Saturday, September 30, 2023

# You will find 12,392 new solutions to the three-body problem

The so-called “three-body problem” has puzzled mathematicians for three centuries. Thanks to Newton’s laws, it is actually possible to easily describe the movement of two bodies in one orbit and to calculate in detail how the gravity of each of them will affect each other in the future. However, the problem becomes much more complex when a third object is added. So complex that it becomes insoluble.

The truth is that there is no one-size-fits-all solution to the problem, which depends on an enormous number of variables. The laws of physics state that by knowing the initial state of a system, it is possible to predict any future state of that system using the appropriate laws. But it is practically impossible to know the initial state in a system consisting of three bodies orbiting each other. It is a chaotic system in which anything is possible and the solution to which simply cannot be expressed in a formula.

For this reason, mathematicians have resorted to the “trick” of setting initial conditions themselves (which do not have to correspond to the real ones) and looking for possible solutions to these particular configurations. In 2017, for example, researchers found 1,223 new solutions to the three-body problem, doubling the number of previously known possibilities.

Now Ivan Hristov of Sofia University in Bulgaria and his colleagues have managed to “discover” thousands of new possible orbits, and all of them “work” by applying Newton’s laws. To achieve this, the team ran an optimized version of the algorithm used in the 2017 paper on a supercomputer and discovered 12,392 new solutions. According to Hristov, if he repeated the search with even more powerful hardware, he could find “up to five times more.” The study can now be viewed on the pre-publication server arXiv.

All solutions assume an initial state in which the three bodies are stationary, then enter free fall and allow gravity to attract them towards each other. Then their momentum carries them side by side before they slow, stop, and pull each other back together. The team found that assuming there was no friction, the pattern would repeat indefinitely.

Solutions to the three-body problem are of great interest to astronomers because they can describe how any three celestial objects (be they stars, planets, or moons) can maintain a stable orbit. However, it remains to be seen how stable the new solutions are when small influences from other distant bodies and other real-world perturbations are also taken into account.

According to Hristov, the astronomical significance of these solutions “will be better known after studying stability, which is very important.” Regardless of whether they are stable or unstable, they are of great theoretical interest. “They have a very nice spatial and temporal structure.”

Of course, most, if not all, of the 12,392 solutions found by Hristov and his colleagues require such precise initial conditions that they are unlikely to ever occur in nature. And if they actually occurred, many of them would be unstable, and after a complex gravitational interaction, the three-body system would split into a binary system (with only two bodies), while the third, the least massive of the three, would be lost in the immensity of the space.

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