Lagrangian point
The Lagrange points, or libration points (from the Latin librare " fluctuate" and " keep the balance " ) are the equilibrium points of the restricted three-body problem. The general three-body problem in celestial mechanics is solved numerically. With the restriction that the third body has negligible mass, Leonhard Euler and Joseph -Louis Lagrange found five analytical solutions: In the points mentioned by Lagrange L1 to L5 third body (eg, research satellites ) resting force-free. These are the zeros of the gravitational field in that rotating reference system in which to rest, the two major celestial bodies (eg sun and planet ). That is, the gravitational forces of the two bodies on the test body to be just canceled by the centrifugal force ( due to the rotation of the reference frame ). In a non- rotating reference system, the Lagrange points are synchronized with the two celestial bodies on circular orbits around their common center of gravity.
- 2.1 Lagrange points L1 to L3
- 2.2 Lagrange points L4 and L5
- 2.3 Derivation of the Lagrange libration points by
Location of the Lagrange points
All five Lagrange points lie in the orbital plane of the two heavy body. Three are located on the line connecting the two bodies, the fourth and fifth form, with the two bodies in each case, the corners of an equilateral (up to relativistic corrections) triangle. For the above graphics in blue and yellow pair of celestial bodies is below the earth and sun as an example.
L1
The inner Lagrangian point L1 is located between the two observed bodies on the line joining them. A body that orbits the sun inside Earth's orbit would normally have a higher web speed as the earth. However, (the two forces act opposite ) is a result of the attraction of the earth, the attraction of the sun on the body weakened, which is sufficient synchronous rotational speed for the balance of power in L1. This point is located about 1.5 million kilometers from the earth towards the sun.
The inner Lagrangian point L1 in the Earth-Sun system serves as a " base " to observe the Sun. As early as 1978 there broke on the probe ISEE -3 to circle him until 1982. She was the first probe to orbit a Lagrangian point. Since 1995, the solar observation satellite orbiting SOHO him with a bunch of twelve measuring instruments. From the perspective of the co-moving with the earth's motion reference system SOHO orbits the Lagrange point once within six months at a distance of about 600,000 km, in order not to be disturbed in the communication of the sun and do not become too large the cost of path corrections. The Advanced Composition Explorer ( ACE) for the study of particles from all possible sources in the universe ( among others, the sun) orbiting the L1 since early 1998. The Genesis spacecraft with instruments to study the solar wind and to capture its particle was there from 2001-2004 positioned.
L2
L2 is behind the smaller of the two big bodies on the line joining them. Cause a similar effect as in the case of the L1. Normally, outside the Earth's orbit, the orbital period would be longer than that of the Earth. The additional attraction of the earth ( forces of the sun and earth on the body are rectified ) but results in a shorter orbital period, which in turn is equal to the orbital period of the Earth in L2. This point is located about 1.5 million kilometers from the Earth's orbit.
The L2 point of the Earth-Sun system offers advantages for space telescopes. As one body in the L2 maintains the orientation relative to the sun and the earth, where the attenuation of solar radiation is much simpler than on a orbit. The WMAP spacecraft ( Wilkinson Microwave Anisotropy Probe ), which examined the cosmic background radiation from the Big Bang was in an orbit around the L2 point of the Earth-Sun system. The ESA stationed in September 2009, the infrared telescope Herschel and Planck telescope to study the background radiation there. In January 2014, the astrometry spacecraft Gaia ESA also reached an orbit around the L2.
L3
L3 is ( from the smaller bodies of view ) behind the larger body on the line joining them just outside the orbit of the smaller of the two bodies. In the case of the Sun-Earth Lagrange point, the third is on the opposite side of us the sun, a little further away from the Sun than the Earth. In this point, the ( rectified ) combined forces of attraction of the earth and sun cause again a circulation time which is equal to the Earth.
The L3 point was in science fiction books and comics a popular place for a hypothetical (for us because the sun is not visible ) " Counter-Earth ". Since the mass of an earthlike " Counter-Earth " would be no longer be neglected in the system, it was here in a slightly different three-body problem and L3 would be for reasons of symmetry exactly on the orbit of the Earth. In principle, the definition would be " vanishingly small mass " not met.
L4 and L5
These two Lagrange points are located at the third point of an equilateral triangle whose base is the line connecting the two great bodies. L4 is in the direction of rotation of the smaller of the two big body in front of him, L5 behind him. The L4 and L5 points are therefore 60 ° before and 60 ° behind the orbiting around the central body body ( approximately ) in its orbit.
In contrast to L1, L2 and L3, L4 and L5 are stable, that is near her body can without orbit correction reside permanently. Therefore, at these points can be expected natural objects. Indeed located near L4 and L5 a variety of clouds of dust and small bodies, and in particular the orbits of the major planets. Asteroids or moons that are in the close vicinity of these points are called by astronomers also trojans or trojan moons. In orbit around the Earth L4 is the 2010 discovered 2010 TK7.
Examples of L4 and L5
Jupiter Trojans
In the vicinity of the points L4 and L5 of Jupiter to keep ( for the first time in so-called Jupiter ) Trojan on, a group of asteroids. They have the same orbital period as Jupiter, but he hurry on average 60 ° before and after and while orbiting the points L4 and L5 periodically in wide arcs. To date, over 3600 and 2000 Trojans are in L4 and L5 known and recognized in the asteroid lists of the Minor Planet Center, the total number is estimated to be several tens of thousands. The first Trojan, ( 588 ) Achilles, was discovered in 1906 by Max Wolf. By far the greatest Trojans should be in 1907 discovered ( 624) Hektor, an irregularly shaped asteroid of 370 km × 195 km stretch.
Trojans of other planets
In 1990, a Mars Trojan was discovered in the libration point L5 of Mars, the Eureka was baptized. Meanwhile, it has been discovered four more Mars Trojans, including one in the L4 point. The end of 2001 were also found 60 ° Neptunian a Trojan. Taken with the 4 -m telescope at Cerro Tololo, the 230 - km - body received the provisional name 2001 QR322, but was " saved " after a year. It orbits the sun - just like Neptune - in 166 Earth years. Then in 2010 was also the first time a Neptune Trojans in the Lagrangian L5, 60 ° before Neptune, proved 2008 LC18.
Erdbegleiter
For the earth was discovered by astronomers of Athabasca University in Canada in 2010 until now only known Trojan asteroid 2010 TK7. The discovery was published in July 2011. He moves around the Lagrangian point L4.
In the 1950s, clouds of dust were found in the L4 and L5 points of the Sun-Earth system. In the L4 and L5 points of the system Earth-Moon also very faint dust clouds were found that Kordylewskischen clouds that are pronounced even weaker than the faint Gegenschein. However, there are some asteroids (ie, a mean orbital period of one year) move on a so-called horseshoe orbit with the earth around the sun. The transition from a trojan to a horseshoe track is fluid: If the distance of a Trojan horse to the L4 or L5 points is too large, then it is even on the Earth's orbit beyond the Earth opposite point and then move towards the other Lagrange point. In particular, the path of the on 9 January 2002 with the help of automatic monitoring sky LINEAR ( Lincoln Near Earth Asteroid Research) discovered asteroid 2002 AA29 (an object with less than 100 m in diameter ) is remarkable. It orbits the sun on one of the earth's orbit very similar orbit, where he saw the co-moving with the earth's motion reference system along the Earth's orbit over the course of 95 years, an arc of nearly 360 ° describes, he swings back again in another 95 years. The shape of the arch is reminiscent of a horseshoe, hence the name horseshoe orbit. The most stable currently known horseshoe orbit of a Erdbegleiters has 2010 SO16.
Co-orbital moons ( Trojan moons )
Other Trojans are in the moon system of Saturn. Thus the Saturn 's moon Tethys, the small moons Telesto in his L4- L5 and Calypso in his point and the Saturn 's moon Dione has Helene moons in his L4- L5 and Polydeuces in his point.
Simplified model of Lagrange
Lagrange points L1 to L3
They can be derived by calculation, if one arranges the three masses in a line and for the rotation around the common center of gravity, the sum of the forces
- From the centrifugal effect of rotation around the common center of gravity and
- From the gravitational attraction to each other
Equal to 0 sets.
Lagrange points L4 and L5
If you can rotate with the same mass around each other on a common circular path three bodies are the center of mass and the Gravizentrum the arrangement in the center of the orbit. At a certain distance dependent on the angular velocity of the masses of each of the three bodies is free of force and the system is in equilibrium. The direct gravitational effect of the three bodies on each other is then compensated for when the three bodies are taking on the circle equidistant from each other. This can be the case only in an equilateral triangle. Where the angle of the individual pages is equal to each other, and is 60 °.
Altering the masses, then the common focal point around which rotates the system shifted to the heaviest mass back. The property that the triangle is equilateral, and consequently the angle of the masses to one another 60 °, but not affected thereby.
Due to the distortion of spacetime by the gravitational effect of the general theory of relativity space is distorted and the location of the libration points disturbed ( non-Euclidean geometry with a different from 180 ° angle sum in triangles ).
Derivation of Lagrange libration points by
In comparably large masses, three bodies move in a rotation system generally chaotic around each other. The situation is different when the mass of the three bodies, either equal to or one of the three body is very small compared to the other two. Lagrange studied the latter case. The former, however, is best suitable for entry into the understanding of the effect that in the latter case leads to equilibrium:
Lagrange went into its derivation assumes that one of the bodies is to have a vanishingly small mass, so that the center of mass is determined only by the two heavier bodies and lies between them; also assume that the two heavier significantly different mass, ie, essentially the medium-duty (planet ) orbiting the heaviest ( sun). In addition, assume that this center of gravity is, even if one of the two massive bodies of significantly severe ( sun ) is significantly shifted out of its center. This means, among other things, that the most massive body ( sun) must be clearly " thrown " around the common center of gravity around due to the interaction with the second- heaviest body (planet ). Right then and proportional to this displacement of the center of gravity, it happens that the two massive bodies at the center of gravity can act on the smallest body in the system under consideration past from opposite directions - similar to the initially observed rotation system with three equal masses, except that the angle under the " sun " by acting on the considered small body at the center of mass is extremely small (but still not equal to 0).
Well turns out that in the case of relatively large mass ratios, firstly back a stable path of three bodies comes about and secondly the structure independent of the specific mass ratio always remains that equilateral triangle (only that it is the a center of gravity close to the sun rather than right in the middle three body orbits ).
The model is not applicable readily to multi- planetary systems like our solar system. The deflection of the sun around its center point is determined at us mainly from Jupiter. This planet is it also has accumulated quite a few as a single mass particles around its Lagrange points L4 and L5 around. All other planetary direct the sun in relation to only a fraction of from, so that the movement of the sun from the point of view is overlaid with a chaotic function high amplitude relative to the Lagrange model. Through statistical effects (different rotational frequencies ) and linear superposition, the Lagrange points can act but also in the smaller planet.
Stability of the Lagrange points
The first three Lagrange points are stable with respect to deviations perpendicular to the line connecting the two large bodies, while deviations in the direction of the connecting line are unstable with respect. The easiest way to see the basis of the L1 point. On a test mass that is removed from L1 along one of the red arrows perpendicular from the connecting line, a force back to the equilibrium point (in the y direction: effective attractive force). Reason for this is that the horizontal force components of the two major body cancel each other out, while adding their vertical force components. (!, The blue arrows ), in contrast, an object of the L1 point to one of the other two body moves a little closer, then the gravitational force of the body, which he came closer, bigger: He leaves so the equilibrium point ( in the x- direction: repulsive effective force ).
The points L1 and L2 are therefore indeed unstable, but nevertheless useful, since small correction maneuvers of a satellite enough to keep him there. Without this, he would be removed from these points.
In contrast, possible to L4 and L5 to stable orbits, provided that the mass ratio of the two major body is greater than 24.96. If a befindlicher at these points lower body slightly deflected, so bring him the Coriolis force from the perspective of the reference system, in which lie the Lagrange points, in a kidney-shaped orbit around this point. So he now remains without correction maneuvers in the vicinity of these points.
Constancy of places
Despite the variability of the overall system and although they change the other details of the effective potential shown in the last picture, take the points L1 to L5 always the same location.