Lagrange Point Calculator
Enter primary mass, secondary mass, and orbital separation to find all five Lagrange equilibrium points and Hill sphere radius for any two-body gravitational system.
⚖️ What Are Lagrange Points?
Lagrange points (named after mathematician Joseph-Louis Lagrange) are the five special positions in the orbital plane of a two-body gravitational system where a small third object experiences zero net force in the rotating reference frame of the two larger bodies. At these points, the gravitational pull of the primary, the gravitational pull of the secondary, and the centrifugal force in the rotating frame combine to produce a perfect equilibrium. A spacecraft or asteroid placed exactly at a Lagrange point will remain stationary relative to the two larger bodies without any propulsion.
There are five Lagrange points, each with different geometries and stability properties. L1 lies between the two bodies, along the line connecting them, closer to the secondary. The Sun-Earth L1 is where SOHO and the DSCOVR satellite monitor the solar wind. L2 is on the opposite side of the secondary from the primary, the same distance away as L1. This is where the James Webb Space Telescope, Planck, Herschel, and Gaia have all operated. L3 sits on the far side of the primary, directly opposite the secondary, and is the most unstable of the five points. L4 and L5 are off the primary-secondary line, forming equilateral triangles with both bodies at 60 degrees ahead and behind the secondary in its orbit. These are the only two conditionally stable Lagrange points, hosting Jupiter's famous Trojan asteroid swarms and Earth's Trojan asteroid 2010 TK7.
The Hill sphere radius, rH = a(mu/3)^(1/3), equals the L1 and L2 distance from the secondary body to leading order. It represents the region where the secondary body's own gravity dominates over tidal forces from the primary. Objects orbiting within the Hill sphere can maintain stable orbits around the secondary; the Moon, for example, lies well within Earth's Hill sphere of about 1.5 million km. Beyond the Hill sphere, the Sun's tidal forces dominate and objects are pulled away.
Lagrange points are of critical importance in space mission design. Placing a spacecraft at L1 gives it a direct, unobstructed view of the Sun from a semi-stable position near Earth. L2 is ideal for space telescopes because the Sun, Earth, and Moon are always in the same direction, allowing a single sunshield to block all three heat sources simultaneously. L4 and L5 have been proposed as sites for future space colonies or fuel depots because of their long-term gravitational stability.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Sun-Earth System (JWST and SOHO Orbits)
Sun (333,000 M⊕) and Earth (1 M⊕) separated by 1.000 AU
Example 2 — Earth-Moon System
Earth (1 M⊕) and Moon (0.01230 M⊕) separated by 0.002570 AU (384,472 km)
Example 3 — Sun-Jupiter System (Trojan Asteroids)
Sun (333,000 M⊕) and Jupiter (317.8 M⊕) separated by 5.203 AU
❓ Frequently Asked Questions
🔗 Related Calculators
What are the five Lagrange points and where are they located?
Lagrange points are the five positions in a two-body gravitational system where a small third object can orbit in a fixed configuration relative to the two larger bodies. L1 sits between the two bodies. L2 lies beyond the smaller body. L3 sits on the far side of the larger body. L4 and L5 form equilateral triangles with both bodies, leading and trailing the secondary by 60 degrees.
Which Lagrange points are stable and which are unstable?
L4 and L5 are stable (conditionally) when the mass ratio criterion 27mu(1-mu) < 1 is satisfied, which requires the primary to be at least 24.96 times heavier than the secondary. L1, L2, and L3 are all saddle-point unstable, meaning any small perturbation causes objects to drift away. Spacecraft at L1 and L2 require periodic station-keeping to remain in place.
What is the Hill sphere and how does it relate to L1?
The Hill sphere is the region around a secondary body where its own gravity dominates over tidal forces from the primary. Its radius r_H = a(mu/3)^(1/3) equals the L1 and L2 distance from the secondary body (to first approximation). Objects within a body's Hill sphere can have stable bound orbits; for Earth the Hill sphere radius is about 1.5 million km.
Why does the James Webb Space Telescope orbit at L2?
The Sun-Earth L2 point, about 1.5 million km beyond Earth, keeps the Sun, Earth, and Moon all in the same direction behind the telescope at all times. This allows JWST's sunshield to block all three heat sources simultaneously, cooling the telescope to below 50 K, which is essential for detecting the faint infrared signals from distant galaxies. L2 also provides a stable gravitational environment with minimal station-keeping.
What makes Jupiter's L4 and L5 points so populated with asteroids?
Jupiter's L4 and L5 points are genuinely stable because Jupiter's mass ratio (mu = 9.53e-4) satisfies the 27mu(1-mu) < 1 criterion with a large margin. Objects there have been trapped since the early solar system. Over 12,000 Trojan asteroids are known at Jupiter's L4 and L5, more than in the entire main asteroid belt. Their stability has persisted for billions of years.
How accurate is the Lagrange point approximation formula?
The formula r_H = a(mu/3)^(1/3) is an exact leading-order result from the restricted three-body problem. It is highly accurate for small mass ratios (mu < 0.001, such as Sun-Earth or Sun-Jupiter), with errors below 0.1%. For larger ratios like Earth-Moon (mu = 0.012), the approximation overestimates the L1 and L2 distances by about 5 percent compared to the exact numerical solution.
What is the mass ratio threshold for L4 and L5 stability?
L4 and L5 are stable when 27mu(1-mu) < 1, which corresponds to mu < 0.03852, or equivalently M/m > 24.96. For the Sun-Earth system mu = 3e-6, far below the threshold. Earth's mass ratio with the Moon (mu = 0.012) also satisfies stability. A binary system with equal masses (mu = 0.5) gives 27 x 0.25 = 6.75 > 1, so L4 and L5 are unstable.
Where exactly are the Sun-Earth Lagrange points in kilometres?
L1 is about 1.496 million km from Earth toward the Sun (0.01 AU), hosting SOHO and DSCOVR. L2 is the same distance beyond Earth away from the Sun, where JWST and the upcoming LIFE telescope will orbit. L3 is directly opposite Earth on the far side of the Sun, at approximately 1 AU. L4 and L5 are along Earth's orbit, 60 degrees ahead and behind, each 1 AU from both the Sun and Earth.
Can Earth have Trojan asteroids at its L4 and L5 points?
Yes. Earth's mass ratio is well within the stability limit. In 2010 astronomers discovered 2010 TK7, the first known Earth Trojan asteroid, at Earth's L4 point. It is about 300 metres in diameter. Earth's Trojans are harder to observe than Jupiter's because they are always close to the Sun in our sky, visible only near twilight.
What is the significance of L3 in science fiction and popular culture?
L3, the point directly opposite the smaller body on the far side of the primary, is often imagined in science fiction as the location of a hidden counter-Earth permanently invisible from our planet. In reality L3 is highly unstable and any object placed there would drift away within decades due to gravitational perturbations from Venus, Jupiter, and other planets.