Zero-Velocity Surface

Author: CislunarSpace

Website: https://cislunarspace.cn

Definition

In the Circular Restricted Three-Body Problem (CR3BP), the zero-velocity surface is obtained by setting velocity to zero in the Jacobi integral expression, yielding the equation 2Ω(x,y,z) = C. This surface divides space into accessible and forbidden regions, with its topology changing as the Jacobi constant C varies. When C > C₁, the spacecraft is confined near either primary; when C₁ > C > C₂, transfer through L₁ is possible; when C₂ > C > C₃, the spacecraft can reach exterior space through L₂.

Key Properties

  • The surface topology changes at critical values C₁, C₂, C₃ corresponding to the Jacobi constants at libration points L₁, L₂, L₃.
  • For large C values, the forbidden region forms closed "bottlenecks" around each primary, preventing transfer.
  • As C decreases, channels open sequentially at L₁, L₂, and L₃, enabling increasingly free motion.
  • The zero-velocity curves on the configuration plane are also known as Hill regions.

References

  • Szebehely, V. Theory of Orbits: The Restricted Problem of Three Bodies. Academic Press, 1967.