Mathematicians Found 12,000 Solutions to the Notoriously Hard Three-Body Problem
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Three-body problems—the analytical expressions of three celestial bodies in a stable orbit—have beguiled mathematicians for centuries.
In 2017, Chinese mathematicians discovered more than a thousand solutions by running 16 million orbits into a supercomputer.
Now, scientists from Sofia University in Bulgaria have expanded on that work and found an additional 12,000 solutions. They say that as many as five times more could be discovered.
When it comes to the movements of meteorites, stars, planets, and black holes, calculating the motion of these bodies on long timescales is incredibly difficult due a massive number of variables that simply can’t be defined by mathematical equations. This is what’s known as an “n-body problem.” In the late 17th century, the work of Johannes Kepler and Isaac Newton helped solve the two-body problem, but when a third party enters this mix, things get complicated.
In the late 19th century, mathematicians Heinrich Bruns and Henri Poincaré stated that the three-body problem couldn’t be expressed analytically due the problem’s chaotic nature. Mathematicians tame this chaos by making approximations of this celestial motion. For example, sometimes the mass of a small body will be largely ignored due to the more influential mass of the larger bodies, which is known as a restricted three-body problem. Over time, mathematicians devised specialized three-body solutions, but that number has grown exponentially in the age of supercomputers.
In 2017, for example, researchers from China discovered 1,223 solutions to the three-body problem by testing 16 million orbits using a supercomputer. Now, researchers from Sofia University in Bulgaria have further expanded upon that 2017 algorithm and discovered more than 12,000 additional solutions.
The lead researcher Ivan Hristov told New Scientist that access to even more powerful supercomputers could discover five times as many solutions to the infamously tricky three-body problem. His team’s work was posted to the preprint server arXiv in late August.
Understanding the extremely subtle movements among three orbital bodies is important for space travel—when space agencies need to send satellites, landers, or spacecraft to a distant planet, they need to know how all the pieces of a system will play together. Hristov’s three-body solutions start with three bodies at rest before entering free fall and being pulled by one another’s gravity. The “solutions” were instances when these three bodies found a way to maintain an orbit around one another.
Many of these solutions look like a jumbled mess of lines. In fact, other than a few known solutions (including one that introduced the concept of Lagrange points), it’s highly unlikely that any of Hristov’s three-body solutions exists in nature. That’s because, in most cases, three-body problems quickly become two-body ones as the object with the smallest mass is ejected from the system.
But the study of this chaotic math still gives astronomers a deeper understanding about how the universe works—even though it’s mathematically impossible to describe.
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