A fresh, abstract approach to solving the statistical model for the famous “three-body problem”. This promising idea tackles chaos between co-orbiting bodies by treating space as an eight-dimensional region.
Order out of chaos
Isaac Newton’s three laws of motion can elegantly describe the basic physics of how things interact in the universe. These equations can be used to describe how Earth orbits the sun or how the moon orbits the Earth.
However, when Newton tried to introduce a third object to a pair of orbiting objects — such as a relationship between Earth, moon, and the sun — his equations broke down.
Immediately, a power struggle ensues that throws the entire system into chaos. For instance, if you imagine three stars of similar size on a collision course, even if you knew their precise velocities and locations, there’s no way to predict how exactly the stars’ fates will unfold. The third star could loop furiously around the center of gravity of the other stellar pair, before escaping into space. Or maybe the star that gets ejected falls back toward the pair again.
In any event, there’s no way to know for sure — and that’s the only certainty we have since Henry Poincaré demonstrated mathematically in the 19th century that there is no equation that could accurately predict the positions of all bodies at all future moments. Essentially, Poincaré showed that a general solution to the problem is essentially impossible due to chaotic dynamics This conundrum, known as the “three-body problem” has remained unsolvable to this day.
But that doesn’t mean that today scientists don’t know how to predict, at least to a point and with some margin of error, how three bodies will interact. Rather than trying to solve the three-body problem by force, scientists over the years have developed methods that approximate the probability of certain outcomes.
The most recent such endeavor is also the most spectacular. Barak Kol, a physicist at the Hebrew University of Jerusalem, simplified the probabilistic framework of the three-body problem by renouncing conventional 3D space. Instead, the physicist opted for an abstract realm known as “phase space”, where each spot represents one possible configuration of the three stars (position, velocity, mass), resulting in an eight-dimensional playground.
In a chaotic system such as the three-body problem, there is never just one possible outcome. But using Kol’s framework, it is possible to explore every chaotic path by statistically computing the volume inside the phase space for each chaotic motion.
Previously, Dr. Nicholas Stone, also of the Hebrew University of Jerusalem, published a 2019 study in which they described an approximate solution to the three-body problem by focusing on the boundary of the chaotic space where the three-body system transitions to the motion of a predictable two-body system — in other words, just about when the third body is ejected.
Kol’s framework is quite different, involving “holes” in the chaotic system. While Stone imagined a chaotic region as a balloon, Kol’s approach involves discrete holes like Swiss cheese. These holes represent regions where chaos is more likely to switch on and off.
Finally, Kol introduces the concept of chaotic absorptivity, which describes the odds that a stable stellar pair will plunge into chaos if a third star is introduced.
The method described in the study, which appeared in Celestial Mechanics and Dynamical Astronomy, can be used to answer all sorts of statistical questions. For instance, the framework can be used to assess when a trio will eject a member or what the odds are that this third member will be ejected out of the system at certain speeds.
In the universe, it is quite common to find three-member systems, such as trios of stars and black holes. If scientists are ever to fully understand how these systems formed or will continue to evolve into the future, they have to tackle the dreaded three-body problem. Kol’s framework, which still requires a lot of testing, is highly promising so far. In computer simulations that performed millions of iterations, this abstract approach accurately nailed the outcomes of simulated forecasts.