GFDL:Lagrangian point

File:Lagrange points.jpg

A contour plot of the effective potential (the Hill's Surfaces) of a two-body system (the Sun and Earth here), showing the five Lagrange points.

The Lagrangian points (pronounced Template:IPA or Template:IPA); also Lagrange point, L-point, or libration point), are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be stationary relative to two larger objects (such as a satellite with respect to the Earth and Moon). The Lagrange points mark positions where the combined gravitational pull of the two large masses provides precisely the centripetal force required to rotate with them. They are analogous to geosynchronous orbits in that they allow an object to be in a "fixed" position in space rather than an orbit in which its relative position changes continuously.

A more precise but technical definition is that the Lagrangian points are the stationary solutions of the circular restricted three-body problem. For example, given two massive bodies in circular orbits around their common center of mass, there are five positions in space where a third body, of comparatively negligible mass, could be placed which would then maintain its position relative to the two massive bodies. As seen in a rotating reference frame with the same period as the two co-orbiting bodies, the gravitational fields of two massive bodies combined with the centrifugal force are in balance at the Lagrangian points, allowing the third body to be stationary with respect to the first two bodies.

History and concepts

In 1772, the famed Italian-French mathematician Joseph-Louis Lagrange was working on the famous three-body problem when he discovered an interesting quirk in the results. Originally, he had set out to discover a way to easily calculate the gravitational interaction between arbitrary numbers of bodies in a system, because Newtonian mechanics conclude that such a system results in the bodies orbiting chaotically until there is a collision, or a body is thrown out of the system so that equilibrium can be achieved. The logic behind this conclusion is that a system with one body is trivial, as it is merely static relative to itself; a system with two bodies is very simple to solve for, as the bodies orbit around their common center of gravity. However, once more than two bodies are introduced, the mathematical calculations become very complicated. A situation arises where you would have to calculate every gravitational interaction between every object at every point along its trajectory.

Lagrange, however, wanted to make this simpler. He did so with a simple conclusion: The trajectory of an object is determined by finding a path that minimizes the action over time. This is found by subtracting the potential energy from the kinetic energy. With this way of thinking, Lagrange re-formulated the classical Newtonian mechanics to give rise to Lagrangian mechanics. With his new system of calculations, Lagrange’s work led him to hypothesize how a third body of negligible mass would orbit around two larger bodies which were already in a near-circular orbit. In a frame of reference that rotates with the larger bodies, he found five specific fixed points where the third body experiences zero net force as it follows the circular orbit of its host bodies (planets). These points were named “Lagrangian points” in Lagrange's honor. It took over a hundred years before his mathematical theory was observed with the discovery of the Trojan asteroids in the 1900s at the Lagrange points of the Sun-Jupiter system.

In the more general case of elliptical orbits, there are no longer stationary points in the same sense: it becomes more of a Lagrangian “area”. The Lagrangian points constructed at each point in time as in the circular case form stationary elliptical orbits which are similar to the orbits of the massive bodies. This is due to Newton's second law (<math>d\mathbf{p}/dt=\mathbf{F}</math>), where p = mv (p the momentum, m the mass, and v the velocity) is invariant if force and position are scaled by the same factor. A body at a Lagrangian point orbits with the same period as the two massive bodies in the circular case, implying that it has the same ratio of gravitational force to radial distance as they do. This fact is independent of the circularity of the orbits, and it implies that the elliptical orbits traced by the Lagrangian points are solutions of the equation of motion of the third body.

The Lagrangian points

File:Lagrange very massive.svg

A diagram showing the five Lagrangian points in a two-body system with one body far more massive than the other (e.g. the Sun and the Earth). In such a system L₃–L₅ will appear to share the secondary's orbit, although in fact they are situated slightly outside it.

The five Lagrangian points are labeled and defined as follows:

L₁

The L₁ point lies on the line defined by the two large masses M₁ and M₂, and between them. It is the most intuitively understood of the Lagrangian points: the one where the gravitational attractions of the two other objects effectively cancel each other out.

Example: An object which orbits the Sun more closely than the Earth would normally have a shorter orbital period than the Earth, but that ignores the effect of the Earth's own gravitational pull. If the object is directly between the Earth and the Sun, then the effect of the Earth's gravity is to weaken the force pulling the object towards the Sun, and therefore increase the orbital period of the object. The closer to Earth the object is, the greater this effect is. At the L₁ point, the orbital period of the object becomes exactly equal to the Earth's orbital period.

The Sun–Earth L₁ is ideal for making observations of the Sun. Objects here are never shadowed by the Earth or the Moon. The Solar and Heliospheric Observatory (SOHO) is stationed in a Halo orbit at the L₁ and the Advanced Composition Explorer (ACE) is in a Lissajous orbit, also at the L₁ point. The Earth–Moon L₁ allows easy access to lunar and earth orbits with minimal delta-v, and would be ideal for a half-way manned space station intended to help transport cargo and personnel to the Moon and back.

L₂

The L₂ point lies on the line defined by the two large masses, beyond the smaller of the two.

Example: On the side of the Earth away from the Sun, the orbital period of an object would normally be greater than that of the Earth. The extra pull of the Earth's gravity decreases the orbital period of the object, and at the L₂ point that orbital period becomes equal to the Earth's.

Sun–Earth L₂ is a good spot for space-based observatories. Because an object around L₂ will maintain the same orientation with respect to the Sun and Earth, shielding and calibration are much simpler. The Wilkinson Microwave Anisotropy Probe is already in orbit around the Sun–Earth L₂. The future Herschel Space Observatory and Gaia probe as well as the proposed James Webb Space Telescope will be placed at the Sun–Earth L₂. Earth–Moon L₂ would be a good location for a communications satellite covering the Moon's far side.

If M₂ is much smaller than M₁, then L₁ and L₂ are at approximately equal distances r from M₂, equal to the radius of the Hill sphere, given by:

<math>r \approx R \sqrt[3]{\frac{M_2}{3 M_1}}</math>

where R is the distance between the two bodies.

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 <math>\sqrt{3}\approx 1.73</math>.

Examples:

• Sun and Earth: 1,500,000 km from the Earth
• Earth and Moon: 61,500 km from the Moon

L₃

The L₃ point lies on the line defined by the two large masses, beyond the larger of the two.

Example: L₃ in the Sun–Earth system exists on the opposite side of the Sun, a little outside the Earth's orbit but slightly closer to the Sun than the Earth is. Here, the combined pull of the Earth and Sun again causes the object to orbit with the same period as the Earth. The Sun–Earth L₃ point was a popular place to put a "Counter-Earth" in pulp science fiction and comic books - though of course, once space based observation was possible via satellites and probes, it was shown to hold no such object. In actual fact, Sun–Earth L₃ is highly unstable, because the gravitational forces of the other planets outweigh that of the Earth (Venus, for example, comes within 0.3 AU of L₃ every 20 months).

L₄ and L₅

File:L4 diagram.svg

Gravitational accelerations at L₄

The L₄ and L₅ points lie at the third corners of the two equilateral triangles in the plane of orbit whose common base is the line between the centres of the two masses, such that the point is ahead of (L₄), or behind (L₅), the smaller mass with regard to its orbit around the larger mass.

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 the barycentre of the system; additionally, the geometry of the triangle ensures that the resultant acceleration is to the distance from the barycentre in the same ratio as for the two massive bodies. The barycentre being both the centre of mass and centre of rotation of the system, this resultant force is exactly that required to keep a body at the Lagrange point in orbital equilibrium with the rest of the system. (Indeed, the third body need not have negligible mass; the general triangular configuration was discovered by Lagrange in work on the 3-body problem.)

L₄ and L₅ are sometimes called triangular Lagrange points or Trojan points. The name Trojan points comes from the Trojan asteroids at the Sun-Jupiter L₄ and L₅ points, which themselves are named after characters from Homer's Iliad (the legendary siege of Troy). Asteroids at the L4 point, which leads Jupiter are referred to as the 'Greek camp' while at the L5 point they are referred to as the 'Trojan camp', and asteroids there are (largely) named after characters from the respective sides of the war.

Examples:

• The Sun–Earth L₄ and L₅ points lie 60° ahead of and 60° behind the Earth as it orbits the Sun. They contain interplanetary dust.
• The Earth–Moon L₄ and L₅ points lie 60° ahead of and 60° behind the Moon as it orbits the Earth. They also contain interplanetary dust in what is called Kordylewski cloud.
• The Sun–Jupiter L₄ and L₅ points are occupied by the Trojan asteroids.
• Neptune has Trojan Kuiper Belt Objects at its L₄ and L₅ points.
• Saturn's moon Tethys has two much smaller satellites at its L₄ and L₅ points named Telesto and Calypso, respectively.
• Saturn's moon Dione has smaller moons Helene and Polydeuces at its L₄ and L₅ points, respectively.
• The giant impact hypothesis suggests that an object named Theia formed at L₄ or L₅ and crashed into the Earth after its orbit destabilized, forming the moon.

Stability

The first three Lagrangian points are technically stable only in the plane perpendicular to the line between the two bodies. This can be seen most easily by considering the L₁ point. A test mass displaced perpendicularly from the central line would feel a force pulling it back towards the equilibrium point. This is because the lateral components of the two masses' gravity would add to produce this force, whereas the components along the axis between them would balance out. However, if an object located at the L₁ point drifted closer to one of the masses, the gravitational attraction it felt from that mass would be greater, and it would be pulled closer. (The pattern is very similar to that of tidal forces.)

Although the L₁, L₂, and L₃ points are nominally unstable, it turns out that it is possible to find stable periodic orbits around these points, at least in the restricted three-body problem. These perfectly periodic orbits, referred to as "halo" orbits, do not exist in a full n-body dynamical system such as the solar system. However, quasi-periodic (i.e. bounded but not precisely repeating) orbits following Lissajous curve trajectories do exist in the n-body system. These quasi-periodic Lissajous orbits are what all Lagrangian point missions to date have used. Although they are not perfectly stable, a relatively modest effort at station keeping can allow a spacecraft to stay in a desired Lissajous orbit for an extended period of time. It also turns out that, at least in the case of Sun–Earth L₁ missions, it is actually preferable to place the spacecraft in a large amplitude (100,000–200,000 km) Lissajous orbit instead of having it sit at the Lagrangian point, because this keeps the spacecraft off the direct Sun–Earth line and thereby reduces the impacts of solar interference on the Earth–spacecraft communications links. 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.

In contrast to the collinear Lagrangian points, the triangular points (L₄ and L₅) are stable equilibria (cf. attractor), provided the ratio of M₁/M₂ is greater than 24.96. This is the case for the Sun/Earth and, by a smaller margin, the Earth/Moon systems. When a body at these points is perturbed, it moves away from the point, but the Coriolis effect then acts, and bends the object's path into a stable, kidney bean‐shaped orbit around the point (as seen in the rotating frame of reference). However, in the Earth-Moon case the problem of stability is greatly complicated by the appreciable solar gravitational influence.

Lagrangian point missions

The Lagrangian point orbits have unique characteristics that have made them a good choice for performing some kinds of missions. NASA has operated a number of spacecraft in orbit around the Sun-Earth L₁ and L₂ points, including

Mission	Lagrangian point
Advanced Composition Explorer (ACE)	L1
Genesis	L1
International Sun/Earth Explorer 3 (ISEE-3)	L1
Solar and Heliospheric Observatory (SOHO)	L1
Wilkinson Microwave Anisotropy Probe (WMAP)	L2

The James Webb Space Telescope is also planned to be placed in orbit around L₂.

The L5 Society was a precursor of the National Space Society, and promoted the possibility of establishing a colony and manufacturing facility in orbit around the L₄ or L₅ points in the Earth–Moon system (see Space colonization).

Natural examples

In the Sun–Jupiter system several thousand asteroids, collectively referred to as Trojan asteroids, are in orbits around the Sun–Jupiter L₄ and L₅ points. Other bodies can be found in the Sun–Saturn, Sun–Mars, Sun-Neptune, Jupiter–Jovian satellite, and Saturn-Saturnian satellite systems. There are no known large bodies in the Sun–Earth system's Trojan points, but clouds of dust surrounding the L₄ and L₅ points were discovered in the 1950s. Clouds of dust, called Kordylewski clouds, even fainter than the notoriously weak gegenschein, are also present in the L₄ and L₅ of the Earth–Moon system.

The Saturnian moon Tethys has two smaller moons in its L₄ and L₅ points, Telesto and Calypso. The Saturnian moon Dione also has two Lagrangian co-orbitals, Helene at its L₄ point and Polydeuces at L₅. The moons wander azimuthally about the Lagrangian points, with Polydeuces describing the largest deviations, moving up to 32 degrees away from the Saturn–Dione L₅ point. Tethys and Dione are hundreds of times more massive than their "escorts" (see the moons' articles for exact diameter figures; masses are not known in several cases), and Saturn is far more massive still, which makes the overall system stable.

Other co-orbitals

The Earth's companion object 3753 Cruithne is in a relationship with the Earth which is somewhat Trojan-like, but different from a true Trojan. This asteroid occupies one of two regular solar orbits, one of them slightly smaller and faster than the Earth's orbit, and the other slightly larger and slower. The asteroid periodically alternates between these two orbits due to close encounters with Earth. When the asteroid is in the smaller, faster orbit and approaches the Earth, it loses orbital energy to the Earth and moves into the larger, slower orbit. It then falls farther and farther behind the earth, and eventually Earth approaches it from the other direction. Then the asteroid gains orbital energy from the Earth, and the asteroid moves back into the smaller orbit, thus beginning the cycle anew. The cycle has no noticeable impact on the length of the year, because Earth's mass is over 20 billion (2 × 10¹⁰) times more than 3753 Cruithne.

Epimetheus and Janus, satellites of Saturn, have a similar relationship, though they are of similar masses and so actually exchange orbits with each other periodically. (Janus is roughly 4 times more massive, but still light enough for its orbit to be altered.) Another similar configuration is known as orbital resonance, in which orbiting bodies tend to have periods of a simple integer ratio, due to their interaction.

In fiction

• The Lagrange points are mentioned in science fiction from time to time (most often hard science fiction), but, due to the general lack of public familiarity with them, they are rarely used as a plot device or reference.

• Lagrange points are mentioned most famously in the science fiction film 2010: The Year We Make Contact, where the Discovery spacecraft is located on a Lagrange point.

• The Lagrange points are mentioned in the anime saga Mobile Suit Gundam, where clusters of space colonies (called "Sides") are located at the five Lagrange points of Earth, in addition to resource satellites and space fortresses. Lagrange Points have been mentioned in several other Gundam series as well.

• The L5 Lagrange point is mentioned in L5: First City in Space, an early IMAX 3D movie.

• In William Gibson's novel Neuromancer, much of the action takes place in the L5 "archipelago", the location of many space stations.

• Lagrange points also play a role in the Larry Niven/Jerry Pournelle classic The Mote in God's Eye.

• The eponymous interplanetary relay station in George O. Smith's "Venus Equilateral" stories was located in the L₄ point of the Sun-Venus system.

• In Robert Forward's Rocheworld the locations for Lagrange points around a binary planet are disscussed in contrast to typical system.

• The planet Troas in the stories "Sucker Bait" by Isaac Asimov and "Question and Answer" by Poul Anderson was located in the L₅ point of a fictional binary star system.

• In the Star Trek: The Next Generation episode, "The Survivors", the Enterprise is surprised by an enemy ship that had been hiding in a Lagrange point.

• The space station Babylon 5 is described to be located "at the L-5 point in a binary star system between a moon and a barren, lifeless planet." [1]

• In the Independence War computer games, Lagrange points are used as the only locations for jump-points.

• In the Battletech game series, a star's Nadir and Zenith are the standard hyperspace jump points for most interstellar spacecraft. Lagrange points (usually the L4 and L5 points) are sometimes used to enter a system closer to planets, almost always for small-scale military or pirate operations due to the risk of catastrophic misjumps.

• In the PC video game Star Wars: X-Wing, Lagrange points are mentioned in the briefings of some missions that revolve around attacking objects placed at them.

• In the Xbox video game Halo: Combat Evolved (2001) and sequel Halo 2 (2004), Halo Megastructures play key locations throughout the games. In Halo: CE and Halo 2, the Halo structures are in L1 Lagrange points between the Gas Giants (and a moon) Threshold and Substance, respectively.

• In the Halo novels the Lagrange points are the only places where a human ship can safely make a slipspace jump.

• In the Robert A. Heinlein novel The Number of the Beast, two of the main characters engage in a discussion of adding planets to the solar system at Lagrange points.

• In Hideo Kojima's video game Policenauts, the setting of the game, an O'Neill model space colony, is located at the L5 Lagrange point.

• In the sci-fi series Stargate Atlantis there was a defensive satellite located at a Lagrangian point in the solar system in which Atlantis was located.[2]

• In the Hugo Award-winning novel A Deepness in the Sky by Vernor Vinge, a temporary human habitat is built at the L1 point between the planet Arachna and its primary star, a highly variable dwarf called the On/Off Star.

• In Peter F Hamilton's Night's Dawn Trilogy, a ZTT jump drive cannot be used in a strong gravitational field. In the first book of the trilogy, The Reality Dysfunction, the main characters cannot escape from a gas giant's gravity well before their pursuers catch up with them. Instead, they race to the Lagrange point between the gas giant and one of its moons in order to activate their drive. Successful execution of this untried and reckless maneuver gains captain Joshua Calvert the nickname "LaGrange" Calvert. In the second book The Neutronium Alchemist, a visit is paid to the supposed home planet of the Kiint, Jobis, which features three moons orbiting the Lagrange One point, rotating around a common centre.

• In the TV series Quatermass II, the hostile aliens live on a small asteroid "no more than half a mile across" at a "theoretical point of equilibrium" on the dark side of the earth, although neither L2 or Lagrange are mentioned by name (the term "Bieber Variation" is used instead).

• In Arthur Clarke and Stephen Baxter's novel Sunstorm L1 point plays a crucial role in the building of a shield that has the purpose of saving Earth from a storm of energy from the Sun.

• In the manga series Battle Angel Alita: Last Order, the ex-colony ship turned space station Leviathan 1 is at the L2 point in the Earth/Moon system.

• In 1991, Konami released a science fiction RPG for the NES in Japan called Lagrange Point.

See also

• List of objects at Lagrangian points
• Lunar space elevator
• Home on Lagrange (The L5 Song)
• L5 Society
• Colonization of the Sun (Lagrange Points)

External links

• Explanation of Lagrange points – Prof. Neil J. Cornish
• A NASA explanation - also attributed to Neil J. Cornish
• Explanation of Lagrange points – Prof. John Baez
• Geometry and calculations of Lagrange points – Dr J R Stockton
• Locations of Lagrange points, with approximations – Dr. David P. Stern