Lissajous orbit

In practice, any orbits around Lagrangian points L₁, L₂, or L₃ are dynamically unstable, meaning small departures from equilibrium grow over time. As a result, spacecraft in these Lagrangian point orbits must use their propulsion systems to perform orbital station-keeping. Although they are not perfectly stable, a modest effort of station keeping keeps a spacecraft in a desired Lissajous orbit for a long time.

In the absence of other influences, orbits about Lagrangian points L₄ and L₅ are dynamically stable so long as the ratio of the masses of the two main objects is greater than about 24.96. The natural dynamics keep the spacecraft (or natural celestial body) in the vicinity of the Lagrangian point without use of a propulsion system, even when slightly perturbed from equilibrium. These orbits can however be destabilized by other nearby massive objects. For example, orbits around the L₄ and L₅ points in the Earth–Moon system can last only a few million years instead of billions because of perturbations by the other planets in the Solar System.

Spacecraft using Lissajous orbits

Several missions have used Lissajous orbits: ACE at Sun–Earth L1, SOHO at Sun–Earth L1, DSCOVR at Sun–Earth L1, WMAP at Sun–Earth L2, and also the Genesis mission collecting solar particles at L1. On 14 May 2009, the European Space Agency (ESA) launched into space the Herschel and Planck observatories, both of which use Lissajous orbits at Sun–Earth L2.

ESA's Gaia mission also uses a Lissajous orbit at Sun–Earth L2.

In 2011, NASA transferred two of its THEMIS spacecraft from Earth orbit to Lunar orbit by way of Earth–Moon L1 and L2 Lissajous orbits.

In June 2018, Queqiao, the relay satellite for China's Chang'e 4 lunar lander mission, entered orbit around Earth-Moon L2.