Lagrange point

The three collinear Lagrange points (L₁, L₂, L₃) were discovered by the Swiss mathematician Leonhard Euler around 1750, a decade before the Italian-born Joseph-Louis Lagrange discovered the remaining two.{{Cite book |title=Dynamical Systems, the Three-Body Problem, and Space Mission Design |last1=Koon |first1=Wang Sang |first2=Martin W. |last2=Lo |author-link2=Martin Lo |first3=Jerrold E. |last3=Marsden |author-link3= Jerrold E. Marsden |first4=Shane D. |last4=Ross |url=http://www.cds.caltech.edu/~marsden/books/Mission_Design.html |date=2006 |page=9 |access-date=9 June 2008 |archive-url=https://web.archive.org/web/20080527145955/http://www.cds.caltech.edu/~marsden/books/Mission_Design.html |archive-date=27 May 2008 |url-status=dead }} (16MB){{cite book|author-link=Leonhard Euler|first=Leonhard|last=Euler |url=http://www.math.dartmouth.edu/~euler/docs/originals/E327.pdf |title=De motu rectilineo trium corporum se mutuo attrahentium |date=1765}}

In 1772, Lagrange published an "Essay on the three-body problem". In the first chapter he considered the general three-body problem. From that, in the second chapter, he demonstrated two special constant-pattern solutions, the collinear and the equilateral, for any three masses, with circular orbits.{{Cite book |last=Lagrange |first=Joseph-Louis |author-link=Joseph-Louis Lagrange |title=Œuvres de Lagrange |date=1867–1892 |publisher=Gauthier-Villars |chapter=Tome 6, Chapitre II: Essai sur le problème des trois corps |pages=229–334 |chapter-url=http://gallica.bnf.fr/ark:/12148/bpt6k229225j/f231.image.r=Oeuvres+de+Lagrange.langFR |language=fr }}

{{Also|List of objects at Lagrange points}}

The five Lagrange points are labeled and defined as follows:

The {{L1|nolink=yes}} point lies on the line defined between the two large masses M₁ and M₂. It is the point where the gravitational attraction of M₂ and that of M₁ combine to produce an equilibrium. An object that orbits the Sun more closely than Earth would typically have a shorter orbital period than Earth, but that ignores the effect of Earth's gravitational pull. If the object is directly between Earth and the Sun, then Earth's gravity counteracts some of the Sun's pull on the object, increasing the object's orbital period. The closer to Earth the object is, the greater this effect is. At the {{L1|nolink=yes}} point, the object's orbital period becomes exactly equal to Earth's orbital period. {{L1|nolink=yes}} is about 1.5 million kilometers, or 0.01 au, from Earth in the direction of the Sun.{{cite web|last1=Cornish|first1=Neil J. |date=1998 |title=The Lagrange Points |publisher=WMAP Education and Outreach |url=http://www.physics.montana.edu/faculty/cornish/lagrange.pdf| archive-url =https://web.archive.org/web/20150907090116/http://www.physics.montana.edu/faculty/cornish/lagrange.pdf|archive-date=7 September 2015|url-status=dead|access-date=15 December 2015}}

The {{L2|nolink=yes}} point lies on the line through the two large masses beyond the smaller of the two. Here, the combined gravitational forces of the two large masses balance the centrifugal force on a body at {{L2|nolink=yes}}. On the opposite side of Earth from the Sun, the orbital period of an object would normally be greater than Earth's. The extra pull of Earth's gravity decreases the object's orbital period, and at the {{L2|nolink=yes}} point, that orbital period becomes equal to Earth's. Like L₁, L₂ is about 1.5 million kilometers or 0.01 au from Earth (away from the sun). An example of a spacecraft designed to operate near the Earth–Sun L₂ is the James Webb Space Telescope.{{cite web |url=http://www.stsci.edu/jwst/overview/design/orbit|archive-url=https://wayback.archive-it.org/all/20140203174537/http://www.stsci.edu/jwst/overview/design/orbit|url-status=dead|archive-date=3 February 2014 |title=L2 Orbit|publisher=Space Telescope Science Institute|access-date=28 August 2016}} Earlier examples include the Wilkinson Microwave Anisotropy Probe and its successor, Planck.

The {{L3|nolink=yes}} point lies on the line defined by the two large masses, beyond the larger of the two. Within the Sun–Earth system, the {{L3|nolink=yes}} point exists on the opposite side of the Sun, a little outside Earth's orbit and slightly farther from the center of the Sun than Earth is. This placement occurs because the Sun is also affected by Earth's gravity and so orbits around the two bodies' barycenter, which is well inside the body of the Sun. An object at Earth's distance from the Sun would have an orbital period of one year if only the Sun's gravity is considered. But an object on the opposite side of the Sun from Earth and directly in line with both "feels" Earth's gravity adding slightly to the Sun's and therefore must orbit a little farther from the barycenter of Earth and Sun in order to have the same 1-year period. It is at the {{L3|nolink=yes}} point that the combined pull of Earth and Sun causes the object to orbit with the same period as Earth, in effect orbiting an Earth+Sun mass with the Earth-Sun barycenter at one focus of its orbit.

File:L4 diagram.svg

The {{L4|nolink=yes}} and {{L5|nolink=yes}} points lie at the third vertices of the two equilateral triangles in the plane of orbit whose common base is the line between the centers of the two masses, such that the point lies 60° ahead of ({{L4|nolink=yes}}) or behind ({{L5|nolink=yes}}) the smaller mass with regard to its orbit around the larger mass.

The triangular points ({{L4|nolink=yes}} and {{L5|nolink=yes}}) are stable equilibria, provided that the ratio of {{sfrac|M₁|M₂}} is greater than 24.96.Actually {{sfrac|25 + 3{{sqrt|69}}|2}} ≈ {{val|24.9599357944}} {{OEIS|A230242}} This is the case for the Sun–Earth system, the Sun–Jupiter system, and, by a smaller margin, the Earth–Moon system. When a body at these points is perturbed, it moves away from the point, but the factor opposite of that which is increased or decreased by the perturbation (either gravity or angular momentum-induced speed) will also increase or decrease, bending the object's path into a stable, kidney bean-shaped orbit around the point (as seen in the corotating frame of reference).{{cite web|url= https://wmap.gsfc.nasa.gov/media/ContentMedia/lagrange.pdf |title=The Lagrange Points |date=1998|publisher=NASA}}, Neil J. Cornish, with input from Jeremy Goodman

The points {{L1|nolink=yes}}, {{L2|nolink=yes}}, and {{L3|nolink=yes}} are positions of unstable equilibrium. Any object orbiting at {{L1|nolink=yes}}, {{L2|nolink=yes}}, or {{L3|nolink=yes}} will tend to fall out of orbit; it is therefore rare to find natural objects there, and spacecraft inhabiting these areas must employ a small but critical amount of station keeping in order to maintain their position.

{{Main|List of objects at Lagrange points}}

Due to the natural stability of {{L4|nolink=yes}} and {{L5|nolink=yes}}, it is common for natural objects to be found orbiting in those Lagrange points of planetary systems. Objects that inhabit those points are generically referred to as 'trojans' or 'trojan asteroids'. The name derives from the names that were given to asteroids discovered orbiting at the Sun–Jupiter {{L4|nolink=yes}} and {{L5|nolink=yes}} points, which were taken from mythological characters appearing in Homer's Iliad, an epic poem set during the Trojan War. Asteroids at the {{L4|nolink=yes}} point, ahead of Jupiter, are named after Greek characters in the Iliad and referred to as the "Greek camp". Those at the {{L5|nolink=yes}} point are named after Trojan characters and referred to as the "Trojan camp". Both camps are considered to be types of trojan bodies.

As the Sun and Jupiter are the two most massive objects in the Solar System, there are more known Sun–Jupiter trojans than for any other pair of bodies. However, smaller numbers of objects are known at the Lagrange points of other orbital systems:

• The Sun–Earth {{L4|nolink=yes}} and {{L5|nolink=yes}} points contain interplanetary dust and at least two asteroids, {{mpl|2010 TK|7}} and {{mpl|2020 XL|5}}.{{Cite web |url=https://www.space.com/12443-earth-asteroid-companion-discovered-2010-tk7.html |title=First Asteroid Companion of Earth Discovered at Last |first=Charles Q. |last=Choi |website=Space.com |date=27 July 2011 }}{{Cite web |url=https://www.nasa.gov/home/hqnews/2011/jul/HQ_11-247_WISE_Trojan.html |title=NASA - NASA's Wise Mission Finds First Trojan Asteroid Sharing Earth's Orbit |website=www.nasa.gov }}{{cite journal

|first1 = Man-To |last1 = Hui

|first2 = Paul A. |last2 = Wiegert |author-link2=Paul Wiegert

|first3 = David J. |last3 = Tholen |author-link3=David J. Tholen

|first4 = Dora |last4 = Föhring

|title = The Second Earth Trojan 2020 XL5

|journal = The Astrophysical Journal Letters

|date = November 2021

|volume = 922

|issue = 2

|pages = L25

|doi = 10.3847/2041-8213/ac37bf

|arxiv = 2111.05058

|bibcode = 2021ApJ...922L..25H

|s2cid = 243860678

|doi-access = free

}}

• The Earth–Moon {{L4|nolink=yes}} and {{L5|nolink=yes}} points contain concentrations of interplanetary dust, known as Kordylewski clouds.{{cite journal|title=Celestial mechanics and polarization optics of the Kordylewski dust cloud in the Earth-Moon Lagrange point L5 - Part I. Three-dimensional celestial mechanical modelling of dust cloud formation |first1=Judit |last1=Slíz-Balogh |first2=András |last2=Barta |first3=Gábor |last3=Horváth |journal=Monthly Notices of the Royal Astronomical Society |volume=480 |issue=4 |pages=5550–5559 |date=2018 |doi=10.1093/mnras/sty2049|doi-access=free |arxiv=1910.07466 |bibcode=2018MNRAS.480.5550S }}{{cite journal|title=Celestial mechanics and polarization optics of the Kordylewski dust cloud in the Earth-Moon Lagrange point L5. Part II. Imaging polarimetric observation: new evidence for the existence of Kordylewski dust cloud |first1=Judit |last1=Slíz-Balogh |first2=András |last2=Barta |first3=Gábor |last3=Horváth |journal=Monthly Notices of the Royal Astronomical Society |volume=482 |issue=1 |pages=762–770 |date=2019 |doi=10.1093/mnras/sty2630|arxiv=1910.07471 |bibcode=2019MNRAS.482..762S |doi-access=free }} Stability at these specific points is greatly complicated by solar gravitational influence.{{cite journal|url=http://www.rfreitas.com/Astro/SearchIcarus1980.htm |title=A Search for Natural or Artificial Objects Located at the Earth–Moon Libration Points |first1=Robert |last1=Freitas |first2=Francisco |last2=Valdes |journal=Icarus |volume=42 |issue=3 |pages=442–447 |date=1980 |doi=10.1016/0019-1035(80)90106-2 |bibcode=1980Icar...42..442F}}
• The Sun–Neptune {{L4|nolink=yes}} and {{L5|nolink=yes}} points contain several dozen known objects, the Neptune trojans.{{cite web | title=List Of Neptune Trojans | publisher=Minor Planet Center | url=http://www.minorplanetcenter.org/iau/lists/NeptuneTrojans.html |access-date=27 October 2010| archive-url = https://web.archive.org/web/20110725075646/http://www.minorplanetcenter.org/iau/lists/NeptuneTrojans.html |archive-date=25 July 2011 | url-status=live}}
• Mars has four accepted Mars trojans: 5261 Eureka, {{mpl|1999 UJ|7}}, {{mpl|1998 VF|31}}, and {{mpl|2007 NS|2}}.
• Saturn's moon Tethys has two smaller moons of Saturn in its {{L4|nolink=yes}} and {{L5|nolink=yes}} points, Telesto and Calypso. Another Saturn moon, Dione also has two Lagrange co-orbitals, Helene at its {{L4|nolink=yes}} point and Polydeuces at {{L5|nolink=yes}}. The moons wander azimuthally about the Lagrange points, with Polydeuces describing the largest deviations, moving up to 32° away from the Saturn–Dione {{L5|nolink=yes}} point.
• One version of the giant impact hypothesis postulates that an object named Theia formed at the Sun–Earth {{L4|nolink=yes}} or {{L5|nolink=yes}} point and crashed into Earth after its orbit destabilized, forming the Moon.{{cite journal |first1=Edward |last1=Belbruno |author-link1=Edward Belbruno |first2=J. Richard |last2=Gott III |author-link2=J. Richard Gott |title=Where Did The Moon Come From? |journal=The Astronomical Journal |volume=129 |issue=3 |pages=1724–1745 |date=2005 |doi=10.1086/427539 |arxiv=astro-ph/0405372 |bibcode=2005AJ....129.1724B |s2cid=12983980 }}
• In binary stars, the Roche lobe has its apex located at {{L1|nolink=yes}}; if one of the stars expands past its Roche lobe, then it will lose matter to its companion star, known as Roche lobe overflow.{{cite journal |title=Equipotential Surfaces and Lagrangian Points in Nonsynchronous, Eccentric Binary and Planetary Systems |last1=Sepinsky |first1=Jeremy F. |last2=Willems |first2=Bart |last3=Kalogera |first3=Vicky |author-link3=Vicky Kalogera |journal=The Astrophysical Journal |volume=660 |issue=2 |pages=1624–1635 |date=May 2007 |doi=10.1086/513736 |arxiv=astro-ph/0612508 |bibcode=2007ApJ...660.1624S |s2cid=15519581 }}

Objects which are on horseshoe orbits are sometimes erroneously described as trojans, but do not occupy Lagrange points. Known objects on horseshoe orbits include 3753 Cruithne with Earth, and Saturn's moons Epimetheus and Janus.

File:Lagrangian points equipotential.png in the plane containing the orbit (grey rubber-sheet model with purple contours of equal potential).{{Cite journal|title=The Roche Problem: Some Analytics|first=Zakir F.|last=Seidov|date=1 March 2004|journal=The Astrophysical Journal|volume=603|issue=1|pages=283–284|doi=10.1086/381315|arxiv=astro-ph/0311272|bibcode=2004ApJ...603..283S|s2cid=16724058}}
Click for animation. ]]

Lagrange points are the constant-pattern solutions of the restricted three-body problem. For example, given two massive bodies in orbits around their common barycenter, there are five positions in space where a third body, of comparatively negligible mass, could be placed so as to maintain its position relative to the two massive bodies. This occurs because the combined gravitational forces of the two massive bodies provide the exact centripetal force required to maintain the circular motion that matches their orbital motion.

Alternatively, when seen in a rotating reference frame that matches the angular velocity of the two co-orbiting bodies, at the Lagrange points the combined gravitational fields of two massive bodies balance the centrifugal pseudo-force, allowing the smaller third body to remain stationary (in this frame) with respect to the first two.

The location of L₁ is the solution to the following equation, gravitation providing the centripetal force:

$$\backslash frac\{M\_1\}\{(R-r)^2\}-\backslash frac\{M\_2\}\{r^2\}=\backslash left(\backslash frac\{M\_1\}\{M\_1+M\_2\}R-r\backslash right)\backslash frac\{M\_1+M\_2\}\{R^3\}$$

where r is the distance of the L₁ point from the smaller object, R is the distance between the two main objects, and M₁ and M₂ are the masses of the large and small object, respectively. The quantity in parentheses on the right is the distance of L₁ from the center of mass. The solution for r is the only real root of the following quintic function

$$x^5\; +\; (\backslash mu\; -\; 3)\; x^4\; +\; (3\; -\; 2\backslash mu)\; x^3\; -\; (\backslash mu)\; x^2\; +\; (2\backslash mu)\; x\; -\; \backslash mu\; =\; 0$$

where $$\backslash mu\; =\; \backslash frac\{M\_2\}\{M\_1+M\_2\}$$

is the mass fraction of M₂ and $$x\; =\; \backslash frac\{r\}\{R\}$$

is the normalized distance. If the mass of the smaller object (M₂) is much smaller than the mass of the larger object (M₁) then {{L1|nolink=yes}} and {{L2|nolink=yes}} are at approximately equal distances r from the smaller object, equal to the radius of the Hill sphere, given by:

$$r\; \backslash approx\; R\; \backslash sqrt[3]\{\backslash frac\{\backslash mu\}\{3\}\}$$

We may also write this as:

$$\backslash frac\{M\_2\}\{r^3\}\backslash approx\; 3\backslash frac\{M\_1\}\{R^3\}$$

Since the tidal effect of a body is proportional to its mass divided by the distance cubed, this means that the tidal effect of the smaller body at the L{{sub|1}} or at the L{{sub|2}} point is about three times of that body. We may also write:

$$\backslash rho\_2\backslash left(\backslash frac\{d\_2\}\{r\}\backslash right)^3\backslash approx\; 3\backslash rho\_1\backslash left(\backslash frac\{d\_1\}\{R\}\backslash right)^3$$

where ρ{{sub|1}} and ρ{{sub|2}} are the average densities of the two bodies and d{{sub|1}} and d{{sub|2}} are their diameters. The ratio of diameter to distance gives the angle subtended by the body, showing that viewed from these two Lagrange points, the apparent sizes of the two bodies will be similar, especially if the density of the smaller one is about thrice that of the larger, as in the case of the Earth and the Sun.

This distance can be described as being such that the orbital period, corresponding to a circular orbit with this distance as radius around M₂ in the absence of M₁, is that of M₂ around M₁, divided by {{sqrt|3}} ≈ 1.73:

$$T\_\{s,M\_2\}(r)\; =\; \backslash frac\{T\_\{M\_2,M\_1\}(R)\}\{\backslash sqrt\{3\}\}.$$

File:L2 rendering.jpg–Earth system]]

The location of L₂ is the solution to the following equation, gravitation providing the centripetal force:

$$\backslash frac\{M\_1\}\{(R+r)^2\}+\backslash frac\{M\_2\}\{r^2\}=\backslash left(\backslash frac\{M\_1\}\{M\_1+M\_2\}R+r\backslash right)\backslash frac\{M\_1+M\_2\}\{R^3\}$$

with parameters defined as for the L₁ case. The corresponding quintic equation is

$$x^5\; +\; x^4\; (3\; -\; \backslash mu)\; +\; x^3\; (3\; -\; 2\backslash mu)\; -\; x^2\; (\backslash mu)\; -\; x\; (2\backslash mu)\; -\; \backslash mu\; =\; 0$$

Again, if the mass of the smaller object (M₂) is much smaller than the mass of the larger object (M₁) then L₂ is at approximately the radius of the Hill sphere, given by:

$$r\; \backslash approx\; R\; \backslash sqrt[3]\{\backslash frac\{\backslash mu\}\{3\}\}$$

The same remarks about tidal influence and apparent size apply as for the L{{sub|1}} point. For example, the angular radius of the Sun as viewed from L₂ is arcsin({{sfrac|{{val|695.5e3}}|{{val|151.1e6}}}}) ≈ 0.264°, whereas that of the Earth is arcsin({{sfrac|6371|{{val|1.5e6}}}}) ≈ 0.242°. Looking toward the Sun from L₂ one sees an annular eclipse. It is necessary for a spacecraft, like Gaia, to follow a Lissajous orbit or a halo orbit around L₂ in order for its solar panels to get full sun.

The location of L₃ is the solution to the following equation, gravitation providing the centripetal force:

$$\backslash frac\{M\_1\}\{\backslash left(R-r\backslash right)^2\}+\backslash frac\{M\_2\}\{\backslash left(2R-r\backslash right)^2\}=\backslash left(\backslash frac\{M\_2\}\{M\_1+M\_2\}R+R-r\backslash right)\backslash frac\{M\_1+M\_2\}\{R^3\}$$

with parameters M₁, M₂, and R defined as for the L₁ and L₂ cases, and r being defined such that the distance of L₃ from the center of the larger object is R − r. If the mass of the smaller object (M₂) is much smaller than the mass of the larger object (M₁), then:{{Cite web|url=https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamics-fall-2009/lecture-notes/MIT16_07F09_Lec18.pdf |title=Widnall, Lecture L18 - Exploring the Neighborhood: the Restricted Three-Body Problem}}

$$r\backslash approx\; R\backslash tfrac\{7\}\{12\}\backslash mu.$$

Thus the distance from L₃ to the larger object is less than the separation of the two objects (although the distance between L₃ and the barycentre is greater than the distance between the smaller object and the barycentre).

{{Further|Trojan (celestial body)}}

The reason these points are in balance is that at {{L4|nolink=yes}} and {{L5|nolink=yes}} the distances to the two masses are equal. Accordingly, the gravitational forces from the two massive bodies are in the same ratio as the masses of the two bodies, and so the resultant force acts through the barycenter of the system. Additionally, the geometry of the triangle ensures that the resultant acceleration is to the distance from the barycenter in the same ratio as for the two massive bodies. The barycenter being both the center of mass and center of rotation of the three-body system, this resultant force is exactly that required to keep the smaller body at the Lagrange point in orbital equilibrium with the other two larger bodies of the system (indeed, the third body needs to have negligible mass). The general triangular configuration was discovered by Lagrange working on the three-body problem.

File:Radial acceleration Earth-Moon Lagrangian.svg

The radial acceleration a of an object in orbit at a point along the line passing through both bodies is given by:

$$a\; =\; -\backslash frac\{G\; M\_1\}\{r^2\}\backslash sgn(r)+\backslash frac\{G\; M\_2\}\{(R-r)^2\}\backslash sgn(R-r)+\backslash frac\{G\backslash bigl((M\_1+M\_2)\; r-M\_2\; R\backslash bigr)\}\{R^3\}$$

where r is the distance from the large body M₁, R is the distance between the two main objects, and sgn(x) is the sign function of x. The terms in this function represent respectively: force from M₁; force from M₂; and centripetal force. The points L₃, L₁, L₂ occur where the acceleration is zero — see chart at right. Positive acceleration is acceleration towards the right of the chart and negative acceleration is towards the left; that is why acceleration has opposite signs on opposite sides of the gravity wells.

File:Roche_potential.stl of the Roche potential of two orbiting bodies, rendered half as a surface and half as a mesh]]

Although the {{L1|nolink=yes}}, {{L2|nolink=yes}}, and {{L3|nolink=yes}} points are nominally unstable, there are quasi-stable periodic orbits called halo orbits around these points in a three-body system. A full n-body dynamical system such as the Solar System does not contain these periodic orbits, but does contain quasi-periodic (i.e. bounded but not precisely repeating) orbits following Lissajous-curve trajectories. These quasi-periodic Lissajous orbits are what most of Lagrangian-point space missions have used until now. Although they are not perfectly stable, a modest effort of station keeping keeps a spacecraft in a desired Lissajous orbit for a long time.

For Sun–Earth-{{L1|nolink=yes}} missions, it is preferable for the spacecraft to be in a large-amplitude ({{convert|100000|–|200000|km|abbr=on|disp=or}}) Lissajous orbit around {{L1|nolink=yes}} than to stay at {{L1|nolink=yes}}, because the line between Sun and Earth has increased solar interference on Earth–spacecraft communications. Similarly, a large-amplitude Lissajous orbit around {{L2|nolink=yes}} keeps a probe out of Earth's shadow and therefore ensures continuous illumination of its solar panels.

The {{L4}} and {{L5}} points are stable provided that the mass of the primary body (e.g. the Earth) is at least 25 times the mass of the secondary body (e.g. the Moon),{{cite web|last1=Fitzpatrick|first1=Richard|title=Stability of Lagrange Points|url=http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node126.html|website=Newtonian Dynamics|publisher=University of Texas}}{{cite web|last1=Greenspan|first1=Thomas|title=Stability of the Lagrange Points, L4 and L5|url=http://www.math.cornell.edu/~templier/junior/final_paper/Thomas_Greenspan-Stability_of_Lagrange_points.pdf|date=7 January 2014|access-date=28 February 2018|archive-date=18 April 2018|archive-url=https://web.archive.org/web/20180418011225/http://www.math.cornell.edu/~templier/junior/final_paper/Thomas_Greenspan-Stability_of_Lagrange_points.pdf|url-status=dead}} The Earth is over 81 times the mass of the Moon (the Moon is 1.23% of the mass of the Earth{{cite journal |last1=Pitjeva |first1=Elena V. |author-link1=Elena V. Pitjeva |last2=Standish |first2=E. Myles |author-link2=E. Myles Standish |title=Proposals for the masses of the three largest asteroids, the Moon-Earth mass ratio and the Astronomical Unit |journal=Celestial Mechanics and Dynamical Astronomy |date=1 April 2009 |volume=103 |issue=4 |pages=365–372 |doi=10.1007/s10569-009-9203-8 |bibcode=2009CeMDA.103..365P |s2cid=121374703 |url=https://zenodo.org/record/1000691 }}). Although the {{L4|nolink=yes}} and {{L5|nolink=yes}} points are found at the top of a "hill", as in the effective potential contour plot above, they are nonetheless stable. The reason for the stability is a second-order effect: as a body moves away from the exact Lagrange position, Coriolis acceleration (which depends on the velocity of an orbiting object and cannot be modeled as a contour map) curves the trajectory into a path around (rather than away from) the point.Cacolici, Gianna Nicole, et al., "[http://math.arizona.edu/~gabitov/teaching/141/math_485/Final_Report/Lagrange_Final_Report.pdf Stability of Lagrange Points: James Webb Space Telescope"], University of Arizona. Retrieved 17 Sept. 2018. Because the source of stability is the Coriolis force, the resulting orbits can be stable, but generally are not planar, but "three-dimensional": they lie on a warped surface intersecting the ecliptic plane. The kidney-shaped orbits typically shown nested around {{L4|nolink=yes}} and {{L5|nolink=yes}} are the projections of the orbits on a plane (e.g. the ecliptic) and not the full 3-D orbits.

File:Lagrange points of planets relative to sun.svg

This table lists sample values of L₁, L₂, and L₃ within the Solar System. Calculations assume the two bodies orbit in a perfect circle with separation equal to the semimajor axis and no other bodies are nearby. Distances are measured from the larger body's center of mass (but see barycenter especially in the case of Moon and Jupiter) with L₃ showing a negative direction. The percentage columns show the distance from the orbit compared to the semimajor axis. E.g. for the Moon, L₁ is {{val|326400|u=km}} from Earth's center, which is 84.9% of the Earth–Moon distance or 15.1% "in front of" (Earthwards from) the Moon; L₂ is located {{val|448900|u=km}} from Earth's center, which is 116.8% of the Earth–Moon distance or 16.8% beyond the Moon; and L₃ is located {{val|-381700|u=km}} from Earth's center, which is 99.3% of the Earth–Moon distance or 0.7084% inside (Earthward) of the Moon's 'negative' position.

{{anchor|Spacecraft and missions}}{{anchor|Lagrangian spacecraft and missions}}

{{Main|List of objects at Lagrange points}}

File:ACE at L1.png in an orbit around Sun–Earth {{L1|nolink=yes}}]]

File:The orbits of Gaia and Webb ESA23998736.png (yellow) and James Webb Space Telescope (blue) orbits around Sun–Earth {{L2|nolink=yes}}]]

Sun–Earth {{L1|nolink=yes}} is suited for making observations of the Sun–Earth system. Objects here are never shadowed by Earth or the Moon and, if observing Earth, always view the sunlit hemisphere. The first mission of this type was the 1978 International Sun Earth Explorer 3 (ISEE-3) mission used as an interplanetary early warning storm monitor for solar disturbances.{{cite web |url=http://solarsystem.nasa.gov/missions/profile.cfm?MCode=ISEEICE |title=ISEE-3/ICE |work=Solar System Exploration |publisher=NASA |access-date=8 August 2015 |archive-url=https://web.archive.org/web/20150720021218/http://solarsystem.nasa.gov/missions/profile.cfm?MCode=ISEEICE |archive-date=20 July 2015 |url-status=dead }} Since June 2015, DSCOVR has orbited the L₁ point. Conversely, it is also useful for space-based solar telescopes, because it provides an uninterrupted view of the Sun and any space weather (including the solar wind and coronal mass ejections) reaches L₁ up to an hour before Earth. Solar and heliospheric missions currently located around L₁ include the Solar and Heliospheric Observatory, Wind, Aditya-L1 Mission and the Advanced Composition Explorer. Planned missions include the Interstellar Mapping and Acceleration Probe(IMAP) and the NEO Surveyor.

Sun–Earth {{L2|nolink=yes}} is a good spot for space-based observatories. Because an object around {{L2|nolink=yes}} will maintain the same relative position with respect to the Sun and Earth, shielding and calibration are much simpler. It is, however, slightly beyond the reach of Earth's umbra,Angular size of the Sun at 1 AU + 1.5 million kilometres: 31.6′, angular size of Earth at 1.5 million kilometres: 29.3′ so solar radiation is not completely blocked at L₂. Spacecraft generally orbit around L₂, avoiding partial eclipses of the Sun to maintain a constant temperature. From locations near L₂, the Sun, Earth and Moon are relatively close together in the sky; this means that a large sunshade with the telescope on the dark-side can allow the telescope to cool passively to around 50 K – this is especially helpful for infrared astronomy and observations of the cosmic microwave background. The James Webb Space Telescope was positioned in a halo orbit about L₂ on 24 January 2022.

Sun–Earth {{L1|nolink=yes}} and {{L2|nolink=yes}} are saddle points and exponentially unstable with time constant of roughly 23 days. Satellites at these points will wander off in a few months unless course corrections are made.

Sun–Earth {{L3|nolink=yes}} was a popular place to put a "Counter-Earth" in pulp science fiction and comic books, despite the fact that the existence of a planetary body in this location had been understood as an impossibility once orbital mechanics and the perturbations of planets upon each other's orbits came to be understood, long before the Space Age; the influence of an Earth-sized body on other planets would not have gone undetected, nor would the fact that the foci of Earth's orbital ellipse would not have been in their expected places, due to the mass of the counter-Earth. The Sun–Earth {{L3|nolink=yes}}, however, is a weak saddle point and exponentially unstable with time constant of roughly 150 years. Moreover, it could not contain a natural object, large or small, for very long because the gravitational forces of the other planets are stronger than that of Earth (for example, Venus comes within 0.3 AU of this {{L3|nolink=yes}} every 20 months).{{cite web|last=DUNCOMBE|first=R. L.|title=Appendix E. Report on Numerical Experiment on the Possible Existence of an "Anti-Earth"|url=http://files.ncas.org/condon/text/appndx-e.htm|work=1968|publisher=U.S. NAVAL OBSERVATORY|access-date=24 October 2013|quote=The separation of [a Counter-Earth] from the line joining the Earth and the Sun shows a variation with increasing amplitude in time, the effect being most pronounced for the largest assumed mass. During the 112 years covered by the integration the separation becomes large enough in all cases that Clarion should have been directly observed, particularly at times of morning or evening twilight and during total solar eclipses.}}

A spacecraft orbiting near Sun–Earth {{L3|nolink=yes}} would be able to closely monitor the evolution of active sunspot regions before they rotate into a geoeffective position, so that a seven-day early warning could be issued by the NOAA Space Weather Prediction Center. Moreover, a satellite near Sun–Earth {{L3|nolink=yes}} would provide very important observations not only for Earth forecasts, but also for deep space support (Mars predictions and for crewed missions to near-Earth asteroids). In 2010, spacecraft transfer trajectories to Sun–Earth {{L3|nolink=yes}} were studied and several designs were considered.{{Cite journal|title=Spacecraft trajectories to the {{L3|nolink=yes}} point of the Sun–Earth three-body problem |journal=Celestial Mechanics and Dynamical Astronomy |last1=Tantardini|first1=Marco |last2=Fantino|first2=Elena |first3=Yuan |last3=Ren |first4=Pierpaolo |last4=Pergola |first5=Gerard |last5=Gómez |first6=Josep J. |last6=Masdemont |date=2010 |doi=10.1007/s10569-010-9299-x |volume=108 |issue=3 |pages=215–232|bibcode = 2010CeMDA.108..215T |s2cid=121179935 |url=https://hal.archives-ouvertes.fr/hal-00568378/file/PEER_stage2_10.1007%252Fs10569-010-9299-x.pdf }}

Earth–Moon {{L1|nolink=yes}} allows comparatively easy access to lunar and Earth orbits with minimal change in velocity and this has as an advantage to position a habitable space station intended to help transport cargo and personnel to the Moon and back. The SMART-1 mission [http://www.moontoday.net/news/viewsr.html?pid=14345 SMART-1: On Course for Lunar Capture | Moon Today – Your Daily Source of Moon News] {{webarchive|url=https://web.archive.org/web/20051102120549/http://www.moontoday.net/news/viewsr.html?pid=14345 |date=2 November 2005 }} passed through the L₁ Lagrangian Point on 11 November 2004 and passed into the area dominated by the Moon's gravitational influence.

Earth–Moon {{L2|nolink=yes}} has been used for a communications satellite covering the Moon's far side, for example, Queqiao, launched in 2018,{{cite magazine |last=Jones |first=Andrew |title=Chang'e-4 relay satellite enters halo orbit around Earth-Moon L2, microsatellite in lunar orbit |url=https://spacenews.com/change-4-relay-satellite-enters-halo-orbit-around-earth-moon-l2-microsatellite-in-lunar-orbit/ |magazine=SpaceNews |date=14 June 2018}} and would be "an ideal location" for a propellant depot as part of the proposed depot-based space transportation architecture.{{cite web |last1=Zegler |first1=Frank |title=Evolving to a Depot-Based Space Transportation Architecture |url=http://www.ulalaunch.com/uploads/docs/Published_Papers/Exploration/DepotBasedTransportationArchitecture2010.pdf |work=AIAA SPACE 2010 Conference & Exposition |publisher=AIAA |access-date=25 January 2011 |first2=Bernard |last2=Kutter |date=2 September 2010 |page=4 |quote=L₂ is in deep space far away from any planetary surface and hence the thermal, micrometeoroid, and atomic oxygen environments are vastly superior to those in LEO. Thermodynamic stasis and extended hardware life are far easier to obtain without these punishing conditions seen in LEO. L₂ is not just a great gateway—it is a great place to store propellants. ... L₂ is an ideal location to store propellants and cargos: it is close, high energy, and cold. More importantly, it allows the continuous onward movement of propellants from LEO depots, thus suppressing their size and effectively minimizing the near-Earth boiloff penalties. |archive-url=https://web.archive.org/web/20140624125633/http://www.ulalaunch.com/uploads/docs/Published_Papers/Exploration/DepotBasedTransportationArchitecture2010.pdf |archive-date=24 June 2014 |url-status=dead }}

Earth–Moon {{L4|nolink=yes}} and {{L5|nolink=yes}} are the locations for the Kordylewski dust clouds.{{cite magazine |last=Kordylewski |first=Kazimierz |author-link=Kazimierz Kordylewski |title=Photographische Untersuchungen des Librationspunktes L₅ im System Erde-Mond |url=https://adsabs.harvard.edu/full/1961AcA....11..165K |magazine=Acta Astronomica, Vol. 11, p.165 |date=1961|volume=11 |page=165 |bibcode=1961AcA....11..165K }} The L5 Society's name comes from the L₄ and L₅ Lagrangian points in the Earth–Moon system proposed as locations for their huge rotating space habitats. Both positions are also proposed for communication satellites covering the Moon alike communication satellites in geosynchronous orbit cover the Earth.{{cite magazine |last=Hornig |first=Andreas |title=TYCHO: Supporting Permanently Crewed Lunar Exploration with High-Speed Optical Communication from Everywhere |url=https://ideas.esa.int/servlet/hype/IMT?userAction=Browse&templateName=&documentId=e3f5e00bb992b91a4378e46176b02376 |magazine=ESA |date=1 May 2022}}{{cite magazine |last=Hornig |first=Andreas |title=TYCHO mission to Earth-Moon libration point EML-4 @ IAC 2013 |url=https://www.youtube.com/watch?v=7RJSLFP7yyA |magazine=IAC2013 |date=6 October 2013}}

Scientists at the B612 Foundation were{{Cite web |last=Foust |first=Jeff |date=20 June 2017 |url=https://spacenews.com/b612-studying-smallsat-missions-to-search-for-near-earth-objects/ |title=B612 studying smallsat missions to search for near Earth objects |website=SpaceNews.com |access-date=6 May 2025 }} planning to use Venus's L₃ point to position their planned Sentinel telescope, which aimed to look back towards Earth's orbit and compile a catalog of near-Earth asteroids.{{cite web|url=https://b612foundation.org/sentinel-mission/ |archive-url=https://web.archive.org/web/20120630034544/http://b612foundation.org/sentinel-mission/ |url-status=dead |archive-date=30 June 2012 |title=The Sentinel Mission |publisher=B612 Foundation |access-date=1 February 2014}}

In 2017, the idea of positioning a magnetic dipole shield at the Sun–Mars {{L1|nolink=yes}} point for use as an artificial magnetosphere for Mars was discussed at a NASA conference.{{Cite web|url=https://phys.org/news/2017-03-nasa-magnetic-shield-mars-atmosphere.html|title=NASA proposes a magnetic shield to protect Mars' atmosphere|website=phys.org}} The idea is that this would protect the planet's atmosphere from the Sun's radiation and solar winds.

{{Portal|Astronomy|Spaceflight}}

{{div col|small=no|colwidth=15em}}

• Co-orbital configuration
• Euler's three-body problem
• Gegenschein
• Interplanetary Transport Network
• Klemperer rosette
• L₅ Society
• Lagrange point colonization
• Lagrangian mechanics
• List of objects at Lagrange points
• Lunar space elevator
• Oberth effect

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{{Reflist|group=note}}

{{Reflist}}

{{Commons category|Lagrange points}}

• Joseph-Louis, Comte Lagrange, from Œuvres, Tome 6, « Essai sur le Problème des Trois Corps »—[https://web.archive.org/web/20160506112535/http://www.ltas-vis.ulg.ac.be/cmsms/uploads/File/Lagrange_essai_3corps.pdf Essai (PDF)]; source [http://gallica.bnf.fr/ark:/12148/bpt6k229225j.image.r=Lagrange%2C+Joseph-Louis%2C.langEN.f231.pagination Tome 6 (Viewer)]
• "Essay on the Three-Body Problem" by J.-L. Lagrange, translated from the above, in [http://www.merlyn.demon.co.uk/essai-3c.htm merlyn.demon.co.uk] {{Webarchive|url=https://web.archive.org/web/20190623015513/http://www.merlyn.demon.co.uk/essai-3c.htm |date=23 June 2019 }}.
• [http://eulerarchive.maa.org/pages/E304.html Considerationes de motu corporum coelestium]—Leonhard Euler—transcription and translation at [http://www.merlyn.demon.co.uk/euler304.htm merlyn.demon.co.uk] {{Webarchive|url=https://web.archive.org/web/20200803041106/http://www.merlyn.demon.co.uk/euler304.htm |date=3 August 2020 }}.
• [https://web.archive.org/web/20030605101737/http://www.physics.montana.edu/faculty/cornish/lagrange.html Explanation of Lagrange points]—Neil J. Cornish
• [https://web.archive.org/web/20200229000635/https://www.haydenplanetarium.org/tyson/essays/2002-04-the-five-points-of-lagrange.php The Five Points of Lagrange] by Neil deGrasse Tyson

{{Commons category|Lagrange points}}

• [https://drive.google.com/file/d/1GaTwyQkbb38RGDFoyNJZc7ZJnUJE0Mns/view?usp=sharing ZIP file]—J R Stockton - Includes translations of Lagrange's Essai and of two related papers by Euler
• [https://www.esa.int/Enabling_Support/Operations/What_are_Lagrange_points What are Lagrange points?]—European Space Agency page, with good animations
• [https://map.gsfc.nasa.gov/mission/observatory_l2.html A NASA explanation]—also attributed to Neil J. Cornish
• [http://math.ucr.edu/home/baez/lagrange.html Explanation of Lagrange points]—John Baez
• [http://www.phy6.org/stargaze/Slagrang.htm Locations of Lagrange points, with approximations]—David Peter Stern
• [http://orbitsimulator.com/formulas/LagrangePointFinder.html An online calculator to compute the precise positions of the 5 Lagrange points for any 2-body system]—Tony Dunn
• [http://www.astronomycast.com/physics/ep-76-lagrange-points/ Astronomy Cast—Ep. 76: "Lagrange Points"] by Fraser Cain and Pamela L. Gay
• [http://www.nasa.gov/mission_pages/WISE/news/wise20110727.html Earth, a lone Trojan discovered]
• See the Lagrange Points and Halo Orbits subsection under the section on Geosynchronous Transfer Orbit in [https://solarsystem.nasa.gov/basics/chapter5-1#critical NASA: Basics of Space Flight, Chapter 5]

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Category:Trojans (astronomy)