7.5.2 The Galaxian Age-Metallicity Relation

In Figure  we plot ${{\rm [Mg/H]}}$, ${{\rm [Fe/H]}}$, and ${{\rm [Mg/Fe]}}$ versus ${\log {\rm age}_{\rm Mg}}$. Both ${{\rm [Mg/H]}}$ and ${{\rm [Fe/H]}}$ are highly (anti-)correlated with ${\log {\rm age}_{\rm Mg}}$. Part of the age-metallicity relation may be due to measurement errors. This is because the lines of constant age and metallicity are not quite perpendicular to each other in the ${ {\rm Mg}_2}$-$\log(M/L)$ and $\log { <{\rm Fe}>}$-$\log(M/L)$ diagrams (see Fig. ). To quantify this effect, Monte-Carlo simulations are needed. This is planned for a future extension of this work.

${{\rm [Mg/Fe]}}$ is not significantly correlated with ${\log {\rm age}_{\rm Mg}}$. For the combined HydraI+Coma sample, we find ${P_{\rm no\;corr.}}= 26$%.

 

A fit to the Mg and Fe age-metallicity relations for the combined HydraI+Coma sample gives

$$\arraycolsep=2pt % \begin{array}{lllll} {{\rm [Mg/H]}}= & - &... ... fit}}=0.19 \quad & N=152 \\ & \pm & 0.10 & & \\ \end{array}$$ (7.27)

and

$$\arraycolsep=2pt % \begin{array}{lllll} {{\rm [Fe/H]}}= & - &... ...m fit}}=0.19 \quad & N=83 \\ & \pm & 0.14 & & \\ \end{array}$$ (7.28)

The `mixed' relation with ${{\rm [Fe/H]}}$ and ${\log {\rm age}_{\rm Mg}}$, which is what is plotted in Fig. (c-d), gives

$$\arraycolsep=2pt % \begin{array}{lllll} {{\rm [Fe/H]}}= & - &... ... fit}}=0.19 \quad & N=152 \\ & \pm & 0.20 & & \\ \end{array}$$ (7.29)

We also tried to include a ${\log\sigma}$ term in the age-metallicity relations. Still for the combined HydraI+Coma sample, a fit gives

$$\arraycolsep=2pt % \begin{array}{lllllll} {{\rm [Mg/H]}}= & ... ...quad & N=152 \\ & \pm & 0.11 & \pm & 0.07 & & \\ \end{array}$$ (7.30)

and

$$\arraycolsep=2pt % \begin{array}{lllllll} {{\rm [Fe/H]}}= & ... ...\quad & N=83 \\ & \pm & 0.27 & \pm & 0.14 & & \\ \end{array}$$ (7.31)

The ${\log\sigma}$ terms are highly significant. The two relations look similar, but there is the important difference, that while ${{\rm [Mg/H]}}$ is correlated with ${\log\sigma}$ ( ${P_{\rm no\;corr.}}= 0.22$%), ${{\rm [Fe/H]}}$ is not significantly correlated with ${\log\sigma}$ ( ${P_{\rm no\;corr.}}= 36$%). Both ${\log {\rm age}_{\rm Mg}}$ and ${\log {\rm age}_{\rm Fe}}$ are correlated with ${\log\sigma}$ ( ${P_{\rm no\;corr.}}< 0.01$% and ${P_{\rm no\;corr.}}= 0.04$%, respectively). There is also the difference, that for the Mg relation the scatter decreases when we add a ${\log\sigma}$ term, while it increases for the Fe relation. We also fitted the `mixed' relation with ${{\rm [Fe/H]}}$ and ${\log {\rm age}_{\rm Mg}}$. The result is ${{\rm [Fe/H]}}= (1.66 \pm 30\,000)\log\sigma - (0.99 \pm 1.9)\,{\log {\rm age}_{\rm Mg}}- 2.78$, with ${\sigma_{\rm fit}}=0.36$. As can be seen from the bootstrap uncertainties, the relation is not well defined.

The above restates the result from Worthey, Trager, & Faber (1995), that (a) there is an age-metallicity relation with a large span in age, and (b) galaxies of higher velocity dispersion follow an age-metallicity relation at higher metallicity (or older age). These authors used the index C₂4668 and several Balmer line indices (probably ${ {\rm H}_{\beta}}$, ${{\rm H}_{\gamma_{\rm A}}}$, and ${{\rm H}_{\gamma_{\rm F}}}$) to derive mean metallicities and ages, not ${ {\rm Mg}_2}$ and ${\log(M/L_{\rm r})}$ (or ${ <{\rm Fe}>}$ and ${\log(M/L_{\rm r})}$) as we did. It is therefore encouraging that our result is in qualitative agreement with their result.

Worthey et al. report that they were not able to establish the slope nor the zero point of this age-metallicity-sigma relation. No doubt they could have made a fit to their data, so what they mean is probably that the different indices give different ages and metallicities. For example, they find that an Mg index gives a significantly different age than an Fe index. While we also find our two ages to be significantly different, the size of this difference is small. In accordance with this, the coefficients for ` ${\log {\rm age}}$' in Eq. () and () are not significantly different. We are not able to establish the true zero point.

We can now revisit two problems raised earlier, namely the interpretation of (a) the intrinsic scatter in the ${ {\rm Mg}_2}$-$\sigma$ relation, and (b) the similar intrinsic scatter in the FP in Gunn r, Johnson B, and Johnson U.

If we take the Mg-version of the age-metallicity-sigma relation (Eq. ) at face value and insert it in the analytical approximation to the predictions from the Vazdekis et al. models for ${ {\rm Mg}_2}$ (J97; Eq. ), we can eliminate either ${{\rm [Mg/H]}}$ or ${\log {\rm age}_{\rm Mg}}$. We get

$$\arraycolsep=2pt % \begin{array}{llllll} { {\rm Mg}_2}= & - &... ...rm [Mg/H]}}+ d \\ & \pm & 0.01 & & \pm & 0.01 \\ \end{array}$$ (7.32)

where the constants c and d depend on ${\log\sigma}$. This means that due to the relation between age and metallicity for a given sigma the ${ {\rm Mg}_2}$ index changes very little as either age or metallicity changes. Using these relations, the intrinsic scatter in the ${ {\rm Mg}_2}$-$\sigma$ relation of 0.024 translates into a ${\log {\rm age}}$ variation of 0.8 dex or a ${{\rm [M/H]}}$ variation of 0.6 dex, both at a given ${\log\sigma}$. This is much larger than the estimates obtained without the age-metallicity-sigma relation taken into account, 0.2 dex and 0.13 dex. Worthey et al. (1995) also reached the conclusion that when taking into account the age-metallicity relation, the intrinsic scatter in the ${ {\rm Mg}_2}$-$\sigma$ relation allowed for a larger variation in age than 15%. If we had excluded the 5 (out of 155) galaxies that have very large residuals from the ${ {\rm Mg}_2}$-$\sigma$ relation, the intrinsic scatter would be 0.014, which translates into either 0.45 dex in ${\log {\rm age}}$ or 0.35 dex in ${{\rm [M/H]}}$. The corresponding numbers without taking into account the age-metallicity-sigma relation are 0.13 dex and 0.08 dex, respectively.

In a similar manner, we insert the Mg-version of the age-metallicity-sigma relation (Eq. ) into the analytical approximations to the predictions from the Vazdekis et al. models for $\log(M/L)$ (Eq. -). When eliminating either ${{\rm [Mg/H]}}$ or ${\log {\rm age}_{\rm Mg}}$, the result is

$$\arraycolsep=2pt % \begin{array}{llllll} {\log(M/L_{\rm r})}=... ... [Mg/H]}}+ d_3 \\ & \pm & 0.04 & & \pm & 0.11 \\ \end{array}$$ (7.33)

where the constants c_i and d_i depend on ${\log\sigma}$. It is seen that the coefficients for ${\log {\rm age}_{\rm Mg}}$ and ${{\rm [Mg/H]}}$ vary much less with passband than when the age-metallicity-sigma relation is not taken into account, see Eq. ()-(), p. . Therefore, if we explain the intrinsic scatter in the FP (interpreted as the ${\left( M/L \right)}\propto M^b$ relation) by either an age variation at a given sigma and metallicity, or a metallicity variation at a given sigma and age, the scatter in $\log(M/L)$ is not expected to be very different in the different passbands, in agreement with the observations.

The intrinsic scatter in the ${\left( M/L \right)}\propto M^b$ relation is 0.103 dex in Gunn r. This translates into a variation in ${\log {\rm age}}$ of 0.24 dex, or a variation in ${{\rm [M/H]}}$ of 0.19 dex. This is substantially less than the variation needed to explain the intrinsic scatter in the ${ {\rm Mg}_2}$-$\sigma$ relation in the same way. Since we do not have a detailed understanding of the origin of these two relations, it might well be, that galaxy formation and evolution made ${ {\rm Mg}_2}$ be less well determined from $\sigma$ than ${\left( M/L \right)}$ from mass.