The Science of Kerbal Space Program

Spheres of Influence

In Kerbal

Under Stable Orbits, there was brief mention of the idea of a sphere of influence in the calculations for conic motions. Spheres of influence are useful when combined with conic approximation to define region where a specific body is used as the focus for the path of a smaller body. Outside of the sphere, a more massive object is the focus of the object's movement.

The sphere of influence of an object is defined as

$$r_{SOI}=R({m_1/m_2})^{2/5}$$
where $r_{SOI}$ is the radius of the sphere of influence, $R$ is the distance between the two competing bodies and $m_1$ and $m_2$ are the masses of the smaller influencing body (such as a planet) and the larger influencing body (such as a star), respectively. For simple orbital path calculations, this is sufficient, and makes for easy simulation. Under this, the sphere of influence of Earth comes out to

or roughly 145 Earth radii. Within this range, objects tend to be influenced more by the gravitational force of Earth than the pull of the sun.

Reality

Sometimes, you need more than a simple approximation though. As any physics student knows, gravity extends infinitely, though at large distances decreases to the point of being nearly nonexistent. That nearly is important, though, because there still is a gravitational force. The practical result of this is that rather than tracking one force as one would in a sphere of influence, a computer is normally used to calculate the net force on a body over time. This requires integrating six differential equations for every body being tracked.

There is a reason though that a game that is meant to be played by the averge person uses spheres of influence instead of a more accurate gravity calculation: the power and time required for the calculations required is not something that a typical home computer can run though, and so simplifications must be made. With the exception of special cases, spheres of influence make for good approximations.