<<An astronomical body's Hill sphere
is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sphere of its own. Any object within that distance would tend to become a satellite of the moon, rather than of the planet itself.
The radius RH
of the Hill sphere for a moon is, approximately equal to (1-e) (where e is the moon's eccentricity) times the moon's semi-major axis am
divided by the planet's radius (am
) times the moon's radius Rm
Hill sphere radius RH ~ (1-e) x (am/Rp) x Rm
An astronaut could not orbit the Space Shuttle (with mass of 104 tonnes), where the orbit is 300 km above the Earth, because its Hill sphere is only 120 cm in radius, much smaller than the shuttle itself. A sphere of this size and mass would be denser than lead. In fact, in any low Earth orbit, a spherical body must be more dense than lead in order to fit inside its own Hill sphere, or else it will be incapable of supporting an orbit. A spherical geostationary satellite, however, would only need to be more than 6% of the density of water to support satellites of its own.
Within the Solar System, the planet with the largest Hill radius (and thus capable of the most distant moons like Neso) is Neptune, with 116 million km, or 0.775 au; its great distance from the Sun amply compensates for its small mass relative to Jupiter (whose own Hill radius measures 53 million km). An asteroid from the asteroid belt will have a Hill sphere that can reach 220 000 km (for 1 Ceres), diminishing rapidly with decreasing mass. The Hill sphere of (66391) 1999 KW4
, a Mercury-crosser asteroid that has a moon (S/2001 (66391) 1), measures 22 km in radius.
A typical extrasolar "hot Jupiter", HD 209458 b, has a Hill sphere radius of 593,000 km, about 8 times its physical radius of approx 71,000 km. Even the smallest close-in extrasolar planet, CoRoT-7b, still has a Hill sphere radius (61,000 km) six times its physical radius (approx 10,000 km). Therefore, these planets could have small moons close in, although not within their respective Roche limits.>>