Hill Sphere Calculator Formula
Understand the math behind the hill sphere calculator. Each variable explained with a worked example.
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) / 1000Variables
| Variable | Description | Default |
|---|---|---|
semi_major_axis | Orbital Distance (a)(m) | 149597870700 |
body_mass | Orbiting Body Mass(kg) | 5.972e+24 |
primary_mass | Primary 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.
- 01r_H = a (m / 3M)^(1/3)
- 02m / (3M) = 5.972e24 / (3 × 1.989e30) = 1.001e-6
- 03(1.001e-6)^(1/3) = 0.01000
- 04r_H = 1.496e11 × 0.01000 ≈ 1.5 million km
Frequently Asked Questions
Can a moon orbit at the edge of the Hill sphere?
In practice no. Prograde moons are stable only within about r_H/2, and retrograde moons within about r_H.
How does the Hill sphere relate to the Roche limit?
The Hill sphere is where an orbiter can exist; the Roche limit is how close it can approach before tidal forces tear it apart.
Does Jupiter have a large Hill sphere?
Yes, roughly 53 million km in radius, allowing it to hold many moons.
Ready to run the numbers?
Open Hill Sphere Calculator