The Lagrangian points (pronounced /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 stationary relative to 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 rotate with them. They are analogous to geostationary orbits in that they allow an object to be in a "fixed" position in space rather than an orbit in which its relative position changes continuously.
More technically and precisely, Lagrangian points are the stationary solutions of the circular restricted three-body problem. For example, given two massive bodies in circular orbits around their common center of mass, there are five positions in space where a third body, of comparatively negligible mass, could be placed which would then maintain its position relative to the two massive bodies. As seen in a rotating reference frame with the same period as the two co-orbiting bodies, the gravitational fields of two massive bodies combined with the centrifugal force are in balance at the Lagrangian points, allowing the third body to be stationary with respect to the first two bodies.
History and concepts
The three collinear Lagrange points were first discovered by Leonhard Euler around 1750.
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. A situation arises where you would have to calculate every gravitational interaction between every pair of objects at every point along its 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. With his new system of calculations, Lagrange’s work led him to 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). These points were named “Lagrangian points” in Lagrange's honor. It took over a hundred years before his mathematical theory was observed with the discovery of the Trojan asteroids at the Lagrange points of the Sun–Jupiter system in 1906.
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 Lagrangian points
The five Lagrangian points are labeled and defined as follows:
L 1
The L 1 point lies on the line defined by the two large masses M 1 and M 2 , and between them. It is the most intuitively understood of the Lagrangian points: the one where the gravitational attraction of M 2 partially cancels M 1 gravitational attraction.
The Sun–Earth L 1 is ideal for making observations of the Sun. Objects here are never shadowed by the Earth or the Moon. The Solar and Heliospheric Observatory (SOHO) is stationed in a Halo orbit at L 1 , and the Advanced Composition Explorer (ACE) is in a Lissajous orbit, also at the L 1 point. WIND is also at L1.
The Earth–Moon L 1 allows easy access to lunar and earth orbits with minimal change in velocity and would be ideal for a half-way manned space station intended to help transport cargo and personnel to the Moon and back.
L 2
The L 2 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 the smaller mass.
The Sun–Earth L 2 is a good spot for space-based observatories. Because an object around L 2 will maintain the same orientation 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 Wilkinson Microwave Anisotropy Probe, Herschel Space Observatory and Planck space observatory are already in orbit around the Sun–Earth L 2 . The Gaia probe and James Webb Space Telescope will be placed at the Sun–Earth L 2 . Earth–Moon L 2 would be a good location for a communications satellite covering the Moon's far side.
If the mass of the smaller object (M 2 ) is much smaller than the mass of the larger object (M 1 ) then L 1 and L 2 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 M 2 in the absence of M 1 , is that of M 2 around M 1 , divided by
.
Examples
- Sun and Earth: 1,500,000 km (930,000 mi) from the Earth
- Earth and Moon: 61,500 km (38,200 mi) from the Moon
L 3
The L 3 point lies on the line defined by the two large masses, beyond the larger of the two.
The Sun–Earth L 3 point was a popular place to put a "Counter-Earth" in pulp science fiction and comic books — though of course, once space based observation was possible via satellites and probes, it was shown to hold no such object. Actually, the Sun–Earth L 3 is highly unstable, because the gravitational forces of the other planets outweigh that of the Earth (Venus, for example, comes within 0.3 AU of L 3 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, such a Counter-Earth would frequently be visible from Earth.
L 4 and L 5
The L 4 and L 5 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 (L 5 ) or ahead of (L 4 ) the smaller mass with regard to its orbit around the larger mass.
The reason these points are in balance is that, at L 4 and L 5 , the distances to the two masses are equal. Accord
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