What is the Fundamental plane for Elliptical Galaxies?

Question

An alternate explanation rather than from wikipedia is preferable.

Answer 1

I am not an expert, but it seems that the fundamental plane is a relation among characteristic quantities of the galaxy, showing a correlation that is the analogous of the Tully-Fisher relation for spiral galaxies.

You can start from the virial theorem: $\frac{M}{R}\sim v^2$

($M$ mass contained in the radius $R$, $v$ velocity dispersion of the stars, also indicated with $\sigma$ in these cases) meaning that the stars behave like an isothermal sphere.

Assuming that all galaxies have the same mass-to-light ratio $L/M$, and that all galaxies have the same surface brightness

$\Sigma=L/R^2$

(with $L$ luminosity), you obtain the Tully-Fisher relation:

$L\sim v^4$

Of course, all galaxies do not have the same surface brightness. So if you take $\Sigma=L/R^2$ and substitute that into the virial theorem (keeping the mass-to-light ratio assumption), then a relation between luminosity, surface brightness, and velocity dispersion is obtained:

$L \sim \sigma^4 \Sigma^{-1}$

You can plot the stars distributions in elliptical galaxy according to this relation, and will note that the stars do not distribute randomly, but are more concentrated along the fundamental plane (more or less the same happens for the HR diagram):

(Here $R_e$ is the so called effective radius).

For completeness, another illuminating picture from here:

References here and here.