Zero-velocity surface

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.

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:

${\displaystyle C=\omega ^{2}(x^{2}+y^{2})+2\left({\frac {\mu _{1}}{r_{1}}}+{\frac {\mu _{2}}{r_{2}}}\right)-\left({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}\right)}$

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.[2]

For a given value of ${\displaystyle C}$, points on the surface

${\displaystyle C=\omega ^{2}(x^{2}+y^{2})+2\left({\frac {\mu _{1}}{r_{1}}}+{\frac {\mu _{2}}{r_{2}}}\right)}$

require that ${\displaystyle {\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}=0}$. 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.[3]

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.[2]

The concept can be generalized to more complex problems, for example with masses in elliptic orbits,[4] the general planar three-body problem,[5] the four-body problem with solar wind drag,[6] or in rings.[7]

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 ${\displaystyle C}$ is changed.[8] Since trajectories are confined by the surfaces, a trajectory that seeks to escape (or enter) a region with minimal energy will typically pass close to the Lagrange point, which is used in low-energy transfer trajectory planning.

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.[9]

• Hill sphere
• Low-energy transfer
• Orbital mechanics