## What Makes Lagrange Points Special Locations In Space

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Although the L1, L2, and L3 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.

L2 is one of the so-called Lagrangian points, discovered by mathematician Joseph Louis Lagrange. Lagrangian points are locations in space where gravitational forces and the orbital motion of a body balance each other. Therefore, they can be used by spacecraft to 'hover'. L2 is located 1.5 million kilometres directly 'behind' the Earth as viewed from the Sun. It is about four times further away from the Earth than the Moon ever gets and orbits the Sun at the same rate as the Earth.

When the Chicago Society for Space Studies was started 43 years ago (then known as the Chicago Society for Space Settlement), there was much talk about using the Lagrange points for space development. Lagrange points are a family of locations in space where the tug of gravity from the Sun and the Earth balance out. Consequently, any spacecraft located at one of these points can remain in place for a long time. Lagrange 1 (L1), Lagrange 2 (L2) and Lagrange 3 (L3) are along a line contacting the two large masses: the Earth and the Sun. These three points are unstable versus the stable Lagrange 4 (L4) and Lagrange 5 (L5) points, which are the apexes of two equilateral triangles.

In 1772 Joseph-Louis Lagrange demonstrated that, for planets on circular orbits, there are five special locations where if you start small bodies in precisely those positions they will continue to orbit around the central object in a fixed pattern, maintaining their relative positions to both the planet and the Sun. These five points became known as the Langrange points.

These five special locations are known as the Lagrange Points, discovered by Leonhard Euler and Joseph-Louis Lagrange in the 18th century. Not only are they of special interest in celestial mechanics, they also have important applications for space missions and the placing of satellites. We even find asteroids caught around some of the Lagrange points: so called "Trojan asteroids".

Euler found three such locations, but Lagrange's analysis was more thorough and he discovered an additional two less obvious points. Not that Euler should feel cheated however, as he had already lost sight in both eyes and completed the entire solution in his head! But neither of them could have had any idea how important their work would become 200 years in the distant future, with ourspacecraft now routinely voyaging through the inky blackness of the solar system.

Figure 2 shows a map of the gravity field of the Sun-Earth restricted three body problem. The contours show that the steepest gradients surround the Earth and Sun, with the five Earth Lagrange Points located in equilibrium regions with relatively gentle gradient. L1-L3 are unstable saddle points, and spacecraft positioned here will always drift away from the equilibrium. L4 and L5 are stableequilibria, and objects can orbit here indefinitely. The blue arrows show that L4 and L5 are actually atop a potential hill - it is the additional effect of the "Coriolis force" that makes them stable.

Lagrange points are locations in space where gravitational forces and the orbital motion of a body balance each other. They were discovered by French mathematician Joseph Lagrange in 1772 in his gravitational studies of the 'Three body problem': how a third, small body would orbit around two orbiting large ones. There are five Lagrangian points in the Sun-Earth system