The Three Body Problem
• a mechanical system with three
objects
• needed to explain the movement of
celestial objects due to gravitational
attraction
• famous in math, physics and classical
mechanics
• first addressed by Newton in 1687 (he
solved two-body problem)
Joseph Louis Lagrange
• mathematician in late 1700s
• the three body problem cannot be
solved exactly, but he found that if
the third body is very small
compared to the other two, can
get useful approximate solutions
• Earth, Sun, Moon system is
example of three-body system
In 1889, King Oscar II of Norway and
Sweden celebrated his sixtieth birthday.
As part of the festivities, a much
celebrated competition was held — in
mathematics.
French mathematician Henri Poincaré
addressed the three body problem: given
three celestial objects, describe how they
move around one another and, given their
starting position and speed, predict their
position and speed in the future.
He won but then realized he’d made an
error and couldn’t fix it so he was
awarded a prize for inventive and
revolutionary ideas instead.
The Butterfly
Effect
Years later, Poincaré realized that small differences in
where the 3 bodies start, produce very great differences
in where they end up. Prediction is impossible!
The Three Body Problem
mathematicians in the 1950s finally managed
an elegant proof proving that the three body
problem is impossible to solve
Edward Lorenz 1960
Edward Lorenz was a mathematician and
meteorologist at the Massachusetts
Institute of Technology who loved the
study of weather. He combined math and
meteorology to construct a computer
model of weather.
It didn’t work!
The source of the problem was rounding from 6 decimal
places to just 3. But we don’t measure temperature or wind
to this detail!
This “sensitive dependence on initial conditions” is now
known as the butterfly effect.
In 1963 he concluded that
weather can’t be accurately
predicted. His ideas gave rise
to a new field of math called
chaos theory that studies the
behavior and condition of
dynamic systems that are
highly sensitive to initial
conditions – like a three body
system!
The Three Body Problem
Lagrange Points
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For the Sun-Earth-Moon system, the Sun's
mass is so dominant that it can be treated as a
fixed object.
The Earth-Moon system can then be treated as
a two-body system.
18th century mathematicians discovered that
there were five special points where
gravitational equilibrium could be maintained.
An object placed at any one of these five points
would stay there. These are called Lagrange
Points.
L1, L2 and L3 unstable
equilibrium points.
However, in practice
these points are very
useful since a spacecraft
can be made to execute a
small orbit around a point
with a very small
expenditure of energy.
The orbits around L1 and
L2 are often called "halo
orbits." L3 is on the
opposite side of the Earth
from the Moon is not
easy to use.
If the spacecraft is placed between Sun and Earth, the Earth's
gravity pulls it in the opposite direction and cancels some of the
pull of the Sun. With a weaker pull towards the Sun, the spacecraft
then needs less speed to maintain its orbit.
If the distance is about 4 times the distance to the Moon or 1/100
the distance to the Sun, the spacecraft will need just one year to go
around the Sun and will keep its position. That position is L1.
L1 is a very good position for monitoring
the solar wind, which reaches it about one
hour before reaching Earth. In 1978 the
"International Sun-Earth Explorer-3" (ISEE3) was placed at L1. In 1994, WIND, was
placed in that position. More recently the
solar wind at L1 has been monitored by
the solar observatory SOHO and by ACE.
DSCVR launched 2/11/15 and placed at L1 for
space weather prediction because what happens
at the sun doesn’t stay at the sun.
L2 is at about the same distance from Earth as
L1 but on the night side, away from the Sun. The
L2 point has been chosen by NASA as the future
site of a large infra-red observatory, the "Next
Generation Space Telescope," renamed in honor
of a late NASA director The James Webb
Observatory. Launch set for 2018.
L4 and L5 are stable
equilibrium points
for an orbit with
respect to the Earth
and Moon. Because
of small departures,
a satellite needs an
effective restoring
force to bring it
back to the stable
point.
Attention has been given to two
stable points L4 and L5, located in
the Moon's orbit. These positions
have been studied as possible sites
for artificial space colonies in some
very distant future.
L4 and L5 lie at 60 degrees ahead
of and behind Earth and are
resistant to gravitational
perturbations. Because of this
stability, objects such as dust and
asteroids tend to accumulate in
these regions.
At L4 or L5, a spacecraft is truly
stable.
What are Lagrange
points? /
Operations / Our
Activities / ESA
https://www.youtube.com/watch?v
=z52WWLE8bBo&feature=player_e
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