Three-body orbit

A three-body orbit is a trajectory governed by the gravitational three-body problem. For three point masses at positions ${\mathbf{r}}_i$, Newton's law of universal gravitation givesBoth attracting bodies act simultaneously; switching between separate Kepler orbits is instead a patched conic approximation. In the restricted three-body problem, one mass is negligible, so its trajectory responds to the two primary bodies without appreciably changing their motion. Taking their mutual orbit to be circular gives the circular restricted three-body problem.^[ref]

For example, an asteroid influenced by the Sun and a planet can deviate from any single fixed Kepler orbit. Around a synchronously rotating Moon, a maintained trajectory involving the planet–moon system is likewise distinct from a simple moon-centred stationary Kepler orbit. The Hill sphere estimates where the moon's local gravitational dominance can support that simpler approximation; it is not a sharp boundary beyond which all three-body trajectories are impossible.