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 $\Omega$, the epicyclic frequency $\kappa$ is given by

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

This quantity can be used to examine the 'boundaries' of an accretion disc: when $\kappa ^{2}$ 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 $6GM/c^{2}$.

For a Keplerian disk, $\kappa =\Omega$.