Lagrange Points of the Earth-Moon System

A mechanical system with three objects, say the Earth, Moon and Sun, constitutes a three-body problem. The three-body problem is famous in both mathematics and physics circles, and mathematicians in the 1950s finally managed an elegant proof that it is impossible to solve. However, approximate solutions can be very useful, particularly when the masses of the three objects differ greatly.

For the Sun-Earth-Moon system, the Sun's mass is so dominant that it can be treated as a fixed object and the Earth-Moon system treated as a two-body system from the point of view of a reference frame orbiting the Sun with that system. 18th century mathematicians Leonhard Euler and Joseph-Louis Lagrange discovered that there were five special points in this rotating reference frame where a gravitational equilibrium could be maintained. That is, an object placed at any one of these five points in the rotating frame would stay there, with the effective forces with respect to this frame canceling. Such an object would then orbit the Sun, maintaining the same relative position with respect to the Earth-Moon system. These five points were named Lagrange points and numbered from L1 to L5.

The Lagrange points L4 and L5 constitute stable equilibrium points, so that an object placed there would be in a stable orbit with respect to the Earth and Moon. With small departures from L4 or L5, there would be an effective restoring force to bring a satellite back to the stable point. 拉格朗日点L4和L5构成稳定的平衡点，因此将物体放置在那里，它相对于地球和月球将处于稳定的轨道中。从L4或L5小范围偏离时，会有一个有效的恢复力将卫星带回稳定点。

The L5 point was the focus of a major proposal for a colony in "The High Frontier" by Gerard K. O'Neill and a major effort was made in the 1970's to work out the engineering details for creating such a colony. There was an active "L5 Society" that promoted the ideas of O'Neill. The L4 and L5 points make equilateral triangles with the Earth and Moon.

The Lagrange points L1, L2 and L3 would not appear to be so useful because they are unstable equilibrium points. Like balancing a pencil on its point, keeping a satellite there is theoretically possible, but any perturbing influence will drive it out of equilibrium. However, in practice these Lagrange points have proven to be very useful indeed since a spacecraft can be made to execute a small orbit about one of these Lagrange points with a very small expenditure of energy. They have provided useful places to "park" a spacecraft for observations. These orbits around L1 and L2 are often called "halo orbits". L3 is on the opposite side of the Earth from the Moon, so is not so easy to use.

Three-Body Equipotential Surfaces

A mechanical system with three objects, say the Earth, Moon and Sun, constitutes a three-body problem. The three-body problem is famous in both mathematics and physics circles, and mathematicians in the 1950s finally managed an elegant proof that it is impossible to solve. However, approximate solutions can be very useful, particularly when the masses of the three objects differ greatly.

One of the contributions of Lagrange was to plot contours of equal gravitational potential energy for systems where the third mass was very small compared to the other two. Below is a sketch of such equipotential contours for a system like the Earth-Moon system. The equipotential contour that makes a figure-8 around both masses is important in assessing scenarios were one partner loses mass to the other. These equipotential loops form the basis for the concept of the Roche lobe.

Contours of Equal Gravitational Potential

One of Lagrange's observations from the potential contours was that there were five points at which the third body could be at equilibrium, points which are now referred to as Lagrange points.

The Lagrange Points for a system like the Earth-Moon system

The Lagrange points L₁, L₂, and L₃ are unstable equilibrium points. Like standing a pencil on its point, it is possible to achieve equilbrium, but any displacement away from that equilibrium would lead to forces that take it further away from equilibrium. Remarkably, the Lagrange points L₄ and L₅ are stable equilibrium points for the small mass in the three-body system and this three-body geometry could be maintained as M₂ orbited about M₁.

Joseph Louis Lagrange

Lagrange was and 18th century mathematician who tackled the famous "three-body problem" in the late 1700s. The problem cannot be solved exactly, but he found that in the case where the third body is very small compared to the other two, some useful approximate solutions could be found.