread, and indeed it falls outside the general trend in most of the present calibrations.

6.2. [FORMULA]

The first definition of the [FORMULA] index was given in Da Costa & Armandroff (1990), where a calibration in terms of the CG scale was also given: [FORMULA]. The same index (measured on the absolute RGBs corrected with the LDZ HB luminosity-metallicity relation) is plotted, in Fig. 9, as a function of the metallicity on both scales, and the solid lines represent our calibrations. The top panel shows the quadratic relation on the CG scale, whose rms is 0.14 dex. The bottom panel of Fig. 9 shows the relation on the CG scale. In this case, a quadratic fit is not able to reproduce the trend of the observational data. A better result can be obtained by making a variable change, i.e. using the variable [FORMULA]; in this case, a linear relation is found, and its rms is 0.15 dex. This measure of the residual scatter has been computed after transforming back to metallicity, so the reliability of the index can be compared to that of the other ones. Again, the index can be calibrated on both scales with a comparable accuracy. The dashed curve in the upper panel of Fig. 9 shows the original relation obtained by DA90: there is a small discrepancy at the high-metallicity end, which can be explained by the different 47 Tuc fiducial line that was adopted by DA90 (cf. below the discussion on [FORMULA]).

As already recalled, we checked the effect of adopting another distance scale, by repeating our measurements and fits, and adopting the C99 distance scale. For the CG metallicity scale, we obtain the quadratic relation whose coefficients are listed in Table 6, and whose rms is 0.15 dex. The bottom panel of Fig. 9 shows the relation on the CG scale. Again, a quadratic fit is not able to reproduce the trend of the observational data. Making the already discussed variable substitution, the linear relation in z has an rms of 0.16 dex, so the two metallicity scales yield almost comparable results.

6.3. [FORMULA]

Using the same "standard" GC branches of DA90, Lee et al. (1993) defined a new index, [FORMULA], to be used for the farthest population II objects. It was also calibrated in terms of the CG scale: [FORMULA]. A new calibration was also given recently in Caldwell et al. (1998): [Fe/H][FORMULA], where [FORMULA]. The index and our calibrations (solid lines) are plotted, in Fig. 10, on both metallicity scales. Again, the measurements were made in the absolute CMD, assuming the LDZ distance scale. Our quadratic calibration vs. the CG scale has a residual rms scatter of 0.13 dex, which is the same of the linear relation on the CG metallicity vs. z.

The Lee et al. relation (dashed line) predicts slightly too larger metallicities on the CG scale, for [FORMULA]. This can also be interpreted as if the DA90 47 Tuc branch were [FORMULA] mag bluer than ours. Indeed, if one looks at Fig. 5 of DA90, one can easily see that some weight is given to the brightest RGB star, which is brighter than the trend defined by the previous ones. The result is a steeper branch, which also justifies the DA90 slightly bluer RGB fiducial. Since our metal richest point is defined by two clusters, and since the two measured parameters agree very well, we are confident that our calibration is reliable. In any case, the discrepancy between the two scales is no larger than [FORMULA] dex. It is also reassuring that the Caldwell et al. (1998) relation (pluses) is closer to the present calibration, since the former is based on a larger set of clusters. This might be an indication that the Lee et al. relation is actually inaccurate at the metal rich end, due to the small set of calibrating clusters.

As before, we obtained a further calibration also using the C99 [FORMULA] vs. [Fe/H] relation; the quadratic fit on the CG scale has a residual rms scatter of 0.13 dex, while the z variable can be fitted with a straight line, with an rms of 0.14 dex.

6.4. The [FORMULA] family and [FORMULA]

For any [FORMULA] index, the quadratic relations vs. the CG metallicity do not improve the rms and they are not plotted in the figures. The coefficients are listed in Table 6.

The best metallicity estimates of the "[FORMULA] family" are obtained with the [FORMULA] index. The errors on [FORMULA] are just slightly larger than the standard uncertainties of the spectroscopic determinations. The solid lines of Fig. 11 show the calibrations that we obtain. The quadratic equation on the CG scale, and the linear one on the CG scale, are obtained with residual scatters of 0.16 dex.

The rest of the indices in this family, and [FORMULA], lack the precision of the other abundance indicators. This is due to the fact that the error on any [FORMULA] index is proportional to the uncertainty on the color of the RGB (which depends on the reddening), times its local slope where the reference point is measured. Since the RGB slope increases going away from the tip (i.e. towards bluer colors), we expect that the scatter on the [FORMULA] indices will also increase as the color of the reference point gets bluer. Indeed, Table 6 shows that in most cases the rms uncertainties are [FORMULA] dex for these indices. The residual scatter is largest for the [FORMULA] index, which is the most affected by the uncertainties on the reddening.

The [FORMULA] and [FORMULA] parameters have been earlier calibrated, on the CG scale, by Carretta & Bragaglia (1998). Using their quadratic relation for [FORMULA], and both their linear and quadratic relations for [FORMULA], the corresponding rms of the residuals in metallicity are 0.21 dex and [FORMULA] dex, respectively. Our new and the old calibrations are therefore compatible, within the (albeit large) uncertainties.

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© European Southern Observatory (ESO) 2000

Online publication: April 3, 2000
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