Mean ages and metal abundances are estimated for the stellar populations in a sample of 115 E and S0 galaxies in the central 64 ^ { \prime } \times 70 ^ { \prime } of the Coma cluster .
The estimates are based on the absorption line indices { { Mg } _ { 2 } } , { < \hskip { -3.0 pt } { Fe } \hskip { -3.0 pt } > } and { { H } { \beta } _ { G } } , and the mass-to-light ratios ( M/L ) .
Single stellar population models from Vazdekis et al .
were used to transform from the measured line indices and M/L ratios to mean ages and mean metal abundances ( [ Mg/H ] and [ Fe/H ] ) .
The non-solar abundance ratios [ Mg/Fe ] were taken into account by assuming that for a given age and iron abundance , a [ Mg/Fe ] different from solar will affect the { { Mg } _ { 2 } } index but not the M/L ratio or the { < \hskip { -3.0 pt } { Fe } \hskip { -3.0 pt } > } and { { H } { \beta } _ { G } } indices .
The derived ages and abundances are the luminosity weighted mean values for the stellar populations in the galaxies .

By comparing the mean ages derived from the { { Mg } _ { 2 } } - { { H } { \beta } _ { G } } diagram to those derived from the { { Mg } _ { 2 } } -M/L diagram , we estimate the variations of the fraction of dark matter .
Alternatively , the difference between the two estimates of the mean age may be due to variations in the initial mass function or to any non-homology of the galaxies .

The distributions of the derived mean ages and abundances show that there are real variations in both the mean ages and in the abundances .
We find an intrinsic rms scatter of [ Mg/H ] , [ Fe/H ] and [ Mg/Fe ] of 0.2 dex , and an intrinsic rms scatter of the derived ages of 0.17 dex .
The magnesium abundances [ Mg/H ] and the abundance ratios [ Mg/Fe ] are both strongly correlated with the central velocity dispersions of the galaxies , while the iron abundances [ Fe/H ] are uncorrelated with the velocity dispersions .
Further , [ Mg/H ] and [ Fe/H ] are strongly anti-correlated with the mean ages of the galaxies .
This in not the case for [ Mg/Fe ] .

We have tested whether the slopes of the scaling relations between the global parameters for the galaxies ( the { { Mg } _ { 2 } } - \sigma relation , the { < \hskip { -3.0 pt } { Fe } \hskip { -3.0 pt } > } - \sigma relation , the { { H } { \beta } _ { G } } - \sigma relation and the Fundamental Plane ) are consistent with the relation between the ages , the abundances and the velocity dispersions .
We find that all the slopes , except the slope of the Fundamental Plane , can be explained in a consistent way as due to a combination between variations of the mean ages and the mean abundances as functions of the velocity dispersions .
The slope of the Fundamental Plane is “ steeper ” than predicted from the variations in the ages and abundances .

Because of the correlation between the mean ages and the mean abundances , substantial variations in the ages and the abundances are possible while maintaining a low scatter of all the scaling relations .
When this correlation is taken into account , the observed scatter of the scaling relations is consistent with the rms scatter in derived the ages and abundances at a given velocity dispersion .