•Stability due to Rotation – large scales
–See Toomre (1964) for details
–Considered axisymmetric perterbations
–Consider small region of size L on a disk with surface
density μ (mass then is ~μL2), rotating at Ω, distance d from the center.
Imagine a perturbation that locally increases the surface
density.
–Angular momentum
conserved, so the angular velocity must also increase,
creating a centrifugal force in the rotating frame. If this force can
send the mass back to its original position, system is
stable.
–Stability: L > Lcrit = 2Gμ/3Ω2, so requires large scales. Order of magnitude, Ω2R = V2/R ~ GμR2/R2 = Gμ, so Lcrit ~ R
–