The Lagrangian points (pronounced /ləˈgrɒndʒiən/, French IPA: [lagʀɑ̃ʒjɑ̃]; 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). In Astronomy libration (from the Latin verb librare "to balance to sway" cf In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star Gravitation is a natural Phenomenon by which objects with Mass attract one another This article is about artificial satellites For natural satellites also known as moons see Natural satellite. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 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. The centripetal force is the external force required to make a body follow a curved path They are analogous to geostationary 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 geostationary orbit (GEO is a Geosynchronous orbit directly above the Earth 's Equator (0° Latitude) with a period equal to the Earth's
A more precise but technical definition is that the Lagrangian points are the stationary solutions of the circular restricted three-body problem . The n -body problem is the problem of finding given the initial positions masses and velocities of n bodies their subsequent motions as determined by 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. In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object 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. A rotating frame of reference is a special case of a Non-inertial reference frame that is rotating relative to an Inertial reference frame. A gravitational field is a model used within Physics to explain how gravity exists in the universe 
In 1772, the Italian-French mathematician Joseph-Louis Lagrange was working on the famous three-body problem when he discovered an interesting quirk in the results. Year 1772 ( MDCCLXXII) was a Leap year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or a The n -body problem is the problem of finding given the initial positions masses and velocities of n bodies their subsequent motions as determined by 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. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that 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 pair of objects at every point along its trajectory.
Lagrange, however, wanted to make this simpler. He did so with a simple hypothesis: The trajectory of an object is determined by finding a path that minimizes the action over time. In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation This is found by subtracting the potential energy from the kinetic energy. Potential energy can be thought of as Energy stored within a physical system The kinetic energy of an object is the extra Energy which it possesses due to its motion With this way of thinking, Lagrange re-formulated the classical Newtonian mechanics to give rise to Lagrangian mechanics. Lagrangian mechanics is a re-formulation of Classical mechanics that combines Conservation of momentum with Conservation of energy. 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. A hypothesis (from Greek) consists either of a suggested explanation for a phenomenon (an event that is observable or of a reasoned proposal suggesting a possible 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”. In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a 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 (), 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. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Mathematics and Theoretical physics, an invariant is a property of a system which remains unchanged under some transformation. 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 five Lagrangian points are labeled and defined as follows:
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 attraction of M2 partially cancels M1 gravitational attraction.
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 L1, and the Advanced Composition Explorer (ACE) is in a Lissajous orbit, also at the L1 point. The Solar and Heliospheric Observatory ( SOHO) is a Spacecraft that was launched on a Lockheed Martin Atlas IIAS launch vehicle on December A halo orbit is a periodic three-dimensional Orbit near the L1 L2 or L3 Lagrange points in the three-body problem of Advanced Composition Explorer (ACE is a Space exploration mission being conducted as part of the Explorer program to study Matter In situ In Orbital mechanics, a Lissajous orbit is a quasi-periodic orbital trajectory that an object can follow around a collinear Libration point ( Lagrangian point The Earth–Moon L1 allows easy access to lunar and earth orbits with minimal change in velocity and would be ideal for a half-way manned space station intended to help transport cargo and personnel to the Moon and back.
The L2 point lies on the line defined by the two large masses, beyond the smaller of the two. Here, the gravitational forces of the two large masses balance the centrifugal force on the smaller mass.
The 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, Gaia probe, and James Webb Space Telescope will be placed at the Sun–Earth L2. The Herschel Space Observatory ("Herschel" is a European Space Agency (ESA mission originally proposed in 1982 by a consortium of European scientists that included Gaia is a European Space Agency (ESA Astrometry space mission and a successor to the ESA Hipparcos mission The James Webb Space Telescope ( JWST) is a planned space Infrared observatory the successor to the aging Hubble Space Telescope. Earth–Moon L2 would be a good location for a communications satellite covering the Moon's far side. A communications satellite (sometimes abbreviated to comsat) is an artificial Satellite stationed in space for the purposes of Telecommunications.
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 distances r from the smaller object, equal to the radius of the Hill sphere, given by:
where R is the distance between the two bodies. A Hill sphere is roughly the volume around an Astronomical body (such as a Planet) where it dominates in attraction of Satellites to that body rather
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 . The orbital period is the time taken for a given object to make one complete Orbit about another object
The L3 point lies on the line defined by the two large masses, beyond the larger of the two. The Sun (Sol is the Star at the center of the Solar System. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001
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 lies behind (L5) or ahead of (L4) the smaller mass with regard to its orbit around the larger mass. Properties The area of an equilateral triangle with sides of length a\\!
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. In Mathematics, the resultant of two Monic polynomials P and Q over a field k is defined as the product A ratio is an expression which compares quantities relative to each other 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. A dynamic equilibrium occurs when two opposing Processes proceed at the same rate (Indeed, the third body need not have negligible mass; the general triangular configuration was discovered by Lagrange in work on the 3-body problem. The n -body problem is the problem of finding given the initial positions masses and velocities of n bodies their subsequent motions as determined by )
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). Homer ( Ancient Greek:, Homēros) is a legendary ancient Greek epic Poet, traditionally said to be the author of the epic poems the The Iliad ( Greek: Ἰλιάς (Ancient Ιλιάδα (Modern is together with the Odyssey, one of two ancient Troy ( Greek: grc Τροία Troia, also, Ilion; Latin: Trōia, Īlium, Hittite: Wilusa or 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'. List of Trojan asteroids (Trojan campThis is a list of Jupiter 's Trojan asteroids that lie in the elongated curved regions around the leading L4 Lagrangian point List of Trojan asteroids (Greek campThis is a list of Jupiter 's Trojan asteroids that lie in the elongated curved regions around the trailing L5 Lagrangian point These asteroids are (largely) named after characters from the respective sides of the war.
The first three Lagrangian points are technically stable only in the plane perpendicular to the line between the two bodies. In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent 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. The tidal force is a secondary effect of the Force of Gravity and is responsible for the Tides It arises because the gravitational acceleration experienced )
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. The Solar System consists of the Sun and those celestial objects bound to it by Gravity. However, quasi-periodic (i. e. bounded but not precisely repeating) orbits following Lissajous curve trajectories do exist in the n-body system. In Mathematics, a Lissajous curve ( Lissajous figure or Bowditch curve) is the graph of the system of Parametric equations These quasi-periodic Lissajous orbits are what all Lagrangian point missions to date have used. In Orbital mechanics, a Lissajous orbit is a quasi-periodic orbital trajectory that an object can follow around a collinear Libration point ( Lagrangian point 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. In Astrodynamics orbital station-keeping is a term used to describe a particular set of Orbital maneuvers used to keep a spacecraft in assigned Orbit 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, thereby reducing the impact 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. The Interplanetary Transport Network (ITN is a collection of Gravitationally determined pathways through the Solar system that require very little Energy
In contrast to the collinear Lagrangian points, the triangular points (L4 and L5) are stable equilibria (cf. attractor), provided that the ratio of M1/M2 is greater than 24. An attractor is a set to which a Dynamical system evolves after a long enough time 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 bends the object's path into a stable, kidney bean‐shaped orbit around the point (as seen in the rotating frame of reference). In physics the Coriolis effect is an apparent deflection of moving objects when they are viewed from a Rotating frame of reference. However, in the Earth–Moon case, the problem of stability is greatly complicated by the appreciable solar gravitational influence. 
This section non-mathematically (intuitively) explains the five Lagrangian points using the Earth–Moon system.
Lagrangian points L2 through L5 only exist in rotating systems, as in the monthly orbiting of the Moon about the Earth. At these points, an outward (fictitious, as explained below) centrifugal force is balanced by the attractive gravitational forces of the Moon and Earth.
Imagine using your hand to spin a stone at the end of a string. The string provides a tension force that continuously accelerates the stone towards the center. To an ant standing on the stone, however, it seems as if there is a force trying to fling him directly outward from the center. This apparent or fictitious force is called the centrifugal force . This same effect is present in the Earth–Moon system, where the role of the string is played by the summed (or net) effect of the two attractive gravities, and the stone is an asteroid or a spacecraft. The Earth–Moon system rotates about its combined center of mass, or barycenter. Because the Earth is much heavier than the Moon, this point is located within the Earth (about a thousand miles below the surface). Any object gravitationally held by the rotating Earth–Moon system will sense a centrifugal force directed away from the barycenter, in the same way as does the ant on our stone.
Unlike the other Lagrangian points, L1 would exist even in a non-rotating (static or inertial) system. In Physics, an inertial frame of reference is a Frame of reference which belongs to a set of frames in which Physical laws hold in the same and simplest Rotation slightly pushes L1 away from the (heavier) Earth towards the (lighter) Moon. L1 is slightly unstable (see stability above) because drifting towards the Moon or Earth increases one gravitational attraction while decreasing the other, causing more drift.
At Lagrangian points L2, L3, L4, and L5, a satellite feels an outward centrifugal force, away from the barycenter, that exactly balances the attractive gravity of the Earth and Moon. L2 and L3 are slightly unstable because small changes in satellite position more strongly affect gravity than the balancing centrifugal force. Stability at L4 and L5 depends crucially on the satellite being pulled in three different directions, namely the outward centrifugal force away from the barycenter, balancing the inward gravitational forces towards the Moon and Earth.
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
|Advanced Composition Explorer (ACE)|
|International Sun/Earth Explorer 3 (ISEE-3)|
|Solar and Heliospheric Observatory (SOHO)|
|Wilkinson Microwave Anisotropy Probe (WMAP)|
ESA's Herschel Space Observatory (formerly called Far Infrared and Sub-millimetre Telescope or FIRST) in 2008 and the James Webb Space Telescope are also planned to be placed in orbit around L2. The National Aeronautics and Space Administration ( NASA, ˈnæsə is an agency of the United States government, responsible for the nation's public space program Advanced Composition Explorer (ACE is a Space exploration mission being conducted as part of the Explorer program to study Matter In situ The Genesis spacecraft was the first ever attempt to collect a sample of Solar wind, and the first " Sample return mission " to return from beyond the The International Cometary Explorer (ICE Spacecraft was originally known as International Sun/Earth Explorer 3 (ISEE-3 satellite launched August 12, 1978 The Solar and Heliospheric Observatory ( SOHO) is a Spacecraft that was launched on a Lockheed Martin Atlas IIAS launch vehicle on December The Herschel Space Observatory ("Herschel" is a European Space Agency (ESA mission originally proposed in 1982 by a consortium of European scientists that included The James Webb Space Telescope ( JWST) is a planned space Infrared observatory the successor to the aging Hubble Space Telescope.  ESA's Planck satellite planned for launch in 2008 will be placed in orbit around L2. The Planck satellite is a Spacecraft built in the Cannes Mandelieu Space Center, that is designed to observe the anisotropies of the cosmic microwave
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). The L5 Society was founded in 1975 by Carolyn and Keith Henson to promote the space colony ideas of Dr The National Space Society (NSS is an international nonprofit 501(c(3, educational and scientific organization specializing in Space advocacy. Space colonization (also called space settlement, space humanization, Space habitation, etc
The Earth–Moon L2 point has been proposed as a location for a communication satellite covering the far side of the Moon. NASA TN_D-4059
The Deep Space Climate Observatory was intended to be positioned at L1, but has been put into indefinite storage after being built. Deep Space Climate Observatory (DSCOVR (formerly known as Triana) is a NASA satellite proposed in 1998 by then- Vice President Al Gore for
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. Asteroids, sometimes called Minor planets or planetoids', are bodies—primarily of the inner Solar System —that are smaller than planets but Recent observations suggest that the Sun–Neptune L4 and L5 points, known as the Neptune Trojans, may be very thickly populated, containing large bodies an order of magnitude more numerous than the Jupiter Trojans. Neptune ( English|AmE] ] is the eighth and farthest Planet from the Sun in the Solar System. As of May 2008, there are six known Neptune trojans (named by analogy to the Trojan asteroids which have the same orbital period as Neptune. Other bodies can be found in the Sun–Mars 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. The 1950s Decade refers to the years of 1950 to 1959 inclusive Clouds of dust, called Kordylewski clouds, even fainter than the notoriously weak gegenschein, may also be present in the L4 and L5 of the Earth–Moon system. Kordylewski clouds are large concentrations of dust that may exist at the L4 and L5 Lagrangian points of the Earth-Moon system Gegenschein ( very roughy like GAY-guhn-shine German for "counter shine" is a faint brightening of the night sky in the region of the Antisolar
The Saturnian moon Tethys has two smaller moons in its L4 and L5 points, Telesto and Calypso. TemplateInfobox Planet.--> Tethys (ˈtiːθɨs, /ˈtɛθɨs/, or TemplateInfobox Planet.--> Telesto (tɨˈlɛstoʊ, or as Greek TemplateInfobox Planet.--> Calypso (kəˈlɪpsoʊ, or as in Greek The Saturnian moon Dione also has two Lagrangian co-orbitals, Helene at its L4 point and Polydeuces at L5. TemplateInfobox Planet.--> Dione (daɪˈoʊni, or as in Greek TemplateInfobox Planet.--> Helene (ˈhɛlɨni, sometimes, or as 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. Azimuth ( is a mathematical concept defined as the angle usually measured in degrees (° between a reference plane and a 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.
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. TemplateInfobox Planet. --> 3753 Cruithne (ˈkrɪnjə from Old Irish ˈkrɪθnɛ Modern 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 gains orbital energy from the Earth and moves up 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 gives up orbital energy to the Earth, and drops 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. TemplateInfobox Planet.--> Epimetheus (ˌɛpɨˈmiːθiəs,, or Saturn X redirects here For the spurious moon reported in 1905 see Themis (moon (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 Celestial mechanics, an orbital resonance occurs when two Orbiting bodies exert a regular periodic gravitational influence on each other usually due to their A ratio is an expression which compares quantities relative to each other
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