Vertical Orbit

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

Definition

A Vertical Orbit is a three-dimensional periodic orbit family surrounding libration points, belonging to an important category of Libration Point Orbits (LPO). Unlike the planar motion of Lyapunov orbits, vertical orbits have a significant periodic oscillation component in the $z$ -direction while maintaining a certain range of motion in the $xOy$ plane, forming a unique three-dimensional spatial configuration. Vertical orbits exist near the L1, L2, and L3 collinear libration points as well as the L4 and L5 triangular libration points of the Earth-Moon system.

Core Elements

Dynamical Characteristics of Vertical Orbits

Vertical orbits possess the following properties within the CR3BP framework:

Three-dimensional motion: Vertical orbits have non-zero amplitudes in the $x$ , $y$ , and $z$ directions; the $z$ -direction oscillation is the source of their name and the most significant distinction from Lyapunov orbits
Periodicity: Vertical orbits are strictly periodic, closing precisely in the rotating coordinate system
Symmetry: Standard vertical orbits are symmetric with respect to the $xOz$ plane
Amplitude parameterization: The vertical orbit family can be parameterized by $z$ -direction amplitude; different amplitudes correspond to different energy levels (Jacobi constant) and orbital periods

Classification of Vertical Orbits

Based on the libration point locations in the Earth-Moon system, vertical orbits can be classified into the following families:

Orbit Family	Parent Libration Point	Characteristics
V1 (Vertical L1)	L1	Located between Earth and Moon, moderate $z$ -direction amplitude
V2 (Vertical L2)	L2	Located on the far side of the Moon, large $z$ -direction amplitude
V3 (Vertical L3)	L3	Located on the far side of the Earth from the Moon, long period
V4/V5 (Vertical L4/L5)	L4/L5	Located near the triangular libration points, good stability

Orbital Parameter Characteristics

Using the Earth-Moon system as an example, the main parameter ranges for the vertical orbit family are as follows (based on the dynamic catalog statistics by Guzzetti et al.):

Orbit Family	Jacobi Constant Range	Period Range (days)	Stability Index
V1	Approximately 2.5 ~ 3.0	Approximately 24	Relatively high
V2	Approximately 0.8 ~ 3.0	Approximately 16	Moderate
V3	Approximately 2.5 ~ 3.0	Relatively long	Relatively low

There is a nonlinear relationship between the Jacobi constant and orbital period of vertical orbits; as amplitude increases, orbital energy decreases (Jacobi constant decreases) and the period changes accordingly.

Relationship Between Vertical Orbits and Other Libration Point Orbits

Vertical orbits belong to the same libration point orbit family as Lyapunov orbits, Halo orbits, and Axial orbits:

Relationship with Lyapunov orbits: Lyapunov orbits lie strictly in the $xOy$ plane and can be viewed as the degenerate form of vertical orbits when the $z$ -direction amplitude approaches zero
Relationship with Halo orbits: Halo orbits are symmetric with respect to the $xOz$ plane, while vertical orbits also have significant $y$ -direction amplitudes; both have three-dimensional characteristics but with different geometric configurations
Relationship with Axial orbits: Both Axial orbits and vertical orbits have three-dimensional characteristics, but their dominant oscillation directions differ

Stability Analysis

The stability of vertical orbits varies depending on the associated libration point:

Vertical orbits near L1, L2, and L3 typically have unstable modes and require station-keeping maneuvers
Vertical orbits near L4 and L5 may have members with good long-term stability due to the inherent stability of the triangular libration points
The stability index is a key metric for evaluating the local stability of vertical orbits; an index closer to 1 indicates a more stable orbit

Application Value

Vertical orbits have the following potential applications in cislunar space missions:

Scientific observation platform: The three-dimensional characteristics of vertical orbits enable unique observation geometries, suitable for space science and astronomical observation missions
Communication relay: L1/L2 vertical orbits can serve as Earth-Moon communication relays, covering both the near and far sides of the Moon
Orbit transfer corridors: The stable and unstable manifolds of vertical orbits can serve as low-energy transfer corridors connecting different regions of cislunar space
Mission design reference: As an important component of the libration point orbit classification system, vertical orbits are fundamental to understanding cislunar dynamical structures and designing complex mission trajectories

References

Guzzetti D, Bosanac N, Howell K C. A framework for efficient trajectory comparisons in the Earth-Moon design space[C]. AAS/AIAA Space Flight Mechanics Meeting, 2014.
Doedel E J, Romanov V A, Paffenroth R C, et al. Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem[J]. International Journal of Bifurcation and Chaos, 2007, 17(8): 2625-2677.
Richardson D L. Analytic construction of periodic orbits about the collinear points[J]. Celestial Mechanics, 1980, 22(3): 241-253.