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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.[1]
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 :
References
^p161, Astrophysical Flows, Pringle and King 2007
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