Epicyclic frequency
Introduction 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 , the epicyclic frequency  is given by: , where R is the radial co-ordinate.This quantity can be used to examine the 'boundaries' of an accretion disc: when  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 .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 :
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