Short answers, pulled from the story.
The sphere of influence is a region of space shaped like an oblate spheroid where one celestial body exerts the main gravitational influence on any orbiting object. This area defines the boundaries within which a planet dominates the orbits of surrounding objects such as moons despite the presence of the much more massive but distant Sun.
Scientists use several base models to calculate the radius including the Hill sphere and the Laplace sphere as the most common frameworks. Updated models by researchers like Gleb Chebotaryov offer dynamic alternatives to traditional methods while a general equation describes the radius using the semimajor axis of the smaller object's orbit around the larger body.
Mission planners rely on the patched conic approximation to estimate trajectories of bodies moving between neighborhoods of different celestial objects. Within this framework the sphere of influence serves as the boundary where trajectory switches from one mass field to another once an object leaves a planet's SOI and the Sun becomes the primary or only gravitational influence until entering another body's domain.
Jupiter exhibits a primary sphere of influence of 48.2 million kilometers despite being much closer to the Sun than Neptune. Mercury shows a radius of just 72,700 kilometers due to its proximity to our star while the Moon has a measured radius of 39,993 kilometers when reported relative to Earth.
Physicists derive the sphere of influence by analyzing perturbation ratios between tidal forces and main body gravity. The surface separating these regions occurs where the perturbation ratio equals one for both frames defining the transition point between competing gravitational influences in multi-body systems.