The gravitational motion of three interacting bodies under their coupled Newtonian mechanics equations. When one body's mass is negligible, the restricted three-body problem neglects its influence on the other two while retaining both forces on its trajectory.
A three-body orbit is a trajectory governed by the gravitational three-body problem. For three point masses at positions , 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]
Introduction to Celestial Mechanics by Richard Fitzpatrick. Online Circular restricted three-body problem, equations 9.5–9.9 and the negligible-third-mass assumption
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.
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The three-body problem is a classic problem in physics and mathematics that involves predicting the motion of three celestial bodies as they interact with one another through gravitational forces. The challenge of the three-body problem arises from the fact that while the gravitational interactions between two bodies can be described by simple equations (the two-body problem), adding a third body leads to a complex and chaotic system that generally cannot be solved analytically.