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 $\Omega$ , the epicyclic frequency $\kappa$ is given by

$\kappa ^{{2}}\equiv {\frac {2\Omega }{R}}{\frac {d}{dR}}(R^{2}\Omega )$ , where R is the radial co-ordinate.^[1]

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 3x the event horizon - at $6GM/c^{{2}}$ .

For a Keplerian disk, $\kappa =\Omega$ .

References

↑ p161, Astrophysical Flows, Pringle and King 2007

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