Lagrange points

Draft

The general three-body problem has no closed-form solution. One restricted case does have exact equilibrium solutions, and five of them turn out to be extraordinarily useful places to put a spacecraft.

The circular restricted three-body problem

Assume two massive bodies on circular orbits about their common barycentre, and a third body of negligible mass that responds to both but perturbs neither. Work in a frame rotating with the primaries at their orbital rate $n$ , with distances normalised so the primary separation is 1 and $\mu = m_2/(m_1 + m_2)$ .

In that rotating frame the equations of motion pick up centrifugal and Coriolis terms:

\ddot{x} - 2\dot{y} = \frac{\partial U}{\partial x},\qquad \ddot{y} + 2\dot{x} = \frac{\partial U}{\partial y},\qquad \ddot{z} = \frac{\partial U}{\partial z}

with the effective potential

U = \frac{x^2+y^2}{2} + \frac{1-\mu}{r_1} + \frac{\mu}{r_2}

The first term is centrifugal, the other two gravitational. The Lagrange points are the stationary points of $U$ : the places where gravity and centrifugal effects balance exactly, so a body placed there at rest in the rotating frame stays there.

Coriolis does no work and so never appears in $U$ , but it is the term that makes $L_4$ and $L_5$ stable. Keep it in mind; it is the whole story below.

The Jacobi constant

The system is not conservative in the inertial sense, but the rotating frame admits one integral of the motion:

Jacobi constant

C_J = 2U - v^2 = x^2 + y^2 + \frac{2(1-\mu)}{r_1} + \frac{2\mu}{r_2} - v^2

Since $v^2 \ge 0$ , the surface $C_J = 2U$ bounds where a spacecraft of that energy can go. These zero-velocity surfaces open and close as $C_J$ decreases, and the order in which they open at $L_1$ , then $L_2$ , then $L_3$ defines the energy thresholds for transfer between regions. Low-energy transfer design, including the ballistic capture trajectories used by several lunar missions, is built on exactly this structure.

Where the five points are

The three collinear points lie on the line through the primaries. Their positions satisfy a quintic that has no useful closed form, but a first-order expansion in the mass parameter gives

r_{L1,L2} \approx R\left(\frac{\mu}{3}\right)^{1/3}

for the distance from the smaller primary, where $R$ is the primary separation.

The two triangular points are exact and require no approximation at all: $L_4$ and $L_5$ sit at the third vertex of an equilateral triangle with the two primaries, $L_4$ leading the secondary by 60° and $L_5$ trailing it by 60°.

Point	Location	Sun-Earth	Earth-Moon
$L_1$	Between the primaries	1.48 × 10⁶ km sunward of Earth	~58 000 km from the Moon, Earth side
$L_2$	Beyond the secondary	1.50 × 10⁶ km anti-sunward of Earth	~64 500 km beyond the Moon
$L_3$	Beyond the primary, opposite the secondary	~1 AU, on the far side of the Sun	~381 700 km, opposite the Moon
$L_4$	60° ahead of the secondary	On Earth’s orbit, leading	Equilateral, leading
$L_5$	60° behind the secondary	On Earth’s orbit, trailing	Equilateral, trailing

For Sun-Earth, $\mu = 3.003 \times 10^{-6}$ and the approximation gives $1.50 \times 10^6$ km, which matches the true $L_2$ distance closely and overestimates $L_1$ by about 1%. The asymmetry is real: $L_1$ is genuinely nearer than $L_2$ , because the solar gravity gradient does not act symmetrically about Earth.

For Earth-Moon the mass ratio is far larger ( $\mu = 0.01215$ ) and the first-order expansion is poor. It gives 61 500 km against true values of roughly 58 000 and 64 500 km. Use the quintic, not the expansion, whenever $\mu$ exceeds about $10^{-3}$ .

Stability

Linearising about each point separates the collinear and triangular cases completely.

The collinear points are saddle points of $U$ and are linearly unstable. The instability is exponential with a characteristic e-folding time of roughly 23 days for Sun-Earth $L_2$ . A spacecraft left alone there drifts away, slowly at first and then not slowly.

That instability is a feature rather than a defect. Because the unstable manifold is a well-defined direction, corrections can be applied along it and they are tiny: Sun-Earth $L_2$ station keeping costs on the order of a few metres per second per year. The same manifold structure allows a departure from the region for almost no propellant, which is what low-energy transfer design exploits.

The triangular points are stable, but only when the primaries are sufficiently unequal. The linear analysis gives the Gascheau condition:

Triangular point stability criterion

\frac{m_1}{m_2} > \frac{25 + \sqrt{621}}{2} \approx 24.96

The stabilising agent is the Coriolis force. $L_4$ and $L_5$ are actually local maxima of the effective potential, so a body displaced from one begins to slide away, and Coriolis then curves that motion into a closed path around the point. Nothing about the potential alone would predict stability.

System	Mass ratio	$L_4$ / $L_5$ stable	Observed
Sun-Jupiter	1047	Yes	Thousands of Trojan asteroids
Sun-Earth	333 000	Yes	Asteroid 2010 TK7 and dust concentrations
Sun-Neptune	19 400	Yes	Neptune Trojans
Earth-Moon	81.3	Yes	Kordylewski dust clouds, disputed for decades
Pluto-Charon	8.1	No	No Trojans, as predicted

Real systems add solar perturbations, eccentricity and radiation pressure, so “stable” in the linear CR3BP sense does not guarantee indefinite residence for a spacecraft. It does mean station keeping is cheap or unnecessary over mission timescales.

Orbit families

Spacecraft are almost never placed exactly at a collinear point. They are flown in periodic or quasi-periodic orbits around it, for three reasons: the point itself is unstable, a spacecraft exactly at Sun-Earth $L_2$ sits in Earth’s shadow, and a spacecraft exactly at Sun-Earth $L_1$ sits in front of the solar disc as seen from Earth, so its downlink competes with solar radio noise.

Lyapunov orbit: Planar periodic orbit in the plane of the primaries. The simplest family.
Halo orbit: Three-dimensional periodic orbit, bifurcating from the Lyapunov family above a threshold amplitude. Used where a fixed out-of-plane excursion is required.
Lissajous orbit: Quasi-periodic: in-plane and out-of-plane frequencies are incommensurate, so the path never closes. Cheaper to reach than a halo, at the cost of a drifting geometry.
Near-rectilinear halo orbit: A halo family member with very close periapsis passage to the secondary. Nearly stable, and the orbit chosen for the lunar Gateway.

Missions

Mission	Point	Orbit	Why there
SOHO	Sun-Earth $L_1$	Halo	Uninterrupted view of the Sun, upstream of Earth’s magnetosphere
ACE, DSCOVR, Wind	Sun-Earth $L_1$	Halo / Lissajous	Solar wind measured about an hour before it reaches Earth
JWST	Sun-Earth $L_2$	Halo	Sun, Earth and Moon all on one side, so a single sunshade gives passive cryogenic cooling
Gaia	Sun-Earth $L_2$	Lissajous	Thermally quiet, and a stable environment for microarcsecond astrometry
Euclid	Sun-Earth $L_2$	Halo	Same thermal and stray-light argument
Planck, Herschel	Sun-Earth $L_2$	Lissajous	Passive cooling; both missions now ended
Queqiao relay	Earth-Moon $L_2$	Halo	Continuous line of sight to both the lunar far side and Earth
Gateway	Earth-Moon	NRHO	Near-stable, permanent Earth communications, modest delta-v to the lunar surface

The $L_2$ concentration is not a coincidence. It is the only place in the inner solar system where the Sun, Earth and Moon stay within a single narrow cone, so one shade blocks all three and an instrument can radiate to deep space continuously. JWST’s operating temperature below 50 K is a direct consequence of that geometry, not of any active cooler on the cold side.

$L_3$ has no missions. It is permanently on the far side of the Sun, so it has no communications path, and it is unstable. It survives mainly in fiction as the location of a hidden counter-Earth.

References

Szebehely, V. Theory of Orbits: The Restricted Problem of Three Bodies, Academic Press, 1967. The standard reference on the CR3BP.
Koon, W. S., Lo, M. W., Marsden, J. E. and Ross, S. D. Dynamical Systems, the Three-Body Problem and Space Mission Design, Springer, 2011.
Farquhar, R. W. “The Control and Use of Libration-Point Satellites”, NASA TR R-346, 1970. Origin of the halo orbit concept.
Gascheau, G. “Examen d’une classe d’équations différentielles…”, Comptes Rendus, 1843. First statement of the 24.96 stability criterion.
Zimovan, E. M., Howell, K. C. and Davis, D. C. “Near Rectilinear Halo Orbits and Their Application in Cis-Lunar Space”, IAA-AAS-DyCoSS3, 2017.