Hill Sphere Radius Calculator: Gravitational Sphere of Influence and Stable Moon Orbits
Calculate the Hill sphere radius of a planet, moon or exoplanet from its orbit and the two masses, with eccentricity, the stable limits for prograde and retrograde moons (Domingos et al. 2006), a check of any satellite distance, presets for solar-system bodies and a reference table.
How it works
r_H ≈ a (1 − e) × ∛( m ÷ 3M ) · Stable prograde moons: a_sat ≲ 0.4895 (1 − 1.0305 e − 0.2738 e_sat) × a ∛( m ÷ 3M ) · Stable retrograde moons: a_sat ≲ 0.9309 (1 − 1.0764 e − 0.9812 e_sat) × a ∛( m ÷ 3M )
The Hill sphere is the region around a planet (or a moon, or an asteroid) where its own gravity holds a satellite more strongly than the tidal pull of the star it orbits. It comes from the restricted three-body problem: at the distance r_H the planet’s pull, the star’s tidal pull and the effect of the planet’s own orbit balance, which gives the cube-root formula. Distance from the star counts in full, but mass only as its cube root, so a small planet far out can have a larger Hill sphere than a large one close in. For an eccentric orbit the radius is taken at the closest approach, a(1 − e), because a satellite has to survive every pass. The sphere is an upper bound, not a guarantee. Numerical simulations by Domingos, Winter and Yokoyama (2006) found that prograde moons stay bound to about 0.49 Hill radii for a circular planetary orbit and retrograde moons to about 0.93, with both limits shrinking as the planet’s or the moon’s orbit becomes more eccentric. This is why the outer irregular moons of Jupiter and Saturn are mostly retrograde. Masses use the IAU ratio of 332,946 Earth masses per solar mass. The formula assumes the body is much lighter than its primary. For two stars or a binary of similar masses, use the Roche lobe (Eggleton formula) instead.
Worked example
Earth orbits at 1 AU (149,597,871 km) with eccentricity 0.0167 and a mass 1 ÷ 332,946 that of the Sun. The cube root of (3.0035 × 10⁻⁶ ÷ 3) is 0.010004, and the periapsis distance is 149,597,871 × (1 − 0.0167) = 147,099,586 km, so the Hill radius is about 1,471,566 km, 0.0098 AU or 231 Earth radii. Prograde moons are stable out to 0.4895 × (1 − 1.0305 × 0.0167) × 1,496,559 = 719,958 km (48.9% of the Hill radius), retrograde moons to 1,368,103 km. The Moon, at 384,400 km, sits at 26.1% of the Hill radius, well inside that limit.
Sources: Wikipedia: Hill sphere (formula and solar-system Hill radii) · Domingos, Winter & Yokoyama (2006), Stable satellites around extrasolar giant planets, MNRAS 373, 1227 (abstract with fitted limits)
Frequently asked questions
How big is Earth’s Hill sphere?
About 1.47 million km, roughly 0.01 AU or 3.8 times the Moon’s distance. Spacecraft at the Sun–Earth L1 and L2 points, 1.5 million km away, sit right at its edge.
How is the Hill sphere different from the Roche limit?
The Roche limit is the inner boundary where a moon is torn apart by the planet’s tides. The Hill sphere is the outer boundary where the moon is pulled away by the star. Moons orbit between the two.
Why are distant moons usually retrograde?
Retrograde orbits stay stable almost twice as far out (about 0.93 Hill radii versus 0.49 for prograde). Moons captured far from their planet survive mainly on retrograde orbits, as with Phoebe at Saturn and most of Jupiter’s outer moons.
Which planet has the largest Hill sphere?
Neptune, at about 115 million km, followed by Uranus and Saturn. Its distance from the Sun outweighs Jupiter’s much larger mass because distance counts in full and mass only as a cube root.
Does the Moon have its own Hill sphere?
Yes. Relative to Earth it extends about 58,000 km, so a spacecraft orbiting the Moon far beyond that would be pulled away by Earth.
Related calculators
Textbook celestial mechanics with simplifying assumptions (point masses, the restricted three-body problem, no other perturbers). Real satellite stability depends on resonances and other bodies; use numerical integrations for research.