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 Schwarzchild black hole, the Innermost Stable Circular Orbit (ISCO) occurs at 3x the event horizon - at \(6GM/c^{2}\).

For a Keplerian disk, \(\kappa = \Omega\).