Newton's three-body problem explained - Fabio Pacucci

In 2009, two researchers ran a simple
experiment.

They took everything we know about our
solar system and calculated

where every planet would be up to 5
billion years in the future.

To do so they ran over 2000 numerical
simulations

with the same exact initial conditions
except for one difference:

the distance between Mercury and the Sun,
modified by less than a millimeter

from one simulation to the next.

Shockingly, in about 1 percent of their
simulations,

Mercury’s orbit changed so drastically
that it could plunge into the Sun

or collide with Venus.

Worse yet,

in one simulation it destabilized
the entire inner solar system.

This was no error; the astonishing variety
in results

reveals the truth that our solar system
may be much less stable than it seems.

Astrophysicists refer to this astonishing
property of gravitational systems

as the n-body problem.

While we have equations that can
completely predict the motions

of two gravitating masses,

our analytical tools fall short when
faced with more populated systems.

It’s actually impossible to write down
all the terms of a general formula

that can exactly describe the motion
of three or more gravitating objects.

Why? The issue lies in how many unknown
variables an n-body system contains.

Thanks to Isaac Newton, we can write
a set of equations

to describe the gravitational force
acting between bodies.

However, when trying to find a general
solution for the unknown variables

in these equations,

we’re faced with a mathematical
constraint: for each unknown,

there must be at least one equation
that independently describes it.

Initially, a two-body system appears to
have more unknown variables

for position and velocity than
equations of motion.

However, there’s a trick:

consider the relative position and
velocity of the two bodies

with respect to the center of
gravity of the system.

This reduces the number of unknowns
and leaves us with a solvable system.

With three or more orbiting objects in the
picture, everything gets messier.

Even with the same mathematical trick
of considering relative motions,

we’re left with more unknowns than
equations describing them.

There are simply too many variables
for this system of equations

to be untangled into a general solution.

But what does it actually look like for
objects in our universe

to move according to analytically
unsolvable equations of motion?

A system of three stars––

like Alpha Centauri could come crashing
into one another or, more likely,

some might get flung out of orbit
after a long time of apparent stability.

Other than a few highly improbable
stable configurations,

almost every possible case is
unpredictable on long timescales.

Each has an astronomically large range
of potential outcomes,

dependent on the tiniest of differences
in position and velocity.

This behaviour is known as chaotic
by physicists,

and is an important characteristic
of n-body systems.

Such a system is still deterministic—
meaning there’s nothing random about it.

If multiple systems start from the exact
same conditions,

they’ll always reach the same result.

But give one a little shove at the start,
and all bets are off.

That’s clearly relevant for human space
missions,

when complicated orbits need to
be calculated with great precision.

Thankfully, continuous advancements
in computer simulations

offer a number of ways
to avoid catastrophe.

By approximating the solutions with
increasingly powerful processors,

we can more confidently predict the motion
of n-body systems on long time-scales.

And if one body in a group of three is so
light it exerts no significant force on the other two,

the system behaves, with very good
approximation, as a two-body system.

This approach is known as the “restricted
three-body problem.”

It proves extremely useful in describing,
for example,

an asteroid in the Earth-Sun
gravitational field,

or a small planet in the field of a
black hole and a star.

As for our solar system, you’ll
be happy to hear

that we can have reasonable confidence
in its stability

for at least the next several
hundred million years.

Though if another star, launched from
across the galaxy, is on its way to us,

all bets are off.