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Orbital eccentricity (view source)
Revision as of 21:20, February 10, 2008
, 21:20, February 10, 2008New article
'''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 section]]s.
For the general case of an [[ellipse]] having semi-major axis a and distance c from the center to either focus:
''e'' = c/a.
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:
''e'' = 1.
Therefore, for any closed orbit, 0 <= ''e'' <= 1.
== Practical application ==
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
''e'' = (Q - q)/(Q + q)
or
''e'' = 1 - 2/((Q/q) + 1)
By the same token, a and ''e'' can predict Q and q.
Q/q = (1+e)/(1-e)
and
Q + q = 2a
Therefore
Q - q(1+e)/(1-e) = 0
and
Q + q = 2a.
Subtracting the first equation from the second yields
q(1 + (1+e)/(1-e)) = 2a
From the above:
q = a(1-e)
and
Q = a(1+e).
For ''e'' = 0, Q = q = a = r, the orbital radius, as one would expect.
[[Category:Astronomy]]
== Definitions ==
In geometry, eccentricity (''e'') is a concept universally applicable to [[conic section]]s.
For the general case of an [[ellipse]] having semi-major axis a and distance c from the center to either focus:
''e'' = c/a.
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:
''e'' = 1.
Therefore, for any closed orbit, 0 <= ''e'' <= 1.
== Practical application ==
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
''e'' = (Q - q)/(Q + q)
or
''e'' = 1 - 2/((Q/q) + 1)
By the same token, a and ''e'' can predict Q and q.
Q/q = (1+e)/(1-e)
and
Q + q = 2a
Therefore
Q - q(1+e)/(1-e) = 0
and
Q + q = 2a.
Subtracting the first equation from the second yields
q(1 + (1+e)/(1-e)) = 2a
From the above:
q = a(1-e)
and
Q = a(1+e).
For ''e'' = 0, Q = q = a = r, the orbital radius, as one would expect.
[[Category:Astronomy]]