Past exam of the formation of Structure in the Universe course of the University of Cambridge / 2016 / Paper 1 / 2 / d / Solution

To test radial stability of an axisymmetric circular orbit, let $\Phi (r)$ be the time-independent gravitational potential per unit stellar mass in the midplane. Axial symmetry conserves the specific angular momentum $j=r^2\dot{\phi }$. The radial equation isA circular orbit at $r_0$ obeys $j_0^2=r_0^3{\Phi }^{\prime }(r_0)$. Choose the guiding circular orbit with the same conserved angular momentum as the perturbed star. Setting $r=r_0+x$ and expanding at fixed $j=j_0$ giveswhere the radial epicyclic frequency isHere ${\Omega }^2(r)={\Phi }^{\prime }(r)/r$ and $j_c=r^2\Omega$ refer to the family of circular orbits. They are differentiated as functions of radius only after the perturbation equation has been expanded at conserved stellar $j$.

For a radially stable circular orbit, ${\kappa }^2>0$, so to linear orderThus the orbit performs small simple harmonic radial oscillations about its guiding radius.

Symmetry alone is insufficient to guarantee stability: ${\kappa }^2<0$ gives exponentially growing displacement, and ${\kappa }^2=0$ is marginal. The intended statement assumes a stable unperturbed orbit.

For a Kepler potential $\kappa =\Omega$; for a flat circular-speed curve $\kappa =\sqrt{2}\Omega$; for solid-body rotation $\kappa =2\Omega$. Conservation of $j$ also gives $\dot{\phi }\simeq {\Omega }_0(1-2x/r_0)$, explaining the accompanying azimuthal epicycle rather than a purely radial motion in the inertial plane.