The Three-Body Problem theory

Among the greatest of history’s mathematicians, physicists, and astronomers, Isaac Newton helped to shed light on the many inner workings and mechanisms of our universe, ushering humanity into a new era of heightened scientific knowledge and understanding. Yet while his accomplishments helped answer and explain many of the questions that had puzzled the scientific community for centuries, Newton also raised a number of questions that remain unanswered even today. Perhaps one of the most famous and mysterious of these is the Three-Body Problem.

The Two-Body Problem

The origins of the problem draw root from Newton’s astronomical exploration into whether or not long-term stability is possible within our solar system, specifically in the system encompassing the Sun, the Moon, and the Earth. This can determine the likelihood of a catastrophic collapse occurring within the solar system itself.

In his book Principia Mathematica, Newton first posed and solved a relatively simple Two-Body Problem: “How will two masses move in space if the only force on them is their mutual gravitational attraction?” As Newton framed the question, it was a matter of solving a system of differential equations – equations that can predict the future motion of an object given its present position and velocity. Therefore, the orbits followed by the bodies in the system would act as the solution.

Newton was able to completely solve the Two-Body Problem: the two bodies would follow two separate conic (elliptical) orbits, sharing a single focus point situated at the center of mass of the entire system. In the case of the Sun and the Earth, the mass of the Sun is so much greater than the mass of the Earth that the center of mass of the two bodies is located within the Sun itself. Thus, the Earth follows an orbit that revolves around the Sun.

The Three-Body Problem

By simply adding a third body to the system, Newton created the infamous Three-Body Problem. However, when considering the motions of three masses in space instead of two in the same circumstances, the problem becomes almost impossibly difficult. The solution still lies in deriving the orbits of all three bodies – in other words solving the system of three differential equations – yet even with centuries of combined work by some of humanity’s brightest minds and the power of modern supercomputers, the solution to the Three-Body Problem remains an absolute mystery to us.

Despite this, however, partial solutions have been discovered. For instance, Swiss mathematician Leonhard Euler solved a special case of the Three-Body Problem, in which three masses lie equally spaced on a single line. French mathematician Joseph-Louis Lagrange, on the other hand, solved another special case in which the three objects always lie on the vertices of an equilateral triangle. Yet none of these even approach the level of generalization and completeness of Newton’s solution to the Two-Body Problem.

But why does adding another object to the system complicate matters so much? The answer lies in the chaotic and intricate dynamics governing a three-body system. The sheer instability and volatility of the system makes finding an explicit formula – a function that completely describes the motion of the objects given known variables – nearly impossible.

Numerical Integration and Approximations

Despite being unable to completely solve the Three-Body Problem, modern-day computers have allowed us to approximate the solutions to remarkable accuracy. Using the method of numerical integration, we can essentially generate a multitude of finite segments to mimic the orbits, instead of calculating the exact orbits, thus simplifying the process significantly. This method also makes modeling the motions of far more than three celestial objects possible, an essential component of space mission planning.

A Precarious and Catastrophic Future?

Newton’s ultimate goal in delving into the Three-Body Problem was to determine just how stable our Solar System is – and thus, how close are we to a catastrophic breakup of the Solar System?

At the present moment, it is impossible to precisely model the orbits of all the celestial objects within the Solar System – after all, if we cannot even solve the Three-Body Problem, how can we possibly even approach a solution to the Ten-Body, Twenty-Body Problem? Yet using numerical integration, we can, again, approximate the orbits, and better understand the stability of the Solar System.

In 2009, astronomers at the Paris Observatory did just that. They ran thousands of computer simulations with a digital model of the Solar System, modeling the motions of the planets and other objects billions of years into the future – but with a twist: they initialized the simulation by moving Mercury from its orbit by a single meter. 

The results of the test were ominous. While the Solar System remained stable in most of the simulations, 1% of cases saw Mercury gradually straying from its elliptical orbit, and wreaking havoc among the planets. In some cases, it smashed into Venus, and in others, caused Mars to spin away from the Solar System and led to the collision between Venus and Earth. It could be the end of the Solar System so far as we know.

This discovery shocked mathematicians and astronomers alike – all of a sudden, it seemed that the Solar System was far more volatile than we had thought. For centuries, scientists had believed that the model of planets revolving around the Sun was a highly stable one, yet add in the gravitational interactions between the planets, and the picture is entirely different. After all, if displacing Mercury by only a single meter can cause calamity, what would happen if we were impacted by a more drastic external factor? Are we already speeding towards our inevitable destruction? These are questions that science cannot yet answer.

Nonetheless, it is important to note that this is mere speculation, lacking concrete evidence. Besides, the time scale of these celestial events are far beyond our lifetimes, so it may be the least of our present worries. Yet mathematicians and astronomers are still searching for answers – for the key to unlocking the complex and intricate mechanisms of our Solar System, to truly grasp our future: the solution to the Three-Body Problem.

Works Cited:

"The Three-Body Problem." Scientific American, www.scientificamerican.com/article/the-three-body-problem/.

"Three-Body Problem." Scholarpedia, www.scholarpedia.org/article/Three_body_problem#Equations.

Overbye, Dennis. "New Math Shows When Solar Systems Become Unstable." Quanta Magazine, 16 May 2023, www.quantamagazine.org/new-math-shows-when-solar-systems-become-unstable-20230516/.