7.1.1 Data Overview

The HydraI data have already been described in detail in the previous chapters. In summary, the photometry (Chapter -) is based on observations made in April 1994 with the Danish 1.5 meter telescope at ESO, La Silla, equipped with the DFOSC instrument. Global photometric parameters for 64 galaxies in Gunn r and Johnson B and 22 galaxies in Johnson U are listed in Table  (p. ) for Gunn r, and Table  (p. ) for Johnson B and U. The spectroscopy (Chapter ) is in part from the above-mentioned observing run, and in part from JFK95b and J97. In addition data from Lucey & Carter (1988) were used for three galaxies. Global spectroscopic parameters for 51 galaxies are listed in Table  (p. ). The number of galaxies with both photometry and spectroscopy is 45; for details see Table  (p. ). The photometry is estimated to be complete to ${m_{\rm r_T}}= 14\hbox{$.\!\!^{\rm m}$ }5$, corresponding to ${M_{\rm r_T}}= -20\hbox{$.\!\!^{\rm m}$ }1$. (The calculation of distance dependent quantities is explained below.) Of these galaxies, 80% have spectroscopy. The sample covers the central part of the cluster, with a maximum center distance of 57', corresponding to 1.3 Mpc. This sample is larger than any other complete sample of HydraI galaxies used for investigations of the FP.

The Coma sample consists of 116 galaxies. The data are given in Appendix  (p. ). Of the 116 galaxies, D120 and D121 are omitted from the analysis, as their photometric parameters are highly uncertain, due to the small angular distance between the galaxies. The photometry is from JF94. We use the seeing corrected values that are given in JFK95a. Velocity dispersions and ${ {\rm Mg}_2}$ values are from the literature (Davies et al. 1987; Dressler 1987; Lucey et al. 1991b; Guzmán et al. 1992) as compiled into a consistent system by JFK95b, and from Jørgensen (1997b, in prep.). ${ <{\rm Fe}>}$ indices are available for a subsample of 44 of the galaxies (Jørgensen 1997b, in prep.) - this subsample is not magnitude limited. The photometry is estimated to be complete to ${m_{\rm r_T}}= 15\hbox{$.\!\!^{\rm m}$ }05$ corresponding to ${M_{\rm r_T}}= -20\hbox{$.\!\!^{\rm m}$ }75$. Of these galaxies, 93% have a measured $\sigma$ and ${ {\rm Mg}_2}$. The sample covers the central $64' \times 70'$ of the cluster, with a maximum center distance of 47', corresponding to 1.7 Mpc. This sample is larger than any other complete sample of Coma galaxies used for investigations of the FP. For comparison, the Coma sample used in JFK96 has 79 galaxies. Their Coma sample is a subsample of the Coma sample used in this study.

Table  also list typical measurement errors for the different quantities. These are all based on external comparisons. The exception is $\log { <{\rm Fe}>}$, where the typical uncertainties are based on comparisons between data from different observing runs using different instruments, but not from other authors. The HydraI and Coma samples do not have enough galaxies in common with e.g. Faber (1994) to enable an actual external comparison. The errors in ${\log{r_{\rm e}}}$ and $\log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$ are highly anti-correlated, with a linear correlation coefficient r of about -0.95. The uncertainty on a function $f = f({\log{r_{\rm e}}},{\log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}})$ is calculated as

$$\sigma_f \left[ \left( \frac{\partial f}{\partial {\log{r_{\rm e}}}} \righ... ...}}}} \right)^2 \sigma^2_{\log{< \hspace{-1pt} I \hspace{-1pt}>_{\rm e}}}\right.$$ =

$$+ \left. 2 r \sigma_{\log{r_{\rm e}}}\sigma_{\log{< \hspace{-1pt}... ...og{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}}} \right) \right]^{1/2} \enspace ,$$ (7.1)

which is the usual propagation of errors formula with the covariance term included (e.g. Bevington & Robinson 1992), since Cov( ${\log{r_{\rm e}}}$, $\log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$) = $r \sigma_{\log{r_{\rm e}}}\sigma_{\log{< \hspace{-1pt} I \hspace{-1pt}>_{\rm e}}}$ (e.g. Press et al. 1992). As an important example, the combination $(\log {r_{\rm e}}- \beta \log {< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}) \approx (\log {r_{\rm e}}+ 0.82 \log {< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}})$ enters the FP, and as can be seen from Table , this combination has a much smaller uncertainty than ${\log{r_{\rm e}}}$ and $\log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$ individually have. Where nothing else is stated, the typical error bars shown on the figures in this chapter are based on the values listed in Table .

The distance dependent quantities (absolute magnitudes, ${r_{\rm e}}$ in kpc, and cluster center distances in Mpc) were calculated as follows. Coma was assumed to be at rest with respect to the CMB frame at $cz = 7200\,{\rm km}\,{\rm s}^{-1}$, and the distance to HydraI relative to Coma was calculated from the FP zero point difference $\Delta\gamma = 0.232$ (Eq. ). $H_0 = 50\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$ and q₀ = 0.5 was assumed. The formulas used are given in Sect.  (p. ). The derived redshift for HydraI (Eq. ) is $cz = 4143\,{\rm km}\,{\rm s}^{-1}$. For comparison, the CMB redshift is $cz = 4050\,{\rm km}\,{\rm s}^{-1}$. The distance moduli for Coma and HydraI are

$$(m-M)_{\rm Coma} = 35\hbox{$.\!\!^{\rm m}$ }80 \, ; \quad (m-M)_{\rm HydraI} = 34\hbox{$.\!\!^{\rm m}$ }60 \enspace .$$ (7.2)

The conversion of ${r_{\rm e}}$ from arcsec to kpc (Eq. , p. ) is

$$\log r_{\rm e,kpc} = \log r_{\rm e,arcsec} - \left\{ \begin{a... ...\ 0.407 & \mbox{for HydraI} \\ \end{array}\right. \enspace .$$ (7.3)

Cluster center distances in Mpc can be calculated using a similar expression. Note that the assumed value of H₀ does not affect the comparison of the two clusters, since their relative distance is determined from the FP. The dependence on q₀ is negligible.

 

Table: Properties of the HydraI and Coma Samples

Galaxies with photometry in Gunn r
	HydraI	Coma
Number of galaxies with $\sigma$	45	114
Number of galaxies with ${ {\rm Mg}_2}$	42	113
Number of galaxies with ${ <{\rm Fe}>}$	42	44
Maximum distance from cluster center$^{\rm a}$	57' = 1.3 Mpc	47' = 1.9 Mpc
Median distance from cluster center$^{\rm a}$	13' = 0.3 Mpc	14' = 0.6 Mpc
Completeness	80% to ${m_{\rm r_T}}= 14\hbox{$.\!\!^{\rm m}$ }5$	93% to ${m_{\rm r_T}}= 15\hbox{$.\!\!^{\rm m}$ }05$
	i.e. ${M_{\rm r_T}}= -20\hbox{$.\!\!^{\rm m}$ }1$	i.e. ${M_{\rm r_T}}= -20\hbox{$.\!\!^{\rm m}$ }75$
Number of galaxies$^{\rm a}$ fainter than completeness	6	9
Number of galaxies$^{\rm a}$ fainter than Coma limit$^{\rm b}$	17	9
Typical Measurement Errors$^{\rm c}$
	HydraI	Coma
Uncertainty on ${\log{r_{\rm e}}}$	0.028	0.045
Uncertainty on $\log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$	0.039	0.064
Correlation$^{\rm d}$ between $\Delta \log {r_{\rm e}}$ and $\Delta \log {< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$	-0.94	-0.94
Uncertainty on $(\log {r_{\rm e}}+ 0.82 \log {< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}})$$^{\rm e}$	0.011	0.020
Uncertainty on ${\log\sigma}$	0.036	0.031
Uncertainty on ${ {\rm Mg}_2}$	0.013	0.013
Uncertainty on $\log { <{\rm Fe}>}$	0.030	0.037
Galaxies with photometry in Johnson B
	HydraI
Number of galaxies with $\sigma$	45
Number of galaxies with ${ {\rm Mg}_2}$	42
Number of galaxies with ${ <{\rm Fe}>}$	42
Maximum distance from cluster center$^{\rm a}$	57' = 1.3 Mpc
Median distance from cluster center$^{\rm a}$	13' = 0.3 Mpc
Galaxies with photometry in Johnson U
	HydraI
Number of galaxies with $\sigma$	19
Number of galaxies with ${ {\rm Mg}_2}$	18
Number of galaxies with ${ <{\rm Fe}>}$	18
Maximum distance from cluster center$^{\rm a}$	13' = 0.3 Mpc
Median distance from cluster center$^{\rm a}$	6' = 0.1 Mpc

 

Notes: All galaxies are classified as E or S0. The distance dependent quantities are based on $H_0 = 50\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$; further explanation is given in the text. $^{\rm a}$ This refers to galaxies with a measured $\sigma$. $^{\rm b}$ The galaxies fainter than the Coma completeness limit ( ${M_{\rm r_T}}= -20\hbox{$.\!\!^{\rm m}$ }75$) are plotted as open symbols throughout this chapter. $^{\rm c}$ The typical measurement errors are estimated from external comparisons, except for $\log { <{\rm Fe}>}$, where comparisons between different observing runs using different instruments are used. The uncertainties on the photometric parameters refer to Gunn r, but we assume that they are also representative for Johnson B and U. $^{\rm d}$ The number given is the linear correlation coefficient (Pearson's r). $^{\rm e}$ This is the combination that enters the FP. It has a low uncertainty because the errors in ${r_{\rm e}}$ and ${< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$ are highly anti-correlated.