7.5 Ages, Metallicities, and Abundance Ratios

In this section we use the single-age single-metallicity stellar population models from Vazdekis et al. (1996) to infer luminosity weighted mean ages, metallicities, and abundance ratios from the data. We use the models with a bi-modal IMF with high mass slope $\mu = 1.35$.

It is not surprising that from a single observational quantity, such as (B-r), it is not possible to determine both age and metallicity (e.g. Worthey 1994). For example, a color of $(B-r) = 1\hbox{$.\!\!^{\rm m}$ }15$, which is typical for E and S0 galaxies, can be matched by both a low age and high metallicity, say 3 Gyr and Z = 0.05, an intermediate age and solar metallicity, say 8 Gyr and Z = 0.02, and a high age and low metallicity, say 13-17 Gyr and Z = 0.008. The numbers quoted are from the Vazdekis et al. models.

Given two observational quantities we could hope to determine both the age and metallicity. Unfortunately, for a number of such color-color, color-index, and index-index diagrams the effects of age and metallicity are nearly degenerate (e.g. Worthey 1994, Faber et al. 1995).

It turns out, that in the two-dimensional ${ {\rm Mg}_2}$-$\log(M/L)$ and $\log { <{\rm Fe}>}$-$\log(M/L)$ diagrams the effects of age and metallicity are not degenerate. This was mentioned by Faber et al. (1995).

We assume homology and are then able to calculate ${\log(M/L_{\rm r})}$ from the data (Eq. ; cf. Sect. , p. ). Our data are plotted in the ${ {\rm Mg}_2}$- ${\log(M/L_{\rm r})}$ and $\log { <{\rm Fe}>}$- ${\log(M/L_{\rm r})}$ diagrams in Fig. . Overplotted are the predictions from the Vazdekis et al. models. The first thing to note is that the models span the data quite well, the measurement errors taken into account. We are free to shift the values of $\log(M/L)$ up and down since they depend on the two unknown quantities H₀ and the fraction of dark matter. The used values of $H_0 = 50\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$ and $M_{\rm total} = 10 M_{\rm luminous}$ give a good match to the data, and we do not apply any offset.

 

The Vazdekis et al. models predict $\log(M/L)$, ${ {\rm Mg}_2}$, and ${ <{\rm Fe}>}$ for a grid of 45 (age,[M/H]) points, with the 15 age values ranging from 1.00 to 17.38 Gyr, and the three [M/H] values being -0.4, 0.0, and 0.4. Recall that ${{\rm [M/H]}}\equiv \log(Z/Z_\odot)$, with $Z_\odot = 0.02$. Note, that not all the age values are shown on Fig. . The inverse problem, i.e. given observed values of $\log(M/L)$ and ${ {\rm Mg}_2}$ (or $\log(M/L)$ and ${ <{\rm Fe}>}$) determine the corresponding age and metallicity, is a question of interpolation in an irregular grid. That the grid is irregular is apparent in Fig. .

We did this irregular interpolation in three steps. First, a Delaunay triangulation of the irregular grid points was established. Second, a large (say 1000 $\times$ 1000) regular grid of interpolated or extrapolated values of both ${\log {\rm age}}$ and [M/H] was calculated, with the grid being chosen to span the data. The interpolation and extrapolation was done using the Akima's quintic polynomials. Third, a standard bilinear interpolation was used to get the final values of ${\log {\rm age}}$ and [M/H]. These calculations were done using IDL (Interactive Data Language). For details, see the help pages for triangulate and trigrid. Estimates of uncertainties on ${\log {\rm age}}$ and [M/H] were obtained by in turn keeping $\log(M/L)$ and ${ {\rm Mg}_2}$ (or ${ <{\rm Fe}>}$) constant while varying the other by plus/minus the observational error, calculating ${\log {\rm age}}$ and [M/H] for those four points, and taking half the min-max variation as the estimate of uncertainty. For the interpolation in the $\log { <{\rm Fe}>}$- ${\log(M/L_{\rm r})}$ diagram, we omitted the three galaxies with an uncertainty on $\log { <{\rm Fe}>}$ larger than 0.065.

To discuss the results from the interpolation in the ${ {\rm Mg}_2}$- ${\log(M/L_{\rm r})}$ and $\log { <{\rm Fe}>}$- ${\log(M/L_{\rm r})}$ diagrams, we introduce the following notation

    

$${{\rm [Mg/H]}} \equiv \mbox{${{\rm [M/H]}}$\space inferred from ${ {\rm Mg}_2}$ --$\log(M/L)$\space diagram}$$ (7.20)

$${\log {\rm age}_{\rm Mg}} \equiv \mbox{${\log {\rm age}}$\space inferred from ${ {\rm Mg}_2}$ --$\log(M/L)$\space diagram}$$ (7.21)

$${{\rm [Fe/H]}} \equiv \mbox{${{\rm [M/H]}}$\space inferred from $\log { <{\rm Fe}>}$ --$\log(M/L)$\space diagram}$$ (7.22)

$${\log {\rm age}_{\rm Fe}} \equiv \mbox{${\log {\rm age}}$\space inferred from $\log { <{\rm Fe}>}$ --$\log(M/L)$\space diagram}$$ (7.23)

We also define the two differential quantities

  

$${{\rm [Mg/Fe]}} \equiv {{\rm [Mg/H]}}- {{\rm [Fe/H]}}$$ (7.24)

$${\Delta \log {\rm age}_{\rm Mg/Fe}} \equiv {\log {\rm age}_{\rm Mg}}- {\log {\rm age}_{\rm Fe}}$$ (7.25)

The uncertainties on ${{\rm [Mg/Fe]}}$ and ${\Delta \log {\rm age}_{\rm Mg/Fe}}$ were calculated in the same way as for ${{\rm [Mg/H]}}$, ${\log {\rm age}_{\rm Mg}}$, ${{\rm [Fe/H]}}$, and ${\log {\rm age}_{\rm Fe}}$, i.e. by half the min-max variation over the four points described above. This takes into account the correlation between the errors caused by the fact that $\log(M/L)$ appears in both diagrams.

The above notation indicates our first order assumptions: we assume that the metallicity inferred from ${ {\rm Mg}_2}$ gives the magnesium abundance [Mg/H], and that the metallicity inferred from ${ <{\rm Fe}>}$ gives the iron abundance [Fe/H]. This is despite the fact that the Vazdekis et al. models have solar abundance ratios, including [Mg/Fe] = 0. That these are reasonable approximations is supported by the work of Tripicco & Bell (1995), and Weiss et al. (1995), as described in Sect.  (p. ). In summary, Tripicco & Bell (1995) found that the ${ {\rm Mg}_2}$ index depended strongly on the magnesium abundance, and that the ${ <{\rm Fe}>}$ index depended strongly on the iron abundance, although is was just as sensitive to changes in the total metallicity. Weiss et al. (1995) found from isochrones with [Mg/Fe] $\ne$ 0 that the effect on e.g. the luminosity of changing the abundance ratios while keeping the total metallicity constant was small.

If the models provided an adequate description of the data, the metallicity difference ${{\rm [Mg/Fe]}}$ and the age difference ${\Delta \log {\rm age}_{\rm Mg/Fe}}$ should be zero within their errors caused by the observational errors. However, this is not the case, and we already suspected this discrepancy from the $\log { <{\rm Fe}>}$- ${ {\rm Mg}_2}$ diagram, Fig.  (p. ). The metallicity difference ${{\rm [Mg/Fe]}}$ is large compared with the two metallicities, typically 70%, while the age difference ${\Delta \log {\rm age}_{\rm Mg/Fe}}$ is small compared with the two ages, typically only 5%. We regard the differences in ages as an indication of the limitations in the method, and use our ${{\rm [Mg/Fe]}}$ as an estimate of the true ${{\rm [Mg/Fe]}}$. Note, that ${\Delta \log {\rm age}_{\rm Mg/Fe}}$ has to be non-zero since ${{\rm [Mg/Fe]}}$ is non-zero, because $\log(M/L)$ appears in both diagrams. The Vazdekis et al. models for ages > 5 Gyr can be approximated by ${\log(M/L_{\rm r})}= 0.63 \, {\log {\rm age}}+ 0.26 \, {{\rm [M/H]}}- 0.16$ (J97; Eq. ). Applying this to the two diagrams, we get $0 = 0.63 \,{\Delta \log {\rm age}_{\rm Mg/Fe}}+ 0.26 \,{{\rm [Mg/Fe]}}$, or ${{\rm [Mg/Fe]}}= -2.42 \,{\Delta \log {\rm age}_{\rm Mg/Fe}}$. A fit to the combined HydraI+Coma sample (N=83) gives ${{\rm [Mg/Fe]}}= (-2.48 \pm 0.09) \,{\Delta \log {\rm age}_{\rm Mg/Fe}}+ 0.00$, in agreement with this. The scatter is small, ${\sigma_{\rm fit}}= 0.09$.

Note, that since ${{\rm [Mg/H]}}$, ${{\rm [Fe/H]}}$, and the ages all depend on $\log(M/L)$, and since we can only determine $\log(M/L)$ to within an offset (cf. above), these four quantities are also only determined to within an offset. For ${{\rm [Mg/Fe]}}$, on the other hand, the effect of this unknown offset cancels out, at least to some extend.

 

In Figure  we show histograms over the distribution of ${{\rm [Mg/H]}}$, ${{\rm [Fe/H]}}$, ${{\rm [Mg/Fe]}}$, ${\log {\rm age}_{\rm Mg}}$, ${\log {\rm age}_{\rm Fe}}$, and ${\Delta \log {\rm age}_{\rm Mg/Fe}}$. ${\Delta \log {\rm age}_{\rm Mg/Fe}}$ is included just to show that also the absolute range in in this quantity is rather small.

We tested the HydraI and the Coma data against each other by means of Kolmogorov-Smirnov tests. This test gives the probability ${P_{\rm same\;distr.}}$ that the two samples are drawn from the same underlying distribution. For the HydraI sample (N=41) and the Coma sample with ${ <{\rm Fe}>}$ data (N=42) we find ${P_{\rm same\;distr.}}$ in the range 26%-98% for the above six quantities. In other words, we do not find any significant differences between the HydraI and the Coma samples. For the HydraI sample and the full Coma sample (N=111) we find ${P_{\rm same\;distr.}}= 15$% for ${{\rm [Mg/H]}}$ and 41% for ${\log {\rm age}_{\rm Mg}}$; again we find no significant differences between the HydraI and the Coma samples.