Solving the angular momentum equation

(1)

where h is the specific angular momentum, G is the gravitational constant, M is the mass of the central body, and a is the semimajor axis of the orbit, for the eccentricity e gives

(2)

Plugging the gravitational energy equation

(3)

where m is the mass of the orbiting body, into (2) gives

(4)

Circular Orbit, Eccentricity Perturbations, Elliptical Orbit, Hyperbolic Orbit, Parabolic Orbit