Conclusions

We presented a new set of models for the integrated absorption line indices and UBV magnitudes of single stellar populations. The models are based on the Jones (1999) spectral library, for which we determined a new set of homogeneous and accurate stellar parameters and a new set of fitting functions for Lick indices. Adopting theoretical isochrones from the Padova group, a set of reliable calibrations relating fundamental stellar parameters and colors, stellar abundances from the literature, which characterize the abundance pattern of the library stars, and sensitivity tables describing how line indices vary as a function of elemental abundance variations, we produced a new set of model predictions for Lick indices in single stellar populations. The models were extensively compared with superb data for Galactic clusters and nearby galaxies. Our main results can be summarized as follows:

1) Stellar parameters (effective temperature, surface gravity, and iron abundance) were determined for the 624 stars from the Jones (1999) library, based on semi-empirical calibrations of photometric and spectroscopic indices and fundamental stellar parameters. These parameters were contrasted with previous determinations from the literature and were found to be devoid of significant systematic effects. All stars for which substantial deviations from determinations by other groups were found were carefully examined and the best set of parameters was chosen. We show that accurate stellar parameters lie in the core of our ability to predict accurately the integrated indices of single stellar populations. The abundance pattern of the stellar library was characterized by surveying the literature for determinations of abundances of such key elements as iron, oxygen, carbon, nitrogen, magnesium, titanium, silicon, and calcium.

2) The above stellar parameters were combined with equivalent widths of line indices measured in the spectra of Jones library stars. This new equivalent width system is based on flux-calibrated spectra smoothed to the same resolution as that of the Lick/IDS system. Index measurements in the new system are substantially more accurate than those upon which the old Lick/IDS system is based. The two systems differ by small zero-point shifts. Polynomial fitting functions describing the behavior of 16 line indices as a function of effective temperature, surface gravity and iron abundance were computed. We compare our new fitting functions to those by Worthey et al. (1994) and found small but important improvements for some indices, which are mostly due to the best quality of the spectra and of the stellar parameters adopted.

3) The fitting functions are combined with theoretical isochrones from the Padova group in order to produce predictions of integrated line indices for single stellar populations. These are compared with high S/N data from four Galactic clusters spanning a representative range of ages, metallicities, and abundance patterns. We successfully match essentially all 16 line indices for the known input parameters (age, metallicity, abundance pattern, mass function) for each of the clusters considered. Our predictions for Fe indices, the Mg indices, C$_2$4668, the CN indices, and Ca4227 match the cluster data for [Fe/H], [Mg/H], [C/H], [N/H], and [Ca/H] within 0.1 dex of the known cluster values. Spectroscopic ages based on all Balmer line indices agree with those based on CMD analyses to within 1-2 Gyr for all clusters. We showed that the Ca4227 index is strongly influenced by CN lines. It is also shown that consistent results can only be obtained for Mg$_2$ and Mg $b$ if the appropriate (dwarf-depleted) mass function for the clusters is adopted, which means that a combination of these indices can potentially be used to set constraints on the IMF in the low mass regime.

4) Combining our single stellar population models with the abundance ratios for the library stars and the sensitivities of line indices to elemental abundances from Korn et al. (2005), we computed models for single stellar populations with several different abundance patterns. We devised a method that employs these models in order to estimate mean stellar ages and abundances of iron, magnesium, calcium, nitrogen and carbon. These models and method were tested against the observations of Galactic clusters with known abundance patterns with very satisfactory results. The models predict the cluster elemental abundances in agreement with the known values to within $\sim$ 0.1 dex. Spectroscopic ages agree with those based on analysis of cluster CMD data to within 1-2 Gyr, for all Balmer line indices considered. Using our model predictions for variable abundance ratios, we found that models with [Fe/H] = -0.1 $\pm$ 0.1 and a mild $\alpha$-enhancement, [$\alpha$/Fe] $\sim$ +0.1, are a better match to the data on NGC 6528 than those with higher enhancement. Matching the G band and CN indices, we found [C/Fe] $\sim$ -0.1 and [N/Fe] $\sim$ +0.5 as mean values for the stars in the core of NGC 6528. This mean abundance pattern resembles that of other well-known Galactic globular clusters such as 47 Tuc, M 71, and M 5, among others. This might suggest the carbon-nitrogen abundance dichotomy that characterizes these clusters is also present in NGC 6528.

5) The very good match obtained to data on Galactic clusters encouraged us to apply our models to observations of nearby galaxies. Initially, models were compared with the high quality measurements of $<Fe>$, Mg $b$, and $H\beta$ by Trager et al. (2000) for a sample of nearby galaxies. We reproduce their results, finding mean [Mg/Fe] $\sim$ +0.3, a spread in mean ages between 2.5 and 14 Gyr. We also found that [Fe/H] seems to decrease and [Mg/Fe] to increase when one goes from strong to low-$H\beta$ galaxies, in agreement with previous findings.

6) Our models were compared to high S/N measurements taken on stacked SDSS spectra of early-type galaxies from Eisenstein et al. (2003). Applying our method to determine mean ages and abundance ratios from Lick index measurements, we were able to estimate the abundances of iron, carbon, nitrogen, magnesium and calcium in these galaxies. We found that, while iron abundances are slightly below solar, the galaxies are enhanced relative to iron in all other elements. Iron abundances and all abundance ratios are shown to be positively correlated with luminosity. Unlike previous studies, we find that [Ca/Fe] is slightly enhanced in the sample studied. Among the elements studied, nitrogen is the one displaying the most conspicuous behavior, as it is the one whose enhancement is most strongly correlated with luminosity. This result might be subject to systematics due to uncertainties in the response of CN formation to carbon abundance variations, but we nevertheless speculate that it might be indicating the presence of a strong contribution by a secondary production mode. If this interpretation is correct, our result poses a constraint on the lower limit for the timescale for star formation in early-type galaxies (40-200 Myr). Magnesium is also seen to be enhanced and its enhancement is also correlated with luminosity, which is consistent with our finding that lower luminosity galaxies form stars for longer time periods (see below). More work is clearly needed to interpret these abundance ratio results.

7) Comparing the spectroscopic ages inferred from $H\beta$ and higher-order Balmer lines we found a systematic trend whereby the latter, especially $H\delta$, yield systematically younger ages than the former. Moreover, this discrepancy is stronger for lower luminosity galaxies. This is in strong contrast with the results from our analysis of cluster data, where we found that spectroscopic ages from all Balmer lines were remarkably consistent. We examined four possible scenarios to account for the observations: abundance-ratio effects, contamination by a small fraction of young/intermediate-age stars, by blue stragglers, and by metal-poor stellar populations with a blue horizontal branch. We argue that the metal-poor scenario cannot match the data and that the blue-straggler and abundance-ratio scenarios require extreme conditions to do so. Therefore, we conclude that the most likely explanation is a spread in the ages of the stellar populations in early-type galaxies. The implications are two-fold. On one hand, if this scenario is confirmed, stellar population synthesis will have evolved to a stage where it is now able to constrain not only the mean ages of stars in galaxies, but also their distribution. On the other, it implies that small amounts of star formation have occurred in the recent past in these nearby early-type galaxies. Our results suggest that on average lower luminosity galaxies formed stars until more recently than their more luminous counterparts, which lends further support to the ``downsizing'' scenario. Extending the accuracy of models and data blue-ward is the best way of further testing these results, posing stronger and more refined constraints on the history of star formation in early-type galaxies.

This work was initiated when I was a postdoc at Lick Observatory, as a member of the DEEP group, first as a Gemini Fellow, then as a CNPq fellow. I would like to thank my parents, Ennio and Norma, without whose relentless support this work would never have become a reality. Ingrid Gnerlich is thanked for her permanent encouragement. I would also like to thank Sandy Faber, Jim Rose, Bob O'Connell, Beatriz Barbuy, David Koo, Bob Rood, and Ruth Peterson for inspiration, encouragement, and continuous support. Nico Cardiel is thanked for initial suggestions on the calculation of fitting functions, Daniel Thomas for making available the Korn et al. sensitivities in advance of publication, and Achim Weiss for expert advice on evolutionary tracks. Many thanks go also to Jenny Graves, who provided the $C_2$4668 measurements in Indo-US spectra and implemented the models presented here in a slick IDL code. An anonymous referee is thanked for a thorough and very helpful reading of the first version of this paper. This work has made extensive use of the Simbad and ADS databases. The author acknowledges financial support from the National Science Foundation through grant GF-1002-99, from the Association of Universities for Research in Astronomy, Inc., under NSF cooperative agreement AST 96-13615, from the NSF, through grant AST 00-71198 to the University of California, Santa Cruz, from CNPq/Brazil, under grant 200510/99-1, and finally from HST Treasury Program grant GO-09455.05-A to the University of Virginia.