7.4.4 Color Relations

$\sigma$, ${ {\rm Mg}_2}$, and ${ <{\rm Fe}>}$ are all measured in the central part of the galaxy. Specifically, the HydraI spectroscopical parameters, which originates from four different observing runs, were measured within apertures of equivalent radius $1\hbox{$.\!\!^{\prime\prime}$ }3$- $2\hbox{$.\!\!^{\prime\prime}$ }35$, corresponding to 0.5-0.9 kpc (here and below we assume $H_0 = 50\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$). The measurements were aperture corrected to an aperture of radius 1.19 kpc, i.e. the standard JFK95b aperture of diameter 1.19 h^-1 kpc (note the h^-1 factor), cf. Sect.  and . The HydraI effective colors ${(B-r)_{\rm e}}$, ${(U-r)_{\rm e}}$, and ${(U-B)_{\rm e}}$ on the other hand, are measured within a radius of 0.8-53 kpc, typically 3.6 kpc. The fact that tight correlations exist between e.g. central velocity dispersion and global color has been interpreted by Burstein et al. (1988), Franx & Illingworth (1990), and Bender et al. (1993) as an indication that the galaxy to galaxy variations in radial gradients in colors and line indices are small. However, if the size of the gradient is correlated with the central value, this conclusion does not necessarily hold. Since the relative uncertainties on the radial gradients currently available in the literature are quite large, it is not yet clear whether the radial gradients are correlated with e.g. central velocity dispersion or global color.

For the 45 galaxies in the HydraI sample we have the color ${(B-r)_{\rm e}}$, and for 19 of these galaxies we also have the colors ${(U-r)_{\rm e}}$ and ${(U-B)_{\rm e}}$. All the colors correlate strongly with ${\log\sigma}$. We find the following color-$\sigma$ relations

$$\arraycolsep=2pt % \begin{array}{lllllll} {\rm HydraI} & : & ... ...}=0.062 \quad & N=18 \\ & & & \pm & 0.126 & & \\ \end{array}$$ (7.18)

We have excluded the galaxy R293, since the spectrum of this galaxy has very low S/N (only 13 per Å; cf. Table , p. ), and since it deviates strongly from the relation defined by the other galaxies. The galaxy R295/E437G15, which also deviates substantially, was not excluded - however, the fit is not sensitive to this. The color-$\sigma$ relations are shown in Fig. (a-c).

Franx & Illingworth (1990) found the relations $(B-R) = (0.70\pm0.3) \log \sigma + 0.22$, and $(U-R) = (1.42\pm0.3) \log \sigma - 0.92$. It is not specified which photometric system the R-magnitudes are on. By comparing with our data, we find that it could well be Johnson R and most likely not Kron-Cousins R. The difference between the two is $R_{\rm J} - R_{\rm C} = -0\hbox{$.\!\!^{\rm m}$ }25$, based on $R_{\rm J} - R_{\rm C} = -0.12 (B - R_{\rm C}) -0.07$ (Davis et al. 1985) and a typical E/S0 color of $(B - R_{\rm C}) = 1\hbox{$.\!\!^{\rm m}$ }5$. It is seen from Fig. (a-b) that the dot-dashed lines matches our data reasonably well, while a line $\approx$ $0\hbox{$.\!\!^{\rm m}$ }25$ above would match our data less well, especially for (B-r). We conclude that the R-magnitudes that Franx & Illingworth used for their color-$\sigma$ relations are probably close to Johnson R, but that the zero point could be a bit off.

Under the assumption that the color-$\sigma$ relations from Franx & Illingworth refer to Johnson R, we can transform their (B-R) relation by means of Eq. (, p. ) to the relation ${(B-r)}= (0.63 \pm 0.27) \log\sigma - 0.22$. This relation is shown as the dot-dashed line in Fig. (a). To transform their (U-R) relation, we combine the relation $(U - R_{\rm J}) = 1.07 (U - R_{\rm C}) + 0.08$ from Davis et al. (1985) and the relation $r - R_{\rm C} = 0.354$ from Jørgensen (1994) to give the relation $(U - R_{\rm J}) = 1.07 (U - r) + 0.46$. Then the Franx & Illingworth (U-R)-$\sigma$ relation becomes ${(U-r)}= (1.33 \pm 0.28) \log\sigma - 1.29$. This relation is shown as the dot-dashed line in Fig. (b). Finally, we subtract their two color-$\sigma$ relations to give ${(U-B)}= 0.72 \log\sigma -1.14$. This relation is shown as the dot-dashed line in Fig. (c).

The ${\log\sigma}$ coefficients in the Franx & Illingworth color-$\sigma$ relations are determined as the geometrical mean of the coefficients from two least squares fits, one in each direction. When we fit our data in the same way, we get ${(B-r)_{\rm e}}= (0.30 \pm 0.07) \log \sigma + {\rm const}$, and ${(U-r)_{\rm e}}= (0.83 \pm 0.28) \log \sigma + {\rm const}$. These slopes do not differ much from the slopes obtained from our normal fitting method, cf. Eq. (). The slopes from Franx & Illingworth are in rough agreement with the slopes that we find. The slope differences (`our'-`their') are $-0.33 \pm 0.28$ for (B-r) and $-0.50 \pm 0.40$ for (U-r).

Bender et al. (1993) established the relation ${ {\rm Mg}_2}= 0.20 \log\sigma -0.166$ for their DHG sample. They found that the relation ${(B-V)}= 1.12 { {\rm Mg}_2}+ 0.615$ matched their DHG sample well. This relation was established by Burstein et al. (1988) for 276 bright ellipticals. Bender et al. combined the two relations to ${(B-V)}= 0.224 \log\sigma + 0.429$. By means of the relation (B-V)= 0.673 (B-r)+ 0.184 from Jørgensen (1994), this can be transformed to ${(B-r)}= 0.333 \log\sigma + 0.364$. This is shown as the dotted line in Fig. (a). The slope is in agreement with our data, but not the zero point, we find a mean difference of $0\hbox{$.\!\!^{\rm m}$ }064 \pm 0\hbox{$.\!\!^{\rm m}$ }007$ (with the differences calculated as $\Delta = {(B-r)_{\rm e}}-0.333 \log\sigma - 0.364)$.

 

The colors are also well correlated with ${ {\rm Mg}_2}$. We find the following relations

$$\arraycolsep=2pt % \begin{array}{lllllll} {\rm HydraI} & : & ... ...}}=0.058 \quad & N=18 \\ & & & \pm & 1.41 & & \\ \end{array}$$ (7.19)

No galaxies were excluded from the fits. R293 does not deviate from the color- ${ {\rm Mg}_2}$ relations, and excluding it has very little effect. The effect of omitting R295/E437G15 is larger, but still not significant, the slope of the ${(B-r)_{\rm e}}$- ${ {\rm Mg}_2}$ relations changes from $1.32\pm0.74$ to $1.45\pm0.65$. The color- ${ {\rm Mg}_2}$ relations are shown in Fig. (d-f).

Burstein et al. (1988) established the relation ${(B-V)}= 1.12 { {\rm Mg}_2}+ 0.615$ for 276 bright ellipticals, as mentioned above. This can be transformed into ${(B-r)}= 1.66 { {\rm Mg}_2}+ 0.640$. This is shown as the dotted line in Fig. (d). The slope is in agreement with the slope that we find. The zero point is not in agreement with our data, we find a mean residual from their relation of $0\hbox{$.\!\!^{\rm m}$ }054 \pm 0\hbox{$.\!\!^{\rm m}$ }008$.

We would like to use stellar population models to estimate the variation in age and metallicity needed to reproduce the three observed color- ${ {\rm Mg}_2}$ relations. Unfortunately, the Vazdekis et al. (1996) models (with a bi-modal IMF with high mass slope $\mu = 1.35$) are not able to reproduce any of them - galaxies with large ${ {\rm Mg}_2}$ values (say $\sim 0.3$) are predicted to have much redder colors than what is actually observed. The failure to reproduce the color- ${ {\rm Mg}_2}$ relations could be due to problems reproducing the colors and/or problems reproducing ${ {\rm Mg}_2}$. Several models cannot reproduce the ${ {\rm Mg}_2}$-Mgb relation, which could indicate a problem in reproducing ${ {\rm Mg}_2}$. Some models do not get the colors right. Worthey (1994) noted that his (B-V) colors were too red by $0\hbox{$.\!\!^{\rm m}$ }05$ when compared to globular clusters. This might be related to the problems we observe with the Vazdekis et al. models. However, we also observe problems for the (U-r) color, which is independent of B. Systematic differences between the different models are known to exist. For example, Borges et al. (1995) find that their (B-V) colors are $0\hbox{$.\!\!^{\rm m}$ }1$- $0\hbox{$.\!\!^{\rm m}$ }03$ more blue than those of Worthey (1994).

Finally, we note that $\log { <{\rm Fe}>}$ is also weakly correlated with the ${(B-r)_{\rm e}}$, as expected from the weak correlations with ${\log\sigma}$ and ${ {\rm Mg}_2}$. For the HydraI sample we find ${P_{\rm no\;corr.}}$ = 7.0%. No significant correlations can be seen with ${(U-r)_{\rm e}}$ and ${(U-B)_{\rm e}}$, but this is most likely due to the small sample size (N = 17).