# Sphere of influence

The sphere of influence, or SOI, indicates the spherical space around a celestial body in which it has sole gravitational influence on a craft, Kerbal, or other object. This differs from real-world physics in which gravitational forces from multiple bodies may simultaneously have significant effects on an object.

This decision by the game developers at Squad allows simplifying the n-body problem to a one-body problem. In addition to giving predictability to contouring trajectories of an object in flight, it is easier for the player to understand. Although the two-body problem is also solvable, it isn't used in Kerbal Space Program because the orbits of the celestial bodies can't be changed.

The general equation describing the radius of the sphere (rSOI) of a smaller celestial body orbiting around a larger one:

${\displaystyle r_{SOI}=a\left({\frac {m}{M}}\right)^{2/5}}$
Where:
• ${\displaystyle a}$ is the semi-major axis of the smaller object's orbit around the larger body.
• ${\displaystyle m}$ and ${\displaystyle M}$ are the masses of the smaller and the larger object, respectively.

## One-body problem

Kerbal Space Program defines around each celestial body of the Kerbol System a spherical area in which this celestial body is the only body whose attraction has any influence on objects in flight. The trajectory is thus almost completely predictable.

All spheres of influence are arranged in a hierarchy to determine what SOI an object is currently in. The hierarchy from lowest to highest order is moons, then planets or dwarf planets, and lastly the star Kerbol's sphere of influence. The lowest order body whose SOI a craft is in has sole gravitational influence over it. This means there are three possible cases:

1. If a craft is within the SOI of a moon, any gravitational influence from its parent planet and from Kerbol will be ignored.
2. If a craft is within the SOI of a planet but outside the SOI of any moon of that planet, gravitational influence of all its moons and of Kerbol will be ignored.
3. If a craft is not within the SOI of any planet or any moon, only Kerbol's gravitational influence will be in effect.

Again, this model of sole gravitational influence is not realistic but it makes a more playable game.

Additionally, planets and moons orbit on predefined paths around their higher-order bodies. Physics calculations are not performed on celestial bodies, nor can forces be applied to them.

## N-body problem

Each body with mass exerts a mutual attractive force known as gravitation. In reality, gravitational fields are infinite in distance, although field strength decreases rapidly with distance. Nevertheless, the gravitational field of every object affects every other object's motion across the entire universe.

A prime example of a three-body system is the Earth-Moon system together with the Sun. Each object in the vicinity is influenced by the attractions of these three celestial bodies, all of which influence trajectories differently as the object moves relative to each. The calculation of such trajectories is very complex; results generally are only approximate. But such calculations had to be handled for the success of the moon landings which inspire much of KSP.

## Two-body problem

The two-body problem is basically a simplified version of the n-body problem. Each body pair is observed which then orbits around their common center of mass, barycenter. For example a moon would not orbit the center of the planet which would result in a small orbit of the planet.

## Effects

These simplifications result in particular trajectories, such as Lagrange points, not existing in-game.

Furthermore, trajectories have consistent curvatures because they can not be disturbed by changing amounts of influence while travelling relative to multiple bodies. Otherwise a supposedly stable trajectory could suddenly, after some time, turn out to be unstable and crash a rocket on a celestial body or be catapulted out of the system. This is because the game has to constantly check if which sphere of influence the craft is. If that change is not detected fast enough it might be skipping the change completely and the craft is not entering the sphere of influence. It might as well be deep inside the other sphere of influence which results in incorrect orbital parameters, because the object followed the trajectory around the parent body.

It is also possible for an object to have a trajectory which is mathematically a section of a ellipse, with eccentricity less than 1, but is treated as escaping because the apoapsis is outside the body's SOI.

## Restricted three-body problem

Victor Szebehely outlined The Restricted 3 Body Problem in his book "Theory of Orbits", Academic Press, 1967. On pages 133 to 138 he gives a power series expansion from a 5th order polynomial in terms of ${\displaystyle v=\mu /\left(3*(1-\mu )\right)^{1/3}}$ to the 6th power of ${\displaystyle v}$. As ${\displaystyle \mu }$ gets smaller, the series converges more quickly. In this case ${\displaystyle \mu }$ is the mass fraction of the orbiting body. ${\displaystyle \mu =M2/(M1+M2)}$ where ${\displaystyle M2}$ is mass of orbiting body and ${\displaystyle M1}$ is the mass of central body.

#### Examples:

Distance to L2 from orbiting body:

• ${\displaystyle v(1+v/3-v^{2}/9-31v^{3}/81-119v^{4}/243-v^{5}/9)+O(v^{7})}$

Distance to L1 from orbiting body:

• ${\displaystyle v(1-v/3-v^{2}/9-23v^{3}/81+151v^{4}/243-v^{5}/9)+O(v^{7})}$

Note: Here L1 is the region between M1 and M2. L2 is the region on the far side of M2. Those checking out Szebehely's book will find that he names Lagrangian points vice versa. This problem predicted the Trojan moons existence before they were first seen.

## Trivia

• For the moons Mun, Tylo, Vall and Laythe, the SOIs are smaller than the semi-major axes of their synchronous orbit. Hence, synchronous orbits over these moons are not possible.
• Despite the exact solvability of all path equations, calculating them in-game gives unexpectedly changing and struggling trajectories. This is due to the technical limits of inaccuracy with floating point values in the computer.