Hill Sphere Radius Calculator
The Hill sphere marks how far a planet's gravity holds sway over a satellite before the star it orbits takes over, so it sets the outer limit for stable moons. Its radius follows a well-known formula that balances the planet's gravity against the star's tidal pull. This calculator takes the orbital semi-major axis, the eccentricity and the masses of the two bodies (only their ratio matters), then returns the Hill radius, the mass-ratio cube-root factor and the perihelion distance used in the estimate. Lengths come out in whatever unit you enter for the semi-major axis.
Hill sphere formula
perihelion = semi-major axis * (1 - eccentricity)
cube-root factor = cube root of (smaller mass / (3 * larger mass))
Hill radius = perihelion * cube-root factor
The Hill radius balances the smaller body's gravity against the primary's tidal force. Evaluated at perihelion (semi-major axis times one minus eccentricity) it gives a conservative bound. The mass ratio appears under a cube root, so even a large mass advantage extends the sphere only modestly.
Hill sphere context
- Satellites orbiting within the Hill sphere are gravitationally bound to the smaller body.
- The sphere is smallest at perihelion, where the primary's tidal pull is strongest.
- Only the ratio of the two masses matters, so any consistent mass unit works.
- Stable prograde moons typically orbit within roughly a third to half of the Hill radius.
- The formula is the standard restricted three-body Hill radius estimate.
Hill sphere: frequently asked questions
What is the Hill sphere?
The Hill sphere is the region around a body, such as a planet, within which its gravity dominates over the larger body it orbits, so it can hold onto satellites. A moon orbiting inside the Hill sphere is gravitationally bound to the planet rather than being stripped away by the star.
How is the Hill sphere radius calculated?
The standard estimate is the orbital semi-major axis times one minus the eccentricity, multiplied by the cube root of the smaller body's mass divided by three times the larger body's mass. This calculator evaluates that formula from your inputs.
Why does the mass ratio use a cube root?
The Hill radius comes from balancing the smaller body's gravity against the tidal pull of the primary, which scales with the cube of distance. Solving that balance leaves the mass ratio under a cube root, so a body needs a very large mass advantage to greatly extend its Hill sphere.
Why include the eccentricity term?
The Hill sphere is smallest at the perihelion of the orbiting body's path, where it is closest to the primary and the tidal force is strongest. Using the semi-major axis times one minus eccentricity evaluates the radius at that tightest point, giving a conservative bound.
What units does this use?
Enter the semi-major axis in any length unit and the result comes out in that same unit. The masses can be in any unit as long as both use the same one, since only their ratio matters. The eccentricity is dimensionless between 0 and just below 1.
Official sources
- NASA: Basics of Space Flight, Gravitation.
- NASA Jet Propulsion Laboratory: Planetary Physical Parameters.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.