fundamental plane
Elliptical galaxies, despite their boring smooth looks, turn out to be remarkably orderly. Three of their measurable properties — how big they are, how brightly the central region glows, and how fast their stars buzz around randomly — are not free to take any values. They are locked together so tightly that, if you plot ellipticals in a 3D graph of these three quantities, they all fall onto a single thin, tilted sheet. That sheet is the fundamental plane.
The three properties are the galaxy's size (its effective radius), its surface brightness (how concentrated its light is), and its velocity dispersion, written sigma — the spread of random stellar speeds, which you read off how broad the galaxy's spectral lines are. A simpler, earlier version using just two of these (luminosity and velocity dispersion) is the Faber-Jackson relation, where brighter ellipticals have faster-moving stars. The fundamental plane is the tighter, three-way version, and its very existence reflects that ellipticals obey the virial theorem — their structure is governed by a balance between gravity pulling in and random stellar motions pushing out.
The fundamental plane matters as the ellipticals' answer to the spirals' Tully-Fisher relation, and like it, it serves as a distance indicator: measure an elliptical's velocity dispersion and surface brightness, read its true size off the plane, and compare with its apparent size to get the distance. Beyond distances, the thinness of the plane, and the small ways galaxies deviate from it, encode how ellipticals formed and how their stars, gas, and dark matter are distributed — valuable clues about galaxy assembly through mergers.
Two ellipticals with stars whirling at the same velocity dispersion of, say, 250 km/s and the same surface brightness will, by the fundamental plane, have nearly the same physical size — so any difference in their apparent size on the sky tells you which one is farther away.
Three properties of ellipticals, locked onto one thin sheet by gravity and stellar motion.
The plane is 'tilted' relative to what the simple virial theorem alone predicts; that tilt is real and reflects how the ratio of mass to light changes from galaxy to galaxy, so the relation is empirical and not derivable from gravity in one line.