Solutions to the Three Body Problem (Script) (Patreon)

The three body problem is famous for being impossible to solve. But actually it's been solved many times, and in ingenious ways. Some of those solutions are incredibly useful, and some are incredibly bizarre.

Physics, and arguably all of science, changed forever in 1687 when Isaac Newton published his Principia. Within it were equations of motion and gravity that transformed our erratic-seeming cosmos into a perfectly tuned machine of clockwork predictability. Given the current positions and velocities of the bodies of the solar system, Newton’s equations could be used in principle be used to calculate their locations at any distant time, future or past. I say “in principle” because the reality isn’t so simple. Despite the beauty of Newton’s equations, they lead to a simple solution for planetary motion in only one case - when two and only two bodies orbit each other sans any other gravitational influence in the universe. Add just one more body and in most cases all motion becomes fundamentally chaotic - there exists no simple solution. This is the three-body problem, and we’ve been trying to solve it for over 300 years.

What does it mean to find a solution to the three-body problem? Newton’s laws of motion and his law of universal gravitation give us a set of differential equations. In some cases these can be solved with Newton’s other great invention - calculus - to give a simple equation. Plug numbers into that equation and its solved. Those numbers are the starting positions and velocities of your gravitating bodies, plus a value for time. The equations will then tell you the state of the system at that time, no matter how far in the past or future. We call such a simple, exactly-solvable equation an analytic expression. That just means it can be written out with a finite number of mathematical operations and functions.

In the case of two gravitating bodies, the solutions to Newton’s laws are just the equations for the path traveled by the bodies - be it the parabola of a thrown ball, the circle or ellipse of a planetary orbit, or the hyperbola of an interstellar comet - in general, conic sections - the shapes you get when you slice up a cone. These equations are so simple that Johanne Kepler figured out much about the elliptical solution for planetary motion 70 years before Newton’s laws were even known.

And after the Principia was published many sought simple, analytic solutions for more complex systems, with systems of three gravitating bodies being the natural next step. But the additional influence of even a single extra body appeared to make an exact solution impossible. The three body problem became the obsession of many great mathematicians - but the following three centuries, solutions have been found for very specialized cases. Why? Well, in the late 1800s, mathematicians Ernst Bruns and Henri Poincaré convincingly asserted that no general analytic solution exists.

The reality of the three-body problem is that the evolution of almost all starting configurations is dominated by chaotic dynamics. Future states are highly dependent on small changes in the initial state. Orbits tend towards wild and unpredictable patterns, and almost inevitably one of the bodies is eventually ejected from the system. But despite the apparent hopelessness, there was much profit in learning to predict the gravitational motion of many bodies. For most of the three centuries since Newton, predicting the motion of the planets and the moon was critical for nautical navigation. Now it’s essential to space travel.

So how do we do it? Well, just because the three body problem for the most part has no useful analytic solution, approximate solutions can be found. For example, if the bodies are far enough apart then  we can approximate a many-body system as a series of two-body systems. For example, each planet of our solar system can be thought of as a separate two-body system with the Sun. That gives you a series of simple elliptical orbits, but those orbits eventually shift due to the interactions between the planets.

Another useful approximation is when one of the three bodies has a very low mass compared to the other two. We can ignore the miniscule gravitational influence of the smaller body exerts and assume that it moves within the completely solvable two-body orbit of its larger companions. We call this the reduced three-body problem. It works very well for tiny things like artificial satellites around the Earth. It can also be used to approximate the orbits of the moon relative to the Earth and Sun, or the Earth relative to the Sun and Jupiter.

These approximate solutions are useful, but ultimately fail to predict perfectly. Even the smallest planetary bodies have some mass, and the solar system as a whole has many massive constituents. The Sun, Jupiter and Saturn alone are automatically a three-body system with no analytic solution, before we even add in the Earth.

But the absence of an analytic solution doesn’t mean the absence of any solution. To get an accurate prediction for most three-body systems you need to break the motion of the system into many pieces, and solve them one at a time. A sufficiently small section of any gravitational trajectory can be approximated with an exact, analytical solution - perhaps a straight line or a segment of two-body path around the center of mass of the entire system, assuming everything else stays fixed. If you break up the problem into tiny enough paths segments or time-steps, then the small motions of all bodies in the system can be updated step by step.

This method of solving differential equations one step at a time is called numerical integration, and when applied to the motion of many bodies it’s an N-body simulation.

With modern computers, N-body simulations can accurately predict the motion of the planets into the distant future or solve for millions of objects to simulate the formation and evolution of galaxies. But these numerical solutions didn’t begin with the invention artificial computers these calculations had to be done by hand - in fact by many hands. 

The limitations of approximate solutions, the laboriousness of pre-computer numerical integration, and also the legendary status of the three-body problem inspired generations of physicists and mathematicians to continue to seek exact, analytic solutions. And some succeeded - albeit in very specialized cases. The first was Leonhard Euler, who found a family of solutions for three bodies orbiting a mutual center of mass, where all bodies remain in a straight line - essentially in permanent eclipse. Joseph-Louis Lagrange found solutions in which the three bodies form an equilateral triangle. In fact, for any two bodies orbiting each other, the Euler and Lagrange’s solutions define 5 additional orbits for a third body that can be described with simple equations. These are the only perfectly analytical solutions to the three body problem that exist. Place a low-mass object on any of these 5 orbits and it will stay there indefinitely, tracking the Earth’s orbit around the Sun. We now call these the Lagrange points, and they’re useful places to park our spacecraft.

There was a bit of a gap after Euler and Lagrange because to discover new specialized three-body solutions, we had to search the vast space of possible orbits using computers. The key was to find three-body systems that had periodic motion - they evolve - sometimes in complex ways - back to their starting configuration. In the 70s, Michel Henon and Roger Broucke found a family of solutions involving two masses bouncing back and forth in the center of a third body’s orbit.  In the 90s Cris Moore discovered a stable figure-8 orbit of three equal masses. The numerical discovery of the figure-8 solution was proved mathematically by Alain Chenciner and Richard Montgomery, and insights gained from that proof led to a boom in the discovery of new periodic three body orbits.

Some of these periodic solutions are incredibly complex, but Montgomery came up with a fascinating way to depict them in the absence of simple equations. It’s called the shape-sphere, and it works like this. Imagine the bodies in 3-body system are the vertices of a triangle, whose center is the center of mass of the system. The evolution of the system can be expressed through the changing shape of that triangle. We throw away certain information - the size of the triangle and its orientation, keeping only information about the relative lengths of the edges, or equivalently the angles between the edges.

Now we map that information on the surface of a sphere. We only need the 2-D surface because if we know 2 internal angles of the triangle we also know the 3rd. The equator of the sphere represents both angles being zero- that’s a fully collapsed triangle - the 3-bodies are in a straight line, as in Euler’s solutions. The poles are equilateral triangles - so, Lagrange’s solutions. All other orbits move on this sphere as the triangle defined by the orbits evolves. It turns out that the periodic motion on the shape-sphere appears much simpler and easier to analyze than the motion of the bodies themselves.

Now hundreds of stable 3-body orbits are known - although it should be noted that besides the Euler and Lagrange solutions, none of these are likely to occur in nature, and so their practical use may be limited.

Very recently, a new approach to solving the three-body problem has appeared, which transforms the chaotic nature of three-body interactions into useful tool, rather than a liability. Nicholas Stone and Nathan Leigh published this in Nature in December 2019. The thing about chaotic motion is that the state of the system seems to get randomly shuffled over time. The motion is actually perfectly deterministic - defined between one instant and the next - but can be thought of as approximately random over long intervals. Such a pseudo-random system will, over time, explore all possible configurations consistent with some basic properties like the energy and angular momentum of the system. The system explores what we call a phase space - the space of possible arrangements of position and velocity. Well, for a pseudo-random system, statistical mechanics lets us calculate the probability of the system being in any part of that phase space at any one time.

How is this useful? Well, eventually, almost all three-body systems eject one of the bodies, leaving a nice, stable two-body system - a binary pair. Stone and Leigh found that they could identify the regions of phase space where these ejections were likely - and by doing so they could map the range of likely orbital properties for the two objects left behind after the ejection. This looks to be incredibly useful for understanding the evolution of dense regions of the universe, where three-body systems of stars or black holes may form and then disintegrate very frequently.

One last thing on the three-body problem. Henri Poincare thought the general case couldn’t be solved. In fact he was wrong. In 1906, not so long after Poincare stern proclamation, 

Finnish mathematician Karl Sundman found a solution to the general three-body problem. It was a converging infinite series that added together an endless chain of terms to solve the orbital calculation. Because the series converged, which successive terms diminished to effectively nothing, so in principle the equation could be written out on paper. However the convergence of Sundman’s series so slow that it would 10^8 million terms to converge for a typical calculation in celestial mechanics. That’s a lot of sheets of paper.

So there you have it - the three-body problem is perfectly solved uselessly, or for seemingly useless and bizarre orbits. And it can be approximately solved for all useful and practical purposes - with enough precision to work just fine. Good to know next time you’re in a chaotic orbit, trying to astronavigate around two other gravitating denizens of space time.