Newton’s three-body problem explained - Fabio Pacucci

Translator: Reviewer: Daban Q. Jaff

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 2,000 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.