Lagrange points - 2D rotating-frame map

The rotating frame

In units where the primary separation is 1 and the angular rate is 1, a test mass moves as x″ − 2y′ = ∂Ω/∂x, y″ + 2x′ = ∂Ω/∂y with Ω = (x²+y²)/2 + (1−μ)/r₁ + μ/r₂ and μ = m₂/(m₁+m₂). The five equilibria are the zeros of ∇Ω: three on the axis (found by bisection) and two equilateral points at (1/2−μ, ±√3/2). The Jacobi constant C = 2Ω − v² is conserved and the region 2Ω < C is forbidden to the particle.

Stability

Linearising gives λ⁴ + (4 − Ω_xx − Ω_yy)λ² + (Ω_xxΩ_yy − Ω_xy²) = 0. L1 to L3 have a real positive root (a saddle, e-folding in a fraction of an orbit); L4 and L5 are stable when μ < (1 − √69/9)/2 ≈ 0.03852 (Routh), thanks to the Coriolis force. Trojan asteroids and the Earth-Moon dust clouds live there.

Honest simplifications: planar and circular (no orbital eccentricity, no inclination), point masses, no solar radiation pressure, no third perturber, no station-keeping manoeuvres. Real halo orbits around L1 and L2 are three-dimensional and are not drawn here. The 2D map uses only the physics of the CR3BP.

• Press Sun-Earth L1 escape: the cloud leaves along the saddle directions within a few orbits.
• Press Earth-Moon L4 cloud, then Sun-Jupiter L5 Trojans: the cloud keeps librating.
• Zoom with the view slider to see the potential contours around each point.