Nudge a test object near each of the five Lagrange points and watch what happens: near L1, L2, or L3 it drifts away no matter what; near L4 or L5 it loops back - as long as the two bodies' mass ratio stays below a real threshold. Drag the mass-ratio slider past that threshold and even L4/L5 lose their grip.
Red markers (L1, L2, L3) always drift away when perturbed. Green markers (L4, L5) loop back and hold - unless the mass-ratio slider passes Routh's critical ratio (about 0.0385), at which point they turn red and drift away too.
Drag to orbit and scroll or pinch to zoom. Pick a real system with the preset buttons, drag the slider for any ratio in between, then press "Perturb all 5 points" and watch the trails.
Lagrange Point Stability (L1-L5) 3D Explorer
Three of the five Lagrange points always fail: nudge anything sitting at L1, L2, or L3 and it drifts away, regardless of the two bodies involved. The other two, L4 and L5, hold - but only because of the Coriolis force, and only below a real mass-ratio threshold this explorer lets you cross.
Drag the mass-ratio slider to any value between two named real systems, or press a preset button to jump straight to Sun-Earth, Sun-Jupiter, Earth-Moon, or Pluto-Charon. Then press "Perturb all 5 points" to nudge a small test object at each of the five equilibrium points at once and watch which trails hold and which escape.
- Watch all five Lagrange points recomputed live for the current mass ratio - L1, L2, L3 on the line joining the two bodies; L4 and L5 at the exact equilateral-triangle points, 60 degrees ahead and behind
- Perturb all five points at once with a real numerical integration (Runge-Kutta 4) of the rotating-frame equations of motion - not a canned animation
- Drag the mass-ratio slider (Sun-Earth to Pluto-Charon and beyond) and watch L4/L5 change color from stable green to unstable red at Routh's critical ratio
- Jump to four real named systems with one click: Sun-Earth, Sun-Jupiter, Earth-Moon, Pluto-Charon
- Read the live mass-ratio readout and a full figures table in the facts panel, including the Sun-Earth-scaled L1 distance in kilometers
- Drag to orbit, scroll or pinch to zoom
- Runs fully in the browser with the vendored three.js engine - no account, no upload
| System | Mass ratio (mu) | L4/L5 stability |
|---|---|---|
| Sun-Earth | about 3.0e-6 | stable (Trojan-capable) |
| Sun-Jupiter | about 9.5e-4 | stable - thousands of real Trojan asteroids since 1906 |
| Earth-Moon | about 0.0121 | stable in the idealized three-body sense |
| Routh's critical ratio | 0.0385 (about 24.96x mass difference) | the stability boundary itself |
| Pluto-Charon | about 0.105 to 0.117 (measurement range) | unstable - above the critical ratio |
Why L4 and L5 usually hold
The counterintuitive part is L4 and L5 themselves: they are not valleys in the gravity-plus-rotation potential, they are local hilltops. A ball on a real hilltop rolls off with the slightest push. What actually holds a nudged object near L4 or L5 is the Coriolis force - the same effect that curves a spinning top's fall into a slow, stable precession instead of a topple - and that curving effect is only strong enough to win below Routh's critical mass ratio. L1, L2, and L3 get no such help: they are saddle points of the same potential, so any small displacement runs downhill and never comes back, for any mass ratio at all.
Real-world proof: Trojan asteroids and station-keeping spacecraft
This is why Jupiter's L4 and L5 have hosted thousands of real Trojan asteroids since 588 Achilles was discovered there in 1906, while Sun-Earth L1 and L2 - despite being useful parking spots for real missions like SOHO, DSCOVR, and the James Webb Space Telescope - need periodic station-keeping thruster burns to stay put, because nothing holds a spacecraft there on its own.
For where the five Sun-Earth Lagrange points sit and which real spacecraft fly near them, see the Lagrange Points 3D Explorer - that page covers location, not stability. For a general three-body sandbox with equal masses and a figure-8 orbit, see the Three-Body Problem 3D Explorer - that page has no Lagrange points at all. This page's story is the mass-ratio-dependent stability behind all five points.
Everything renders on your device with WebGL. The 3D engine loads once (about 0.7 MB) and is cached.
This is an educational approximation of the circular restricted three-body problem: the two bodies are assumed on exactly circular orbits and the perturbed test object is assumed massless, so it never pulls back on either body. Within that named, standard approximation, the perturbation trails are a real RK4 integration of the rotating-frame equations of motion, and the L4/L5 triangle points are an exact closed-form result for any mass ratio - neither is faked for effect. Sizes and the separation distance are compressed for readability and are not to scale.
Frequently Asked Questions
Why are L1, L2, and L3 always unstable?
They are saddle points of the effective gravity-plus-rotation potential in the rotating frame - the potential decreases in one direction and increases in the other, like a mountain pass. Any small displacement runs downhill and keeps going, for any mass ratio at all.
Why are L4 and L5 stable if they are hilltops, not valleys?
L4 and L5 are genuinely local maxima of the same effective potential, so naive intuition says they should be unstable too. What actually holds a nudged object near them is the Coriolis force, which curves any velocity the object picks up back around the point - similar to how a spinning top's fall curves into a slow precession instead of a topple.
What is Routh's critical mass ratio?
It is the mass-ratio threshold, about 0.0385 (the larger body must be more than about 24.96 times the smaller body's mass), below which the Coriolis stabilization of L4 and L5 actually works. Above that ratio, L4 and L5 become unstable too, just like L1, L2, and L3.
Do real Trojan asteroids prove L4/L5 stability?
Yes. The Sun-Jupiter mass ratio (about 9.5e-4) is far below the critical ratio, and thousands of Trojan asteroids have been found orbiting near Jupiter's L4 and L5 since 588 Achilles was discovered there in 1906 - direct observational confirmation spanning more than a century.
Does the mass-ratio slider change real physics or just this scene?
It changes real physics. The scene recomputes the actual Lagrange point positions and re-integrates the actual rotating-frame equations of motion for whatever mass ratio the slider is set to - the same equations that describe the real Sun-Earth, Sun-Jupiter, Earth-Moon, or Pluto-Charon systems when the slider matches their real ratios.
Is this the same as the Lagrange Points or Three-Body Problem pages?
No. The Lagrange Points 3D Explorer shows where the five Sun-Earth points sit and which real spacecraft fly near them, with no stability treatment. The Three-Body Problem 3D Explorer is a general equal-mass figure-8 sandbox with no Lagrange points. This page's story is why three of the five points always fail and two of them hold, and exactly when even those two let go.
Is this a full physics simulator?
No. It is an educational approximation of the circular restricted three-body problem - both bodies are assumed on exactly circular orbits and the perturbed test object is assumed massless. Within that standard, named approximation, the integration itself is real, not scripted for effect.