Engineering & Science
Hill Sphere Radius (rH) & Orbital Moon Stability Calculator
Calculate George William Hill’s pericenter gravitational dominance sphere r_H = a * (1 - e) * (m_planet / (3 * M_star))^(1/3) (km) for planetary moons, James Webb L1/L2 halo orbits, and exoplanets.
Why Earth keeps the Moon—and why James Webb orbits 1.5 million km away at L2!
How far from a planet of mass m_p orbiting a star of mass M_s (semi-major axis a, eccentricity e) can a moon or spacecraft orbit before solar tidal gravity rips it away? American dynamical astronomer George William Hill (1878) solved the restricted three-body problem around Lagrange points L1 and L2 to prove that a planet’s gravitational dominance zone at periapsis spans the Hill Radius: r_H = a * (1 - e) * (m_p / (3 * M_s))^(1 / 3)!
Notice two remarkable insights: (1) Earth (a = 149,597,870.7 km, e = 0.0167, m_p = 5.9722e24 kg, Sun M_s = 1.9885e30 kg) has a Hill radius of r_H = 1,471,546 km—which is why L1/L2 observatories (JWST, SOHO) sit ~1.5 million km from Earth, and why our Moon at 384,400 km (26.1% of r_H, well inside the long-term prograde stability ceiling ~0.33–0.50 * r_H!) stays stably bound! (2) Even massive Jupiter (m_p = 1.898e27 kg), if migrated inward as a "Hot Jupiter" to a = 7,500,000 km (0.05 AU, e = 0), shrinks its Hill sphere to just 512,009 km!
| Planetary System (Star Mass Ms = 1.9885e30 kg) | Semi-Major Axis a (km) & e | Hill Sphere Radius r_H (km) | Prograde Moon Stability Limit (~r_H / 3) |
|---|---|---|---|
| Earth–Sun System (m = 5.9722e24 kg) | 149,597,871 km, e = 0.0167 | 1,471,546 km (~0.01 AU) | 490,515 km (Moon = 384,400 km Safe!) |
| Hot Jupiter at 0.05 AU (m = 1.898e27 kg) | 7,500,000 km, e = 0.0000 | 512,009 km | 170,670 km (Tidal Stripping Zone) |
| Parabolic/Hyperbolic Escape (e >= 1) | Bound-Orbit Guard | "Invalid Orbital Inputs" | Requires a > 0 and 0 <= e < 1 |
Pericenter Hill Sphere Radius r_H (km) = a * (1 - e) * (m_p / (3 * M_s))^(1/3)
Evaluates cube-root three-body tidal boundary in kilometers for bound elliptical orbits 0 <= e < 1.
=IF(OR(B2 <= 0, B3 < 0, B3 >= 1, B4 <= 0, B5 <= B4), "Invalid Orbital Inputs", ROUND(B2 * (1 - B3) * ((B4 / (3 * B5)) ^ (1 / 3)), 0))Conservative Prograde Satellite Stability Ceiling (~r_H / 3 in km)
Long-term numerical integrations show prograde moons remain stable over Gyr timescales inside ~1/3 of the Hill radius.
=IF(OR(B2 <= 0, B3 < 0, B3 >= 1, B4 <= 0, B5 <= B4), "Invalid Orbital Inputs", ROUND((B2 * (1 - B3) * ((B4 / (3 * B5)) ^ (1 / 3))) / 3, 0))Try the example
Use the original values or change them to test your understanding.