Lagrange points in the Sun–Earth system (not to scale). This view is from the north, so that Earth's orbit is counterclockwise.Acontour plot of theeffective potential due to gravity and thecentrifugal 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) and away from them (blue). Counterintuitively, the L4 and L5 points are thehigh points of the potential. At the points themselves these forces are balanced. An example of a spacecraft at Sun-Earth L2, the Wilkinson Microwave Anisotropy Probe, or WMAP WMAPEarth
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, thegravitational forces of the two large bodies and thecentrifugal force balance each other.[2] This can make Lagrange points an excellent location for satellites, asorbit 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, L1 to L5, all in the orbital plane of the two large bodies. There are five Lagrange points for the Sun–Earth system, and fivedifferent Lagrange points for the Earth–Moon system. L1, L2, and L3 are on the line through the centers of the two large bodies, while L4 and L5 each act as the thirdvertex of anequilateral triangle formed with the centers of the two large bodies.
When the mass ratio of the two bodies is large enough, the L4 and L5 points are stable points, meaning that objects can orbit them and that they have a tendency to pull objects into them. Several planets havetrojan asteroids near their L4 and L5 points with respect to the Sun;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 L1, between the Sun and Earth, and L2, on the same line at the opposite side of the Earth; both are well outside the Moon's orbit. Currently, anartificial satellite called theDeep Space Climate Observatory (DSCOVR) is located at L1 to study solar wind coming toward Earth from the Sun and to monitor Earth's climate, by taking images and sending them back.[3] TheJames Webb Space Telescope, a powerful infrared space observatory, is located at L2.[4] 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 L1 and L2 Lagrange points are located about 1,500,000 km (930,000 mi) from Earth.
The European Space Agency's earlierGaia telescope, and its newly launchedEuclid, also occupy orbits around L2. Gaia keeps a tighterLissajous orbit around L2, while Euclid follows ahalo 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.
The three collinear Lagrange points (L1, L2, L3) were discovered by the Swiss mathematicianLeonhard Euler around 1750, a decade before the Italian-bornJoseph-Louis Lagrange discovered the remaining two.[5][6]
In 1772, Lagrange published an "Essay on thethree-body problem". In the first chapter he considered the general three-body problem. From that, in the second chapter, he demonstrated two specialconstant-pattern solutions, the collinear and the equilateral, for any three masses, withcircular orbits.[7]
The L1 point lies on the line defined between the two large massesM1 andM2. It is the point where the gravitational attraction ofM2 and that ofM1 combine to produce an equilibrium. An object thatorbits theSun more closely thanEarth 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, thenEarth'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 point, the object's orbital period becomes exactly equal to Earth's orbital period. L1 is about 1.5 million kilometers, or 0.01au, from Earth in the direction of the Sun.[1]
The L2 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. 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 point, that orbital period becomes equal to Earth's. Like L1, L2 is about 1.5 million kilometers or 0.01au from Earth (away from the sun). An example of a spacecraft designed to operate near the Earth–Sun L2 is theJames Webb Space Telescope.[8] Earlier examples include theWilkinson Microwave Anisotropy Probe and its successor,Planck.
The L3 point lies on the line defined by the two large masses, beyond the larger of the two. Within the Sun–Earth system, the L3 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 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.
The L4 and L5 points lie at the third vertices of the twoequilateral 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) or behind (L5) the smaller mass with regard to its orbit around the larger mass.
The triangular points (L4 and L5) are stable equilibria, provided that the ratio ofM1/M2 is greater than 24.96.[note 1] 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).[9]
The points L1, L2, and L3 are positions ofunstable equilibrium. Any object orbiting at L1, L2, or L3 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 ofstation keeping in order to maintain their position.
Due to the natural stability of L4 and L5, 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 and L5 points, which were taken from mythological characters appearing inHomer'sIliad, anepic poem set during theTrojan War. Asteroids at the L4 point, ahead of Jupiter, are named after Greek characters in theIliad and referred to as the "Greek camp". Those at the L5 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 and L5 points contain interplanetary dust and at least two asteroids,2010 TK7 and2020 XL5.[10][11][12]
The Earth–Moon L4 and L5 points contain concentrations ofinterplanetary dust, known asKordylewski clouds.[13][14] Stability at these specific points is greatly complicated by solar gravitational influence.[15]
Saturn's moonTethys has two smaller moons of Saturn in its L4 and L5 points,Telesto andCalypso. Another Saturn moon,Dione also has two Lagrange co-orbitals,Helene at its L4 point andPolydeuces at L5. The moons wanderazimuthally about the Lagrange points, with Polydeuces describing the largest deviations, moving up to 32° away from the Saturn–Dione L5 point.
One version of thegiant impact hypothesis postulates that an object namedTheia formed at the Sun–Earth L4 or L5 point and crashed into Earth after its orbit destabilized, forming the Moon.[17]
Objects which are onhorseshoe orbits are sometimes erroneously described as trojans, but do not occupy Lagrange points. Known objects on horseshoe orbits include3753 Cruithne with Earth, and Saturn's moonsEpimetheus andJanus.
Visualization of the relationship between the Lagrange points (red) of a planet (blue) orbiting a star (yellow) counterclockwise, and theeffective potential in the plane containing the orbit (grey rubber-sheet model with purple contours of equal potential).[19] Click for animation.
Lagrange points are the constant-pattern solutions of the restrictedthree-body problem. For example, given two massive bodies in orbits around their commonbarycenter, there are five positions in space where a third body, of comparatively negligiblemass, 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 thecircular motion that matches their orbital motion.
The location of L1 is the solution to the following equation, gravitation providing the centripetal force:wherer is the distance of the L1 point from the smaller object,R is the distance between the two main objects, andM1 andM2 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 forr is the onlyrealroot of the followingquintic function
whereis the mass fraction ofM2 andis the normalized distance. If the mass of the smaller object (M2) is much smaller than the mass of the larger object (M1) then L1 and L2 are at approximately equal distancesr from the smaller object, equal to the radius of theHill sphere, given by:
We may also write this as:Since thetidal 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 L1 or at the L2 point is about three times of that body. We may also write:whereρ1 andρ2 are the average densities of the two bodies andd1 andd2 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 theorbital period, corresponding to a circular orbit with this distance as radius aroundM2 in the absence ofM1, is that ofM2 aroundM1, divided by√3 ≈ 1.73:
The location of L2 is the solution to the following equation, gravitation providing the centripetal force:with parameters defined as for the L1 case. The corresponding quintic equation is
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 theHill sphere, given by:
The same remarks about tidal influence and apparent size apply as for the L1 point. For example, the angular radius of the Sun as viewed from L2 is arcsin(695.5×103/151.1×106) ≈ 0.264°, whereas that of the Earth is arcsin(6371/1.5×106) ≈ 0.242°. Looking toward the Sun from L2 one sees anannular eclipse. It is necessary for a spacecraft, likeGaia, to follow aLissajous orbit or ahalo 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:with parametersM1,M2, andR defined as for the L1 and L2 cases, andr being defined such that the distance of L3 from the center of the larger object isR − r. If the mass of the smaller object (M2) is much smaller than the mass of the larger object (M1), then:[20]
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).
The reason these points are in balance is that at L4 and L5 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 thebarycenter of the system. Additionally, the geometry of the triangle ensures that theresultant acceleration is to the distance from the barycenter in the sameratio as for the two massive bodies. The barycenter being both thecenter 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 orbitalequilibrium 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 thethree-body problem.
Net radial acceleration of a point orbiting along the Earth–Moon line
The radial accelerationa of an object in orbit at a point along the line passing through both bodies is given by:wherer is the distance from the large bodyM1,R is the distance between the two main objects, and sgn(x) is thesign function ofx. The terms in this function represent respectively: force fromM1; force fromM2; 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.
STL 3D model of the Roche potential of two orbiting bodies, rendered half as a surface and half as a mesh
Although the L1, L2, and L3 points are nominally unstable, there are quasi-stable periodic orbits calledhalo orbits around these points in a three-body system. A fulln-bodydynamical system such as theSolar System does not contain these periodic orbits, but does contain quasi-periodic (i.e. bounded but not precisely repeating) orbits followingLissajous-curve trajectories. These quasi-periodicLissajous orbits are what most of Lagrangian-point space missions have used until now. Although they are not perfectly stable, a modest effort ofstation keeping keeps a spacecraft in a desired Lissajous orbit for a long time.
For Sun–Earth-L1 missions, it is preferable for the spacecraft to be in a large-amplitude (100,000–200,000 km or 62,000–124,000 mi) Lissajous orbit around L1 than to stay at L1, because the line between Sun and Earth has increased solarinterference on Earth–spacecraft communications. Similarly, a large-amplitude Lissajous orbit around L2 keeps a probe out of Earth's shadow and therefore ensures continuous illumination of its solar panels.
TheL4 andL5 points are stable provided that the mass of the primary body (e.g. the Earth) is at least 25[note 1] times the mass of the secondary body (e.g. the Moon),[21][22] The Earth is over 81 times the mass of the Moon (the Moon is 1.23% of the mass of the Earth[23]). Although the L4 and L5 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)[22] curves the trajectory into a path around (rather than away from) the point.[22][24] 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 and L5 are the projections of the orbits on a plane (e.g. the ecliptic) and not the full 3-D orbits.
Sun–planet Lagrange points to scale (Click for clearer points.)
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 seebarycenter 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 is326400 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 located448900 km from Earth's center, which is 116.8% of the Earth–Moon distance or 16.8% beyond the Moon; and L3 is located−381700 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.
Sun–Earth L2 is a good spot for space-based observatories. Because an object around L2 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'sumbra,[26] 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 forinfrared astronomy and observations of thecosmic microwave background. TheJames Webb Space Telescope was positioned in a halo orbit about L2 on 24 January 2022.
Sun–Earth L1 and L2 aresaddle points and exponentially unstable withtime constant of roughly 23 days. Satellites at these points will wander off in a few months unless course corrections are made.[9]
Sun–Earth L3 was a popular place to put a "Counter-Earth" inpulpscience fiction andcomic 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, however, is a weak saddle point and exponentially unstable with time constant of roughly 150 years.[9] 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 every 20 months).[27]
A spacecraft orbiting near Sun–Earth L3 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 theNOAASpace Weather Prediction Center. Moreover, a satellite near Sun–Earth L3 would provide very important observations not only for Earth forecasts, but also for deep space support (Mars predictions and for crewed missions tonear-Earth asteroids). In 2010, spacecraft transfer trajectories to Sun–Earth L3 were studied and several designs were considered.[28]
Earth–Moon L1 allows comparatively easy access to lunar and Earth orbits with minimal change in velocity and this has as an advantage to position a habitablespace station intended to help transport cargo and personnel to the Moon and back. TheSMART-1 mission[29] passed through the L1 Lagrangian Point on 11 November 2004 and passed into the area dominated by the Moon'sgravitational influence.
Earth–Moon L2 has been used for acommunications satellite covering the Moon's far side, for example,Queqiao, launched in 2018,[30] and would be "an ideal location" for apropellant depot as part of the proposed depot-based space transportation architecture.[31]
Earth–Moon L4 and L5 are the locations for theKordylewski dust clouds.[32] TheL5 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 ingeosynchronous orbit cover the Earth.[33][34]
In 2017, the idea of positioning amagnetic dipole shield at the Sun–Mars L1 point for use as an artificial magnetosphere for Mars was discussed at a NASA conference.[37] The idea is that this would protect the planet's atmosphere from the Sun's radiation and solar winds.
^abCornish, Neil J. (1998)."The Lagrange Points"(PDF). WMAP Education and Outreach. Archived fromthe original(PDF) on 7 September 2015. Retrieved15 December 2015.
^"ISEE-3/ICE".Solar System Exploration. NASA. Archived fromthe original on 20 July 2015. Retrieved8 August 2015.
^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′
^DUNCOMBE, R. L."Appendix E. Report on Numerical Experiment on the Possible Existence of an "Anti-Earth"".1968. U.S. NAVAL OBSERVATORY. Retrieved24 October 2013.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.
^Zegler, Frank; Kutter, Bernard (2 September 2010)."Evolving to a Depot-Based Space Transportation Architecture"(PDF).AIAA SPACE 2010 Conference & Exposition. AIAA. p. 4. Archived fromthe original(PDF) on 24 June 2014. Retrieved25 January 2011.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.
