4 The new calibrations

Calibrations obtained from the procedure described above are given in Table 3. These calibrations supersede all counterparts that have previously been derived from Taylor's [Fe/H] catalogs. The notation "[Fe/H](q)'' is used in Table 3 to designate metallicities derived from the calibrations. The metallicity limits within which the calibrations apply are given (with other information) in Table 3's footnotes.

4.1 The DDO calibration

The values of S in the $\delta$CN calibration require comment. [34, Taylor (1991]) presented a calibration in which S depends on color. By contrast, [45, Twarog & Anthony-Twarog (1996]) found no evidence for a color dependence that is statistically significant. As part of the new analysis, preliminary solutions were performed to investigate this problem. The results of the solutions revealed three intervals in the color (45-48), with S and Z being essentially constant within each interval but differing between intervals. For the final results given in Table 3, t tests show that the values of S and Z differ between the first and second intervals with confidence levels of at least 99.9%. The same is true for the second and third intervals.

The first three lines of Table 3 contain results for the three color intervals. The fourth line contains averages for use near the boundary between the first and second color intervals. In the same way, the fifth line applies for the boundary between the second and third intervals. These latter relations are based on a guess that relatively smooth transitions between color intervals are more likely than abrupt changes between them.

Somewhat to the author's surprise, S decreases as one goes from the first color interval to the second, but then increases again as one goes from the second interval to the third. The reason for this kind of variation is not known. Given the results of the statistical tests quoted above, however, the existence of the variation seems to be reasonably well established.

4.2 Accidental errors for [Fe/H](q)

Values of the following rms error are given in the fifth column of Table 3:

$$\sigma_{Fq}= S [v(q)]^{0.5}.$$ (3)

These are the errors that apply to values of [Fe/H](q). Equation (3) is derived from Eq. (10.14) of [22, Kendall & Stuart (1977]).

One would like to know how well the errors listed in Table 3 compare to the errors given for values of [Fe/H] in Taylor's catalogs. The largest of the Table 3 errors are for $\delta M_1$ and [M/H], and are quite comparable to the rms error range for a single determination of [Fe/H] [35, (Taylor 1994, 1999b]). The smallest of the Table 3 errors are for G and $\langle$Fe$\rangle$, and would be typical for stars with values of [Fe/H] that have been determined several times. The small sizes of the latter errors suggest that the isoabundance relations for G and $\langle$Fe$\rangle$ are correct.

It is also of interest to find out whether net values of $\sigma_{Fq}$ can be decreased by averaging results from two (or more) calibrations. This is possible only if the datum from each calibration is an independent sample of underlying random effects. That condition is not met if there are internal correlations in the data; if (say) $F(Q)-{\rm [Fe/H]}$ and $f(q)-{\rm [Fe/H]}$ are correlated, f(q) and F(Q) are effectively identical samples of underlying random effects, and their average conveys no more information than f(q) or F(Q) alone. To check for correlations of this sort, the two-error least-squares algorithm described in Sect. 3.2 was applied to residuals from the Table 3 relations. For the following parameter pairs, correlations with a confidence level of 3.5$\sigma$ or better were found:

1. D and [M/H],

2. $\delta$CN and $\langle$Fe$\rangle$, and

3. $\delta$CN and G.

No corresponding correlation was obtained for G and $\langle$Fe$\rangle$. However, the number of stars for which both parameters are available is relatively small. Larger numbers of data could be used to test these parameters against $\delta$CN. Since correlations were found when both G and $\langle$Fe$\rangle$ were tested in this way, it appears safest to assume that the G and $\langle$Fe$\rangle$ residuals are correlated.

Recall now that the M₁ calibration is intended only to answer a question about $\mu$ Leo (see Sect. 2). In the present context, that calibration may be set aside. Apparently results for the other three evolved-star calibrations cannot be meaningfully averaged. The same appears to be true for results for the two calibrations for dwarfs. To avoid misleading appearances, it is probably best not to average results from two or more calibrations at any point in an analysis.