Drunk solution to the chaotic three-body problem

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The problem of three bodies is one of the oldest problems in physics: it is about the movements of systems of three bodies – such as the sun, Earth and the moon – and how their orbits change and evolve due to their mutual gravity. The three-body problem has been the focus of scientific study ever since Newton.

When a massive object comes close to another, their relative motion follows a path dictated by their mutual gravitational attraction, but as they move and change their position along their paths, the forces between them change, depending on their mutual positions. which in turn affects their trajectory. For two bodies (eg, the Earth moving around the Sun without the influence of other bodies), the Earth’s orbit would continue to follow a specific curve (an ellipse), which can be described exactly mathematically. But under the influence of a third object, the complex interactions lead to the three-body problem – the system becomes chaotic and unpredictable, and the development of the system over long time scales cannot be predicted. Although this phenomenon has been known for over 400 years, ever since Newton and Kepler, there is still a lack of a nice mathematical description of the problem of three bodies.

In the past, physicists – including Newton himself – have tried to solve the three-body problem; in 1889 King Oscar II of Sweden even offered an award to celebrate his 60th birthday to anyone who could provide a general solution. In the end, it was the French mathematician Henri PoincarĂ© who won the competition. He destroyed any hope of a complete solution by proving that such interactions are chaotic, in the sense that the end result is essentially random; in fact, his discovery opened up a new field of scientific research, called chaos theory.

The absence of a solution to the three-body problem means that scientists can not predict what happens during a close interaction between a binary system (formed by two stars orbiting the Earth and the Sun) and a third star, except by simulate it on a computer and follow the development step-by-step. These simulations show that when such an interaction occurs, it proceeds in two phases: First, a chaotic phase in which all three bodies pull violently into each other until one star is ejected far from the other two, which then settles into a ellipse. If the third star is in a bound orbit, it eventually comes back down towards the binary, after which the first phase reappears. This triple dance ends when one of the stars in the second phase escapes on an unbound orbit, never to return.

In a paper accepted for publication in Physical review X this month, Ph.D. students Yonadav Barry Ginat and Professor Hagai Perets of the Technion-Israel Institute of Technology used this coincidence to provide a statistical solution to the entire two-phase process. Instead of predicting the actual outcome, they calculated the probability of a given outcome of each phase-1 interaction. Although chaos implies that a complete solution is impossible, its random nature allows the calculation of the probability that a triple interaction ends in one particular way rather than another. Then the whole series of close approaches could be modeled using the theory of random walks, sometimes called “the course of drinking.” The term gets its name from mathematicians who think about how a drunk would go and consider it a random process – with each step, the drunk does not realize where they are and takes the next step in a random direction.

The triple system behaves in essentially the same way. After each close encounter, one of the stars is emitted at random (but with the three stars together still retaining the overall energy and momentum of the system). This series of close encounters could be considered a drunken bolt walk. Like a drunken bolt’s step, one star is shot out at random, comes back, and another (or the same star) is thrown out in a probably different random direction (corresponding to another step taken by the drunk) and comes back, and so on, until a star is completely flung out and never returns (akin to a drunk falling into a ditch).

Another way of thinking about this is to notice the similarities of describing the weather, which also exhibits the same phenomenon of chaos that Poincaré discovered; that is why the weather is so hard to predict. Meteorologists therefore have to resort to probability predictions (think of the time when a 70 percent chance of rain ended up as radiant sunshine in reality). In addition, to predict the weather in a week, meteorologists must take into account the probability of all possible weather types in the intervening days, and only by putting them together can they get a proper long-term outlook.

What Ginat and Perets showed in their research was how this could be done for the three-body problem: They calculated the probability of each phase-2 binary-single configuration (the probability of finding different energies, for example), and then composed all of the individual phases using the theory of random walks to find the final probability of any possible outcome, as well as by calculating long-term weather forecasts.

“We found on the model a random walk in 2017 when I was a bachelor’s student,” said Mr. Ginat, “I took a course taught by Prof. Perets, and there I was to write an essay on the three-body problem… We published it not then, but when I started on a PhD, we decided to expand the essay and publish it. “

The three-body problem was studied independently by research groups in recent years, including Nicholas Stone of the Hebrew University of Jerusalem, in collaboration with Nathan Leigh, then at the American Museum of Natural History, and Barak Kol, also of the Hebrew University. Now, with the current study of Ginat and Perets, whole, multi-stage, three-body interaction is fully resolved statistically.

“This has important implications for our understanding of gravitational systems, and in particular cases where many encounters between three stars occur, as in dense clusters of stars,” said Prof. Perets. “In such areas, many exotic systems are formed through three-body encounters, leading to collisions between stars and compact objects such as black holes, neutron stars and white dwarfs, which also produce gravitational waves that have only been directly detected in the last few years. solution could serve as an important step in modeling and predicting the formation of such systems. “

The random walk model can also do more: So far, studies of the three-body problem treat the individual stars as idealized point particles. In reality, of course, they are not, and their internal structure can affect their movement, for example in tides. Tides on Earth are caused by the moon and change the shape of the planet slightly. Friction between the water and the rest of the planet dissipates some of the tidal energy as heat. Energy, however, is conserved, so this heat must come from the moon’s energy in its motion around the Earth. Similarly, for the three-body problem, tides can draw orbital energy out of the motion of the three-bodies.

“The random walking model stands for such phenomena naturally,” said Mr. Ginat. “All you have to do is remove the tidal heat from the total energy in each step, and then compose all the steps. We found that we were also able to calculate the outcome probabilities in this case.” As it turns out, a drunkard’s walk can even shed light on some of the most fundamental questions in physics.


New theory deals with centuries-old physics problems


More information:
Yonadav Barry Ginat et al., Analytical, statistically approximate solution of dissipative and non-dissipative binary-single star encounters, Physical review X (2021). DOI: 10.1103 / PhysRevX.11.031020

Provided by Technion – Israel Institute of Technology

Citation: Drunken solution to the chaotic three-body problem (2021, December 28) retrieved December 30, 2021 from https://phys.org/news/2021-12-druken-solution-chaotic-three-body-problem.html

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