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 Ω , the epicyclic frequency κ is given by

This quantity can be used to examine the 'boundaries' of an accretion disc - when κ 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 3x the event horizon - at 6 G M / c 2 .

For a Keplerian disk, κ = Ω .