Lagrangian point

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"Lagrange Point" redirects here. For the video game, see Lagrange Point (video game).
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 effective potential in the plane containing the orbit (grey rubber-sheet model with purple contours of equal potential).[1]
Click for animation.

The Lagrangian points (/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. A satellite at L1 would have the same angular velocity of the earth with respect to the sun and hence it would maintain the same position with respect to the sun as seen from the earth. Without the earth's gravitational influence, a satellite of the sun, at the distance of L1, would have to move at a higher angular velocity than that of the earth.

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 that matches 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 smaller third body to be relatively stationary with respect to the first two.[1]

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.[2][3]

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 circular orbits.[4] 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 F={\mathrm {d}}p/{\mathrm {d}}t), 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's 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.[5]

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 center 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}}{3M_{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.[6] 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 scientific 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,[7] so solar radiation is not completely blocked. 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.[8]

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

{\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}}{3M_{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[9]

Spacecraft at Sun–Earth L2

Spacecraft at the Sun–Earth L2 are in a Lissajous orbit until decommissioned, when they are sent into a heliocentric graveyard orbit.

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}}{12M_{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[14] 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).

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.[15]

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 (L4) of 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 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 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.[16][17]
  • 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.[18]
  • 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 62,000–124,000 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.[note 1][19] 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.[20] 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.[21]

Intuitive explanation

Lagrangian points can be explained intuitively using the Earth–Moon system.[22]

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.

List of missions to Lagrangian points

Orbits around Lagrangian points offer unique advantages that have made them a good choice for performing certain spacecraft missions. For example the Sun–Earth L1 point is useful for observations of the Sun, as the Sun is always visible without obstructions by the Earth or the Moon. The Sun–Earth L2 point is useful for astronomy missions using space observatories, as 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.

Missions to Lagrangian points generally orbit the points rather than occupy them directly.

Color key:

{|
     – Mission at Lagrangian point completed successfully (or partially successfully)          – Unflown, cancelled or failed mission      – Mission en route or in progress (including mission extensions)      – Planned mission

Past and present missions

Mission Lagrangian point Agency Description
"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—both the satellites and the landing were never realized.[23]
Space colonization and manufacturing Earth–Moon L4 or L5 Gerard K. O'Neill / L5 Society First proposed in 1974 by Gerard K. O'Neill.[24]
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.[25] Now in a heliocentric orbit and could be reactivated in 2014 when it makes a flyby of the Earth–Moon system.[26]
Hiten Earth–Moon L4 and L5 JAXA 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. As of 2013 operational.[27]
Solar and Heliospheric Observatory (SOHO) Sun–Earth L1 ESA, NASA Orbiting near the L1 since 1996. As of 2013 operational.[28]
WIND Sun–Earth L1 NASA Arrived at L1 in 2004 with fuel for 60 yrs. As of 2013 operational.[29]
Wilkinson Microwave Anisotropy Probe (WMAP) Sun–Earth L2 NASA Arrived at L2 in 2001. Mission ended 2010,[30] then sent to solar orbit outside L2.[31]
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.[32][33]
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.[34]
Chang'e 2 Sun–Earth L2 CNSA Original mission ended, left L2 point for 4179 Toutatis at April 15, 2012.[35]
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.[36][37]
Gravity Recovery and Interior Laboratory (GRAIL) Sun–Earth L1 NASA Passed L1 on its way to lunar orbit.[38]
Gaia Sun–Earth L2 ESA Launched on 19 December 2013.[39][40]

Future and proposed missions

Mission Lagrangian point Agency Description
Deep Space Climate Observatory (DSCOVR) Sun–Earth L1 NASA On hold.[citation needed]
LISA Pathfinder (LPF) Sun–Earth L1 ESA, NASA As of 2012 launch is scheduled for July 2015.[41]
Solar-C Sun–Earth L1 JAXA Possible mission after 2010.[citation needed]
James Webb Space Telescope (JWST) Sun–Earth L2 NASA, ESA, CSA As of 2013 launch is planned for October 2018.[42]
Euclid Sun–Earth L2 ESA, NASA As of 2013 planned for launch in 2020.[43]
Wide Field Infrared Survey Telescope (WFIRST) Sun–Earth L2 NASA, USDOE As of 2013 in a 'pre-formulation' phase until at least early 2016, possible launch in the early 2020s.[44]
Exploration Gateway Platform Earth–Moon L2[45] NASA Proposed in 2011.[46]

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

Notes

  1. ↑ Actually {\tfrac {25+3{\sqrt {69}}}{2}}

References

  1. ↑ "Lagrange Points" by Enrique Zeleny, Wolfram Demonstrations Project.
  2. ↑ Koon, W. S.; M. W. Lo, J. E. Marsden, and S. D. Ross (2006). Dynamical Systems, the Three-Body Problem, and Space Mission Design. p. 9.  (16MB)
  3. ↑ Leonhard Euler, De motu rectilineo trium corporum se mutuo attrahentium (1765)
  4. ↑ (French) Lagrange, Joseph-Louis (1867–92). "Tome 6, Chapitre II: Essai sur le problème des trois corps". Oeuvres de Lagrange. Gauthier-Villars. pp. 229–334. 
  5. ↑ Cornish, Neil J. "The Lagrangian Points". Department of Physics, Bozeman Campus, Montana State University, USA. Retrieved 29 July 2011. 
  6. ↑ Farquhar, R. W.: "The Control and Use of Libration-Point Satellites", Ph.D. Dissertation, Dept. of Aeronautics and Astronautics, Stanford University, Stanford, CA, 1968
  7. ↑ Angular size of the Sun at 1 AU + 930000 miles: 31.6', angular size of Earth at 930000 miles: 29.3'
  8. ↑ Zegler, Frank; Bernard Kutter (2010-09-02). "Evolving to a Depot-Based Space Transportation Architecture". AIAA SPACE 2010 Conference & Exposition. AIAA. p. 4. Retrieved 2011-01-25. "L2 is in deep space far away from any planetary surface and hence the thermal, micrometeoroid, and atomic oxygen environments are vastly superior to those in LEO. Thermodynamic stasis and extended hardware life are far easier to obtain without these punishing conditions seen in LEO. L2 is not just a great gateway- it is a great place to store propellants. ... L2 is an ideal location to store propellants and cargos: it is close, high energy, and cold. More importantly, it allows the continuous onward movement of propellants from LEO depots thus suppressing their size and effectively minimizing the near-earth boiloff penalties." 
  9. ↑ Zegler, Frank; Bernard Kutter (2010-09-02). "Evolving to a Depot-Based Space Transportation Architecture". AIAA SPACE 2010 Conference & Exposition. AIAA. p. 4. Retrieved 2011-08-30. "We can create an energy savings account by moving propellant to the earth-moon Lagrange points—especially L2. Located 60,000 km beyond the Moon, propellant or cargo cached at L2 is very nearly at earth escape energy. It takes only a small nudge to dislodge it from Earth's gravitational grasp. This has been known for decades and L2 is often called a gateway to the solar system."
  10. ↑ "MISSION COMPLETE! WMAP FIRES ITS THRUSTERS FOR THE LAST TIME". 
  11. ↑ "Herschel Space Telescope Shut Down For Good, ESA Announces". 
  12. ↑ Lakdawalla, Emily (14 June 2012). "Chang'E 2 has departed Earth's neighborhood for.....asteroid Toutatis!?". Retrieved 15 June 2012. 
  13. ↑ "Update on yesterday's post about Chang'E 2 going to Toutatis". Planetary Society. 15 June 2012. Retrieved 26 June 2012. 
  14. ↑ STEREO mission description by NASA, http://www.nasa.gov/mission_pages/stereo/main/index.html#.UuG0NxDb-kk
  15. ↑ Tantardini, Marco; Fantino, Elena; Yuan Ren, Pierpaolo Pergola, Gerard Gómez and Josep J. Masdemont (2010). "Spacecraft trajectories to the L3 point of the Sun–Earth three-body problem". Celestial Mechanics and Dynamical Astronomy (Springer). 
  16. ↑ Space.com: First Asteroid Companion of Earth Discovered at Last
  17. ↑ NASA—NASA's Wise Mission Finds First Trojan Asteroid Sharing Earth's Orbit
  18. ↑ "List Of Neptune Trojans". Minor Planet Center. Archived from the original on 2011-08-23. Retrieved 2010-10-27. 
  19. ↑ The Lagrange Points PDF, Neil J. Cornish with input from Jeremy Goodman
  20. ↑ "A Search for Natural or Artificial Objects Located at the Earth–Moon Libration Points" by Robert Freitas and Francisco Valdes, Icarus 42, 442-447 (1980)
  21. ↑ "Solar and planetary destabilization of the Earth–Moon triangular Lagrangian points" by Jack Lissauer and John Chambers, Icarus, vol. 195, issue 1, May 2008, pp. 16-27.
  22. ↑ Tyson, Neil deGrasse (2007-01-16). Death by Black hole: and other cosmic quandaries. W. W. Norton & Company. ISBN 978-0-393-06224-3. 
  23. ↑ P. E. Schmid (June 1968). "Lunar Far-Side Communication Satellites" (PDF). NASA. Retrieved 2008-07-16. 
  24. ↑ O'Neill, Gerard K. (September 1974). "The Colonization of Space". Physics Today (American Institute of Physics) 27 (9): 32–40. doi:10.1063/1.3128863. 
  25. ↑ "Solar System Exploration: ISEE-3/ICE". NASA. Retrieved 2010-09-28. 
  26. ↑ Lakdawalla, Emily (October 3, 2008). "It's Alive!". The Planetary Science Weblog. 
  27. ↑ "ACE Mission". Caltech ACE Science Center. Retrieved 2013-03-18. 
  28. ↑ "SOHO's Orbit: An Uninterrupted View of the Sun". NASA. Retrieved 2010-09-28. 
  29. ↑ "WIND Spacecraft". NASA. Retrieved 2010-09-28. 
  30. ↑ "WMAP Facts". NASA. Retrieved 2013-03-18. 
  31. ↑ http://map.gsfc.nasa.gov/news/events.html WMAP Ceases Communications
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  33. ↑ "Herschel space telescope finishes mission". BBC news. 29 April 2013. 
  34. ↑ "Last command sent to ESA's Planck space telescope". European Space Agency. October 23, 2013. Retrieved October 23, 2013. 
  35. ↑ "嫦娥二号再接新任务预计年底飞掠小行星". 现代快报. Retrieved 2012-06-27. 
  36. ↑ Fox, Karen C. "First ARTEMIS Spacecraft Successfully Enters Lunar Orbit". The Sun-Earth Connection: Heliophysics. NASA. 
  37. ↑ Hendrix, Susan. "Second ARTEMIS Spacecraft Successfully Enters Lunar Orbit". The Sun-Earth Connection: Heliophysics. NASA. 
  38. ↑ "GRAIL | Gravity Recovery and Interior Laboratory". NASA. Retrieved 2011-09-16. 
  39. ↑ "Worldwide launch schedule". Spaceflight Now. 27 November 2013. 
  40. ↑ "ESA Gaia home". ESA. Retrieved 23 October 2013. 
  41. ↑ "LISA Pathfinder factsheet". ESA. 11 June 2012. Retrieved 26 June 2012. 
  42. ↑ "JWST factsheet". ESA. 2013-09-04. Retrieved 2013-09-07. 
  43. ↑ "NASA Officially Joins ESA's 'Dark Universe' Mission". JPL/NASA. 24 January 2013. Retrieved 12 April 2013. 
  44. ↑ Paul Hertz (2013-06-04), NASA Astrophysics presentation to American Astronomical Society, retrieved 2013-09-10 
  45. ↑ NASA teams evaluating ISS-built Exploration Platform roadmap
  46. ↑ Bergin, Chris (December 2011). "Exploration Gateway Platform hosting Reusable Lunar Lander proposed". NASA Spaceflight.com. Retrieved 2011-12-05. 

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