= Schwarzschild launch-orbit recurrence
{c}
{title2=$\Phi_r/(2\pi)\simeq(1-6GM/(c^2R))^{-1/2}$}
For a small radial excursion about a stable circular <timelike geodesic> in <Schwarzschild spacetime>, the inverse radius oscillates with azimuthal frequency $\sqrt{1-6GM/(c^2R)}$. 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.
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