Lagrange Points: The Five Parking Spots of the Three-Body Problem

A problem with no closed-form solution

The general three-body problem — three masses interacting purely through gravity — has no closed-form analytical solution. Henri Poincaré showed in 1890 that its trajectories are chaotic: infinitesimally different starting conditions diverge into wildly different futures. The restricted three-body problem sidesteps this by assuming the third body is so light it doesn't perturb the two large ones. In 1772, Joseph-Louis Lagrange found that this restricted system has exactly five points where the combined gravitational pull of both masses and the centrifugal force of the rotating frame cancel out — a small object placed there stays fixed relative to both bodies, orbiting the system's centre of mass in perfect lockstep.

In the frame co-rotating with the two primaries, everything reduces to the gradient of a single effective potential:

Φ_eff(x,y) = −GM₁/r₁ − GM₂/r₂ − ½ω²(x²+y²)
The five Lagrange points are exactly where ∇Φ_eff = 0

Five points where forces balance

All five sit along or near the orbit of the two primaries — Sun and Earth, for a concrete example. L1 sits between the two bodies, about 1.5 million km sunward of Earth — an unobstructed view of the Sun, but unstable, home to SOHO and DSCOVR. L2 sits the same distance on the far side of Earth from the Sun, permanently shadowed — unstable too, but it's where JWST, Gaia, Planck and Herschel all park. L3 sits opposite Earth, on the far side of the Sun, permanently hidden and essentially unusable. L4 and L5 sit 60° ahead of and behind Earth respectively, forming equilateral triangles with the Sun and Earth — and, remarkably, these two are stable.

Why L4 and L5 are stable and L1–L3 are not

This looks backwards at first: L1–L3 are saddle points of Φ_eff, and a ball on a saddle rolls away under any nudge, which is exactly what happens — perturbations there grow exponentially, and spacecraft need periodic stationkeeping burns to stay put. L4 and L5, meanwhile, sit at local maxima of the effective potential, which sounds even less promising. But potential energy alone doesn't decide stability here — the Coriolis force in the rotating frame does. Linearising the equations of motion around L4/L5 gives a stability condition on the mass ratio:

μ = M₂/(M₁+M₂) < μ_c = (1 − √69/9) / 2 ≈ 0.03852

For Sun–Earth, μ ≈ 3×10⁻⁶; for Sun–Jupiter, μ ≈ 9.5×10⁻⁴; for Earth–Moon, μ ≈ 0.012 — all comfortably below the threshold, so all three systems have stable L4/L5 points. Objects that stray near them don't fall in or drift away; they settle into slow looping tadpole orbits, or the wider horseshoe orbits that arc all the way from L4 past L3 to L5 and back over centuries.

Real missions and the edge of the Hill sphere

JWST reached Sun–Earth L2 thirty days after its 2021 launch and now flies a huge halo orbit around it, roughly 800,000 km across, keeping the Sun, Earth and Moon all behind its tennis-court sunshield at once while staying clear of Earth's shadow. SOHO and DSCOVR sit at L1 for the opposite reason — an uninterrupted view of the Sun, with about an hour's warning before a solar storm reaches Earth. Jupiter's L4 and L5 points have captured roughly 12,000 known Trojan asteroids — primordial material NASA's Lucy mission is now visiting — and even Earth has one, the 300-metre asteroid 2010 TK₇, librating around L4.

It's not a coincidence that L1 and L2 sit near the edge of Earth's Hill sphere, the region r_H ≈ a(M₂/3M₁)^(1/3) within which Earth's gravity, not the Sun's, dominates — roughly 1.5 million km, almost exactly the L1/L2 distance. Beyond that radius, a third body belongs to the Sun, not to Earth.

Frequently asked questions

Try it live

Everything above runs in your browser — open Lagrange Points and place test masses at L1 through L5, then perturb them to see which equilibria pull back and which drift away. Nothing is installed, nothing is uploaded.