Lagrange points: the five equilibria and the forbidden regions

Positions of the five Lagrange points of the circular restricted problem, the Jacobi constants C1 to C4 and the linear stability of L4 and L5 (Routh criterion), from the two masses. For a chosen Jacobi constant, the zero-velocity curve separates accessible from forbidden regions.

In the frame that rotates with the pair, the five equilibria exist because of the centrifugal term (x² + y²)/2. L4 and L5 lie at distance 1 from each of the two bodies. C1 > C2 > C3 > C4 for every admissible μ.

Circular restricted problem

two massive bodies on a circular orbit around their barycentre, and a third of negligible mass that feels their gravity without perturbing their orbit.

Effective potential (Ω)

the sum of the gravitational potential of the two bodies and the centrifugal potential of the rotating frame.

Jacobi constant (C)

C = 2Ω − v², the single first integral of the restricted problem.

Zero-velocity curve

the boundary 2Ω = C, where the velocity of the third body is zero. Beyond it, v² would be negative.

Neck

a passage between two accessible regions, open when C is below the constant of the corresponding collinear point: below C1 between the two massive bodies, below C2 to the exterior beyond the secondary, below C3 on the far side of the primary. Below C4, the forbidden region has vanished.

Routh criterion

the condition 27 μ (1 − μ) < 1 for linear stability of L4 and L5. The threshold is μ = (9 − √69)/18 ≈ 0.0385209.

• Circular restricted problem: the two massive bodies follow a circle around their barycentre. On an eccentric orbit the Jacobi constant is no longer conserved.
• No trajectory is integrated: the zero-velocity curve bounds where the body can go, not where it goes.
• The stability of L4 and L5 is linear (Routh criterion), with no long-term analysis under perturbations.
• The collinear points are obtained by bisection, down to a bracket width of 10⁻¹⁵ separation. Over 1004 values of μ: 51 iterations at worst, residual |dΩ/dx| ≤ 8 × 10⁻¹⁵ at the roots.
• The zero-velocity curve is traced by marching squares on 241 × 161 nodes. Normal departure from the exact isoline: 0.43 drawing unit at worst over five settings, less than half a cell. This departure diverges as C tends to C4, where the forbidden region contracts onto the minimum of the potential.
• The Earth/Moon mass ratio and the lunar separation are prefilled examples; the calculation is dimensionless.

Halo and Lissajous orbits, captures and escapes are not computed. The lower bound μ = 10⁻⁷ is a display limit: below it, the L1 and L2 necks close onto the secondary within less than one screen point, and a local zoom would break that frame. At mid-domain, the forbidden band goes from 2.67 cells at μ = 10⁻⁵ to 0.58 cell at 10⁻⁷; below one cell, around 5 × 10⁻⁷, it is no longer drawn as a single stroke. The bisection also converges below this bound.

• m2 ≥ 0model · refusal

  The mass parameter μ = M2/(M1 + M2) must lie between 10⁻⁷ and 0.5. Upper bound: convention that the first body is the more massive. Lower bound: display limit of the figure.

  Instead · If M2 exceeds M1, swap the two masses. Below μ = 10⁻⁷, the positions can be computed by hand from the Hill-radius expansion, a (μ/3)^(1/3).

• 2.5 ≤ c ≤ 4.2model · refusal

  The Jacobi constant must stay in the useful domain of the pair entered, shown under the figure. The declared bounds are its envelope over all admissible μ; the actual domain, narrower, shrinks as μ decreases.

  Instead · Choose a value between the four constants shown under the figure.

Nothing attracts towards the five points: the gravity of the two bodies and the centrifugal force balance there. L1 (between the two bodies), L2 (beyond the secondary) and L3 (on the far side of the primary) are saddles of the effective potential, unstable whatever the masses; L4 and L5 are its minima, and their linear stability depends on the Coriolis force. These equilibria exist only in the rotating frame, where the two massive bodies are fixed. The map is therefore not a gravity map.

Mass ratio 81.3005691 (DE440 header constants), hence μ = 0.0121506; separation 384 399 km. L1 is 58 019 km from the Moon and L2 64 515 km, roughly one Hill radius away; L3 is 381 674 km from the Earth. C1 = 3.18834, C2 = 3.17216, C3 = 3.01215 and C4 = 2.98800. 27 μ (1 − μ) = 0.324 080 < 1: L4 and L5 are linearly stable.

1. Szebehely, V., Theory of Orbits: The Restricted Problem of Three Bodies, 1967.

   Location: Academic Press, New York: effective potential, Jacobi constant, zero-velocity curves, the five libration points and the Routh stability condition · accessed 2026-08-26

2. Park, R. S., Folkner, W. M., Williams, J. G. & Boggs, D. H., The JPL Planetary and Lunar Ephemerides DE440 and DE441, 2021.

   Location: AJ 161:105, DE440 header constants: Earth/Moon mass ratio 81.3005691 · accessed 2026-08-26 · DOI 10.3847/1538-3881/abd414

Provisional · Sources and assumptions are provided; the documentary review is incomplete.

Physical model. The result follows the significant figures of the least precise input.

revision 1 · reviewed on 26 August 2026

Revision log

1. revision 1 · 2026-08-26 · scope change

   Creation of the tool: Lagrange points of the circular restricted problem (L1 to L3 by bisection, L4 and L5 in closed form), Jacobi constants, zero-velocity curve and Routh criterion 27 μ (1 − μ) < 1.

   First version, with no earlier value. Check values: L1 at 58 018.99 km and L2 at 64 514.74 km from the Moon (Earth-Moon example, mass ratio 81.3005691, separation 384 399 km).