Rosette orbit ($r(\theta )=r(\theta +2\pi /\nu )$)

A planar orbit whose radius oscillates as its polar angle increases, producing successive lobes around the force center. For nonconstant $r$ with fundamental angular radial period $2\pi /\nu$ and nonzero specific angular momentum, the full position and velocity repeat precisely when $\nu$ is a rational number, provided the radius is bounded above and away from zero. If $\nu =p/q$ in lowest terms, $p$ radial periods advance the angle by $2\pi q$; $dt=r^{2}d\theta /h$ then gives a finite time period. This proves closure rather than merely a spatial self-intersection.