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.

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 Butterfly Effect

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 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! Edward Lorenz 1960

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 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" (ISEE- 3) 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

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