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 [math]\displaystyle{ \Omega }[/math], the epicyclic frequency [math]\displaystyle{ \kappa }[/math] is given by

[math]\displaystyle{ \kappa^{2} \equiv \frac{2 \Omega}{R}\frac{d}{dR}(R^2 \Omega) }[/math], where R is the radial co-ordinate.

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

For a Keplerian disk, [math]\displaystyle{ \kappa = \Omega }[/math].