# Lissajous orbit

In orbital mechanics, a **Lissajous orbit** (pronounced [li.sa.ʒu]), named after Jules Antoine Lissajous, is a quasi-periodic orbital trajectory that an object can follow around a Lagrangian point of a three-body system without requiring any propulsion. Lyapunov orbits around a Lagrangian point are curved paths that lie entirely in the plane of the two primary bodies. In contrast, Lissajous orbits include components in this plane and perpendicular to it, and follow a Lissajous curve. Halo orbits also include components perpendicular to the plane, but they are periodic, while Lissajous orbits are usually not.

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In practice, any orbits around Lagrangian points L_{1}, L_{2}, or L_{3} are dynamically unstable, meaning small departures from equilibrium grow over time.[1] 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 L_{4} and L_{5} are dynamically stable so long as the ratio of the masses of the two main objects is greater than about 25.[2] 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.[3] These orbits can however be destabilized by other nearby massive objects. For example, orbits around the L_{4} and L_{5} points in the Earth–Moon system can last only a few million years instead of billions because of perturbations by the planets.[4]