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Topic: Lagrange points

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	Lagrange Points
		Of the five Lagrange points, three are unstable and two are stable.
		The stable Lagrange points - labelled L4 and L5 - form the apex of two equilateral triangles that have the large masses at their vertices.
		The L1 point of the Earth-Sun system affords an uninterrupted view of the sun and is currently home to the Solar and Heliospheric Observatory Satellite SOHO.
	www.physics.montana.edu /faculty/cornish/lagrange.html (761 words)

	WMAP Observatory_Lagrange Points
		The Italian-French mathematician Joseph-Louis Lagrange discovered five special points in the vicinity of two orbiting masses where a third, smaller mass can orbit at a fixed distance from the larger masses.
		The L2 point of the Earth-Sun system is home to the WMAP spacecraft and (perhaps by the year 2011) the James Webb Space Telescope.
		The L1 and L2 points are unstable on a time scale of approximately 23 days, which requires satellites parked at these positions to undergo regular course and attitude corrections.
	map.gsfc.nasa.gov /m_mm/ob_techorbit1.html (751 words)

	Lagrangian point - Wikipedia, the free encyclopedia
		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.
		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 Halo: CE and Halo 2, the Halo structures are in L1 Lagrange points between the Gas Giants (and a moon) Threshold and Substance, respectively.
	en.wikipedia.org /wiki/Lagrange_point (2936 words)

	Math Forum - Ask Dr. Math
		It is also true that given three points, (x1,y1), (x2,y2), and (x3,y3), with different x-coordinates, either the three points lie in a line, or else there is exactly one parabola (second degree polynomial y = ax^2 + bx + c) through the three points.
		Lagrange multipliers are used in "constrained optimization": you want to find the largest value of a function of several values, e.g.
		Lagrange was a very great and prolific mathematician, and he did a lot of work, so you see his name in many different areas of mathematics.
	mathforum.org /library/drmath/view/63984.html (1266 words)

	Astronomy Answers: Lagrange Points
		If two celestial bodies orbit around a common center of gravity, then there are five points that move with the bodies and in which the gravity of the two bodies and the centripetal force of the orbit of the point around the center of gravity are exactly balanced.
		For Lagrange points L₄ and L₅, the motion perpendicular to the orbital plane is an oscillation with a period equal to the period of the Kepler orbits of the two objects and the Lagrange point.
		The orbital periods near those Lagrange points are equal to 3.331 and 1.048 times the period of the system, i.e., that many sidereal months.
	www.astro.uu.nl /~strous/AA/en/reken/lagrange.html (1397 words)

	NASA - Home On Lagrange
		Lagrange believed that in a two-body system, such as the Earth and the Sun, there would be five points nearby where an object could be sent and remain in place.
		Lagrange point 2 is located the same distance from the Earth as SOHO, but in the opposite direction.
		Lagrange points 4 and 5 are positioned on the Earth's orbit, about 92 million miles in front of and behind the planet as it travels around the Sun.
	www.nasa.gov /missions/solarsystem/f-lagrange.html (550 words)

	Odyssey August 2001: The LaGrange Points and You
		Points 4 and 5, on the other hand, lie close to the Moon's orbital track about the Earth - at the point of an equilateral triangle, one side of which is the line connecting the Earth and Moon (not the barycenter and the Moon).
		The easiest of the points to understand, and perhaps the only one that most people would have thought of intuitively, would be the L1 point, which lies on the line between the two bodies.
		Although the L4 and L5 points are also synchronous to the surface of the Moon, they are so much further away than the L1 and L2 points that they are of less use as communications points (except that they see both near and far sides of the moon so they can fill out the coverage).
	www.oasis-nss.org /articles/2001/08/lagrange.html (1856 words)

	BBC - h2g2 - Lagrange Points - A947333
		Lagrange was one of the first people to study this problem, which is known as the 'three-body problem'.
		Lagrange discovered that there are five points called the Lagrange (or Lagrangian) points where the asteroid can share approximately the same orbit as the planet, with the same orbital period.
		Rumours of a supposed planet at the L3 point, where it would be invisible from the Earth, cannot be true, as the L3 point has a 'stability lifetime' of around 150 years.
	www.bbc.co.uk /dna/h2g2/alabaster/A947333 (1158 words)

	ESA - Space Science - What are Lagrange points?
		Lagrange points are locations in space where gravitational forces and the orbital motion of a body balance each other.
		At a certain point, the spacecraft’s orbital period equals that of Earth’s.
		Since the position of this Lagrange point lies behind the Sun, any objects which may be orbiting there cannot be seen from Earth.
	www.esa.int /esaSC/SEMM17XJD1E_index_0.html (820 words)

	Biography of Lagrange
		Lagrange studied independently and was appointed professor at the Royal Artillery School in Turin at the age of 19.
		Lagrange won an award for the best essay on the liberation of the moon with his analysis of a special case of the three-body problem, which uses the Lagrange points.
		Lagrange points are where gravitational forces and the speed of motion of a body are in balance.
	www.andrews.edu /~calkins/math/biograph/biolagra.htm (884 words)

	Artemis Project: The Lagrange Points
		Lagrange won the award with his analysis of a special case of the three-body problem.
		An object placed at these points, 60 degrees ahead of and behind the moon at the radius of its orbit, will remain at the same point with respect to the moon.
		Because of the inclination of the moon's orbit around the earth and the influence of the sun's gravity field, L4 and L5 are not stable points.
	www.asi.org /adb/m/03/12/lagrange-points.html (384 words)

	Lagrange Points of the Earth-Moon System (Site not responding. Last check: 2007-10-12)
		The Lagrange points L4 and L5 constitute stable equilibrium points, so that an object placed there would be in a stable orbit with respect to the Earth and Moon.
		The L5 point was the focus of a major proposal for a colony in "The High Frontier" by Gerard K. O'Neill and a major effort was made in the 1970's to work out the engineering details for creating such a colony.
		However, in practice these Lagrange points have proven to be very useful indeed since a spacecraft can be made to execute a small orbit about one of these Lagrange points with a very small expenditure of energy.
	hyperphysics.phy-astr.gsu.edu /Hbase/mechanics/lagpt.html (516 words)

	Lagrangian Points
		The sister-site From Stargazers to Starships discusses Lagrangian points in more detail than is done here, among other things deriving the distance of L1 (the derivation of L2 is almost identical) and also the equilibrium points L4 and L5.
		The L1 point is a very good position for monitoring the solar wind, which reaches it about one hour before reaching Earth.
		The L2 point has been chosen by NASA as the future site of a large infra-red observatory, the "Next Generation Space Telescope", renamed in honor of a late NASA director The James Webb Observatory.
	www-istp.gsfc.nasa.gov /Education/wlagran.html (901 words)

	Lagrange points (Henry Spencer)
		The three in-line points are either: L3 Earth L1 Moon L2 or L1 Earth L2 Moon L3 depending on whether you believe the space people or the astronomers.
		Newsgroups: alt.sci.planetary From: henry@spsystems.net (Henry Spencer) Subject: Re: Lagrange Points around Jupiter Date: Mon, 7 Feb 2000 20:19:52 GMT In article , Matthew F Funke wrote: >...a space station in synchronous orbit around Europa would be kind of nifty...
		In the long term, probably not; the L4 and L5 points are the only stable ones even in an ideal system, and around Jupiter, there are likely to be enough disturbances from the other Galilean satellites and Europa's slightly-elliptical orbit to mess them up in the long run.
	yarchive.net /space/orbits/lagrange_points.html (643 words)

	Lagrange Points, Schema-Root news (Site not responding. Last check: 2007-10-12)
		This time, it is considering sending a new spacecraft built from spare parts to the Lagrange point L1 - an area in space where the gravity of the sun and...
		This Lagrange point is a place in space where the Earth's and the sun's gravity roughly balance to provide a point of near-equilibrium where a spacecraft can...
		The instrument was to orbit the Sun at the Lagrange point, where the combined - and oppositely directed - gravitational forces of the sun and the Earth yield...
	schema-root.org /science/physics/orbits/lagrange_points (998 words)

	Lagrange points (Site not responding. Last check: 2007-10-12)
		The restricted coplanar three-body problem has five solutions called Lagrange Points if the mass of the largest body is at least 24 times greater than the mass of the second-largest body, and the mass of the third body is negligible compared to the other two.
		The Lagrange Points L1, L2, and L3 all lie on the line that passes through the Earth and Moon.
		The map shows these three points to be in "saddles" or mountain passes, so if a spacecraft runs out of station-keeping propellant and is perturbed away from them, it will drift to regions unknown.
	home.comcast.net /~rubyredinger/lagrange.html (410 words)

	Lagrange Interpolation
		Lagrange interpolation is a way to pass a polynomial of degree N-1 through N points.
		In the applet below you can modify each of the points (by dragging it to the desired position) and the number of points by clicking at the number shown in the lower left corner of the applet.
		Lagrange polynomials are the interpolating polynomials that equal zero in all given points, save one.
	www.cut-the-knot.org /Curriculum/Calculus/LagrangeInterpolation.shtml (179 words)

	The Lagrange Points (Site not responding. Last check: 2007-10-12)
		They are called Lagrange Points in honour of the French-Italian mathematician Joseph Lagrange, who discovered them while studying the three-body problem in which two of the masses are very much heavier than the third.
		These Lagrange Points occur as stationary solutions to the equations of motion when the relative positions of the three bodies are fixed.
		The stability of Lagrange Points L1 and L2 is an important consideration for some of the space missions.
	www.frontlineonnet.com /fl1818/18180800.htm (189 words)

	The Use of Lagrange Points in Lunar Exploration (Site not responding. Last check: 2007-10-12)
		All of the orbiting bodies in the solar system (and throughout the universe, for that matter) are surrounded by special points where the forces on an orbiting object are balanced.
		The most salient feature of trajectories exploiting Lagrange points is the efficiency of transfers made to, from, and between them.
		One of the first steps to exploiting Lagrange points is to chart out where these points lie in relation to the bodies in the solar system.
	ccar.colorado.edu /asen5050/projects/projects_2003/cain (3614 words)

	Gravity Simulations--Lagrange Points (Site not responding. Last check: 2007-10-12)
		Lagrange Points (L1-L5): A simulation of up to 7 gravitational masses: the sun, the earth, and 5 satelites located at the Lagrange points (L4, L5, L1, L2, and L3).
		In this case, all lagrange points are unstable.
		In this case, points L4 and L5 are stable whereas L1, L2, and L3 are unstable.
	www.princeton.edu /~rvdb/JAVA/astro/galaxy/Galaxy1.html (452 words)

	Gravitorium - Lagrange Points (Site not responding. Last check: 2007-10-12)
		These are points where the force of attraction of the Earth and the Moon 'balance out' in various ways.
		You'll see that over about a month, not all the points are stable in the long term.
		The L4 and L5 points, however, are stable in the same way that a marble is stable at the bottom of a bowl - push it away from the centre and it will come back.
	www.rightword.com.au /products/gravitorium/lagrange.html (97 words)

	Lagrange Points
		They were discovered by French mathematician Louis Lagrange in 1772 in his gravitational studies of the 3-body problem: how a third, small body would orbit around two orbiting large ones.
		A spacecraft at one of these points has to use frequent, small rocket firings or other means to remain in the area.
		An object at L4 or L5 is truly stable, like a ball in a bowl: when gently pushed away, it orbits the Lagrange point without drifting farther and farther, and without the need of frequent rocket firings.
	www.freemars.org /l5/aboutl5.html (447 words)

	Lagrange Points for Two Similar Masses
		Most treatments of Lagrange points assume two orbiting masses and a test point at one of the five Lagrange points.
		The Lagrange points are at the curved boundaries between fl and white.
		If we decompose the acceleration at such a point into two orthogonal components, one parallel to the line between the masses, then the acceleration perpendicular to that line is constant due symmetry and the fact that sum of those two masses is constant.
	www.cap-lore.com /MathPhys/LagrangePoints/Lagrange.html (772 words)

	Imperial College Astrophysics: Herschel Space Observatory
		The Lagrange Stationary Points were discovered by the Italian-born French scientist Joseph-Louis Lagrange in the 18th century.
		These points exist where the gravitational pull of the two bodies exactly equals the centripetal force required to maintain the orbit.
		L4/5 points are often home to asteroids, which are known as Trojans, because of the three large asteroids called Achilles, Agamemnon and Hector found at the Jupiter-Sun L4 & L5 points.
	astro.imperial.ac.uk /Research/Infrared/Herschel/sci_lagrange.shtml (419 words)

	Joseph Louis Lagrange - Wikipedia, the free encyclopedia
		Turin, baptised in the name of Giuseppe Lodovico Lagrangia) was an Italian mathematician and astronomer who made important contributions to all fields of analysis and number theory and to classical and celestial mechanics as arguably the greatest mathematician of the 18th century.
		Lagrange established the theory of differential equations, and provided many new solutions and theorems in number theory, including Wilson's theorem.
		The second volume contains a long paper embodying the results of several papers in the first volume on the theory and notation of the calculus of variations; and he illustrates its use by deducing the principle of least action, and by solutions of various problems in dynamics.
	en.wikipedia.org /wiki/Joseph_Louis_Lagrange (3545 words)

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