 Zero-velocity surface

In a rotating restricted three-body model, the Jacobi constant can be written $C=2\Omega -|\dot{x}{|}^2$, where $\Omega$ is the effective potential. The zero-velocity surface $2\Omega =C$ bounds the accessible region: motion requires $2\Omega \ge C$.

The surface is an energy constraint, not a physical wall or a trajectory. As $C$ changes, narrow connections can open near Lagrange points, permitting passage between previously separated regions. In a planar section these boundaries appear as zero-velocity curves. Actual passage still depends on the dynamics within the allowed region.