Epicyclic Frequency

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
: $\kappa^ \equiv \frac\frac(R^2 \Omega)$, where R is the radial co-ordinate.p161, Astrophysical Flows, Pringle and King 2007

This quantity can be used to examine the 'boundaries' of an accretion disc: when $\kappa^$ 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 $6GM/c^$.

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 :
$\begin \ddot r - r \dot \theta^2 &= -\partial_r \Phi \\r \ddot \theta + 2 \dot r\dot\theta &= 0 \\ \ddot z &= -\partial_z \Phi \end$

The second line implies that the specific angular momentum is conserved. We can then define an effective potential $\Phi_ = \Phi - \frac r^2\dot\theta^2 = \Phi + \frac$ and so :
$\begin\ddot r &= -\partial_r \Phi_\\ \ddot z &= - \partial_z \Phi_\end$

We can apply a small perturbation $\delta\vec r = \delta r \vec e_r + \delta z \vec e_z$ to the circular orbit : $\vec r = r_0 \vec e_r + \delta \vec r$
So, $\ddot + \delta \ddot = -\vec \nabla \Phi_(\vec r + \delta \vec r)\approx-\vec \nabla \Phi_ (\vec r) - \partial_r^2 \Phi_(\vec r)\delta r - \partial_z^2 \Phi_(\vec r)\delta z$

And thus :
$\begin \delta \ddot r &= - \partial_r^2 \Phi_ \delta r = -\Omega_r^2 \delta r\\\delta \ddot z &= - \partial_r^2 \Phi_ \delta z = -\Omega_z^2 \delta z\end$
We then note $\kappa^2 = \Omega_r^2 = \partial_r^2\Phi_ = \partial_r^2\Phi + \frac$
In a circular orbit $h_c^2=r^3 \partial_r \Phi$. Thus :
$\kappa^2 = \partial_r^2\Phi + \frac\partial_r \Phi$
The frequency of a circular orbit is $\Omega_c^2 = \frac 1r \partial_r \Phi$ which finally yields :
$\kappa^2=4\Omega_c^2 + 2r\Omega_c \frac{dr}$

References

Fluid dynamics
Astrophysics
Equations of astronomy