# Sphere of influence (astrodynamics)

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.

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

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

where

${\displaystyle a}$ is the semimajor axis of the smaller object's (usually a planet's) orbit around the larger body (usually the Sun).
${\displaystyle m}$ and ${\displaystyle M}$ 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 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

The table shows the values of the sphere of gravity of the bodies of the solar system in relation to the Sun.[1]:

Mercury 0.112 46
Venus 0.616 102
Earth 0.924 145
Moon 0.0661 38
Mars 0.576 170
Jupiter 48.2 687
Saturn 54.6 1025
Uranus 51.8 2040
Neptune 86.8 3525

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 ${\displaystyle \theta }$ from the massive body. A more accurate formula is given by

${\displaystyle r_{SOI}(\theta )\approx a\left({\frac {m}{M}}\right)^{2/5}{\frac {1}{\sqrt[{10}]{1+3\cos ^{2}(\theta )}}}}$

Averaging over all possible directions we get

${\displaystyle {\overline {r_{SOI}}}=0.9431a\left({\frac {m}{M}}\right)^{2/5}}$

## Derivation

Consider two point masses ${\displaystyle A}$ and ${\displaystyle B}$ at locations ${\displaystyle r_{A}}$ and ${\displaystyle r_{B}}$, with mass ${\displaystyle m_{A}}$ and ${\displaystyle m_{B}}$ respectively. The distance ${\displaystyle R=|r_{B}-r_{A}|}$ separates the two objects. Given a massless third point ${\displaystyle C}$ at location ${\displaystyle r_{C}}$, one can ask whether to use a frame centered on ${\displaystyle A}$ or on ${\displaystyle B}$ to analyse the dynamics of ${\displaystyle C}$.

Geometry and dynamics to derive the sphere of influence

Let's consider a frame centered on ${\displaystyle A}$. The gravity of ${\displaystyle B}$ is denoted as ${\displaystyle g_{B}}$ and will be treated as a perturbation to the dynamics of ${\displaystyle C}$ due to the gravity ${\displaystyle g_{A}}$ of body ${\displaystyle A}$. Due their gravitational interactions, point ${\displaystyle A}$ is attracted to point ${\displaystyle B}$ with acceleration ${\displaystyle a_{A}={\frac {Gm_{B}}{R^{3}}}(r_{B}-r_{A})}$, this frame is therefore non-inertial. To quantify the effects of the perturbations in this frame, one should consider the ratio of the perturbations to the main body gravity i.e. ${\displaystyle \chi _{A}={\frac {|g_{B}-a_{A}|}{|g_{A}|}}}$. The perturbation ${\displaystyle g_{B}-a_{A}}$ is also known as the tidal forces due to body ${\displaystyle B}$. It is possible to construct the perturbation ratio ${\displaystyle \chi _{B}}$ for the frame centered on ${\displaystyle B}$ by interchanging ${\displaystyle A\leftrightarrow B}$.

Frame A Frame B
Main acceleration ${\displaystyle g_{A}}$ ${\displaystyle g_{B}}$
Frame acceleration ${\displaystyle a_{A}}$ ${\displaystyle a_{B}}$
Secondary acceleration ${\displaystyle g_{B}}$ ${\displaystyle g_{A}}$
Perturbation, tidal forces ${\displaystyle g_{B}-a_{A}}$ ${\displaystyle g_{A}-a_{B}}$
Perturbation ratio ${\displaystyle \chi }$ ${\displaystyle \chi _{A}={\frac {|g_{B}-a_{A}|}{|g_{A}|}}}$ ${\displaystyle \chi _{B}={\frac {|g_{A}-a_{B}|}{|g_{B}|}}}$

As ${\displaystyle C}$ gets close to ${\displaystyle A}$, ${\displaystyle \chi _{A}\rightarrow 0}$ and ${\displaystyle \chi _{B}\rightarrow \infty }$, and vice versa. The frame to choose is the one that has the smallest perturbation ratio. The surface for which ${\displaystyle \chi _{A}=\chi _{B}}$ separates the two regions of influence. In general this region is rather complicated but in the case that one mass dominates the other, say ${\displaystyle m_{A}\ll m_{B}}$, it is possible to approximate the separating surface. In such a case this surface must be close to the mass ${\displaystyle A}$, denote ${\displaystyle r}$ as the distance from ${\displaystyle A}$ to the separating surface.

Frame A Frame B
Main acceleration ${\displaystyle g_{A}={\frac {Gm_{A}}{r^{2}}}}$ ${\displaystyle g_{B}\approx {\frac {Gm_{B}}{R^{2}}}+{\frac {Gm_{B}}{R^{3}}}r\approx {\frac {Gm_{B}}{R^{2}}}}$
Frame acceleration ${\displaystyle a_{A}={\frac {Gm_{B}}{R^{2}}}}$ ${\displaystyle a_{B}={\frac {Gm_{A}}{R^{2}}}\approx 0}$
Secondary acceleration ${\displaystyle g_{B}\approx {\frac {Gm_{B}}{R^{2}}}+{\frac {Gm_{B}}{R^{3}}}r}$ ${\displaystyle g_{A}={\frac {Gm_{A}}{r^{2}}}}$
Perturbation, tidal forces ${\displaystyle g_{B}-a_{A}\approx {\frac {Gm_{B}}{R^{3}}}r}$ ${\displaystyle g_{A}-a_{B}\approx {\frac {Gm_{A}}{r^{2}}}}$
Perturbation ratio ${\displaystyle \chi }$ ${\displaystyle \chi _{A}\approx {\frac {m_{B}}{m_{A}}}{\frac {r^{3}}{R^{3}}}}$ ${\displaystyle \chi _{B}\approx {\frac {m_{A}}{m_{B}}}{\frac {R^{2}}{r^{2}}}}$

The distance to the sphere of influence must thus satisfy ${\displaystyle {\frac {m_{B}}{m_{A}}}{\frac {r^{3}}{R^{3}}}={\frac {m_{A}}{m_{B}}}{\frac {R^{2}}{r^{2}}}}$ and so ${\displaystyle r=\left({\frac {m_{A}}{m_{B}}}\right)^{2/5}R}$ is the radius of the sphere of influence of body ${\displaystyle A}$