Why Can't We Know the Orbits of Three Celestial Bodies?

In the science fiction novel The Three-Body Problem, aliens live in a triple star system. Their planet has no predictable days, nights, or seasons. Because they can't make sense of natural phenomena, science never advances. With barely enough technology to survive, they set out to find a planet with a stable orbit and relocate.

The fictional premise is rooted in a real scientific challenge. When two celestial bodies orbit each other, their paths are easily predicted using Newtonian mechanics and the law of universal gravitation. This is called the two-body problem. Add one more body to the mix, and something surprising happens. What seems like a minor change turns out to be an entirely different class of problem—far more complex than anything two bodies can produce.

Isaac Newton himself attempted to solve the three-body problem in the Principia. After him, Joseph-Louis Lagrange, Pierre-Simon Laplace, and Leonhard Euler all struggled to find a general solution. It wasn't until the 19th century that Henri Poincaré proved that no general solution exists. The orbits of three celestial bodies simply cannot be expressed as a single equation that works in all cases.

Solutions do exist, but only under very specific conditions. One well-known example is the Lagrangian point. When two bodies A and B orbit each other, there are five points in space where gravitational pull and centrifugal force balance out perfectly. If a third object C sits at one of these points—and its mass is negligibly small compared to A and B—the system can still be solved. This is exactly how the Trojan asteroids around Jupiter formed. They drifted into the Lagrangian points of the Sun and Jupiter and never left. Hundreds of thousands of them now cluster there. Lagrangian points also exist in the Sun-Earth and Earth-Moon systems, and we already use them—for satellites, space telescopes, and as waypoints for spacecraft changing orbits.

Outside these special cases, the best available approach is approximation. Algorithms can get very close to the true solution, but they can never fully eliminate error. For short timescales, that error is small enough to ignore. For the distant future, prediction becomes impossible. The three bodies are extraordinarily sensitive to their initial conditions—tiny differences in mass, position, or velocity compound into vastly different outcomes over time. This is the butterfly effect, the same phenomenon at the heart of chaos theory.

If scientists had simply accepted the three-body problem as unsolvable, space exploration would never have happened. Calculating a probe's trajectory requires solving it. But the impact went further than that. Wrestling with the three-body problem produced chaos theory, phase space, Lagrangian points, numerical analysis, and dynamical systems theory—tools that now shape weather forecasting, financial modeling, medicine, ecology, and more. Some of the most useful scientific frameworks we have today grew directly out of a problem that, strictly speaking, has no solution.

Disclosure: I'm participating in a paid supporter activity for SOAK, a science content platform, and this post was written as part of that activity.