Lagrange point

Lagrange points arise in the idealized restricted three-body problem: two massive bodies orbit their common centre of mass, while a third body has so little mass that it does not significantly alter their motion. In a reference frame rotating with the two large bodies, five positions have a balance of gravitational and centrifugal effects. A small object placed at one of these points can keep the same relative position to the larger pair in the ideal model. This does not mean gravity is absent, nor that an object there is motionless in an inertial frame. Real spacecraft usually follow orbits around a point rather than sitting exactly on it.

L1, L2 and L3 lie on the line through the two large bodies: L1 between them, L2 beyond the smaller body, and L3 beyond the larger body on the opposite side. L4 and L5 lie in the orbital plane, each forming an equilateral triangle with the two primary bodies; they lead and trail the smaller primary in its orbit. In the Sun–Earth system, L1 offers a useful vantage for observing the Sun, while L2, beyond Earth as seen from the Sun, can keep a telescope in a convenient orientation relative to Earth and the Sun. L4 and L5 are associated with populations of co-orbiting asteroids in some systems.

The geometry of an equilibrium point does not guarantee that a displaced object will return to it. L1, L2 and L3 are unstable: small deviations tend to grow, so spacecraft using nearby orbits need periodic trajectory corrections. L4 and L5 are stable against small disturbances only when the mass ratio of the two primaries satisfies a threshold; they are not unconditionally stable in every two-body system. This difference explains why natural objects can accumulate near some triangular points, whereas spacecraft around collinear points require active station-keeping.

Lagrange regions provide useful observing conditions, but they are not fuel-free parking places for every mission. A spacecraft near Sun–Earth L1 can monitor the Sun and solar wind before disturbances reach Earth. Sun–Earth L2 supports space telescopes that need a relatively unobstructed view of deep space while keeping the Sun, Earth and Moon on the shielded side. The James Webb Space Telescope orbits the Sun near L2, about 1.5 million kilometres from Earth; it does not remain stationary precisely at the point. These applications exploit the geometry and orbital dynamics of the system, with mission design balancing observation needs, communications and correction costs.