Lagrange Point Calculator

L2 is 1.5 million km from Earth, on the far side from the Sun. A telescope at L2 always has the Sun, Earth, and Moon in the same direction (behind it), allowing a single sun-shield to block all three sources of heat and light. This enables extreme cryogenic cooling needed for infrared telescopes. JWST, Gaia, Planck, Herschel, and WMAP all operated near Sun-Earth L2.

L4 and L5 are stable for mass ratios M1/M2 > 24.96 (the Sun-Earth ratio of 333,000 satisfies this easily). Objects at L4 and L5 oscillate around the equilibrium point in complex tadpole or horseshoe orbits due to perturbations from other planets. L1, L2, and L3 are unstable — objects drift away exponentially without station keeping.

A halo orbit is a three-dimensional periodic orbit around an unstable Lagrange point (L1 or L2). Spacecraft cannot simply park at L1/L2 because these points are unstable — slight perturbations grow. Instead, they orbit around the Lagrange point in a halo orbit, requiring small periodic station-keeping burns (typically 2-10 m/s per year) to maintain the orbit.

Trojan asteroids orbit at L4 (60 degrees ahead) and L5 (60 degrees behind) of a planet. Jupiter has over 10,000 known trojans, the most of any planet. Mars, Earth, Uranus, and Neptune also have trojans. The Trojan War namesakes — Greek heroes at Jupiter's L4 and Trojans at L5 — give the name to these populations. NASA's Lucy mission (2021) will visit Jupiter trojans.

The Hill sphere is the region around a smaller body where its gravitational influence dominates over the larger body's tidal force. For Earth: Hill sphere radius = 1.5 million km. The Moon, at 384,400 km, is well within Earth's Hill sphere. Satellites placed beyond the Hill sphere eventually escape to heliocentric orbits. The L1 and L2 distances approximately equal the Hill sphere radius.

In theory, yes. Physicist Gerard O'Neill (1970s) proposed large cylindrical space habitats at Earth-Moon L4 and L5, using the stable equilibrium to avoid significant station keeping. These habitats (later called O'Neill cylinders or Stanford tori) would rotate to provide artificial gravity. The L4/L5 Society (now the National Space Society) advocated for this vision in the 1970s-80s.

L3 is on the opposite side of M1 from M2 (beyond M1, not M2). In the Sun-Earth system, L3 is on the far side of the Sun from Earth, about 1 AU from the Sun. It is unstable and difficult to reach (it is always behind the Sun from Earth's perspective). L3 is sometimes called the 'anti-Earth' point and appears in science fiction as the location of a mirror-Earth hidden by the Sun.

For mass ratio M2/(M1+M2) < 0.0385 (approximately 1/26), L4 and L5 are stable against small perturbations. This threshold is called the Routh criterion. For the Sun-Jupiter system (ratio = 1/1048), L4/L5 are stable, allowing trojan asteroids. For Earth-Moon (ratio = 1/82), L4/L5 are stable but orbital resonances can slowly remove objects over geological timescales.

Sun-Earth L1: SOHO (1995), ACE (1997), WIND (1994), DSCOVR (2015) — all solar wind/space weather monitors. Sun-Earth L2: WMAP (2001), Herschel (2009), Planck (2009), Gaia (2013), JWST (2021). Earth-Moon L2: proposed lunar Gateway (2024+) habitat. Sun-Earth L4/L5: STEREO spacecraft briefly visited these regions. These missions require 2-10 m/s/year station keeping.