Lagrange point

In celestial mechanics, the Lagrange points (), also called the 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. Normally, the two massive bodies exert an unbalanced gravitational force at a point, altering the orbit of any other celestial body at that point. At the Lagrange points, the gravitational forces of the two large bodies and the centrifugal pseudo-force balance each other. 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 sunshield to protect the telescope from the light and heat of the Sun, Earth and Moon simultaneously with no need to rotate the sunshield. The Nancy Grace Roman Space Telescope is also scheduled to be launched to L2 no earlier than 30 August 2026. The L1 and L2 Lagrange points are located about 1,500,000 km (930,000 mi) 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 benefits 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.

History 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 (L4 and L5). 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.

Lagrange points

The five Lagrange points are labeled and defined as follows:

L1 point The L1 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 L1 point, the object's orbital period becomes exactly equal to Earth's orbital period. L1 is about 1.5 million kilometers, or 0.01 au, from Earth in the direction of the Sun.

L2 point The L2 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 L2. 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 L2 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.

L3 point The L3 point lies on the line defined by the two large masses, beyond the larger of the two. Within the Sun–Earth system, the L3 point exists on the opposite side of the Sun, a little outside Earth's orbit and slightly farther from the barycenter of the Earth–Sun system 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 L3 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. For the above argument, it is important to distinguish between the center of the Sun and the Earth-Sun barycenter. The distance between the L3 point and the center of the Sun is actually smaller than the distance between the center of the Earth and the center of the Sun.

L4 and L5 points

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