The Three Body Problem

Up to now, we have considered orbits involving two objects -- ie, the Earth orbiting the Sun, the Moon orbiting the Earth, a comet orbiting the Sun.

For this situation, one can easily solve for the orbital properties, and (if m << M) the orbiting body has conserved quantities: energy and angular momentum.

But there are more than two objects in the solar system!

What happens when 3 objects interact? Forces add:

So, for object m:

Now, energy and angular momentum for each object are no longer constant!

Instead, the total angular momentum and total energy are conserved.

And analytic solutions for the orbit no longer exist. This situation is known as the three body problem.

What are some examples of 3-body systems?

Let's look at an example. Let's say the Earth, Sun, and Jupiter are all in a line, like this (we say that Jupiter is at opposition):

For the Earth, how does the force from Jupiter compare to the force from the Sun?

In other words, the Earth doesn't feel much from Jupiter....

So who cares about the 3-body problem?