Lagrangian Points

The Lagrangian points are the five locations in an orbital system where the combined gravitational force of two large masses is exactly canceled out by the centrifugal force arising from the rotating reference frame.

At these five points, the net force on a third body (of negligible mass) is 0, allowing the third object to be completely stationary relative to the two other masses. That is, when placed at any of these points, the third body stays perfectly still in the rotating frame.

The first image shows the fields due to the first mass, the second mass, and the rotating reference frame. When added together, these fields generate the effective field shown in the second image. The five Lagrangian points are indicated with gray spheres.

The first three Lagrangian points (labeled L1, L2, and L3) lie in line with the two larger bodies and are considered metastable equilibria. L4 and L5 lie 60° ahead of and behind the second body in its orbit and are considered stable equilibria.

Lagrangian points offer unique advantages for space research, and the Lagrangian points of the Sun-Earth system are currently home to four different satellites.

Mathematica code posted here.

Additional sources not linked above: [1] [2] [3] [4] [5]