Hill Sphere
The volume of space where a planet's gravity wins out over the Sun's.
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Out at Earth's orbit, the Sun's pull on you would be about 46,000× weaker than it is on the Sun's surface — but it's still strong. So why does the Moon orbit Earth and not the Sun? Because right next to Earth, you're SO close to Earth that its 80× smaller mass still wins. The boundary where that flips — where you're far enough away that the Sun is back in charge — is the Hill sphere.
Inside Earth's Hill sphere (a roughly spherical region about 1.5 million km across), an object can hold a stable orbit around Earth. Outside it, the Sun's gravity reaches in too strongly and yanks the object away. The Moon, at 384,000 km out, sits about a quarter of the way to that edge — comfortably stable. The James Webb Space Telescope orbits the Sun–Earth L2 point, also ~1.5 million km out, and needs small station-keeping burns (~2.5 m/s per year) because L2 is a dynamically unstable saddle point — not because it sits at the edge of Earth's Hill sphere.
Every planet has one. They scale with the planet's mass and its distance from the Sun. Mercury — small and close — has a Hill sphere only about 90 Mercury-radii across, which is why Mercury has no moon (any captured object would be ripped away). Jupiter — huge and far — has a Hill sphere about 740 Jupiter-radii across, big enough to hold 95 known moons plus extensive Trojan asteroid swarms.
The Hill radius formula approximates the boundary as r_H ≈ a × (1−e) × (m / 3M)^(1/3), where `a` is the planet's semi-major axis, `e` its eccentricity, `m` its mass, and `M` the Sun's mass. The (1−e) factor accounts for perihelion shrinkage — a more eccentric orbit means the planet swings closer to the Sun and its authority shrinks accordingly. The cube root means doubling a planet's mass only enlarges its Hill sphere by ~26%; doubling its orbital distance enlarges it by 100%.
Real numbers for the solar system: Mercury 220,000 km (90 R_Mercury). Venus 1.0 M km (170 R_Venus). Earth 1.5 M km (236 R_Earth). Mars 1.1 M km (320 R_Mars — bigger than Earth's in radii because Mars is small and farther from the Sun). Jupiter 53 M km (740 R_Jupiter — by far the largest in the system). Saturn 65 M km (1100 R_Saturn). Uranus 70 M km. Neptune 116 M km. Pluto ~6.0 M km (~5000 R_Pluto — the largest Hill sphere measured in its own body radii in the solar system, because Pluto is far from the Sun and tiny).
The Hill sphere is the outer limit; the inner limit is the Roche limit, where the planet's tidal forces would shatter a satellite. Stable moons live in the band between the two. Mercury's case is the strict one — its Hill sphere is small enough that any captured body would be pulled within Roche distance and disrupted, so Mercury has stayed moonless for 4.5 billion years.