Editing Lagrange point (section)

{{Short description|Equilibrium points near two orbiting bodies}}
{{For|the video game|Lagrange Point (video game){{!}}''Lagrange Point'' (video game)}}
{{Use American English|date=May 2025}}
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[[File:Lagrange points simple.svg|thumb|upright=1.35|Lagrange points in the Sun–Earth system (not to scale). This view is from the north, so that Earth's orbit is counterclockwise.]]
[[File:Lagrange points2.svg|thumb|A [[contour plot]] of the [[effective potential]] due to gravity and the [[centrifugal force]] of a two-body system in a rotating frame of reference. The arrows indicate the downhill gradients of the potential around the five Lagrange points, toward them ({{red|red}}) and away from them ({{blue|blue}}). Counterintuitively, the L<sub>4</sub> and L<sub>5</sub> points are the [[Maxima and minima|high points]] of the potential. At the points themselves these forces are balanced.]]
[[File:Animation of Wilkinson Microwave Anisotropy Probe trajectory.gif |thumb |upright=1.35|right| An example of a spacecraft at Sun-Earth L2, the Wilkinson Microwave Anisotropy Probe, or WMAP<br>{{legend2|magenta|[[Wilkinson Microwave Anisotropy Probe|WMAP]]}} {{space}} {{legend2|RoyalBlue|[[Earth]]}}]]
{{Astrodynamics}}

In [[celestial mechanics]], the '''Lagrange points''' ({{IPAc-en|l|ə|ˈ|ɡ|r|ɑː|n|dʒ}}; also '''Lagrangian points''' or '''libration points''') are points of [[equilibrium (mechanics)|equilibrium]] for small-mass objects under the [[gravity|gravitational]] influence of two massive [[orbit|orbiting]] bodies. Mathematically, this involves the solution of the [[restricted three-body problem]].<ref name="Lagrange Cornish" />

Normally, the two massive bodies exert an unbalanced gravitational force at a point, altering the orbit of whatever is at that point. At the Lagrange points, the [[gravitation]]al forces of the two large bodies and the [[centrifugal force]] balance each other.<ref>{{Cite web |url=https://scienceworld.wolfram.com/physics/LagrangePoints.html |title=Lagrange Points |first=Eric W. |last=Weisstein |author-link=Eric W. Weisstein |website=Eric Weisstein's World of Physics }}</ref> This can make Lagrange points an excellent location for satellites, as [[Orbital station-keeping|orbit corrections]], and hence fuel requirements, needed to maintain the desired orbit are kept at a minimum. 

For any combination of two orbital bodies, there are five Lagrange points, L<sub>1</sub> to L<sub>5</sub>, all in the orbital plane of the two large bodies. There are five Lagrange points for the Sun–Earth system, and five ''different'' Lagrange points for the Earth–Moon system. L<sub>1</sub>, L<sub>2</sub>, and L<sub>3</sub> are on the line through the centers of the two large bodies, while L<sub>4</sub> and L<sub>5</sub> each act as the third [[Vertex (geometry)|vertex]] of an [[equilateral triangle]] formed with the centers of the two large bodies.

When the mass ratio of the two bodies is large enough, the L<sub>4</sub> and L<sub>5</sub> points are stable points,<!--not gravity wells--> meaning that objects can orbit them and that they have a tendency to pull objects into them. Several planets have [[trojan (astronomy)|trojan asteroids]] near their L<sub>4</sub> and L<sub>5</sub> points with respect to the Sun; [[Jupiter#Interaction with the Solar System|Jupiter]] has more than one million of these trojans. 

Some Lagrange points are being used for space exploration. Two important Lagrange points in the Sun-Earth system are L<sub>1</sub>, between the Sun and Earth, and L<sub>2</sub>, on the same line at the opposite side of the Earth; both are well outside the Moon's orbit. Currently, an [[artificial satellite]] called the [[Deep Space Climate Observatory]] (DSCOVR) is located at L<sub>1</sub> to study solar wind coming toward Earth from the Sun and to monitor Earth's climate, by taking images and sending them back.<ref>{{cite web |url=https://solarsystem.nasa.gov/missions/DSCOVR/in-depth/ |title=DSCOVR: In-Depth |website=NASA Solar System Exploration |publisher=NASA |access-date=27 October 2021}}</ref> The [[James Webb Space Telescope]], a powerful infrared space observatory, is located at L<sub>2</sub>.<ref>{{Cite web |url=https://webb.nasa.gov/content/about/orbit.html |title=About Orbit |website=NASA |access-date=1 January 2022 }}</ref> This allows the satellite's sunshield to protect the telescope from the light and heat of the Sun, Earth and Moon simultaneously with no need to rotate the sunshield. The L<sub>1</sub> and L<sub>2</sub> Lagrange points are located about {{cvt|1,500,000|km|mi}} from Earth.

The European Space Agency's earlier [[Gaia (spacecraft)|Gaia]] telescope, and its newly launched [[Euclid (spacecraft)|Euclid]], also occupy orbits around L<sub>2</sub>. Gaia keeps a tighter [[Lissajous orbit]] around L<sub>2</sub>, while Euclid follows a [[halo orbit]] similar to JWST. Each of the space observatories benefit from being far enough from Earth's shadow to utilize solar panels for power, from not needing much power or propellant for station-keeping, from not being subjected to the Earth's magnetospheric effects, and from having direct line-of-sight to Earth for data transfer.