Radial epicyclic frequency (${\kappa }_r$)

The radial epicyclic frequency describes small radial oscillations about a circular orbit. For specific angular momentum $h(r)$, ${\kappa }_r^2=r^{-3}d(h^2)/dr$.

For inviscid cylindrical rotation without a magnetic field, an axisymmetric radial interchange conserves each parcel's specific angular momentum $l=r^2\Omega$. Its restoring coefficient is ${\kappa }^2=r^{-3}d(l^2)/dr$. Increasing $l^2$ gives centrifugal stability; decreasing $l^2$ gives an exponentially growing interchange mode when radial displacement is allowed by the boundaries. A vanishing derivative is marginal, not a strictly restoring condition.

// Target: astrophysics.bigb

For a nonnegative, radially nonincreasing spherical mass density, ${\Omega }^2=GM(r)/r^3$ and ${\kappa }^2={\Omega }^2+4\pi G\rho (r)$. Since $\rho (r)$ does not exceed the mean enclosed density, the radial epicyclic frequency lies between $\Omega$ and $2\Omega$. General axisymmetric disks need not satisfy this bound.

For $\Phi =-GM/r+\alpha {\omega }^2{r}^2/2$, nearly circular motion has ${\Omega }^2=GM/r^3+\alpha {\omega }^2$ and ${\kappa }^2=GM/r^3+4\alpha {\omega }^2$. Their generally nonrational ratio makes small-eccentricity orbits precess even if $\alpha$ was added slowly. This illustrates adiabatic actions do not preserve orbital closure.

Epicyclic motion is the small oscillation of a displaced fluid element or particle about a circular orbit. Its natural radial frequency is the radial epicyclic frequency.

For $k_y=0$ and $k_z\ne 0$, pressure elimination gives ${\dot{v}}_x=2\Omega \chi v_y-\nu k^2{v}_x$ and ${\dot{v}}_y=-2(\Omega -A)v_x-\nu k^2{v}_y$, where $\chi =k_z^2/k^2$. The inviscid frequency is the radial epicyclic frequency times $\sqrt{\chi }$. With $w=v_x/\sqrt{\chi }$, $(\Omega -A)|w{|}^2+\Omega |v_y{|}^2$ is conserved when viscosity vanishes. In a Keplerian flow it bounds kinetic-energy amplification by four, whereas with zero rotation the same system has the algebraic lift-up effect.

// Target: turbulence.bigb

A small planar velocity kick on a stellar circular orbit produces a local epicyclic motion in the source's rotating reference frame. With outward $x$, forward $y$, and initial displacements zero, Hill equations give $x/(\gamma a)=2\cos \theta (1-\cos nt)+\sin \theta \sin nt$ and $y/(\gamma a)=\cos \theta (4\sin nt-3nt)+2\sin \theta (\cos nt-1)$. Tangential kicks combine an epicycle with Keplerian shear; a radial kick gives a closed leading-order epicyclic ellipse.

The azimuthal advance over one complete small radial oscillation is $\Delta \psi =2\pi \Omega /\kappa$. The oscillatory azimuthal correction averages to zero at first order. This angle compares successive radial turning points of the same type and describes apsidal precession when it differs from a full revolution.

In a Cartesian frame rotating with an epicyclic guiding center, a small planar oscillation obeys $x=X\cos (\kappa t+\chi )$ and $y=-(2\Omega X/\kappa )\sin (\kappa t+\chi )$. The ellipse is elongated azimuthally when the radial epicyclic frequency satisfies $\kappa <2\Omega$. Conservation of angular momentum makes the motion around the guiding centre retrograde.

The epicyclic guiding center is the circular-orbit location about which a particle performs epicyclic motion. In a Keplerian shearing sheet, the conserved canonical momentum is $p_y=\dot{y}+2\Omega x=\Omega x_0/2$. The guiding center therefore labels the particle's first-order specific angular momentum offset, even while its instantaneous radial coordinate oscillates.

For epicyclic motion in a Keplerian shearing sheet, write $x-x_0=\mathrm{Re}(A{e}^{-i\Omega t})$. The energy of the radial harmonic oscillator per unit mass is $E_{\mathrm{epi}}={\Omega }^2|A{|}^2/2$. The full horizontal Jacobi energy in a shearing sheet also includes the negative guiding-center term $-3{\Omega }^2{x}_0^2/8$.