What are Lagrange points?

Lagrange points (L1–L5) are five locations in a two-body gravitational system where the combined gravitational forces and centrifugal force balance out. An object placed there stays roughly fixed relative to the two main bodies, making them ideal parking spots for spacecraft.

Which Lagrange points are stable?

L4 and L5 are conditionally stable when the primary is at least 24.96 times more massive than the secondary. L1, L2, and L3 are unstable — a spacecraft must use station-keeping thrusters (or a halo orbit) to stay near them.

How accurate are the L1 and L2 approximations?

L1 and L2 use the Hill-sphere approximation r ≈ R·(m/3M)^(1/3), which is accurate to better than 1% when the mass ratio m/M is small (e.g., Earth/Sun ≈ 3×10⁻⁶). For large mass ratios the approximation degrades — for those cases treat the result as an estimate.

Why is L3 on the far side of the primary?

L3 sits just beyond the primary on the opposite side of the orbit from the secondary. Its distance from the secondary is approximately 2R (the full orbital diameter). It is rarely used for missions because it is hard to communicate with from Earth.

L4 leads the secondary by 60° along its orbit, and L5 trails it by 60°. Both form equilateral triangles with the two primary bodies, so their distance from each body equals the orbital separation R. Jupiter's Trojan asteroids cluster at the Sun-Jupiter L4 and L5 points.

Lagrange Point Calculator

Enter the masses of two gravitating bodies and their separation to find the distances to all five Lagrange points.

System Parameters

Primary Mass (M)

Sun = 1 M☉ = 1.989×10³⁰ kg

Secondary Mass (m)

Earth = 1 M⊕ = 5.972×10²⁴ kg

Orbital Separation (R)

1 AU ≈ 1.496×10⁸ km (Sun–Earth distance)

Lagrange Point Distances

Enter parameters and press Calculate.

Summary

Lagrange points are five positions in a two-body orbital system where a small object can maintain a stable position relative to both massive bodies. L1, L2, and L3 lie on the line connecting the two bodies; L4 and L5 form equilateral triangles with them. This calculator solves for all five positions using the mass ratio and the orbital separation.

How it works

Enter the mass of the primary body (e.g., the Sun) in kilograms or solar masses.
Enter the mass of the secondary body (e.g., Earth) in kilograms or solar masses.
Enter the orbital separation (semi-major axis) between the two bodies in km or AU.
Select your preferred units for each input.
The calculator uses the Hill-sphere approximation for L1 and L2, a small-mass-ratio approximation for L3, and equilateral-triangle geometry for L4 and L5.
Results show each point's distance from the secondary body and its position along the orbital axis.

Use cases

Locate the Sun-Earth L2 point where the James Webb Space Telescope orbits.
Find the Sun-Earth L1 point used by solar-wind monitoring spacecraft.
Compute Trojan asteroid positions at L4 and L5 for any planet.
Teach orbital mechanics and the restricted three-body problem.
Plan mission trajectories to gravitational equilibrium points.
Compare Lagrange geometry across different planetary systems.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu