Lagrange Point Calculator
Calculate the locations of the five Lagrange points for a two-body gravitational system. Enter values for instant results with step-by-step formulas.
Calculator
Adjust values & calculateFormula
The collinear Lagrange points L1, L2, and L3 lie along the line connecting the two masses. L1 and L2 are approximately one Hill sphere radius from the smaller body. L4 and L5 form equilateral triangles with the two bodies. The mass ratio mu = M2/(M1+M2) determines all positions. L4/L5 are stable when mu < 0.0385 (Routh criterion).
Last reviewed: December 2025
Worked Examples
Example 1: Sun-Earth Lagrange Points
Example 2: Earth-Moon Lagrange Points
Background & Theory
The Lagrange Point Calculator applies the following established principles and formulas. Astronomy and space science rely on a set of precisely defined physical relationships that allow distances, sizes, motions, and energies of celestial objects to be calculated from observational data. Kepler's three laws of planetary motion, derived empirically in the early seventeenth century, describe elliptical orbits, equal areas swept in equal times, and the harmonic law Tยฒ = aยณ, where T is the orbital period in Earth years and a is the semi-major axis in astronomical units (AU). This relationship holds for any object orbiting the Sun and can be generalized using Newton's law of gravitation. Distances in astronomy are expressed in multiple units: one light-year equals approximately 9.461 ร 10ยนโต meters, one parsec equals 3.086 ร 10ยนโถ meters or about 3.26 light-years, defined as the distance at which one AU subtends one arcsecond of parallax. Angular size is calculated as ฮธ = 206,265 ร (d / D) arcseconds, where d is the physical diameter and D is the distance. The stellar magnitude system uses Pogson's formula: m1 โ m2 = โ2.5 ร log10(F1 / F2), where F represents flux. Each magnitude step corresponds to a flux ratio of approximately 2.512, meaning a first-magnitude star is 100 times brighter than a sixth-magnitude star. Hubble's Law relates recessional velocity to distance: v = Hโd, where the Hubble constant Hโ is approximately 70 km/s/Mpc. Escape velocity from any body is given by v = โ(2GM/r), yielding 11.2 km/s for Earth. Orbital period for a circular orbit follows T = 2ฯโ(rยณ/GM). Luminosity and distance are linked by the inverse square law: F = L / (4ฯdยฒ). Stars are classified by spectral type using the mnemonic OBAFGKM, corresponding to surface temperatures from approximately 30,000 K (O-type) to under 3,500 K (M-type). Each type reflects characteristic absorption spectra tied to ionization states of elements in the stellar photosphere.
History
The history behind the Lagrange Point Calculator traces back through the following developments. The history of astronomy is one of progressive scale โ each era expanding humanity's conception of the universe's size and structure. The Copernican revolution of 1543, when Nicolaus Copernicus published De revolutionibus orbium coelestium, displaced Earth from the center of the cosmos and placed the Sun at the center of the planetary system. Decades later, Galileo Galilei turned a Dutch-invented telescope toward the sky in 1609, discovering the moons of Jupiter, the phases of Venus, and the cratered surface of the Moon โ observations that provided compelling evidence for the heliocentric model and led to his conflict with the Catholic Church. Johannes Kepler, working from Tycho Brahe's meticulous naked-eye observations, derived his three laws of planetary motion between 1609 and 1619. Isaac Newton unified celestial and terrestrial mechanics with his law of universal gravitation in 1687, explaining the cause behind Kepler's empirical laws and enabling precise prediction of planetary positions. The eighteenth and nineteenth centuries brought systematic sky surveys, stellar parallax measurements, and the discovery that the Milky Way is itself a galaxy among many. Edwin Hubble's 1929 observations using the 100-inch Hooker Telescope at Mount Wilson demonstrated that galaxies are receding from us at velocities proportional to their distance โ the first direct evidence for an expanding universe and the empirical basis for Big Bang cosmology. NASA was founded in 1958 following the Sputnik shock, and the Apollo 11 mission landed humans on the Moon on July 20, 1969. The Hubble Space Telescope, launched in 1990, revolutionized observational astronomy by operating above Earth's atmosphere and producing imagery from ultraviolet to near-infrared wavelengths. The first confirmed exoplanet around a Sun-like star was detected in 1995 by Michel Mayor and Didier Queloz using the radial velocity method. The James Webb Space Telescope, launched in December 2021 and fully operational by 2022, extended infrared observations to probe the earliest galaxies formed after the Big Bang.
Frequently Asked Questions
Formula
L1,L2 distance from M2 ~ R x (M2 / 3M1)^(1/3); L4,L5 form equilateral triangles with M1 and M2
The collinear Lagrange points L1, L2, and L3 lie along the line connecting the two masses. L1 and L2 are approximately one Hill sphere radius from the smaller body. L4 and L5 form equilateral triangles with the two bodies. The mass ratio mu = M2/(M1+M2) determines all positions. L4/L5 are stable when mu < 0.0385 (Routh criterion).
Worked Examples
Example 1: Sun-Earth Lagrange Points
Problem: Calculate the five Lagrange points for the Sun-Earth system. Sun mass = 1.989 x 10^30 kg, Earth mass = 5.972 x 10^24 kg, distance = 1.496 x 10^11 m.
Solution: Mass ratio mu = 5.972e24 / (1.989e30 + 5.972e24) = 3.003e-6\nHill radius = R x (mu/3)^(1/3) = 1.496e11 x (1.001e-6)^(1/3) = 1.496e11 x 0.01 = 1.50e9 m\nL1: ~1.494e11 m from Sun (~1.50 million km from Earth toward Sun)\nL2: ~1.498e11 m from Sun (~1.50 million km from Earth away from Sun)\nL3: ~1.496e11 m from Sun (opposite side)\nL4/L5: At 60 degrees ahead/behind Earth in its orbit
Result: L1: 1.50 Gm from Earth | L2: 1.50 Gm from Earth | L4/L5: Equilateral, stable
Example 2: Earth-Moon Lagrange Points
Problem: Calculate Lagrange points for Earth-Moon system. Earth mass = 5.972 x 10^24 kg, Moon mass = 7.342 x 10^22 kg, distance = 384,400 km.
Solution: Mass ratio mu = 7.342e22 / (5.972e24 + 7.342e22) = 0.01215\nHill radius = 3.844e8 x (0.01215/3)^(1/3) = 3.844e8 x 0.1587 = 6.10e7 m\nL1: ~3.234e8 m from Earth (326,000 km, between Earth and Moon)\nL2: ~4.454e8 m from Earth (449,000 km, beyond Moon)\nL4/L5: Equilateral triangle points, 384,400 km from both bodies\nStability: mu = 0.012 < 0.0385 so L4/L5 are stable
Result: L1: 58,000 km from Moon | L2: 65,000 km beyond Moon | L4/L5: Stable
Frequently Asked Questions
What are Lagrange points and why are they important?
Lagrange points are five special positions in a two-body gravitational system where the combined gravitational pull of the two large masses provides exactly the centripetal force needed for a smaller object to orbit with them. Named after mathematician Joseph-Louis Lagrange who discovered them in 1772, these points allow spacecraft to maintain a relatively stable position with minimal fuel expenditure. The James Webb Space Telescope orbits the Sun-Earth L2 point, about 1.5 million km from Earth, where it stays shielded from the Sun while maintaining constant communication with Earth. The SOHO solar observatory sits at L1, continuously monitoring the Sun. Understanding Lagrange points is fundamental to mission planning for space telescopes, communication relays, and future space habitats.
What is the difference between stable and unstable Lagrange points?
Lagrange points L1, L2, and L3 are unstable equilibrium points, meaning that any small perturbation will cause an object to drift away, like a ball balanced on top of a hill. Spacecraft at these points require regular station-keeping maneuvers using thrusters. In contrast, L4 and L5 are conditionally stable, meaning objects displaced slightly will oscillate around the point rather than drifting away, like a ball in a bowl. However, L4 and L5 stability requires that the mass ratio between the two primary bodies satisfies the Routh criterion: the smaller mass must be less than about 3.85% of the larger mass. The Sun-Earth system easily satisfies this condition, which is why Jupiter has thousands of Trojan asteroids collected at its L4 and L5 points.
What spacecraft are currently at Lagrange points?
Several major space missions utilize Lagrange points. At Sun-Earth L1: SOHO (solar observatory since 1996), DSCOVR (Earth observation), and the Advanced Composition Explorer (ACE) studying solar wind. At Sun-Earth L2: the James Webb Space Telescope (launched 2021), the Gaia space observatory mapping a billion stars, the Euclid telescope studying dark energy, and the Planck satellite that mapped the cosmic microwave background. L2 is particularly valuable for space telescopes because it keeps the Sun, Earth, and Moon behind the spacecraft, providing a stable thermal environment and unobstructed view of deep space. The Chinese Chang'e 4 mission used a relay satellite at Earth-Moon L2 to communicate with its far-side lunar lander.
How is the mass ratio important in Lagrange point calculations?
The mass ratio mu, defined as M2/(M1+M2), is the fundamental parameter that determines the geometry and stability of all five Lagrange points. For the collinear points L1, L2, and L3, the distance from the smaller body scales approximately as the cube root of mu/3, known as the Hill sphere radius. When mu is very small (as in the Sun-Earth system where mu is approximately 3 times 10 to the negative 6), L1 and L2 are close together near the smaller body, and L3 is almost diametrically opposite on the other side of the larger body. The L4 and L5 points always form equilateral triangles regardless of the mass ratio, but their stability depends on mu being less than the Routh critical value of approximately 0.0385. Nearly all natural two-body systems in our solar system satisfy this criterion.
Can we build space colonies at Lagrange points?
Lagrange points, particularly L4 and L5, have long been considered prime candidates for future space colonies. Physicist Gerard O'Neill famously proposed large rotating habitats at Earth-Moon L5 in the 1970s, and the L5 Society was founded to advocate for this vision. L4 and L5 are attractive because their stability means structures would require minimal energy to maintain position, and they provide constant access to solar energy. However, significant challenges remain: transporting construction materials from Earth is prohibitively expensive, though lunar or asteroid mining could provide resources. The Sun-Earth L4 and L5 points are too far from Earth for practical colonization, but the Earth-Moon L4 and L5 points, about 384,000 km from Earth, are more accessible. Current technology could theoretically support small research stations at these points within decades.
Why might my result differ from another tool or reference?
Differences typically arise from rounding conventions, the specific version of a formula (for example, simple vs compound interest), or unit inconsistencies between inputs. Check that both tools are using the same formula variant and the same units. The References section links to the authoritative source behind the formula used here.
References
Reviewed by Daniel Agrici, Founder & Lead Developer ยท Editorial policy