Lissajous orbit

Short description: Quasi-periodic orbital trajectory

Animation of WMAP's trajectory

Oblique view

Viewed from Earth

Earth * WMAP

Part of a series on

Astrodynamics

Orbital mechanics

Orbital parameters

* Apsis
* Argument of periapsis
* Azimuth
* Eccentricity
* Inclination
* Mean anomaly
* Orbital nodes
* Semi-major axis
* True anomaly

Types of two-body orbits, by
eccentricity

* Circular orbit
* Elliptic orbit

Transfer orbit

* (Hohmann transfer orbit
* Bi-elliptic transfer orbit)

* Parabolic orbit
* Hyperbolic orbit
* Radial orbit
* Decaying orbit

Equations

* Dynamical friction
* Escape velocity
* Kepler's equation
* Kepler's laws of planetary motion
* Orbital period
* Orbital velocity
* Surface gravity
* Specific orbital energy
* Vis-viva equation

Celestial mechanics

Gravitational influences

* Barycenter
* Hill sphere
* Perturbations
* Sphere of influence

N-body orbits

Lagrangian points

* (Halo orbits)

* Lissajous orbits
* Lyapunov orbits

Engineering and efficiency

Preflight engineering

* Mass ratio
* Payload fraction
* Propellant mass fraction
* Tsiolkovsky rocket equation

Efficiency measures

* Gravity assist
* Oberth effect

* v
* t
* e

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 with minimal 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.

In practice, any orbits around Lagrangian points L1, L2, or L3 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 L4 and L5 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 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.[4]

## Contents

* 1 Spacecraft using Lissajous orbits
* 2 Fictional appearances
* 3 See also
* 4 Notes
* 5 References
* 6 External links

## Spacecraft using Lissajous orbits

Several missions have used Lissajous orbits: ACE at Sun–Earth L1,[5] SOHO at Sun–Earth L1, DSCOVR at Sun–Earth L1,[6] WMAP at Sun–Earth L2,[7] and also the Genesis mission collecting solar particles at L1.[8] 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.[9]

ESA's Gaia mission also uses a Lissajous orbit at Sun–Earth L2.[10]

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.[11]

In June 2018, Queqiao, the relay satellite for China's Chang'e 4 lunar lander mission, entered orbit around Earth-Moon L2.[12][lower-alpha 1]

## Fictional appearances

In the 2005 science fiction novel Sunstorm by Arthur C. Clarke and Stephen Baxter, a huge shield is constructed in space to protect the Earth from a deadly solar storm. The shield is described to have been in a Lissajous orbit at L1. In the story a group of wealthy and powerful people shelter opposite the shield at L2 so as to be protected from the solar storm by the shield, the Earth and the Moon.

In the 2017 science fiction novel Artemis by Andy Weir, a Lissajous orbit is used as a transfer point for routine travel to and from the Moon.

## See also

* Libration point orbit

## Notes

1. ↑ Possibly a halo orbit. Sources disagree.

## References

1. ↑ "ESA Science & Technology: Orbit/Navigation". European Space Agency. 14 June 2009. http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=34699.
2. ↑ "A230242 – Decimal expansion of (25+3*sqrt(69))/2". https://oeis.org/A230242.
3. ↑ Vallado, David A. (2007). Fundamentals of Astrodynamics and Applications (3rd ed.). Springer New York. (paperback), (hardback). ISBN 978-1-881883-14-2.
4. ↑ Lissauer, Jack J.; Chambers, John E. (2008). "Solar and planetary destabilization of the Earth–Moon triangular Lagrangian points". Icarus 195 (1): 16–27. doi:10.1016/j.icarus.2007.12.024. Bibcode: 2008Icar..195...16L. https://zenodo.org/record/1259033.
5. ↑ "ACE Mission". https://izw1.caltech.edu/ACE/ace_mission.html.
6. ↑ Bergin, Chris (2015-02-11). "SpaceX Falcon 9 successfully launches the DSCOVR spacecraft" (in en-US). https://www.nasaspaceflight.com/2015/02/spacex-falcon-9-dscovr-mission/.
7. ↑ "WMAP Observatory: Trajectory and Orbit". https://map.gsfc.nasa.gov/mission/observatory_orbit.html.
8. ↑ "Genesis: Search for Origins | JPL | NASA". https://solarsystem.nasa.gov/genesismission/.
9. ↑ "Herschel: Orbit/Navigation". ESA. http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=34699.
10. ↑ "Gaia's Lissajous Type Orbit". ESA. http://sci2.esa.int/interactive/media/flashes/5_5_1.htm.
11. ↑ ARTEMIS: The First Mission to the Lunar Libration Orbits
12. ↑ Jones, Andrew (14 June 2018). "Chang'e-4 relay satellite enters halo orbit around Earth-Moon L2, microsatellite in lunar orbit". https://spacenews.com/change-4-relay-satellite-enters-halo-orbit-around-earth-moon-l2-microsatellite-in-lunar-orbit/.

## External links

* Koon, W. S.; M. W. Lo; J. E. Marsden; S. D. Ross (2006). Dynamical Systems, the Three-Body Problem, and Space Mission Design. http://www.cds.caltech.edu/~marsden/books/Mission_Design.html.
* Koon, Wang Sang (2000). "Dynamical Systems, the Three-Body Problem, and Space Mission Design". Berlin: World Scientific. pp. 1167–1181. http://www.gg.caltech.edu/~mwl/publications/papers/dynamicalThreeBody.pdf.

* v
* t
* e

Gravitational orbits

Types|

| General|

* Box
* Capture
* Circular
* Elliptical / Highly elliptical
* Escape
* Horseshoe
* Hyperbolic trajectory
* Inclined / Non-inclined
* Lagrangian point
* Osculating
* Parabolic trajectory
* Parking
* Prograde / Retrograde
* Synchronous
* semi
* sub
* Transfer orbit

|

Geocentric|

* Geosynchronous
* Geostationary
* Geostationary transfer
* Graveyard
* High Earth
* Low Earth
* Medium Earth
* Molniya
* Near-equatorial
* Orbit of the Moon
* Polar
* Sun-synchronous
* Tundra

About other points|

* Mars
* Areosynchronous
* Areostationary
* Lagrange points
* Distant retrograde
* Halo
* Lissajous
* Lunar
* Sun
* Heliocentric
* Earth's orbit
* Heliosynchronous

Parameters|

* Shape
* Size

* e Eccentricity
* a Semi-major axis
* b Semi-minor axis
* Q, q Apsides

|

Orientation|

* i Inclination
* Ω Longitude of the ascending node
* ω Argument of periapsis
* ϖ Longitude of the periapsis

Position|

* M Mean anomaly
* ν, θ, f True anomaly
* E Eccentric anomaly
* L Mean longitude
* l True longitude

Variation|

* T Orbital period
* n Mean motion
* v Orbital speed
* t0 Epoch

Maneuvers|

* Bi-elliptic transfer
* Collision avoidance (spacecraft)
* Delta-v
* Delta-v budget
* Gravity assist
* Gravity turn
* Hohmann transfer
* Inclination change
* Low energy transfer
* Oberth effect
* Phasing
* Rocket equation
* Rendezvous
* Transposition, docking, and extraction

Orbital mechanics|

* Celestial coordinate system
* Characteristic energy
* Escape velocity
* Ephemeris
* Equatorial coordinate system
* Ground track
* Hill sphere
* Interplanetary Transport Network
* Kepler's laws of planetary motion
* Lagrangian point
* n-body problem
* Orbit equation
* Orbital state vectors
* Perturbation
* Retrograde motion
* Specific orbital energy
* Specific relative angular momentum
* Two-line elements

* List of orbits

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