Lagrange 5

The L4 and L5 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 (L5) or ahead of (L4) the smaller mass with regard to 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 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 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. (Indeed, the third body need not have negligible mass; the general triangular configuration was discovered by Lagrange in work on the 3-body problem.)

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). Asteroids at the L4 point, which leads Jupiter, are referred to as the 'Greek camp', while at the L5 point they are referred to as the 'Trojan camp'. These asteroids are (largely) named after characters from the respective sides of the Trojan War.


 * Examples:


 * The Sun–Earth L4 and L5 points lie 60° ahead of and 60° behind the Earth as it orbits the Sun. They contain interplanetary dust.
 * The Earth–Moon L4 and L5 points lie 60° ahead of and 60° behind the Moon as it orbits the Earth. They may contain interplanetary dust in what is called Kordylewski clouds.
 * The Sun–Jupiter L4 and L5 points are occupied by the Trojan asteroids.
 * Neptune has Trojan Kuiper Belt Objects at its L4 and L5 points.
 * Saturn's moon Tethys has two much smaller satellites at its L4 and L5 points named Telesto and Calypso, respectively.
 * Saturn's moon Dione has smaller moons Helene and Polydeuces at its L4 and L5 points, respectively.
 * 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.