Lagrangian Point - Intuitive Explanation

Intuitive Explanation

Lagrangian points can be explained intuitively using the Earth–Moon system.

Lagrangian points L₂ through L₅ exist only in rotating systems, such as in the monthly orbiting of the Moon about the Earth. At these points, the combined attraction from the two masses is equivalent to what would be exerted by a single mass at the barycenter of the system, sufficient to cause a small body to orbit with the same period.

Imagine a person swinging a stone at the end of a string. The string provides a tension force that continuously accelerates the stone toward the center. To an ant standing on the stone, however, it seems as if there is an opposite force trying to fling it directly away from the center. This apparent force is called the centrifugal force. It is actually simply the outward radial component of the stone's inertia caused by its swing. This same effect is present at the Lagrangian points in the Earth–Moon system, where the analogue of the string is the summed (or net) gravitational attraction of the two masses, and the stone is an asteroid or a spacecraft. The Earth–Moon system and the spacecraft all rotate about this combined center of mass, or barycenter. Because the Earth is much heavier than the Moon, the barycenter is located within the Earth (about 1,700 km/1,100 mi below the surface). Any object gravitationally held by the rotating Earth–Moon system will be attracted to the barycenter to an equal and opposite degree as its tendency to fly off into space.

Unlike the other Lagrangian points, L₁ would exist even in a non-rotating (static or inertial) system. In a rotating system, L₁ is a bit farther from the (less massive) Moon and closer to the (more massive) Earth than it would be in a non-rotating system. L₁ is slightly unstable (see stability, above) because drifting towards the Moon or Earth increases one gravitational attraction while decreasing the other, causing more drift.

At Lagrangian points L₂, L₃, L₄, and L₅, a spacecraft's inertia to move away from the barycenter is balanced by the attraction of gravity toward the barycenter. L₂ and L₃ are slightly unstable because small changes in position upset the balance between gravity and inertia, allowing one or the other force to dominate, so that the spacecraft either flies off into space or spirals in toward the barycenter. Stability at L₄ and L₅ is explained by gravitational equilibrium: if the object were moved into a tighter orbit, it would orbit faster which would counteract the increase in gravity; if the object moves into a wider orbit, the gravity is lower, but it loses speed. The net result is that the object appears constantly to hover or orbit around the L₄ or L₅ point.

The easiest way to understand the resulting stability is to say L₁, L₂, and L₃ positions are as stable as a ball balanced on the tip of a wedge would be stable: any disturbance will toss it out of equilibrium. The L₄, and L₅ positions are stable as a ball at the bottom of a bowl would be stable: small perturbations will move it out of place, but it will drift back toward the center of the bowl.

Note that from the perspective of the smaller-mass object — from the moon, in the preceding example — a spacecraft might appear to orbit in an irregular path about the L₄ or L₅ point, but from the perspective above the orbital plane, it becomes clear that both the smaller mass and the spacecraft are orbiting the larger mass (or more precisely, all of the objects are in orbit around the barycenter of the system); they simply have overlapping orbital paths. This point of view difference is illustrated clearly by animations in the 3753 Cruithne and Coriolis effect articles.