5 Updating T91: The results of the new analysis

5.1 Zero points and scale factors

It is of interest to see how different the results of the new analysis are from their T91 counterparts. Zero points and scale factors will be considered first. A selection of values of S and Z (recall Eq. (1)) is given in Table 3, with T91 results in the third column and new results in the fourth column. Table 3 also contains the first values of Z derived for the Luck et al. data.

  

Table 3: T91 and revised corrections

All numbers in Cols. 3-4 are in dex.
$^{\mathrm{a}}$ Symbols are defined in Eq. (1). If a value of S does not differ significantly from 0, it is assumed to be $\equiv 0$ and is not quoted.
$^{\mathrm{b}}$ The ``T91'' entry was derived by Taylor 1996.
$^{\mathrm{c}}$ This entry is for data based on ``spectroscopic'' gravities.
$^{\mathrm{d}}$ This entry is for data based on ``physical'' gravities.

From the Table 3 data that have T91 counterparts, one gets the overall impression that little has changed. Note especially the entries in the table for the photometric and McWilliam data. Since those data are added at stage 3, the tabular entries show how much the overall zero point established at stage 2 has changed. Despite a marked increase in the size of the data base, that change appears to be no more than 0.01-0.02 dex. Apparently there is good reason to hope that future changes will be no larger than this.

5.2 The overall zero point: An estimate of accidental error

To estimate the rms error of the overall zero point, one may use the stage 2 results and either the photometric or the McWilliam data. Z is known with better precision for the McWilliam data, so those data will be used here (note the first and second lines of Table 3). To three decimal places, the rms error of Z for the McWilliam data is 0.014 dex. If the McWilliam data and the stage 2 data contribute about equally to this error, then $\sigma \sim 0.010$ dex for the overall zero point. This result is essentially unchanged from T91.

5.3 Values of $\sigma_0$: Sizes and possible sources

For values of $\sigma_0$, there is a greater change than the one found for systematic effects. In T91, the "default error'' per datum is found to be $0.16\ \pm\ 0.02$ dex. Typical counterpart values obtained here are in the range 0.10- 0.12 dex. These smaller errors are obtained from a noticeably larger data base than was available for the T91 analysis. There is therefore good reason to hope that the revised errors are more reliable than the T91 errors. Testing of the revised errors is planned as further data become available.

One would like to know whether the derived errors can be explained by an appeal to obvious sources. Line-to-line scatter is clearly one such source. Estimates of error from this source can be derived from data in Table 7 of Gratton & Sneden (1990). For photographic spectra, the estimated error contribution is 0.06 dex. For Reticon and CCD spectra, the corresponding number is 0.02 dex.

Errors in assigned temperatures also contribute to $\sigma_0$. To understand this error source, one should recall that $\sigma_0$ has been derived by comparing data from various stars. When such comparisons are made, systematic errors from a temperature calibration should largely cancel out because they should be about the same for all stars. On the other hand, if photometric indices are used to assign temperatures (as they often are here), the errors introduced by such indices will not cancel out because they vary randomly from star to star. An estimated size for this second kind of error is therefore required.

The error size may be obtained from the following relation:

$$\sigma_{\rm 0,T} = \sigma_{\theta}\vert{\rm d}[{\rm Fe/H}]/\rm d\theta\vert.$$ (3)

To obtain a specific result from this equation, a star in the level 2 data base with $T_{\rm eff} = 5000$ K will be considered. For such a star, $\rm \vert d[{\rm Fe/H}]/d\theta\vert$ is typically 3.3 (see Eq. (4) of Taylor 1998a). A value of $\sigma_{\theta}$ may be obtained from errors derived for photometric indices (see Table III of Taylor et al. 1987 and Appendix B of Taylor 1996). For level 2 data, the mean value of $\sigma_{\theta}$ is about 0.009. The resulting value of $\sigma_{\rm 0,T}$ is then 0.03 dex.

If this error is added in quadrature to the line-scatter errors, the results are about 0.07 dex for data from photographic plates and about 0.04 dex for data from Reticons and CCDs. Recalling that the derived value of $\sigma_0$ is about 0.10 dex for level 2 data, one can see at once that some important error source is being missed. Identifying that source turns out to be something of a problem. In principle, the missing error could be produced by inconsistencies in the way that spectra for diverse program stars are reduced. This explanation is not very attractive because it is hard to name a specific inconsistency that might plausibly be causing the problem. Nevertheless, it might be worthwhile to look into this possibility by exposing and reducing a series of spectrograms for a given star.