AI authored to showcase ironpad capabilities.
In celestial mechanics, the Lagrange points are five positions in a two-body gravitational system where a small object can remain stationary relative to the two large bodies.
We work in a co-rotating frame that spins with the two primary bodies (M_1, M_2). In this frame, both primaries are stationary and the dynamics of a test particle are governed by an effective potential that combines gravity and centrifugal effects:
\Phi(x, y) = -\frac{1-\mu}{r_1} - \frac{\mu}{r_2} - \frac{1}{2}(x^2 + y^2)
where \mu = M_2 / (M_1 + M_2) is the mass ratio, and r_1, r_2 are distances to each primary. The full setup, including the one force this potential cannot capture, gets a proper treatment below the simulation.
| Point | Location | Stability |
|---|---|---|
| L1 | Between the two bodies | Unstable (saddle) |
| L2 | Beyond the smaller body | Unstable (saddle) |
| L3 | Beyond the larger body | Unstable (saddle) |
| L4 | 60° ahead in orbit | Stable (for \mu < 0.0385) |
| L5 | 60° behind in orbit | Stable (for \mu < 0.0385) |
L4 and L5 form equilateral triangles with the two primaries; these are the famous Trojan points where Jupiter's Trojan asteroids collect.
The white diamond shows a spacecraft in a Lissajous orbit around L2, the same type of orbit JWST uses. L2 is unstable, so the spacecraft doesn't sit on the point; it orbits around it in a quasi-periodic pattern. Real missions require periodic station-keeping burns to prevent escape.
Use the Frame slider to switch between the co-rotating view (where L-points are fixed) and the inertial view (where everything orbits)!
The general three-body problem has no closed-form solution; Poincaré settled that. The circular restricted three-body problem is the version that gives you real leverage: two massive primaries on circular orbits about their common barycenter, plus a third body so small it feels their gravity without perturbing them.
The standard nondimensionalization strips the problem to a single parameter. Set the separation, total mass, and orbital rate to 1; all that remains is the mass ratio \mu = M_2/(M_1+M_2). Earth-Moon is \mu \approx 0.01215. Sun-Earth is \mu \approx 3.0 \times 10^{-6}. One number describes the whole system; the simulation above only needed MU.
The rotating frame is the load-bearing trick. Spin your coordinates at the orbital rate and both primaries freeze: M_1 at (-\mu, 0), M_2 at (1-\mu, 0). The price is two fictitious forces. Centrifugal depends only on position, so it folds into the effective potential \Phi from the introduction. Coriolis depends on velocity, so it cannot be folded into any potential; that asymmetry becomes the entire story at L4 and L5.
The equations of motion in the rotating frame:
\ddot{x} - 2\dot{y} = -\frac{\partial \Phi}{\partial x}, \qquad \ddot{y} + 2\dot{x} = -\frac{\partial \Phi}{\partial y}
The 2\dot{y} and 2\dot{x} terms are Coriolis; they appear verbatim in the simulation's derivs function. The five Lagrange points are exactly the places where \nabla \Phi = 0: the forces cancel, and a particle at rest in the rotating frame stays at rest. L1, L2, and L3 sit on the x-axis, and solving for them means finding roots of a quintic with no closed form. So we bisect.
Look at the effective potential near L1, L2, or L3 and you find a saddle: downhill along the axis joining the primaries, uphill perpendicular to it. A marble placed there rolls away. The instability is not subtle, either: for Sun-Earth L1 and L2, a position error grows by a factor of e roughly every 23 days.
So why do we put flagship observatories there? Because unstable is not the same as expensive. A saddle also supports near-periodic halo and Lissajous orbits that loop around the point without ever sitting on it. A spacecraft rides one of these and only pays to cancel the small component of drift along the unstable direction. The arithmetic is startlingly cheap: JWST budgets roughly 2-4 m/s of delta-v per year for station-keeping, with correction burns about every 21 days. For a mental model, that is a car that stays on the road with one small nudge of the wheel every three weeks.
One more directional trick: JWST can only thrust away from the Sun (turning its cold side sunward would cook the optics), so its orbit is biased toward the near side of L2 and every burn fires in the one allowed direction. Choosing the sign of your instability is half of mission design.
The white diamond in the simulation does a crude version of this: drift past a threshold from L2 and it cancels radial velocity and kicks back inward. Real station-keeping is a scheduled optimization problem, but the physics being fought is identical.
Plot the effective potential and L4 and L5 are maxima: the tops of hills, not saddles. Every instinct from rolling-marble physics says these should be the most unstable places in the system. And yet Jupiter's L4 and L5 hold thousands of Trojan asteroids, four billion years and counting.
The resolution is the force we could not fold into the potential. Coriolis does the stabilizing. A particle that starts sliding off the L4 hill picks up velocity; the Coriolis force, always at right angles to the motion, bends the trajectory sideways; the particle curls around the summit instead of leaving it. Nothing holds the particle in place. It is perpetually deflected into loops around the peak, called tadpole orbits after their shape.
The deflection only wins if it has time to act, and that comes down to the mass ratio. Linearize the dynamics at L4 and the motion is stable precisely when 27\mu(1-\mu) < 1, the Gascheau-Routh criterion (Gascheau proved it in 1843). Solving the quadratic gives \mu_{\text{crit}} \approx 0.03852, which corresponds to a primary-to-secondary mass ratio of about 24.96. A heavier secondary than that, and the hilltop sheds particles faster than Coriolis can curl them back.
Almost every pair you care about clears the bar easily: Sun-Jupiter by about 40x, Earth-Moon by about 3.2x. The interesting failure is Pluto-Charon, at a mass ratio near 8: no stable tadpoles, no Trojans possible. The cell below does the arithmetic.
There is hardware parked at most of these points right now. A quick catalog.
Sun-Earth L1 sits 1.5 million km sunward, and its value is simple: the Sun never sets there. NASA/ESA's SOHO has watched the Sun from an L1 halo orbit continuously since 1996. DSCOVR and ACE sit there as upstream sentries: solar wind reaches L1 roughly 30-60 minutes before Earth, enough warning to safe a power grid.
Sun-Earth L2 is the same distance in the opposite direction, prime real estate for a different reason: the Sun, Earth, and Moon all crowd into one small patch of sky. The James Webb Space Telescope hides all three behind a single sunshield and lets its optics passively cool to about 40 K, which is the whole game for infrared astronomy. Gaia, Euclid, and Planck chose it for the same thermal quiet. The white diamond in the simulation shows this kind of Lissajous orbit.
The Sun-Earth L3 point (on the far side of the Sun) is a staple of sci-fi as the hiding place for a "Counter-Earth." In reality it is the least useful point of the five: unstable, permanently out of direct radio contact, and further perturbed by Venus and Jupiter. Nothing has ever been stationed there.
Jupiter's L4 and L5 points harbor over 12,000 known Trojan asteroids. NASA's Lucy mission is touring several of them because they are dynamically pristine, parked undisturbed since the solar system formed. The Sun-Earth L4/L5 regions have also been proposed as sites for future space colonies (the O'Neill cylinder concept).
In the co-rotating frame, the primaries and L-points are fixed; you see the dynamics relative to the system. In the inertial frame, everything orbits the barycenter; you see the true paths through space. Switch between them using the slider to build intuition for both perspectives!
The collinear points (L1, L2, L3) are all unstable; spacecraft must perform periodic thruster burns to remain nearby. They typically follow Lissajous or halo orbits around the Lagrange point rather than sitting exactly on it.
The triangular points (L4, L5) are linearly stable when the mass ratio satisfies \mu < \mu_{\text{crit}} \approx 0.0385 (the Gascheau/Routh criterion). This covers all planet-Sun pairs in our solar system.