A sphere of influence (SOI) in astrodynamics and astronomy is the oblate-spheroid-shaped region around a celestial body where the primary gravitational influence on an orbiting object is that body. This is usually used to describe the areas in the Solar System where planets dominate the orbits of surrounding objects (such as moons) despite the presence of the much more massive (but distant) Sun. In the patched conic approximation, used in estimating the trajectories of bodies moving between the neighbourhoods of different masses using a two body approximation (ellipses and hyperbolae), the SOI is taken as the boundary where the trajectory switches which mass field it is influenced by.
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The general equation describing the radius of the sphere
where
In the patched conic approximation, once an object leaves the planet's SOI, the primary/only gravitational influence is the Sun (until the object enters another body's SOI). Because the definition of rSOI relies on the presence of the Sun and a planet, the term is only applicable in a three-body or greater system and requires the mass of the primary body to be much greater than the mass of the secondary body. This changes the three-body problem into a restricted two-body problem.
Table of selected SOI radii
These are all taken relative to the Sun, except for the Moon, which is relative to the Earth.
Increased accuracy on the SOI
The Sphere of influence is, in fact, not quite a sphere. The distance to the SOI depends on the angular distance
Averaging over all possible directions we get