Appendix B: An improved derivation of rms errors for class W values of [Fe/H]

In the first two iterations of the catalog, rms errors for class W data were derived by comparing them with class N data (see Appendix B of T94). This process is far from optimal because the class N rms errors are substantially larger than their class W counterparts. As a result, the noise introduced by the former yields an indeterminate result for the latter (see Sect. 3.8 of T94). When T94 was published, there was no apparent way around this problem. Now, however, there are more class W data than previously, so an improved way of deriving their errors may be applied.

Suppose that for star i of N stars in total, there are M class W data, with $M \geq 2$. Let the weight for datum F_j be given by

w_j = (V_w + v_j)^-1 (B.1)

(see Eq. (10.16) of Kendall & Stuart 1977). In this equation, v_j is a variance produced by EW error, while V_w is a "frame-to-frame variance'' for which an initial guess is made. The calculated values of w_j are used to obtain a weighted average F_M of the values of F_j. A statistic Q is then calculated:

$$Q_i=\sum_{j=1}^M w_j (F_j - F_M)^2.$$ (B.2)

Q is $\chi^2$ distributed with $\nu \equiv M-1$ degrees of freedom (see Lampton et al. 1976, Appendix, Sect. III). $\nu$ is the associated number of degrees of freedom.

The values of Q_i and $\nu_i$ are summed over the data for all N stars. This procedure is repeated with a number of trial values of V_w. The correct value of V_w is the one for which

$$\sum_{i=1}^N Q_i = \sum_{i=1}^N \nu_i.$$ (B.3)

The $\chi^2$ distribution may be used to calculate an rms error U_W for V_w. An equivalent number of degrees of freedom $\nu_W$ is then obtained from the following definition:

$$U_W^2 = V_w^2 (2/\nu_W)$$ (B.4)

(see Eq. (A.4) of T94).

As Eq. (B.1) shows, this algorithm requires knowledge of values of v_j. Those variances are calculated as follows:

v_j = v_j0 n_j^-1, (B.5)

with n_j being the number of contributing lines. Values of v_j0 are sometimes given in contributing papers, but a default value of (0.072 dex)² (from Favata et al. 1997) is commonly used instead. The results of the calculation are not at all sensitive to this choice because V is substantially larger than v_j in almost all cases.