Newton’s three-body problem explained - Fabio Pacucci

Mathematical equations can predict the motions of two gravitating masses. But when a third or more are introduced, our analytical tools fall short.

It is impossible to write down all the terms of a general formula that can accurately describe the motion of three or more gravitating objects. The problem is in knowing how many unknown variables an n-body system contains.

Thanks to Isaac Newton, we can write a set of equations to describe the gravitational force that acts between bodies.

But, when we try to find a general solution for the unknown variables in these equations, we are faced with a mathematical limitation.

For every unknown, there must be one equation that describes it. A two-body system is solvable because its gravitational attraction influences the paths they take in such a way that a simple mathematical formula can describe it. But three or more orbiting objects leaves us with more variables than equations that can describe them.

A system of three stars could come crashing or flung out of orbit after a long time of apparent stability.

Almost every possible configuration is unpredictable on long timescales. A slight difference in position and velocity could have a large range of potential outcomes. Physicists describe this behaviour as chaotic - an important characteristic of n-body systems. But it is still deterministic; it’s not random.

Advancements in computer simulations help to avoid disaster. By approximating solutions with powerful processors, we can better predict the motion of n-body systems on long time scales.  

When one body in a group of three is so light it has no significant force on the other two, the system behaves as a two-body system. This is known as the “restricted three-body problem.”

The restricted three-body system is useful to describe an asteroid in the Earth-Sun gravitational field or a small planet in the field of a black hole and a star.