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Energy Diagrams and Orbits

Let's write the total energy of a particle moving in a gravitational field in a clever way:

 (20) (21) (22) (23) (24)

Where
 (25)

is the (radial) potential energy plus the transverse kinetic energy (related to the constant angular momentum of the particle). If we plot the effective potential (and its pieces) we get a one-dimensional radial energy plot.

By drawing a constant total energy on this plot, the difference between and is the radial kinetic energy, which must be positive. We can determine lots of interesting things from this diagram.

In this figure, we show orbits with a given angular momentum and four generic total energies . These orbits have the following characteristics and names:

1. . This is a hyperbolic orbit.
2. . This is a parabolic orbit. This orbit defines escape velocity as we shall see later.
3. . This is generally an elliptical orbit (consistent with Kepler's First Law).
4. . This is a circular orbit. This is a special case of an elliptical orbit, but deserves special mention.

Note well that all of the orbits are conic sections. This interesting geometric connection between forces and conic orbits was a tremendous motivation for important mathematical work two or three hundred years ago.

Next: Escape Velocity, Escape Energy Up: Gravity Previous: Gravitational Potential Energy   Contents
Robert G. Brown 2004-04-12