Index Measurements

All Lick indices of interest were measured in the stacked spectra using lick_ew, an idl routine written by G. Graves (for a description, see Graves & Schiavon 2006, in preparation). Before we can compare these measurements with our models, we need to correct the line indices for the effect of $\sigma$-broadening. That of course requires measuring first the velocity dispersions directly in the stacked spectra. We proceeded as follows. Velocity dispersions were measured through Fourier cross correlation using the IRAF rv.fxcor routine. The templates adopted were model single stellar population spectra calculated for a range of ages and metallicities, as described in Paper I. The choice of template is very important and can introduce important systematic effects if not carefully done. For each spectrum, a first guess of $\sigma$ is made, the indices are corrected and initial values of age and [Fe/H] are determined, if these values do not agree with those of the initial single stellar population template, a new template is adopted with the estimated age and [Fe/H] and the process is iterated until convergence is attained.

After velocity dispersions are determined, the indices can be corrected to their standard $\sigma =0$ values. These corrections were estimated again using model spectra of single stellar populations with appropriate ages and metallicities. Listings of such corrections for a set of representative ages and velocity dispersions are provided in Tables A through A in the Appendix.

The last step before we can compare models and data is the correction of Balmer line indices for the effect of emission line in-fill. Balmer line emission was estimated from the equivalent width of the [OII] $\lambda$ 3727 ${\rm\AA}$ line adopting a ratio between that line and $H\alpha$ given by EW[OII]/EW($H\alpha$) = 6. This value was taken from Yan et al. (2006), who studied the emission line properties of a large sample of SDSS galaxies. They found that most line-emitting red galaxies in SDSS tend to present LINER-like line ratios (see also, e.g., Phillips et al. 1986 and Rampazzo et al. 2005). Emission EWs for higher-order Balmer lines are obtained from EW($H\alpha$) by assuming standard values for the Balmer decrement (in the absence of reddening) and continuum fluxes measured in the stacked spectra. In this way, one obtains EW($H\beta$)/EW($H\alpha$)=0.36, EW($H\gamma$)/EW($H\alpha$)=0.19, and EW($H\delta$)/EW($H\alpha$)=0.13. Corrected indices and the velocity dispersions measured in the stacked spectra are listed in Table 28. The equivalent widths of the [OII] line, measured according to the definition of Fisher et al. (1998), are also provided in that Table

Before attempting quantitative estimates of mean ages and metal abundances of SDSS early-type galaxies, we compare the indices, measured as described above, with our model predictions in Figure 29. In all plots, error bars are smaller than symbol sizes and galaxy luminosity increases from left to right. In the upper left panel, data are compared with solar-scaled models in the $<Fe>$-$H\beta$ plane. Because these indices are essentially insensitive to abundance ratio effects, this plot allows us to obtain a first estimate of mean age and [Fe/H]. The result is that the stacked spectra indicate approximately the same mean age ($\sim$ 8 Gyr), regardless of luminosity. On the other hand, [Fe/H] is just below solar, and seems to be weakly correlated with luminosity.

In the remaining panels, SDSS early-type galaxies are compared with models in index-index planes that are sensitive to the abundances of magnesium, carbon, and nitrogen. In all these diagrams, solar-scaled (gray lines) models are plotted along with models computed with the abundances of a few key elements enhanced by +0.3 dex. In the upper right panel, the data are compared with solar-scaled and magnesium-enhanced models for an age of 8 Gyr, in the $<Fe>$-Mg $b$ plane. This plot suggests that early-type galaxies are enhanced in magnesium, with [Mg/Fe] slightly below +0.3. A slight correlation between Mg-enhancement and luminosity is apparent. In the lower left panel, data are compared with solar-scaled and carbon-enhanced models in the $<Fe>$-C$_2$4668 plane. Again in this case there is a clear indication of carbon-enhancement in early-type galaxies, with a more clear trend of carbon-enhancement as a function of luminosity than in the case of magnesium enhancement. Finally, in the lower right plot, data and models are compared in the $<Fe>$-CN$_1$ diagram. Because the CN$_1$ index is sensitive to both carbon and nitrogen enhancements, three models are displayed: solar scaled (gray) and carbon-enhanced (thin) models, plus models where both carbon and nitrogen are enhanced (thick). One can see that, while carbon-enhanced models do a good job of matching the C$_2$4668 index, the same is not true for CN$_1$ data, which are stronger than predicted by [C/Fe]=+0.3 models. In fact, matching CN$_1$ data requires that the abundance of nitrogen be also enhanced--thick lines do reach the high CN$_1$ values observed. One can also note that the correlation between enhancement and luminosity here is even stronger than in the case of the other plots.

With these qualitative results in mind, we apply our method described in Section 4.4 in order to obtain quantitative estimates of mean ages and metal abundances of the stellar populations of galaxies in the Eisenstein et al. (2003) sample, on the basis of the line indices measured in their spectra. The results are listed in Table 29 and shown in Figure 30, where resulting abundance ratios and mean ages are plotted as a function of mean $r$-band absolute magnitude. As expected from the discussion above, abundance ratios vary strongly as a function of mean luminosity. We will return to this issue in Section 6.2.4, but first discuss the mean ages of early-type galaxies in the next section.