Schwarzschild launch-orbit recurrence (${\Phi }_r/(2\pi )\simeq (1-6GM/(c^{2}R){)}^{-1/2}$)

For a small radial excursion about a stable circular timelike geodesic in Schwarzschild spacetime, the inverse radius oscillates with azimuthal frequency $\sqrt{1-6GM/(c^{2}R)}$. A satellite launched tangentially returns to its launch radius once per radial period. A later exact encounter with a point station at the launch azimuth requires the exact radial-to-azimuthal rotation ratio to be rational. An irrational ratio instead gives dense, arbitrarily close recurrence, which still threatens objects of finite size. The linearized rotation ratio is an approximation; exact orbital closure uses the actual orbit's radial advance.