There is a beautifully simple result concerning the total mechanical energy for an object in a closed orbit in a central gravitational field. The result holds for any elliptical orbit but for simplicity we shall derive the result for a circular orbit and then generalize by replacing the radius in orbit by the semi-major axis as we did when we argued for Newton's derivation of Kepler's third law.
The total mechanical energy for a planet with mass, m in a circular
orbit with radius, r, around a body with mass M can be written
We can eliminate v by equating the net force in circular motion to the force of gravity
Inserting this in Eq. 1 yields
Note that as we should expect for a closed orbit is less than zero. It is this last expression for which can be generalized to the general elliptical case:
Where a is the semi major axis of the elliptical orbit. So the total mechanical energy is constant and takes on similar forms for circular and elliptical orbits. In the circular orbit, since there speed is constant, we furthermore have that kinetic energy and potential energy are constants of motion. Specifically we see that;
In contrast the kinetic and potential energy in elliptical orbits are not constant but vary so that one is large when the other is small and vise versa. It is easy to derive the following equations:
Where is the eccentricity of the elliptical orbit. We see that when e=0 corresponding to a circular orbit then we re-find the corresponding expressions listed above. The larger the eccentricity, e, the larger is variation of the potential and kinetic energies during each period of the motion.