Question 1

Why is the three-body problem unsolvable?

Accepted Answer

The three-body problem has no general closed-form solution. In 1887, Poincaré proved that the system is non-integrable: it has too few conserved quantities (only energy, momentum, angular momentum) relative to its 18 degrees of freedom. Trajectories exhibit sensitive dependence on initial conditions — infinitesimally different starting points diverge exponentially, making long-term prediction impossible. This was one of the first demonstrations of deterministic chaos in physics.

Question 2

What are Lagrange points?

Accepted Answer

Lagrange points L1–L5 are equilibrium positions in the restricted three-body problem (where the third body is massless). L1/L2/L3 lie on the line connecting the two main bodies and are unstable — small perturbations cause the third body to drift away. L4 and L5 form equilateral triangles with the two main bodies and are stable (when the mass ratio m1/m2 > 24.96). Trojan asteroids cluster near Jupiter's L4 and L5. The James Webb Space Telescope orbits the Sun-Earth L2 point.

Question 3

What is the figure-8 three-body orbit?

Accepted Answer

The figure-8 choreography, discovered analytically by Chenciner and Montgomery in 2000, is a remarkable periodic solution where three equal-mass bodies chase each other around a figure-8 shaped curve in the plane. All three bodies follow identical paths but displaced in time by one-third of the period. Unlike most three-body orbits, it is stable against small perturbations. Dozens of other periodic 'choreographies' have since been discovered numerically.

Question 4

What is the Hill sphere?

Accepted Answer

The Hill sphere is the region around a body where it dominates the gravitational attraction over a more massive nearby body. For a planet of mass m orbiting a star of mass M at semi-major axis a: r_Hill ≈ a × (m/3M)^(1/3). Earth's Hill sphere radius is ~1.5 million km (about 4 times the Moon's distance), meaning the Moon is safely bound within it. The Moon at ~384,000 km sits deep inside Earth's Hill sphere and is thus stable against solar perturbations.

🧪 Three-Body Gravitational Chaos

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