Newtons threebody 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 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.