Orbital Energy 

What is the energy of an orbiting object?
Consider a circular orbit (r=a) and m << M.

So E = K + U. We know U. What is K (or, what is speed)?

and we have Kepler's 3rd Law:

so chugging through the algebra, we get:

or the general formula for circular velocity:

So for, example, look at the Hubble Space Telescope. It is in a low-Earth orbit (600 km above the surface). So,


Now, back to energy:
Note that E < 0. This is called a bound orbit.
We have derived this for circular orbits, but it can also be derived for any bound orbit.

Note two important things:

Let's do two thought experiments.

Orbital speed

Let's equate the two different expressions for orbital energy:
Doing a little algebra we get
which gives the object's speed at any point along the orbit. This is the vis-viva equation, and is more generally