Lagrangian point

"Lagrange Point" redirects here. For the video game, see Lagrange Point (video game).

This article is about three-body libration points. For two-body libration points, see Geostationary orbit#Earth orbital libration points.

A contour plot of the effective potential due to gravity and the centrifugal force of a two-body 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 L₄ and L₅ 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 gravitational potential in the plane containing the orbit (grey rubber-sheet model with purple contours of equal potential).
Click for animation.

The Lagrangian points (pron.: /ləˈɡrɑːndʒiən/; also Lagrange points, L-points, or libration points) are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be part of a constant-shape pattern with two larger objects (such as a satellite with respect to the Earth and Moon). 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.

Lagrangian points are the constant-pattern solutions of the restricted three-body problem. For example, given two massive bodies in orbits around their common center of mass, 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 matching the angular velocity of the two co-orbiting bodies, the gravitational fields of two massive bodies combined with the satellite's acceleration are in balance at the Lagrangian points, allowing the third body to be relatively stationary with respect to the first two bodies.

History and concepts

The three collinear Lagrange points (L1, L2, L3) were discovered by Leonhard Euler a few years before Lagrange discovered the remaining two.

In 1772, Joseph Louis 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 conic section orbits. Thence, if one mass is made negligible, one immediately gets the five positions now known as the Lagrange Points; but Lagrange himself apparently did not note that.

In the more general case of elliptical orbits, there are no longer stationary points in the same sense: it becomes more of a Lagrangian “area”. The Lagrangian points constructed at each point in time, as in the circular case, form stationary elliptical orbits which are similar to the orbits of the massive bodies. This is due to Newton's second law (Force = Mass times Acceleration, or $\mathbf{F}=d\mathbf{p}/dt$), where p = mv (p the momentum, m the mass, and v the velocity) is invariant if force and position are scaled by the same factor. A body at a Lagrangian point orbits with the same period as the two massive bodies in the circular case, implying that it has the same ratio of gravitational force to radial distance as they do. This fact is independent of the circularity of the orbits, and it implies that the elliptical orbits traced by the Lagrangian points are solutions of the equation of motion of the third body.^{[citation needed]}

Early in the 20th century, Trojan asteroids were discovered at the L₄ and L₅ Lagrange points of the Sun–Jupiter system.

Terminology

The five Sun–Earth Lagrangian points are called SEL1–SEL5, and similarly those of the Earth–Moon system EML1–EML5, etc.

The Lagrangian points

A diagram showing the five Lagrangian points in a two-body system with one body far more massive than the other (e.g. the Sun and the Earth). In such a system, L₃–L₅ will appear to share the secondary's orbit, although in fact they are situated slightly outside it.

The three-body problem can be visualized as a combination of the potentials due to the gravity of the two primary bodies along with the centrifugal effect from their rotation (Coriolis effects are dynamic and not shown). The Lagrange points can then be seen as the five places where the gradient on the resultant surface is zero (shown as blue lines) indicating that the forces are in balance there.

The five Lagrangian points are labeled and defined as follows:

L₁

The L₁ point lies on the line defined by the two large masses M₁ and M₂, and between them. It is the most intuitively understood of the Lagrangian points: the one where the gravitational attraction of M₂ partially cancels M₁ gravitational attraction.

Example: An object which orbits the Sun more closely than the Earth would normally have a shorter orbital period than the Earth, but that ignores the effect of the Earth's own gravitational pull. If the object is directly between the Earth and the Sun, then the Earth's gravity weakens the force pulling the object towards the Sun, 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 the Earth's orbital period. L₁ is about 1.5 million kilometers from the Earth.

The location of L₁ is the solution to the following equation balancing gravitation and centrifugal force:

$\frac{M_1}{(R-r)^2}=\frac{M_2}{r^2}+\left(\frac{M_1}{M_1+M_2}R-r\right)\frac{M_1+M_2}{R^3}$

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. (The quantity in parenthesis on the right is the distance of L₁ from the centre of mass.) Solving this for r involves solving a quintic, 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:

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

This distance can be described as being such that the orbital period, corresponding to a circular orbit with this distance as radius around M₂ in the absence of M₁, is that of M₂ around M₁, divided by $\sqrt{3}\approx 1.73$:

$T_{s,M_2}(r) = \frac{T_{M_2,M_1}(R)}{\sqrt{3}}.$

The Sun–Earth L₁ is suited for making observations of the Sun–Earth system. Objects here are never shadowed by the 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. The feasibility of this orbit was the result of a PhD thesis by the astrodynamicist Robert W. Farquhar. Subsequently the Solar and Heliospheric Observatory (SOHO) was stationed in a Halo orbit at L₁, and the Advanced Composition Explorer (ACE) in a Lissajous orbit, also at the L₁ point. WIND is also at L₁.

The Earth–Moon L₁ allows comparatively easy access to lunar and earth orbits with minimal change in velocity and has this as an advantage to position a half-way manned space station intended to help transport cargo and personnel to the Moon and back.

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.

The satellite ACE in an orbit around L₁

The L₁ point is also important for research satellites.

L₂

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

The L₂ point lies on the line 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.

Example: On the side of the Earth away from the Sun, the orbital period of an object would normally be greater than that of the Earth. The extra pull of the Earth's gravity decreases the orbital period of the object, and at the L₂ point that orbital period becomes equal to the Earth's.

The Sun–Earth L₂ is a good spot for space-based observatories. Because an object around L₂ will maintain the same 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. The space observatories Herschel and Planck are already in orbit around the Sun–Earth L₂, as was Chang'e 2 (until April 2012) and the Wilkinson Microwave Anisotropy Probe (until October 2010). The Gaia probe and James Webb Space Telescope are to be placed at the Sun–Earth L₂. Earth–Moon L₂ would be a good location for a communications satellite covering the Moon's far side. Earth–Moon L₂ would be "an ideal location" for a propellant depot as part of the proposed depot-based space transportation architecture.

The location of L₂ is the solution to the following equation balancing gravitation and centrifugal force:

$\frac{M_1}{(R+r)^2}+\frac{M_2}{r^2}=\left(\frac{M_1}{M_1+M_2}R+r\right)\frac{M_1+M_2}{R^3}$

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:

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

Examples

• Sun and Earth: 1,500,000 km (930,000 mi) from the Earth
• Earth and Moon: 60,000 km (37,000 mi) from the Moon

L₃

The L₃ point lies on the line defined by the two large masses, beyond the larger of the two.

Example: L₃ in the Sun–Earth system exists on the opposite side of the Sun, a little outside the Earth's orbit but slightly closer to the Sun than the Earth is. (This apparent contradiction is because the Sun is also affected by the 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 the Earth and Sun again causes the object to orbit with the same period as the Earth.

The location of L₃ is the solution to the following equation balancing gravitation and centrifugal force:

$\frac{M_1}{(R-r)^2}+\frac{M_2}{(2R-r)^2}=\left(\frac{M_2}{M_1+M_2}R+R-r\right)\frac{M_1+M_2}{R^3}$

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:

$r \approx R \frac{7M_2}{12 M_1}$

The Sun–Earth L₃ point 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 an object, large or small, for very long. This is because the gravitational forces of the other planets are stronger than that of the Earth (Venus, for example, comes within 0.3 AU of this L₃ every 20 months). In addition, because Earth's orbit is elliptical and because the barycenter of the Sun–Jupiter system is unbalanced relative to Earth (that is, the Sun orbits the Sun–Jupiter center of mass, which is outside of the Sun itself), such a Counter-Earth would frequently be visible from Earth.

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.

One example of asteroids which visit an L₃ point is the Hilda family whose orbit brings them to the Sun–Jupiter L₃ point.

L₄ and L₅

Gravitational accelerations at L₄

The L₄ and L₅ points lie at the third corners of the two equilateral triangles in the plane of orbit whose common base is the line between the centers of the two masses, such that the point lies behind (L₅) or ahead of (L₄) the smaller mass with regard to its orbit around the larger mass.

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 system, this resultant force is exactly that required to keep a body at the Lagrange point in orbital equilibrium with the rest of the system. (Indeed, the third body need not have negligible mass). The general triangular configuration was discovered by Lagrange in work on the 3-body problem.

L₄ and L₅ are sometimes called triangular Lagrange points or Trojan points. The name Trojan points comes from the Trojan asteroids at the Sun–Jupiter L₄ and L₅ points, which themselves are named after characters from Homer's Iliad (the legendary siege of Troy). Asteroids at the L₄ point, which leads Jupiter, are referred to as the "Greek camp", while those at the L₅ point are referred to as the "Trojan camp". These asteroids are (largely) named after characters from the respective sides of the Trojan War.

Examples

• The Sun–Earth L₄ and L₅ points lie 60° ahead of and 60° behind the Earth as it orbits the Sun. The regions around these points contain interplanetary dust and at least one asteroid, 2010 TK₇, detected October 2010 by WISE and announced July 2011. Watch NASA's animated clip..
• The Earth–Moon L₄ and L₅ points lie 60° ahead of and 60° behind the Moon as it orbits the Earth. They may contain interplanetary dust in what is called Kordylewski clouds; however, the Hiten spacecraft's Munich Dust Counter (MDC) detected no increase in dust during its passes through these points.
• The region around the Sun–Jupiter L₄ and L₅ points are occupied by the Trojan asteroids.
• The region around the Sun–Neptune L₄ and L₅ points have trojan objects.
• Saturn's moon Tethys has two much smaller satellites at its L₄ and L₅ points named Telesto and Calypso, respectively.
• Saturn's moon Dione has smaller moons Helene and Polydeuces at its L₄ and L₅ points, respectively.
• 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 the Earth after its orbit destabilized, forming the Moon.

Stability

Although the L₁, L₂, and L₃ points are nominally unstable, it turns out that it is possible to find (unstable) periodic orbits around these points, at least in the restricted three-body problem. These periodic orbits, referred to as "halo" orbits, do not exist in a full n-body dynamical system such as the Solar System. However, quasi-periodic (i.e., bounded but not precisely repeating) orbits following Lissajous-curve trajectories do exist in the n-body system. These quasi-periodic Lissajous orbits are what most of Lagrangian-point 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–Earth-L₁ missions, it is actually preferable to place the spacecraft in a large-amplitude (100,000–200,000 km or 62,000–120,000 mi) Lissajous orbit, instead of having it sit at the Lagrangian point, because this keeps the spacecraft off the direct Sun–Earth line, thereby reducing the impact of solar interference on Earth–spacecraft communications. 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.

In contrast to the collinear Lagrangian points, the triangular points (L₄ and L₅) are stable equilibria, provided that the ratio of M₁/M₂ is greater than 24.96. 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 rotating frame of reference). However, in the Earth–Moon case, the problem of stability is greatly complicated by the appreciable solar gravitational influence. In 2008 it was found that the L₄ and L₅ points in the Earth–Moon system would be stable for 1000 million years, even with perturbations from the sun, but because of smaller perturbations by the planets, orbits around these points can only last a few million years.

Intuitive explanation

Lagrangian points can be explained intuitively using the Earth–Moon system.

Lagrangian points L₂ through L₅ exist only in rotating systems, such as in the monthly orbiting of the Moon about the Earth. At these points, the combined attraction from the two masses is equivalent to what would be exerted by a single mass at the barycenter of the system, sufficient to cause a small body to orbit with the same period.

Imagine a person swinging a stone at the end of a string. The string provides a tension force that continuously accelerates the stone toward the center. To an ant standing on the stone, however, it seems as if there is an opposite force trying to fling it directly away from the center. This apparent force is called the centrifugal force. It is actually simply the outward radial component of the stone's inertia caused by its swing. This same effect is present at the Lagrangian points in the Earth–Moon system, where the analogue of the string is the summed (or net) gravitational attraction of the two masses, and the stone is an asteroid or a spacecraft. The Earth–Moon system and the spacecraft all rotate about this combined center of mass, or barycenter. Because the Earth is much heavier than the Moon, the barycenter is located within the Earth (about 1,700 km or 1,100 mi below the surface). Any object gravitationally held by the rotating Earth–Moon system will be attracted to the barycenter to an equal and opposite degree as its tendency to fly off into space.

L₁ is a bit farther from the (less massive) Moon and closer to the (more massive) Earth than the point where total gravity is zero. L₁ is slightly unstable (see stability, above) because drifting towards the Moon or Earth increases one gravitational attraction while decreasing the other, causing more drift.

At Lagrangian points L₂, L₃, L₄, and L₅, a spacecraft's inertia to move away from the barycenter is balanced by the attraction of gravity toward the barycenter. L₂ and L₃ are slightly unstable because small changes in position upset the balance between gravity and inertia, allowing one or the other force to dominate, so that the spacecraft either flies off into space or spirals in toward the barycenter. Stability at L₄ and L₅ is explained by gravitational equilibrium: if the object were moved into a tighter orbit, it would orbit faster which would counteract the increase in gravity; if the object moves into a wider orbit, the gravity is lower, but it loses speed. The net result is that the object appears constantly to hover or orbit around the L₄ or L₅ point.

The easiest way to understand the resulting stability is to say L₁, L₂, and L₃ positions are as stable as a ball balanced on the tip of a wedge would be stable: any disturbance will toss it out of equilibrium. The L₄, and L₅ positions are stable as a ball at the bottom of a bowl would be stable: small perturbations will move it out of place, but it will drift back toward the center of the bowl.

Note that from the perspective of the smaller-mass object — from the moon, in the preceding example — a spacecraft might appear to orbit in an irregular path about the L₄ or L₅ point, but from the perspective above the orbital plane, it becomes clear that both the smaller mass and the spacecraft are orbiting the larger mass (or more precisely, all of the objects are in orbit around the barycenter of the system); they simply have overlapping orbital paths. This point of view difference is illustrated clearly by animations in the 3753 Cruithne and Coriolis effect articles.

Lagrangian point missions

The Lagrangian point orbits have unique characteristics that have made them a good choice for performing some kinds of missions. These missions generally orbit the points rather than occupy them directly.

Past and present missions

Mission	Lagrangian point	Agency	Status
Hiten	Earth–Moon L4 and L5	Japanese Space Agency	Swung through L4 and L5 in 1992; found no increase in dust levels
Advanced Composition Explorer (ACE)	Sun–Earth L1	NASA	Launched 1997. Has fuel to orbit near the L1 until 2024. Operational
Solar and Heliospheric Observatory (SOHO)	Sun–Earth L1	ESA, NASA	Orbiting near the L1 since 1996. Operational
WIND	Sun–Earth L1	NASA	Arrived at L1 in 2004 with fuel for 60 yrs. Operational
International Sun–Earth Explorer 3 (ISEE-3)	Sun–Earth L1	NASA	Original mission ended, left L1 point in September 1982.
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 Observatory	Sun–Earth L2	ESA	Arrived at L2 July 2009. Ceased operation on 29 April 2013 and will be moved to a heliocentric orbit.
Planck Space Observatory	Sun–Earth L2	ESA	Arrived at L2 July 2009. Operational
Chang'e 2	Sun–Earth L2	CNSA	Original mission ended, left L2 point for 4179 Toutatis at April 15, 2012
ARTEMIS mission extension of THEMIS	Earth–Moon L1 and L2	NASA	Operational
GRAIL	Sun–Earth L1	NASA	Passed L1, arrived at the Moon

Future and proposed missions

Mission	Lagrangian point	Agency	Status
Deep Space Climate Observatory	Sun–Earth L1	NASA	On hold[citation needed]
LISA Pathfinder (LPF)	Sun–Earth L1	ESA, NASA	Launch Date 2014
Solar-C	Sun–Earth L1	Japan Aerospace Exploration Agency	Possible mission after 2010[citation needed]
Gaia	Sun–Earth L2	ESA	Planned for August 2013
James Webb Space Telescope	Sun–Earth L2	NASA, ESA, Canadian Space Agency	Working on 2018 launch
Euclid	Sun–Earth L2	ESA	Proposed for launch in 2019
Wide Field Infrared Survey Telescope	Sun–Earth L2	NASA, U.S. Department of Energy	Proposed for launch in 2020
"Lunar Far-Side Communication Satellites"	Earth–Moon L2	NASA	Proposed in 1968
Exploration Gateway Platform	Earth–Moon L2	NASA	Proposed in 2011
Space colonization and manufacturing	Earth–Moon L4 or L5	L5 Society	Proposed in 1974
Capture of asteroid 1999 AO10	Earth–Moon L2	NASA	Proposed in 2012

Natural examples

In the Sun–Jupiter system several thousand asteroids, collectively referred to as Jupiter trojans or "Trojan asteroids", are in orbits around the Sun–Jupiter L₄ and L₅ points. 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. Mars has three known co-orbital asteroids (5261 Eureka, 1999 UJ₇, and 1998 VF₃₁), all at its Lagrangian points. There is one known trojan for Earth as of July 2011. Clouds of dust, called Kordylewski clouds, even fainter than the notoriously weak gegenschein, may also be present in the L₄ and L₅ of the Earth–Moon system.

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 degrees 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.

Other co-orbitals

2010 TK₇ is the first Earth trojan asteroid discovered. It was discovered by the Wide-field Infrared Survey Explorer in October, 2010. After evaluation of the collected data the trojan character of its motion was published in July 2011. 2010 TK₇ has a diameter of 300 meters. Its path is around the Sun–Earth L₄ Lagrangian point.

The Earth's companion object 3753 Cruithne is in a relationship with the Earth which is somewhat trojan-like, but different from a true trojan. This asteroid occupies one of two regular solar orbits, one of them slightly smaller and faster than the Earth's orbit, and the other slightly larger and slower. The asteroid periodically alternates between these two orbits due to close encounters with Earth. When the asteroid is in the smaller, faster orbit and approaches the Earth, it gains orbital energy from the Earth and moves up into the larger, slower orbit. It then falls farther and farther behind the Earth, and eventually Earth approaches it from the other direction. Then the asteroid gives up orbital energy to the Earth, and drops back into the smaller orbit, thus beginning the cycle anew. The cycle has no noticeable impact on the length of the year, because Earth's mass is over 20 billion (2×10¹⁰) times more than 3753 Cruithne.

Epimetheus and Janus, satellites of Saturn, have a similar relationship, though they are of similar masses and so actually exchange orbits with each other periodically. (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.

See also

• Euler's three-body problem
• List of objects at Lagrangian points
• Lunar space elevator
• Home on Lagrange (The L5 Song)
• L5 Society
• Horseshoe orbit
• Hill sphere
• Lagrange point colonization
• Lissajous orbit
• Lagrangian mechanics
• Gegenschein
• Co-orbital configuration
• Trojan asteroids
• Venus Equilateral

External links

• 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