New vs. Old Fitting Functions

We restrict our comparison to previous work on Lick/IDS fitting functions to those computed by G. Worthey and collaborators, because they are available for all the indices studied here and are based on a very comprehensive spectral library. Moreover, they are the most widely used fitting functions in stellar population synthesis work.

In order to assess the impact of our new fitting functions on model predictions we proceeded as follows. We computed integrated line indices for models 1 through 5 in Table 24 adopting our own fitting functions and those of Worthey et al. (1994) and Worthey & Ottaviani (1997) (henceforth simply Worthey et al.). In this way we isolate the effect on model predictions due only to the adoption of our new fitting functions.

The two sets of model predictions are compared in Figures 7a through 7d for all indices. The indices computed adopting the Worthey et al. fitting functions were brought into our system of EWs using the zero-points listed in Table 1. In each panel arrows indicate in which sense model metallicity varies, to help the reader identify the models with different [Fe/H].

The overall agreement between the two sets of computations is good. Not unexpectedly, most of the differences are found at low metallicity, where both sets of fitting functions are more uncertain. Amongst the Balmer lines, the most important differences are found for $H\beta$. This index is more metallicity-dependent when the Worthey et al. fitting functions are adopted. This is a very interesting result that serves to illustrate how improvements in the accuracy of stellar data (both stellar parameters and spectra) can cause a noticeable improvement in model predictions. In Figure 8 we compare the input data used in the computation of both sets of fitting functions. In the upper panel, $H\beta$ and $T_{\rm eff}$ from the Worthey et al. (1994) are plotted against each other for G and K giants and the same plot is repeated in the lower panel using our data. Stars in two ranges of metallicity are plotted in order to highlight the dependence of $H\beta$ on this parameter. Stars with [Fe/H] $>$ 0 are plotted with solid squares and stars with [Fe/H] $<$ -0.3 with open squares. A dependence of $H\beta$ on metallicity, whereby at fixed $T_{\rm eff}$ the index becomes stronger for higher [Fe/H], can be seen in both data-sets, but is far more clear-cut in our data than in those of Worthey et al. (1994). As a result, we can estimate the dependence of $H\beta$ on metallicity in stellar spectra more accurately. We find that $H\beta$ in the spectra of GK giants responds to variations in [Fe/H] roughly twice as strongly than predicted by Worthey et al. (1994) in the sense that, we repeat, $H\beta$ becomes stronger for higher metallicity. On the other hand, we know that higher metallicity systems tend to have cooler turn-offs, which tends to produce weaker $H\beta$. Therefore, the two above effects tend to cancel out, with the net result that the index in integrated spectra of stellar populations becomes less sensitive to [Fe/H] than predicted by former models. As a result, the new fitting functions show that $H\beta$ is a better age indicator (i.e., less sensitive to [Fe/H]) than previously thought.

The Fe indices are extremely important because they are mostly sensitive to the abundance of iron (Tripicco & Bell 1995), thus providing a close estimate of the mean [Fe/H] of an integrated stellar population. In Figure 7 we compare our model predictions to those based on the Worthey et al. fitting functions for all the Fe indices modelled here. Agreement between the two sets of fitting functions is good for Fe4383 and Fe5335. The most important differences are found for Fe5270 at metallicities below solar. In Figure 9 we compare the two sets of fitting functions for dwarfs with [Fe/H]=-0.4. Our data and fitting functions are represented respectively by the solid squares and thick solid line. The open squares and thin line indicate Worthey et al. data and fitting functions. Only dwarfs with [Fe/H] = -0.4 $\pm$ 0.15 are plotted. As in the case of Figure 8 the quality of the new stellar data is quite superior, as can be seen by the lower scatter in the solid squares. That of course makes it far easier to compute an accurate fitting function for the index. It can be seen that our new set of fitting functions provides a better description of the data for mildly metal-poor dwarfs. The latter accounts for roughly 2/3 of the mismatch seen in Figure 7. The rest of the mismatch is due to smaller differences in the fitting functions for giant stars.

Another interesting case is that of indices that are strongly sensitive to surface gravity, such as Mg$_2$, Mg $b$, and Ca4227, for which the Worthey et al. al fitting functions yield higher values for solar metallicity at all ages. This is because the line strengths in the spectra of giants are stronger according to the Worthey et al. fitting functions. This point is illustrated in Figure 10 where the two sets of fitting functions are compared with Mg$_2$ data for M 67 stars in an Mg$_2$-magnitude diagram. The data come from Paper III, whereas the isochrones were computed by combining the two sets of fitting functions with the Girardi et al. (2000) isochrone for an age of 3.5 Gyr and solar metallicity. The latter was shown to provide an excellent match to the color-magnitude diagram of the cluster (see Paper III for details). It can be seen from this Figure that, when the Worthey et al. fitting functions are adopted the index is over-predicted by $\sim$ 0.05 mag throughout most of the red-giant sequence and also at the horizontal branch (thick lines). A similar behavior is seen for Mg $b$ and Ca4227.

It is also important to point out that the two Mg indices have a markedly different sensitivity to IMF variations. While Mg$_2$ is strongly sensitive to the contribution of K dwarfs, Mg $b$ is nearly insensitive. This can be understood by looking at Figure 11, where we plot measurements of the two indices in our library star spectra as a function of $T_{\rm eff}$ for dwarf and giant stars. For K stars (5500 $\mathrel{\copy\simgreatbox}$ $T_{\rm eff}$ $\mathrel{\copy\simgreatbox}$ 4000 K), both indices respond to $T_{\rm eff}$ and $\log g$ in essentially the same way. In particular, they tend to be stronger in K dwarfs, because both the Mg II lines and the MgH band-head included in the Mg$_2$ passband are stronger for higher surface gravities (Barbuy, Erdelyi-Mendes & Milone 1992). At lower $T_{\rm eff}$, presumably because the Mg II lines saturate, the indices cease to increase for lower temperatures and its dependence on $\log g$ also becomes weaker. In the M-star regime ($T_{\rm eff}$ $\mathrel {\copy \simlessbox }$ 4000 K) the two indices behave in drastically different ways. While Mg $b$ becomes much stronger in giants than in dwarfs, Mg$_2$ is very little dependent on surface gravity. The reason for this behavior is that, as pointed out in Paper III, Mg $b$ is severely affected by a TiO band, which is so strong in the spectra of M giants that Mg $b$ becomes essentially a TiO indicator (see Figure 3 in Paper III, for details). Because TiO bands are very strongly sensitive to $\log g$ being stronger in giants than in dwarfs of the same $T_{\rm eff}$ (Schiavon & Barbuy 1999, Schiavon 1998), the Mg $b$ index becomes much stronger in the former than in the latter. The Mg$_2$ index, on the other hand, is far less influenced by TiO lines, because they affect both the pseudo-continuum and index passband in similar ways. This result has an interesting ramification, namely, that Mg$_2$ is an IMF-sensitive index, and Mg $b$ is nearly unaffected by IMF variations. This can be understood by looking at Figure 11. The Mg$_2$ index is IMF-sensitive because it is much stronger in dwarf stars, so that it tends to be stronger for dwarf-enriched IMFs. The same is not true for Mg $b$, because the index is so strong in cool giants that its sensitivity to the contribution by K-dwarfs is washed away. As a result, when used in combination, the Mg$_2$ and Mg $b$ indices can be used to constrain both the magnesium abundance and the shape of the IMF in the low-mass regime. We return to this topic in Section 5.2.2.

There is a caveat here that needs to be highlighted. When we first computed the model predictions with our fitting functions we obtained too weak Mg$_2$ values for stars in the lower giant branch ( $12.5 \mathrel{\copy\simlessbox}$ V $\mathrel{\copy\simlessbox}11.5$ in M67, cf. Figure 10). That region of the diagram is inhabited by K stars with intermediate surface gravities ( $3.0 \mathrel{\copy\simlessbox}\log g \mathrel{\copy\simlessbox}3.6$), which are scarce in our spectral library. Therefore, our fitting functions are poorly constrained in this region of stellar parameter space. For that reason, we decided to interpolate our predictions for gravity-sensitive indices, using index-magnitude diagrams such as the one shown in Figure 10 to check the quality of the interpolations.

In Summary, we conclude that our fitting functions are generally in good agreement with those of Worthey et al. (1994). The differences found are mostly due to the better quality of our data and the higher accuracy of our stellar parameters. The latter validates our efforts to refine the stellar parameter determinations, as described in Section 2.3.