Energy Considerations in Planetary and Satellite Motion

Consider the motion of a body orbiting circularly around a much more massive body . The Gravitational Potential Energy of the system is:

The Kinetic Energy of the system is:

as is considered to be at rest. By making Newton’s Law of Gravitation equivalent to Centripetal Force:

We can substitute this with Kinetic Energy:

So the total Mechanical Energy is:

The equation above is still valid for elliptical orbits when we replace with the semimajor axis .

The interpretation of this constant and negative energy is that the system is closed, and is always bound to . For to escape , it would need a positive energy as tends towards infinity. However, both and tend towards 0. If we supply enough energy so that it is positive or 0, then the orbit is no longer elliptical. When , it is parabolic, and when , it is hyperbolic.

The equation also tells us that we cannot change the speed without changing the radius. If we want to increase the speed of a satellite, it increasing kinetic energy. As a result the total energy is less negative and the radius increases.