# Celestial Mechanics

## The Three-body Problem

No closed general solution has been found for the problem of systems of three or more bodies whose motions are controlled by their mutual gravitational attractions in a form analogous to the generalized Kepler's Laws for the two-body problem.

However, in 1772 Joseph Lagrange (1736-1813) found a special stable solution known as the Restricted Three-Body Problem. If the second body in the three-body system has a mass M2 less than 0.04M1 where M1 is the mass of the most massive Body 1, then there are five stability points in the orbital plane of Bodies 1 and 2. Three of these points, L1, L2, and L3 lie on the line joining Bodies 1 and 2. The stability of particles placed at these points is minimal; slight perturbations will cause them to move away from these points indefinitely. The points L4 and L5, respectively 60° ahead of and 60° behind Body 2 in its orbit around the system's center of mass, are more stable; particles placed there will, if slightly perturbed, go into orbits around these points.

Lagrange's solution became relevant to the solar system in 1906 when Max Wolf (1863–1952) discovered the asteroid Achilles in Jupiter's orbit but about 60° ahead of it (near the L4 point of the solution). Several hundred such asteroids are now known; they are called the Trojan asteroids, since they are named for heroes of the Trojan War. Following the three-body problem, the Sun is Body 1, Jupiter is Body 2, and the asteroids Achilles, Agamemnon, Ajax, Diomedes, Odysseus, and other asteroids named after Greek heroes cluster around the L4 point of Jupiter's orbit, forming the "Greek camp." The asteroids Anchises, Patroclus, Priam, Aneas, that are named for Trojan heroes cluster around the L5 point (60° behind Jupiter in its orbit), forming the "Trojan camp." The L4 and L5 points of the orbits of Earth, Mars, and Saturn around the Sun and of the Moon's orbits around Earth have been searched for the presence of small bodies ranging in size from asteroids to interplanetary dust without confirmed success. In Saturn's satellite system, with Saturn as Body 1 and its satellite Dione as Body 2, Saturn's small satellite Helene orbits Saturn in Dione's orbit near the L4 point; with Saturn's satellite Tethys as Body 2, Saturn's satellite Telesto orbits in Tethys' orbit close to the leading L4 point and Calypso orbits close to the-following L5 point.