7.4.1 The ${ {\rm Mg}_2}$-$\sigma$ Relation

The existence of a tight correlation between ${ {\rm Mg}_2}$ and ${\log\sigma}$ is well established (e.g. Burstein et al. 1988, Bender et al. 1993). It is remarkable that the stellar population ( ${ {\rm Mg}_2}$) is so closely connected with the structural properties ($\sigma$).

For our samples, ${ {\rm Mg}_2}$ has been measured for 42 of the HydraI galaxies and 113 of the Coma galaxies. We test if the two samples follow the same ${ {\rm Mg}_2}$-$\sigma$ relation as follows. First we fit the two samples individually. We compare the slopes and find a non-significant difference of $\Delta({\rm slope}) = 0.009 \pm 0.043$. Second we fit the two samples together, with the zero point for each sample left free. We find a non-significant difference of $\Delta(\mbox{zero point}) = -0.002 \pm 0.006$. We conclude that the galaxies in the central parts of the HydraI cluster and the Coma cluster follow the same ${ {\rm Mg}_2}$-$\sigma$ relation. We then fit the two samples together and get

$$\arraycolsep=2pt % \begin{array}{lllllll} {\rm HydraI+Coma} &... ...=0.028 \quad & N=155 \\ & & & \pm & 0.014 & & \\ \end{array}$$ (7.16)

${ {\rm Mg}_2}$ versus ${\log\sigma}$ is shown in Figure .

 

The slope that we find, $0.189 \pm 0.014$, is in agreement with other studies in the literature: 0.175 (Burstein et al. 1988), 0.20 (Bender et al. 1993), $0.196 \pm 0.016$ (JFK96), and $0.196 \pm 0.009$ (J97). Note that the Bender et al. value is for all types of `dynamically hot galaxies' (DHGs), from dwarf spheroidals to giant ellipticals, with ${M_{\rm B_T}}$ in the impressive range -8 to -24 mag. For comparison, our 155 galaxies fall in the categories `giant ellipticals' ( ${M_{\rm B_T}}$$\le$ $-20\hbox{$.\!\!^{\rm m}$ }5$; 66 galaxies), `intermediate ellipticals' ( $-20\hbox{$.\!\!^{\rm m}$ }5$< ${M_{\rm B_T}}$$\le$ $-18\hbox{$.\!\!^{\rm m}$ }5$; 88 galaxies), and `bright dwarf ellipticals' ( ${M_{\rm B_T}}$> $-18\hbox{$.\!\!^{\rm m}$ }5$; 1 galaxy) in the classification scheme of Bender et al. A typical E/S0 color of $(B-r) = 1\hbox{$.\!\!^{\rm m}$ }15$ was used.

We estimate the intrinsic scatter ${\sigma_{\rm int}}$ by subtracting the uncertainties on ${ {\rm Mg}_2}$ and ${\log\sigma}$ (see Table ) in quadrature from ${\sigma_{\rm fit}}$. We find ${\sigma_{\rm int}}= (0.028^2-[(0.189 \cdot 0.032)^2+0.013^2])^{1/2} = 0.024$. A large part of the scatter is caused by the five very deviating galaxies, which are marked in Fig. . For all but D62, we have a possible explanation why they deviate, cf. the caption to the figure. If these five galaxies are omitted, the scatter ${\sigma_{\rm fit}}$ decreases from 0.028 to 0.020. The latter value corresponds to ${\sigma_{\rm int}}= 0.014$. Bender et al. (1993) found ${\sigma_{\rm int}}= 0.018$ for their DHG sample. ${ {\rm Mg}_2}$ depends on both age and metallicity. For the Vazdekis et al. (1996) models with a bi-modal IMF with high mass slope $\mu = 1.35$, the predictions can be approximated by ${ {\rm Mg}_2}= 0.12 \, {\log {\rm age}}+ 0.19 \, {{\rm [M/H]}}+ 0.14$ (J97; Eq. ) for ages > 5 Gyr. If the intrinsic scatter ( ${\sigma_{\rm int}}= 0.024$) in the ${ {\rm Mg}_2}$-$\sigma$ relation is due to scatter in age alone, the scatter in ${\log {\rm age}}$ at a given ${\log\sigma}$ is 0.20 dex (46% in age). If it is due to scatter in metallicity alone, the scatter in ${{\rm [M/H]}}$ is 0.13 dex (30% in Z). These considerations are revisited in Sect.  (p. ) in the light of the age-metallicity-sigma relation. Bender et al. (1993) found the allowed scatter in age or Z to be 15% for both. This value was based on their ${\sigma_{\rm int}}= 0.018$ and different population synthesis models than the ones we use.