Vertical Orbit

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

A vertical orbit is a family of periodic orbits near a collinear libration point dominated by out-of-plane (z-direction) oscillation. Under linearization, the center subspace of a collinear point decomposes into two normal oscillations, in-plane (ωp) and out-of-plane (ωv); the periodic solution that excites only the out-of-plane center mode is the vertical orbit (Belló 2010). The literature also calls it the vertical Lyapunov orbit: like the planar Lyapunov family, its existence is guaranteed by Lyapunov's center theorem, and the two are sibling families of the same rank (Parker & Anderson 2014, Alessi 2009).

Parker & Anderson 2014 describe it as piercing the plane at the Lagrange point itself: the orbit's out-of-plane oscillation crosses up and down through the libration point as its center.

Geometry and Symmetry

A vertical orbit is doubly symmetric: symmetric about the xz-plane, and tracing the same path in the upper and lower half-planes (Parker & Anderson 2014, Grebow 2008). Its yz projection is a figure eight (Grebow 2008); a nonlinear vertical orbit thus has appreciable in-plane motion, and the claim that the xy projection collapses to a point holds only in the linearized/Hill limit (Gómez 2001). As the family continues, large-amplitude vertical orbits can enclose both primaries and bend toward the lunar north and south poles (Grebow 2008).

Relation to Other Families

• The vertical orbit is the Lissajous limit as in-plane amplitude → 0, with Lissajous tori winding around it (Belló 2010, Folta 2014); it is connected to the planar Lyapunov family by families of quasi-periodic tori (Guzzetti 2016: at JC∈[3.15, 3.17] the two L2 families are linked by same-energy tori).
• The halo family passes near the vertical orbits as it continues; in Hill-problem analysis the L2 halo family terminates at a vertical collision orbit (Gómez 2001).
• At L4/L5, the axial family bifurcates from the vertical Lyapunov family and is no longer symmetric (He 2026).

History and Range of Existence

Moulton had already pointed out the existence of vertical orbits in 1920 (cited by Grebow 2008). Vertical families exist near the collinear points L1/L2/L3 as well as the triangular points L4/L5 (Folta 2015, Vaquero & Howell 2014, He 2026).

Applications

• Polar coverage: vertical orbits below 100,000 km altitude bend toward the poles and serve lunar south pole coverage constellations (Grebow 2008).
• Libration-point-to-libration-point transfers: homoclinic connections of L2 vertical orbits, and L2↔L3 transfers via vertical-orbit manifolds (Haapala 2013).
• Space domain awareness: the L1/L2 vertical families are listed as candidate observer orbit families for cislunar space domain awareness (Klonowski 2024).

Terminology Variants

Related Concepts

• Lyapunov Orbit
• Axial Orbit
• Halo Orbit
• Lissajous Orbit
• Butterfly Orbit

References

• Gómez et al., 2001, Dynamics and Mission Design Near Libration Points, Vol. I
• Grebow et al., 2008, Multibody orbit architectures for lunar south pole coverage
• Alessi et al., 2009, Leaving the Moon by means of invariant manifolds of libration point orbits
• Belló et al., 2010, Invariant manifolds, Lagrangian trajectories and space mission design
• Haapala & Howell, 2013, Homoclinic connections of vertical orbits and L2↔L3 transfers
• Parker & Anderson, 2014, Low-Energy Lunar Trajectory Design
• Folta et al., 2014, Earth–Moon libration point orbit stationkeeping: theory, modeling, and operations
• Folta et al., 2015, An Earth-Moon system trajectory design reference catalog
• Vaquero & Howell, 2014, Leveraging resonant-orbit manifolds to design transfers between libration-point orbits
• Guzzetti et al., 2016, Rapid trajectory design in the Earth–Moon ephemeris system via an interactive catalog of periodic and quasi-periodic orbits
• Klonowski et al., 2024, Cislunar space domain awareness architecture design and analysis for cooperative agents
• He et al., 2026, A review of cislunar constellation design and optimization