epicyclic frequency

{{Short description|Characteristic of accretion discs}}

{{one source |date=March 2024}}

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 $\backslash Omega$, the epicyclic frequency $\backslash kappa$ is given by

: $\backslash kappa^\{2\}\; \backslash equiv\; \backslash frac\{2\; \backslash Omega\}\{R\}\backslash frac\{d\}\{dR\}(R^2\; \backslash Omega)$, where R is the radial co-ordinate.p161, Astrophysical Flows, Pringle and King 2007

This quantity can be used to examine the 'boundaries' of an accretion disc: when $\backslash kappa^\{2\}$ 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, $\backslash kappa\; =\; \backslash Omega$.

== 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 $\backslash Phi\; (r,z)\; =\; \backslash Phi\; (r,-z)$.

Starting from the equations of movement in cylindrical coordinates :

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

We can apply a small perturbation $\backslash delta\backslash vec\; r\; =\; \backslash delta\; r\; \backslash vec\; e\_r\; +\; \backslash delta\; z\; \backslash vec\; e\_z$ to the circular orbit : $$\backslash vec\; r\; =\; r\_0\; \backslash vec\; e\_r\; +\; \backslash delta\; \backslash vec\; r$$

So, $$\backslash ddot\{\backslash vec\; r\}\; +\; \backslash delta\; \backslash ddot\{\backslash vec\; r\}\; =\; -\backslash vec\; \backslash nabla\; \backslash Phi\_\{eff\}(\backslash vec\; r\; +\; \backslash delta\; \backslash vec\; r)\backslash approx-\backslash vec\; \backslash nabla\; \backslash Phi\_\{eff\}\; (\backslash vec\; r)\; -\; \backslash partial\_r^2\; \backslash Phi\_\{eff\}(\backslash vec\; r)\backslash delta\; r\; -\; \backslash partial\_z^2\; \backslash Phi\_\{eff\}(\backslash vec\; r)\backslash delta\; z$$

And thus :

We then note $$\backslash kappa^2\; =\; \backslash Omega\_r^2\; =\; \backslash partial\_r^2\backslash Phi\_\{eff\}\; =\; \backslash partial\_r^2\backslash Phi\; +\; \backslash frac\{3\; h^2\}\{r^4\}$$

In a circular orbit $h\_c^2=r^3\; \backslash partial\_r\; \backslash Phi$. Thus :

$$\backslash kappa^2\; =\; \backslash partial\_r^2\backslash Phi\; +\; \backslash frac\{3\}\{r\}\backslash partial\_r\; \backslash Phi$$

The frequency of a circular orbit is $\backslash Omega\_c^2\; =\; \backslash frac\; 1r\; \backslash partial\_r\; \backslash Phi$ which finally yields :

$$\backslash kappa^2=4\backslash Omega\_c^2\; +\; 2r\backslash Omega\_c\; \backslash frac\{d\backslash Omega\_c\}\{dr\}$$