Точки Лагранжа: равновесия в системе двух тел

Equilibrium points near two orbiting bodies
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In celestial mechanics, the Lagrange points (/l//ə/'/ɡ//r//ɑː//n//dʒ/; also Lagrangian points or libration points) are points of equilibrium for small mass objects under the gravitational influence of two massive orbiting bodies. Mathematically, this involves the solution of the restricted three body problem. This can make Lagrange points an excellent location for satellites, as orbit corrections, and hence fuel requirements, needed to maintain the desired orbit are kept at a minimum. For any combination of two orbital bodies, there are five Lagrange points, L1 to L5, all in the orbital plane of the two large bodies. There are five Lagrange points for the Sun–Earth system, and five different Lagrange points for the Earth–Moon system. L1, L2, and L3 are on the line through the centers of the two large bodies, while L4 and L5 each act as the third vertex of an equilateral triangle formed with the centers of the two large bodies. When the mass ratio of the two bodies is large enough, the L4 and L5 points are stable points meaning that objects can orbit them, and that they have a tendency to pull objects into them. Several planets have trojan asteroids near their L4 and L5 points with respect to the Sun; Jupiter has more than one million of these trojans. Some Lagrange points are being used for space exploration. Two important Lagrange points in the Sun Earth system are L1, between the Sun and Earth, and L2, on the same line at the opposite side of the Earth; both are well outside the Moon's orbit. Currently, an artificial satellite called the Deep Space Climate Observatory (DSCOVR) is located at L1 to study solar wind coming toward Earth from the Sun and to monitor Earth's climate, by taking images and sending them back. The James Webb Space Telescope, a powerful infrared space observatory, is located at L2. This allows the satellite's large sunshield to protect the telescope from the light and heat of the Sun, Earth and Moon. The L1 and L2 Lagrange points are located about from earth. The European Space Agency's earlier Gaia telescope, and its newly launched Euclid, also occupy orbits around L2. Gaia keeps a tighter Lissajous orbit around L2, while Euclid follows a halo orbit similar to JWST. Each of the space observatories benefit from being far enough from Earth's shadow to utilize solar panels for power, from not needing much power or propellant for station keeping, from not being subjected to the Earth's magnetospheric effects, and from having direct line of sight to Earth for data transfer.

The three collinear Lagrange points (L1, L2, L3) were discovered by the Swiss mathematician Leonhard Euler around 1750, a decade before the Italian born Joseph Louis Lagrange discovered the remaining two. In 1772, Lagrange published an "Essay on the three body problem". In the first chapter he considered 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 circular orbits.

The point lies on the line defined between the two large masses M1 and M2. It is the point where the gravitational attraction of M2 and that of M1 combine to produce an equilibrium. An object that orbits the Sun more closely than Earth would typically have a shorter orbital period than Earth, but that ignores the effect of Earth's gravitational pull. If the object is directly between Earth and the Sun, then Earth's gravity counteracts some of the Sun's pull on the object, increasing the object's orbital period. The closer to Earth the object is, the greater this effect is. At the point, the object's orbital period becomes exactly equal to Earth's orbital period. is about 1.5 million kilometers, or 0.01 au, from Earth in the direction of the Sun.

The point lies on the line through the two large masses beyond the smaller of the two. Here, the combined gravitational forces of the two large masses balance the centrifugal force on a body at On the opposite side of Earth from the Sun, the orbital period of an object would normally be greater than Earth's. The extra pull of Earth's gravity decreases the object's orbital period, and at the point, that orbital period becomes equal to Earth's. Like L1, L2 is about 1.5 million kilometers or 0.01 au from Earth (away from the sun). An example of a spacecraft designed to operate near the Earth–Sun L2 is the James Webb Space Telescope. Earlier examples include the Wilkinson Microwave Anisotropy Probe and its successor, Planck.

The point lies on the line defined by the two large masses, beyond the larger of the two. Within the Sun–Earth system, the point exists on the opposite side of the Sun, a little outside Earth's orbit and slightly farther from the center of the Sun than Earth is. This placement occurs because the Sun is also affected by Earth's gravity and so orbits around the two bodies' barycenter, which is well inside the body of the Sun. An object at Earth's distance from the Sun would have an orbital period of one year if only the Sun's gravity is considered. But an object on the opposite side of the Sun from Earth and directly in line with both "feels" Earth's gravity adding slightly to the Sun's and therefore must orbit a little farther from the barycenter of Earth and Sun in order to have the same 1 year period. It is at the point that the combined pull of Earth and Sun causes the object to orbit with the same period as Earth, in effect orbiting an Earth+Sun mass with the Earth Sun barycenter at one focus of its orbit.

The and points lie at the third vertices 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 60° ahead of or behind the smaller mass with regard to its orbit around the larger mass.

The triangular points ( and ) are stable equilibria, provided that the ratio of is greater than 24.96. 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 corotating frame of reference). The points , , and are positions of unstable equilibrium. Any object orbiting at , , or will tend to fall out of orbit; it is therefore rare to find natural objects there, and spacecraft inhabiting these areas must employ a small but critical amount of station keeping in order to maintain their position.

Due to the natural stability of and , it is common for natural objects to be found orbiting in those Lagrange points of planetary systems. Objects that inhabit those points are generically referred to as 'trojans' or 'trojan asteroids'. The name derives from the names that were given to asteroids discovered orbiting at the Sun–Jupiter and points, which were taken from mythological characters appearing in Homer's Iliad, an epic poem set during the Trojan War. Asteroids at the point, ahead of Jupiter, are named after Greek characters in the Iliad and referred to as the "Greek camp". Those at the point are named after Trojan characters and referred to as the "Trojan camp". Both camps are considered to be types of trojan bodies. As the Sun and Jupiter are the two most massive objects in the Solar System, there are more known Sun–Jupiter trojans than for any other pair of bodies. However, smaller numbers of objects are known at the Lagrange points of other orbital systems:
The Sun–Earth and points contain interplanetary dust and at least two asteroids, 2010 TK and 2020 XL. The Earth–Moon and points contain concentrations of interplanetary dust, known as Kordylewski clouds. Stability at these specific points is greatly complicated by solar gravitational influence. The Sun–Neptune and points contain several dozen known objects, the Neptune trojans. Mars has four accepted Mars trojans: 5261 Eureka, 1999 UJ, 1998 VF, and 2007 NS. Saturn's moon Tethys has two smaller moons of Saturn in its and points, Telesto and Calypso. Another Saturn moon, Dione also has two Lagrange co orbitals, Helene at its point and Polydeuces at The moons wander azimuthally about the Lagrange points, with Polydeuces describing the largest deviations, moving up to 32° away from the Saturn–Dione point. One version of the giant impact hypothesis postulates that an object named Theia formed at the Sun–Earth or point and crashed into Earth after its orbit destabilized, forming the Moon. In binary stars, the Roche lobe has its apex located at ; if one of the stars expands past its Roche lobe, then it will lose matter to its companion star, known as Roche lobe overflow. Objects which are on horseshoe orbits are sometimes erroneously described as trojans, but do not occupy Lagrange points. Known objects on horseshoe orbits include 3753 Cruithne with Earth, and Saturn's moons Epimetheus and Janus.

Lagrange points are the constant pattern solutions of the restricted three body problem. For example, given two massive bodies in orbits around their common barycenter, 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. This occurs because the combined gravitational forces of the two massive bodies provide the exact centripetal force required to maintain the circular motion that matches their orbital motion. Alternatively, when seen in a rotating reference frame that matches the angular velocity of the two co orbiting bodies, at the Lagrange points the combined gravitational fields of two massive bodies balance the centrifugal pseudo force, allowing the smaller third body to remain stationary (in this frame) with respect to the first two.

The reason these points are in balance is that at and 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 three body system, this resultant force is exactly that required to keep the smaller body at the Lagrange point in orbital equilibrium with the other two larger bodies of the system (indeed, the third body needs to have negligible mass). The general triangular configuration was discovered by Lagrange working on the three body problem.

Although the , , and points are nominally unstable, there are quasi stable periodic orbits called halo orbits around these points in a three body system. A full n body dynamical system such as the Solar System does not contain these periodic orbits, but does contain quasi periodic (i. e. bounded but not precisely repeating) orbits following Lissajous curve trajectories. These quasi periodic Lissajous orbits are what most of Lagrangian point space missions have used until now. Although they are not perfectly stable, a modest effort of station keeping keeps a spacecraft in a desired Lissajous orbit for a long time. For Sun–Earth missions, it is preferable for the spacecraft to be in a large amplitude Lissajous orbit around than to stay at , because the line between Sun and Earth has increased solar interference on Earth–spacecraft communications. Similarly, a large amplitude Lissajous orbit around keeps a probe out of Earth's shadow and therefore ensures continuous illumination of its solar panels. The and points are stable provided that the mass of the primary body (e. g. the Earth) is at least 25 The Earth is over 81 times the mass of the Moon (the Moon is 1.23% of the mass of the Earth). Although the and points are found at the top of a "hill", as in the effective potential contour plot above, they are nonetheless stable. The reason for the stability is a second order effect: as a body moves away from the exact Lagrange position, Coriolis acceleration (which depends on the velocity of an orbiting object and cannot be modeled as a contour map) Because the source of stability is the Coriolis force, the resulting orbits can be stable, but generally are not planar, but "three dimensional": they lie on a warped surface intersecting the ecliptic plane. The kidney shaped orbits typically shown nested around and are the projections of the orbits on a plane (e. g. the ecliptic) and not the full 3 D orbits.

This table lists sample values of L1, L2, and L3 within the Solar System. Calculations assume the two bodies orbit in a perfect circle with separation equal to the semimajor axis and no other bodies are nearby. Distances are measured from the larger body's center of mass (but see barycenter especially in the case of Moon and Jupiter) with L3 showing a negative direction. The percentage columns show the distance from the orbit compared to the semimajor axis. E. g. for the Moon, L1 is 326400 from Earth's center, which is 84.9% of the Earth–Moon distance or 15.1% "in front of" (Earthwards from) the Moon; L2 is located 448900 from Earth's center, which is 116.8% of the Earth–Moon distance or 16.8% beyond the Moon; and L3 is located 381700 from Earth's center, which is 99.3% of the Earth–Moon distance or 0.7084% inside (Earthward) of the Moon's 'negative' position. +Lagrangian points in Solar System Body pair Semimajor axis, SMA (×109 m) L1 (×109 m) 1 − L1/SMA (%) L2 (×109 m) L2/SMA − 1 (%) L3 (×109 m) 1 + L3/SMA (%) Earth–Moon 0.3844 0.32639 15.09 0.4489 16.78 0.38168 0.7084 Sun–Mercury 57.909 57.689 0.3806 58.13 0.3815 57.909 0.000009683 Sun–Venus 108.21 107.2 0.9315 109.22 0.9373 108.21 0.0001428 Sun–Earth 149.598 148.11 0.997 151.1 1.004 149.6 0.0001752 Sun–Mars 227.94 226.86 0.4748 229.03 0.4763 227.94 0.00001882 Sun–Jupiter 778.34 726.45 6.667 832.65 6.978 777.91 0.05563 Sun–Saturn 1426.7 1362.5 4.496 1492.8 4.635 1426.4 0.01667 Sun–Uranus 2870.7 2801.1 2.421 2941.3 2.461 2870.6 0.002546 Sun–Neptune 4498.4 4383.4 2.557 4615.4 2.602 4498.3 0.003004

Sun–Earth is suited for making observations of the Sun–Earth system. Objects here are never shadowed by Earth or the Moon and, if observing Earth, always view the sunlit hemisphere. The first mission of this type was the 1978 International Sun Earth Explorer 3 (ISEE 3) mission used as an interplanetary early warning storm monitor for solar disturbances. Since June 2015, DSCOVR has orbited the L1 point. Conversely, it is also useful for space based solar telescopes, because it provides an uninterrupted view of the Sun and any space weather (including the solar wind and coronal mass ejections) reaches L1 up to an hour before Earth. Solar and heliospheric missions currently located around L1 include the Solar and Heliospheric Observatory, Wind, Aditya L1 Mission and the Advanced Composition Explorer. Planned missions include the Interstellar Mapping and Acceleration Probe(IMAP) and the NEO Surveyor. Sun–Earth is a good spot for space based observatories. Because an object around 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 at L2. Spacecraft generally orbit around L2, avoiding partial eclipses of the Sun to maintain a constant temperature. From locations near L2, the Sun, Earth and Moon are relatively close together in the sky; this means that a large sunshade with the telescope on the dark side can allow the telescope to cool passively to around 50 K – this is especially helpful for infrared astronomy and observations of the cosmic microwave background. The James Webb Space Telescope was positioned in a halo orbit about L2 on January 24, 2022. Sun–Earth and are saddle points and exponentially unstable with time constant of roughly 23 days. Satellites at these points will wander off in a few months unless course corrections are made.

Earth–Moon allows comparatively easy access to Lunar and Earth orbits with minimal change in velocity and this has as an advantage to position a habitable space station intended to help transport cargo and personnel to the Moon and back. The SMART 1 Mission passed through the L1 Lagrangian Point on 11 November 2004 and passed into the area dominated by the Moon's gravitational influence. Earth–Moon has been used for a communications satellite covering the Moon's far side, for example, Queqiao, launched in 2018, and would be "an ideal location" for a propellant depot as part of the proposed depot based space transportation architecture. Earth–Moon and are the locations for the Kordylewski dust clouds. The L5 Society's name comes from the L4 and L5 Lagrangian points in the Earth–Moon system proposed as locations for their huge rotating space habitats. Both positions are also proposed for communication satellites covering the Moon alike communication satellites in geosynchronous orbit cover the Earth.

Scientists at the B612 Foundation were planning to use Venus's L3 point to position their planned Sentinel telescope, which aimed to look back towards Earth's orbit and compile a catalogue of near Earth asteroids.

In 2017, the idea of positioning a magnetic dipole shield at the Sun–Mars point for use as an artificial magnetosphere for Mars was discussed at a NASA conference. The idea is that this would protect the planet's atmosphere from the Sun's radiation and solar winds.