January 1889 marked the 60th birthday of Sweden’s King Oscar II. To commemorate the milestone, the monarch – who studied mathematics in his youth and even founded the journal Acta Mathematica (still considered one of the most prestigious in the field) – decided to hold a scientific competition. He offered a prize to whoever could solve the intractable three-body problem – accounting for the trajectories of three-body systems.
Isaac Newton was the first to formulate mathematical principles that made it possible to accurately predict the motion of two massive celestial bodies in close proximity to each other, when he published his “Principia” in 1687. This achievement reinforced the notion of a mechanistic universe operating as a giant clock. But Newton soon discovered that when another body was added to the system, he failed to find an accurate general solution.
The “three-body problem” remained without a mathematical solution for some 200 years, despite the best efforts of scientists. This is where Oscar II came to the rescue. His competition was won by the French mathematician Henri Poincaré, who was awarded a gold medal and 2,500 Swedish kronor. His solution was published in the Royal Mathematical Journal.
But then Poincaré discovered a miscalculation. He hurried to purchase all the editions of the journal containing the error – which cost him 3,500 kronor – and the following year published a revised version. He proved, to the chagrin of the king and proponents of the mechanistic conception of the universe, that the interactions between the three bodies are fundamentally chaotic and therefore a deterministic mathematical solution to the problem can’t be found (i.e., there is no formula that solves it).
This proof is considered one of the foundations of chaos theory.
The absence of a deterministic solution to the “three-body problem” means that scientists cannot predict what happens during a close interaction between two orbiting bodies such as the Earth and the Moon, and a third object approaching them.
Now, though, 121 years after the publication of Poincaré’s findings, doctoral student Yonadav Barry Ginat and Prof. Hagai Perets from the Technion – Israel Institute of Technology, Haifa, claim to have achieved a complete statistical solution to the problem.
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Computer simulations of three-body systems show that they evolve in a two-phase process: in the first, chaotic phase, the three bodies are in close proximity to each other and exert equally intense gravitational forces on each other, that change continually due to the three bodies’ relative motion. Eventually, one celestial body is ejected from the system, and two are left to orbit each other in an elliptical, deterministic trajectory. If the third body is on a bound orbit, it eventually comes back down toward the other two, whereupon the first phase reoccurs.
This three-way dance ends when, in the second phase, one of the bodies escapes on an unbound orbit, never to return.
How a drunk person walks
Although a complete solution to the “three-body problem” is not possible due to the chaotic nature of the process, it is possible to calculate the probability that a triple interaction ends in a certain way – for example, which object will be ejected, at what speed, etc. Over the years, solutions have been proposed that have used different methods to arrive at an as-accurate-as-possible calculation of this probability.
Like a drunkard’s walk, after phase 1 occurs, one object is ejected randomly, returns, and so forth, until one is completely ejected to never return (and the drunk person falls asleep or into a ditch)
The two researchers from the Technion’s physics department used tools from a branch of mathematics known as random walk theory – sometimes referred to as “drunkard’s walk,” since it began with mathematicians studying how drunk people move. Mathematicians understood this as a random process, since a drunkard seemingly takes every step randomly. However, it is possible to estimate, for instance, the distance a drunk will travel after several steps (this is a statistical solution, which results in an average distance of about 10 steps away from the starting position for every hundred steps taken).
The tripartite system behaves, fundamentally, in a similar way: Like a drunkard’s walk, after phase 1 occurs, one object is ejected randomly, returns, and so forth, until one is completely ejected to never return (and the drunk person falls asleep or into a ditch).
Instead of predicting the actual result of each three-body interaction, Ginat and Perets calculated the probability of each possible outcome at every phase of the interaction, and then combined all the individual phases – using random walk theory – to calculate the final probability of each possible outcome.
The two began thinking about the random walk model in 2017, when Ginat was an undergraduate student in one of Perets’ courses and was writing an essay on the three-body problem. Their solution was published recently in the journal Physical Review X.
According to Perets, “This is a major problem for understanding any situation where there are high-density clusters of stars. Until the 1970s, there was no solution. However, with advancements in computing power, numerical solutions were attempted” – i.e., throwing data into the simulation and seeing what happened.
“We recovered the results of millions of simulations that were conducted during previous studies, and offered an analytical solution – which means that one no longer needs to run simulations,” Perets said.
This is not the first time Israeli researchers have proposed a solution to the “three-body problem.” In recent years, two prominent proposals also came from local researchers: Dr. Nicholas Stone and Prof. Barak Kol, both from the Hebrew University.
“Stone and Kol’s proposals are relevant only to the last phase, when one body is ejected from the system, without dealing with the prior phases,” Perets explained. “But there are interactions that occur during the mixing, and if you only look at the last phase, you lose information.”
He did acknowledge, though, that in certain aspects – for example, the identity of the body ejected from the system – Kol’s method is more accurate, at a deviation rate of about 1 percent, compared to about 5 percent in his and Ginat’s proposal. But he stressed that “our method offers full predictions, and in particular what the trajectory of the remaining binary system will look like, which is critical to understanding the future development of the system.
“So, our solution is more general and contains all the important parameters of the system,” Perets added. “We think this is the end solution to this centuries-old problem. It is not a full solution, because a full solution is impossible. But statistically, it is complete.”
The method Perets and Ginat used is called the “density-based method,” used by researchers since the mid-’70s. This method is based on calculating the density of states in the phase space (the space of configurations of all three bodies and their velocities).
To illustrate the complex concept, Kol suggested that we observe dice: “The probability that the result of a cast die will be, for example, three or more is four out of six, since there are four such states out of six possibilities.” Kol’s method, which he proposed a few months ago, requires a calculation of flux within the phase space, however.
Here, Kol suggested that we think of a container with a small hole in its sides that is filled with gas and contains one painted particle. The chance of the painted particle coming out of the tank in a given unit of time increases as the gas flow through the hole increases.
For Kol, “the two methods seek a statistical solution, in different ways. Ginat and Perets have taken the density-based approach an important step forward in the sense that they include the intermediate phases of the bound problem, i.e., the state in which a celestial object is temporarily ejected.
“These are two different, not rival, methods,” Kol added. “This is a collaborative work toward the same purpose. We all want to understand nature, and everyone chooses their own path. This is the correct approach to scientific practice.”