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 :

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

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

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

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 :

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

We then note

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

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 $\Omega _{c}^{2}={\frac {1}{r}}\partial _{r}\Phi$ which finally yields :

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