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 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,.

Derivation

An astrophysical disk can be modeled as a fluid with negligible mass compared to the central object (e.g. a star) and with negligible pressure. We can suppose an axial symmetry such that. Starting from the equations of movement in cylindrical coordinates : The second line implies that the specific angular momentum is conserved. We can then define an effective potential and so : We can apply a small perturbation to the circular orbit : So, And thus : We then note In a circular orbit. Thus : The frequency of a circular orbit is which finally yields :