The Base Models

The fitting functions presented in Section 3 were combined with theoretical isochrones in order to produce predictions of integrated indices of single stellar populations. The isochrones employed were those from the Padova group for both the solar-scaled (Girardi et al. 2000) and $\alpha$-enhanced cases (Salasnich et al. 2000). There are several other groups producing state-of-the-art stellar evolutionary tracks and theoretical isochrones (e.g., Charbonnel et al. 1999, Yi et al. 2001, Kim et al. 2002, Pietrinferni et al. 2004, Jimenez et al. 2004) and it is very important to study the dependence of the results on the stellar evolution prescriptions. This will be discussed in a future paper. Absorption line indices and UBV absolute magnitudes were computed for the parameters listed in Table 24. In column (1) of Table 24 we list a model reference number, in columns (2) and (3) we list the mass fraction of elements heavier than He (Z) and that of He (Y). The iron abundance, overall metallicity, and mean $\alpha$-enhancement for the mixture adopted by the Padova group, given by [Fe/H], [Z/H] and [$\alpha$/Fe], are listed in columns (4) through (6). The mean $\alpha$-enhancement of the spectral library is listed in Column (7). Finally, column (8) contains the range of ages encompassed by each model set.

Throughout this paper we refer to the models summarized in Table 24 as our base models, which result from the mere combination of our fitting functions derived in Section 3 and the Padova isochrones. We note that, except for models 3 through 5, the $\alpha$-enhancement of the spectral library is inconsistent with that of the theoretical isochrones adopted (we assume here that a $\sim$ 0.1 dex mismatch is negligible). This condition is not unique to our base models. In fact, other well-known stellar population synthesis models in the literature (e.g., Worthey 1994, Vazdekis 1999, Bruzual & Charlot 2003, Le Borgne et al. 2004, Lee & Worthey 2005), which are based on similar combinations of theoretical isochrones and empirical stellar libraries, are afflicted by the same inconsistency. In principle, this issue can and has been addressed via adoption of the response functions of Tripicco & Bell (1995), Houdashelt et al. (2002), or Korn et al. (2005), as discussed above (e.g., Trager et al. 2000, Thomas et al. 2003a, Thomas et al. 2004, Lee & Worthey 2005). These models, however, are corrected for an assumed abundance pattern of the stars that make up the stellar library and therefore are also lacking in consistency. To our knowledge, the only attempts so far at full consistency between theoretical isochrones and stellar library are those of Coelho (2004) and this work. The former is based on synthetic spectra and the latter are discussed in Section 4.3.2.

Our computations were performed as follows. If is the line index in the integrated spectrum of a model single stellar population, it is given by

when and are defined in terms of an equivalent width, and

when and are defined in terms of a magnitude. In equations (1) and (2), is the index computed from our fitting functions for the stellar parameters corresponding to the -th evolutionary stage. is the number of evolutionary stages in the theoretical isochrone adopted. is the relative number of stars at the -th position in the isochrone, which is given by the initial mass function (IMF) of the stellar population. For simplicity, we adopt a power-law mass function, given by

$$\phi_i = \int A \,\, m_i^{1-x} dm$$ (3)

where $m_i$ is the mass at the -th evolutionary stage in the isochrone and $A$ is a normalization constant which is chosen so that the entire stellar population has 1 $M_\odot$. The integration is performed within a narrow interval centered on $m_i$. For a Salpeter IMF, $x = 1.35$.

The term $f_i$ in equations (1) and (2) gives the weight in flux for stars at the th position of the isochrone. It is computed from interpolation between broad band absolute magnitudes to the index central wavelength. Absolute magnitudes in the U, B, and V bands were computed using the calibrations described in Paper I. Integrated magnitudes for single stellar populations in these bands were computed according to equation 2, with $f_i = 1$ and making equal to the absolute magnitude of the -th position in the isochrone. The results are provided in tables in the Appendix. Tables A2 and A3 provide Lick index predictions, and Tables A4 and A5 list predictions for UBV magnitudes/colors. In the following section we compare our predictions to those obtained when the fitting functions of Worthey et al. (1994) are employed.