Lagrange Points (CR3BP)

Circular Restricted Three-Body Problem Simulator

Astronomical presets

Primary body mass

Mass of the more massive body (e.g. Sun, Earth)

1.000e-61.000e+6

Secondary body mass

Mass of the less massive body (e.g. Earth, Moon)

1.000e-6333100

Orbital distance

Distance between the two primary bodies

0.0011.000e+6

The CR3BP model assumes perfectly circular orbits (eccentricity e = 0). For elliptical orbits, the points would pulsate dynamically.

Orbital Map — Lagrange Points

Lagrange Point Positions

x [UA]

y [UA]

Stability

—

Routh criterion: μ_c ≈ 0.0385

Hill Radius (R_H)

—

Mass parameter (μ)

—

μ = M₂ / (M₁ + M₂)

Fundamentals & Explanation

What are Lagrange Points?

In celestial mechanics, Lagrange points (or libration points) are five special positions in a two-body orbital system where a third object of negligible mass can maintain relative equilibrium — orbiting with the system without falling toward either mass. They were identified by Joseph-Louis Lagrange in 1772 while solving the restricted three-body problem.

This simulator uses the CR3BP (Circular Restricted Three-Body Problem) model: circular orbits, constant masses, and a rotating reference frame where M₁ and M₂ remain stationary. This is the standard tool used by agencies like NASA and ESA to plan missions such as JWST and SOHO.

The mass parameter μ

The entire system is governed by a single dimensionless number, μ, representing the fraction of total mass contributed by the secondary body:

\mu = \frac{M_2}{M_1 + M_2}

The orbital distance is normalized to R = 1. In this frame, M₁ sits at x₁ = −μ and M₂ at x₂ = 1 − μ, with the barycenter at the origin. All results are then converted back to physical units (AU, km, etc.).

System	μ	What it means
Sun – Earth	3.003 × 10⁻⁶	Earth is 333,000× less massive than the Sun
Earth – Moon	0.01215	The Moon is ~1.2% of the Earth-Moon system mass
Sun – Jupiter	9.533 × 10⁻⁴	Jupiter holds 99.9% of all planetary mass

The five equilibrium points

Lagrange proved there are exactly five equilibrium solutions. Three are collinear (along the M₁–M₂ axis) and two are triangular (forming equilateral triangles with the two masses).

Collinear points: L1, L2, L3

Their positions come from solving the effective potential equilibrium equation, which has no closed-form algebraic solution. The simulator solves it numerically to high precision (tolerance 10⁻¹²) using the Newton-Raphson iterative method:

f(x) = x - \frac{(1-\mu)(x+\mu)}{|x+\mu|^3} - \frac{\mu(x-1+\mu)}{|x-1+\mu|^3} = 0

L1 — Between M₁ and M₂. The unstable equilibrium where the gravitational pull of both masses cancels out.
L2 — Beyond M₂ on the far side from M₁. Used by telescopes requiring permanent solar shielding.
L3 — On the opposite side of M₂'s orbit, slightly beyond M₁.

Triangular points: L4 and L5

Their coordinates are exact and require no numerical computation — the equilateral geometry fully determines them:

L_{4,5} = \left(\frac{1}{2} - \mu,\; \pm\frac{\sqrt{3}}{2}\right)

L4 is 60° ahead of M₂ in its orbit (the "Greek" point in the Sun-Jupiter system) and L5 is 60° behind (the "Trojan" point).

Stability

L1, L2, and L3 are always unstable: any perturbation causes an object to drift away exponentially. Spacecraft at these points require periodic thruster corrections (station-keeping).

L4 and L5 are conditionally stable thanks to the Coriolis force in the rotating frame. They are stable only when the secondary mass is small enough — Routh's criterion (1875):

\mu < \mu_c = \frac{1}{2}\left(1 - \frac{\sqrt{69}}{9}\right) \approx 0.03852

All three preset systems satisfy this criterion. Sun–Jupiter (μ ≈ 9.5×10⁻⁴) satisfies it with ample margin, which explains why over 12,000 confirmed Trojan asteroids orbit stably at its L4 and L5 points.

Hill Sphere

Defines the region where M₂ gravitationally dominates over M₁. Any satellite — natural or artificial — orbiting M₂ must remain within this sphere to avoid being captured by M₁:

R_H \approx R \cdot \sqrt[3]{\frac{\mu}{3}}

This is an approximation valid for small μ. The Moon orbits at ~384,400 km, well within Earth's Hill sphere (~1.5 million km). Jupiter has the largest Hill sphere in the solar system: ~742 million km.

Real-world applications

Sun-Earth L1: The SOHO solar observatory has been monitoring the solar wind in real time from this point since 1996.
Sun-Earth L2: The James Webb Space Telescope (JWST) has operated here since 2022, permanently shielded from solar heat.
Sun-Jupiter L4/L5: Over 12,000 confirmed Trojan and Greek asteroids share Jupiter's orbit at these points.
Sun-Earth L4/L5: NASA's STEREO-A/B mission used them to obtain stereoscopic views of the Sun from opposite sides.

Physical constants used

All simulator values are based on international standards:

Constant	Value	Source
M_☉ (Sun)	1.989 × 10³⁰ kg	IAU 2015 Res. B3
M_⊕ (Earth)	5.972 × 10²⁴ kg	NASA NSSDCA
M_☾ (Moon)	7.346 × 10²² kg	NASA NSSDCA
M_J (Jupiter)	1.898 × 10²⁷ kg	JPL SSD
1 AU	1.496 × 10¹¹ m	IAU 2012 Res. B2

Masses in kg are derived as GM/G using CODATA 2018 (G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²). The AU has been defined exactly as 149,597,870,700 m since the IAU 2012 resolution.

Model limitations

Assumes perfectly circular orbits (e = 0). Real orbits are elliptical: Jupiter has e = 0.049.
Ignores perturbations from third bodies (Saturn significantly perturbs the Sun-Jupiter system).
The third body has negligible mass and does not affect M₁ or M₂.
For elliptical orbits, the L4/L5 points oscillate dynamically — see the ER3BP model.