The Three-Body Problem Explained: Why Three Objects Are So Hard to Predict

The simple law behind a difficult problem

Newton's law of universal gravitation states that any two objects with mass attract each other. The strength of that attraction depends on their masses and the distance between them:

In this formula, $F$ is the gravitational force, $G$ is the gravitational constant, $m_1$ and $m_2$ are the two masses, and $r$ is the distance between them.

With only two objects, such as a star and a planet, the resulting motion can be described relatively neatly. Under ideal conditions, the orbit is a circle, ellipse, parabola, or hyperbola. Once the starting position and velocity are known, the future path can be calculated.

What changes when a third object is added?

Add one more object and every body now attracts two others. The first object changes the motion of the second and third, while their changing positions immediately alter the force acting on the first.

The distances, directions, forces, accelerations, and velocities all change together. There is no longer one fixed orbit that can be calculated independently of the rest of the system.

The equations governing the system are known. The real difficulty is that there is no single practical closed-form solution that works for every possible set of starting conditions in the way the standard two-body solution does.

Why three bodies are much harder than two

Consider the Sun, Earth, and Moon. The Sun strongly attracts both Earth and the Moon. Earth keeps the Moon in orbit, but it also changes the Moon's path around the Sun. The Moon is much less massive than Earth, yet it still causes Earth to move slightly.

A computer handles these connected effects by dividing time into very small intervals. During each interval, it repeats a cycle of calculations:

The smaller the time interval $\Delta t$, the more accurately the calculation can usually follow the real motion. However, predicting years or centuries of movement may require millions of repeated steps.

Chaos does not mean randomness

Many three-body systems are extremely sensitive to their initial conditions. Two systems may begin with positions and velocities that differ by an almost invisible amount, yet their future paths can eventually become completely different.

In a simplified description, the growth of a small difference can be represented as:

Here, ${\delta }_0$ represents the initial difference between two systems, while $\delta \left(t\right)$ represents the difference after time has passed.

A short history of the problem

Does the problem have no solution?

Saying that the three-body problem is “unsolved” can be misleading. Scientists can calculate the motion of a specific system with very high accuracy over a useful period of time. What is missing is one simple general formula that immediately describes every possible three-body system for all time.

Special exact and periodic solutions also exist. In some of them, all three bodies follow repeating paths. However, these patterns usually require carefully selected initial positions and velocities.

Lagrange points: useful special solutions

An important simplified model is the restricted three-body problem. Two massive bodies, such as the Sun and Earth, orbit each other while the third object is assumed to be too small to noticeably affect them.

This model contains five special regions called Lagrange points. At these locations, the gravitational effects of the two large bodies and the orbital motion of the smaller object combine in a useful way.

These points are not simply motionless parking places. Spacecraft usually follow controlled orbits around them and occasionally adjust their course. The James Webb Space Telescope, for example, operates in an orbit around the Sun–Earth L2 point.

What can this mathematics help us solve?