Epicyclic frequency explained

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 (2 \Omega)/(R) (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 three times the event horizon, at 6 GM/c ^{2} .

For a Keplerian disk,

\kappa = \Omega .

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

\Phi (r,z) = \Phi (r, - z) . 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

\Phi _{eff} = \Phi - (1)/(2) r ^{• 2 \theta} ^{2} = \Phi - (h ^{2})/(2 r ^{2}) and so :

We can apply a small perturbation

\delta\vec r = \delta r \vec e _{r} + \delta z \vec e _{z} to the circular orbit : $$\backslash vec\; r\; =\; r\_0\; \backslash vec\; e\_r\; +\; \backslash delta\; \backslash vec\; r$$So, $$\backslash ddot\; +\; \backslash delta\; \backslash ddot\; =\; -\backslash vec\; \backslash nabla\; \backslash Phi\_(\backslash vec\; r\; +\; \backslash delta\; \backslash vec\; r)\backslash approx-\backslash vec\; \backslash nabla\; \backslash Phi\_\; (\backslash vec\; r)\; -\; \backslash partial\_r^2\; \backslash Phi\_(\backslash vec\; r)\backslash delta\; r\; -\; \backslash partial\_z^2\; \backslash Phi\_(\backslash vec\; r)\backslash delta\; z$$

And thus : We then note $$\backslash kappa^2\; =\; \backslash Omega\_r^2\; =\; \backslash partial\_r^2\backslash Phi\_\; =\; \backslash partial\_r^2\backslash Phi\; +\; \backslash frac$$In a circular orbit

h ^{2= r} _{c} ^{3} \partial _{r} \Phi . Thus : $$\backslash kappa^2\; =\; \backslash partial\_r^2\backslash Phi\; +\; \backslash frac\backslash partial\_r\; \backslash Phi$$The frequency of a circular orbit is \Omega ^{2} _{c} = (1 r)/(\partial) _{r} \Phi which finally yields :$$\backslash kappa^2=4\backslash Omega\_c^2\; +\; 2r\backslash Omega\_c\; \backslash frac$$

References

1. p161, Astrophysical Flows, Pringle and King 2007