Hill Sphere Radius Calculator

Hill Sphere

The Hill sphere (also called the Roche sphere or sphere of influence) is the region around an astronomical body where its gravity dominates over that of the larger body it orbits. A satellite stays bound only if its orbit lies inside this sphere — beyond it, the parent body’s gravity strips the satellite away.

Formula

r_H ≈ a × (1 − e) × ∛(m / (3 × M))

Where:

• a = semi-major axis of the smaller body’s orbit around the larger one
• e = orbital eccentricity (0 for circular orbit)
• m = mass of the smaller body
• M = mass of the larger (parent) body

Worked Example: Earth

• a = 1 AU = 1.496 × 10⁸ km
• e ≈ 0.0167
• m_Earth = 5.972 × 10²⁴ kg, M_Sun = 1.989 × 10³⁰ kg
• r_H ≈ 1.496 × 10⁸ × 0.9833 × ∛(5.972 × 10²⁴ / (3 × 1.989 × 10³⁰))
• r_H ≈ 1.496 × 10⁸ × 0.9833 × 0.0100 ≈ 1.47 × 10⁶ km

That is about 4 lunar distances — the Moon, at about 384 000 km, sits comfortably inside Earth’s Hill sphere.

Hill Sphere of Selected Bodies

These are computed with the (1 − e) factor from the formula above, so they give the Hill radius at perihelion, where the parent’s tide is strongest and the sphere is at its tightest. Many references quote the value at mean distance instead, dropping the (1 − e), which runs a few percent higher for the near-circular planets and a lot higher for eccentric bodies. Pluto is the one to watch: at perihelion its Hill radius is 5.8 million km, but at mean distance the same formula gives 7.7. If a number you find elsewhere disagrees with this table by 5 to 30 percent, that convention is almost always why.

Why It Matters

A satellite’s orbit must be substantially smaller than the Hill radius — typically less than half — to remain stable against perturbations. Anything beyond ~½ r_H is gradually pulled away by the parent body or shed entirely by the three-body dynamics.

Practical Examples

• The Moon orbits at ~26% of Earth’s Hill radius: very stable.
• Triton orbits at ~0.3% of Neptune’s Hill radius (354,800 km out of 115 million): extremely deep inside, and stable despite being retrograde.
• Distant Centaur asteroids cross multiple planets’ Hill spheres and have chaotic orbits.

Caveats

The formula above is an approximation that assumes m ≪ M and a Keplerian two-body system. For highly eccentric orbits, three-body resonances, or systems with comparable masses (binary stars), more careful analytical or numerical methods are required. The Hill sphere is also a stability bound, not a hard wall — bodies on retrograde orbits can stay stable a bit beyond r_H while prograde orbits become unstable a bit inside it.

Hill Sphere vs Roche Limit

These two concepts share the name “Roche” but mean opposite things.

• Hill sphere: outer limit, beyond this, satellites escape.
• Roche limit: inner limit, closer than this, tidal forces tear satellites apart. Stable moons live between these two bounds.