Hill Sphere Calculator Formula

Understand the math behind the hill sphere calculator. Each variable explained with a worked example.

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Formulas Used

Hill Sphere Radius

hill_radius = semi_major_axis * pow(body_mass / (3 * primary_mass), 1/3)

Hill Radius (km)

hill_radius_km = semi_major_axis * pow(body_mass / (3 * primary_mass), 1/3) / 1000

Variables

VariableDescriptionDefault
semi_major_axisOrbital Distance (a)(m)149597870700
body_massOrbiting Body Mass(kg)5.972e+24
primary_massPrimary Body Mass(kg)1.989e+30

How It Works

The Hill Sphere

The Hill sphere approximates the region around a body where its gravity dominates over tidal forces from a larger primary.

Formula

r_H = a (m / 3M)^(1/3)

Moons must orbit well within the Hill sphere for long-term stability (typically within about one-third of r_H).

Worked Example

Find Earth's Hill sphere radius with respect to the Sun.

semi_major_axis = 149597870700body_mass = 5.972e+24primary_mass = 1.989e+30
  1. 01r_H = a (m / 3M)^(1/3)
  2. 02m / (3M) = 5.972e24 / (3 × 1.989e30) = 1.001e-6
  3. 03(1.001e-6)^(1/3) = 0.01000
  4. 04r_H = 1.496e11 × 0.01000 ≈ 1.5 million km

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Open Hill Sphere Calculator