Zero-Velocity Surface (ZVS)

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

From the Jacobi integral $v^2 = 2\Omega - C$ , setting $v=0$ yields the surface in configuration space

\Omega(x,y,z) = C/2,

i.e. a level surface of the effective potential $\Omega$ . It is called the zero-velocity surface (ZVS), or Hill surface. Its $xy$ -plane sections are the zero-velocity curves (ZVC) or Hill curves (Szebehely 1967, §4.7; Vallado 2022, §12.7.3).

Zero velocity means: a spacecraft on this surface has zero speed in the synodic frame: all of its rotating-frame kinetic energy has been traded for potential energy. The surface is, for that $C$ , the boundary of where the spacecraft can reach.

Physical meaning: allowed and forbidden regions

From $v^2 = 2\Omega - C$ :

Where $2\Omega > C$ , $v^2 > 0$ : allowed (real velocity).
Where $2\Omega < C$ , $v^2 < 0$ : forbidden region (imaginary velocity).
Where $2\Omega = C$ , $v = 0$ : the ZVS itself.

A spacecraft on pure CR3BP dynamics cannot enter the forbidden region; to cross it one must apply $\Delta v$ to reduce $C$ below the relevant threshold. In other words, fixing $C$ fixes the spacecraft's playpen in configuration space: the most direct qualitative conclusion of the restricted problem (Szebehely 1967, §4.7).

Topology evolution with $C$

As $C$ decreases from large values, the ZVS opens successively at the five libration points, taking the accessible region through five topological stages (Earth-Moon: $C_1=3.18834$ , $C_2=3.17216$ , $C_3=3.01215$ , $C_4=C_5=2.98800$ ; Parker & Anderson 2014, Table 2-2):

Energy range	ZVS shape	Accessibility
$C > C_1$	three disconnected closed surfaces	spacecraft confined to one of: near Earth, near Moon, or exterior; mutually inaccessible
$C_2 < C < C_1$	open at $L_1$	Earth and Moon regions connected; exterior still closed
$C_3 < C < C_2$	open at $L_1$ and $L_2$	cislunar region connected to deep space beyond $L_2$
$C_4 < C < C_3$	open at $L_1$ , $L_2$ , $L_3$	only two small forbidden islands remain near $L_4$ , $L_5$
$C \le C_4=C_5$	ZVS vanishes	entire space accessible

The openings at $C_1$ , $C_2$ , $C_3$ , $C_4$ correspond to the saddle structure of $\Omega$ at the respective libration points: geometrically, the saddles of $\Omega$ are the energy thresholds.

Necks and transfer channels

The openings at $L_1$ , $L_2$ , $L_3$ are called necks or energy channels:

$L_1$ neck: connects the Earth and Moon regions; the geometric choke point of low-energy Earth-Moon transfers (Halo insertion, manifold-based transfers).
$L_2$ neck: connects the interior cislunar region to deep space beyond the Moon; the departure gate for lunar-far-side relay orbits (e.g. NRHO) and for cislunar-to-Sun-Earth manifold splicing.
$L_3$ neck: connects the Earth-Moon system to exterior space on the anti-Earth side; rarely used operationally.

Dynamics near a neck is governed by the unstable/stable invariant manifolds of the corresponding libration point: whether two periodic orbits at the same $C$ have manifolds that intersect in phase space determines whether a zero-fuel transfer between them exists (Parker & Anderson 2014, §2.6).

Sections and visualization

$xy$ section (most common): the Hill curves give the in-plane accessible outline. For Earth-Moon, $C=3.18$ is the curve just touching at $L_1$ ; $C=3.0$ leaves only two small islands at $L_{4,5}$ (Parker & Anderson 2014, Fig. 2-3; Vallado 2022, Fig. 12-14).
$xz$ section: reveals that out-of-plane motion is also restricted: the forbidden layer is thin near the primaries and thicker far away (Vallado 2022, Fig. 12-16; Lundberg et al. 1985).
Three-dimensional picture: plotting $\Omega$ as a surface $z=\Omega(x,y)$ shows infinite peaks at the primaries, the lowest basins at $L_{4,5}$ , and saddle-shaped passes at $L_{1,2,3}$ ; this is the classical Deprit illustration (Szebehely 1967, Fig. 4.30).

Applications

Transfer feasibility: comparing $C$ at the start and end of a candidate transfer tells whether $\Delta v$ is needed to cross a $C_i$ threshold, a zeroth-order filter at the concept-study stage.
Minimum-energy budget: the lower bound on the $\Delta v$ required to take $C$ from LEO to below $C_1$ follows from $\Delta C = -2\,\mathbf v\cdot\Delta\mathbf v$ (see Jacobi integral).
Manifold splicing: invariant manifolds of periodic orbits evolve at fixed $C$ ; the size of a ZVS neck determines how far a manifold can stretch and whether it intersects the next segment's manifold (Koon et al. 2011).
Forbidden-region avoidance: some missions (e.g. lunar pulsar-navigation constellations) deliberately keep $C$ above $C_1$ to lock spacecraft near Earth or Moon and prevent drift into the libration-point neighborhoods.

Common confusions

The ZVS is not the graph of $\Omega$ : the ZVS is one specific level surface $\Omega=C/2$ , whereas a 3D plot of $\Omega$ itself (the Deprit illustration) is only a visualization aid.
The ZVS depends on $C$ : each spacecraft has its own $C$ and therefore its own ZVS. When orbit families are overlaid on a Hill-curve plot, they correspond to different values of $C$ .
No crossing at fixed $C$ : a spacecraft's trajectory in configuration space cannot cross its own ZVS: crossing requires $\Delta v$ and hence a change in $C$ .

References

Szebehely V. Theory of Orbits: The Restricted Problem of Three Bodies. Academic Press, 1967, Ch. 4 (Curves of Zero Velocity).
Vallado D. A. Fundamentals of Astrodynamics and Applications. 5th ed., 2022, §12.7.3.
Parker J. S., Anderson R. L. Low-Energy Lunar Trajectory Design. JPL, 2014, Ch. 2.
Lundberg J. S., Szebehely V., Whipple C. Surfaces of zero velocity in the restricted problem of three bodies. Celestial Mechanics, 1985.
Koon W. S., Lo M. W., Marsden J. E., Ross S. D. Dynamical Systems, the Three-Body Problem and Space Mission Design. 2nd ed., 2011.
Oshima K. A hidden barrier surface complementary to the zero velocity surface in the circular restricted three-body problem. 2024.