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In 2009, two researchers ran a simple 
experiment.

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They took everything we know about our
solar system and calculated

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

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To do so they ran over 2000 numerical 
simulations

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with the same exact initial conditions
except for one difference:

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

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from one simulation to the next.

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Shockingly, in about 1 percent of their 
simulations,

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

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or collide with Venus.

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Worse yet,

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in one simulation it destabilized
the entire inner solar system.

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This was no error; the astonishing variety
in results

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

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Astrophysicists refer to this astonishing
property of gravitational systems

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as the n-body problem.

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While we have equations that can 
completely predict

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the motions of two gravitating masses,

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

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

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

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

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

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to describe the gravitational force 
acting between bodies.

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

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in these equations,

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

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

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

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for position and velocity than 
equations of motion.

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However, there’s a trick:

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consider the relative position and 
velocity of the two bodies

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with respect to the center of 
gravity of the system.

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

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

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Even with the same mathematical trick 
of considering relative motions,

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

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There are simply too many variables
for this system of equations

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to be untangled into a general solution.

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

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to move according to analytically 
unsolvable equations of motion?

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A system of three stars––

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

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

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Other than a few highly improbable 
stable configurations,

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almost every possible case is 
unpredictable on long timescales.

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Each has an astronomically large range
of potential outcomes,

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dependent on the tiniest of differences
in position and velocity.

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This behaviour is known as chaotic 
by physicists,

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and is an important characteristic 
of n-body systems.

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

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If multiple systems start from the exact
same conditions,

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they’ll always reach the same result.

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

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That’s clearly relevant for human space
missions,

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when complicated orbits need to 
be calculated with great precision.

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Thankfully, continuous advancements
in computer simulations

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offer a number of ways
to avoid catastrophe.

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By approximating the solutions with 
increasingly powerful processors,

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

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And if one body in a group of three
is so light

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it exerts no significant force 
on the other two,

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

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

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It proves extremely useful in describing,
for example,

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an asteroid in the Earth-Sun 
gravitational field,

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

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

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that we can have reasonable confidence
in its stability

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for at least the next several 
hundred million years.

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

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all bets are off.