page_title: Zero-velocity surface
section_title: Three-body problem
hierarchical_section_title: Zero-velocity surface / Three-body problem
caption_attribution_description: English: 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 an excluded region for the trajectory, enclosed by a zero-velocity surface that cannot be crossed.
context_page_description: The zero-velocity surface is a concept that relates to the N-body problem of gravity. It represents a surface a body of given energy cannot cross, since it would have zero velocity on the surface. It was first introduced by George William Hill. The zero-velocity surface is particularly significant when working with weak gravitational interactions among orbiting bodies.
context_section_description: 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  is the rotation rate,  the particle's location in the rotating coordinate system,  the distances to the bodies, and  their masses times the gravitational constant.
  For a given value of , points on the surface 
  require that . That is, the particle will not be able to cross over this surface (since the squared velocity would have to become negative). This is the zero-velocity surface of the problem.
  Note that this means zero velocity in the rotating frame: in a non-rotating frame the particle is seen as rotating with the other bodies. The surface also only predicts what regions cannot be entered, not the shape of the trajectory within the surface.
caption: 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 an excluded region for the trajectory, enclosed by a zero-velocity surface that cannot be crossed.
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