# Lagrange Points Simulator - Restricted Three-Body Problem, Zero-Velocity Curves

> Lagrange point simulator: find L1–L5 for Sun–Earth, Earth–Moon and Sun–Jupiter, map the effective potential and zero-velocity curves, launch test particles.

- URL: https://qilingtools.com/en/physics/lagrange-points
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Pick Sun–Earth, Earth–Moon, Sun–Jupiter and more (or set your own mass ratio μ). The tool solves the quintic equations with Newton's method to place L1, L2 and L3 exactly and lists every point's distance from both bodies and its Jacobi constant. The background is the effective-potential heat map with contour lines in the rotating frame. The red line is the zero-velocity curve for your chosen Jacobi constant, and the shaded area is forbidden. Click or drag on the canvas to launch test particles, which follow the full rotating-frame equations (Coriolis terms included) with an adaptive integrator. Built-in demos show tadpole and horseshoe orbits, L1/L2 periodic orbits and a Trojan swarm.

## How to use

1. Choose a system, or pick Custom μ and drag the mass-ratio slider
2. Read the coordinates, distances, Jacobi constants and linear stability of L1–L5 in the table
3. Drag the Jacobi constant slider or tap C_L1…C_L4 to see the zero-velocity curve open up at each critical value
4. Click the canvas to drop a particle at rest, or press and drag to give it a velocity, or run one of the demos

## Features

- Quintic equations solved numerically for L1/L2/L3, equilateral L4/L5, with real distances in km
- Effective-potential heat map, contours, critical contours and zero-velocity curve with forbidden-region shading
- Rotating-frame particle integration (Coriolis included) with adaptive Dormand–Prince steps and a Jacobi drift readout
- Demos: tadpole orbit, horseshoe orbit, L1/L2 Lyapunov orbits and a Trojan swarm
- Linear-stability eigenvalues and the Routh criterion μ < 0.03852
- Notes on real missions: SOHO, JWST, Queqiao and Lucy

## FAQ

### What are Lagrange points?

When two bodies orbit each other, there are five points in the frame rotating with them where gravity and the centrifugal effect balance exactly, so a small object placed there stays fixed relative to both bodies. L1, L2 and L3 lie on the line through the two bodies; L4 and L5 form equilateral triangles with them. The Sun–Earth L1 and L2 points are about 1.5 million km sunward and anti-sunward of Earth.

### Why are L1, L2 and L3 unstable while L4 and L5 can be stable?

The collinear points are saddle points of the effective potential energy. A nudge along the line grows, which is why SOHO and JWST need regular station-keeping burns. L4 and L5 are maxima of the effective potential energy, but the Coriolis force turns a drifting object back so it circles the point. That only works if the Routh criterion μ < 0.03852 holds, meaning the larger body must outweigh the smaller by about 25 times. Sun–Jupiter and Earth–Moon pass; Pluto–Charon does not.

### What are zero-velocity curves and the Jacobi constant for?

Energy isn't conserved in the rotating frame, but the Jacobi integral C = 2Ω − v² is. Since v² ≥ 0, a particle can only be where 2Ω ≥ C, and the boundary is the zero-velocity curve. Above C_L1 a particle around the small body is trapped in a bubble. At C_L1 a neck opens at L1 and material can flow between the bodies. Lower still, the gates at L2 and L3 open in turn. This is how mass transfer between binary stars works.

### What are tadpole and horseshoe orbits?

Displace a particle slightly from L4 and it librates slowly around the point; seen in the rotating frame its path looks like a tadpole. Displace it further and the path sweeps past L3 and encloses both L4 and L5 in a horseshoe. Saturn's moons Janus and Epimetheus swap places on horseshoe orbits, and some near-Earth asteroids follow horseshoe paths relative to Earth.

### Why is the halo orbit demo only planar?

A true halo orbit is three-dimensional and needs motion out of the orbital plane. This tool is planar, so it computes the matching planar Lyapunov orbit: differential correction finds the symmetric periodic solution that crosses the x-axis at right angles. Because L1 and L2 are unstable, rounding errors grow exponentially and the particle leaves after a few laps, which is why real spacecraft need station-keeping.

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