Lissajous Orbit

Author: CislunarSpace
Website: https://cislunarspace.cn

Definition

A Lissajous orbit is a quasi-periodic orbit surrounding a libration point that does not close but remains bounded within a finite region. Its name comes from the Lissajous figures studied by French physicist Jules Antoine Lissajous -- in the rotating reference frame, the projection of a Lissajous orbit resembles a Lissajous curve. Unlike Halo orbits, Lissajous orbits are not symmetric and do not precisely close.

Key Elements

Dynamic Characteristics of Lissajous Orbits

Key characteristics of Lissajous orbits in the CR3BP framework include:

Quasi-periodicity: The orbit does not close, gradually sweeping through a torus-like region in the synodic frame, resembling a "ball of yarn"
Asymmetry: Unlike the symmetry of Halo orbits, Lissajous orbits lack symmetry about the $xOz$ plane
Non-resonant frequencies: The in-plane oscillation frequency $\omega_{xy}$ and $z$ -direction oscillation frequency $\omega_z$ of Lissajous orbits do not satisfy a resonance relation, i.e., $\omega_z / \omega_{xy} \neq 1$
Bounded motion: Although not closing, the orbit always remains in a finite region near the libration point

Differences Between Lissajous and Halo Orbits

Feature	Halo Orbit	Lissajous Orbit
Periodicity	Precisely periodic, closed	Quasi-periodic, not closed
Symmetry	Symmetric about $xOz$ plane	No symmetry
Frequency relation	$\omega_z / \omega_{xy} = 1$	$\omega_z / \omega_{xy} \neq 1$
Orbit shape	Three-dimensional ring	Three-dimensional quasi-periodic winding
Control requirement	Station-keeping required	Station-keeping required (more complex)

Lissajous orbits can be viewed as a "generalization" of Halo orbits -- when the frequency ratio between in-plane and $z$ -direction motion is not 1, the periodic orbit degenerates into a quasi-periodic orbit.

Linear Approximation of Lissajous Orbits

In the linearized framework near the libration point, the motion of a Lissajous orbit can be decomposed into three modes:

x(t) = A_{xy} \cos(\omega_{xy} t + \phi_{xy}) + \text{nonlinear corrections}

z(t) = A_z \cos(\omega_z t + \phi_z) + \text{nonlinear corrections}

where $A_{xy}$ and $A_z$ are the in-plane and $z$ -direction amplitudes, and $\phi_{xy}$ and $\phi_z$ are initial phases. Since $\omega_{xy}$ and $\omega_z$ are incommensurable, the orbit never closes.

Stability and Control

Like Halo orbits, Lissajous orbits are also unstable and require station-keeping control. However, because Lissajous orbits do not close, control strategies are more complex:

Target orbit definition: Since the orbit does not close, the "target orbit" is not a precise closed curve but a time-evolving reference trajectory
Control frequency: Typically requires more frequent control maneuvers
Fuel consumption: Station-keeping for Lissajous orbits is typically slightly higher than for Halo orbits of comparable size

Application Value

Lissajous orbits have unique applications in space missions:

SOHO satellite: ESA/NASA's Solar and Heliospheric Observatory operates in a Lissajous orbit near the Sun-Earth L1 point, the most famous application of Lissajous orbits
Relay communications: Lissajous orbits near the Earth-Moon L2 point can provide communication relay for the lunar far side
Orbital design flexibility: The frequency ratio of Lissajous orbits can be freely chosen, offering more design freedom than Halo orbits
Easier injection: In some cases, the $\Delta V$ required to enter a Lissajous orbit is less than for a Halo orbit

References

Richardson D L. Analytic construction of periodic orbits about the collinear points[J]. Celestial Mechanics, 1980, 22(3): 241-253.
Gomez G, Masdemont J, Simo C. Lissajous orbits around halo orbits[J]. Advances in the Astronautical Sciences, 1998.
Qiao C, Long X, Yang L, et al. Orbital parameter characterization and objects cataloging for Earth-Moon collinear libration points[J]. Chinese Journal of Aeronautics, 2025.