M Giants

M giants dominate the integrated light of old stellar populations at wavelengths redder than $\sim$ 6500 ${\rm\AA}$ (Schiavon & Barbuy 1999, Schiavon, Barbuy & Bruzual 2000). In the optical, even though their total contribution to the integrated light is not dominant, they affect the equivalent widths of key absorption features, such as Mg $b$ and $H\beta$ (Paper III). Therefore, it is important to correctly estimate their stellar parameters in order to account for their contribution to the integrated light. However, this is not an easy task, because despite the tremendous progress made in the last decade or so towards the understanding of the atmospheres of these stars, fundamental stellar parameters are known for very few M giants (metallicities for virtually none of those in the field). Likewise, the representation of M giants in current stellar libraries is scarce.

Our procedure to determine the stellar parameters of M giants was the following. We searched the literature for determinations of $T_{\rm eff}$ of library stars from a fundamental method, such as angular diameter measurements (Ridgway et al. 1980, Dyck, van Belle & Thompson 1998, Perrin et al. 1998), or from the infrared flux method (Blackwell, Lynas-Gray & Petford 1991, Alonso et al. 1999), which is known to be fairly model (thus, metallicity) independent. This sample was further supplemented by stars for which $T_{\rm eff}$ determinations from any of those two methods were not available, but for which they could be determined from their $(V-K)$ colors, adopting the relation by Perrin et al. (1998).

The above stars were used to define a standard relation between $T_{\rm eff}$ and the equivalent width of a TiO band measured in our spectra. The latter was used to infer the $T_{\rm eff}$ of stars for which fundamental determinations are currently lacking.

The stars used to define the standard relation between $T_{\rm eff}$ and EW of TiO are listed in Table 2. In Table 3 we show the pseudo-continua and passbands adopted in the measurement of the EWs of the TiO bands in the library spectra. We fitted a 4th order polynomial to the relation between EW(TiO) and $T_{\rm eff}$, whose coefficients are given in Table 4. The latter was applied to infer the $T_{\rm eff}$ of the remaining library M giants.

TiO bands are known to be strongly sensitive to metallicity, especially when they are not saturated. Since the metallicities of all M giants in the library are unknown, we were forced to ignore this effect. Another caveat concerns the strong photometric and spectral variability characteristic of M giants, some of them possibly being variables of Mira type. As a consequence, a given $T_{\rm eff}$ determined either from photometry or from the strength of a TiO band is strongly dependent on the epoch of the observation. Therefore, the $T_{\rm eff}$ determined by our method should be taken with caution.

In order to improve the reliability of our fitting functions in the M giant regime, we enlarged our sample by inclusion of M giants from Worthey et al. (1994) but only in cases for which independent $T_{\rm eff}$ determinations from one of the methods above were available in the literature. These stars are listed in Table 5.