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6. Eccentricity in termsof energy. |
If you aren't one of those reclusive, super abstract pure mathemetician types, you may have found yourself a little bit dissatified with the concept of eccentricity as the punchline to orbital motion. Well, the authors of this paper are highly sensitive to the concrete needs of the disgruntled. In this section we will express this abstract concept of eccentricity in terms of the physical concept of energy. Thus, we will be able to predict the orbit of the body based on its easily measurable total energy.
The kinetic energy of the orbiting body can be found with equation [5.4]
The potential energy of a body under the influence of gravity can be obtained by negating the work required to move that body to infinity. Thus,
The total energy of a body is the sum of the potential and kinetic energies. This leads us to
Now if we use equation [5.21] for when q = 0
in conjunction with equation [6.3] to eliminate the parameter r and then solve for e we will arrive at
Finally, if we substitute this into equation [5.21] we find ourselves in a state of utter serendipity
We can now determine the particular type of orbit that our body will follow based solely on its total energy. We know that for E < 0 we will have an elliptical orbit. This is when the orbiting body does not have enough energy to free itself from the fixed body and it is referred to as a "bound" system. When E = 0 the body has just enough energy to escape and it will follow a parabolic orbit. When E > 0 the body has an excess of escape energy and it will follow a hyperbolic orbit.
