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Orbital eccentricity (view source)
Revision as of 22:55, February 25, 2008
, 22:55, February 25, 2008Force PNG rendering
A [[circle]] is a "degenerate" ellipse. In a circle, the two foci converge at the center. Therefore
A [[circle]] is a "degenerate" ellipse. In a circle, the two foci converge at the center. Therefore
<math>\mathrm{c} = 0</math>
<math>\mathit{c} = 0\!</math>
and
and
<math>\mathit{e} = 0</math>.
<math>\mathit{e} = 0\!</math>.
A [[parabola]] is an extreme case of an ellipse and is the first open conic section. For any parabola:
A [[parabola]] is an extreme case of an ellipse and is the first open conic section. For any parabola:
<math>\mathit{e}=1</math>
<math>\mathit{e}=1\!</math>
Therefore, for any closed orbit,
Therefore, for any closed orbit,
and
and
<math>\mathit{Q} + \mathit{q} = \mathit{2a}</math>
<math>\mathit{Q} + \mathit{q} = \mathit{2a}\!</math>
Therefore
Therefore
and
and
<math>\mathit{Q} + \mathit{q} = \mathit{2a}</math>
<math>\mathit{Q} + \mathit{q} = \mathit{2a}\!</math>
Subtracting the first equation from the second yields
Subtracting the first equation from the second yields
From the above:
From the above:
<math>q = a(1-e)</math>
<math>q = a(1-e)\!</math>
and
and
<math>Q = a(1+e)</math>
<math>Q = a(1+e)\!</math>
For <math>\mathit{e} = 0</math>, <math>\mathit{Q} = \mathit{q} = \mathit{a} = \mathit{r}</math>, the orbital radius, as one would expect.
For <math>\mathit{e} = 0\!</math>, <math>\mathit{Q} = \mathit{q} = \mathit{a} = \mathit{r}\!</math>, the orbital radius, as one would expect.
[[Category:Astronomy]]
[[Category:Astronomy]]