Hill sphere

The Hill sphere is a common model for the calculation of a gravitational sphere of influence. It is the most commonly used model to calculate the spatial extent of gravitational influence of an astronomical body (m) in which it dominates over the gravitational influence of other bodies, particularly a primary (M). It is sometimes confused with other models of gravitational influence, such as the Laplace sphere or being called the Roche sphere, the latter causing confusion with the Roche limit. It was defined by the American astronomer George William Hill, based on the work of the French astronomer Édouard Roche.

To be retained by a more gravitationally attracting astrophysical object—a planet by a more massive star, a moon by a more massive planet—the less massive body must have an orbit that lies within the gravitational potential represented by the more massive body's Hill sphere. That moon would, in turn, have a Hill sphere of its own, and any object within that distance would tend to become a satellite of the moon, rather than of the planet itself.

One simple view of the extent of the Solar System is that it is bounded by the Hill sphere of the Sun (engendered by the Sun's interaction with the galactic nucleus or other more massive stars). A more complex example is the one at right, the Earth's Hill sphere, which extends between the Lagrange points L₁ and L₂, which lie along the line of centers of the Earth and the more massive Sun. The gravitational influence of the less massive body is least in that direction, and so it acts as the limiting factor for the size of the Hill sphere; beyond that distance, a third object in orbit around the Earth would spend at least part of its orbit outside the Hill sphere, and would be progressively perturbed by the tidal forces of the more massive body, the Sun, eventually ending up orbiting the latter.

For two massive bodies with gravitational potentials and any given energy of a third object of negligible mass interacting with them, one can define a zero-velocity surface in space which cannot be passed, the contour of the Jacobi integral. When the object's energy is low, the zero-velocity surface completely surrounds the less massive body (of this restricted three-body system), which means the third object cannot escape; at higher energy, there will be one or more gaps or bottlenecks by which the third object may escape the less massive body and go into orbit around the more massive one. If the energy is at the border between these two cases, then the third object cannot escape, but the zero-velocity surface confining it touches a larger zero-velocity surface around the less massive body at one of the nearby Lagrange points, forming a cone-like point there. At the opposite side of the less massive body, the zero-velocity surface gets close to the other Lagrange point. This limiting zero-velocity surface around the less massive body is its Hill "sphere".