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 ${\displaystyle \Omega }$, 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, ${\displaystyle \kappa =\Omega }$.

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 : $${\displaystyle {\vec {r}}=r_{0}{\vec {e}}_{r}+\delta {\vec {r}}}$$ So, $${\displaystyle {\ddot {\vec {r}}}+\delta {\ddot {\vec {r}}}=-{\vec {\nabla }}\Phi _{eff}({\vec {r}}+\delta {\vec {r}})\approx -{\vec {\nabla }}\Phi _{eff}({\vec {r}})-\partial _{r}^{2}\Phi _{eff}({\vec {r}})\delta r-\partial _{z}^{2}\Phi _{eff}({\vec {r}})\delta z}$$

And thus : $${\displaystyle {\begin{aligned}\delta {\ddot {r}}&=-\partial _{r}^{2}\Phi _{eff}\delta r=-\Omega _{r}^{2}\delta r\\\delta {\ddot {z}}&=-\partial _{r}^{2}\Phi _{eff}\delta z=-\Omega _{z}^{2}\delta z\end{aligned}}}$$ We then note $${\displaystyle \kappa ^{2}=\Omega _{r}^{2}=\partial _{r}^{2}\Phi _{eff}=\partial _{r}^{2}\Phi +{\frac {3h^{2}}{r^{4}}}}$$ In a circular orbit ${\displaystyle h_{c}^{2}=r^{3}\partial _{r}\Phi }$. Thus : $${\displaystyle \kappa ^{2}=\partial _{r}^{2}\Phi +{\frac {3}{r}}\partial _{r}\Phi }$$ The frequency of a circular orbit is ${\displaystyle \Omega _{c}^{2}={\frac {1}{r}}\partial _{r}\Phi }$ which finally yields : $${\displaystyle \kappa ^{2}=4\Omega _{c}^{2}+2r\Omega _{c}{\frac {d\Omega _{c}}{dr}}}$$