The Five Lagrange Points

Locating L1 to L5, linear stability, the saddles of L1–L3, Routh’s criterion for L4 and L5, and the Trojans

Heidelberg, 22 February 1906

Max Wolf, photographing the sky for asteroids at the Königstuhl observatory, found one that did not behave like the others. It moved round the Sun at Jupiter's distance, in Jupiter's period — and it stayed about sixty degrees ahead of Jupiter, week after week. It was later named 588 Achilles, the first of the Trojan asteroids. Its position had been predicted 134 years earlier. In 1772 Joseph-Louis Lagrange, working on the general problem of three bodies, had shown that a small body could keep a fixed place relative to two larger ones if the three formed an equilateral triangle. Nobody had expected nature to use it. Thousands of Jupiter Trojans are known today, in two swarms leading and trailing the planet. NASA's Lucy), launched on 16 October 2021, is flying to visit several of them.

In the rotating frame an equilibrium is a point where a spacecraft placed at rest stays at rest. With zero velocity the Coriolis terms vanish, so the condition is simply

\nabla U = 0

: gravity of the two primaries exactly balanced by the centrifugal term.

The collinear points. On the

x

axis,

\partial U/\partial x = 0

is a fifth-degree equation with one real root in each of three intervals: L1 between the primaries, L2 beyond the small one, L3 beyond the big one. Leonhard Euler found them around 1750 and published them in 1767. The triangular points L4 and L5 sit at the third vertex of an equilateral triangle on the two primaries, leading and trailing the small one by 60° — Lagrange, 1772.

x - \frac{(1-\mu)(x+\mu)}{|x+\mu|^3} - \frac{\mu\,(x-1+\mu)}{|x-1+\mu|^3} = 0,\qquad |x_{L1,L2} - (1-\mu)| \approx \left(\frac{\mu}{3}\right)^{1/3}

The collinear equilibrium condition, and the Hill-radius approximation for L1 and L2 when μ is small.

System	μ	L1 from the small body	L2 from the small body	Hill radius
Sun–Earth	3.0404 × 10⁻⁶	1,497,621 km	1,507,683 km	1,502,668 km
Earth–Moon	1.2151 × 10⁻²	58,019 km	64,515 km	61,274 km
Sun–Jupiter	9.5368 × 10⁻⁴	51,893,930 km	54,309,334 km	53,117,820 km

Sun–Earth L1 is 3.90 times the Moon's distance from the Earth and exactly 1.00% of the way to the Sun. L1 and L2 straddle the Hill radius, L1 a little inside, L2 a little outside. L3 lies on the far side of the Sun, just 265 km closer to it than the Earth is — permanently hidden behind the Sun from the Earth, which is why it has never been used.

💡Stability: saddles and centres

Linearise the equations of motion about each point: the

6 \times 6

matrix has the identity in its upper right block, the Hessian of

U

in its lower left, and the Coriolis coupling. Its eigenvalues decide everything (second lab).

L1, L2, L3 each have one positive real eigenvalue: they are saddles. A displacement grows like $e^{\lambda t}$ . For Sun–Earth L1 and L2 the e-folding time is 23.0 and 23.4 days; for Earth–Moon L1 and L2, 1.5 and 2.0 days. L3 is only weakly unstable (e-folding about 56 years for the Sun and the Earth).
L4 and L5 are maxima of the potential, yet they are stable — the Coriolis force turns a body that starts to slide off the hill sideways, into a loop. This works only if the big primary is heavy enough: Edward Routh showed in 1875 that the condition is $\mu < 0.03852$ , a mass ratio above 24.96. The Earth and the Moon (81:1) and the Sun and Jupiter (1047:1) pass; a binary with $\mu = 0.05$ does not.

📐Worked Example: one kilometre off L1, and a Trojan for 2,000 years

The third lab releases a spacecraft at rest just 1 km from Sun–Earth L1. It is more than 1,000 km away after 166 days, 100,000 km after 272 days and a million kilometres after 320 days — exponential growth with the 23-day time constant, until the Earth or the Sun takes over. A test particle released 0.01 AU outward from Sun–Jupiter L4 behaves completely differently. Over 2,000 years its distance from the Sun stays between 5.018 and 5.392 AU, and its angle ahead of Jupiter swings between 39.5° and 87.9° around L4's 60°, with a period of about 152 years. Seen in the rotating frame its path is a long curved loop — a tadpole orbit — which is how the real Trojans move.

Newton was the greatest genius that ever existed, and the most fortunate, for we cannot find more than once a system of the world to establish.

– Joseph-Louis Lagrange, as reported by his contemporaries

Point	Occupants	Why there
Sun–Earth L1	ISEE-3 (1978), SOHO (1995), ACE (1997), DSCOVR (2015), Aditya-L1 (2024)	An unbroken view of the Sun; solar wind measured before it reaches the Earth
Sun–Earth L2	Planck, Gaia, JWST (arrived 24 January 2022), Euclid	Sun, Earth and Moon all on one side: a cold, stable view of the sky
Earth–Moon L2	Queqiao (2018), relay for China's far-side lander	In view of the Earth and of the Moon's far side at once
Sun–Earth L4/L5	2010 TK7, the first Earth Trojan found (by WISE)	Natural — stable for small bodies
Sun–Jupiter L4/L5	Thousands of Trojan asteroids, Lucy's targets	Natural — stable for 4.5 billion years

↓But there is a problem

Every spacecraft in that table's first three rows lives at an unstable point. Released at L1 itself, it would be gone within a year. In 1966 a graduate student at Stanford, Robert Farquhar, proposed something better than sitting at the point: a large, closed loop around it, tilted out of the plane like a halo. In 1978 a NASA spacecraft flew the first one.