A Novel Collocation Method to Solve the PeriodicOrbits: Application to the Circular RestrictedThree-Body Problem

We develop a novel collocation method to solve for stable closed orbits in a static/periodic potential field. Periodic orbits in the Circular Restricted Three-Body Problem (CR3BP) form the foundation of dynamics in binary-star systems, but are usually computed via costly shooting methods or long N -body integrations; a fast, direct solver for closed orbits can therefore help find stability regions and provide high-quality initial conditions for follow-up simulations. We apply it to the CR3BP, finding closed trajectories that may resemble planet orbits in a binary star system or pressure-less streamlines in tidally distorted accretion disks, consistent with orbits found by more traditional methods. In tests withM1M1+M2=0.8and n = 64128 collocation points, the solver converges in 513 Newton steps to residual norms 1013with wall-clock times of 0.20.8 s per orbit. For the same setup, closed circumbinary (P-type) orbits require < 0.3, implying ap/abin (1+)2/3 1.3. Around the primary, closed circumstellar (S-type) orbits require > 1.8, implying ap/abin (1 + )2/3M11/3 0.47.