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The Lagrangian points (also Lagrange point, L-point, or libration point), are the five positions in interplanetary space 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 geosynchronous 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.

## The Lagrangian points

The five Lagrangian points are labeled and defined as follows:

### L1

The L1 point lies on the line defined by the two large masses M1 and M2, and between them.

Example: An object which orbits the Sun more closely than the Earth would normally have a shorter orbital period than the Earth, but that ignores the effect of the Earth's own gravitational pull. If the object is directly between the Earth and the Sun, then the effect of the Earth's gravity is to weaken the force pulling the object towards the Sun, and therefore increase the orbital period of the object. The closer to Earth the object is, the greater this effect is. At the L1 point, the orbital period of the object becomes exactly equal to the Earth's orbital period.

The Sun–Earth L1 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 the L1 and the Advanced Composition Explorer (ACE) is in a Lissajous orbit, also at the L1 point. The Earth–Moon L1 allows easy access to lunar and earth orbits with minimal delta-v, and would be ideal for a half-way manned space station intended to help transport cargo and personnel to the Moon and back.

### L2

The L2 point lies on the line defined by the two large masses, beyond the smaller of the two.

Example: On the side of the Earth away from the Sun, the orbital period of an object would normally be greater than that of the Earth. The extra pull of the Earth's gravity decreases the orbital period of the object, and at the L2 point that orbital period becomes equal to the Earth's.

Sun–Earth L2 is a good spot for space-based observatories. Because an object around L2 will maintain the same orientation with respect to the Sun and Earth, shielding and calibration are much simpler. The Wilkinson Microwave Anisotropy Probe is already in orbit around the Sun–Earth L2. The future Herschel Space Observatory as well as the proposed James Webb Space Telescope will be placed at the Sun–Earth L2. Earth–Moon L2 would be a good location for a communications satellite covering the Moon's far side.

If M2 is much smaller than M1, then L1 and L2 are at approximately equal distances r from M2, equal to the radius of the Hill sphere, given by:

$r \approx R \sqrt[3]{\frac{M_2}{3 M_1}}$

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 M2 in the absence of M1, is that of M2 around M1, divided by $\sqrt{3}\approx 1.73$.

Examples:

• Sun and Earth: 1,500,000 km from the Earth
• Earth and Moon: 61,500 km from the Moon

### L3

The L3 point lies on the line defined by the two large masses, beyond the larger of the two.

Example: L3 in the Sun–Earth system exists on the opposite side of the Sun, a little farther away from the Sun than the Earth is, where the combined pull of the Earth and Sun again causes the object to orbit with the same period as the Earth. The Sun–Earth L3 point was a popular place to put a "Counter-Earth" in pulp science fiction and comic books.

### L4 and L5

The L4 and L5 points lie at the third point of an equilateral triangle whose base is the line between the two masses, such that the point is ahead of, or behind, the smaller mass in its orbit around the larger mass.

The reason these points are in balance is that at L4 and L5, 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 barycentre of the system. The barycentre being both the centre of mass and centre of rotation of the system, this resultant force is exactly that required to keep a body at the Lagrange point in orbital equilibrium with the rest of the system.

L4 and L5 are sometimes called triangular Lagrange points or Trojan points. The name Trojan points comes from the Trojan asteroids at the Sun-Jupiter L4 and L5 points, which themselves are named after characters from Homer's Iliad (the legendary siege of Troy).

Examples: The Sun–Earth L4 and L5 points lie 60° ahead of and 60° behind the Earth in its orbit around the Sun. They contain interplanetary dust. The Sun–Jupiter L4 and L5 points are occupied by the Trojan asteroids.
The giant impact hypothesis suggests that an object named Theia formed at L4 or L5 and crashed into the Earth after its orbit destabilized, forming the moon.