Lagrangian point(Redirected from Lagrange points)
In celestial mechanics, the Lagrangian points (// also Lagrange points, L-points, or libration points) are positions in an orbital configuration of two large bodies, wherein a small object, affected only by the gravitational forces from the two larger objects, will maintain its position relative to them. The Lagrange points mark positions where the combined gravitational pull of the two large masses provides precisely the centripetal force required to orbit at the same angular velocity (essentially, the speed of the orbit) and thus remain in the same relative position. 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 through the two large bodies; 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 satellites 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.
In 1772, Lagrange published an "Essay on the three-body problem". 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 orbits.
The five Lagrangian points are labeled and defined as follows:
The L1 point lies on the line defined by the two large masses M1 and M2, and between them. It is the most intuitively understood of the Lagrangian points: the one where the gravitational attraction of M2 partially cancels M1's gravitational attraction.
- 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 L1 point, the orbital period of the object becomes exactly equal to Earth's orbital period. L1 is about 1.5 million kilometers from Earth, or 0.01 au, 1/100th the distance to the Sun.
The L2 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 L2.
- 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 L2 point that orbital period becomes equal to Earth's. Like L1, L2 is about 1.5 million kilometers or 0.01 au from Earth.
The L3 point lies on the line defined by the two large masses, beyond the larger of the two.
- L3 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 L3 point, the combined pull of Earth and Sun again causes the object to orbit with the same period as Earth.
The L4 and L5 points lie at the third corners of the two equilateral triangles in the plane of orbit whose common base is the line between the centers of the two masses, such that the point lies behind (L5) or ahead (L4) of the smaller mass with regard to its orbit around the larger mass.
The triangular points (L4 and L5) are stable equilibria, provided that the ratio of M1/ is greater than 24.96.[note 1] 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).
In contrast to L4 and L5, where stable equilibrium exists, the points L1, L2, and L3 are positions of unstable equilibrium. Any object orbiting at L1, L2, or L3 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 pointsEdit
It is common to find objects at or orbiting the L4 and L5 points of natural orbital systems. These are commonly called "trojans". In the 20th century, asteroids discovered orbiting at the Sun–Jupiter L4 and L5 points were named after characters from Homer's Iliad. Asteroids at the L4 point, which leads Jupiter, are referred to as the "Greek camp", whereas those at the L5 point are referred to as the "Trojan camp".
Other examples of natural objects orbiting at Lagrange points:
- The Sun–Earth L4 and L5 points contain interplanetary dust and at least one asteroid, 2010 TK7, detected in October 2010 by Wide-field Infrared Survey Explorer (WISE) and announced during July 2011.
- The Earth–Moon L4 and L5 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.
- Recent observations suggest that the Sun–Neptune L4 and L5 points, known as the Neptune trojans, may be very thickly populated, containing large bodies an order of magnitude more numerous than the Jupiter trojans.
- Several asteroids also orbit near the Sun-Jupiter L3 point, called the Hilda family.
- The Saturnian moon Tethys has two smaller moons in its L4 and L5 points, Telesto and Calypso. The Saturnian moon Dione also has two Lagrangian co-orbitals, Helene at its L4 point and Polydeuces at L5. The moons wander azimuthally about the Lagrangian points, with Polydeuces describing the largest deviations, moving up to 32° away from the Saturn–Dione L5 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 L4 or L5 points and crashed into Earth after its orbit destabilized, forming the Moon.
- Mars has four known co-orbital 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 trojan-like, 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 (×1010) times more than that of 3753 Cruithne. 2
- 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 L1; if a star overflows its Roche lobe, then it will lose matter to its companion star.
Lagrangian points are the constant-pattern solutions of the restricted three-body 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 co-orbiting bodies, the gravitational fields of two massive bodies combined providing the centripetal force at the Lagrangian 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:
where r is the distance of the L1 point from the smaller object, R is the distance between the two main objects, and M1 and M2 are the masses of the large and small object, respectively. Solving this for r involves solving a quintic function, but if the mass of the smaller object (M2) is much smaller than the mass of the larger object (M1) then L1 and L2 are at approximately equal distances r from the smaller object, equal to the radius of the Hill sphere, given by:
This distance can be described as being such that the orbital period, corresponding to a circular orbit with this distance as radius around M2 in the absence of M1, is that of M2 around M1, divided by √ ≈ 1.73:
The location of L2 is the solution to the following equation, gravitation providing the centripetal force:
with parameters defined as for the L1 case. Again, if the mass of the smaller object (M2) is much smaller than the mass of the larger object (M1) then L2 is at approximately the radius of the Hill sphere, given by:
The location of L3 is the solution to the following equation, gravitation providing the centripetal force:
with parameters defined as for the L1 and L2 cases except that r now indicates how much closer L3 is to the more massive object than the smaller object. If the mass of the smaller object (M2) is much smaller than the mass of the larger object (M1) then:
L4 and L5Edit
The reason these points are in balance is that, at L4 and L5, the distances to the two masses are equal. Accordingly, the gravitational forces from the two massive bodies are in the same ratio as the masses of the two bodies, and so the resultant force acts through the 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 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 system. (Indeed, the third body need not have negligible mass.) The general triangular configuration was discovered by Lagrange in work on the three-body problem.
The radial acceleration a of an object in orbit at a point along the line passing through both bodies is given by:
Where r is the distance from the large body M1 and sgn(x) is the sign function of x. The terms in this function represent respectively: force from M1; force from M2; and centrifugal force. The points L3, L1, L2 occur where the acceleration is zero — see chart at right.
Although the L1, L2, and L3 points are nominally unstable, there are (unstable) periodic orbits called "halo" orbits around these points in a three-body system. A full n-body dynamical system such as the Solar System 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. Also, for Sun–Earth-L1 missions, it is preferable for the spacecraft to be in a large-amplitude (100,000–200,000 km or 62,000–124,000 mi) Lissajous orbit around L1 than to stay at L1, because the line between Sun and Earth has increased solar interference on Earth–spacecraft communications. Similarly, a large-amplitude Lissajous orbit around L2 keeps a probe out of Earth's shadow and therefore ensures continuous illumination of its solar panels.
The L4 and L5 points are stable provided that the mass of the primary body (e.g. the Earth) is at least 25[note 1] 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 L4 and L5 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.
Solar System valuesEdit
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 to the semimajor axis. E.g. for the Moon, L1 is located 400 km from Earth's center, which is 84.9% of the Earth-Moon distance or 15.1% in front of the Moon; L2 is located 326900 km from Earth's center, which is 116.8% of the Earth-Moon distance or 16.8% beyond the Moon; and L3 is located 448700 km from Earth's center, which is 99.3% of the Earth-Moon distance or 0.7084% in front of the Moon's 'negative' position. −381
|Body pair||Semimajor axis (SMA)||L1||1 − L1/SMA (%)||L2||L2/SMA − 1 (%)||L3||1 + L3/SMA (%)|
|Earth-Moon||×108 m 3.844||×108 m 3.2639||15.09||×108 m 4.489||16.78||×108 m −3.8168||0.7084|
|Sun-Mercury||×1010 m 5.7909||×1010 m 5.7689||0.3806||×1010 m 5.813||0.3815||×1010 m −5.7909||009683 0.000|
|Sun-Venus||×1011 m 1.0821||×1011 m 1.072||0.9315||×1011 m 1.0922||0.9373||×1011 m −1.0821||1428 0.000|
|Sun-Earth||×1011 m 1.496||×1011 m 1.4811||0.997||×1011 m 1.511||1.004||×1011 m −1.496||1752 0.000|
|Sun-Mars||×1011 m 2.2794||×1011 m 2.2686||0.4748||×1011 m 2.2903||0.4763||×1011 m −2.2794||01882 0.000|
|Sun-Jupiter||×1011 m 7.7834||×1011 m 7.2645||6.667||×1011 m 8.3265||6.978||×1011 m −7.7791||63 0.055|
|Sun-Saturn||×1012 m 1.4267||×1012 m 1.3625||4.496||×1012 m 1.4928||4.635||×1012 m −1.4264||67 0.016|
|Sun-Uranus||×1012 m 2.8707||×1012 m 2.8011||2.421||×1012 m 2.9413||2.461||×1012 m −2.8706||546 0.002|
|Sun-Neptune||×1012 m 4.4984||×1012 m 4.3834||2.557||×1012 m 4.6154||2.602||×1012 m −4.4983||004 0.003|
Sun–Earth L1 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 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 telescopes, because it provides an uninterrupted view of the Sun and any space weather (including the solar wind and coronal mass ejections) reaches L1 a few hours before Earth. Solar telescopes currently located around L1 include the Solar and Heliospheric Observatory and Advanced Composition Explorer.
Sun–Earth L2 is a good spot for space-based observatories. Because an object around L2 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. (Real 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 and observations of the cosmic microwave background. The James Webb Space Telescope is due to be positioned at L2.
Sun–Earth L3 was a popular place to put a "Counter-Earth" in pulp science fiction and comic books. Once space-based observation became possible via satellites and probes, it was shown to hold no such object. The Sun–Earth L3 is unstable and could not contain a natural 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 L3 every 20 months).
A spacecraft orbiting near Sun–Earth L3 would be able to closely monitor the evolution of active sunspot regions before they rotate into a geoeffective position, so that a 7-day early warning could be issued by the NOAA Space Weather Prediction Center. Moreover, a satellite near Sun–Earth L3 would provide very important observations not only for Earth forecasts, but also for deep space support (Mars predictions and for manned mission to near-Earth asteroids). In 2010, spacecraft transfer trajectories to Sun–Earth L3 were studied and several designs were considered.
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.
Earth–Moon L1 allows comparatively easy access to Lunar and Earth orbits with minimal change in velocity and this has as an advantage to position a half-way manned space station intended to help transport cargo and personnel to the Moon and back.
Earth–Moon L2 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 depot-based space transportation architecture.
Scientists at the B612 Foundation are planning to use Venus's L3 point to position their planned Sentinel telescope, which aims to look back towards Earth's orbit and compile a catalogue of near-Earth asteroids.
In 2017, Nasa proposed the idea of positioning a magnetic dipole shield at the Sun-Mars L1 point for use as an artificial magnetosphere for Mars. The idea is that this would protect the planet's atmosphere from the Sun's radiation and solar winds.
Lagrangian spacecraft and missionsEdit
Spacecraft at Sun–Earth L1Edit
International Sun Earth Explorer 3 (ISEE-3) began its mission at the Sun–Earth L1 before leaving to intercept a comet in 1982. The Sun–Earth L1 is also the point to which the Reboot ISEE-3 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).
Deep Space Climate Observatory (DSCOVR), launched on 11 February 2015, began orbiting L1 on 8 June 2015 to study the solar wind and its effects on Earth. DSCOVR is unofficially known as GORESAT, because it carries a camera always oriented to Earth and capturing full-frame photos of the planet similar to the Blue Marble. This concept was proposed by then-Vice President of the United States Al Gore in 1998 and was a centerpiece in his film An Inconvenient Truth.
LISA Pathfinder (LPF) was launched on 3 December 2015, and arrived at L1 on 22 January 2016, where, among other experiments, it will test the technology needed by (e)LISA to detect gravitational waves. LISA Pathfinder uses an instrument consisting of two small gold alloy cubes.
Spacecraft at Sun–Earth L2Edit
- 1 October 2001 – October 2010: Wilkinson Microwave Anisotropy Probe
- November 2003 – April 2004: WIND, then it returned to Earth orbit before going to L1 where it still remains
- July 2009 – 29 April 2013: Herschel Space Telescope
- 3 July 2009 – 21 October 2013: Planck Space Observatory
- 25 August 2011 – April 2012: Chang'e 2, from where it travelled to 4179 Toutatis and then into deep space
- January 2014 – 2018: Gaia Space Observatory
- 2020: Euclid Space Telescope
- 2021: James Webb Space Telescope will use a halo orbit
- 2024: Wide Field Infrared Survey Telescope (WFIRST) will use a halo orbit
- 2028: Advanced Telescope for High Energy Astrophysics (ATHENA) will use a halo orbit
Spacecraft at Earth–Moon L2Edit
Past and current missionsEdit
|Unflown or planned mission||Mission en route or in progress (including mission extensions)||Mission at Lagrangian point completed successfully (or partially successfully)|
|International Sun–Earth Explorer 3 (ISEE-3)||Sun–Earth L1||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 L1 Sun–Earth point. After the original mission ended, it was commanded to leave L1 in September 1982 in order to investigate comets and the Sun. 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.|
|Advanced Composition Explorer (ACE)||Sun–Earth L1||NASA||Launched 1997. Has fuel to orbit near L1 until 2024. Operational as of 2016[update].[needs update]|
|Deep Space Climate Observatory (DSCOVR)||Sun–Earth L1||NASA||Launched on 11 February 2015. Planned successor of the Advanced Composition Explorer (ACE) satellite. Operational as of 2016[update].[needs update]|
|LISA Pathfinder (LPF)||Sun–Earth L1||ESA, NASA||Launched one day behind revised schedule (planned for the 100th anniversary of the publication of Einstein's General Theory of Relativity), on 3 December 2015. Arrived at L1 on 22 January 2016. LISA Pathfinder was deactivated on 30 June 2017.|
|Solar and Heliospheric Observatory (SOHO)||Sun–Earth L1||ESA, NASA||Orbiting near L1 since 1996. Operational as of 2017[update].|
|WIND||Sun–Earth L1||NASA||Arrived at L1 in 2004 with fuel for 60 years. Operational as of 2017[update].|
|Wilkinson Microwave Anisotropy Probe (WMAP)||Sun–Earth L2||NASA||Arrived at L2 in 2001. Mission ended 2010, then sent to solar orbit outside L2.|
|Herschel Space Telescope||Sun–Earth L2||ESA||Arrived at L2 July 2009. Ceased operation on 29 April 2013; will be moved to a heliocentric orbit.|
|Planck Space Observatory||Sun–Earth L2||ESA||Arrived at L2 July 2009. Mission ended on 23 October 2013; Planck has been moved to a heliocentric parking orbit.|
|Chang'e 2||Sun–Earth L2||CNSA||Arrived in August 2011 after completing a lunar mission before departing en route to asteroid 4179 Toutatis in April 2012.|
|Chang'e 5-T1 service module||Earth–Moon L2||CNSA||Arrived in a Lissajous orbit with period of 14 days around the EM-L2 point on November 27, 2014 after its primary mission, the re-entry demonstration of the sample return capsule was completed. It left the EM-L2 orbit on January 4, 2015 and returned to lunar orbit on January 13.|
|ARTEMIS mission extension of THEMIS||Earth–Moon L1 and L2||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.|
|WIND||Sun–Earth L2||NASA||Arrived at L2 in November 2003 and departed April 2004.|
|Gaia Space Observatory||Sun–Earth L2||ESA||Launched 19 December 2013. Operational as of 24 January 2017.[needs update]|
|Queqiao||Earth–Moon L2||CNSA||Launched on 21 May 2018, en route to L2 halo orbit.|
Future and proposed missionsEdit
|"Lunar Far-Side Communication Satellites"||Earth–Moon L2||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.
|Space colonization and manufacturing||Earth–Moon L4 or L5||—||First proposed in 1974 by Gerard K. O'Neill and subsequently advocated by the L5 Society.|
|EQUULEUS||Earth-Moon L2||University of Tokyo, JAXA||6U CubeSat, launch planned in 2019 as a secondary payload onboard SLS Exploration Mission 1.|
|James Webb Space Telescope (JWST)||Sun–Earth L2||NASA, ESA, CSA||As of 2018[update], launch is planned for 2021.|
|Euclid||Sun–Earth L2||ESA, NASA||As of 2013[update], launch is planned in 2020.|
|Aditya-L1||Sun–Earth L1||ISRO||Launch planned for 2021; 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.|
|Demonstration and Experiment of Space Technology
for INterplanetary voYage (DESTINY)
|Earth–Moon L2||JAXA||Candidate for JAXA's next "Competitively-Chosen Medium-Sized Focused Mission",|
possible launch in the early 2020s.
|Exploration Gateway Platform||Earth–Moon L2||NASA||Proposed in 2011.|
|Wide Field Infrared Survey Telescope (WFIRST)||Sun–Earth L2||NASA, USDOE||As of 2013[update], in a "pre-formulation" phase until at least early 2016; possible launch in the early 2020s.|
|LiteBIRD||Sun–Earth L2||JAXA, NASA||As of 2015[update], one of two finalists for JAXA's next "Strategic Large Mission"; would be launched in 2024 if selected.|
|Planetary Transits and Oscillations of stars (PLATO)||Sun–Earth L2||ESA||Planned for launch in 2024 for an initial six-year mission.|
|Space Infrared Telescope for Cosmology
and Astrophysics (SPICA)
|Sun–Earth L2||JAXA, ESA, SRON||As of 2015[update], awaiting approval from both Japanese and European side, launch proposed for 2025.|
|Advanced Telescope for High Energy Astrophysics
|Sun–Earth L2||ESA||Launch planned for 2028.|
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|Wikimedia Commons has media related to Lagrange points.|
- Joseph-Louis, Comte Lagrange, from Oeuvres Tome 6, "Essai sur le Problème des Trois Corps"—Essai (PDF); source Tome 6 (Viewer)
- "Essay on the Three-Body Problem" by J-L Lagrange, translated from the above, in http://www.merlyn.demon.co.uk/essai-3c.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 2-body 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