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
, the epicyclic frequency
κ
is given by This quantity can be used to examine the 'boundaries' of an accretion disc: when
κ
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
For a Keplerian disk,
κ =
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
Starting from the equations of movement in cylindrical coordinates :
θ ¨
The second line implies that the specific angular momentum is conserved.
We can then define an effective potential
θ ˙
{\displaystyle {\begin{aligned}{\ddot {r}}&=-\partial _{r}\Phi _{eff}\\{\ddot {z}}&=-\partial _{z}\Phi _{eff}\end{aligned}}}
We can apply a small perturbation
to the circular orbit :
{\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
κ
In a circular orbit
κ
The frequency of a circular orbit is
which finally yields :
κ
{\displaystyle \kappa ^{2}=4\Omega _{c}^{2}+2r\Omega _{c}{\frac {d\Omega _{c}}{dr}}}