Hill Sphere Radius Calculator
The Hill sphere marks how far a planet's gravity can hold onto moons before the parent star wins the tug of war. Any satellite must orbit well inside this radius to stay bound. This calculator applies the standard Hill radius approximation using the orbital semi-major axis, the eccentricity, and the masses of the two bodies. Enter your values in consistent units and the result comes back in the same length unit as the semi-major axis.
Hill sphere radius formula
r_H = a * (1 - e) * cbrt(m / (3 * M))
where a is the semi-major axis, e the eccentricity, m the smaller mass, M the parent mass
The radius comes out in the same length unit you use for a. Only the mass ratio m/M matters, so the masses can be in any consistent unit.
Worked example
For Earth orbiting the Sun (a = 149,600,000 km, e = 0.0167, m = 5.972e24 kg, M = 1.989e30 kg): r_H = 149,600,000 * (1 - 0.0167) * cbrt(5.972e24 / (3 * 1.989e30)) = about 1,471,400 km. That is roughly 0.0098 of the semi-major axis, comfortably enclosing the Moon's orbit.
Hill sphere radius: frequently asked questions
What is the Hill sphere?
The Hill sphere is the region around a body, such as a planet orbiting a star, within which it dominates the gravitational attraction of satellites. A moon must orbit inside its planet's Hill sphere to remain bound; outside it, the star's pull would strip the moon away. It sets the practical outer limit for stable satellites.
What is the Hill sphere formula?
A standard approximation is r_H = a times (1 minus e) times the cube root of m divided by 3M, where a is the orbital semi-major axis, e the orbital eccentricity, m the smaller body's mass, and M the parent body's mass. The (1 minus e) factor uses the closest approach (perihelion) distance, where the Hill sphere is smallest.
Is the Hill sphere the same as the sphere of influence?
They are related but not identical. The Hill sphere describes long-term orbital stability against a third body, while the patched-conic sphere of influence used in mission design has a slightly different definition and a somewhat smaller radius. For questions about whether a moon can stay bound, the Hill sphere is the relevant measure.
Sources
- The Hill radius approximation is a standard result in celestial mechanics. Planetary masses and orbital elements: NASA Planetary Fact Sheet.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.