celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...

, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive

orbiting In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...

bodies. Mathematically, this involves the solution of the restricted

three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...

in which two bodies are far more massive than the third. Normally, the two massive bodies exert an unbalanced gravitational force at a point, altering the orbit of whatever is at that point. At the Lagrange points, the gravitational forces of the two large bodies and the

centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is parallel ...

balance each other. This can make Lagrange points an excellent location for satellites, as few orbit corrections are needed to maintain the desired orbit. Small objects placed in orbit at Lagrange points are in equilibrium in at least two directions relative to the center of mass of the large bodies. For any combination of two orbital bodies there are five Lagrange points, L₁ to L₅, all in the orbital plane of the two large bodies. There are five Lagrange points for the Sun–Earth system, and five ''different'' Lagrange points for the Earth–Moon system. L₁, L₂, and L₃ are on the line through the centers of the two large bodies, while L₄ and L₅ each act as the third

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

of an

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

formed with the centers of the two large bodies. When the mass ratio of the two bodies is large enough, the L₄ and L₅ points are stable points meaning that objects can orbit them, and that they have a tendency to pull objects into them. Several planets have trojan asteroids near their L₄ and L₅ points with respect to the Sun;

Jupiter Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but slightly less than one-thousandt ...

has more than one million of these trojans.

Artificial satellite A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioisoto ...

s, for example the

James Webb Space Telescope The James Webb Space Telescope (JWST) is a space telescope which conducts infrared astronomy. As the largest optical telescope in space, its high resolution and sensitivity allow it to view objects too old, distant, or faint for the Hubble ...

, have been placed at L₁ and L₂ with respect to the

Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared rad ...

and

Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surf ...

, and with respect to the Earth and the

Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width ...

. The Lagrange points have been proposed for uses in space exploration.

History

The three collinear Lagrange points (L₁, L₂, L₃) were discovered by

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

around 1750, a decade before

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia In 1772, Lagrange published an "Essay on the

". In the first chapter he considered the general three-body problem. From that, in the second chapter, he demonstrated two special constant-pattern solutions, the collinear and the equilateral, for any three masses, with

circular orbit A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle. Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is ...

Lagrange points

The five Lagrange points are labelled and defined as follows:

point

The point lies on the line defined between the two large masses ''M''₁ and ''M''₂. It is the point where the gravitational attraction of ''M''₂ and that of ''M''₁ combine to produce an equilibrium. An object that

orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such a ...

s the

more closely than

would normally have a shorter orbital period than Earth, but that ignores the effect of Earth's own gravitational pull. If the object is directly between Earth and the Sun, then

Earth's gravity The gravity of Earth, denoted by , is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation). It is a vector quantit ...

counteracts some of the Sun's pull on the object, and therefore increases the orbital period of the object. The closer to Earth the object is, the greater this effect is. At the point, the orbital period of the object becomes exactly equal to Earth's orbital period. is about 1.5 million kilometers from Earth, or 0.01 au.

point

The point lies on the line through the two large masses, beyond the smaller of the two. Here, the gravitational forces of the two large masses balance the centrifugal effect on a body at . On the opposite side of Earth from the Sun, the orbital period of an object would normally be greater than that of Earth. The extra pull of Earth's gravity decreases the orbital period of the object, and at the point that orbital period becomes equal to Earth's. Like L₁, L₂ is about 1.5 million kilometers or 0.01 au from Earth. An example of a spacecraft at L₂ is the

, designed to operate near the Earth–Sun L₂. Earlier examples include the

Wilkinson Microwave Anisotropy Probe The Wilkinson Microwave Anisotropy Probe (WMAP), originally known as the Microwave Anisotropy Probe (MAP and Explorer 80), was a NASA spacecraft operating from 2001 to 2010 which measured temperature differences across the sky in the cosmic mic ...

and its successor, ''

Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical p ...

''.

point

The point lies on the line defined by the two large masses, beyond the larger of the two. Within the Sun–Earth system, the point exists on the opposite side of the Sun, a little outside Earth's orbit and slightly closer to the center of the Sun than Earth is. This placement occurs because the Sun is also affected by Earth's gravity and so orbits around the two bodies'

barycenter In astronomy, the barycenter (or barycentre; ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important co ...

, which is well inside the body of the Sun. An object at Earth's distance from the Sun would have an orbital period of one year if only the Sun's gravity is considered. But an object on the opposite side of the Sun from Earth and directly in line with both "feels" Earth's gravity adding slightly to the Sun's and therefore must orbit a little farther from the barycenter of Earth and Sun in order to have the same 1-year period. It is at the point that the combined pull of Earth and Sun causes the object to orbit with the same period as Earth, in effect orbiting an Earth+Sun mass with the Earth-Sun barycenter at one focus of its orbit.

and points

The and points lie at the third vertices of the two

s in the plane of orbit whose common base is the line between the centers of the two masses, such that the point lies 60° ahead of () or behind () the smaller mass with regard to its orbit around the larger mass.

Stability

The triangular points ( and ) are stable equilibria, provided that the ratio of is greater than 24.96.Actually (25 + 3)/2 ≈ This is the case for the Sun–Earth system, the Sun–Jupiter system, and, by a smaller margin, the Earth–Moon system. When a body at these points is perturbed, it moves away from the point, but the factor opposite of that which is increased or decreased by the perturbation (either gravity or angular momentum-induced speed) will also increase or decrease, bending the object's path into a stable,

kidney bean The kidney bean is a variety of the common bean (''Phaseolus vulgaris''). It resembles a human kidney and thus is named after such. Red kidney beans should not be confused with other red beans, such as adzuki beans. Classification There are ...

-shaped orbit around the point (as seen in the corotating frame of reference)., Neil J. Cornish, with input from Jeremy Goodman The points , , and are positions of

unstable equilibrium In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form ...

. Any object orbiting at , , or will tend to fall out of orbit; it is therefore rare to find natural objects there, and spacecraft inhabiting these areas must employ a small but critical amount of station keeping in order to maintain their position.

Natural objects at Lagrange points

Due to the natural stability of and , it is common for natural objects to be found orbiting in those Lagrange points of planetary systems. Objects that inhabit those points are generically referred to as '

trojans Trojan or Trojans may refer to: * Of or from the ancient city of Troy * Trojan language, the language of the historical Trojans Arts and entertainment Music * ''Les Troyens'' ('The Trojans'), an opera by Berlioz, premiered part 1863, part 189 ...

' or 'trojan asteroids'. The name derives from the names that were given to asteroids discovered orbiting at the Sun–

and points, which were taken from mythological characters appearing in

Homer Homer (; grc, Ὅμηρος , ''Hómēros'') (born ) was a Greek poet who is credited as the author of the '' Iliad'' and the '' Odyssey'', two epic poems that are foundational works of ancient Greek literature. Homer is considered one of ...

's ''

Iliad The ''Iliad'' (; grc, Ἰλιάς, Iliás, ; "a poem about Ilium") is one of two major ancient Greek epic poems attributed to Homer. It is one of the oldest extant works of literature still widely read by modern audiences. As with the '' Odys ...

'', an

epic poem An epic poem, or simply an epic, is a lengthy narrative poem typically about the extraordinary deeds of extraordinary characters who, in dealings with gods or other superhuman forces, gave shape to the mortal universe for their descendants. ...

set during the

Trojan War In Greek mythology, the Trojan War was waged against the city of Troy by the Achaeans (Greeks) after Paris of Troy took Helen from her husband Menelaus, king of Sparta. The war is one of the most important events in Greek mythology and ha ...

. Asteroids at the point, ahead of Jupiter, are named after Greek characters in the ''Iliad'' and referred to as the "

Greek camp This is a list of Jupiter trojans that lie in the Greek camp, an elongated curved region around the leading Lagrangian point (), 60 ° ahead of Jupiter in its orbit. All the asteroids at Jupiter's point have names corresponding to participants ...

". Those at the point are named after Trojan characters and referred to as the "

Trojan camp This is a list of Jupiter trojans that lie in the Trojan camp, an elongated curved region around the trailing Lagrangian point, 60° behind Jupiter. All the asteroids at the trailing point have names corresponding to participants on the Trojan ...

". Both camps are considered to be types of trojan bodies. As the Sun and Jupiter are the two most massive objects in the Solar System, there are more Sun–Jupiter trojans than for any other pair of bodies. However, smaller numbers of objects are known at the Lagrange points of other orbital systems: * The Sun–Earth and points contain interplanetary dust and at least two asteroids, and . * The Earth–Moon and points contain concentrations of

interplanetary dust The interplanetary dust cloud, or zodiacal cloud (as the source of the zodiacal light), consists of cosmic dust (small particles floating in outer space) that pervades the space between planets within planetary systems, such as the Solar Syste ...

, known as

Kordylewski cloud Kordylewski clouds are large concentrations of dust that exist at the and Lagrangian points of the Earth–Moon system. They were first reported by Polish astronomer Kazimierz Kordylewski in the 1960s, and confirmed to exist in October 2018. D ...

s. Stability at these specific points is greatly complicated by solar gravitational influence. * The Sun–

Neptune Neptune is the eighth planet from the Sun and the farthest known planet in the Solar System. It is the fourth-largest planet in the Solar System by diameter, the third-most-massive planet, and the densest giant planet. It is 17 time ...

and points contain several dozen known objects, the

Neptune trojan Neptune trojans are bodies that orbit the Sun near one of the stable Lagrangian points of Neptune, similar to the trojans of other planets. They therefore have approximately the same orbital period as Neptune and follow roughly the same orbital ...

s. *

Mars Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, only being larger than Mercury. In the English language, Mars is named for the Roman god of war. Mars is a terrestrial planet with a thin atmos ...

has four accepted

Mars trojan The Mars trojans are a group of trojan objects that share the orbit of the planet Mars around the Sun. They can be found around the two Lagrangian points 60° ahead of and behind Mars. The origin of the Mars trojans is not well understood. One ...

5261 Eureka 5261 Eureka is the first Mars trojan discovered. It was discovered by David H. Levy and Henry Holt at Palomar Observatory on 20 June 1990. It trails Mars (at the ) at a distance varying by only 0.3 AU during each revolution (with a secular tre ...

, , , and . * Saturn's moon Tethys has two smaller moons of Saturn in its and points, Telesto and Calypso. Another Saturn moon, Dione also has two Lagrange co-orbitals, Helene at its point and Polydeuces at . The moons wander

azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematicall ...

ally about the Lagrange points, with Polydeuces describing the largest deviations, moving up to 32° away from the Saturn–Dione point. * One version of the

giant impact hypothesis The giant-impact hypothesis, sometimes called the Big Splash, or the Theia Impact, suggests that the Moon formed from the ejecta of a collision between the proto-Earth and a Mars-sized planet, approximately 4.5 billion years ago, in the Hadean ...

postulates that an object named

Theia In Greek mythology, Theia (; grc, Θεία, Theía, divine, also rendered Thea or Thia), also called Euryphaessa ( grc, Εὐρυφάεσσα) "wide-shining", is one of the twelve Titans, the children of the earth goddess Gaia and the sky god ...

formed at the Sun–Earth or point and crashed into Earth after its orbit destabilized, forming the Moon. * In

binary star A binary star is a system of two stars that are gravitationally bound to and in orbit around each other. Binary stars in the night sky that are seen as a single object to the naked eye are often resolved using a telescope as separate stars, in ...

s, the

Roche lobe In astronomy, the Roche lobe is the region around a star in a binary system within which orbiting material is gravitationally bound to that star. It is an approximately teardrop-shaped region bounded by a critical gravitational equipotential, ...

has its apex located at ; if one of the stars expands past its Roche lobe, then it will lose matter to its

companion star A binary star is a system of two stars that are gravitationally bound to and in orbit around each other. Binary stars in the night sky that are seen as a single object to the naked eye are often resolved using a telescope as separate stars, in ...

, known as Roche lobe overflow. Objects which are on

horseshoe orbit In celestial mechanics, a horseshoe orbit is a type of co-orbital motion of a small orbiting body relative to a larger orbiting body. The osculating (instantaneous) orbital period of the smaller body remains very near that of the larger body, a ...

s are sometimes erroneously described as trojans, but do not occupy Lagrange points. Known objects on horseshoe orbits include

3753 Cruithne 3753 Cruithne is a Q-type, Aten asteroid in orbit around the Sun in 1:1 orbital resonance with Earth, making it a co-orbital object. It is an asteroid that, relative to Earth, orbits the Sun in a bean-shaped orbit that effectively describes ...

with Earth, and Saturn's moons

Epimetheus In Greek mythology, Epimetheus (; grc-gre, Ἐπιμηθεύς, , afterthought) was the brother of Prometheus (traditionally interpreted as "foresight", literally "fore-thinker"), a pair of Titans who "acted as representatives of mankind". The ...

and

Janus In ancient Roman religion and myth, Janus ( ; la, Ianvs ) is the god of beginnings, gates, transitions, time, duality, doorways, passages, frames, and endings. He is usually depicted as having two faces. The month of January is named for Janu ...

Physical and mathematical details

Lagrange points are the constant-pattern solutions of the restricted

. For example, given two massive bodies in orbits around their common

, there are five positions in space where a third body, of comparatively negligible

mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different element ...

, could be placed so as to maintain its position relative to the two massive bodies. As seen in a

rotating reference frame A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth. (This article considers onl ...

that matches the

angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...

of the two co-orbiting bodies, the gravitational fields of two massive bodies combined providing the centripetal force at the Lagrange points, allowing the smaller third body to be relatively stationary with respect to the first two.

The location of L₁ is the solution to the following equation, gravitation providing the centripetal force:

\frac-\frac=\left(\frac-r\right)\frac

where ''r'' is the distance of the L₁ point from the smaller object, ''R'' is the distance between the two main objects, and ''M''₁ and ''M''₂ are the masses of the large and small object, respectively. The quantity in parentheses on the right is the distance of L₁ from the center of mass. Solving this for ''r'' involves solving a

quintic function In algebra, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a q ...

, but if the mass of the smaller object (''M''₂) is much smaller than the mass of the larger object (''M''₁) then and are at approximately equal distances ''r'' from the smaller object, equal to the radius of the

Hill sphere The Hill sphere of an astronomical body is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sp ...

, given by:

r \approx R \sqrt /math>

We may also write this as: \frac\approx 3\frac Since the tidal effect of a body is proportional to its mass divided by the distance cubed, this means that the tidal effect of the smaller body at the L or at the L point is about three times of that body. We may also write: \rho_2(d_2/r)^3\approx 3\rho_1(d_1/R)^3 where ρ and ρ are the average densities of the two bodies and d_1 and d_2 are their diameters. The ratio of diameter to distance gives the angle subtended by the body, showing that viewed from these two Lagrange points, the apparent sizes of the two bodies will be similar, especially if the density of the smaller one is about thrice that of the larger, as in the case of the earth and the sun.

This distance can be described as being such that the

orbital period The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets ...

, corresponding to a circular orbit with this distance as radius around ''M''₂ in the absence of ''M''₁, is that of ''M''₂ around ''M''₁, divided by ≈ 1.73:

T_(r) = \frac.

The location of L₂ is the solution to the following equation, gravitation providing the centripetal force:

\frac+\frac=\left(\frac(R+r)\right)\frac

with parameters defined as for the L₁ case. Again, if the mass of the smaller object (''M''₂) is much smaller than the mass of the larger object (''M''₁) then L₂ is at approximately the radius of the

, given by:

r \approx R \sqrt /math>

The same remarks about tidal influence and apparent size apply as for the L point. For example, the angular radius of the sun as viewed from L

₂ is arcsin(/) ≈ 0.264°, whereas that of the earth is arcsin(6371/) ≈ 0.242°. Looking toward the sun from L₂ one sees an

annular eclipse A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby obscuring the view of the Sun from a small part of the Earth, totally or partially. Such an alignment occurs during an eclipse season, approximately every six mon ...

. It is necessary for a spacecraft, like

Gaia In Greek mythology, Gaia (; from Ancient Greek , a poetical form of , 'land' or 'earth'),, , . also spelled Gaea , is the personification of the Earth and one of the Greek primordial deities. Gaia is the ancestral mother—sometimes parthe ...

, to follow a

Lissajous orbit In orbital mechanics, a Lissajous orbit (), named after Jules Antoine Lissajous, is a quasi-periodic orbital trajectory that an object can follow around a Lagrangian point of a three-body system without requiring any propulsion. Lyapunov orbi ...

or a

halo orbit A halo orbit is a periodic, three-dimensional orbit near one of the L1, L2 or L3 Lagrange points in the three-body problem of orbital mechanics. Although a Lagrange point is just a point in empty space, its peculiar characteristic is that it c ...

around L₂ in order for its solar panels to get full sun.

L₃

The location of L₃ is the solution to the following equation, gravitation providing the centripetal force:

\frac+\frac=\left(\fracR+R-r\right)\frac

with parameters ''M''₁, ''M''₂, and ''R'' defined as for the L₁ and L₂ cases, and ''r'' now indicates the distance of L₃ from the position of the smaller object, if it were rotated 180 degrees about the larger object, while positive ''r'' implying L₃ is closer to the larger object than the smaller object. If the mass of the smaller object (''M''₂) is much smaller than the mass of the larger object (''M''₁), then:

r\approx R\frac

and

The reason these points are in balance is that at and the distances to the two masses are equal. Accordingly, the gravitational forces from the two massive bodies are in the same ratio as the masses of the two bodies, and so the resultant force acts through the

of the system; additionally, the geometry of the triangle ensures that the

resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over ...

acceleration is to the distance from the barycenter in the same

ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...

as for the two massive bodies. The barycenter being both the center of mass and center of rotation of the three-body system, this resultant force is exactly that required to keep the smaller body at the Lagrange point in orbital equilibrium with the other two larger bodies of the system (indeed, the third body needs to have negligible mass). The general triangular configuration was discovered by Lagrange working on the

Radial acceleration

The radial acceleration ''a'' of an object in orbit at a point along the line passing through both bodies is given by:

a = -\frac\sgn(r)+\frac\sgn(R-r)+\frac

where ''r'' is the distance from the large body ''M''₁, R is the distance between the two main objects, and sgn(''x'') is the

sign function In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To a ...

of ''x''. The terms in this function represent respectively: force from ''M''₁; force from ''M''₂; and centripetal force. The points L₃, L₁, L₂ occur where the acceleration is zero — see chart at right. Positive acceleration is acceleration towards the right of the chart and negative acceleration is towards the left; that is why acceleration has opposite signs on opposite sides of the gravity wells.

Stability

Although the , , and points are nominally unstable, there are quasi-stable periodic orbits called ''halo orbits'' around these points in a three-body system. A full ''n''-body

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...

such as the

Solar System The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...

does not contain these periodic orbits, but does contain quasi-periodic (i.e. bounded but not precisely repeating) orbits following Lissajous-curve trajectories. These quasi-periodic

s are what most of Lagrangian-point space missions have used until now. Although they are not perfectly stable, a modest effort of station keeping keeps a spacecraft in a desired Lissajous orbit for a long time. For Sun–Earth- missions, it is preferable for the spacecraft to be in a large-amplitude () Lissajous orbit around than to stay at , because the line between Sun and Earth has increased solar

interference Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extra ...

on Earth–spacecraft communications. Similarly, a large-amplitude Lissajous orbit around keeps a probe out of Earth's shadow and therefore ensures continuous illumination of its solar panels. The and points are stable provided that the mass of the primary body (e.g. the Earth) is at least 25 times the mass of the secondary body (e.g. the Moon), and the mass of the secondary is at least 10 times that of the tertiary (e.g. the satellite). The Earth is over 81 times the mass of the Moon (the Moon is 1.23% of the mass of the Earth). Although the and points are found at the top of a "hill", as in the effective potential contour plot above, they are nonetheless stable. The reason for the stability is a second-order effect: as a body moves away from the exact Lagrange position,

Coriolis acceleration In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...

(which depends on the velocity of an orbiting object and cannot be modeled as a contour map) curves the trajectory into a path around (rather than away from) the point. Because the source of stability is the Coriolis force, the resulting orbits can be stable, but generally are not planar, but "three-dimensional": they lie on a warped surface intersecting the ecliptic plane. The kidney-shaped orbits typically shown nested around and are the projections of the orbits on a plane (e.g. the ecliptic) and not the full 3-D orbits.

Solar System values

Lagrange_points_of_planets_relative_to_sun

This table lists sample values of L₁, L₂, and L₃ within the Solar System. Calculations assume the two bodies orbit in a perfect circle with separation equal to the semimajor axis and no other bodies are nearby. Distances are measured from the larger body's center of mass with L₃ showing a negative location. The percentage columns show how the distances compare with the semimajor axis. E.g. for the Moon, L₁ is located from Earth's center, which is 84.9% of the Earth–Moon distance or 15.1% in front of the Moon; L₂ is located from Earth's center, which is 116.8% of the Earth–Moon distance or 16.8% beyond the Moon; and L₃ is located from Earth's center, which is 99.3% of the Earth–Moon distance or 0.7084% in front of the Moon's 'negative' position.

Spaceflight applications

Sun–Earth

Sun–Earth is suited for making observations of the Sun–Earth system. Objects here are never shadowed by Earth or the Moon and, if observing Earth, always view the sunlit hemisphere. The first mission of this type was the 1978 International Sun Earth Explorer 3 (ISEE-3) mission used as an interplanetary early warning storm monitor for solar disturbances. Since June 2015,

DSCOVR Deep Space Climate Observatory (DSCOVR; formerly known as Triana, unofficially known as GoreSat) is a National Oceanic and Atmospheric Administration (NOAA) space weather, space climate, and Earth observation satellite. It was launched by S ...

has orbited the L₁ point. Conversely it is also useful for space-based solar telescopes, because it provides an uninterrupted view of the Sun and any

space weather Space weather is a branch of space physics and aeronomy, or heliophysics, concerned with the time varying conditions within the Solar System, including the solar wind, emphasizing the space surrounding the Earth, including conditions in the m ...

(including the

solar wind The solar wind is a stream of charged particles released from the upper atmosphere of the Sun, called the corona. This plasma mostly consists of electrons, protons and alpha particles with kinetic energy between . The composition of the ...

and

coronal mass ejections A coronal mass ejection (CME) is a significant release of plasma and accompanying magnetic field from the Sun's corona into the heliosphere. CMEs are often associated with solar flares and other forms of solar activity, but a broadly accepted t ...

) reaches L₁ up to an hour before Earth. Solar and heliospheric missions currently located around L₁ include the

Solar and Heliospheric Observatory The Solar and Heliospheric Observatory (SOHO) is a European Space Agency (ESA) spacecraft built by a European industrial consortium led by Matra Marconi Space (now Airbus Defence and Space) that was launched on a Lockheed Martin Atlas IIAS l ...

Wind Wind is the natural movement of air or other gases relative to a planet's surface. Winds occur on a range of scales, from thunderstorm flows lasting tens of minutes, to local breezes generated by heating of land surfaces and lasting a few ...

, and the

Advanced Composition Explorer Advanced Composition Explorer (ACE or Explorer 71) is a NASA Explorer program satellite and space exploration mission to study matter comprising energetic particles from the solar wind, the interplanetary medium, and other sources. Real-time ...

. Planned missions include the Interstellar Mapping and Acceleration Probe (IMAP) and the

NEO Surveyor NEO Surveyor, formerly called Near-Earth Object Camera (NEOCam), then NEO Surveillance Mission, is a planned space-based infrared telescope designed to survey the Solar System for potentially hazardous asteroids.umbra The umbra, penumbra and antumbra are three distinct parts of a shadow, created by any light source after impinging on an opaque object. Assuming no diffraction, for a collimated beam (such as a point source) of light, only the umbra is cast. ...

, so solar radiation is not completely blocked at L₂. Spacecraft generally orbit around L₂, avoiding partial eclipses of the Sun to maintain a constant temperature. From locations near L₂, the Sun, Earth and Moon are relatively close together in the sky; this means that a large sunshade with the telescope on the dark-side can allow the telescope to cool passively to around 50 K – this is especially helpful for

infrared astronomy Infrared astronomy is a sub-discipline of astronomy which specializes in the observation and analysis of astronomical objects using infrared (IR) radiation. The wavelength of infrared light ranges from 0.75 to 300 micrometers, and falls in betwee ...

and observations of the

cosmic microwave background In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...

. The

was positioned in a halo orbit about L₂ on January 24, 2022. Sun–Earth and are saddle points and exponentially unstable with

time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a s ...

of roughly 23 days. Satellites at these points will wander off in a few months unless course corrections are made. Sun–Earth was a popular place to put a "

Counter-Earth The Counter-Earth is a hypothetical body of the Solar System that orbits on the other side of the solar system from Earth. A Counter-Earth or ''Antichthon'' ( el, Ἀντίχθων) was hypothesized by the pre-Socratic Greek philosopher Philola ...

" in

pulp Pulp may refer to: * Pulp (fruit), the inner flesh of fruit Engineering * Dissolving pulp, highly purified cellulose used in fibre and film manufacture * Pulp (paper), the fibrous material used to make paper * Molded pulp, a packaging material ...

science fiction Science fiction (sometimes shortened to Sci-Fi or SF) is a genre of speculative fiction which typically deals with imagination, imaginative and futuristic concepts such as advanced science and technology, space exploration, time travel, Paral ...

and

comic book A comic book, also called comicbook, comic magazine or (in the United Kingdom and Ireland) simply comic, is a publication that consists of comics art in the form of sequential juxtaposed panel (comics), panels that represent individual scenes. ...

s, despite the fact that the existence of a planetary body in this location had been understood as an impossibility once orbital mechanics and the perturbations of planets upon each other's orbits came to be understood, long before the Space Age; the influence of an Earth-sized body on other planets would not have gone undetected, nor would the fact that the foci of Earth's orbital ellipse would not have been in their expected places, due to the mass of the counter-Earth. The Sun–Earth , however, is a weak saddle point and exponentially unstable with time constant of roughly 150 years. Moreover, it could not contain a natural object, large or small, for very long because the gravitational forces of the other planets are stronger than that of Earth (for example,

Venus Venus is the second planet from the Sun. It is sometimes called Earth's "sister" or "twin" planet as it is almost as large and has a similar composition. As an interior planet to Earth, Venus (like Mercury) appears in Earth's sky never f ...

comes within 0.3 AU of this every 20 months). A spacecraft orbiting near Sun–Earth would be able to closely monitor the evolution of active sunspot regions before they rotate into a geoeffective position, so that a seven-day early warning could be issued by the

NOAA The National Oceanic and Atmospheric Administration (abbreviated as NOAA ) is an United States scientific and regulatory agency within the United States Department of Commerce that forecasts weather, monitors oceanic and atmospheric conditio ...

Space Weather Prediction Center The Space Weather Prediction Center (SWPC), named the Space Environment Center (SEC) until 2007, is a laboratory and service center of the US National Weather Service (NWS), part of the National Oceanic and Atmospheric Administration (NOAA), lo ...

. Moreover, a satellite near Sun–Earth would provide very important observations not only for Earth forecasts, but also for deep space support (Mars predictions and for crewed mission to

near-Earth asteroids A near-Earth object (NEO) is any small Solar System body whose orbit brings it into proximity with Earth. By convention, a Solar System body is a NEO if its closest approach to the Sun (perihelion) is less than 1.3 astronomical units (AU). ...

). In 2010, spacecraft transfer trajectories to Sun–Earth were studied and several designs were considered.

Earth–Moon

Earth–Moon allows comparatively easy access to Lunar and Earth orbits with minimal change in velocity and this has as an advantage to position a habitable

space station A space station is a spacecraft capable of supporting a human crew in orbit for an extended period of time, and is therefore a type of space habitat. It lacks major propulsion or landing systems. An orbital station or an orbital space station ...

intended to help transport cargo and personnel to the Moon and back. Earth–Moon has been used for a

communications satellite A communications satellite is an artificial satellite that relays and amplifies radio telecommunication signals via a transponder; it creates a communication channel between a source transmitter and a receiver at different locations on Ear ...

covering the Moon's far side, for example,

Queqiao ''Queqiao'' relay satellite (), also known as the Chang'e 4 Relay, is a communications relay and radio astronomy satellite for the Chang'e 4 lunar farside mission. As part of the Chinese Lunar Exploration Program, the China National Space Admi ...

, launched in 2018, and would be "an ideal location" for a

propellant depot An orbital propellant depot is a cache of propellant that is placed in orbit around Earth or another body to allow spacecraft or the transfer stage of the spacecraft to be fueled in space. It is one of the types of space resource depots that h ...

as part of the proposed depot-based space transportation architecture.

Sun–Venus

Scientists at the B612 Foundation were planning to use

's L₃ point to position their planned Sentinel telescope, which aimed to look back towards Earth's orbit and compile a catalogue of

Sun–Mars

In 2017, the idea of positioning a

magnetic dipole In electromagnetism, a magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the size of the source is reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric ...

shield at the Sun–Mars point for use as an artificial magnetosphere for Mars was discussed at a NASA conference. The idea is that this would protect the planet's atmosphere from the Sun's radiation and solar winds.

Explanatory notes

References

External links

* Joseph-Louis, Comte Lagrange, from Oeuvres Tome 6, "Essai sur le Problème des Trois Corps"�
Essai (PDF)
sourc
Tome 6 (Viewer)
** "Essay on the Three-Body Problem" by J-L Lagrange, translated from the above, i

��

—transcription and translation a
merlyn.demon.co.uk

What are Lagrange points?
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European Space Agency , owners = , headquarters = Paris, Île-de-France, France , coordinates = , spaceport = Guiana Space Centre , seal = File:ESA emblem seal.png , seal_size = 130px , image = Views in the Main Control Room (120 ...

page, with good animations
Explanation of Lagrange points
��Prof. Neil J. Cornish

��also attributed to Neil J. Cornish

��Prof. John Baez

��Dr J R Stockton

��Dr. David Peter Stern

��Tony Dunn * ttp://www.astronomycast.com/physics/ep-76-lagrange-points/ ''Astronomy Cast''—Ep. 76: "Lagrange Points"by Fraser Cain and Dr. Pamela Gay
The Five Points of Lagrange
by

Neil deGrasse Tyson Neil deGrasse Tyson ( or ; born October 5, 1958) is an American astrophysicist, author, and science communicator. Tyson studied at Harvard University, the University of Texas at Austin, and Columbia University. From 1991 to 1994, he was a p ...

Earth, a lone Trojan discovered
* See the Lagrange Points and Halo Orbits subsection under the section on Geosynchronous Transfer Orbit i
NASA: Basics of Space Flight, Chapter 5
{{DEFAULTSORT:Lagrangian Point Trojans (astronomy) Point

History

Lagrange points

point

point

point

and points

Stability

Natural objects at Lagrange points

Physical and mathematical details

L3

and

Radial acceleration

Stability

Solar System values

Spaceflight applications

Sun–Earth

Earth–Moon

Sun–Venus

Sun–Mars

See also

Explanatory notes

References

External links

L₃