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The Lagrangian points ( //; 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.
In 1772, the Italian-French mathematician Joseph Louis Lagrange was working on the famous three-body problem when he discovered an interesting quirk in the results. Originally, he had set out to discover a way to easily calculate the gravitational interaction between arbitrary numbers of bodies in a system, because Newtonian mechanics concludes that such a system results in the bodies orbiting chaotically until there is a collision, or a body is thrown out of the system so that equilibrium can be achieved.
The logic behind this conclusion is that a system with one body is trivial, as it is merely static relative to itself; a system with two bodies is the relatively simple two-body problem, with the bodies orbiting around their common center of mass. However, once more than two bodies are introduced, the mathematical calculations become very complicated. It becomes necessary to calculate the gravitational interaction between every pair of objects at every point along their trajectory.
Lagrange, however, wanted to make this simpler. He did so with a simple hypothesis: The trajectory of an object is determined by finding a path that minimizes the action over time. This is found by subtracting the potential energy from the kinetic energy. With this way of thinking, Lagrange re-formulated the classical Newtonian mechanics to give rise to Lagrangian mechanics.
Common opinion has been that Lagrange himself considered how a third body of negligible mass would orbit around two larger bodies which were already in a near-circular orbit, and found that in a frame of reference that rotates with the larger bodies, there are five specific fixed points where the third body experiences zero net force as it follows the circular orbit of its host bodies (planets). However, that is false.
Actually, Lagrange considered in the first chapter of the Essai 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 ), 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.
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 gravitational attraction.
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 L2 point lies on the line defined by 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.
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 Herschel Space Observatory, Planck space observatory are already, Chang'e 2 was until April 2012, and the Wilkinson Microwave Anisotropy Probe  was until October 2010, in orbit around the Sun–Earth L2. The Gaia probe and James Webb Space Telescope will 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.
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:
where R is the distance between the two bodies.
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 :
The L3 point lies on the line defined by the two large masses, beyond the larger of the two.
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.
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 at the L5 point they are referred to as the "Trojan camp". These asteroids are (largely) named after characters from the respective sides of the Trojan War.
The first three Lagrangian points are technically stable only in the plane perpendicular to the line between the two bodies. This can be seen most easily by considering the L1 point. A test mass displaced perpendicularly from the central line would feel a force pulling it back towards the equilibrium point. This is because the lateral components of the two masses' gravity would add to produce this force, whereas the components along the axis between them would balance out. However, if an object located at the L1 point drifted closer to one of the masses, the gravitational attraction it felt from that mass would be greater, and it would be pulled closer. (The pattern is very similar to that of tidal forces.)
Although the L1, L2, and L3 points are nominally unstable, it turns out that it is possible to find stable periodic orbits around these points, at least in the restricted three-body problem. These perfectly 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/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 (cf. attractor), provided that the ratio of M1/M2 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 rotating frame of reference). However, in the Earth–Moon case, the problem of stability is greatly complicated by the appreciable solar gravitational influence.
Lagrangian points can be explained intuitively using the Earth–Moon system.
Lagrangian points L2 through L5 only exist 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/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.
Unlike the other Lagrangian points, L1 would exist even in a non-rotating (static or inertial) system. In a rotating system, L1 is a bit farther from the (less massive) Moon and closer to the (more massive) Earth than it would be in a non-rotating system. 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.
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.
|Advanced Composition Explorer (ACE)||Sun–Earth L1||NASA||Operational|
|Solar and Heliospheric Observatory (SOHO)||Sun–Earth L1||ESA, NASA||Operational|
|International Sun–Earth Explorer 3 (ISEE-3)||Sun–Earth L1||NASA||Original mission ended, left L1 point|
|Wilkinson Microwave Anisotropy Probe (WMAP)||Sun–Earth L2||NASA||Operational, graveyard orbit|
|Herschel and Planck Space Observatories||Sun–Earth L2||ESA||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|
|Deep Space Climate Observatory||Sun–Earth L1||NASA||On hold|
|LISA Pathfinder (LPF)||Sun–Earth L1||ESA, NASA||Launch Date 2014 |
|Solar-C||Sun–Earth L1||Japan Aerospace Exploration Agency||Possible mission after 2010|
|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|
In the Sun–Jupiter system several thousand asteroids, collectively referred to as 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.
2010 TK7 is the first found Earth trojan asteroid. 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.
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