Lagrange point

Lagrange point, in astronomy, a point in space at which a small body, under the gravitational influence of two large ones, will remain approximately at rest relative to them. In any two-body system, there are five Lagrange points numbered L1 to L5. At these points, the gravitational pull exerted by the two bodies equals the centripetal force needed for an object to maintain a constant distance relative to them.

The existence of such points was deduced by the Italian French mathematician and astronomer Joseph-Louis Lagrange in 1772 as part of his investigation of celestial mechanics’ three-body problem, which attempts to describe the motion of three celestial bodies moving under no influence other than their mutual gravitational attraction. No general solution to the three-body problem exists, but it can be solved in some special cases, including the Lagrangian case, in which the three bodies form an equilateral triangle.

The Lagrange points L1, L2, and L3 lie along a line that passes through the two large bodies. L4 and L5 are each at the vertex of an equilateral triangle that has the large bodies at the other two vertices. L4 is the leading Lagrange point, and L5 is the trailing Lagrange point.

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L1, L2, and L3 are unstable. Spacecraft placed at those points, if disturbed, will eventually move away into their own orbit around the larger of the two bodies, unless they burn fuel to remain in that position. L4 and L5 are stable, as long as the mass ratio between the two masses is greater than 24.96, which is met in the Earth-Sun and Earth-Moon systems. Objects at L4 or L5 orbit around those points, rather than drift away.

The first known objects at Lagrange points were the Trojan asteroids at L4 and L5 in the Jupiter-Sun system. The first Trojan asteroid was discovered in 1906, and as of 2022 more than 12,000 are known. These asteroids with names are named for characters in Homer’s Iliad, the asteroids at L4 for Greek characters and those at L5 for Trojan characters; the exceptions, which were bestowed before the naming convention was established, are Hektor, the lone Trojan in the Greek camp, and Patroclus, the lone Greek in the Trojan camp. The Lucy spacecraft, launched in 2021, is scheduled to visit seven of Jupiter’s Trojan asteroids. Trojan asteroids have also been observed in the orbits of Earth, Mars, Uranus, and Neptune.

In the Earth-Sun system, L1 is between Earth and the Sun about 1.5 million km (900,000 miles) from Earth—about four times farther from Earth than the Moon. Spacecraft at L1 are ideally placed for studying the Sun (Solar and Heliospheric Observatory and Advanced Composition Explorer) or the sunlit side of Earth (Deep Space Climate Observatory). L2 is the same distance from Earth as L1 but is located on the opposite side of Earth from the Sun. L2 is an advantageous position for space observatories, such as Gaia and the James Webb Space Telescope, which can also use its sun shield to block the telescope from the heat and light of the Sun, Earth, and the Moon. L3 is on the opposite side of the Sun from Earth; because of its distant location, no spacecraft orbit here. L4 has both of Earth’s two Trojan asteroids.