Astronomy:Epicyclic frequency

Short description: Characteristic of accretion discs

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 $\mathrm{\Omega }$, the epicyclic frequency $\kappa$ is given by

${\kappa }^2\equiv \frac{2\mathrm{\Omega }}{R}\frac{d}{dR}(R^2\mathrm{\Omega })$, 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, $\kappa =\mathrm{\Omega }$.