Newton's threebody problem explained  Fabio Pacucci

0:08  0:12In 2009, two researchers ran
a simple experiment. 
0:12  0:15They took everything we know
about our solar system 
0:15  0:21and calculated where every planet would be
up to 5 billion years in the future. 
0:21  0:25To do so they ran over 2,000
numerical simulations 
0:25  0:30with the same exact initial conditions
except for one difference: 
0:30  0:35the distance between Mercury and the Sun,
modified by less than a millimeter 
0:35  0:38from one simulation to the next.

0:38  0:41Shockingly, in about 1 percent
of their simulations, 
0:41  0:46Mercury’s orbit changed so drastically
that it could plunge into the Sun 
0:46  0:49or collide with Venus.

0:49  0:50Worse yet,

0:50  0:55in one simulation it destabilized
the entire inner solar system. 
0:55  0:59This was no error;
the astonishing variety in results 
0:59  1:05reveals the truth that our solar system
may be much less stable than it seems. 
1:05  1:10Astrophysicists refer to this astonishing
property of gravitational systems 
1:10  1:12as the nbody problem.

1:12  1:15While we have equations
that can completely predict 
1:15  1:18the motions of two gravitating masses,

1:18  1:24our analytical tools fall short
when faced with more populated systems. 
1:24  1:29It’s actually impossible to write down
all the terms of a general formula 
1:29  1:35that can exactly describe the motion
of three or more gravitating objects. 
1:35  1:42Why? The issue lies in how many unknown
variables an nbody system contains. 
1:42  1:45Thanks to Isaac Newton,
we can write a set of equations 
1:45  1:49to describe the gravitational force
acting between bodies. 
1:49  1:54However, when trying to find a general
solution for the unknown variables 
1:54  1:55in these equations,

1:55  1:58we’re faced with
a mathematical constraint: 
1:58  2:02for each unknown,
there must be at least one equation 
2:02  2:04that independently describes it.

2:04  2:09Initially, a twobody system appears
to have more unknown variables 
2:09  2:13for position and velocity
than equations of motion. 
2:13  2:15However, there’s a trick:

2:15  2:19consider the relative position
and velocity of the two bodies 
2:19  2:23with respect to the center
of gravity of the system. 
2:23  2:27This reduces the number of unknowns
and leaves us with a solvable system. 
2:27  2:33With three or more orbiting objects
in the picture, everything gets messier. 
2:33  2:37Even with the same mathematical trick
of considering relative motions, 
2:37  2:42we’re left with more unknowns
than equations describing them. 
2:42  2:46There are simply too many variables
for this system of equations 
2:46  2:50to be untangled into a general solution.

2:50  2:54But what does it actually look like
for objects in our universe 
2:54  2:59to move according to analytically
unsolvable equations of motion? 
2:59  3:02A system of three stars—
like Alpha Centauri— 
3:02  3:05could come crashing
into one another or, more likely, 
3:05  3:10some might get flung out of orbit
after a long time of apparent stability. 
3:10  3:14Other than a few highly improbable
stable configurations, 
3:14  3:21almost every possible case
is unpredictable on long timescales. 
3:21  3:25Each has an astronomically large range
of potential outcomes, 
3:25  3:30dependent on the tiniest of differences
in position and velocity. 
3:30  3:34This behaviour is known
as chaotic by physicists, 
3:34  3:37and is an important characteristic
of nbody systems. 
3:37  3:42Such a system is still deterministic—
meaning there’s nothing random about it. 
3:42  3:46If multiple systems start
from the exact same conditions, 
3:46  3:48they’ll always reach the same result.

3:48  3:54But give one a little shove at the start,
and all bets are off. 
3:54  3:57That’s clearly relevant
for human space missions, 
3:57  4:02when complicated orbits need
to be calculated with great precision. 
4:02  4:06Thankfully, continuous advancements
in computer simulations 
4:06  4:09offer a number of ways
to avoid catastrophe. 
4:09  4:14By approximating the solutions
with increasingly powerful processors, 
4:14  4:20we can more confidently predict the motion
of nbody systems on long timescales. 
4:20  4:23And if one body in a group
of three is so light 
4:23  4:26it exerts no significant force
on the other two, 
4:26  4:31the system behaves, with very good
approximation, as a twobody system. 
4:31  4:35This approach is known
as the “restricted threebody problem.” 
4:35  4:38It proves extremely useful
in describing, for example, 
4:38  4:42an asteroid in the EarthSun
gravitational field, 
4:42  4:47or a small planet in the field
of a black hole and a star. 
4:47  4:49As for our solar system,
you’ll be happy to hear 
4:49  4:53that we can have reasonable confidence
in its stability 
4:53  4:56for at least the next
several hundred million years. 
4:56  4:58Though if another star,

4:58  5:02launched from across the galaxy,
is on its way to us, 
5:02  5:04all bets are off.
 Title:
 Newton's threebody problem explained  Fabio Pacucci
 Speaker:
 Fabio Pacucci
 Description:

View full lesson: https://ed.ted.com/lessons/newtonsthreebodyproblemexplainedfabiopacucci
In 2009, 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. They ran over 2,000 simulations, and the astonishing variety in results revealed that our solar system may be much less stable than it seems. Fabio Pacucci explores the nbody problem and the motion of gravitating objects.
Lesson by Fabio Pacucci, directed by Hype CG.
 Video Language:
 English
 Team:
 TED
 Project:
 TEDEd
 Duration:
 05:09
lauren mcalpine approved English subtitles for Newton's threebody problem explained  
lauren mcalpine accepted English subtitles for Newton's threebody problem explained  
lauren mcalpine edited English subtitles for Newton's threebody problem explained  
Tara Ahmadinejad edited English subtitles for Newton's threebody problem explained  
Tara Ahmadinejad edited English subtitles for Newton's threebody problem explained 