H8 - the Faber-Jackson relation

for ellipticals, the bulk rotation is very small $\sim 10$ km s $^{- 1}$
the velocity dispersion along the profile is very high $\sim 200 - 300$ km s $^{- 1}$
the lack of bulk motion indicates that they cannot be supported by rotation
they are instead supported by turbulence, the random kinetic energy distribution of stars
the primordial angular momentum was cancelled, possibly by mergers
this ,means that stars have completely relaxed orbits
energy is redistributed among a;; the stars in the galaxy
stars in ellipticals can be thought of as particles of an ideal gas moving in random directions
observations project any random velocity along the line of sight
this produces a Gaussian profile
empirically, the Faber-Jackson relation is obtained:

L = 2 \times 10^{10} L_{⊙} {(\frac{σ_{r}}{200 {km s}^{- 1}})}^{4}

it follows that the velocity dispersion of ellipticals is a distance-independent indicator of luminosity, and hence can be used to measure distances
Keplerian motion cannot be used
considering a closed system in equilibrium, so using the virial theorem:

2 K + U = 0 2 \sum_{i} \frac{1}{2} m_{i} v i^{2} = \sum_{i, j > i} \frac{G m_{i} m_{j}}{| r_{i} - r_{j} |}

to obtain a proportionality, using the mean square velocity $⟨ v^{2} ⟩$ , and characteristic separation $r_{g}$ between stars:

M_{t o t} ⟨ v^{2} ⟩ \propto \frac{G M_{t o t}^{2}}{r_{g}}

⟨ v^{2} ⟩ = σ_{x}^{2} + σ_{y}^{2} + σ_{z}^{2} = 3 σ_{r}^{2} \propto σ_{r}^{2}

σ_{r}^{2} \propto \frac{M_{t o t}}{R_{e}}

σ_{r}^{2} \propto M \sqrt{\frac{I_{e}}{L}} ⟹ σ_{r}^{4} \propto L I_{e}

σ_{r}^{4} \propto L