Lissajous Orbit

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

A Lissajous orbit is a quasi-periodic orbit on the center manifold near a collinear libration point: an in-plane (xy) oscillation combined with an out-of-plane (z) oscillation of a different frequency, with the in-plane amplitude Ax and out-of-plane amplitude Az as two free parameters (Canalias 2008, Renk 2010). Because the two frequencies are generally incommensurate, the orbit never closes but remains in a bounded region around the libration point, tracing a Lissajous figure. Dynamically, Lissajous orbits are two-dimensional tori surrounding the vertical periodic orbits (Folta 2014). The name comes from the figures studied by the French physicist Jules Antoine Lissajous.

Relation to Halo Orbits

When the in-plane and out-of-plane frequencies become equal, a Lissajous orbit degenerates into a periodic halo orbit (Renk 2010, Gómez 2001). The two differ in symmetry: Lissajous orbits remain (quasi-)symmetric about both the xy- and xz-planes, while halo orbits keep only the xz-plane symmetry and lose the xy-plane one (Renk 2010).

Variants

• Square Lissajous orbits: Lissajous orbits whose in-plane and out-of-plane amplitudes are equal (α3 = α4) (Alessi 2010). Note that square is an amplitude constraint: the two frequencies still differ, so there is no equal period property.
• Quasi-halo orbits: quasi-periodic tori surrounding halo orbits (Folta 2014). Generative relation: once the out-of-plane amplitude exceeds a certain lower bound, a Lissajous orbit loses its xy-plane symmetry and develops an exclusion zone around the line of the primaries; it has then become a quasi-halo (Renk 2010). Quasi-halos therefore do not have small out-of-plane amplitudes; they correspond to the large-amplitude end of the Lissajous family.
• High/low z-amplitude modes: in the nonlinear model the z-amplitude is no longer constant but cycles between high and low modes; the entry phase selects which region of the torus is reached. The low z-amplitude mode is also called the nearly-planar mode. ARTEMIS exploited the high z-amplitude mode on the L2 side to accommodate the out-of-plane arrival conditions of ballistic transfers, and the low z-amplitude (nearly-planar) phase on the L1 side to reduce the ΔV of entering low-inclination lunar orbits (Folta 2014). Note that these modes belong to large quasi-halo orbits (and quasi-periodic orbits in general): Folta 2014 explicitly states that Lissajous orbits (the central region of the Poincaré section) do not possess nearly-planar modes.

Parameterization

• Osculating Lissajous elements (Renk 2010): by analogy with Keplerian elements: unstable amplitude A1 (exponentially growing term), stable amplitude A2 (decaying term), in-plane amplitude Ax (Ay scales with Ax and is not listed separately), out-of-plane amplitude Az, in-plane phase Φxy, out-of-plane phase Φz. Setting A1 = A2 = 0 yields a Lissajous orbit.
• Effective phase plane (EPP): the effective phases (Φ, Ψ) map one-to-one onto the state of a Lissajous orbit of given amplitudes, used for two-spacecraft rendezvous and eclipse-avoidance design (Perozzi & Ferraz-Mello 2010).

Applications

• ARTEMIS: P1 and P2 entered Earth–Moon L2 and L1 Lissajous orbits on 2010-08-25 and 2010-10-22 respectively, each via a single Lissajous orbit insertion (LOI) maneuver (Folta 2012). Station-keeping with orbit continuation proved best under ephemeris-model errors, with a floor of about 15 m/s per year against a budget under 25 m/s per year (Folta 2010). After the fact, Poincaré-section analysis showed all three ARTEMIS libration-point orbits to be arcs of large southern quasi-halo orbits (Folta 2014): Lissajous by design intent and quasi-halo by post-hoc classification, both attested in the literature.
• Sun–Earth↔Earth–Moon natural transfers: matching the hyperbolic manifolds of two three-body systems on a Poincaré section enables maneuver-free transfers between their Lissajous orbits, with coupling maneuvers generally below 100 m/s after multiple-shooting refinement (Canalias 2008).
• Eclipse avoidance: the shape controllability of the two-parameter Lissajous family makes eclipse avoidance inexpensive (Alessi 2010).

Terminology Variants

Related Concepts

• Halo Orbit
• Near-Rectilinear Halo Orbit (NRHO)
• Lyapunov Orbit
• Quasi-Periodic Orbit (QPO)
• Weak Stability Boundary Transfer Trajectory

References

• Canalias & Masdemont, 2008, Computing natural transfers between Sun–Earth and Earth–Moon Lissajous libration point orbits
• Renk et al., 2010, Study on Lissajous and quasi-halo orbits
• Alessi et al., 2010, Two-manoeuvres transfers between LEOs and Lissajous orbits in the Earth–Moon system
• Folta et al., 2010, Stationkeeping of Lissajous trajectories in the Earth-Moon system with applications to ARTEMIS
• Folta et al., 2012, ARTEMIS transfer and insertion study
• Folta et al., 2014, Earth–Moon libration point orbit stationkeeping: theory, modeling, and operations
• Perozzi & Ferraz-Mello, 2010, Effective-phase methods for rendezvous and eclipse avoidance on libration-point orbits
• Qiao et al., 2025, Review of cislunar libration-point missions