In celestial mechanics, the Lagrangian points (; also Lagrange points, Lpoints, or libration points) are positions in an orbital configuration of two large bodies where a small object affected only by gravity can maintain a stable position relative to the two large bodies. The Lagrange points mark positions where the combined gravitational pull of the two large masses provides precisely the centripetal force required to orbit with them. There are five such points, labeled L1 to L5, all in the orbital plane of the two large bodies. The first three are on the line connecting the two large bodies and the last two, L4 and L5, each form an equilateral triangle with the two large bodies. The two latter points are stable, which implies that objects can orbit around them in a rotating coordinate system tied to the two large bodies.
Several planets have minor planets near their L4 and L5 points (trojans) with respect to the Sun, with Jupiter in particular having more than a million of these. Artificial satellites have been placed at L1 and L2 with respect to the Sun and Earth, and Earth and the Moon for various purposes, and the Lagrangian points have been proposed for a variety of future uses in space exploration.
Lagrange points in the Sun–Earth system (not to scale)
Contents

History 1

Lagrange points 2

Natural objects at Lagrangian points 3

Mathematical details 4

L1 4.1

L2 4.2

L3 4.3

L4 and L5 4.4

Stability 5

Spaceflight applications 6

Spacecraft at Sun–Earth L1 6.1

Spacecraft at Sun–Earth L2 6.2

List of missions to Lagrangian points 6.3

Past and present missions 6.3.1

Future and proposed missions 6.3.2

See also 7

Notes 8

References 9

External links 10
History
The three collinear Lagrange points (L_{1}, L_{2}, L_{3}) were discovered by Leonhard Euler a few years before Lagrange discovered the remaining two.^{[1]}^{[2]}
In 1772, JosephLouis Lagrange published an "Essay on the threebody problem". In the first chapter he considered the general threebody problem. From that, in the second chapter, he demonstrated two special constantpattern solutions, the collinear and the equilateral, for any three masses, with circular orbits.^{[3]}
Lagrange points
The five Lagrangian points are labeled and defined as follows:
The L_{1} point lies on the line defined by the two large masses M_{1} and M_{2}, and between them. It is the most intuitively understood of the Lagrangian points: the one where the gravitational attraction of M_{2} partially cancels M_{1}'s gravitational attraction.

Explanation: An object that orbits the Sun more closely than Earth 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 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.^{[4]}
The L_{2} 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 .

Explanation: 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 L_{2} point that orbital period becomes equal to Earth's. Like L1, L2 is about 1.5 million kilometers from Earth.
The L_{3} point lies on the line defined by the two large masses, beyond the larger of the two.

Explanation: in the Sun–Earth system exists on the opposite side of the Sun, a little outside Earth's orbit but slightly closer to the Sun than Earth is. (This apparent contradiction is because the Sun is also affected by Earth's gravity, and so orbits around the two bodies' barycenter, which is, however, well inside the body of the Sun.) At the L_{3} point, the combined pull of Earth and Sun again causes the object to orbit with the same period as Earth.
Gravitational accelerations at L_{4}
The 'and L_{5} points lie at the third corners of the two equilateral triangles in the plane of orbit whose common base is the line between the centers of the two masses, such that the point lies behind () or ahead (L_{4}) of the smaller mass with regard to its orbit around the larger mass.
The triangular points (L_{4} and L_{5}) are stable equilibria, provided that the ratio of M_{1}/M_{2} is greater than 24.96.^{[note 1]}^{[5]} 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 momentuminduced speed) will also increase or decrease, bending the object's path into a stable, kidneybeanshaped orbit around the point (as seen in the corotating frame of reference).
In contrast to L_{4} and L_{5}, where stable equilibrium exists, the points L_{1}, L_{2}, and L_{3} are positions of unstable equilibrium. Any object orbiting at one of L_{1}–L_{3} will tend to fall out of orbit; it is therefore rare to find natural objects there, and spacecraft inhabiting these areas must employ station keeping in order to maintain their position.
Natural objects at Lagrangian points
It is common to find objects at or orbiting the L_{4} and L_{5} points of natural orbital systems. These are commonly called "trojans"; in the 20th century, asteroids discovered orbiting at the Sun–Jupiter L_{4} and L_{5} points were named after characters from Homer's Iliad. Asteroids at the L_{4} point, which leads Jupiter, are referred to as the "Greek camp", whereas those at the L_{5} point are referred to as the "Trojan camp".
Other examples of natural objects orbiting at Lagrange points:

The Sun–Earth L_{4} and L_{5} points contain interplanetary dust and at least one asteroid, 2010 TK7, detected in October 2010 by Widefield Infrared Survey Explorer (WISE) and announced during July 2011.^{[6]}^{[7]}

The Earth–Moon L_{4} and L_{5} points may contain interplanetary dust in what are called Kordylewski clouds; however, the Hiten spacecraft's Munich Dust Counter (MDC) detected no increase in dust during its passes through these points. Stability at these specific points is greatly complicated by solar gravitational influence.^{[8]}

Recent observations suggest that the Sun–Neptune L_{4} and L_{5} points, known as the Neptune trojans, may be very thickly populated,^{[9]} containing large bodies an order of magnitude more numerous than the Jupiter trojans.

Several asteroids also orbit near the SunJupiter L_{3} point, called the Hilda family.

The Saturnian moon Tethys has two smaller moons in its L_{4} and L_{5} points, Telesto and Calypso. The Saturnian moon Dione also has two Lagrangian coorbitals, Helene at its L_{4} point and Polydeuces at L_{5}. The moons wander azimuthally about the Lagrangian points, with Polydeuces describing the largest deviations, moving up to 32 degrees away from the Saturn–Dione L_{5} 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.

One version of the giant impact hypothesis suggests that an object named Theia formed at the Sun–Earth L_{4} or L_{5} points and crashed into Earth after its orbit destabilized, forming the Moon.

Mars has four known coorbital asteroids (5261 Eureka, 1999 UJ7, 1998 VF31 and 2007 NS2), all at its Lagrangian points.

Earth's companion object 3753 Cruithne is in a relationship with Earth that is somewhat trojanlike, but that is different from a true trojan. Cruithne occupies one of two regular solar orbits, one of them slightly smaller and faster than Earth's, and the other slightly larger and slower. It periodically alternates between these two orbits due to close encounters with Earth. When it is in the smaller, faster orbit and approaches Earth, it gains orbital energy from Earth and moves up into the larger, slower orbit. It then falls farther and farther behind Earth, and eventually Earth approaches it from the other direction. Then Cruithne gives up orbital energy to 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 (7010200000000000000♠2×10^{10}) times more than that of 3753 Cruithne.

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

In a binary star system, the Roche lobe has its apex located at L_{1}; if a star overflows its Roche lobe, then it will lose matter to its companion star.
Mathematical details
A contour plot of the
effective potential due to gravity and the
centrifugal force of a twobody system in a rotating frame of reference. The arrows indicate the gradients of the potential around the five Lagrange points—downhill toward them (
red) or away from them (
blue). Counterintuitively, the L
_{4} and L
_{5} points are the
high points of the potential. At the points themselves these forces are balanced.
Visualisation of the relationship between the Lagrangian points (red) of a planet (blue) orbiting a star (yellow) anticlockwise, and the
effective potential in the plane containing the orbit (grey rubbersheet model with purple contours of equal potential).
^{[10]}
Click for animation.
Lagrangian points are the constantpattern solutions of the restricted threebody problem. 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, could be placed so as to maintain its position relative to the two massive bodies. As seen in a rotating reference frame that matches the angular velocity of the two coorbiting bodies, the gravitational fields of two massive bodies combined with the minor body's centrifugal force are in balance at the Lagrangian points, allowing the smaller third body to be relatively stationary with respect to the first two.^{[11]}
L_{1}
The location of L_{1} is the solution to the following equation, balancing gravitation and the centrifugal force:
\frac{M_1}{(Rr)^2}=\frac{M_2}{r^2}+\frac{M_1}{R^2}\frac{r\left(M_1+M_2\right)}{R^3}
where r is the distance of the L_{1} 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 L_{1} from the center of mass.) Solving this for r involves solving a quintic function, but if the mass of the smaller object (M_{2}) is much smaller than the mass of the larger object (M_{1}) then L_{1} and L_{2} 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}}
This distance can be described as being such that the orbital period, 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 \sqrt{3}\approx 1.73:

T_{s,M_2}(r) = \frac{T_{M_2,M_1}(R)}{\sqrt{3}}.
L_{2}
The location of L_{2} is the solution to the following equation, balancing gravitation and inertia:
\frac{M_1}{(R+r)^2}+\frac{M_2}{r^2}=\frac{M_1}{R^2}+\frac{r\left(M_1+M_2\right)}{R^3}
with parameters defined as for the L_{1} case. Again, if the mass of the smaller object (M_{2}) is much smaller than the mass of the larger object (M_{1}) then L_{2} is at approximately the radius of the Hill sphere, given by:

r \approx R \sqrt[3]{\frac{M_2}{3 M_1}}
L_{3}
The location of L_{3} is the solution to the following equation, balancing gravitation and the centrifugal force:
\frac{M_1}{(Rr)^2}+\frac{M_2}{(2Rr)^2}=\left(\frac{M_2}{M_1+M_2}R+Rr\right)\frac{M_1+M_2}{R^3}
with parameters defined as for the L_{1} and L_{2} cases except that r now indicates how much closer L_{3} is to the more massive 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{7M_2}{12 M_1}
L_{4} and L_{5}
The reason these points are in balance is that, at L_{4} and L_{5}, 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 acceleration is to the distance from the barycenter in the same ratio as for the two massive bodies. The barycenter being both the center of mass and center of rotation of the threebody 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 system. (Indeed, the third body need not have negligible mass.) The general triangular configuration was discovered by Lagrange in work on the threebody problem.
Stability
Although the L_{1}, L_{2}, and L_{3} points are nominally unstable, it turns out that it is possible to find (unstable) periodic orbits around these points, at least in the restricted threebody problem. These periodic orbits, referred to as "halo" orbits, do not exist in a full nbody dynamical system such as the Solar System. However, quasiperiodic (i.e. bounded but not precisely repeating) orbits following Lissajouscurve trajectories do exist in the nbody system. These quasiperiodic Lissajous orbits are what most of Lagrangianpoint missions to date have used. Although they are not perfectly stable, a relatively modest effort at station keeping can allow a spacecraft to stay in a desired Lissajous orbit for an extended period of time. It also turns out that, at least in the case of Sun–EarthL_{1} missions, it is actually preferable to place the spacecraft in a largeamplitude (100,000–200,000 km or 62,000–124,000 mi) Lissajous orbit, instead of having it sit at the Lagrangian point, because this keeps the spacecraft off the direct line between Sun and Earth, thereby reducing the impact of solar interference on Earth–spacecraft communications. Similarly, a largeamplitude Lissajous orbit around L2 can keep a probe out of Earth's shadow and therefore ensures a better illumination of its solar panels.
Spaceflight applications
Earth–Moon L_{1} allows comparatively easy access to Lunar and Earth orbits with minimal change in velocity and has this as an advantage to position a halfway manned space station intended to help transport cargo and personnel to the Moon and back.
Earth–Moon L_{2} would be a good location for a communications satellite covering the Moon's far side and would be "an ideal location" for a propellant depot as part of the proposed depotbased space transportation architecture.^{[12]}
The satellite
ACE in an orbit around L
_{1}
Sun–Earth L_{1} is suited for making observations of the Sun–Earth system. Objects here are never shadowed by Earth or the Moon. The first mission of this type was the International Sun Earth Explorer 3 (ISEE3) mission used as an interplanetary early warning storm monitor for solar disturbances.
Sun–Earth L_{2} is a good spot for spacebased observatories. Because an object around L_{2} 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,^{[13]} so solar radiation is not completely blocked. From this point, the Sun, Earth and Moon are relatively closely positioned together in the sky, and hence leave a large field of view without interference – this is especially helpful for infrared astronomy.
Sun–Earth L_{3} was a popular place to put a "CounterEarth" in pulp science fiction and comic books. Once spacebased observation became possible via satellites^{[14]} and probes, it was shown to hold no such object. The Sun–Earth L_{3} is unstable and could not contain an object, large or small, for very long. This is because the gravitational forces of the other planets are stronger than that of Earth (Venus, for example, comes within 0.3 AU of this L_{3} every 20 months).
A spacecraft orbiting near Sun–Earth L_{3} would be able to closely monitor the evolution of active sunspot regions before they rotate into a geoeffective position, so that a 7day early warning could be issued by the NOAA Space Weather Prediction Center. Moreover, a satellite near Sun–Earth L_{3} would provide very important observations not only for Earth forecasts, but also for deep space support (Mars predictions and for manned mission to nearEarth asteroids). In 2010, spacecraft transfer trajectories to Sun–Earth L_{3} were studied and several designs were considered.^{[15]}
Scientists at the B612 Foundation are planning to use Venus's L_{3} point to position their planned Sentinel telescope, which aims to look back towards Earth's orbit and compile a catalogue of nearEarth asteroids.^{[16]}
Missions to Lagrangian points generally orbit the points rather than occupy them directly.
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.
Spacecraft at Sun–Earth L_{1}
International Sun Earth Explorer 3 (ISEE3) began its mission at the Sun–Earth L_{1} before leaving to intercept a comet in 1982. The Sun–Earth L_{1} is also the point to which the Reboot ISEE3 mission was attempting to return the craft as the first phase of a recovery mission (as of September 25, 2014 all efforts have failed and contact was lost).^{[17]}
Solar and Heliospheric Observatory (SOHO) is stationed in a halo orbit at L_{1}, and the Advanced Composition Explorer (ACE) in a Lissajous orbit. WIND is also at L_{1}.
Deep Space Climate Observatory (DSCOVR), launched on 11 February 2015, began orbiting L_{1} on 8 June 2015 to study the solar wind and its effects on Earth.^{[18]} DISCOVR is unofficially known as GORESAT, because it carries a camera always oriented to Earth and capturing fullframe photos the planet similar to the Blue Marble. This concept was proposed by thenVice President Al Gore in 1998^{[19]} and was a centerpiece in his film An Inconvenient Truth.^{[20]}
Spacecraft at Sun–Earth L_{2}
Spacecraft at the Sun–Earth L_{2} point are in a Lissajous orbit until decommissioned, when they are sent into a heliocentric graveyard orbit.
List of missions to Lagrangian points
Color key:

– Unflown or planned mission


– Mission en route or in progress (including mission extensions)


– Mission at Lagrangian point completed successfully (or partially successfully)


Past and present missions
Mission

Lagrangian point

Agency

Description

International Sun–Earth Explorer 3 (ISEE3)

Sun–Earth L_{1}

NASA

Launched in 1978, it was the first spacecraft to be put into orbit around a libration point, where it operated for four years in a halo orbit about the L_{1} Sun–Earth point. After the original mission ended, it was commanded to leave L_{1} in September 1982 in order to investigate comets and the Sun.^{[25]} Now in a heliocentric orbit, an unsuccessful attempt to return to halo orbit was made in 2014 when it made a flyby of the Earth–Moon system.^{[26]}^{[27]}

Advanced Composition Explorer (ACE)

Sun–Earth L_{1}

NASA

Launched 1997. Has fuel to orbit near the L1 until 2024. Operational as of 2013.^{[28]}

Deep Space Climate Observatory (DSCOVR)

Sun–Earth L_{1}

NASA

Launched on 11 February 2015. At L1 and undergoing testing as of 12 June 2015. ^{[29]}

Solar and Heliospheric Observatory (SOHO)

Sun–Earth L_{1}

ESA, NASA

Orbiting near the L1 since 1996. Operational as of 2013.^{[30]}

WIND

Sun–Earth L_{1}

NASA

Arrived at L1 in 2004 with fuel for 60 yrs. Operational as of 2013.^{[31]}

Wilkinson Microwave Anisotropy Probe (WMAP)

Sun–Earth L_{2}

NASA

Arrived at L2 in 2001. Mission ended 2010,^{[32]} then sent to solar orbit outside L2.^{[33]}

Herschel Space Observatory

Sun–Earth L_{2}

ESA

Arrived at L2 July 2009. Ceased operation on 29 April 2013; will be moved to a heliocentric orbit.^{[34]}^{[35]}

Planck Space Observatory

Sun–Earth L_{2}

ESA

Arrived at L2 July 2009. Mission ended on 23 October 2013; Planck has been moved to a heliocentric parking orbit.^{[36]}

Chang'e 2

Sun–Earth L_{2}

CNSA

Arrived in August 2011 after completing a lunar mission before departing en route to asteroid 4179 Toutatis in April 2012.^{[37]}

ARTEMIS mission extension of THEMIS

Earth–Moon L_{1} and L_{2}

NASA

Mission consists of two spacecraft, which were the first spacecraft to reach Earth–Moon Lagrangian points. Both moved through Earth–Moon Lagrangian points, and are now in lunar orbit.^{[38]}^{[39]}

Gaia

Sun–Earth L_{2}

ESA

Launched on 19 December 2013.^{[40]}^{[41]}

Future and proposed missions
Mission

Lagrangian point

Agency

Description

"Lunar FarSide Communication Satellites"

Earth–Moon L_{2}

NASA

Proposed in 1968 for communications on the far side of the Moon during the Apollo program, mainly to enable an Apollo landing on the far side—neither the satellites nor the landing were ever realized.^{[42]}

Space colonization and manufacturing

Earth–Moon L_{4} or L_{5}

—

First proposed in 1974 by Gerard K. O'Neill^{[43]} and subsequently advocated by the L5 Society.

LISA Pathfinder (LPF)

Sun–Earth L_{1}

ESA, NASA

As of 2012, launch is scheduled for July 2015.^{[44]}

SolarC

Sun–Earth L_{1}

JAXA

Possible mission after 2010.

Aditya

Sun–Earth L_{1}

ISRO

Launch planned for 2016–17; it will be going to a point 1.5 million kilometers away from Earth, from where it will observe the Sun constantly and study the solar corona, the region around the Sun's surface.^{[45]}

James Webb Space Telescope (JWST)

Sun–Earth L_{2}

NASA, ESA, CSA

As of 2013, launch is planned for October 2018.^{[46]}

Euclid

Sun–Earth L_{2}

ESA, NASA

As of 2013, launch is planned in 2020.^{[47]}

Demonstration and Experiment of Space Technology for INterplanetary voYage (DESTINY)

Earth–Moon L_{2}

JAXA

Candidate for JAXA's second "Competitively Chosen Mclass mission", possible launch in the early 2020s.^{[48]}

Wide Field Infrared Survey Telescope (WFIRST)

Sun–Earth L_{2}

NASA, USDOE

As of 2013, in a "preformulation" phase until at least early 2016; possible launch in the early 2020s.^{[49]}

LiteBIRD

Sun–Earth L_{2}^{[50]}

JAXA, NASA

As of 2015, one of two finalists for strategic Lclass mission 2; would be launched in 2022 if selected.^{[51]}

Space Infrared Telescope for Cosmology and Astrophysics (SPICA)

Sun–Earth L_{2}

JAXA, ESA, SRON

As of 2015, awaiting approval from both Japanese and European side, launch proposed for 2025.^{[52]}

Exploration Gateway Platform

Earth–Moon L_{2}^{[53]}

NASA

Proposed in 2011.^{[54]}

Advanced Telescope for High Energy Astrophysics (ATHENA)

Sun–Earth L_{2}

ESA

Launch planned for 2028.^{[55]}

See also
Notes

^ Actually \tfrac{25+\sqrt{621}}{2} ≈ 24.9599357944
References

^ (16MB)

^ Leonhard Euler, De motu rectilineo trium corporum se mutuo attrahentium (1765)

^

^

^ The Lagrange Points PDF, Neil J. Cornish with input from Jeremy Goodman

^ First Asteroid Companion of Earth Discovered at LastSpace.com:

^ NASA—NASA's Wise Mission Finds First Trojan Asteroid Sharing Earth's Orbit

^ "A Search for Natural or Artificial Objects Located at the Earth–Moon Libration Points" by Robert Freitas and Francisco Valdes, Icarus 42, 442447 (1980)

^

^ ZF Seidov, "The Roche Problem: Some Analytics", The Astrophysical Journal, 603:283284, 2004 March 1

^ "Lagrange Points" by Enrique Zeleny, Wolfram Demonstrations Project.

^

^ Angular size of the Sun at 1 AU + 930000 miles: 31.6', angular size of Earth at 930000 miles: 29.3'

^ STEREO mission description by NASA, http://www.nasa.gov/mission_pages/stereo/main/index.html#.UuG0NxDbkk

^

^ The Sentinel Mission, B612 Foundation. Retrieved Feb 2014.

^ "ISEE3 is in Safe Mode". Space College. 25 September 2014. "The ground stations listening to ISEE3 have not been able to obtain a signal since Tuesday the 16th"

^ http://www.nesdis.noaa.gov/news_archives/DSCOVR_L1_orbit.html

^ http://www.usatoday.com/story/tech/2015/02/07/goresatgoresatellitedeepspaceclimate/23013283/

^ Mellow, Craig (August 2014). "Al Gore's Satellite". Air & Space/Smithsonian. Retrieved December 12, 2014.

^

^

^

^

^

^

^

^

^

^

^

^

^ http://map.gsfc.nasa.gov/news/events.html WMAP Ceases Communications

^

^

^

^

^

^

^

^

^

^

^

^ http://www.businessstandard.com/article/beyondbusiness/maninspaceandotherplans114111401887_1.html

^

^

^

^

^

^

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^ NASA teams evaluating ISSbuilt Exploration Platform roadmap

^

^
External links

JosephLouis, Comte Lagrange, from Oeuvres Tome 6, "Essai sur le Problème des Trois Corps"—Essai (PDF); source Tome 6 (Viewer)

"Essay on the ThreeBody Problem" by JL Lagrange, translated from the above, in http://www.merlyn.demon.co.uk/essai3c.htm.

Considerationes de motu corporum coelestium—Leonhard Euler—transcription and translation at http://www.merlyn.demon.co.uk/euler304.htm.

What are Lagrange points?—European Space Agency page, with good animations

Explanation of Lagrange points—Prof. Neil J. Cornish

A NASA explanation—also attributed to Neil J. Cornish

Explanation of Lagrange points—Prof. John Baez

Geometry and calculations of Lagrange points—Dr J R Stockton

Locations of Lagrange points, with approximations—Dr. David Peter Stern

An online calculator to compute the precise positions of the 5 Lagrange points for any 2body system—Tony Dunn

Astronomy cast—Ep. 76: Lagrange Points Fraser Cain and Dr. Pamela Gay

The Five Points of Lagrange by Neil deGrasse Tyson

Earth, a lone Trojan discovered






Shape/Size



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