Lagrange point

In celestial mechanics, the Lagrange points (also Lagrangian points, L-points, or libration points) are points near two large orbiting bodies. Normally, the two objects exert an unbalanced gravitational force at a point, altering the orbit of whatever is at that point. At the Lagrange points, the gravitational forces of the two large bodies and the centrifugal force balance each other. This can make Lagrange points an excellent location for satellites, as few orbit corrections are needed to maintain the desired orbit. Small objects placed in orbit at Lagrange points are in equilibrium in at least two directions relative to the center of mass of the large bodies.
There are five such points, labeled L1 to L5, all in the orbital plane of the two large bodies, for each given combination of two orbital bodies. For instance, there are five Lagrangian points L1 to L5 for the Sun–Earth system, and in a similar way there are five "different" Lagrangian 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 third vertex of an equilateral triangle formed with the centers of the two large bodies. L4 and L5 are stable, which implies that objects can orbit around them in a rotating coordinate system tied to the two large bodies.
The L4 and L5 points are stable gravity wells and have a tendency to pull objects into them. Several planets have trojan asteroids near their L4 and L5 points with respect to the Sun. Jupiter has more than a million of these trojans. Artificial satellites have been placed at L1 and L2 with respect to the Sun and Earth, and with respect to the Earth and the Moon. The Lagrangian points have been proposed for uses in space exploration.
History.
The three collinear Lagrange points (L1, L2, L3) were discovered by Leonhard Euler a few years before Joseph-Louis Lagrange discovered the remaining two.
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.
Lagrange points.
The five Lagrange points are labeled and defined as follows:
point.
The point lies on the line defined by the two large masses "M"1 and "M"2, and between them. It is the point where the gravitational attraction of "M"2 partially cancels that of "M"1. An object that orbits the Sun more closely than Earth would normally have a shorter orbital period than Earth, but that ignores the effect of Earth's own 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, and therefore increases the orbital period of the object. The closer to Earth the object is, the greater this effect is. At the point, the orbital period of the object becomes exactly equal to Earth's orbital period. is about 1.5 million kilometers from Earth, or 0.01 au, 1/100th the distance to the Sun.
point.
The point lies on the line through the two large masses, beyond the smaller of the two. Here, the gravitational forces of the two large masses balance the centrifugal effect on a body at . On the opposite side of Earth from the Sun, the orbital period of an object would normally be greater than that of Earth. The extra pull of Earth's gravity decreases the orbital period of the object, and at the point that orbital period becomes equal to Earth's. Like L1, L2 is about 1.5 million kilometers or 0.01 au from Earth.
point.
The point lies on the line defined by the two large masses, beyond the larger of the two. Within the Sun–Earth system, the point exists on the opposite side of the Sun, a little outside Earth's orbit and slightly closer to 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 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.
and points.
The and points lie at the third corners 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 behind () or ahead () of the smaller mass with regard to its orbit around the larger mass.
Point stability.
The triangular points ( and ) are stable equilibria, provided that the ratio of is greater than 24.96. 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).
The points , , and are positions of unstable equilibrium. Any object orbiting at , , or will tend to fall out of orbit; it is therefore rare to find natural objects there, and spacecraft inhabiting these areas must employ station keeping in order to maintain their position.
Natural objects at Lagrange points.
Due to the natural stability of and , 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 and points, which were taken from mythological characters appearing in Homer's "Iliad", an epic poem set during the Trojan War. Asteroids at the point, ahead of Jupiter, are named after Greek characters in the "Iliad" and referred to as the "Greek camp". Those at the 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 Sun-Jupiter trojans than for any other pair of bodies. However, smaller numbers of objects are known at the Langrage points of other orbital systems:
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.
Mathematical details.
Lagrangian 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. As seen in a rotating reference frame that matches the angular velocity of the two co-orbiting bodies, the gravitational fields of two massive bodies combined providing the centripetal force at the Lagrangian points, allowing the smaller third body to be relatively stationary with respect to the first two.
The location of L1 is the solution to the following equation, gravitation providing the centripetal force:
where "r" is the distance of the L1 point from the smaller object, "R" is the distance between the two main objects, and "M"1 and "M"2 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.) Solving this for "r" involves solving a quintic function, but if the mass of the smaller object ("M"2) is much smaller than the mass of the larger object ("M"1) then and are at approximately equal distances "r" from the smaller object, equal to the radius of the Hill sphere, given by:
We may also write this as:
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 or at the L point is about three times that of the larger body. We may also write:
where ρ and ρ are the average densities of the two bodies and formula_5 and formula_6 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 "M"2 in the absence of "M"1, is that of "M"2 around "M"1, divided by ≈ 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. Again, if the mass of the smaller object ("M"2) is much smaller than the mass of the larger object ("M"1) then L2 is at approximately the radius of the Hill sphere, given by:
The same remarks about tidal influence and apparent size apply as for the L point. For example, the angular radius of the sun as viewed from L is arcsin(/) ≈ 0.264°, whereas that of the earth is arcsin(6371/ ≈ 0.242°. Looking toward the sun from L one sees an annular eclipse. It is necessary for a spacecraft, like Gaia, to follow a Lissajous orbit or a halo orbit around L in order for its solar panels to get full sun.
L3.
The location of L3 is the solution to the following equation, gravitation providing the centripetal force:
with parameters "M"1,2 and "R" defined as for the L1 and L2 cases, and "r" now indicates the distance of L3 from the position of the smaller object, if it were rotated 180 degrees about the larger object, while positive r implying L3 is closer to the larger object than the smaller object. If the mass of the smaller object ("M"2) is much smaller than the mass of the larger object ("M"1) then:
and.
The reason these points are in balance is that, at and , 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 need not have negligible mass.) The general triangular configuration was discovered by Lagrange in work on the three-body problem.
Radial acceleration.
The radial acceleration "a" of an object in orbit at a point along the line passing through both bodies is given by:
where "r" is the distance from the large body "M"1 and sgn("x") is the sign function of "x". The terms in this function represent respectively: force from "M"1; force from "M"2; and centrifugal force. The points L3, L1, L2 occur where the acceleration is zero — see chart at right.
Stability.
Although the , , and 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- missions, it is preferable for the spacecraft to be in a large-amplitude () Lissajous orbit around than to stay at , because the line between Sun and Earth has increased solar interference on Earth–spacecraft communications. Similarly, a large-amplitude Lissajous orbit around keeps a probe out of Earth's shadow and therefore ensures continuous illumination of its solar panels.
The and 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). The Earth is over 81 times the mass of the Moon (the Moon is 1.23% of the mass of the Earth). Although the and 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. 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 and are the projections of the orbits on a plane (e.g. the ecliptic) and not the full 3-D orbits.
Solar System values.
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 with L3 showing a negative location. The percentage columns show how the distances compare to the semimajor axis. E.g. for the Moon, L1 is located from Earth's center, which is 84.9% of the Earth–Moon distance or 15.1% in front of the Moon; L2 is located from Earth's center, which is 116.8% of the Earth–Moon distance or 16.8% beyond the Moon; and L3 is located from Earth's center, which is 99.3% of the Earth–Moon distance or 0.7084% in front of the Moon's 'negative' position.
Spaceflight applications.
Sun–Earth.
Sun–Earth 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. 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, and the Advanced Composition Explorer. Planned missions include the Interstellar Mapping and Acceleration Probe (IMAP).
Sun–Earth is a good spot for space-based observatories. Because an object around 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, 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 is due to be positioned at L2.
Sun–Earth was a popular place to put a "Counter-Earth" in pulp science fiction and comic books. Once space-based observation became possible via satellites and probes, it was shown to hold no such object. The Sun–Earth is unstable and could not contain a natural object, large or small, for very long. This is because the gravitational forces of the other planets are stronger than that of Earth (Venus, for example, comes within 0.3 AU of this every 20 months).
A spacecraft orbiting near Sun–Earth would be able to closely monitor the evolution of active sunspot regions before they rotate into a geoeffective position, so that a 7-day early warning could be issued by the NOAA Space Weather Prediction Center. Moreover, a satellite near Sun–Earth would provide very important observations not only for Earth forecasts, but also for deep space support (Mars predictions and for manned mission to near-Earth asteroids). In 2010, spacecraft transfer trajectories to Sun–Earth were studied and several designs were considered.
Missions to Lagrangian points generally orbit the points rather than occupy them directly.
Another interesting and useful property of the collinear Lagrangian points and their associated Lissajous orbits is that they serve as "gateways" to control the chaotic trajectories of the Interplanetary Transport Network.
Earth–Moon.
Earth–Moon allows comparatively easy access to Lunar and Earth orbits with minimal change in velocity and this has as an advantage to position a half-way manned space station intended to help transport cargo and personnel to the Moon and back.
Earth–Moon has been used for a communications satellite covering the Moon's far side, for example, Queqiao, launched in 2018, and would be "an ideal location" for a propellant depot as part of the proposed depot-based space transportation architecture.
Sun–Venus.
Scientists at the B612 Foundation were 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 catalogue of near-Earth asteroids.
Sun–Mars.
In 2017, the idea of positioning a magnetic dipole shield at the Sun–Mars point for use as an artificial magnetosphere for Mars was discussed at a NASA conference. The idea is that this would protect the planet's atmosphere from the Sun's radiation and solar winds.
Lagrangian spacecraft and missions.
Spacecraft at Sun–Earth L1.
International Sun Earth Explorer 3 (ISEE-3) began its mission at the Sun–Earth L1 before leaving to intercept a comet in 1982. The Sun–Earth L1 is also the point to which the Reboot ISEE-3 mission was attempting to return the craft as the first phase of a recovery mission (as of September 25, 2014 all efforts have failed and contact was lost).
Solar and Heliospheric Observatory (SOHO) is stationed in a halo orbit at , and the Advanced Composition Explorer (ACE) in a Lissajous orbit. WIND is also at . Currently slated for launch in late 2024, the Interstellar Mapping and Acceleration Probe will be placed near .
Deep Space Climate Observatory (DSCOVR), launched on 11 February 2015, began orbiting L1 on 8 June 2015 to study the solar wind and its effects on Earth. DSCOVR is unofficially known as GORESAT, because it carries a camera always oriented to Earth and capturing full-frame photos of the planet similar to the Blue Marble. This concept was proposed by then-Vice President of the United States Al Gore in 1998 and was a centerpiece in his 2006 film "An Inconvenient Truth".
LISA Pathfinder (LPF) was launched on 3 December 2015, and arrived at on 22 January 2016, where, among other experiments, it tested the technology needed by (e)LISA to detect gravitational waves. LISA Pathfinder used an instrument consisting of two small gold alloy cubes.
Spacecraft at Sun–Earth L2.
Spacecraft at the Sun–Earth L2 point are in a Lissajous orbit until decommissioned, when they are sent into a heliocentric graveyard orbit.