Gravitational Potential Energy

How do we define gravitational potential energy (U)? Let's push a planet away from the sun. What is the work involved in doing that? In vector notation that work is:

So inserting the gravitational force law, we get

Integrating this:

So let's push the planet all the way to infinity,
and define the gravitational energy there to be 0:

Then we see that the gravitational potential energy is

How does this help us?

Total Energy

Remember that kinetic energy (energy of motion) is

And that total energy (kinetic + potential) of the system is conserved!

K + U = constant

Now, let's look at two orbiting bodies:

then the total energy of the system is

and if

(Why?)