Astronomy:Epicyclic frequency

Short description: Characteristic of accretion discs

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

${\kappa }^2\equiv \frac{2\mathrm{\Omega }}{R}\frac{d}{dR}(R^2\mathrm{\Omega })$, 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, $\kappa =\mathrm{\Omega }$.

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 $\mathrm{\Phi }(r,z)=\mathrm{\Phi }(r,-z)$. Starting from the equations of movement in cylindrical coordinates : $$\begin{array}{rl}\ddot{r}-r{\dot{\theta }}^2& =-{\partial }_r\mathrm{\Phi }\\ r\ddot{\theta }+2\dot{r}\dot{\theta }& =0\\ \ddot{z}& =-{\partial }_z\mathrm{\Phi }\end{array}$$

The second line implies that the specific angular momentum is conserved. We can then define an effective potential ${\mathrm{\Phi }}_{\text{eff}}=\mathrm{\Phi }-\frac{1}{2}{r}^2{\dot{\theta }}^2=\mathrm{\Phi }+\frac{h^2}{2r^2}$ and so : $$\begin{array}{rl}\ddot{r}& =-{\partial }_r{\mathrm{\Phi }}_{\text{eff}}\\ \ddot{z}& =-{\partial }_z{\mathrm{\Phi }}_{\text{eff}}\end{array}$$

We can apply a small perturbation $\delta \overrightarrow{r}=\delta r{\overrightarrow{e}}_r+\delta z{\overrightarrow{e}}_z$ to the circular orbit : $$\overrightarrow{r}=r_0{\overrightarrow{e}}_r+\delta \overrightarrow{r}$$ So, $$\ddot{\overrightarrow{r}}+\delta \ddot{\overrightarrow{r}}=-\overrightarrow{\mathrm{\nabla }}{\mathrm{\Phi }}_{\text{eff}}(\overrightarrow{r}+\delta \overrightarrow{r})\approx -\overrightarrow{\mathrm{\nabla }}{\mathrm{\Phi }}_{\text{eff}}(\overrightarrow{r})-\underset{r}{\overset{2}{\partial }}{\mathrm{\Phi }}_{\text{eff}}(\overrightarrow{r})\delta r-\underset{z}{\overset{2}{\partial }}{\mathrm{\Phi }}_{\text{eff}}(\overrightarrow{r})\delta z$$

And thus : $$\begin{array}{rl}\delta \ddot{r}& =-\underset{r}{\overset{2}{\partial }}{\mathrm{\Phi }}_{\text{eff}}\delta r=-{\mathrm{\Omega }}_r^2\delta r\\ \delta \ddot{z}& =-\underset{r}{\overset{2}{\partial }}{\mathrm{\Phi }}_{\text{eff}}\delta z=-{\mathrm{\Omega }}_z^2\delta z\end{array}$$ We then note $${\kappa }^2={\mathrm{\Omega }}_r^2=\underset{r}{\overset{2}{\partial }}{\mathrm{\Phi }}_{\text{eff}}=\underset{r}{\overset{2}{\partial }}\mathrm{\Phi }+\frac{3h^2}{r^4}$$ In a circular orbit $h_c^2=r^3{\partial }_r\mathrm{\Phi }$. Thus : $${\kappa }^2=\underset{r}{\overset{2}{\partial }}\mathrm{\Phi }+\frac{3}{r}{\partial }_r\mathrm{\Phi }$$ The frequency of a circular orbit is ${\mathrm{\Omega }}_c^2=\frac{1}{r}{\partial }_r\mathrm{\Phi }$ which finally yields : $${\kappa }^2=4{\mathrm{\Omega }}_c^2+2r{\mathrm{\Omega }}_c\frac{d{\mathrm{\Omega }}_c}{dr}$$