Procedure

The polynomial functions describing the relations between the various spectral indices and stellar parameters were computed through a general linear least squares method. The spectral library spans a vast range of stellar types, with $T_{\rm eff}$ varying from $\sim$ 3000 to $\sim$ 15000 K, and $\log g$ and [Fe/H] varying by 6 and 3 orders of magnitude, respectively. Photospheric structure, and with it the dependence of absorption line indices on stellar parameters, varies greatly within this large region of stellar parameter space. As a consequence, it is very hard to devise a single simple mathematical expression capable of accounting for line index behavior in the whole range of stellar parameters spanned by the spectral library. For that reason, we decided to split the library in five major stellar classes and perform the fits separately for each class. The five sub-regions of stellar parameter space we consider are roughly: G-K giants, F-G dwarfs, B-A dwarfs, M giants, and K-M dwarfs. The strict boundaries defining each sub-region vary from index to index, and are given in Tables 7 through 22. Considerable inter-region overlap was adopted when performing the fits, in order to ensure a smooth transition between adjacent sub-regions.

The goal when determining index fitting functions is to find the simplest mathematical representation of the dependence of a given index on stellar parameters that yet is reasonably accurate. Very simple statistical tools come in very handy, but cannot be fully trusted, given the specific limitations of the spectral library in use. It is worth to describe two illustrative examples. The approach chosen by Worthey et al. (1994) was that of considering relevant the terms whose inclusion reduces the overall r.m.s. of the fit by a given fractional amount. The danger of this approach in our case resides in the fact that, for instance, for the giants, the majority of the stars have [Fe/H] $\mathrel{\copy\simgreatbox}$ -0.7, so that the r.m.s. is not very sensitive to the quality of the fit for lower metallicity stars. Another approach is that followed by Cenarro et al. (2002), where an automatic routine searches, among a large collection of terms, those whose coefficients depart (according to a t-test) significantly from zero. The problem with that approach is that, again due to the low density with which the spectral library occupies certain regions of parameter space, it may happen that a given coefficient is statistically significant, but unphysical, which may introduce unrealistic high frequency features in the final fitting function.

We addressed this problem by trying to combine the best from each of the above approaches. We started by following the procedure of Cenarro et al. (2002) where a first fit was attempted adopting a polynomial with 25 terms involving products of different powers of $T_{\rm eff}$, $\log g$ and [Fe/H]. A t-test was then applied to verify and remove terms which were not statistically significant. Then a new fit based on the reduced set of terms was performed and the procedure iterated until only terms with t $\mathrel {\copy \simlessbox }$ 0.01 survived. This was all performed automatically. The next step was to examine the quality of the fits interactively, removing terms that seem unphysical or otherwise unnecessary, while monitoring how their removal affects the final r.m.s. of the fit. We also adopted a $\sigma$-clipping procedure, whereby stars departing by more than (typically) 2-3 $\sigma$ from the solution were removed from the sample and the fit redone. We adopted at most one $\sigma$-clipping iteration for each fit and typically more than 97% of the input stars were preserved at each fitting set. Automatic $\sigma$-clipping was turned off in regions of parameter space where poor statistics, due to the scarcity of input stars, prevented a robust estimate of $\sigma$. That was the case for the fits for dwarfs cooler than $\sim$ 5000 K, giants cooler than $\sim$ 4000 K, all stars hotter than $\sim$ 8000 K, and giants more metal-poor than [Fe/H] $\sim$ -1.0.