GFDL:Lagrangian point - Revision history

Strangelv: /* Stability */ link tinkering

2007-06-19T23:42:29Z

Stability: link tinkering

Strangelv: /* History and concepts */ applying interwiki template and finding a problem

2007-06-14T15:06:28Z

History and concepts: applying interwiki template and finding a problem

Strangelv: /* Lagrangian point missions */ formatting

2007-06-13T12:26:54Z

Lagrangian point missions: formatting

Strangelv: /* See also */ removed section

2007-06-13T11:24:41Z

Strangelv: /* In fiction */ removed section

2007-06-11T21:46:36Z

In fiction: removed section

Strangelv: added move2space and WP tags

2007-05-12T16:48:44Z

added move2space and WP tags

Mdelaney at 07:16, 14 April 2007

2007-04-14T07:16:37Z

Mdelaney: 100 revision(s)

2007-04-12T23:10:38Z

100 revision(s)

Mdelaney: /* The Lagrangian points */

2007-04-12T09:06:47Z

The Lagrangian points

Mdelaney: /* External links */

2007-04-12T08:42:19Z

External links

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 ==Stability==
-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 L<sub>1</sub> 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 L<sub>1</sub> 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 force]]s.)
+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 L<sub>1</sub> 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 L<sub>1</sub> 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 {{exd|tidal force|tidal forces}}.)
-Although the L<sub>1</sub>, L<sub>2</sub>, and L<sub>3</sub> 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 orbit]]s are what all Lagrangian point missions to date have used. Although they are not perfectly stable, a relatively modest effort at [[Orbital stationkeeping|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 L<sub>1</sub> missions, it is actually preferable to place the spacecraft in a large amplitude (100,000–200,000 km) Lissajous orbit instead of having it sit at the Lagrangian point, because this keeps the spacecraft off the direct Sun–Earth line and thereby reduces the impacts of solar interference on the Earth–spacecraft communications links. 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]].
+Although the L<sub>1</sub>, L<sub>2</sub>, and L<sub>3</sub> 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 {{exd|Solar System|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 {{exd|Lissajous orbit|Lissajous orbits}} are what all Lagrangian point missions to date have used. Although they are not perfectly stable, a relatively modest effort at {{exd|Station Keeping|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 L<sub>1</sub> missions, it is actually preferable to place the spacecraft in a large amplitude (100,000–200,000 km) Lissajous orbit instead of having it sit at the Lagrangian point, because this keeps the spacecraft off the direct Sun–Earth line and thereby reduces the impacts of solar interference on the Earth–spacecraft communications links. 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 (L<sub>4</sub> and L<sub>5</sub>) are stable equilibria (cf. [[attractor]]), provided the ratio of M<sub>1</sub>/M<sub>2</sub> is greater than 24.96<ref>Actually <math>\left(25\left(\frac{1-\sqrt{1-4/625}}{2}\right)\right)^{-1}</math></ref><ref name="cornish">[http://map.gsfc.nasa.gov/ContentMedia/lagrange.pdf The Lagrange Points] – Neil J. Cornish with input from Jeremy Goodman</ref>.  This is the case for the Sun/Earth and, by a smaller margin, the Earth/Moon systems.  When a body at these points is perturbed, it moves away from the point, but the [[Coriolis effect]] then acts, and bends 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.<ref>
+In contrast to the collinear Lagrangian points, the triangular points (L<sub>4</sub> and L<sub>5</sub>) are stable equilibria (cf. attractor), provided the ratio of M<sub>1</sub>/M<sub>2</sub> is greater than 24.96<ref>Actually <math>\left(25\left(\frac{1-\sqrt{1-4/625}}{2}\right)\right)^{-1}</math></ref><ref name="cornish">[http://map.gsfc.nasa.gov/ContentMedia/lagrange.pdf The Lagrange Points] – Neil J. Cornish with input from Jeremy Goodman</ref>.  This is the case for the Sun/Earth and, by a smaller margin, the Earth/Moon systems.  When a body at these points is perturbed, it moves away from the point, but the {{exd|Coriolis Force|Coriolis effect}} then acts, and bends 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.<ref>
 http://www.rfreitas.com/Astro/SearchIcarus1980.htm A Search for Natural or Artificial Objects Located at the Earth-Moon Libration Points; Icarus.</ref>

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 ==History and concepts==
-In [[1772]], the famed Italian-French mathematician [[Joseph-Louis Lagrange]] was working on the famous [[Three body problem|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]] conclude that such a system results in the bodies orbiting [[chaos theory|chaotically]] until there is a collision, or a body is thrown out of the system so that [[Mechanical equilibrium|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 very simple to solve for, as the bodies orbit around their common center of gravity. However, once more than two bodies are introduced, the mathematical calculations become very complicated. A situation arises where you would have to calculate every gravitational interaction between every object at every point along its trajectory.
+In [[1772]], the famed Italian-French mathematician {{exd|Joseph-Louis Lagrange|Joseph-Louis Lagrange}} was working on the famous {{exd|Three Body Problem|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]] conclude that such a system results in the bodies orbiting [[chaos theory|chaotically]] until there is a collision, or a body is thrown out of the system so that [[Mechanical equilibrium|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 very simple to solve for, as the bodies orbit around their common center of gravity. However, once more than two bodies are introduced, the mathematical calculations become very complicated. A situation arises where you would have to calculate every gravitational interaction between every object at every point along its trajectory.
 Lagrange, however, wanted to make this simpler. He did so with a simple conclusion: ''The trajectory of an object is determined by finding a path that minimizes the [[action (physics)|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]]. With his new system of calculations, Lagrange’s work led him to [[hypothesis|hypothesize]] how a third body of negligible mass would orbit around two larger bodies which were already in a near-circular orbit.  In a frame of reference that rotates with the larger bodies, he found five specific fixed points where the third body experiences zero net force as it follows the circular orbit of its host bodies (planets).<ref>

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 The Lagrangian point orbits have unique characteristics that have made them a good choice for performing some kinds of missions. [[NASA]] has operated a number of spacecraft in orbit around the Sun-Earth L<sub>1</sub> and L<sub>2</sub> points, including
-{| class="wikitable"
+{| style="border:1px solid black"
-! Mission || Lagrangian point
+|- style="background:#BFBFBF"
 |-
 | [[Advanced Composition Explorer|Advanced Composition Explorer (ACE)]] || <div style="text-align:center;">L<sub>1</sub></div>
-|-
+|- style="background:#EFEFEF"
 | [[Genesis (spacecraft)|Genesis]] || <div style="text-align:center;">L<sub>1</sub></div>
 |-
 | [[International Cometary Explorer|International Sun/Earth Explorer 3 (ISEE-3)]] || <div style="text-align:center;">L<sub>1</sub></div>
-|-
+|- style="background:#EFEFEF"
 | [[Solar and Heliospheric Observatory|Solar and Heliospheric Observatory (SOHO)]] || <div style="text-align:center;">L<sub>1</sub></div>
 |-
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 The [[James Webb Space Telescope]] is also planned to be placed in orbit around L<sub>2</sub>.
-The [[L5 Society]] was a precursor of the [[National Space Society]], and promoted the possibility of establishing a colony and manufacturing facility in orbit around the L<sub>4</sub> or L<sub>5</sub> points in the Earth&ndash;Moon system (see [[Space colonization]]).
+The [[L5 Society]] was a precursor of the [[National Space Society]], and promoted the possibility of establishing a colony and manufacturing facility in orbit around the L<sub>4</sub> or L<sub>5</sub> points in the Earth&ndash;Moon system.
 ==Natural examples==

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 <references/>
-==See also==
 ==External links==

← Older revision		Revision as of 21:46, 11 June 2007
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	[[Epimetheus (moon)\|Epimetheus]] and [[Janus (moon)\|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.		[[Epimetheus (moon)\|Epimetheus]] and [[Janus (moon)\|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 fiction==~~

−	*The Lagrange points are mentioned in [[science fiction]] from time to time (most often [[hard science fiction]]), but, due to the general lack of public familiarity with them, they are rarely used as a plot device or reference.
−
−	*Lagrange points are mentioned most famously in the science fiction film ''[[2010: The Year We Make Contact]]'', where the Discovery spacecraft is located on a Lagrange point.
−
−	*The Lagrange points are mentioned in the [[anime]] saga ''[[Mobile Suit Gundam]]'', where clusters of space colonies (called "Sides") are located at the five Lagrange points of Earth, in addition to resource satellites and space fortresses. Lagrange Points have been mentioned in several other Gundam series as well.
−
−	*The L5 Lagrange point is mentioned in ''L5: First City in Space'', an early [[IMAX]] 3D movie.
−
−	*In William Gibson's novel [[Neuromancer]], much of the action takes place in the L5 "archipelago", the location of many space stations.
−
−	*Lagrange points also play a role in the Larry Niven/Jerry Pournelle classic ''[[The Mote in God's Eye]]''.
−
−	*The eponymous interplanetary [[relay station]] in [[George O. Smith]]'s "[[Venus Equilateral]]" stories was located in the L<sub>4</sub> point of the Sun-Venus system.
−
−	*In Robert Forward's Rocheworld the locations for Lagrange points around a binary planet are disscussed in contrast to typical system.
−
−	*The planet [[Troas (planet)\|Troas]] in the stories "[[Sucker Bait]]" by [[Isaac Asimov]] and "[[Question and Answer]]" by [[Poul Anderson]] was located in the L<sub>5</sub> point of a fictional [[binary star]] system.
−
−	*In the ''[[Star Trek: The Next Generation]]'' episode, "[[The Survivors (TNG episode)\|The Survivors]]", the [[USS Enterprise (NCC-1701-D)\|Enterprise]] is surprised by an enemy ship that had been hiding in a Lagrange point.
−
−	*The space station ''[[Babylon 5]]'' is described to be located "at the L-5 point in a binary star system between a moon and a barren, lifeless planet." [http://www.ralentz.com/old/sci-fi/b5/b5-info.html]
−
−	*In the [[I-War (Independence War)\|Independence War]] computer games, Lagrange points are used as the only locations for [[Jump drive\|jump-points]].
−
−	*In the [[Battletech]] game series, a star's Nadir and Zenith are the standard hyperspace jump points for most interstellar spacecraft. Lagrange points (usually the L4 and L5 points) are sometimes used to enter a system closer to planets, almost always for small-scale military or pirate operations due to the risk of catastrophic misjumps.
−
−	*In the [[Personal computer\|PC]] [[Computer and video games\|video game]] ''[[Star Wars: X-Wing computer game series\|Star Wars: X-Wing]]'', Lagrange points are mentioned in the briefings of some missions that revolve around attacking objects placed at them.
−
−	*In the [[Xbox]] [[Computer and video games\|video game]] ''[[Halo: Combat Evolved]]'' ([[2001]]) and sequel ''[[Halo 2]]'' ([[2004]]), [[Halo (megastructure)\|Halo Megastructures]] play key locations throughout the games. In ''[[Halo: Combat Evolved\|Halo: CE]]'' and ''[[Halo 2]]'', the Halo structures are in L1 Lagrange points between the Gas Giants (and a moon) [[Threshold (fictional planet)\|Threshold]] and Substance, respectively.
−
−	*In the Halo novels the Lagrange points are the only places where a human ship can safely make a slipspace jump.
−
−	*In the [[Robert A. Heinlein]] novel ''[[The Number of the Beast (novel)\|The Number of the Beast]]'', two of the main characters engage in a discussion of adding [[planets]] to the [[solar system]] at Lagrange points.
−
−	*In [[Hideo Kojima]]'s video game ''[[Policenauts]]'', the setting of the game, an [[O'Neill cylinder\|O'Neill model space colony]], is located at the L5 Lagrange point.
−
−	*In the sci-fi series [[Stargate Atlantis]] there was a defensive satellite located at a Lagrangian point in the solar system in which Atlantis was located.[http://en.wikipedia.org/wiki/The_Siege_%28Stargate_Atlantis%29]
−
−	*In the Hugo Award-winning novel [[A Deepness in the Sky]] by [[Vernor Vinge]], a temporary human habitat is built at the L1 point between the planet Arachna and its primary star, a highly [[variable star\|variable]] [[dwarf star\|dwarf]] called the On/Off Star.
−
−	*In [[Peter F Hamilton]]'s [[Night's dawn\|Night's Dawn Trilogy]], a ZTT jump drive cannot be used in a strong gravitational field. In the first book of the trilogy, [[The Reality Dysfunction]], the main characters cannot escape from a gas giant's gravity well before their pursuers catch up with them. Instead, they race to the Lagrange point between the gas giant and one of its moons in order to activate their drive. Successful execution of this untried and reckless maneuver gains captain Joshua Calvert the nickname "LaGrange" Calvert. In the second book [[The Neutronium Alchemist]], a visit is paid to the supposed home planet of the Kiint, Jobis, which features three moons orbiting the Lagrange One point, rotating around a common centre.
−
−	*In the TV series [[Quatermass]] II, the hostile aliens live on a small asteroid "no more than half a mile across" at a "theoretical point of equilibrium" on the dark side of the earth, although neither L2 or Lagrange are mentioned by name (the term "Bieber Variation" is used instead).
−
−	*In Arthur Clarke and Stephen Baxter's novel Sunstorm L1 point plays a crucial role in the building of a shield that has the purpose of saving Earth from a storm of energy from the Sun.
−
−	*In the manga series [[Battle Angel Alita: Last Order]], the ex-colony ship turned space station ''Leviathan 1'' is at the L2 point in the Earth/Moon system.
−
−	*In 1991, Konami released a science fiction RPG for the NES in Japan called [[Lagrange Point (game)\|Lagrange Point.]]

	==Notes and references==		==Notes and references==

← Older revision		Revision as of 16:48, 12 May 2007
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	[[Image:Lagrange_B.jpg\|thumb\|right\|330px\|A contour plot of the effective [[potential]] (the [[Hill's Surfaces]]) of a two-body system (the [[Sun]] and [[Earth]] here), showing the five Lagrange points.]]		[[Image:Lagrange_B.jpg\|thumb\|right\|330px\|A contour plot of the effective [[potential]] (the [[Hill's Surfaces]]) of a two-body system (the [[Sun]] and [[Earth]] here), showing the five Lagrange points.]]

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 <!-- Excluded {{orbits}} as it called {{Navbox_generic}} which called 17 more templates -->
 [[Category:Celestial mechanics]]
 [[Category:Earth orbits]]

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 ==The Lagrangian points==
-[[image:Lagrange very massive.svg|thumb|350px|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 L<sub>3</sub>–L<sub>5</sub> will appear to share the secondary's orbit, although in fact they are situated slightly outside it.]]
+[[Image:Lagrange_A.jpg|thumb|350px|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 L<sub>3</sub>–L<sub>5</sub> will appear to share the secondary's orbit, although in fact they are situated slightly outside it.]]
 The five Lagrangian points are labeled and defined as follows: