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 is not 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 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 give 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 solutions
were 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 for many great mathematicians - but over the following
three centuries, solutions have been found for very few specialized cases. Why? Well, in
the late 1800s, mathematicians Ernst Bruns and Henri Poincaré convincingly asserted
that no general analytic solutions 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 conditions. 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.
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, like those predicted by Kepler.
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 minuscule gravitational influence of the smaller body
and assume that it moves within the completely solvable two-body orbits 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 entire galaxies. But these numerical solutions didn’t begin with the invention artificial
computers. Before that, 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 around 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. So, 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.
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 a 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 - a 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, actually, 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 about the three-body problem. Henri Poincare thought the general case could not
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 is so slow that it would 10^8 million terms to converge
for a typical calculation in celestial mechanics. That is 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.
A few weeks ago, I invited Matt to come to Fermilab to make an awesome crossover video
on the subject of neutrinos. He accepted and the rest, as they say, is history. There were
some great questions in the comments and Matt asked me to answer a few of them.
So here it goes.
Sanskar Jain asks what it means for a neutrino to go with a particular lepton, meaning electron,
muon or tau. It turns out that over short distances and before neutrinos have a chance
to oscillate, they remember how they were made. Neutrinos made in nuclear reactors are
made with electrons and if they interact again, they make only electrons. In particle beams,
neutrinos are made with muons and can subsequently only make muons. In fact, this observation
in 1962 led to the discovery that there were different kinds of neutrinos and, subsequently,
to the 1988 Nobel Prize in physics.
Gede Ge asks why we use argon in our neutrino detectors, and that’s a great question.
The answer is that we don’t always. Neutrino detectors have been made of water, metal,
dry cleaning fluid, even baby oil doped with a chemical called scintillator. We use argon
because it ionizes very easily. That means when a neutrino >>DOES<< interact in the argon,
we can see the path of the particles made in the interaction. From that, we can reconstruct
the collision and learn more about neutrinos.
Nexus void asks how we’ll learn if neutrinos prefer to interact with matter or antimatter.
Actually, what we’ll do is a little different. We want to see if matter or antimatter neutrinos
change their identity at different rates. We do that by performing the experiment with
a beam of neutrinos and then repeating it with a beam of antineutrinos. If they’re
different, we may be on to something. And, if you want to learn more about that, I recommend
my video on leptogenesis on the Fermilab YouTube channel.
Laura Henley notes that we look like we’re best friends. That’s because we could be.
Although we hadn’t met before we started filming, we are kindred spirits, interested
both in cutting edge science and making videos that share the excitement with everyone.
I'm a huge fan of PBS Space Time and, if you like them, you’ll like ours as well. In fact,
I’d like to invite you to subscribe to the Fermilab YouTube channel. Our videos cover
some of the most interesting topics in all of physics. And that’s saying something,
because physics…and Space Time of course…is everything.