2.2.1 The Original Findings

The Fundamental Plane (FP) was discovered independently and simultaneously by Djorgovski & Davis (1987) and Dressler et al. (1987b). It is a relation between ${r_{\rm e}}$, $\sigma$, and ${< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$, and is linear in logarithmic space. Since $L = 2 \pi {< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}{r_{\rm e}}^2$, the FP can also be expressed as a relation between L, $\sigma$, and ${< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$, or between ${r_{\rm e}}$, $\sigma$, and L.

Djorgovski & Davis (1987) found a tight correlation for elliptical galaxies between either a radius or the luminosity on the one hand, and a linear combination of velocity dispersion and mean surface brightness on the other hand. They dubbed this relation the fundamental plane. They found the best-fitting relation involving a radius to be

$$\log a_{\rm e} = 1.39 \log\sigma -0.90 \log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}+ \mbox{constant}$$ (2.4)

in the $r_{\rm G}$ passband (Djorgovski 1985), with the radius $a_{\rm e}$ being the effective semimajor axis from a fit to an r^1/4 profile. $a_{\rm e}$ is related to ${r_{\rm e}}$ through ${r_{\rm e}}= \sqrt{a_{\rm e} b_{\rm e}} = a_{\rm e} \sqrt{1-\varepsilon}$. They found, that the morphological shape parameters (ellipticity, ellipticity gradient, isophotal twist rate, and slope of the surface brightness profile) did not correlate with the residuals from the FP. Djorgovski & Davis found the thickness of the FP to be given by the measurement errors, and that the intrinsic scatter therefore had to be very small, a few percent or less. Their main sample only consisted of E galaxies, but they reported preliminary results that a fundamental plane also existed for S0 galaxies, and that it even might be identical to that for E galaxies.

Dressler et al. (1987b), also known as the 7 Samurai, found the same result, namely that elliptical galaxies describe a plane in the 3-space of $(\log L, \log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}, \log\sigma)$ or $(\log{r_{\rm e}}, \log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}, \log\sigma)$. They found the plane to be given by

$$\log {r_{\rm e}}= 1.325 \log \sigma -0.825 \log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}+ \mbox{constant}$$ (2.5)

in the Johnson B passband.

Dressler et al. also introduced a new photometric diameter ${D_{\rm n}}$, the diameter within which the mean surface brightness is $20.75{^{\rm m} /{\rm arcsec}^{2}}$, in Johnson B. They found $\log {D_{\rm n}}$ to correlate as well with ${\log\sigma}$ as any linear combination of ${\log{r_{\rm e}}}$ and $\log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$, and they where thus able to reformulate the FP as the ${D_{\rm n}}$-$\sigma$ relation, $\log{D_{\rm n}}= 1.333 \log\sigma + \mbox{constant}$. However, they noted that the correlation between ${D_{\rm n}}$ and a combination of ${r_{\rm e}}$ and ${< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$, namely ${D_{\rm n}}\propto {r_{\rm e}}{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}^{0.8}$, had a small residual curvature. Phillipps (1988) demonstrated theoretically that the relation ${D_{\rm n}}\propto {r_{\rm e}}{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}^{0.8}$ is expected for galaxies with r^1/4 profiles and ${{< \hspace{-3pt} \mu \hspace{-3pt}>}_{\rm e}}$ around $21.8 \, {^{\rm m} /{\rm arcsec}^{2}}$ in Johnson B. He also showed, that for the range in ${{< \hspace{-3pt} \mu \hspace{-3pt}>}_{\rm e}}$ actually spanned by (giant) ellipticals, the FP will be seen as a curved line in the $\log {D_{\rm n}}$ versus ${\log\sigma}$ plot. Lucey, Bower, & Ellis (1991a) were the first to demonstrate that the residuals from the ${D_{\rm n}}$-$\sigma$ relation were correlated with ${{< \hspace{-3pt} \mu \hspace{-3pt}>}_{\rm e}}$. They corrected for this by simply adding a linear term in ${{< \hspace{-3pt} \mu \hspace{-3pt}>}_{\rm e}}$ to the ${D_{\rm n}}$-$\sigma$ relation. JFK93 found for a sample of galaxies in the Coma cluster that the ${D_{\rm n}}$-$\sigma$ residuals showed the dependence on ${{< \hspace{-3pt} \mu \hspace{-3pt}>}_{\rm e}}$ that was predicted by Phillipps (1988), a dependence that has a quadratic term. They concluded that the FP is a true improvement of the ${D_{\rm n}}$-$\sigma$ relation. In accordance with this, they found the scatter of the ${D_{\rm n}}$-$\sigma$ relation to be larger than for the FP, namely 17% versus 11% for their sample.

In the following, we will mainly consider the FP, not the ${D_{\rm n}}$-$\sigma$ relation.