# Sphere of influence (astrodynamics)

For the term relating to black holes, see sphere of influence (astronomy).

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 a more general sense, the patched conic approximation is only valid within the SOI.

The general equation describing the radius of the sphere $r_{SOI}$ of a planet:

$r_{SOI} = a\left(\frac{m}{M}\right)^{2/5}$

where

$a$ is the semimajor axis of the smaller object's (usually a planet's) orbit around the larger body (usually the Sun).
$m$ and $M$ are the masses of the smaller and the larger object (usually a planet and the Sun), respectively.

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. It 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 planetary SOI radii

 Body SOI radius SOI radius - (km) (body radii) Mercury $112 \times 10^3$ 45 Venus $616 \times 10^3$ 100 Earth $925 \times 10^3$ 145 Moon $66.1 \times 10^3$ 38 Mars $577 \times 10^3$ 170 Jupiter $48.2 \times 10^6$ 677 Saturn $54.8 \times 10^6$ 901 Uranus $51.7 \times 10^6$ 2025 Neptune $86.7 \times 10^6$ 3866