# Kinematics of Ellipticals

 Remember how we characterize velocity of populations of stars. rotation (v): the net rotational velocity of a group of stars dispersion (sigma): the characteristic random velocity of stars In the disk of our galaxy, v=220 km/s, sigma=30 km/s, so v/sigma ~ 7. This is called a cold disk. Elliptical galaxies have much higher velocity dispersions, 100s of km/s. These are kinematically hot systems. v/sigma ranges (roughly) from 0 to 1.

Would you expect flattened ellipticals to have higher or lower values of v/sigma? Why?

v/sigma actually correlates with luminosity.

• Lower luminosity ellipticals have higher v/sigma -- rotationally supported.
• Higher luminosity ellipticals have lower v/sigma -- pressure supported.

# The Faber-Jackson law

Remember the Tully-Fisher law for disk galaxies: L ~ v4. Can we make a similar law for elliptical galaxies?
The Faber-Jackson law has a lot of scatter: at a given velocity dispersion, there is a range of +/- 2 magnitudes in luminosity. Compare this to the Tully-Fisher relationship, where at a given circular velocity, there is a range of a few tenths of a magnitude in luminosity.

Clearly, there is something messing with the relationship -- a second parameter.

# The Fundamental Plane

In 1987, two teams of astronomers identified the second parameter -- the effective radius. Rather than two parameters correlating (in which case you fit a line), there are three parameters correlating (in which case you fit a plane).

We have 4 things we can measure:

• Luminosity (L)
• Mean surface brightness (<Ie>)
• Velocity dispersion (sigma)
There are only three independant variables here (L, re, and <Ie> are not all independant).

If you plot one versus another, the third introduces scatter, for example:

 Surface brightness versus luminosity Velocity dispersion versus effective radius

But if you plot one versus a combination of the other two, you can see a very tight correlation:

This correlated plane is now referred to as the fundamental plane. Since we have four observables, only three of which are independant, there are different representations of the FP which are all expressing the same thing. Here is another one:

Or in other words

Why would this be useful?

Doing some simple algebra, you can (and will!) show that one implication of the fundamental plane is that mass-to-light ratio depends on galaxy luminosity:

Why would luminous ellipticals have higher mass-to-light ratios?