What Is the Hill Sphere?

The Hill sphere defines the region around an astronomical body where its own gravity dominates over tidal forces from a more massive object it orbits. Inside this sphere a planet can retain satellites, while beyond it the star’s gravity prevails. The concept is named after the American astronomer George William Hill, who studied three-body motion in the 19th century. Knowing the size of the Hill sphere helps astronomers understand why certain moons remain bound to their planets and why spacecraft leaving Earth’s sphere of influence require additional energy to transition to heliocentric trajectories.

The Hill radius depends on the masses involved and the orbital separation. For a planet of mass orbiting a star of mass at semi-major axis , the radius is given by . This relation emerges from balancing the gravitational pull of the planet against the differential pull of the star on an object located along the line connecting them. The cube root illustrates how even a small increase in orbital distance or planetary mass can significantly expand the gravitational sphere of influence.

Because the definition assumes circular orbits and treats the star as overwhelmingly massive, the Hill sphere is an approximation. Real planetary systems contain eccentric orbits, additional planets, and stellar oblateness, all of which nudge the effective region of gravitational dominance. Nevertheless, the Hill radius offers a useful first-order estimate in mission design and planetary dynamics. When designing satellite constellations or planning gravitational assists, engineers ensure that intended paths lie safely inside or outside the Hill sphere, depending on whether capture or escape is desired.

Objects orbiting within about half the Hill radius tend to remain stable over long times. Numerical simulations show that distant satellites beyond this limit may be stripped away by stellar tides or perturbations from other planets. This calculator therefore reports both the Hill radius and the commonly referenced stability limit of of that value. For Earth, the Hill radius is roughly 1.5 million kilometers, yet the Moon orbits at only a quarter of that distance, offering a comfortable margin of safety.

Example Hill Radii in Our Solar System

The table below illustrates how the Hill sphere varies for several planets orbiting a one-solar-mass star. The values draw on average orbital distances and masses to reveal why giant planets can command enormous retinues of moons while small, inner planets struggle to hold onto satellites.

Planet	a (AU)	Hill Radius (106 km)	Stable Zone (106 km)
Mercury	0.39	0.22	0.11
Earth	1.00	1.50	0.75
Jupiter	5.20	53.1	26.6
Neptune	30.0	115.8	57.9

Mercury’s Hill sphere is tiny, barely a couple hundred thousand kilometers, which explains its lack of natural satellites. Earth enjoys a broader domain, yet still modest compared with the gas giants. Jupiter’s huge mass grants it a gravitational reach exceeding fifty million kilometers, ample room for its dozens of moons and captured asteroids. Farther out, Neptune’s slow orbit enlarges its Hill sphere despite its smaller mass relative to Jupiter. These examples show how both mass and orbital radius combine to set the boundary for satellite stability.

In exoplanetary systems, estimating Hill radii assists in predicting whether discovered planets could host moons or maintain ring systems. Astronomers searching for exomoons often compute the Hill sphere to constrain possible orbital distances. A large Hill radius increases the odds that a transiting exomoon might produce a detectable signature. Likewise, planetary scientists use the concept to study Trojan asteroids that occupy stable Lagrange points near the edge of the Hill sphere where stellar and planetary gravitational pulls balance.

Multistar systems complicate the picture. A planet orbiting one star in a binary must contend not only with its host but also with the companion star, which can shrink the effective Hill sphere or distort it into elongated lobes. Spacecraft traveling in such environments require careful trajectory planning to avoid unintended escapes. Additionally, eccentric orbits cause the Hill radius to expand and contract throughout the orbital cycle. At periapsis the radius reaches a minimum, making satellites most vulnerable to tidal stripping when the planet passes closest to its star.

The calculator above simplifies these complexities by using the classic formula and assuming circular orbits around a single star. Users enter masses in solar and Earth units and the orbital semi-major axis in astronomical units. Internally, the script converts to SI units, evaluates the Hill radius, and reports the value in both kilometers and a fraction of the orbital radius. The stable zone is given as half the Hill radius, reflecting common dynamical studies. This straightforward approach provides an educational baseline for understanding gravitational spheres of influence.

While the Hill sphere sets an outer boundary for satellite stability, it does not guarantee that all orbits inside it are safe. Collisions, resonances, and perturbations from other planets can destabilize satellites even well within the nominal limit. For detailed mission planning, analysts employ more sophisticated numerical integrations that account for gravitational harmonics and non-gravitational forces. Nevertheless, the Hill radius remains a foundational concept in celestial mechanics, offering a quick sense of scale for planetary systems and a starting point for deeper exploration.