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.^[1]

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 3x the event horizon - at $6GM/c^{2}$.

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

References

1. p161, Astrophysical Flows, Pringle and King 2007