# Sphere of influence (astrodynamics)

A sphere of influence (SOI) in astrodynamics and astronomy is the oblate spheroid-shaped region where a particular celestial body exerts the main gravitational influence on an orbiting object. 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 bodies 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. It is not to be confused with the sphere of activity which extends well beyond the sphere of influence.

## Models

The most common base models to calculate the sphere of influence is the Hill sphere and the Laplace sphere, but updated and particularly more dynamic ones have been described. The general equation describing the radius of the sphere $r_{\text{SOI}}$ of a planet: $r_{\text{SOI}}\approx a\left({\frac {m}{M}}\right)^{2/5}$ 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 r$_{SOI}$ 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 (with the exception of the Moon which is reported relative to Earth):
| Body | SOI | SOI | SOI | Body Diameter | Body Diameter | Body Mass (10$^{24}$ kg) | Distance from Sun | Distance from Sun | Distance from Sun |
| Body | (10$^6$ km) | (mi) | (radii) | (km) | (mi) | Body Mass (10$^{24}$ kg) | (AU) | (10$^6$ mi) | (10$^6$ km) |
| Mercury | 0.117 | 72,700 | 46 | 4,878 | 3,031 | 0.33 | 0.39 | 36 | 57.9 |
| Venus | 0.616 | 382,765 | 102 | 12,104 | 7,521 | 4.867 | 0.723 | 67.2 | 108.2 |
| Earth + Moon | 0.929 | 577,254 | 145 | 12,742 (Earth) | 7,918 (Earth) | 5.972 (Earth) | 1 | 93 | 149.6 |
| Moon (Luna) | 0.0643 | 39,993 | 37 | 3,476 | 2,160 | 0.07346 | See Earth + Moon | See Earth + Moon | See Earth + Moon |
| Mars | 0.578 | 359,153 | 170 | 6,780 | 4,212 | 0.65 | 1.524 | 141.6 | 227.9 |
| Jupiter | 48.2 | 29,950,092 | 687 | 139,822 | 86,881 | 1900 | 5.203 | 483.6 | 778.3 |
| Saturn | 54.5 | 38,864,730 | 1025 | 116,464 | 72,367 | 570 | 9.539 | 886.7 | 1,427.0 |
| Uranus | 51.9 | 32,249,165 | 2040 | 50,724 | 31,518 | 87 | 19.18 | 1,784.0 | 2,871.0 |
| Neptune | 86.2 | 53,562,197 | 3525 | 49,248 | 30,601 | 100 | 30.06 | 2,794.4 | 4,497.1 |
An important understanding to be drawn from this table is that "Sphere of Influence" here is "Primary". For example, though Jupiter is much larger in mass than say, Neptune, its Primary SOI is much smaller due to Jupiter's much closer proximity to the Sun.

## 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 $\theta$ from the massive body. A more accurate formula is given by $r_{\text{SOI}}(\theta )\approx a\left({\frac {m}{M}}\right)^{2/5}{\frac {1}{\sqrt[{10}]{1+3\cos ^{2}(\theta )}}}$
Averaging over all possible directions we get: ${\overline {r_{\text{SOI}}}}=0.9431a\left({\frac {m}{M}}\right)^{2/5}$

## Derivation

Consider two point masses $A$ and $B$ at locations $r_{A}$ and $r_{B}$, with mass $m_{A}$ and $m_{B}$ respectively. The distance $R=|r_{B}-r_{A}|$ separates the two objects. Given a massless third point $C$ at location $r_{C}$, one can ask whether to use a frame centered on $A$ or on $B$ to analyse the dynamics of $C$.
Consider a frame centered on $A$. The gravity of $B$ is denoted as $g_{B}$ and will be treated as a perturbation to the dynamics of $C$ due to the gravity $g_{A}$ of body $A$. Due to their gravitational interactions, point $A$ is attracted to point $B$ with acceleration $a_{A}={\frac {Gm_{B}}{R^{3}}}(r_{B}-r_{A})$, 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. $\chi _{A}={\frac {|g_{B}-a_{A}|}{|g_{A}|}}$. The perturbation $g_{B}-a_{A}$ is also known as the tidal forces due to body $B$. It is possible to construct the perturbation ratio $\chi _{B}$ for the frame centered on $B$ by interchanging $A\leftrightarrow B$.
| | Frame A | Frame B |
| Main acceleration | $g_{A}$ | $g_{B}$ |
| Frame acceleration | $a_{A}$ | $a_{B}$ |
| Secondary acceleration | $g_{B}$ | $g_{A}$ |
| Perturbation, tidal forces | $g_{B}-a_{A}$ | $g_{A}-a_{B}$ |
| Perturbation ratio $\chi$ | $\chi _{A}={\frac {|g_{B}-a_{A}|}{|g_{A}|}}$ | $\chi _{B}={\frac {|g_{A}-a_{B}|}{|g_{B}|}}$ |
As $C$ gets close to $A$, $\chi _{A}\rightarrow 0$ and $\chi _{B}\rightarrow \infty$, and vice versa. The frame to choose is the one that has the smallest perturbation ratio. The surface for which $\chi _{A}=\chi _{B}$ separates the two regions of influence. In general this region is rather complicated but in the case that one mass dominates the other, say $m_{A}\ll m_{B}$, it is possible to approximate the separating surface. In such a case this surface must be close to the mass $A$, denote $r$ as the distance from $A$ to the separating surface.
| | Frame A | Frame B |
| Main acceleration | $g_{A}={\frac {Gm_{A}}{r^{2}}}$ | $g_{B}\approx {\frac {Gm_{B}}{R^{2}}}+{\frac {Gm_{B}}{R^{3}}}r\approx {\frac {Gm_{B}}{R^{2}}}$ |
| Frame acceleration | $a_{A}={\frac {Gm_{B}}{R^{2}}}$ | $a_{B}={\frac {Gm_{A}}{R^{2}}}\approx 0$ |
| Secondary acceleration | $g_{B}\approx {\frac {Gm_{B}}{R^{2}}}+{\frac {Gm_{B}}{R^{3}}}r$ | $g_{A}={\frac {Gm_{A}}{r^{2}}}$ |
| Perturbation, tidal forces | $g_{B}-a_{A}\approx {\frac {Gm_{B}}{R^{3}}}r$ | $g_{A}-a_{B}\approx {\frac {Gm_{A}}{r^{2}}}$ |
| Perturbation ratio $\chi$ | $\chi _{A}\approx {\frac {m_{B}}{m_{A}}}{\frac {r^{3}}{R^{3}}}$ | $\chi _{B}\approx {\frac {m_{A}}{m_{B}}}{\frac {R^{2}}{r^{2}}}$ |
The distance to the sphere of influence must thus satisfy ${\frac {m_{B}}{m_{A}}}{\frac {r^{3}}{R^{3}}}={\frac {m_{A}}{m_{B}}}{\frac {R^{2}}{r^{2}}}$ and so $r=R\left({\frac {m_{A}}{m_{B}}}\right)^{2/5}$ is the radius of the sphere of influence of body $A$

## Gravity well

Gravity well (or funnel) is a metaphorical concept for a gravitational field of a mass, with the field being curved in a funnel-shaped well around the mass, illustrating the steep gravitational potential and its energy that needs to be accounted for in order to escape or enter the main part of a sphere of influence.
An example for this is the strong gravitational field of the Sun and Mercury being deep within it. At perihelion Mercury goes even deeper into the Sun's gravity well, causing an anomalistic or perihelion apsidal precession which is more recognizable than with other planets due to Mercury being deep in the gravity well. This characteristic of Mercury's orbit was famously calculated by Albert Einstein through his formulation of gravity with the speed of light, and the corresponding general relativity theory, eventually being one of the first cases proving the theory.