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The motion of the sun, Earth, and the moon is one of the oldest problems in physics. The three-body problem has been a focus of scientific inquiry.
When one massive object comes close to another, their relative motion follows a trajectory dictated by their mutual attraction, but as they move along and change their positions, the forces between them, which depend on their mutual positions, also change, which affects their trajectory. For two bodies, the Earth would follow a specific curve, which can be accurately described. The system can't be predicted because it becomes chaotic and unpredictable under the influence of a third object. The three-body problem has been known for over 400 years, but a neat mathematical description is still missing.
In 1889, King Oscar II of Sweden offered a prize to anyone who could provide a general solution to the three-body problem, which was solved by physicists in the past. The French mathematician Henri Poincaré won the competition. He ruined any hope for a full solution by proving that interactions are chaotic, and that the final outcome is random.
Scientists can't predict what will happen during a close interaction between two stars and a third star because of the lack of a solution to the three-body problem. The simulations show that when there is an interaction, there is a chaotic phase in which all three bodies pull on each other violently until one star is ejected far from the other two. If the third star is on a bound path, it eventually comes back down to the binaries. In the second phase, one of the stars escapes and never returns.
The paper was accepted for publication in Physical Review X and used randomness to provide a solution to the entire process. They calculated the probability of each interaction's outcome. The random nature of chaos allows calculation of the probability that a triple interaction ends in one way or another. The theory of random walks could be used to model the entire series of close approaches. The term was given to mathematicians by thinking about how a drunk would walk, with each step, the drunk doesn't realize where they are and takes the next step in some random direction.
The triple system behaves the same way. One of the stars is ejected randomly after each close encounter, but with the three stars collectively still saving the system's energy and momentum. A series of close encounters could be considered a drunkard's walk. Like a drunk's step, a star is ejected randomly, comes back, and another star is ejected to a different direction, and then comes back again, and again, until a star is completely ejected and never returns.
The same phenomenon of chaos that Poincaré discovered is why the weather is so hard to predict. Think about the time when a 70 percent chance of rain ended up as glorious sunshine in reality. To get a proper long-term forecast, the meteorology profession needs to account for the probabilities of all possible types of weather in the intervening days, and only then can they come up with a proper forecast.
Ginat and Perets showed how this could be done by calculating the probability of each phase and then using the theory of random to compose the individual phases.
Mr. Ginat said that they came up with the random walk model while he was an undergrad. When I started my PhD, we decided to expand the essay and publish it, even though we didn't publish it at the time.
The three-body problem has been studied by several research groups in recent years, including Nicholas Stone of the Hebrew University in Jerusalem and Nathan Leigh of the American Museum of Natural History. The entire, multi-stage, three-body interaction is solved statistically with the current study by Ginat and Perets.
Prof. Perets said that this has important implications for our understanding of gravitational systems, and in particular, cases where many encounters between three stars occur, like in dense clusters of stars. In such regions, many exotic systems form through three-body encounters, leading to the collision of stars and compact objects like black holes and white dwarfs, which have been directly detected only in the last few years. Modeling and predicting the formation of such systems could be improved by the statistical solution.
The random walk model can do more than just treat the stars as point particles. They are not and their internal structure might affect their motion in tides. Tides on Earth are caused by the moon. Some of the tidal energy is dissipated by the Friction between the water and the rest of the planet. The moon's energy must come from its motion around the Earth to cause this heat. The tides draw energy out of the three-body problem.
Mr. Ginat said that the random walk model accounts for such phenomena. To compose all the steps, you have to remove the tidal heat from the total energy in each step. We were able to calculate the outcome probabilities in this case as well. Some of the most fundamental questions in physics can be shed light on by a drunkard's walk.
More information is available in the paper "Analytical, Statistical Approximate Solution of Dissipative and Nondissipative Binary-Single Stellar Encounters". There is a book titled "PhysRevX.11.031020."
The Physical Review X is in the journal.
The Drunken solution to the chaotic three-body problem was retrieved on December 28, 2021.
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