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Epicyclic frequency

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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

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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

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  1. ^ p161, Astrophysical Flows, Pringle and King 2007