2.3.2 Weiss, Peletier, & Matteucci (1995)

Weiss et al. (1995) present models based on isochrones with non-solar abundance ratios. Specifically, these authors enhance the $\alpha$-elements (O, Mg, Si, etc.) relative to Fe, and compute models with [Mg/Fe] = 0, 0.45, and 0.62, and total metallicities Z in the range 0.02-0.07.

The conversion to observable quantities is empirical and based on the galactic bulge stars from Rich (1988). For these stars, the line indices ${ {\rm Mg}_2}$ and ${ <{\rm Fe}>}$ are fitted as function of (V-K) (the temperature indicator) and [M/H] $\equiv \log(Z/Z_\odot)$. In order to study the effect of non-solar abundance ratios, they simply replace [M/H] in the ${ {\rm Mg}_2}$ equation by [Mg/H], and [M/H] in the ${ <{\rm Fe}>}$ equation by [Fe/H]. In other words they assume, that ${ {\rm Mg}_2}$ only depends on the magnesium abundance, and that ${ <{\rm Fe}>}$ only depends on the iron abundance. They present evidence that the stars of Rich (1988) have [Mg/Fe] $\approx$ 0, which is a necessary condition for doing the above. We note, that the metallicity and the abundance ratios for the stars in the galactic bulge is not a settled issue. For example, Idiart, de Freitas Pacheco, & Costa (1996) found a mean abundance ratio [Mg/Fe] = 0.45.

The Salpeter (1955) IMF is used. Only ages of 12, 15 and 18 Gyr are given.

One of the results from Weiss et al. is, that the effect on the isochrone of changing [Mg/Fe] while keeping the total metallicity [M/H] constant is small compared with changing [M/H] by the same amount and keeping [Mg/Fe] constant. However, the changes in the former case are not totally negligible. This is shown by model 7 and 7H of Weiss et al. Model 7 has Z = 0.04 and [Mg/Fe] = 0.45, and is calculated in the way described above. Model 7H is a hybrid model, calculated as follows. First an isochrone is calculated for Z = 0.04 and [Mg/Fe] = 0. The resulting values of L and $T_{\rm eff}$ are used to calculate ${ {\rm Mg}_2}$ and ${ <{\rm Fe}>}$, but instead of inserting [Mg/Fe] = 0 in the equations for ${ {\rm Mg}_2}$ and ${ <{\rm Fe}>}$, the value [Mg/Fe] = 0.45 is used. The offsets (``7''-``7H'') are $\Delta({ {\rm Mg}_2},\log{ <{\rm Fe}>})$ = (-0.020,-0.005), (-0.032,-0.004), and (-0.017,-0.002) for ages of 12, 15, and 18 Gyr, respectively. We make the following two points: (1) That these offsets are small is another way of saying that the effect on the isochrones of abundance changes is small. This implies that the effect on ${\left( M/L \right)}$ is small (Weiss et al. do not list the changes in L, only in ${ {\rm Mg}_2}$ and ${ <{\rm Fe}>}$). We use this fact in our analysis (Sect. ) to derive [Mg/H] (and ages) from the ${ {\rm Mg}_2}$-$\log(M/L)$ diagram, and [Fe/H] (and ages) from the $\log { <{\rm Fe}>}$-$\log(M/L)$ diagram, both using the Vazdekis et al. (1996) models which have [Mg/Fe] = 0. To strictly do this, it is necessary that ${\left( M/L \right)}$ is not affected by non-solar abundance ratios. In addition, it is necessary that ${ {\rm Mg}_2}$ depends only on the magnesium abundance, and that ${ <{\rm Fe}>}$ depends only on the iron abundance. This is supported in part by the work of Tripicco & Bell (1995), cf. below. (2) Since these offsets are nevertheless both non-zero and varying with age and probably also with Z and the change in [Mg/Fe] (the latter two possibilities were not tested by Weiss et al.), the above-mentioned assumptions that we make in our analysis are only valid as a first approximation.

The effect of changing [Mg/Fe] while keeping [M/H] constant on ${ {\rm Mg}_2}$ and ${ <{\rm Fe}>}$ is large. This is not a surprise, especially since Weiss et al. assume that ${ {\rm Mg}_2}$ depends only on the magnesium abundance, and that ${ <{\rm Fe}>}$ depends only on the iron abundance. For Z = 0.02, the differences between [Mg/Fe] = 0.45 (model 5) and [Mg/Fe] = 0 (model 1) are $\Delta({ {\rm Mg}_2},\log{ <{\rm Fe}>})$ = (+0.036,-0.033), (+0.047,-0.047), and (+0.041,-0.063) for ages of 12, 15, and 18 Gyr, respectively, with the differences calculated as ``5''-``1''.

We want to compare the large grid of models from Vazdekis et al., which have [Mg/Fe] = 0, with the somewhat smaller grid of models from Weiss et al. with [Mg/Fe] > 0 to illustrate the effect of [Mg/Fe] > 0. It turns out that the Weiss et al. models with [Mg/Fe] = 0 do not give quite the same values of ${ {\rm Mg}_2}$ and $\log { <{\rm Fe}>}$ as those from Vazdekis et al. If we define $\Delta \equiv$ (Vazdekis et al.) - (Weiss et al.), we find $\Delta({ {\rm Mg}_2}) = -0.003$ and $\Delta({\log { <{\rm Fe}>}}) = -0.050$ for Z = 0.02 and age = 15 Gyr, and $\Delta({ {\rm Mg}_2}) = -0.052$ and $\Delta({\log { <{\rm Fe}>}}) = -0.065$ for Z = 0.05 and age = 15 Gyr. Weiss et al. do not have Z = 0.05, so we have calculated the above values by a linear interpolation between their Z = 0.04 and Z = 0.07 values. For the Vazdekis et al. models we used the bimodal $\mu = 1.35$ IMF. However, using the Salpeter IMF instead has very little impact on the offsets. Thus, they must be caused by a difference in the theoretical isochrones and/or the conversion to the observable quantities. We add the found offsets to the Weiss et al. predictions (also for their [Mg/Fe] > 0 models) when we compare with Vazdekis et al. We use the same offset for all the three ages given by Weiss et al. This was done in the same way by J97.