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

. This is a hyperbolic orbit.

. This is a parabolic orbit. This orbit
defines escape velocity as we shall see later.

. This is generally an elliptical orbit
(consistent with Kepler's First Law).

. 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
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Robert G. Brown
20040412