Zero-velocity surface

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.

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 ${\displaystyle \omega }$ is the rotation rate, ${\displaystyle x,y}$ the particle's location in the rotating coordinate system, ${\displaystyle r_{1},r_{2}}$ the distances to the bodies, and ${\displaystyle \mu _{1},\mu _{2}}$ their masses times the gravitational constant.

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