RK4-integrated gravitational N-body simulation with chaotic trajectories, Lagrange points, and the famous figure-8 choreography
In 1887 Poincaré proved that three gravitating bodies have no general closed-form solution. The system has 18 degrees of freedom but only 10 conserved quantities (energy, momentum, angular momentum, centre of mass). This deficit means trajectories are generically non-periodic and chaotic — exponentially sensitive to initial conditions.
Discovered by Chenciner and Montgomery (2000), three equal-mass bodies can chase each other around a stable figure-8 curve, each displaced by one-third of the period. This was found analytically using the calculus of variations. Hundreds of other periodic choreographies have since been found numerically, but the figure-8 is the most famous and one of few known to be linearly stable.
In the restricted three-body problem (third body massless), five equilibrium points exist. L1/L2/L3 are on the axis and unstable — useful for spacecraft but require station-keeping. L4/L5 form equilateral triangles with the primaries and are stable when m₁/m₂ > 24.96, explaining why Jupiter's L4/L5 host thousands of Trojan asteroids. JWST, Herschel, and Planck all orbited L2.
Chaotic three-body orbits have positive Lyapunov exponents: nearby trajectories diverge as e^(λt). The inverse 1/λ gives the Lyapunov time — the timescale over which predictions are meaningful. For the Solar System it is ~5 million years. Numerical integrations (REBOUND, Mercury, IAS15) use adaptive timesteps to maintain energy conservation to one part in 10¹⁰ over billions of years.
| Configuration | Masses | Type | Stable? | Period | Notes |
|---|---|---|---|---|---|
| Figure-8 choreography | 1:1:1 | Periodic | Yes (linearly) | T ≈ 6.32 | Chenciner-Montgomery 2000 |
| Lagrange equilateral (L4/L5) | Any (mass ratio >24.96) | Equilibrium | Yes | Same as primary | Jupiter Trojans, Earth Trojans |
| Euler collinear | Any | Equilibrium | No (unstable) | Depends on masses | L1, L2, L3 Lagrange points |
| Hierarchical triple | m₁≫m₂+m₃ | Quasi-stable | Long-term | Two timescales | Most triple star systems |
| Sun-Earth-Moon | 333000:1:0.0123 | Hierarchical | Yes (Hill stable) | 27.3 days (Moon) | Moon inside Earth Hill sphere |
| Binary + flyby | 1:1:1 | Chaotic exchange | No | — | One body ejected after close pass |
| Broucke-Hénon figure-8 variant | 1:1:1 | Periodic | Marginal | T ≈ 16.4 | Found by Šuvakov & Dmitrašinović 2013 |
| Alpha Centauri system | 1.1:0.9:m_brown | Hierarchical | Yes (outer stable) | 79.9 yr | Proxima Cen orbits at 13,000 AU |
Poincaré proved in 1887 that the system has insufficient conserved quantities relative to its degrees of freedom, making it non-integrable. The trajectories are generically chaotic: tiny uncertainties in initial conditions grow exponentially, making long-term prediction fundamentally impossible — not just computationally hard, but mathematically forbidden.
Lagrange points are positions where a small body experiences balanced gravitational forces from two larger bodies. L1/L2/L3 are unstable but require only small thrusts for station-keeping; their fixed positions relative to Earth make them ideal for observatories. JWST orbits Sun-Earth L2 (~1.5 million km from Earth), remaining in the same relative geometry for unobstructed views of deep space.
Three equal-mass bodies orbiting in a figure-8 choreography, discovered by Chenciner and Montgomery (2000). All three bodies share the same path, displaced by T/3. It is linearly stable — small perturbations lead to bounded oscillations rather than escape. It exists under the exact conditions of equal masses and a specific ratio of period to semi-major axis, making it exquisitely sensitive to initial conditions.
The Hill sphere radius r_H ≈ a(m/3M)^(1/3) defines the region where a planet's gravity dominates over the Sun's tidal forces. Earth's Hill sphere extends ~1.5 million km; the Moon at 384,000 km is safely inside it. Moons beyond ~0.5 r_H are destabilised over long timescales. Neptune's moon Triton, in a retrograde orbit, is believed to be a captured Kuiper Belt Object that passed within Neptune's Hill sphere.
Learn about the three centuries of mathematics from Kepler to Poincaré, and how modern N-body integrators handle the three-body problem in star clusters and planetary systems.
Read: Three-Body Problem & Chaos →