Bound Elliptical Orbits:
When an object is in a periodic orbit about the central body, it is said to be bound. In this case the mechanical energy will always be negative, but constant. The total ME is equal to (Derivation)

Thus

• This is the general relationship for the velocity of any orbiting body as a function of its distance from the central body.
• It is the total mechanical energy which determine the shape a bodies orbit - in particular the orbit's semimajor axis a.

Unbound Hyperbolic Orbits:
If the velocity of an object is increased enough so that the total ME becomes positive, then the object will move along a hyperbolic orbit. Such orbits are unbound in that the orbiting body will never return, and will have an eccentricity greater than one. The equations describing hyperbolic orbits are basically the same except for a few signs associated with a. See Conic Orbits Table)

Parabolic Escape Orbits:
If the total ME energy of a body is zero, then the object will be on an escape trajectory. Here the eccentricity of the orbit becomes one, and the semimajor axis goes to infinity. An elliptical orbit with an infinite semimajor axis of infinity is a parabola. See Escape Velocity.