Three-Body Surf

Two genuinely different problems

Restricted mode prescribes a circular binary with total mass, separation, and angular frequency set to one. The two neighboring massless probes do not affect it. C = x² + y² + 2(1−μ)/r₁ + 2μ/r₂ − (vx² + vy²) is conserved in rotating coordinates. Shaded forbidden regions use the reference probe’s C; the perturbed probe can have a slightly different C.

Full mode integrates all three mutually attracting point masses with G = 1. Figure-eight masses are equal. Hierarchical masses are (0.5, 0.5, 0.02); encounter masses are (0.5, 0.5, 0.3). Initial positions and velocities are centered on the barycenter and center-of-momentum frame. A perturbation shifts the third body's x coordinate before recentering.

Each experiment evolves a reference, a neighboring initial condition, and a tighter-tolerance repeat of the reference. Adaptive RK4 step doubling uses tolerances 10⁻⁸ and 10⁻¹¹. The chart compares position separation against numerical disagreement; the latter is a convergence indicator, not a rigorous error bound. Finite-time separation alone does not establish chaos.

Each historical point uses its own timestamp for the coordinate transformation. The rotating full-system view has a declared arbitrary angular speed Ω = 1; full-system masses are not held fixed. Lagrange points and Jacobi regions apply only to the restricted circular model.

Stop at pair separation 0.025, coordinate magnitude 40, or time 60. The close-encounter cutoff is a numerical boundary, not a collision or merger. Crossing the outer domain is not certified escape. No softening, finite stellar radii, relativity, or collision physics is modeled.

Background: Tong · black-hole geometry · REBOUND · a reference for high-accuracy gravitational integration. This experiment uses its own RK4-based solver, not IAS15.