4 Treatment of abundance ratios

Theory of chemical enrichment predicts that stellar populations created in a short burst of star formation show elevated magnesium-to-iron abundance ratios while extended periods of star formation lead to roughly solar abundance ratios (e.g., Tinsley 1979; Worthey et al. 1992; Worthey 1998; Thomas et al. 1999). This is generally explained by either a delay in the production of Fe-peak elements with respect to $\alpha$-elements due to the different timescales of SN Ia and SN II, or by a star-burst with an initial mass function skewed to massive stars. For example, many observations of bright elliptical galaxies in clusters show indeed elevated magnesium-to-iron ratios indicating a rapid star formation (e.g., Peletier 1989; Worthey et al. 1992; Davies et al. 1993; Kuntschner 2000).

GCs represent simple stellar populations (hereafter SSP, i.e., a unique age and chemical composition) since all the stars of one GC are thought to be created in a single star formation event. But it is not known a priori whether the stars of a given GC are formed out of Fe-deficient gas clouds, which have been only enriched by SN II producing little Fe, or out of an already well mixed interstellar medium harbouring the products from both SN Ia (main producer of Fe) and SN II. For GC systems with a bimodal colour distribution like in NGC 3115, all scenarios that have been proposed to explain the origin of the red metal-rich GCs start from the principle that the red population is formed in a separate star formation event (e.g., Ashman & Zepf 1992; Forbes et al. 1997). In a naive star-formation scenario, where the red clusters form from the well-mixed interstellar medium they should show solar abundance ratios.

Figure 6: Dependence of Mg  b, [MgFe], $\langle$Fe$\rangle$ and H$\beta$ on [Fe/H] and age for models with different [Mg/Fe] as predicted by Thomas et al. (2002a). The thick lines and open diamonds stand for $\rm [Mg/Fe] =+0.3$ whereas the thin lines and plus symbols represent $\rm [Mg/Fe] =0.0$. The respective line-style and average measurement error for the different indices is given on the left hand side of panels  a) and  c). For the [MgFe] index the two model lines (for $\rm [Mg/Fe] =0.0$ and 0.3) overlap almost completely and therefore the index is almost independent of the Mg to Fe abundance ratios.

Most of the currently available stellar population models which can be used to investigate the abundance ratios of extragalactic objects are based on stellar libraries from our own Galaxy. This has the disadvantage that, particularly at sub-solar metallicities, galactic disk stars show super-solar abundance ratios for many $\alpha$-elements (Edvardsson et al. 1993; McWilliam 1997). Therefore, without correction, the model predictions will be biased towards super-solar abundance ratios (e.g., Borges et al. 1995; Kuntschner 2000; Thomas et al. 2002a).

Thomas et al. (2002a) (see also Thomas et al. 2002b) provide new models which take the stellar library biases into account and can predict line-strengths for solar abundance as well as non-solar abundance ratios over a large metallicity range ( $-2.25 \le {\rm [Fe/H]} \le 0.35$). Since our paper is among the first that make use of these new models for studying GC spectra, we first explore systematically how the three parameters age, metallicity and abundance ratio ([Mg/Fe]) affect the absorption line-strengths of SSPs. For this purpose we plot in Fig. 6 the model predictions for the line-strengths of the often used metallicity indicators Mg b, $\langle$Fe$\rangle$ and [MgFe] and the age sensitive Balmer line H$\beta$ as a function of metallicity, age and [Mg/Fe].

Figure 7: Probing the [Mg/Fe] ratios of globular clusters in a Mg  b vs. $\langle$Fe$\rangle$ diagram. Panel  a): Our sample of globular clusters in NGC 3115 is shown as open triangles (blue clusters) and open circles (red clusters). The large filled square represents the centre of NGC 3115 taken from Trager et al. (1998) and the small filled squares represent the data of Fisher et al. (1996) which cover radii up to 40 $^{\prime \prime }$ along the major axis. Panel  b): Milky Way globular clusters observed with the Lick/IDS instrumentation. Overplotted as stars are early-type galaxies in the Fornax cluster with central velocity dispersion $\sigma \ge 75$ km s-1 (Kuntschner 2000). Panel  c): Globular clusters in M 31 observed with the Lick/IDS instrumentation. The filled triangle represents the centre of M 31 taken from Trager et al. (1998). Overplotted in all panels are models by Thomas et al. (2002a) with abundance ratios of $\rm [Mg/Fe]=0.0.$, 0.3, 0.5 as indicated in the left panel. The models span a range in age (3-12 Gyr) and metallicity ( ${\rm [Fe/H]} = -2.35~{\rm to}~ +0.3$).

The effects of non-solar abundance ratios (thick lines) with respect to solar ratio model predictions (thin lines) can be clearly seen for the metallicity sensitive indices Mg b and $\langle$Fe$\rangle$. At a given age and all metallicities (Fig. 6a), the Mg b index is predicted to be stronger for super-solar [Mg/Fe]. The difference between solar ratio and non-solar ratio models increases with increasing metallicity. This behaviour is reversed for the $\langle$Fe$\rangle$ index (see also Trager et al. 2000). A similar effect can be observed when we fix [Fe/H] but vary age (Fig. 6b). Remarkably, the [MgFe] index, the geometric mean of the Mg b and $\langle$Fe$\rangle$ indices, does not show any significant dependence on abundance ratios, at least not within the current framework of the models. This has the unique advantage that [MgFe] can be used as an empirical mean metallicity indicator with negligible dependence on the abundance ratios (see also Kuntschner et al. 2001). Similarly, H$\beta$ is hardly affected by [Mg/Fe] with a small increase in line-strength for larger [Mg/Fe] ratios (Figs. 6c and d).

However, the H$\beta$ index shows a noteworthy complication at low metallicities ( ${\rm [Fe/H]} \le -1.35$) and ages >8 Gyrs. Here the models predict a non-monotonic decrease of the H$\beta$ index with increasing age (see Fig. 6d, models for ${\rm [Fe/H]} = -2.25$ and -1.35). Therefore, in this age and metallicity range a given measurement of H$\beta$ and [MgFe] indices does not correspond to one unique age but can in fact be consistent with a range of ages. The ambiguity arises between a genuine measurement of the turn-off temperature by H$\beta$ and the appearance of blue horizontal branch stars in old, metal poor stellar systems which will start to increase the H$\beta$ index, mimicking a younger age (Lee et al. 2000; Maraston & Thomas 2000; Beasley et al. 2002b). The overall result is a "crossing'' of iso-age lines at ages larger than 8 Gyr and low metallicities. In Sect. 5.2 where we will determine our best age estimates, the above effect will be taken into account.