Appendix A: Zero-point procedures for [Fe/H] analysis

Let f be an unknown true value of log (Fe/H), and let E be a generally nonzero systematic error in f which results from a high-dispersion analysis. The derived value of log (Fe/H) is then f + E. Let $f_{\odot}$ and $E_{\odot}$ apply for the Sun, and let $f_{\rm s}$ and $E_{\rm s}$ apply for a program star. Finally, to focus attention on the simplest case of interest, let $E_{\rm s}$ be the same for all program stars. With these constraints in mind, seven zero-point procedures will be discussed.

1) Differential analysis (relative to the Sun). When this case is satisfied in practice,

$${\rm [Fe/H]} = (f_{\rm s} + E) - (f_{\odot} + E) = f_{\rm s} - f_{\odot}.$$ (A.1)

No subscript is attached to E because it is presumably the same for program stars and the Sun. Note that since E is generally nonzero, there is no constraint of any kind on the quantity $f_{\odot} + E$ to equal a published solar metallicity. To perform this kind of differential analysis, one must measure and analyze the Sun and program stars as identically as possible. In particular, solar and stellar model atmospheres must both be empirical or must both be from the same grid (see Sect. 6.1 of Drake & Smith 1991 for a more extensive discussion of this essential point). For instructive examples of differential analysis, the work of Cayrel de Strobel and her associates may be consulted (see, for example, Chmielewski et al. 1992).

2) Differential analysis with model-atmosphere mismatch. Here,

$${\rm [Fe/H]} (f_{\rm s} + E_{\rm G}) - (f_{\odot} + E_{\rm HM})$$ =

$$(f_{\rm s} - f_{\odot}) + (E_{\rm G} - E_{\rm HM}),$$ = (A.2)

with [Fe/H] now being the derived metallicity instead of the true metallicity. In this case, a model atmosphere from a grid is used to calculate $E_{\rm G} \equiv E_{\rm s}$, while the Holweger-Müller (1974) model is used to calculate $E_{\rm HM} \equiv E_{\odot}$. As a result, the absolute difference $\vert\Delta E\vert$ between them is commonly nonzero (see, for example, Gustafsson 1980; Trimble & Bell 1981, Sect. 5; Cayrel de Strobel 1983; Taylor 1998a, Sect. 6). Examples of this kind of analysis have been given by McWilliam & Geisler (1990) and Gratton & Sneden (1991).

3) Differential analysis with equivalent-width mismatch. In this case,

$${\rm [Fe/H]} = (f_{\rm s} + E_{\rm s}) - (f_{\odot} + E_{\odot}) = (f_{\rm s} - f_{\odot}) + (E_{\rm s} - E_{\odot}).$$ (A.3)

Solar equivalent widths (EWs) from a published solar atlas are used instead of EWs from the spectrograph used to observe the program stars. It is known that $\vert\Delta E\vert \equiv \vert(E_{\rm s} - E_{\odot})\vert$ can be as large as 0.08 dex in this case (see Griffin & Holweger 1989, Sect. 3.2). Larger values of  $\vert\Delta E\vert$ are conceivable. As noted in Sect. 3.2 of the text, this kind of differential analysis is commonplace.

4) Differential analysis relative to a star other than the Sun. The equation for example (1) applies here, but with a star such as Procyon substituted for the Sun. Model-atmosphere and EW mismatch do not occur here in practice because the standard star and the program stars are observed and analyzed in the same way. However, the zero-point process is incomplete. If it is to be completed, the value of [Fe/H] for the standard star relative to the Sun must be known. An example of this kind of analysis is given by Kyrolainen et al. (1986).

5) External zeroing to meteoritic abundances. Here,

$${\rm [Fe/H]} = (f_{\rm s} + E_{\rm s}) - f_{\odot} = (f_{\rm s} - f_{\odot}) + E_{\rm s}.$$ (A.4)

For the sake of argument, it is assumed that $E_{\odot} = 0$ in this case (see, for example, Grevesse et al. 1996). The nature of this technique becomes clear if example (1) is used as a benchmark. To frame example (1), one reasons that since $E_{\rm s}$ is not zero except by rare happenstance, $E_{\odot}$ must be chosen to compensate for $E_{\rm s}$. Here, one reasons that since $E_{\rm s}$ is not zero except by rare happenstance, its uncompensated value will almost always affect derived values of [Fe/H]. For this reason, metallicities zeroed in this way should not be accepted unless their zero points can be checked (and revised if necessary). Note that this problem is caused by a plausible-looking mistake: an accurate value of $f_{\odot}$ is adopted instead of a datum that will cancel the effect of $E_{\rm s}$. Examples of this procedure are given by Liu et al. (1999) and Russell (1995). In an instructive comment, Balachandran & Carney (1996, Sect. 3.3) pinpoint the zero-point problem which is common to this example and the one discussed just below.

6) External zeroing to photospheric solar abundances. Here,

$${\rm [Fe/H]} = (f_{\rm s} + E_{\rm s}) - (f_{\odot} + E_{\odot}) = (f_{\rm s} - f_{\odot}) + (E_{\rm s} - E_{\odot}).$$ (A.5)

The value of $f_{\odot} + E_{\odot}$ is now an absolute solar metallicity from a published analysis. In contrast to example (5), $E_{\odot}$ is cautiously regarded as nonzero because of the history of absolute solar analyses (for example, compare the results of Holweger et al. 1995 and Blackwell et al. 1995 and note the title of Kostik et al. 1996). A good way to gauge external zeroing is to look for differences between parallel procedures used in the stellar and published solar analyses. One such difference is the model-atmosphere mismatch that applies to example (2). An extensive discussion of these and other pertinent problems appears in Taylor (1999b, Sect. 3.1). Examples of this kind of external zeroing are given by Beveridge & Sneden (1994) and Castro et al. (1996).

7) Pseudo-absolute analysis. In this case, only values of $f_{\rm s} + E_{\rm s}$ are given. The reasoning applied to $E_{\rm s}$ is the same here as it is for example (5). In a procedure that is intermediate between this example and example (5), values of $f_{\rm s} + E_{\rm s}$ and $f_{\odot}$ are compared without subtraction (see, for example, Adelman et al. 2000). Examples of this kind of analysis are given by Klochkova & Panchuk (1987, 1990).