EQUILIBRIUM
A FIELD GUIDE TO LAGRANGE POINTS
Ten chapters on the five points where gravity and rotation cancel out — worked with the actual formulas, not just the concept — and on where those points turn out to sit, from the spacecraft parked near Earth to the asteroid swarms trailing Jupiter. Scroll to begin.
BEGIN — CHAPTER 01 →Two bodies solve cleanly. Three don't.
Newton's gravity gives an exact, closed-form answer for two bodies orbiting each other: an ellipse, fully described by Kepler's laws. Add a third body, and in general there is no such formula — the system has to be simulated step by step, forever sensitive to its starting conditions.
In 1772, Joseph-Louis Lagrange found a special case that does solve cleanly: assume the third body is so small it doesn't affect the other two, which continue in ordinary circular orbits around their shared centre of mass. Under that restriction — the "restricted three-body problem" — he found exactly five points where the third body can sit motionless relative to the other two, forever.
Freezing the picture by spinning with it
The trick that makes the problem tractable: switch to a reference frame that rotates at the same rate the two large bodies orbit their common centre of mass. In that frame, both large bodies appear frozen in place, and a third, small object feels three apparent forces — the real gravity of each large body, plus a centrifugal force pushing it outward from the rotation axis.
Combine gravity and the centrifugal term into a single "effective potential." A Lagrange point is simply a place where that combined potential is flat — where its gradient is zero, so nothing pushes the object away in any direction, at least to first order.
Solve ∇Φₑff = 0 and exactly five points fall out
Three lie on the straight line through both large bodies — L1, L2, and L3. Two form equilateral triangles with the pair — L4 and L5, one leading and one trailing the smaller body's orbit.
Drag the mass ratio below, or jump to a real system, and watch where all five points land.
Between the two bodies, tipped toward the small one
L1 sits between the two masses. There, the smaller body's gravity works against the larger one's pull, so the point can orbit in sync with the pair at a slightly smaller radius than an unaided orbit around the large body alone would need.
For the Sun–Earth system, this formula predicts roughly 1.5 million km sunward of Earth — almost exactly where SOHO has watched the Sun continuously since 1995.
Same formula, the opposite side
L2 sits just beyond the smaller body, away from the larger one, at almost exactly the same distance as L1 — the leading-order formula is identical. Physically, the combined gravity of both bodies there is strong enough to hold an object in an orbit matching Earth's one-year period, even though that object sits farther out than Earth.
This is where Webb, Gaia, and Euclid all sit — permanently shadowed from the Sun by Earth, at a distance close enough for high-bandwidth communication but far enough for total thermal isolation.
On the far side of the Sun, almost exactly on Earth's orbit
L3 sits on the opposite side of the larger body from the smaller one — for Sun–Earth, permanently hidden behind the Sun. The correction term is tiny: for Earth's mass, it works out to only a few hundred kilometres farther from the Sun than Earth's own orbit.
Its permanent hiding place behind the Sun made it a favourite of science fiction — a "Counter-Earth" no one could ever see. No such planet exists; spacecraft observing from the sides have long since confirmed there's nothing there.
An exact result, for any mass ratio at all
Unlike L1, L2, and L3, the positions of L4 and L5 need no approximation at all — they sit at the two points that form an equilateral triangle with the two large bodies, exactly, regardless of the mass ratio between them.
Whether an object actually stays near that point, though, depends entirely on the mass ratio — below a sharp threshold, small perturbations curve back around the point rather than growing.
Sitting on a hilltop that curves you back in
L1, L2, and L3 are saddle points of the effective potential — stable in some directions, unstable in others. Any small drift eventually grows, which is why spacecraft parked there fly small looping paths and fire thrusters every few weeks to stay put.
L4 and L5 sit on local hilltops of that same potential — which sounds like it should be even less stable. But an object drifting away from a hilltop there picks up sideways motion, and the Coriolis force curves that motion back around the point rather than letting it escape — provided the mass ratio is small enough. It's one of the more counterintuitive results in orbital mechanics: a maximum of potential energy that is nonetheless dynamically stable.
FIG. 8.1 — SCHEMATIC EQUIPOTENTIAL CONTOURS (ILLUSTRATIVE, NOT COMPUTED)
Every orbiting pair has its own set of five
The formulas apply to any two-body system — not just the Sun and a planet.
From one 1978 comet chaser to a growing fleet
Scrub through the spacecraft that have called a Lagrange point home.