Short answers, pulled from the story.
The Hill sphere defines the region where an astronomical body dominates the attraction of satellites over other massive neighbors. For Earth and Moon systems, calculations show the radius changes depending on orbital position. At perigee, the Moon's Hill radius measures 61,000 kilometers while at apogee that same radius expands to 72,000 kilometers.
American astronomer George William Hill defined the concept based on earlier work by French astronomer Édouard Roche. Their combined efforts established this astronomical model for gravitational dominance. The term honors Hill's specific contribution to understanding restricted three-body problems.
Eccentricity plays a critical role in determining stability limits for any given system. When eccentricity remains negligible, the expression reduces to a simpler form used for most approximations. This relationship simplifies through binomial expansion to leading order in rho over r.
Neptune holds the largest Hill radius in our solar system at 115 million kilometers. Its great distance from the Sun compensates for smaller mass compared to Jupiter. Jupiter's own Hill radius measures only 53 million kilometers despite its massive size.
The shuttle possessed a Hill sphere of only 120 centimeters in radius at that height. A sphere of this size would need density greater than lead to fit inside itself. No stable orbit exists when the calculated radius falls below the object's physical dimensions.