Zero-velocity surface

A zero-velocity surface is a concept that relates to the N-body problem of gravity.

[2] The zero-velocity surface is particularly significant when working with weak gravitational interactions among orbiting bodies.

In the circular restricted three-body problem two heavy masses orbit each other at constant radial distance and angular velocity, and a particle of negligible mass is affected by their gravity.

By shifting to a rotating coordinate system where the masses are stationary a centrifugal force is introduced.

Energy and momentum are not conserved separately in this coordinate system, but the Jacobi integral remains constant: where

That is, the particle will not be able to cross over this surface (since the squared velocity would have to become negative).

[8] The zero-velocity surface is also an important parameter in finding Lagrange points.

These points correspond to locations where the apparent potential in the rotating coordinate system is extremal.

This corresponds to places where the zero-velocity surfaces pinch and develop holes as

Given a group of galaxies which are gravitationally interacting, the zero-velocity surface is used to determine which objects are gravitationally bound (i.e. not overcome by the Hubble expansion) and thus part of a galaxy cluster, such as the Local Group.

Jacobi constant, a Zero Velocity Surface and Curve (also Hill's curve) [ 1 ]
A trajectory (red) in the planar circular restricted 3-body problem that orbits the heavier body a number of times before escaping into an orbit around the lighter body. The contours denote values of the Jacobi integral. The dark blue region is supposed to be the excluded region for the trajectory, enclosed by a zero-velocity surface that cannot be crossed. However, this figure is incorrect because wherever the trajectory touches the zero-velocity surface it should be perpendicular to it.