Enter two masses and separation distance to solve. View L1 to L5 coordinates and distances fast. Check iterations and stability status for your system.
Use these starting values to verify outputs and explore typical systems.
| System | Primary mass (kg) | Secondary mass (kg) | Separation (m) | Typical use |
|---|---|---|---|---|
| Sun-Earth | 1.98847e30 | 5.9722e24 | 1.495978707e11 | Deep-space observatories |
| Earth-Moon | 5.9722e24 | 7.342e22 | 3.844e8 | Transfer staging |
| Jupiter-Europa | 1.898e27 | 4.799e22 | 6.711e8 | Moon vicinity studies |
This tool uses the circular restricted three-body model in a rotating barycentric frame. Define the dimensionless mass ratio mu = m2 / (m1 + m2) and normalize distance by the separation a.
Place the primary at x1 = -mu and the secondary at x2 = 1 - mu. Collinear points (L1-L3) satisfy the 1D equilibrium condition:
The solver applies Newton's method to find each root. Triangular points are analytic: x = 1/2 - mu, y = ±√3/2 (all scaled by a).
Lagrange points are equilibrium locations in the rotating two-body system where a small object can remain in a fixed geometry relative to both masses. They enable low-stationkeeping observatories, transfer gateways, and long-duration monitoring. In practice, missions use these points as centers for halo or Lissajous orbits rather than perfect static positions.
This calculator uses the circular restricted three-body framework: two massive bodies move on circular orbits about their barycenter, while the test particle has negligible mass. The rotating reference frame removes the mean orbital motion, leaving a balance between gravity and effective inertial terms. Inputs are only masses and separation, making the results broadly reusable.
Results are computed in normalized units with separation a = 1. The key parameter is mu = m2/(m1+m2), which controls geometry and stability. The primary is placed at x = −mu and the secondary at x = 1−mu. Coordinates are then scaled back to meters, kilometers, or AU for reporting.
The collinear points lie on the line joining the masses, so the equilibrium condition reduces to a single nonlinear equation in x. Newton’s method iterates x_{n+1}=x_n − f(x_n)/f′(x_n) until the change is tiny. The table reports iteration counts and a convergence flag to help you detect challenging mass ratios or extreme unit choices.
The triangular points form equilateral triangles with the two primaries in the normalized model. Their locations are analytic: x = 1/2 − mu and y = ±√3/2. After scaling by the separation, you receive both coordinates and distances to each primary, which are useful when estimating communication ranges or thermal environments.
L4 and L5 are linearly stable only when mu is below the classic Routh limit (approximately 0.03852). For many planet–star systems, mu is far smaller, so the triangular points are stable; for comparable-mass binaries, they are not. The calculator reports a stability status based on this threshold.
For Sun–Earth inputs (1 AU separation), L1 and L2 typically fall about 1.5 million km from Earth along the Sun–Earth line, a scale used by deep-space telescopes. For Earth–Moon, the collinear points lie tens of thousands of kilometers from the Moon, supporting cislunar transfer designs. Use your exact masses and separation to refine these estimates.
Always verify units, then compare distances to the known separation to confirm plausibility. If convergence is flagged, rerun with only L1–L3 selected or adjust to more realistic starting values. Export CSV for spreadsheets and PDF for reports or mission notes, keeping a consistent output unit across runs.
It needs the two masses and their separation distance. You can enter common mass units and choose output units for coordinates and distances.
No. The underlying model assumes circular orbits. Eccentric systems shift the equilibrium geometry over the orbit, so you would need a time-dependent formulation.
L1, L2, and L3 satisfy a nonlinear equilibrium equation. Newton’s method efficiently finds the root, and the iteration count helps you judge numerical difficulty.
Triangular stability depends on the mass ratio. If mu exceeds about 0.03852, small perturbations grow, so L4 and L5 are not linearly stable.
Coordinates are reported in a rotating barycentric frame where the barycenter is at the origin. The primary and secondary lie on the x-axis.
In real missions, objects orbit around the point in halo or Lissajous trajectories. Stationkeeping is still required, especially near L1 and L2.
CSV is best for plotting and batch comparisons in spreadsheets. PDF is convenient for sharing a concise snapshot of your chosen system and units.
Accurate points support planning, insight, and safer trajectories today.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.