Hill Sphere Calculator
Radius of the region where a planet or moon, rather than the body it orbits, controls what circles around it.
Results
What this tool does
The Hill sphere, worked out by George William Hill in the 19th century, marks how far a planet's own gravity beats the pull of the Sun. For Earth it reaches about 1.5 million km, which is no coincidence: the L1 and L2 points, where SOHO and the James Webb Space Telescope sit, lie almost exactly on its edge. A moon needs to stay well inside: orbits beyond roughly a third to a half of the radius are disturbed by the Sun and do not last, which is why the Moon at 384 400 km is safe. The formula is the cube-root approximation, valid when the orbiting body is much lighter than the central one, and uses the closest point of an eccentric orbit, where the sphere is smallest.
Formula
r_H ≈ a (1 − e) ∛(m ÷ 3M)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
sa | Semi-major axis of the orbit | km |
ec | Orbital eccentricity | — |
mb | Mass of the orbiting body | kg |
mc | Mass of the central body | kg |
RH | Hill radius at closest approach | km |
RA | Same, in astronomical units | AU |
RS | Rough limit for long-lived moons (about half) | km |
RC | Hill radius at the mean distance | km |
Worked example
- Semi-major axis of the orbit149,598,023 km
- Orbital eccentricity0.0167
- Mass of the orbiting body5.972e+24 kg
- Mass of the central body1.989e+30 kg
- Hill radius at closest approach1,471,408 km
- Same, in astronomical units0.00984 AU
- Rough limit for long-lived moons (about half)735,704 km
- Hill radius at the mean distance1,496,398 km
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.