Difference between revisions of "Orbital eccentricity"
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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>\ | + | <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, | ||
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and | and | ||
| − | <math>\mathit{Q} + \mathit{q} = \mathit{2a}</math> | + | <math>\mathit{Q} + \mathit{q} = \mathit{2a}\!</math> |
Therefore | Therefore | ||
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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 | ||
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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]] | ||
Latest revision as of 22:55, February 25, 2008
Orbital eccentricity is the measure of the departure of an orbit from a perfect circle.
Definitions
In geometry, eccentricity (e) is a concept universally applicable to conic sections.
For the general case of an ellipse having semi-major axis a and distance c from the center to either focus:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e=\frac{c}{a}}
A circle is a "degenerate" ellipse. In a circle, the two foci converge at the center. Therefore
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{c} = 0\!}
and
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{e} = 0\!} .
A parabola is an extreme case of an ellipse and is the first open conic section. For any parabola:
Therefore, for any closed orbit,
Practical application
In astrodynamics, any given pair of apsides can predict the semi-major axis and eccentricity of any orbit. Specifically, for periapsis q and apoapsis Q:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=\frac{Q + q}{2}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e=\frac{Q - q}{Q + q}}
or
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e = 1 - \frac{2}{(Q/q) + 1}}
By the same token, a and e can predict Q and q.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q/q = \frac{1+e}{1-e}}
and
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{Q} + \mathit{q} = \mathit{2a}\!}
Therefore
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q - q\frac{1+e}{1-e} = 0}
and
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{Q} + \mathit{q} = \mathit{2a}\!}
Subtracting the first equation from the second yields
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\left (1 + \frac{1+e}{1-e}\right ) = 2a}
From the above:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q = a(1-e)\!}
and
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q = a(1+e)\!}
For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{e} = 0\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{Q} = \mathit{q} = \mathit{a} = \mathit{r}\!} , the orbital radius, as one would expect.