A contour plot of the effective potential of a two-body system (the Sun and Earth here) as viewed from the rotating frame of reference in which Sun and Earth remain stationary. The Mathematical study of potentials is known as Potential theory; it is the study of Harmonic functions on Manifolds This mathematical 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 Objects revolving with the same orbital period as the Earth will begin to move according to the arrows indicating the slopes around the five Lagrange points — downhill toward or away from them, but at the points themselves these forces are balanced. The orbital period is the time taken for a given object to make one complete Orbit about another object

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 [1]. 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 [2]

## History and concepts

The three collinear Lagrange points were first discovered by Euler around 1750. [3]

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). [4] 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 ($\mathbf{F}=d\mathbf{p}/dt$), 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 Lagrangian points

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 attraction of M2 partially cancels M1 gravitational attraction.

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. 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 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 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 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.

### L2

A diagram showing the Sun–Earth L2 point, which lies well beyond the Moon's orbit around the Earth.

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.

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.

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:

$r \approx R \sqrt[3]{\frac{M_2}{3 M_1}}$

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 $\sqrt{3}\approx 1.73$. The orbital period is the time taken for a given object to make one complete Orbit about another object

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. 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

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. [5] 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. The Counter-Earth is a hypothetical body of the Solar system first hypothesized by the Presocratic Philosopher Philolaus to support his non- A comic book (often shortened to simply comic and sometimes called a comic paper or comic magazine) is a Magazine or Book of narrative 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. The VENUS ( V ictoria E xperimental N etwork U nder the S ea project is a cabled sea floor observatory operated by the University 3 AU of L3 every 20 months). The astronomical unit ( AU or au or au or sometimes ua) is a unit of Length based on the distance from the Earth to the

### L4 and L5

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 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.

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 may contain interplanetary dust in what is called Kordylewski clouds. Kordylewski clouds are large concentrations of dust that may exist at the L4 and L5 Lagrangian points of the Earth-Moon system
• 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. Neptune ( English|AmE] ] is the eighth and farthest Planet from the Sun in the Solar System. The Kuiper belt (ˈkaɪpɚ to rhyme with "viper" sometimes called the Edgeworth-Kuiper belt, is a region of the Solar System beyond the planets extending
• Saturn's moon Tethys has two much smaller satellites at its L4 and L5 points named Telesto and Calypso, respectively. TemplateInfobox Planet.--> Tethys (ˈtiːθɨs, /ˈtɛθɨs/, or TemplateInfobox Planet.--> Telesto (tɨˈlɛstoʊ, or as Greek TemplateInfobox Planet.--> Calypso (kəˈlɪpsoʊ, or as in Greek
• Saturn's moon Dione has smaller moons Helene and Polydeuces at its L4 and L5 points, respectively. TemplateInfobox Planet.--> Dione (daɪˈoʊni, or as in Greek TemplateInfobox Planet.--> Helene (ˈhɛlɨni, sometimes, or as
• 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. Theia is the Hypothetical planet that according to the Giant impact hypothesis of the Moon 's formation collided with Earth over four billion

## Stability

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[6][7]. 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. [8]

## Intuitive explanation

This section non-mathematically (intuitively[9]) 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.

## 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

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

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. [10] 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

## 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. 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.

### 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. 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

## In fiction

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

#### L1

• In Arthur Clarke and Stephen Baxter's novel Sunstorm, the 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. Hard science fiction is a category of Science fiction characterized by an emphasis on scientific or technical detail or on scientific accuracy or on both Sir Arthur Charles Clarke, CBE (16 December 1917–19 March 2008 was a British Science fiction Author, Inventor, and Stephen Baxter (born 13 November 1957 is a British Hard science fiction Author. Sunstorm is a 2005 Science fiction Novel co-written by Arthur C
• In the Xbox video game Halo: Combat Evolved (2001) and sequel Halo 2 (2004), Halo Megastructures play key locations throughout the games. The Xbox is a sixth-generation Video game console produced by Microsoft Corporation. A video game is a Game that involves interaction with a User interface to generate visual feedback on a video device. Halo Combat Evolved (commonly known as Halo or Halo 1) is a First-person shooter (FPS Video game developed Year 2001 ( MMI) was a Common year starting on Monday according to the Gregorian calendar. Halo 2 is a Science fiction First-person shooter Video game developed by Bungie Studios. "MMIV" redirects here For the Modest Mouse album see " Baron von Bullshit Rides Again " Halos are fictional Megastructures and Superweapons in the Halo video game series 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. Halo Combat Evolved (commonly known as Halo or Halo 1) is a First-person shooter (FPS Video game developed Halo 2 is a Science fiction First-person shooter Video game developed by Bungie Studios. Halo is a Science fiction Video game franchise created by Bungie and owned and published by Microsoft Game Studios.
• 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. A Deepness in the Sky is a Science fiction Novel by Vernor Vinge. Vernor Steffen Vinge (ˈvɪndʒi (born October 2, 1944 in Waukesha Wisconsin, U For the astronomical object see Variable star. Variable Star is a 2006 novel written by Spider Robinson The term dwarf star refers to a variety of distinct classes of Stars Dwarf star alone generally refers to any Main sequence star a star
• The planet on which Pitch Black is set receives continual daylight: an orrery depicts it at the apparent L1 point between one star and a pair of tightly orbiting stars, receiving periodic eclipses from a pair of gas giants occupying a wider orbit about L1 in the same plane. Pitch Black (also known as The Chronicles of Riddick Pitch Black) is a 2000 Science fiction action movie directed This article is on the mechanical device For the British peerage see Earl of Orrery.
• In Peter F Hamilton's Night's Dawn Trilogy, a ZTT jump drive cannot be used in a strong gravitational field. Peter F Hamilton (born 2 March 1960 is a British Science fiction Author. British author Peter F Hamilton 's The Night's Dawn Trilogy consists of three epic Science fiction novels The Reality Dysfunction 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. The Reality Dysfunction is a Science fiction novel by Peter F 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. The Neutronium Alchemist is a Science fiction novel by Peter F

#### L2

• 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 nor Lagrange are mentioned by name (the term "Bieber Variation" is used instead). You may also be looking for an asteroid designated 1181 Lilith Lilith is a hypothetical natural satellite of Earth Quatermass II is a British science-fiction serial, originally broadcast by BBC Television in the autumn of 1955
• 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. Star Trek The Next Generation ( STTNG or TNG) is an Emmy and Peabody Award-winning science fiction television program created by Gene Roddenberry " The Survivors " is a third-season episode of Star Trek The Next Generation first broadcast on October 9, 1989. The USS Enterprise (NCC-1701-D (or Enterprise -D, to distinguish it from prior starships with the same name) is a 24th century Starship
• 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. Battle Angel Alita Last Order, known in Japan as, is the continuation of the Manga series Battle Angel Alita.
• In John Varley's book Wizard, a religious group called the Coven set up a habitat at L2 to make themselves as remote as possible from the earth. John Herbert Varley (born August 9, 1947 in Austin Texas) is an American Science fiction Author. Wizard is a 1980 Science fiction novel by John Varley. It is the second book in his Gaea Trilogy. L2 later slowly deteriorates into the space version of an unorganized shantytown as anyone with enough cash can set up a home there. "L2 became known as Sargasso Point to the pilots who carefully avoided it; those who had to travel through it called it the Pinball Machine, and they didn't smile. "

#### L3

• In the third season of the TV series Lexx, the planets Fire and Water are found to reside in Earth's L3 point. The Counter-Earth is a hypothetical body of the Solar system first hypothesized by the Presocratic Philosopher Philolaus to support his non- Lexx is a science fantasy television series that follows the adventures of a group of mismatched individuals aboard the organic space craft Lexx. In the sci-fi Television series LEXX, the fictional planet " Fire " is the Afterlife for all Evil In the Television series LEXX, the fictional planet " Water " is the Afterlife for all good Souls and the location
• John Norman's Gor series takes place on a counter earth. John Frederick Lange Jr (born June 3, 1931) better known under his pen name John Norman, is a Professor of Philosophy and an author Gor, the Counter-Earth, is the alternate-world setting for John Norman's Chronicles of Gor, a series of twenty six novels that combine Philosophy

#### L4

• In Larry Niven's novels The Integral Trees and The Smoke Ring, the L4 and L5 points of the Smoke Ring (a ring of breathable air and plant life orbiting a neutron star) have become gathering places for masses of water, living things, and human settlers. Laurence van Cott Niven (born April 30, 1938 Los Angeles California) is a US Science fiction author. The Integral Trees is a 1984 Science fiction novel by Larry Niven (first published as a serial in Analog in 1983 The Smoke Ring is a 1987 Science fiction novel by Larry Niven. A neutron star is a type of remnant that can result from the Gravitational collapse of a massive Star during a Type II, Type Ib or Type
• The eponymous interplanetary relay station in George O. Smith's "Venus Equilateral" stories was located in the L4 point of the Sun–Venus system. Microwave radio relay is a technology for transmitting digital and analog signals, such as long-distance Telephone calls and the relay of George Oliver Smith ( April 9, 1911 - May 27, 1981) (also known as Wesley Long) was an American Science fiction The Venus Equilateral series is a set of 13 Science fiction short stories by George O
• In the Japanese series Mobile Suit Gundam Seed, there is an Island-type colony named Mendel in L4. shortened to Gundam SEED, is an Anime television series and a Manga series by Sunrise. It has been abandoned due to a viral outbreak.
• In the Xbox 360 game Mass Effect a group of inhabited space stations are situated at the L4 and L5 points. The Xbox 360 is the second Video game console produced by Microsoft, and was developed in cooperation with IBM, ATI, and SiS. Mass Effect is an Action role-playing game developed by BioWare for Xbox 360 and Microsoft Windows. A space station is an artificial structure designed for Humans to live in Outer space.

#### L5

• The L5 Lagrange point is mentioned in L5: First City in Space, an early IMAX 3D movie. IMAX (short for Image MAXimum is a Film format created by Canada 's IMAX Corporation that has the capacity to display images of far greater size and
• In William Gibson's novel Neuromancer, much of the action takes place in the L5 "archipelago", the location of many space stations. Neuromancer is a 1984 novel by William Gibson, notable for being the most famous early Cyberpunk novel and winner of the science-fiction "triple
• 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. Troas is a fictional Planet that serves as the setting for the Science fiction Novellas " Sucker Bait " by Isaac Asimov and Sucker Bait is a Science fiction Novella by Isaac Asimov. It was first serialized in the February and March 1954 issues of Isaac Asimov (c January 2 1920 &ndash April 6 1992 ˈaɪzək ˈæzɪmʌv originally Исаак Озимов but now transcribed into Russian as, was a Russian Question and Answer is a Science fiction Novel by Poul Anderson that originally appeared in the June and July 1954 issues of Astounding Poul William Anderson ( November 25, 1926 – July 31, 2001) was an American Science fiction author who wrote during a Golden A binary star is a Star system consisting of two Stars orbiting around their Center of mass.
• 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. Babylon 5 is an American science fiction television series created produced and largely written by J " [1]
• 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. is a Japanese video game designer originally employed at Konami. is a cinematic Adventure game, with a hard science fiction storyline published by Konami, written and directed by Hideo Kojima. The O'Neill cylinder also called Island Three is a Space habitat design proposed by physicist Gerard K
• In Dan Simmons's novel The Rise of Endymion, the headquarters of the Pax Mercantilus is located at the L5 point of the planet Pacem, where CEO Kenzo Isozaki has a splendid view of the planet. shortened to Gundam SEED, is an Anime television series and a Manga series by Sunrise. This is a list of fictional nations and factions from the Cosmic Era (CE timeline of the Gundam Anime Metaseries. The Rise of Endymion is a 1997 Science fiction Novel by Dan Simmons.

#### Unspecified Lagrange Points

• Lagrange points are famously mentioned in Arthur C. Clarke's novel 2010: Odyssey Two and the subsequent science fiction film 2010, where the Discovery spacecraft is located on a Lagrange point. Sir Arthur Charles Clarke, CBE (16 December 1917–19 March 2008 was a British Science fiction Author, Inventor, and 2010 Odyssey Two is a best-selling Science fiction novel by Arthur C 2010 is a Science fiction film released in 1984 directed by Peter Hyams. The movie expands on this, claiming that Discovery is located at a point between Io and Jupiter, which would place it in the L1 point of the Jupiter–Io system.
• 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. (anime in Japanese, is a televised Anime series created by Sunrise. Written and directed by Yoshiyuki Tomino, it premiered in Japan on Nagoya Broadcasting Network Lagrange Points have been mentioned in several other Gundam series as well. The colonization of the Lagrange points depicted in Gundam draws extensively from Gerard O'Neill's book The High Frontier: Human Colonies in Space. The High Frontier Human Colonies in Space is a book by Gerard K
• Lagrange points also play a role in the Larry Niven/Jerry Pournelle classic The Mote in God's Eye. The Mote in God's Eye, by Larry Niven and Jerry Pournelle, is a Science fiction Novel that was first published in 1974
• In Robert Forward's Rocheworld the locations for Lagrange points around a binary planet are discussed in contrast to typical system. This is about the physicist and science fiction writer You may be looking for his son Robert D Rocheworld ( 1985) also known as The Flight of the Dragonfly ( 1984) is a Science fiction
• In Iain M. Banks' The Algebraist Lagrange points and their alternate forms are central to the plot, as possible locations for wormholes. Iain Menzies Banks (born on 16 February 1954 in Dunfermline, Fife) is a Scottish Writer. The Algebraist, a Science fiction Novel by Scottish writer Iain M In Physics, a wormhole is a hypothetical topological feature of Spacetime that is fundamentally a 'shortcut' through Space and Time
• In the Independence War computer games, Lagrange points L4 and L5 are used as the only locations for jump-points. I-War (known as Independence War after the North American release is a Space combat simulator developed by English development house A jump drive (or with a capital J is one of the speculative inventions in Science fiction, a method of traveling faster than light (FTL
• In the Battletech game series, a star's Nadir and Zenith are the standard hyperspace jump points for most interstellar spacecraft. BattleTech is a Wargaming and Science fiction franchise launched by FASA Corporation and currently owned by WizKids. 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 Halo novels the Lagrange points are the only places where a human ship can safely make a slipspace jump. Halo is a Science fiction Video game franchise created by Bungie and owned and published by Microsoft Game Studios.
• 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. A personal computer ( PC) is any Computer whose original sales price size and capabilities make it useful for individuals and which is intended to be operated A video game is a Game that involves interaction with a User interface to generate visual feedback on a video device. The X-Wing computer game series is a series of Space combat simulation computer games set in the
• 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. Robert Anson Heinlein (July 7 1907 – May 8 1988 was an American Novelist and Science fiction Writer. The Number of the Beast is a Science fiction Novel by Robert A A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is The Solar System consists of the Sun and those celestial objects bound to it by Gravity.
• 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. Stargate Atlantis (often abbreviated as SGA) is an American - Canadian Science fiction Television program, part of the [2]
• In 1991, Konami released a science fiction RPG for the NES in Japan called Lagrange Point.
• In the Robotech television series, an effect called an Orbital Warp Blast is created when a spaceship creates "a phenomenon known as the molecular vacuum" at a fictional "Lagrange Point 6, approximately 20,000 kilometers from Mars" (where one does not exist in the real world). Lagrange Point is an RPG game by Konami, released only in Japan in 1991. Robotech is a Science fiction franchise that was launched by an 85-episode adaptation of three different Anime Television series.
• In The Monkeys Thought 'Twas All In Fun by Orson Scott Card a structure called the Trojan Object appears at L4 or L5
• In the Star Trek novel The Abode of Life by Lee Corey, Captain Kirk states the planet Mercan has no Lagrangian points. Orson Scott Card (born August 24, 1951) is a bestselling American Author, Critic, political writer and speaker. George Harry Stine ( March 26, 1928 - November 2, 1997) is widely regarded as the father of model Rocketry Although he did
• In a parody of the science fiction comic Freefall a character refers to a satellite bar located in "the Lagrange point" so that it's always Happy hour there. Freefall is a Webcomic written and drawn by Mark Stanley It is a humorous Science fiction story detailing the misadventures of the spaceship "Happy hour" is a Marketing term for a period of Time in which a Restaurant or bar offers discounts on alcoholic drinks such He then heads for the bar promising to be back before sunset (which never happens).
• In the Ken MacLeod novel The Star Fraction, reference is made to a collection of human habitats located at the Earth-Sun Lagrange point, with semi-permanent residents. Ken MacLeod (born 2 August 1954 an award-winning Scottish Science fiction writer lives in South Queensferry near Edinburgh. Implicitly, it is the L1 point which is referred to.

## Notes and references

1. ^ "Restricted Three-Body Problem", Science World.
2. ^ "Lagrange Points" by Enrique Zeleny, The Wolfram Demonstrations Project.
3. ^ Koon, W. S. ; M. W. Lo, J. E. Marsden, and S. D. Ross (2006). Dynamical Systems, the Three-Body Problem, and Space Mission Design, p. 9.
4. ^ (French) Lagrange, Joseph-Louis (1867-1892). "Tome 6, Chapitre II: Essai sur le problème des trois corps", Oeuvres de Lagrange. Gauthier-Villars, 272-292.
5. ^ This apparent contradiction is because the Sun is also affected by the Earth's gravity, and so orbits around the two bodies' barycentre, which is however well inside the body of the Sun.
6. ^ Actually $\tfrac{25+3\sqrt{69}}{2}$
7. ^ The Lagrange Points – Neil J. Cornish with input from Jeremy Goodman
8. ^ A Search for Natural or Artificial Objects Located at the Earth-Moon Libration Points. Icarus.
9. ^ Tyson, Neil deGrasse, Death by Black Hole, ©2007, ISBN 9780393062243
10. ^ "Nasa unveils Hubble's successor". Neil deGrasse Tyson (born October 5 1958 in New York City) is an Astrophysicist and since 1996 the Frederick P A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e BBC (11 May 2007). Retrieved on 2007-11-20. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 284 - Diocletian was chosen as Roman Emperor. 762 - Bögü Khan of the Uyghurs,