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 ${\displaystyle \Omega }$, the epicyclic frequency ${\displaystyle \kappa }$ is given by

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

This quantity can be used to examine the 'boundaries' of an accretion disc: when ${\displaystyle \kappa ^{2))$ 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 ${\displaystyle 6GM/c^{2))$.

For a Keplerian disk, ${\displaystyle \kappa =\Omega }$.

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 ${\displaystyle \Phi (r,z)=\Phi (r,-z)}$. Starting from the equations of movement in cylindrical coordinates :

$${\displaystyle {\begin{aligned}{\ddot {r))-r{\dot {\theta ))^{2}&=-\partial _{r}\Phi \\r{\ddot {\theta ))+2{\dot {r)){\dot {\theta ))&=0\\{\ddot {z))&=-\partial _{z}\Phi \end{aligned))}$$

The second line implies that the specific angular momentum is conserved. We can then define an effective potential ${\displaystyle \Phi _{eff}=\Phi -{\frac {1}{2))r^{2}{\dot {\theta ))^{2}=\Phi -{\frac {h^{2)){2r^{2))))$ and so :

$${\displaystyle {\begin{aligned}{\ddot {r))&=-\partial _{r}\Phi _{eff}\\{\ddot {z))&=-\partial _{z}\Phi _{eff}\end{aligned))}$$

We can apply a small perturbation ${\displaystyle \delta {\vec {r))=\delta r{\vec {e))_{r}+\delta z{\vec {e))_{z))$ to the circular orbit :

$${\displaystyle {\vec {r))=r_{0}{\vec {e))_{r}+\delta {\vec {r))}$$

$${\displaystyle {\ddot {\vec {r))}+\delta {\ddot {\vec {r))}=-{\vec {\nabla ))\Phi _{eff}({\vec {r))+\delta {\vec {r)))\approx -{\vec {\nabla ))\Phi _{eff}({\vec {r)))-\partial _{r}^{2}\Phi _{eff}({\vec {r)))\delta r-\partial _{z}^{2}\Phi _{eff}({\vec {r)))\delta z}$$

And thus :

$${\displaystyle {\begin{aligned}\delta {\ddot {r))&=-\partial _{r}^{2}\Phi _{eff}\delta r=-\Omega _{r}^{2}\delta r\\\delta {\ddot {z))&=-\partial _{r}^{2}\Phi _{eff}\delta z=-\Omega _{z}^{2}\delta z\end{aligned))}$$

$${\displaystyle \kappa ^{2}=\Omega _{r}^{2}=\partial _{r}^{2}\Phi _{eff}=\partial _{r}^{2}\Phi +{\frac {3h^{2)){r^{4))))$$

${\displaystyle h_{c}^{2}=r^{3}\partial _{r}\Phi }$

$${\displaystyle \kappa ^{2}=\partial _{r}^{2}\Phi +{\frac {3}{r))\partial _{r}\Phi }$$

${\displaystyle \Omega _{c}^{2}={\frac {1}{r))\partial _{r}\Phi }$

$${\displaystyle \kappa ^{2}=4\Omega _{c}^{2}+2r\Omega _{c}{\frac {d\Omega _{c)){dr))}$$