# Lagrange

The restricted coplanar three-body problem has five solutions called Lagrange Points if the mass of the largest body is at least 24 times greater than the mass of the second-largest body, and the mass of the third body is negligible compared to the other two. This works in the case of satellites in the Earth-Moon system, because the Earth is 81.3 times more massive than the Moon, and satellites are but a flea compared to both of them.

To simplify the math, the 384,400 kilometers between the centers of the Earth and Moon are defined as 1.0 Distance Unit (DU). So the positions of L4 and L5 are simply 1.0 DU from both the Earth and the Moon, forming two equilateral triangles, as shown here in this contour map which combines the gravity of the two bodies plus the centrifugal force developed by the monthly rotation of the system.

As the map indicates, L4 and L5 are like bean-shaped valleys. If a satellite is parked there, the sun will perturb it into a bean-shaped "halo orbit" that circles close around the valley floor, and it will stay there forever. These are good places to park multi-trillion dollar artificial space colonies without losing them. Hundreds of asteroids have been found to exist in the Sun-Jupiter L4 and L5 points.

The Lagrange Points L1, L2, and L3 all lie on the line that passes through the Earth and Moon. From L3, the Earth will always block the view of the Moon, so the joke goes that L3 is a great place to build an orbiting hospital to treat people prone to becoming werewolves. L2 is behind the Moon, so it is a great place to build a radio telescope immune to any interference from the noisy radio chatter of the Earth. L1 is between the Earth and Moon, so it is a great place to build a communications relay. The map shows these three points to be in "saddles" or mountain passes, so if a spacecraft runs out of station-keeping propellant and is perturbed away from them, it will drift to regions unknown.

To find the approximate location of these points we create a chart with the origin (0,0) at the center of gravity between the Earth and Moon, and we define a value μ to be the mass of the Moon relative to the mass of the whole system of the Earth plus Moon.

The position of the Earth is at - μ DU, which means the origin point (0,0) is about 1,000 miles below the surface of the Earth. The position of the Moon is at 1 - μ DU.

The position of L3 is -1 + (7/12) μ + (1127/20736) μ^3 + (7889/248832) μ ^4 = -0.992912 DU

To find the next two points we define a new value z = (μ/3)^1/3

The position of L2 is 1 - μ + z + (1/3)z^2 - (1/9)z^3 + (50/81)z^4

= 1.155669 DU

The position of L1 is 1 - μ - z + (1/3)z^2 + (1/9)z^3 - (58/81)z^4

= 0.836005 DU