Sphere of Influence Calculator

Two different questions, two different radii. The Hill sphere asks: where can a moon orbit stably? It comes from tidal forces — the balance between the planet's own gravity and the Sun's tidal pull — giving r_H = a(m₂/3m₁)^1/3. The Laplace SOI asks something narrower: where is it a good approximation to pretend only the planet's gravity exists (ignoring the Sun)? That balance gives r_SOI = a(m₂/m₁)^2/5. SOI is always smaller than Hill sphere — for Earth, Hill ≈ 1.5M km but SOI ≈ 925,000 km; for Jupiter, Hill ≈ 53M km but SOI ≈ 48M km. The Hill/SOI ratio hovers around 1.5 for most planets. Mission designers use the SOI boundary as the handoff point for patched-conic trajectories: inside SOI, integrate Earth-centered; outside, switch to heliocentric. It's not physically exact — there is no sharp boundary — but the error is small enough for preliminary mission design. Full precision (for actual navigation) always uses n-body propagation.

The sphere of influence is the invisible bubble around a planet or moon where its gravity is effectively "in charge." Cross the boundary inward and you can safely pretend the Sun isn't there; the planet alone shapes your path. That is why the Moon has its own small sphere of influence, roughly 66,000 km in radius. During Apollo, trajectory calculations switched from an Earth-centered frame to a Moon-centered one right as the spacecraft crossed it.

Look closely at r = a(m₂/m₁)^2/5: the distance a enters linearly, but mass only to the 2/5 power. Being far from the Sun matters far more than being massive. Pluto is about 25 times less massive than Mercury, yet its sphere of influence (~3 million km) dwarfs Mercury's (~112,000 km) — simply because Pluto orbits roughly 100 times farther out.