What the Hill sphere is
A planet orbiting a star has a region around it where its own gravity dominates the star’s tidal pull. Inside this Hill sphere, moons can orbit the planet; outside, they would be pulled away into orbits around the star. The concept was developed by the American astronomer George William Hill in the 19th century.
The formula
For a planet of mass m orbiting a star of mass M at semi-major axis a with eccentricity e:
r_H ≈ a × (1 − e) × (m ÷ 3M)^(1/3)
The (1 − e) factor uses the closest approach to the star, where the sphere is smallest. The Lagrange points L1 and L2 lie at about the same distance from the planet.
Example: Earth
Earth has about 3.0 × 10⁻⁶ solar masses and orbits at 1 AU with e = 0.0167:
- r_H ≈ 0.983 × (3.0 × 10⁻⁶ ÷ 3)^(1/3) AU ≈ 0.0098 AU;
- that is about 1.47 million km (914,000 miles), close to the distance of the Sun–Earth L1 point where the SOHO observatory sits;
- the Moon, at 384,400 km, orbits at about 26% of that radius, well inside.
Stable orbits
Orbits near the edge of the Hill sphere are not stable over long times. As a rule of thumb, prograde moons (orbiting in the same direction as the planet’s spin) are stable out to about one half of r_H, and retrograde moons a bit further. The outermost moons of Jupiter and Saturn, many of them retrograde, orbit near those limits.
Other bodies
| Body | Hill sphere radius |
|---|---|
| Mercury | about 0.18 million km |
| Earth | about 1.5 million km |
| Mars | about 1.0 million km |
| Jupiter | about 53 million km |
| Neptune | about 115 million km |
Distant giant planets have huge Hill spheres because they are both massive and far from the Sun.
Uses
- Estimating where moons, captured asteroids and spacecraft parking orbits can survive.
- Studying planet formation: a growing planet sweeps material from a zone a few Hill radii wide.
- Moons have their own, much smaller Hill spheres inside their planet’s: the Moon’s is about 60,000 km around it, relative to Earth.