2.3.1 Vazdekis et al. (1996)

The isochrones of the Padova group (Bertelli et al. 1994) is used. The conversion to observable quantities is based on empirical studies. For the line indices the conversion is done by the fitting functions of Worthey et al. (1994), a study based on field and cluster stars. Several IMFs are offered. The unimodal IMF is a plain power law with slope $\mu$ (with $\mu = - \Gamma$ in the above notation). The bimodal IMF is a constant below 0.2 $M_\odot$, a power law with slope $\mu$ above 0.6 $M_\odot$, and a spline in the interval 0.2-0.6 $M_\odot$. The metal abundance ratios are solar, e.g. [Mg/Fe] = 0. Predictions are given for 3 values of the total metallicity Z in the range 0.008-0.05, and 15 values of the age in the range 1-17 Gyr. We prefer to use the total metal abundance relative to solar [M/H] $\equiv \log(Z/Z_\odot)$ (with $Z_\odot = 0.02$) instead of Z, since the observable quantities vary almost linearly in [M/H]. For the same reason, we usually use ${\log {\rm age}}$ instead of age.

Vazdekis et al. show that the bimodal IMF with high mass slope $\mu = 1.35$ gives a reasonable fit to data from Scalo (1986). This is the model we will be using as our basic model. Predictions from this model for the four observables ${ {\rm Mg}_2}$, $\log { <{\rm Fe}>}$, ${\log(M/L_{\rm r})}$, and (B-r) are shown on Fig. . The general trend is, that all these four quantities increase with both age and metallicity.

 

J97 found that the predictions from this model could be well approximated by the following analytical expressions for ages of 5 Gyr or larger

   

$${ {\rm Mg}_2} \approx 0.12 \, {\log {\rm age}}+ 0.19 \, {{\rm [M/H]}}+ 0.14$$ (2.34)

$$\log { <{\rm Fe}>} \approx 0.12 \, {\log {\rm age}}+ 0.25 \, {{\rm [M/H]}}+ 0.34$$ (2.35)

$${\log(M/L_{\rm r})} \approx 0.63 \, {\log {\rm age}}+ 0.26 \, {{\rm [M/H]}}- 0.16$$ (2.36)

These relations are overplotted in Fig. .

Vazdekis et al. give colors involving Kron-Cousins R. These were transformed to Gunn r using the constant offset $(r-R)=0\hbox{$.\!\!^{\rm m}$ }354$ from Jørgensen (1994). Vazdekis et al. only list the mass-to-light ratio in Johnson V, but the mass-to-light ratio in e.g. Gunn r is readily calculated as

$$\frac{ (M/L_{\rm r}) } { (M/L_{\rm r})_\odot } = \frac{ (M/L... ...(V-r)-(M_{{\rm V},\odot}-M_{{\rm r},\odot})\right]} \enspace .$$ (2.37)

We used the following solar absolute magnitudes: $M_{{\rm V},\odot} = 4\hbox{$.\!\!^{\rm m}$ }84$, $M_{{\rm R},\odot} = 4\hbox{$.\!\!^{\rm m}$ }48$ (i.e. $M_{{\rm r},\odot} = 4\hbox{$.\!\!^{\rm m}$ }834$), $M_{{\rm B},\odot} = 5\hbox{$.\!\!^{\rm m}$ }41$, and $M_{{\rm U},\odot} = 5\hbox{$.\!\!^{\rm m}$ }60$. All are from Worthey (1994), except $M_{{\rm B},\odot}$, which is from Gonzáles (1993).