Light-element Indices vs Balmer Lines

In Figures 22d-f we show the cluster data compared with the models in light-element-index vs Balmer-line planes. We first focus on the case of the the indices Mg$_2$ and Mg $b$, which are mostly sensitive to the abundance of magnesium. In the case of Mg $b$, the agreement between models and data is very good, especially for 47 Tuc, for which the 14 Gyr-old model for [Fe/H]=-0.7 falls right on top of the data points in all three panels. This is not surprising, given that [Mg/Fe] for both the models and the cluster (Tables 6 and 25) differ by only $\sim$ 0.1 dex. The same is not true for Mg$_2$, for which the model for the same age and metallicity is too strong by 0.03 mag, which would lead to an underestimate of roughly 0.3 dex in [Mg/H]. We suggest that this mismatch is due to the extreme mass segregation in the cluster cores. Because the core of 47 Tuc is strongly depleted of low-mass stars (e.g., De Marchi & Paresce 1995, Howell et al. 2001, Monkman et al. 2006), Mg$_2$ tends to be weaker than the value predicted for a Salpeter IMF, while Mg $b$, which is not so sensitive to the contribution by low mass stars, is less affected (see discussion in Section 4.2). The most recent determination of the mass function in the core of 47 Tuc was performed by Guhathakurta et al. (2006, in preparation, but see Monkman et al. 2006), who found that, within the cluster core, the mass function below the turn-off is well matched by a power law with $x \sim -4.0$. In Figure 23 we illustrate the effect of mass segregation by comparing calculations performed with a Salpeter IMF ($x = 1.35$, bottom panels) and a dwarf-depleted mass function ($x=-4.0$). It can be seen that dwarf-depleted match Mg$_2$ considerably better than those based on a Salpeter IMF. The predictions for Mg $b$ change very little in comparison, with the models agreeing with the data to within 0.1 ${\rm\AA}$. Because both indices are subject to the influence of elemental abundances that may be somewhat uncertain, we only suggest that the initial inconsistency between the magnesium abundances based on Mg $b$ and Mg$_2$ might be due to mass segregation effects. In any case, the conclusion that a combination of these two indices can be used to constrain the low-mass end of the mass function is robust, provided other variables such as abundance ratios are tightly constrained.

We note that in Figure 23 the dwarf-depleted models have slightly stronger $H\beta$ than those computed with a Salpeter IMF. This can be understood in terms of the relative contribution to the integrated light at $\sim$ 4861 ${\rm\AA}$ by giant, turnoff, and lower main sequence stars. Comparing the numbers for dwarf-depleted and Salpeter-IMF models one finds that, because the initial masses of turnoff and giant stars are essentially the same, the ratio between the contribution by these two types of stars to the integrated light varies only by a factor of 1.5 between the two models (being of course higher in dwarf-depleted models). On the other hand, the ratio between giants and lower main-sequence stars varies by roughly a factor of 6. Because giants have stronger $H\beta$ than lower main-sequence stars, the net result is that dwarf-depleted models have stronger $H\beta$. Of more interest to us is the fact that the dwarf-depleted models over-predict the value of $H\beta$ in 47 Tuc by $\sim$ 0.2 ${\rm\AA}$, so that the spectroscopic age of the cluster according to those models is a bit too old. This is not unexpected. As we mentioned in Section 5.2, the oxygen abundance of 47 Tuc is much higher than that adopted in the models of Figure 23. The effect of adopting the right oxygen abundance for 47 Tuc can be gauged by comparing the thick lines in Figure 23, which correspond to computations performed adopting the $\alpha$-enhanced Padova isochrones, for which [O/Fe]=+0.5. While agreement with $H\beta$ in 47 Tuc is improved, the discrepancy is not entirely removed. This is because, as discussed in Section 4.3.1, the $\alpha$-enhanced Padova isochrones overpredict the temperatures of turn-off and giant stars, as pointed out by Weiss et al. (2006). In Section 4.3.1, we estimated an approximate correction to our model predictions and found that $H\beta$ in old $\alpha$-enhanced models should be weaker by roughly 0.15 ${\rm\AA}$, which would bring our models into very good agreement with 47 Tuc data.

In Figure 22e the data are compared in CN-index vs Balmer-line planes. It can be seen that the clusters have much stronger CN indices than predicted by the models. In the case of 47 Tuc, CN$_1$ (CN$_2$) is stronger by 0.06 (0.08) mag than the model for [Fe/H]=-0.7 and 14 Gyr, so that in all panels the cluster falls near the solar metallicity locus. This is not a new result (e.g., Vazdekis 1999, Paper I, Puzia et al. 2002) neither is it unexpected. As anticipated in the beginning of Section 5, the relative abundances of carbon and nitrogen of models and cluster are largely discrepant (Tables 6 and 25). In Figure 22f, model predictions for C$_2$4668 and Ca4227 are compared with the data. Interestingly, models overpredict the values of both indices. In the case of C$_2$4668, this probably indicates that the mean luminosity-weighted [C/Fe] is below solar, which is not surprising, given the abundances measured in individual stars (Table 25). The case of Ca4227 is interesting. Despite the fact that the relative abundance of calcium in the models is a very close match to that of the clusters, there is a very large mismatch with the data. In the case of 47 Tuc, the 14 Gyr-old, [Fe/H]=-0.7 model over-predicts Ca4227 by $\sim$ 0.6 ${\rm\AA}$, leading to an underestimate in [Ca/H] by more than 0.7 dex. We suspect that this might in part be due to the influence of CN lines on the Ca4227 index. These results clearly indicate that indices sensitive to carbon and nitrogen abundances need to be corrected for abundance-ratio effects. This is the topic of Section 5.2.3.