3 Deriving calibrations

3.1 Reviewing isoabundance relations

All indices that are calibrated here are derived from relations between metallicity parameters and temperature parameters. These relations are intended to be isoabundance relations. In practice, however, they may not quite satisfy this condition. One would therefore like to test the isoabundance relation for (say) index q by using it to calculate

$${\rm [Fe/H]} = f(q) = Sq+Z$$ (1)

while allowing S and Z to vary with color if necessary.

This test is feasible for $\delta$CN and [M/H]. No variation of S and Z with color can be detected for [M/H], so a single relation that is applicable for all pertinent colors is obtained for this index. For $\delta$CN, variation in both S and Z is detected and allowed for (see Sect. 4).

For $\langle$Fe$\rangle$, G, and D, there are not enough data to carry out the test. The formal isoabundance relations for these indices are therefore assumed to be correct. For $\langle$Fe$\rangle$ and G, the results of the analyses offer some support to this assumption (again, see Sect. 4).

The isoabundance relations given for M₁ by Eggen are not adequately documented. A relation was therefore derived by assuming that the mean metallicity of stars in a large sample measured by [9, Eggen (1989a]) is independent of temperature. Eggen did not base the selection of his sample on a metallicity parameter, so this assumption is at least plausible.

The data used to derive an isoabundance relation should have some scatter around the relation. The character of the scatter must be understood if the relation is to be derived correctly. In this case, the scatter turns out to be quite a bit larger than one would predict from plausible measurement error. Presumably the "excess'' scatter is caused by star-to-star metallicity differences. Those differences should have relatively large effects on a blanketed index like M₁, but should have small effects on $(R-I)_{\rm E}$ (see the entry for the similar index $(R-I)_{\rm K}$ in Table III of [44, Taylor et al. 1987]). $(R-I)_{\rm E}$ was therefore treated as an error-free parameter, and a one-error least-squares regression of M₁ on $(R-I)_{\rm E}$ was obtained. The result of this calculation will be given below (see Sect. 4, Table 3, footnote "h'').

  

Table 3: Calibrations: [Fe/H]$(q) = Sq + Z^{\mathrm{a}}$

• [$^{\mathrm{a}}$]Units are magnitudes for $\delta$CN and $\delta M_1$.Values of [Fe/H](q) are in dex.
• [$^{\mathrm{b}}$]Limits in [Fe/H](q) are -0.8 dex and +0.2 dex.
• [$^{\mathrm{c}}$]Data for HD 13530 were not used to derive this equation.
• [$^{\mathrm{d}}$]Values of [Fe/H] with $\sigma \geq 0.11$ dex yield a value of S that is anomalous at better than 99.5% confidence. That value of S is rejected, but the corresponding value of Z is retained, and the large-$\sigma$ values of [Fe/H] help to determine the error quoted in Col. 5. Data for HD 116713 and HD 198700 were not used to derive this equation.
• [$^{\mathrm{e}}$]This relation is an average of the first and second relations given above, and is intended for the transition region in (45-48) between the two.
• [$^{\mathrm{f}}$]This relation is an average of the second and third relations given above, and is intended for the transition region in (45-48) between the two.
• [$^{\mathrm{g}}$] Limits in [Fe/H](q) are -0.25 dex and +0.2 dex.
• [$^{\mathrm{h}}$] Limits in [Fe/H](q) are -0.35 dex and +0.2 dex.
• [$^{\mathrm{j}}$]$\delta M_1 = M_1 + 0.695 - 4.336(R-I)_{\rm E} + 3.120(R-I)_{\rm E}^2$. This calibration was derived only from data in [9, Eggen (1989a)]. Data for HR 2574, HR 4308, and reddened stars are not used to calculate S and Z from [Fe/H] and $\delta M_1$ (see Appendix B of [37, Taylor 1996], and Sect. 5.2 of [39, Taylor 1998]). Limits in [Fe/H](q) are -0.6 dex and +0.2 dex.
• [$^{\mathrm{k}}$]Data for HD 41593 were not used. Limits in [Fe/H](q) are -0.6 dex and +0.45 dex.
• [$^{\mathrm{m}}$]The Buser-Kurucz grid is read within the following limits: $-2.0 \leq [M/H] \leq +0.5$, 4000 K $\leq T_{\rm eff} \leq6000$ K. At $T_{\rm eff} = 6000$ K, the grid is read at $\log g = 4.25$ to allow for some evolution in program stars. Values for $\log g = 4.5$ are read at other effective temperatures. Stellar values of U-B and $[(R-I)_{\rm C} - 0.007$ mag] are compared to the numbers from the grid (see note "h'' of Table 1).

3.2 Choosing a least-squares algorithm for the [Fe/H] relation

The derivation of Eq. (1) may now be considered. Again one must consider the scatter around a calculated relation. This time, three sources of such scatter may be important:

1.

the rms errors of the catalog values of [Fe/H],

2.

measurement error in q, and

3.

intrinsic scatter around f(q).

The nature of the intrinsic scatter is most easily visualized for $\delta$CN. Here, one expects CNO/Fe variations to yield a range of values of $\delta$CN for any given choice of temperature, surface gravity, and [Fe/H]. In the same way, G should be influenced by Ca/Fe variations and their counterparts for other metals. Variations in Fe-line strength should be closely correlated with variations in [Fe/H], but nonetheless there is also intrinsic scatter in the relation between these two parameters. The same is true for blanketing and [Fe/H].

It might be argued that a "structural'' least-squares technique is required here because of the intrinsic scatter. However, since the sources of the scatter affect only q and not [Fe/H], one can presumably regard the net scatter from items (2) and (3) as if it were an additional measurement error in q. This viewpoint permits the use of one of the better-known "functional'' least-squares techniques. The technique used here is a linear, two-error algorithm based on the following parameter:

$$\lambda = v({\rm [Fe/H]})/v(q),$$ (2)

with v denoting variance per datum (the square of the rms error per datum). (See Sect. 1.2 of [25, Madansky 1959] and Eq. (7.7) of [1, Babu & Feigelson 1996].)

For single determinations of [Fe/H], the rms errors that yield $v({\rm [Fe/H]})$ are in the range 0.10-0.13 dex (see Table 2 of [41, Taylor 1999b]). The errors are smaller, of course, for stars with multiple determinations. This range of errors poses a problem, since the adopted algorithm requires $v({\rm [Fe/H]})$ to be the same for all contributing stars. To deal with this problem, the data are analyzed in groups with similar rms errors. The values of S from all the groups are then averaged using inverse-variance weights, with the same procedure being applied to Z.

3.3 Calculating v(q)

Though v(q) is required by the algorithm described above, its value is not known in advance. However, one can derive v(q) by assuming that the net scatter around Eq. (1) is produced by v(q) and a known contribution from $v({\rm [Fe/H]})$. A "data-comparsion'' algorithm for deriving v(q) in this way is summarized in Sect. 4 of Appendix B of [34, Taylor (1991]).

In practice, an initial guess for v(q) is made. An initial version of Eq. (1) is then calculated, and the data-comparison algorithm is applied to the scatter around this equation to obtain an improved estimate for v(q). This procedure is iterated to convergence.