7.5.1 ${{\rm [Mg/H]}}$, ${{\rm [Fe/H]}}$, ${{\rm [Mg/Fe]}}$, and ages versus $\sigma$ and Mass

In Figure  we plot ${{\rm [Mg/H]}}$, ${{\rm [Fe/H]}}$, and ${{\rm [Mg/Fe]}}$ versus ${\log\sigma}$. For the HydraI sample, the following trends are seen: ${{\rm [Mg/H]}}$ increases with ${\log\sigma}$, ${{\rm [Fe/H]}}$ is independent of ${\log\sigma}$, and therefore ${{\rm [Mg/Fe]}}$ increases with ${\log\sigma}$. The Coma sample is compatible with the same pattern, although the trends for ${{\rm [Mg/H]}}$ and ${{\rm [Fe/H]}}$ are somewhat less clear. But still ${{\rm [Mg/Fe]}}$ is highly correlated with ${\log\sigma}$. It should be recalled, that the part of the Coma sample that has ${ <{\rm Fe}>}$ data is not magnitude limited.

${\log\sigma}$ did not directly enter the calculation of ${{\rm [Mg/H]}}$ and ${{\rm [Fe/H]}}$. However, $\log(M/L)$ depends on ${\log\sigma}$ as ${\log(M/L_{\rm r})}= 2 \log \sigma - \log {< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}- \log {r_{\rm e}}- 0.733$ (Eq. ). The Vazdekis et al. models can be approximated by ${\log(M/L_{\rm r})}= 0.63 \, {\log {\rm age}}+ 0.26 \, {{\rm [M/H]}}- 0.16$ (J97; Eq. ). Measurement errors in ${\log\sigma}$ can therefore cause a slope in the ${{\rm [M/H]}}$- ${\log\sigma}$ diagrams of $\Delta{{\rm [M/H]}}/\Delta\log\sigma = 7.7$. However, since the measurement errors in ${\log\sigma}$ are small compared to the range in ${\log\sigma}$, the effect should be small. Further, ${{\rm [Mg/Fe]}}$ is not affected since the effect cancels out.

 

For the combined HydraI+Coma sample (N=83), the correlation between ${{\rm [Mg/Fe]}}$ and ${\log\sigma}$ is very significant, a Spearman rank order test gives ${P_{\rm no\;corr.}}< 0.01$%. A fit to the HydraI+Coma sample with the sum of the absolute residuals in ${{\rm [Mg/Fe]}}$ minimized gives the relation

$$\arraycolsep=2pt % \begin{array}{lllllll} {\rm HydraI+Coma} &... ...t}}=0.25 \quad & N=83 \\ & & & \pm & 0.23 & & \\ \end{array}$$ (7.26)

(A least squares fit gives almost the same, ${{\rm [Mg/Fe]}}= (0.67 \pm 0.20) - 1.34$.) We choose to minimize the residuals in ${{\rm [Mg/Fe]}}$ since we want to predict ${{\rm [Mg/Fe]}}$ from ${\log\sigma}$. We find, that when ${\log\sigma}$ increases by 0.4 dex (e.g. from $\sigma = 100\,{\rm km}\,{\rm s}^{-1}$ to $\sigma \approx 250\,{\rm km}\,{\rm s}^{-1}$), ${{\rm [Mg/Fe]}}$ increases by 0.3 dex. This is in agreement with J97, who found a 0.3-0.4 dex increase for the same ${\log\sigma}$ interval. This was based on the different slopes of the ${ {\rm Mg}_2}$-$\sigma$ and the ${ <{\rm Fe}>}$-$\sigma$ relations and the analytical approximations to the predictions from the Vazdekis et al. models.

In Figure  we plot ${{\rm [Mg/H]}}$, ${{\rm [Fe/H]}}$, and ${{\rm [Mg/Fe]}}$ versus ${\log({\rm Mass})}$. The relations are more noisy than for ${\log\sigma}$. Still, for the combined HydraI+Coma sample, the correlation between ${{\rm [Mg/Fe]}}$ and ${\log({\rm Mass})}$ is significant at the 2 sigma level, we find ${P_{\rm no\;corr.}}= 5.5$%. It seems that the most massive galaxies ( , or ) have a smaller variation in ${{\rm [Mg/H]}}$, ${{\rm [Fe/H]}}$, and ${{\rm [Mg/Fe]}}$ than galaxies with lower mass, but this could be due to the small number of objects. Also galaxies with high velocity dispersion (say $\log \sigma > 2.3$) have a smaller scatter than galaxies with lower velocity dispersion, but the division is not as pronounced as for the mass.

 

What are the implications of ${{\rm [Mg/Fe]}}> 0$? Worthey, Faber, & Gonzáles (1992) found that ${{\rm [Mg/Fe]}}$ was larger than zero in giant ellipticals, and that ${{\rm [Mg/Fe]}}$ reached 0.2-0.3 dex for the average strongest-lined galaxies. They reached this conclusion by comparing data with models in the ${ <{\rm Fe}>}$- ${ {\rm Mg}_2}$ diagram. These authors discussed the following possible explanations for ${{\rm [Mg/Fe]}}> 0$. Magnesium and iron are preferably produced in supernovae (SNe) of type II and Ia, respectively. Therefore, a change in the fraction (SN II)/(SN Ia) will give a change in ${{\rm [Mg/Fe]}}$. The following three scenarios can give ${{\rm [Mg/Fe]}}> 0$.
1. Different time scales for star formation. The progenitor stars of type II SNe are more massive and short-lived than those of type Ia SNe. Therefore, if star formation is fast, a large fraction of the total amount of gas available for star formation will have been processed by type II SNe and locked up in long-lived stars before the first generation of type Ia SNe after $\sim$ 1 Gyr will enrich the interstellar medium (ISM) with iron. At this point, only a small fraction of gas is left for new long-lived stars to be formed out of this iron enriched material.
2. A variable IMF slope. A smaller IMF slope $\mu$ (i.e. a more flat IMF) results in the formation of more massive stars, which give rise to more type II SNe.
3. Selective loss mechanisms. If for some reason a galactic wind of some sort would retain magnesium with greater efficiency than iron, then it is also possible to get ${{\rm [Mg/Fe]}}> 0$. In the standard picture of SN-driven winds, the outcome is ${{\rm [Mg/Fe]}}< 0$, in contradiction to the observations.

Any viable theory for star formation in elliptical galaxies should be able to explain not only ${{\rm [Mg/Fe]}}> 0$ per se, but also the correlation between ${{\rm [Mg/Fe]}}$ and velocity dispersion (or mass).

Also the ages are correlated with ${\log\sigma}$ and ${\log({\rm Mass})}$, see Fig.  and . For the combined HydraI+Coma sample, Spearman rank order tests give ${P_{\rm no\;corr.}}<$ 0.01% for ${\log {\rm age}_{\rm Mg}}$ vs. ${\log\sigma}$, ${P_{\rm no\;corr.}}=$ 0.04% for ${\log {\rm age}_{\rm Fe}}$ vs. ${\log\sigma}$, ${P_{\rm no\;corr.}}<$ 0.01% for ${\log {\rm age}_{\rm Mg}}$ vs. ${\log({\rm Mass})}$, and ${P_{\rm no\;corr.}}<$ 0.01% for ${\log {\rm age}_{\rm Fe}}$ vs. ${\log({\rm Mass})}$. Also for the ages we find that the scatter is smaller for the most massive galaxies ( , or ).

 

 

In Fig.  we plot ${{\rm [Mg/H]}}$, ${{\rm [Fe/H]}}$, ${{\rm [Mg/Fe]}}$, ${\log {\rm age}_{\rm Mg}}$, and ${\log {\rm age}_{\rm Fe}}$ versus total absolute magnitude in Gunn r, ${M_{\rm r_T}}$. Shown on the figure is the line ${M_{\rm r_T}}= -23\hbox{$.\!\!^{\rm m}$ }1$. JF94 found this line to demarcate two classes of E and S0 galaxies. The E and S0 galaxies fainter than this limit were best fitted by a model with 10% of the galaxies being diskless, and 90% of the galaxies being drawn from a uniform distribution of relative disk luminosity $L_{\rm D}/L_{\rm tot}$. The E (no S0 galaxies found!) brighter than this limit were all diskless.

It is seen from Fig.  that also for the five quantities studied here, there is a striking difference in properties for galaxies fainter and brighter than approximately ${M_{\rm r_T}}= -23\hbox{$.\!\!^{\rm m}$ }1$. The brighter galaxies show a smaller scatter than the fainter galaxies. The brighter galaxies have an old stellar population, with ${{\rm [Mg/H]}}$ a bit above average, ${{\rm [Fe/H]}}$ a bit below average, and thus ${{\rm [Mg/Fe]}}$ somewhat above average.