Epicyclic frequency

In astrophysics, particularly the study of accretion disks, the epicyclic frequency is the frequency at which a radially displaced fluid parcel will oscillate. It can be referred to as a "Rayleigh discriminant". When considering an astrophysical disc with differential rotation ${\displaystyle \Omega }$, the epicyclic frequency ${\displaystyle \kappa }$ is given by

${\displaystyle \kappa ^{2}\equiv {\frac {2\Omega }{R}}{\frac {d}{dR}}(R^{2}\Omega )}$, where R is the radial co-ordinate.[1]

This quantity can be used to examine the 'boundaries' of an accretion disc: when ${\displaystyle \kappa ^{2}}$ becomes negative, then small perturbations to the (assumed circular) orbit of a fluid parcel will become unstable, and the disc will develop an 'edge' at that point. For example, around a Schwarzschild black hole, the innermost stable circular orbit (ISCO) occurs at three times the event horizon, at ${\displaystyle 6GM/c^{2}}$.

For a Keplerian disk, ${\displaystyle \kappa =\Omega }$.