It’s been 300 years since Isaac Newton devised the principles that allow us to predict exactly where a planet or other astronomical object would be at any given time, allowing us to launch rockets to Mars or probes to Pluto, for example.

When Newton first discovered how two mass-enhanced bodies interact via gravity, he had cracked the code on how matter moves and interacts with one another in the vast expanse of space and time that we call home.

One thing that the great English physicist, Isaac Newton, never managed to do was figure out how not two, but three objects orbiting each other would be related.

As far as math problems go, the “three-body problem” has been around for three centuries. Newton came up with it, and no one else has been able to solve it. It inspired the Chinese writer Cixin Liu to write one of the most popular science fiction works of the last few years.

The three-body issue is challenging because it is a chaotic system, which implies that making any meaningful predictions requires exceedingly exact knowledge about the initial positions of the three bodies in question.

The “butterfly effect” becomes extremely real in these systems, and even the tiniest error results in a completely different orbit than expected. On the face of the Earth, there is no equation capable of predicting how objects will move or determining whether their orbits will be stable over time.

Due to the lack of a solution to the three-body problem, scientists are currently unable to anticipate what will happen when a binary system (two stars orbiting each other) collides with a nearby third star. The only way to do this is to run a computer simulation of the case and watch the triple system evolve over time.

These simulations reveal that the interactions take place in two stages: first, a chaotic phase in which the three bodies fiercely push each other until one star is ejected away from the other two, and then a stable phase in which the three bodies form an ellipse around each other.

If the third star is in a bound orbit, it can re-approach the remaining ‘pair,’ re-enacting the first phase. This triple dance comes to an end when one of the stars, in the second phase, escapes into an infinite orbit, never to return.

In a recent study published in ‘Physical Review X,’ Yonadav Barry Ginat and Hagai Perets, both from Israel’s Technion Institute of Technology, utilised this unpredictability to propose a statistical solution to the two phases of the process. They calculated the chance of any potential outcome in each phase one contact rather than forecasting the actual event. While chaos suggests that a comprehensive solution is impossible, the random character of chaos allows one to compute the likelihood that a triple interaction will conclude in one of two ways.

The theory of random walks, also known as “the drunkard’s walk,” might then be used to represent the entire series of close approximations. The word was coined by mathematicians who imagined how a drunk would walk and conceived of it as a random process: the drunk does not understand where he is and makes the next step in some random direction with each stride.

Essentially, the triple system behaves in the same way. In effect, one of the stars gets tossed out at random after each near collision. And this pattern of entrances and exits could be compared to a drunkard’s walk. One star is ejected randomly, returns, and another (or the same star) is ejected in a likely different random direction (similar to another drunkard’s step), and so on, until the star is firmly expelled and never returns as if the drunken fell into a ditch.

In other words, Ginat and Perets’ research demonstrates how the same statistical system can be applied to the three-body problem. As a result, they estimated the likelihood of each binary configuration before combining them using the theory of random walks to determine the ultimate probability of any potential occurrence, which is akin to creating long-term weather forecasts.

“We came up with the random walk model in 2017, when I was an undergraduate student,” said Ginat, “I took a course that Prof. Perets taught, and there I had to write an essay on the three-body problem. We didn’t publish it at the time, but when I started a Ph.D., we decided to expand the essay and publish it.”

Different teams have addressed the same problem in recent years, and Ginat and Perets’ solution statistically solves all potential types of interaction.

For Perest, the work “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 collisions between stars and compact objects like black holes, neutron stars and white dwarves, which also produce gravitational waves that have been directly detected only in the last few years. The statistical solution could serve as an important step in modeling and predicting the formation of such systems.”

The random walk model, on the other hand, can accomplish a lot more. Individual stars have been treated as idealized point particles in investigations of the three-body problem until now.

Of course, they aren’t in reality, and their interior structure could have an impact on their movement, such as on tides.

The moon causes tides on Earth, which alter the planet’s form significantly. Part of the tidal energy is dissipated as heat as a result of friction between the ocean and the rest of the earth. However, because energy is conserved, this heat must come from the moon’s energy as it orbits the Earth.

Tides can also extract orbital energy from the motion of the three bodies in the three-body problem.

“The random walk model,” Ginat explains, “takes these phenomena into account in a natural way. All you have to do is remove the heat from the total energy tide in each step and then compose all the steps. Who would have guessed that the unsteady gait of a drunkard could shed light on some of the most fundamental questions in physics.

“The random walk model accounts for such phenomena naturally,” Ginat explained.

“All you have to do is to remove the tidal heat from the total energy in each step, and then compose all the steps. We found that we were able to compute the outcome probabilities in this case, too.

“As it turns out, a drunkard’s walk can sometime shed light on some of the most fundamental questions in physics.”

Source: 10.1103/PhysRevX.11.031020

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