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

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 $$6GM/c^{2}$$.

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

The second line implies that the specific angular momentum is conserved. We can then define an effective potential $$ \Phi_{eff} = \Phi - \frac {1}{2} r^2\dot\theta^2 = \Phi - \frac{h^2}{2r^2} $$ and so : $$ \begin{align}\ddot r &= -\partial_r \Phi_{eff}\\ \ddot z &= - \partial_z \Phi_{eff}\end{align} $$

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, $$ \ddot{\vec r} + \delta \ddot{\vec r} = -\vec \nabla \Phi_{eff}(\vec r + \delta \vec r)\approx-\vec \nabla \Phi_{eff} (\vec r) - \partial_r^2 \Phi_{eff}(\vec r)\delta r - \partial_z^2 \Phi_{eff}(\vec r)\delta z $$

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