Lagrange Point Calculator

In the circular restricted three-body problem, where a negligible mass moves under two bodies in mutual circular orbit, five points exist at which gravitational and centrifugal effects balance in the co-rotating frame. Three of them, L1, L2, and L3, lie on the line joining the two masses, and their positions solve a quintic equation. For a small mass ratio mu, defined as the secondary mass over the total, L1 and L2 sit approximately one Hill radius, R times the cube root of mu divided by 3, on either side of the smaller body. In the Sun-Earth system mu is about 3.0 times 10 to the minus 6 and that distance works out near 1.5 million km. L4 and L5 occupy the far vertices of equilateral triangles with the two masses, 60 degrees ahead of and behind the secondary along its orbit. Stability differs sharply between the two families. L1, L2, and L3 are saddle points with exponential instability; the growth time at Sun-Earth L2 is roughly 23 days, so spacecraft there fly halo or Lissajous orbits and perform station-keeping burns every few weeks, typically costing only a few metres per second per year. L4 and L5 sit at maxima of the effective potential yet are stabilized by the Coriolis force provided the Routh criterion holds, mu below about 0.0385, a condition met by every Sun-planet pair and by Earth and the Moon. The model assumes exactly circular orbits and ignores solar radiation pressure and third-body perturbations, so operational halo orbits are designed by numerical integration rather than these closed forms.

Leonhard Euler identified the three collinear solutions in work published in 1765 while analysing the motion of the Moon under the combined attraction of Earth and Sun. Joseph-Louis Lagrange, then in Berlin, extended the analysis in his prize-winning Essai sur le probleme des trois corps of 1772, showing that two additional equilibrium configurations exist in which the three bodies stand at the corners of an equilateral triangle. Both results were treated as mathematical curiosities for well over a century, because no known object was observed to occupy such a position and no vehicle existed that could be placed at one. Confirmation arrived in February 1906, when Max Wolf at Heidelberg discovered asteroid 588 Achilles leading Jupiter by about 60 degrees, the first member of the Trojan population that now numbers well over ten thousand catalogued objects divided between the L4 and L5 camps of Jupiter. Practical use began in the space age. The International Sun-Earth Explorer 3, launched in 1978, was the first spacecraft placed in a halo orbit about Sun-Earth L1, using trajectory design developed by Robert Farquhar. SOHO took up station at L1 after its 1995 launch, and L2 became the preferred perch for infrared and cosmological observatories, hosting WMAP from 2001, Planck and Herschel from 2009, Gaia from 2013, and the James Webb Space Telescope from 2021.