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 gravitational influence of two massive orbiting bodies. Mathematically, this involves the solution of the restricted three-body problem. 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

gravitation In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...

al forces of the two large bodies and the

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

balance each other. This can make Lagrange points an excellent location for satellites, as orbit corrections, and hence fuel requirements, needed to maintain the desired orbit are kept at a minimum. 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 of an

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

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 List of Solar System objects by size, largest in the Solar System. It is a gas giant with a Jupiter mass, mass more than 2.5 times that of all the other planets in the Solar System combined a ...

has more than one million of these trojans. Some Lagrange points are being used for space exploration. Two important Lagrange points in the Sun-Earth system are L₁, between the Sun and Earth, and L₂, on the same line at the opposite side of the Earth; both are well outside the Moon's orbit. Currently, an

artificial satellite A satellite or an artificial satellite is an object, typically a spacecraft, placed into orbit around a celestial body. They have a variety of uses, including communication relay, weather forecasting, navigation ( GPS), broadcasting, scienti ...

called the Deep Space Climate Observatory (DSCOVR) is located at L₁ to study solar wind coming toward Earth from the Sun and to monitor Earth's climate, by taking images and sending them back. The

James Webb Space Telescope The James Webb Space Telescope (JWST) is a space telescope designed to conduct infrared astronomy. As the largest telescope in space, it is equipped with high-resolution and high-sensitivity instruments, allowing it to view objects too old, Lis ...

, a powerful infrared space observatory, is located at L₂. This allows the satellite's sunshield to protect the telescope from the light and heat of the Sun, Earth and Moon simultaneously with no need to rotate the sunshield. The L₁ and L₂ Lagrange points are located about from Earth. The European Space Agency's earlier

Gaia In Greek mythology, Gaia (; , a poetic form of ('), meaning 'land' or 'earth'),, , . also spelled Gaea (), is the personification of Earth. Gaia is the ancestral mother—sometimes parthenogenic—of all life. She is the mother of Uranus (S ...

telescope, and its newly launched

Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...

, also occupy orbits around L₂. Gaia keeps a tighter Lissajous orbit around L₂, while Euclid follows a halo orbit similar to JWST. Each of the space observatories benefit from being far enough from Earth's shadow to utilize solar panels for power, from not needing much power or propellant for station-keeping, from not being subjected to the Earth's magnetospheric effects, and from having direct line-of-sight to Earth for data transfer.

History

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

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

around 1750, a decade before the Italian-born

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

three-body problem In physics, specifically classical mechanics, the three-body problem is to take the initial positions and velocities (or momenta) of three point masses orbiting each other in space and then calculate their subsequent trajectories using Newton' ...

". 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. In this case, not only the distance, but also the speed, angular speed, Potential energy, potential and kinetic energy are constant. T ...

Lagrange points

The five Lagrange points are labeled 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 (also known as orbital revolution) 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 ...

s the Sun more closely than

Earth Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...

would typically have a shorter orbital period than Earth, but that ignores the effect of Earth's 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 qu ...

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

point

The point lies on the line through the two large masses beyond the smaller of the two. Here, the combined gravitational forces of the two large masses balance the centrifugal force on a body at . On the opposite side of Earth from the Sun, the orbital period of an object would normally be greater than Earth's. The extra pull of Earth's gravity decreases the object's orbital period, 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 (away from the sun). An example of a spacecraft designed to operate near the Earth–Sun L₂ is the

. 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''.

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

and points

The and points lie at the third 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 ≈ This is the case for the Sun–Earth system, the Sun–Jupiter system, and, by a smaller margin, the Earth–Moon system. When a body at these points is perturbed, it moves away from the point, but the factor opposite of that which is increased or decreased by the perturbation (either gravity or angular momentum-induced speed) will also increase or decrease, bending the object's path into a stable, kidney bean-shaped orbit around the point (as seen in the corotating frame of reference)., Neil J. Cornish, with input from Jeremy Goodman The points , , and are positions of unstable equilibrium. Any object orbiting at , , or will tend to fall out of orbit; it is therefore rare to find natural objects there, and spacecraft inhabiting these areas must employ 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' 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 (; , ; possibly born ) was an Ancient Greece, Ancient 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. Despite doubts about his autho ...

's ''

Iliad The ''Iliad'' (; , ; ) 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 ''Odyssey'', the poem is divided into 24 books and ...

'', an

epic poem In poetry, 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. With regard to ...

set during the

Trojan War The Trojan War was a legendary conflict in Greek mythology that took place around the twelfth or thirteenth century BC. The war was waged by the Achaeans (Homer), Achaeans (Ancient Greece, Greeks) against the city of Troy after Paris (mytho ...

. Asteroids at the point, ahead of Jupiter, are named after Greek characters in the ''Iliad'' and referred to as the " Greek camp". Those at the point are named after Trojan characters and referred to as the " Trojan camp". Both camps are considered to be types of trojan bodies. As the Sun and Jupiter are the two most massive objects in the Solar System, there are more known 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, known as Kordylewski clouds. Stability at these specific points is greatly complicated by solar gravitational influence. * The Sun–

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

and points contain several dozen known objects, the Neptune trojans. *

Mars Mars is the fourth planet from the Sun. It is also known as the "Red Planet", because of its orange-red appearance. Mars is a desert-like rocky planet with a tenuous carbon dioxide () atmosphere. At the average surface level the atmosph ...

has four accepted Mars trojans: 5261 Eureka, , , 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 ) is the horizontal angle from a cardinal direction, most commonly north, in a local or observer-centric spherical coordinate system. Mathematically, the relative position vector from an observer ( origin) to a point ...

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 postulates that an object named Theia formed at the Sun–Earth or point and crashed into Earth after its orbit destabilized, forming the Moon. * In binary stars, the Roche lobe has its apex located at ; if one of the stars expands past its Roche lobe, then it will lose matter to its companion star, known as Roche lobe overflow. Objects which are on horseshoe orbits are sometimes erroneously described as trojans, but do not occupy Lagrange points. Known objects on horseshoe orbits include 3753 Cruithne with Earth, and Saturn's moons Epimetheus and Janus.

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

mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...

, could be placed so as to maintain its position relative to the two massive bodies. This occurs because the combined gravitational forces of the two massive bodies provide the exact centripetal force required to maintain the

circular motion In physics, circular motion is movement of an object along the circumference of a circle or rotation along a circular arc. It can be uniform, with a constant rate of rotation and constant tangential speed, or non-uniform with a changing rate ...

that matches their orbital motion. Alternatively, when seen in a

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

that matches the

angular velocity In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...

of the two co-orbiting bodies, at the Lagrange points the combined

gravitational field In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as ...

s of two massive bodies balance the centrifugal pseudo-force, allowing the smaller third body to remain stationary (in this frame) 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(\fracR-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. The solution for ''r'' is the only real

root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...

of the following quintic function

x^5 + (\mu - 3) x^4 + (3 - 2\mu) x^3 - (\mu) x^2 + (2\mu) x - \mu = 0

where

\mu = \frac

is the mass fraction of ''M₂'' and

x = \frac

is the normalized distance. 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 is a common model for the calculation of a Sphere of influence (astrodynamics), gravitational sphere of influence. It is the most commonly used model to calculate the spatial extent of gravitational influence of an astronomical ...

, 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\left(\frac\right)^3\approx 3\rho_1\left(\frac\right)^3 where ''ρ'' and ''ρ'' are the average densities of the two bodies and ''d'' and ''d'' 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(\fracR+r\right)\frac

with parameters defined as for the L₁ case. The corresponding quintic equation is

x^5 + x^4 (3 - \mu) + x^3 (3 - 2\mu) - x^2 (\mu) - x (2\mu) - \mu = 0

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() ≈ 0.242°. Looking toward the Sun from L₂ one sees an annular eclipse. It is necessary for a spacecraft, like

, to follow a Lissajous orbit or a halo orbit 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'' being defined such that the distance of L₃ from the center of the larger object is ''R'' − ''r''. 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\tfrac\mu.

Thus the distance from L₃ to the larger object is less than the separation of the two objects (although the distance between L₃ and the barycentre is greater than the distance between the smaller object and the barycentre).

and

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

resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that 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 th ...

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 In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weight function, weighted relative position (vector), position of the d ...

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 a function that has the value , or according to whether the sign of a given real number is positive or negative, or the given number is itself zer ...

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 (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

such as the

Solar System The Solar SystemCapitalization 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 "Sola ...

does not contain these periodic orbits, but does contain quasi-periodic (i.e. bounded but not precisely repeating) orbits following Lissajous-curve trajectories. These quasi-periodic Lissajous orbits are what most of Lagrangian-point space missions have used until now. Although they are not perfectly stable, a modest effort of station keeping keeps a spacecraft in a desired Lissajous orbit for a long time. For Sun–Earth- missions, it is preferable for the spacecraft to be in a large-amplitude () Lissajous orbit around than to stay at , because the line between Sun and Earth has increased solar

interference 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), The Earth is over 81 times the mass of the Moon (the Moon is 1.23% of the mass of the Earth). Although the and points are found at the top of a "hill", as in the effective potential contour plot above, they are nonetheless stable. The reason for the stability is a second-order effect: as a body moves away from the exact Lagrange position, Coriolis acceleration (which depends on the velocity of an orbiting object and cannot be modeled as a contour map) curves the trajectory into a path around (rather than away from) the point. Because the source of stability is the Coriolis force, the resulting orbits can be stable, but generally are not planar, but "three-dimensional": they lie on a warped surface intersecting the ecliptic plane. The kidney-shaped orbits typically shown nested around and are the projections of the orbits on a plane (e.g. the ecliptic) and not the full 3-D orbits.

Solar System values

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 (but see barycenter especially in the case of Earth-Moon and Sun-Jupiter) with L₃ showing a negative direction. The percentage columns show the distance from the orbit compared to the semimajor axis. E.g. for the Moon, L₁ is from Earth's center, which is 84.9% of the Earth–Moon distance or 15.1% "in front of" (Earthwards from) 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% inside (Earthward) of the Moon's 'negative' position.

Spaceflight applications

Sun–Earth

Sun–Earth is suited for making observations of the Sun–Earth system. Objects here are never shadowed by Earth or the Moon and, if observing Earth, always view the sunlit hemisphere. The first mission of this type was the 1978 International Sun Earth Explorer 3 (ISEE-3) mission used as an interplanetary early warning storm monitor for solar disturbances. Since June 2015, DSCOVR has orbited the L₁ point. Conversely, it is also useful for space-based

solar telescope A solar telescope or a solar observatory is a special-purpose telescope used to observe the Sun. Solar telescopes usually detect light with wavelengths in, or not far outside, the visible spectrum. Obsolete names for Sun telescopes include helio ...

s, because it provides an uninterrupted view of the Sun and any space weather (including the

solar wind The solar wind is a stream of charged particles released from the Sun's outermost atmospheric layer, the Stellar corona, corona. This Plasma (physics), plasma mostly consists of electrons, protons and alpha particles with kinetic energy betwee ...

and coronal mass ejections) reaches L₁ up to an hour before Earth. Solar and heliospheric missions currently located around L₁ include the Solar and Heliospheric Observatory,

Wind Wind is the natural movement of atmosphere of Earth, air or other gases relative to a planetary surface, planet's surface. Winds occur on a range of scales, from thunderstorm flows lasting tens of minutes, to local breezes generated by heatin ...

, Aditya-L1 Mission and the Advanced Composition Explorer. Planned missions include the Interstellar Mapping and Acceleration Probe(IMAP) and the NEO Surveyor. Sun–Earth is a good spot for space-based observatories. Because an object around will maintain the same relative position with respect to the Sun and Earth, shielding and calibration are much simpler. It is, however, slightly beyond the reach of Earth's umbra, so solar radiation is not completely blocked at L₂. Spacecraft generally orbit around L₂, avoiding partial eclipses of the Sun to maintain a constant temperature. From locations near L₂, the Sun, Earth and Moon are relatively close together in the sky; this means that a large sunshade with the telescope on the dark-side can allow the telescope to cool passively to around 50 K – this is especially helpful for

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

and observations of the

cosmic microwave background The cosmic microwave background (CMB, CMBR), or relic radiation, is microwave radiation that fills all space in the observable universe. With a standard optical telescope, the background space between stars and galaxies is almost completely dar ...

. The

was positioned in a halo orbit about L₂ on 24 January 2022. Sun–Earth and are

saddle point In mathematics, a saddle point or minimax point is a Point (geometry), point on the surface (mathematics), surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a Critical point (mathematics), ...

s and exponentially unstable with

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

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" in pulp

science fiction Science fiction (often shortened to sci-fi or abbreviated SF) is a genre of speculative fiction that deals with imaginative and futuristic concepts. These concepts may include information technology and robotics, biological manipulations, space ...

and

comic book A comic book, comic-magazine, or simply comic is a publication that consists of comics art in the form of sequential juxtaposed panel (comics), panels that represent individual scenes. Panels are often accompanied by descriptive prose and wri ...

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 often called Earth's "twin" or "sister" planet for having almost the same size and mass, and the closest orbit to Earth's. While both are rocky planets, Venus has an atmosphere much thicker ...

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 (NOAA ) is an American scientific and regulatory agency charged with forecasting weather, monitoring oceanic and atmospheric conditions, charting the seas, conducting deep-sea exploratio ...

Space Weather Prediction Center. Moreover, a satellite near Sun–Earth would provide very important observations not only for Earth forecasts, but also for deep space support (Mars predictions and for crewed missions to near-Earth asteroids). 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 (or orbital station) is a spacecraft which remains orbital spaceflight, in orbit and human spaceflight, hosts humans for extended periods of time. It therefore is an artificial satellite featuring space habitat (facility), habitat ...

intended to help transport cargo and personnel to the Moon and back. The SMART-1 mission passed through the L₁ Lagrangian Point on 11 November 2004 and passed into the area dominated by the Moon's gravitational influence. 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 (satellite communications), transponder; it creates a communication channel between a source transmitter and a Rad ...

covering the Moon's far side, for example, Queqiao, launched in 2018, and would be "an ideal location" for a propellant depot as part of the proposed depot-based space transportation architecture. Earth–Moon and are the locations for the Kordylewski dust clouds. The L5 Society's name comes from the L₄ and L₅ Lagrangian points in the Earth–Moon system proposed as locations for their huge rotating space habitats. Both positions are also proposed for communication satellites covering the Moon alike communication satellites in geosynchronous orbit cover the Earth.

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 catalog of near-Earth asteroids.

Sun–Mars

In 2017, the idea of positioning a magnetic dipole shield at the Sun–Mars point for use as an artificial magnetosphere for Mars was discussed at a NASA conference. The idea is that this would protect the planet's atmosphere from the Sun's radiation and solar winds.

Explanatory notes

References

External links

ZIP file
��J R Stockton - Includes translations of Lagrange's ''Essai'' and of two related papers by Euler
What are Lagrange points?
��

European Space Agency The European Space Agency (ESA) is a 23-member International organization, international organization devoted to space exploration. With its headquarters in Paris and a staff of around 2,547 people globally as of 2023, ESA was founded in 1975 ...

page, with good animations
A NASA explanation
��also attributed to Neil J. Cornish

��John Baez

��David Peter Stern

��Tony Dunn * ttp://www.astronomycast.com/physics/ep-76-lagrange-points/ ''Astronomy Cast''—Ep. 76: "Lagrange Points"by Fraser Cain and Pamela L. Gay
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
{{Authority control Point Trojans (astronomy)

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

Further reading

External links

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

Further reading

External links

L₃