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2.1 Observable Quantities of E and S0 Galaxies

The observations we normally have at hand of E and S0 galaxies are direct images and longslit spectroscopy along the major axis. Only for very nearby galaxies and with superb instrumentation (e.g. with the Hubble Space Telescope) it is possible to resolve the galaxy into individual stars; in all other cases only the integrated light of the stellar population of the galaxy can be observed.

For each galaxy, we want to determine a characteristic size and surface brightness, and the luminosity (or the total magnitude). This is in the following done by first fitting ellipses to the images of the galaxies. The resulting surface photometry yields among other things the local surface brightness $\mu(r)$ as function of projected radius r (in arcsec). $\mu$ is expressed in units of magnitudes per square arc second ( ${^{\rm m} /{\rm arcsec}^{2}}$), and r is calculated as $r = \sqrt{ab}$, where a and b are the semi-major and semi-minor axes of the elliptical isophote, respectively. The surface photometry for the HydraI galaxies is described in Chapter [*].

Elliptical galaxies are in general well described by the r1/4 law (de Vaucouleurs 1948),

 \begin{displaymath}\mu(r) = \mu_{\rm e} +
8.3268 \left[ \left( \frac{r}{{r_{\rm e}}} \right)^{1/4} - 1 \right] \enspace ,
\end{displaymath} (2.1)

where ${r_{\rm e}}$ is the effective radius and $\mu_{\rm e} = \mu({r_{\rm e}})$ is the local surface brightness at ${r_{\rm e}}$. An example is the nearby ($\sim$10 Mpc) `standard' E1 galaxy NGC3379, for which the residuals from the r1/4 fit are less than 0.08 mag over a 10-mag range in $\mu$ (de Vaucouleurs & Capaccioli 1979, Capaccioli et al. 1990). Small systematic deviations from the r1/4 law have been found by e.g. Caon, Capaccioli, & D'Onofrio (1993). Makino, Akiyama, & Sugimote (1990) found from dynamical arguments that the r1/4 law bared little physical significance. They also found that the r1/4 law provided the best fit, but that generalized r1/m laws with m = 3-10 gave almost as good fits for a range in r of about 100. Here, we will by that token use the r1/4 law as a good fitting function to derive a characteristic galaxy size ${r_{\rm e}}$. It is worth noting, that for studies of the FP, the r1/4 parameters are well suited, even though some E and S0 galaxies show deviations from the r1/4 profile. This is due to the fact that the combination of ${r_{\rm e}}$ and ${< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$ enters the FP is rather insensitive to these deviations. The above-mentioned combination is ``FP'' = ${\log{r_{\rm e}}}+ 0.82 \log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$. Jørgensen (1997c) compared ``FP'' based on asymptotic magnitudes (the least model dependent parameters) with ``FP'' based on (1) fit with an r1/m profile, (2) fit with an r1/4 profile, (3) fit with an r1/4 + exponential disk profile, and (4) the Petrosian (1976) parameters. The differences $\Delta$``FP'' showed a small and comparable rms scatter for the four methods (rms = 0.04, 0.06, 0.07, and 0.05, respectively, based on a sample of 31 galaxies). Thus, ``FP'' is well-determined for several choices of parameters, including the r1/4 parameters. Small systematic offsets ( $\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
$\displaystyle ...) in ``FP'' between determinations from different methods were found. Therefore, parameters based on different methods should not be mixed.

The constant in Eq. ([*]) has been chosen so that half the light of the galaxy is inclosed within ${r_{\rm e}}$. When ${r_{\rm e}}$ has been determined, the mean surface brightness within ${r_{\rm e}}$, denoted ${{< \hspace{-3pt} \mu \hspace{-3pt}>}_{\rm e}}$, can be calculated. From ${r_{\rm e}}$ and ${{< \hspace{-3pt} \mu \hspace{-3pt}>}_{\rm e}}$ the total magnitude can be calculated as ${m_{\rm T}}= {{< \hspace{-3pt} \mu \hspace{-3pt}>}_{\rm e}}- 2.5 \log (2 \pi {r_{\rm e}}^2$), since $\pi {r_{\rm e}}^2$ is the surface within which half the light is found. The derivation of global photometric parameters for the HydraI galaxies is described in Chapter [*].

The surface brightness can be expressed in $L_\odot/{\rm pc}^2$ instead of ${^{\rm m} /{\rm arcsec}^{2}}$, where $L_\odot$ is the luminosity of the Sun in the given passband (e.g. Gunn r). This is done as

 \begin{displaymath}\log {< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}= -0.4 ({{< \hspace{-3pt} \mu \hspace{-3pt}>}_{\rm e}}- k) \enspace ,
\end{displaymath} (2.2)

where the constant k is given by

\begin{displaymath}k = M_\odot + 5 \log \left(\frac{206265\,{\rm pc}}{10\,{\rm p...
...=5\hbox{$.\!\!^{\rm m}$ }58) \\
\enspace .
\end{displaymath} (2.3)

k corresponds to the apparent magnitude of the Sun if placed at the distance where 1 pc subtends an angle of 1 arcsec. The values given are those from JFK96. The values of $M_\odot$ given in parentheses are calculated from k, and are correct to within $0\hbox{$.\!\!^{\rm m}$ }05$.

From the spectroscopy we can obtain a measure of the kinetic energy of the galaxy, namely the line-of-sight velocity dispersion of the stars in the galaxy, $\sigma$.

We determine the strength of different individual absorption lines from the spectroscopy. Due to moderate spectral resolution and velocity broadening it is not possible to determine accurate equivalent widths as in high resolution spectroscopy of single stars. Instead, a so-called line index is calculated from the flux within an index passband centered on the spectral feature relative to the level defined by a pseudocontinuum passband on each side of the line. We use the Lick/IDS line index system (Faber et al. 1985, Worthey et al. 1994), of which examples are ${ {\rm Mg}_2}$ and ${ <{\rm Fe}>}$. These indices will usually depend strongly on the abundance of the element that gives rise to the absorption feature on which they are centered. But in addition, lines from other elements present in either the index passband or in the pseudocontinuum passbands will also have an effect. In a few cases the indices will respond in very unexpected ways to abundances changes. For example, Tripicco & Bell (1995) found the Fe4668 index to be very sensitive to the carbon abundance, but almost insensitive to the iron abundance! This index has later been renamed to C4668 or C24668. The indices we use in this study are more `well-behaved', cf. Sect. [*]. In addition to element abundances the line indices are also sensitive to the mean age of the stellar population. For the indices used in this study, namely ${ {\rm Mg}_2}$ and ${ <{\rm Fe}>}$, older ages give stronger absorption lines, cf. Sect. [*]. Note, that this is not the case for the ${ {\rm H}_{\beta}}$ index. We do not have ${ {\rm H}_{\beta}}$ indices for our samples.

This project is based on central spectroscopical values. Because of the cost in observing time to get spatial information in the the spectra there is a trade off between either having large samples of galaxies with centrally measured parameters or having much smaller samples with spatial information in the spectra. Further, this allows for the use of fiber-fed spectrographs. Several studies have found tight correlations between central quantities and more global quantities. For example, Burstein et al. (1988) and Bender, Burstein, & Faber (1993) found a tight correlation between central ${ {\rm Mg}_2}$ and global (B-V) color. Here `global' means within an aperture of 25 times larger diameter of that used for the central values. These authors concluded from this that variations in radial gradients in colors and line indices from galaxy to galaxy are small. However, if the size of the gradient is correlated with the central value, this conclusion does not necessarily hold. We do not study radial gradients in colors and line indices in this work.

From the spectroscopy the redshift z is determined. The observations are usually transformed from the observer's frame to some standard frame. In this work we will use two frames: (1) The heliocentric frame, in which the Sun is at rest. (2) The CMB frame, which is the frame that is at rest relative to the cosmic microwave (CMB) radiation, i.e. the frame in which the CMB radiation is isotropic. Redshifts in these two frames will be denoted $z_{\rm hel}$ and $z_{\rm CMB}$, respectively.

The spectroscopy for the HydraI galaxies is described in Chapter [*].

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Next: 2.2 The Fundamental Plane Up: 2. Background Previous: 2. Background

Properties of E and S0 Galaxies in the Clusters HydraI and Coma
Master's Thesis, University of Copenhagen, July 1997

Bo Milvang-Jensen (