The three-body problem is one of the oldest problems in physics: It concerns
the motions of systems of three bodies—like the sun, Earth, and the moon—and
how their orbits change and evolve due to their mutual gravity. The
three-body problem has been a focus of scientific inquiry ever since Newton.

When one massive object comes close to another, their relative motion
follows a trajectory dictated by their mutual gravitational attraction, but
as they move along and change their positions along their trajectories, the
forces between them, which depend on their mutual positions, also change,
which, in turn, affects their trajectory. For two bodies (e.g., the Earth
moving around the sun without the influence of other bodies), the orbit of
the Earth would continue to follow a specific curve (an ellipse), which can
be accurately described mathematically. However, under the influence of a
third object, the complex interactions lead to the three-body problem—the
system becomes chaotic and unpredictable, and the system's evolution over
long time scales cannot be predicted. Indeed, while this phenomenon has been
known for over 400 years, ever since Newton and Kepler, a neat mathematical
description for the three-body problem is still lacking.

In the past, physicists—including Newton himself—have tried to solve the
three-body problem; in 1889, King Oscar II of Sweden even offered a prize,
in commemoration of his 60th birthday, to anybody who could provide a
general solution. In the end, it was the French mathematician Henri Poincaré
who won the competition. He ruined any hope for a full solution by proving
that such interactions are chaotic, in the sense that the final outcome is
essentially random; in fact, his finding opened a new scientific field of
research, termed chaos theory.

The absence of a solution to the three-body problem means that scientists
cannot predict what happens during a close interaction between a binary
system (formed of two stars that orbit each other like Earth and the sun)
and a third star, except by simulating it on a computer and following the
evolution step-by-step. These simulations show that when such an interaction
occurs, it proceeds in two phases: First, a chaotic phase during which all
three bodies pull on each other violently until one star is ejected far from
the other two, which then settle down to an ellipse. If the third star is on
a bound orbit, it eventually comes back down toward the binary, whereupon
the first phase ensues once again. This triple dance ends when, in the
second phase, one of the stars escapes on an un-bound orbit, never to
return.

In a paper accepted for publication in Physical Review X this month, Ph.D.
student Yonadav Barry Ginat and Professor Hagai Perets of the
Technion-Israel Institute of Technology used this randomness to provide a
statistical solution to the entire two-phase process. Instead of predicting
the actual outcome, they calculated the probability of any given outcome of
each phase-1 interaction. While chaos implies that a complete solution is
impossible, its random nature allows calculation of the probability that a
triple interaction ends in one particular way rather than another. Then, the
entire series of close approaches could be modeled by using a the theory of
random walks, sometimes called "drunkard's walk." The term got its name from
mathematicians thinking about how a drunk would walk, regarding it as a
random process—with each step, the drunk doesn't realize where they are and
takes the next step in some random direction.

The triple system behaves, essentially, in the same way. After each close
encounter, one of the stars is ejected randomly (but with the three stars
collectively still conserving the overall energy and momentum of the
system). This series of close encounters could be regarded as a drunkard's
walk. Like a drunk's step, a star is ejected randomly, comes back, and
another (or the same star) is ejected to a likely different random direction
(similar to another step taken by the drunk) and comes back, and so forth,
until a star is completely ejected and never returns (akin to a drunk
falling into a ditch).

Another way of thinking about this is to notice the similarities with
describing the weather, which also exhibits the same phenomenon of chaos
that Poincaré discovered; this is why the weather is so hard to predict.
Meteorologists therefore have to recourse to probabilistic predictions
(think about that time when a 70 percent chance of rain ended up as glorious
sunshine in reality). Moreover, to predict the weather a week from now,
meteorologists have to account for the probabilities of all possible types
of weather in the intervening days, and only by composing them together can
they get a proper long-term forecast.

What Ginat and Perets showed in their research was how this could be done
for the three-body problem: They computed 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, much like calculating long-term weather forecasts.

"We came up with the random walk model in 2017, when I was an undergraduate
student," said Mr. 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."

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

"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," said Prof. Perets. "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 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 might affect their
motion, for example, in tides. Tides on Earth are caused by the moon and
change the planet's shape slightly. Friction between the water and the rest
of the planet dissipates some of the tidal energy as heat. Energy is
conserved, however, so this heat must come from the moon's energy in its
motion about the Earth. Similarly for the three-body problem, tides can draw
orbital energy out of the three-bodies' motion.

"The random walk model accounts for such phenomena naturally," said Mr.
Ginat. "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.

## Reference:

Yonadav Barry Ginat et al, Analytical, Statistical Approximate Solution of
Dissipative and Nondissipative Binary-Single Stellar Encounters, Physical
Review X (2021).
DOI: 10.1103/PhysRevX.11.031020

Tags:
Space & Astrophysics