Lagrange Points, Roche Limits & Hill Spheres: Structures & Vocabulary

Lagrange points, Roche limits, and Hill spheres describe three different boundaries in multi-body gravity. Lagrange points are special rotating-frame locations associated with balanced effective forces, the Roche limit marks where tidal forces can disrupt a weak satellite, and the Hill sphere estimates the region where a body's gravity dominates satellite stability against a larger primary.

Lagrange points emerge when gravity and the rotating reference frame allow a small body to remain near a repeating geometry. The Roche limit arises because differential gravity across an extended satellite can exceed its self-gravity or material strength. The Hill sphere comes from competition between a smaller body's gravity and tidal influence from the larger primary.

Spacecraft operate in halo or Lissajous orbits near Sun-Earth Lagrange points, Trojan asteroids cluster near Jupiter's L4 and L5 regions, ring systems often lie inside or near relevant Roche-limit regimes, and satellite populations occupy only a fraction of their planets' Hill spheres.

Five classical Lagrange points exist in the circular restricted three-body problem. L1, L2, and L3 are collinear; L4 and L5 form equilateral triangles. Roche limits depend on density, internal strength, and fluid/rigid assumptions. Hill radius depends mainly on mass ratio and orbital distance.