Libration Point / Lagrangian Point

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

The restricted three-body problem has five equilibrium points — locations where a massless third body maintains a fixed position relative to the two primaries. Three collinear libration points L1, L2, L3 lie on the line joining the primaries (attributed to Euler); two triangular libration points L4, L5 sit at the vertices of equilateral triangles on the primaries' connecting line as base (attributed to Lagrange) (Meyer & Offin 2017, Murray & Dermott 1999). "Libration point" is the dynamics-community term; engineering literature favors "Lagrangian point."

Positions in the Earth–Moon system (Qiao 2025 normalized distances, converted at 384,400 km): L1 is about 58,000 km from the Moon / 326,000 km from Earth; L2 about 64,500 km from the Moon / 449,000 km from Earth; L3 on the Earth's far side from the Moon, about 382,000 km from Earth.

Stability

• Collinear points are unstable: the linearized eigenvalues include a real pair — the motion mode is "saddle × center × center" (Qiao 2025; Meyer & Offin 2017 prove the collinear points are saddle points in the plane). This is exactly why nearby orbits carry hyperbolic invariant manifolds — both the engineering value (low-energy transfer channels) and the cataloging challenge.
• Triangular points are conditionally stable: linearly stable when the mass ratio μ is below Routh's critical value μ₁ ≈ 0.0385; the Earth–Moon system with μ ≈ 0.0122 satisfies this (Murray & Dermott 1999, Meyer & Offin 2017). Strictly, a few resonant mass ratios are exceptions in the planar case, but Earth–Moon μ does not coincide with any of them.

Zero-Velocity Surfaces, Hill's Region, and Neck Passages

The Jacobi integral C constrains the positions accessible to motion; the set ℌ(C) = {x : W(x) ≥ C} is the Hill's region, its boundary the zero-velocity curve (Meyer & Offin 2017). The critical Jacobi values are ordered C_L1 > C_L2 > C_L3 > C_L4,5 (Murray & Dermott 1999): as C decreases from high values, the two zero-velocity surfaces enclosing Earth and Moon first meet at L1 (C = C₁); when C drops slightly below C₁ a narrow corridor opens at L1 — a necessary but not sufficient condition for a spacecraft to transit toward the Moon; whether a transfer succeeds also depends on the initial state (Peng & Zhang 2016).

L1/L2 gateway: a descriptive term for the passage region near L1/L2 through which trajectories travel between the lunar vicinity and the exterior region; used as an itinerary label in natural-trajectory clustering studies (Bosanac 2026).

The L1 Transfer Corridor

Transfers through the L1 corridor (Peng & Zhang 2016): the L1 stable manifold at C ≈ C₁ comes no closer to Earth than about 0.1 Earth–Moon distances, so a connecting arc is needed — first burn at the low-Earth parking orbit, second at the manifold insertion point; without a target lunar orbit the spacecraft is often automatically captured by the Moon upon entry (cheaper than Hohmann), while with a target orbit a third burn near perilune is generally applied. An example transfer takes about 363 days from the stable manifold to L1; the survey table lists ">4100 m/s, tens to hundreds of days." For crewed lunar missions the original text judges these "often do not save energy" and "time-consuming and costly, not advisable."

Applications

Although unstable, collinear-point orbits (halo, quasi-halo, Lissajous) can be maintained at low cost, and their invariant manifolds enable low-energy transfers (Vellutini & Avanzini 2014). Mission examples: Queqiao relay on an Earth–Moon L2 halo orbit supporting far-side communication; ARTEMIS in L1/L2 Lissajous orbits; CAPSTONE validating the L2 southern NRHO (Qiao 2025). For the triangular points see Tadpole Orbit and Horseshoe Orbit.

Terminology Variants

Related Concepts

• Libration Point Orbit (LPO)
• Halo Orbit
• Tadpole Orbit
• Horseshoe Orbit
• Weak Stability Boundary Transfer Trajectory

References

• Murray & Dermott, 1999, Solar System Dynamics
• Vellutini & Avanzini, 2014, Shape-based design of low-thrust trajectories to cislunar Lagrangian points
• Peng & Zhang, 2016, Survey of crewed lunar Earth–Moon transfer trajectory schemes
• Meyer & Offin, 2017, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem
• Qiao et al., 2025, Orbital parameter characterization and objects cataloging for Earth-Moon collinear libration points
• Bosanac, 2026, Clustering natural trajectories in the Earth-Moon circular restricted three-body problem