Difference between revisions of "Orbital eccentricity"
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In astrodynamics, any given pair of [[apsis|apsides]] can predict the [[semi-major axis]] and eccentricity of any orbit. Specifically, for periapsis q and apoapsis Q: | In astrodynamics, any given pair of [[apsis|apsides]] can predict the [[semi-major axis]] and eccentricity of any orbit. Specifically, for periapsis q and apoapsis Q: | ||
| − | a = (Q + q)/2 | + | <math>a = (Q + q)/2</math> |
| − | + | <math>e = (Q - q)/(Q + q)</math> | |
or | or | ||
| − | + | <math>e = 1 - 2/((Q/q) + 1)</math> | |
By the same token, a and ''e'' can predict Q and q. | By the same token, a and ''e'' can predict Q and q. | ||
| − | Q/q = (1+e)/(1-e) | + | <math>Q/q = (1+e)/(1-e)</math> |
and | and | ||
| − | Q + q = 2a | + | <math>Q + q = 2a</math> |
Therefore | Therefore | ||
| − | Q - q(1+e)/(1-e) = 0 | + | <math>Q - q(1+e)/(1-e) = 0</math> |
and | and | ||
| − | Q + q = 2a | + | <math>Q + q = 2a</math> |
Subtracting the first equation from the second yields | Subtracting the first equation from the second yields | ||
| − | q(1 + (1+e)/(1-e)) = 2a | + | <math>q(1 + (1+e)/(1-e)) = 2a</math> |
From the above: | From the above: | ||
| − | q = a(1-e) | + | <math>q = a(1-e)</math> |
and | and | ||
| − | Q = a(1+e) | + | <math>Q = a(1+e)</math> |
For ''e'' = 0, Q = q = a = r, the orbital radius, as one would expect. | For ''e'' = 0, Q = q = a = r, the orbital radius, as one would expect. | ||
[[Category:Astronomy]] | [[Category:Astronomy]] | ||
Revision as of 21:23, February 10, 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:
A circle is a "degenerate" ellipse. In a circle, the two foci converge at the center. Therefore c = 0 and 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:
or
By the same token, a and e can predict Q and q.
and
Therefore
and
Subtracting the first equation from the second yields
From the above:
and
For e = 0, Q = q = a = r, the orbital radius, as one would expect.