Orbital eccentricity

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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:

e=\frac{c}{a}

A circle is a "degenerate" ellipse. In a circle, the two foci converge at the center. Therefore

\mathit{c} = 0\!

and

\mathit{e} = 0\!.

A parabola is an extreme case of an ellipse and is the first open conic section. For any parabola:

\mathit{e}=1\!

Therefore, for any closed orbit,

0 \le e \le 1

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:

a=\frac{Q + q}{2}

e=\frac{Q - q}{Q + q}

or

e = 1 - \frac{2}{(Q/q) + 1}

By the same token, a and e can predict Q and q.

Q/q = \frac{1+e}{1-e}

and

\mathit{Q} + \mathit{q} = \mathit{2a}\!

Therefore

Q - q\frac{1+e}{1-e} = 0

and

\mathit{Q} + \mathit{q} = \mathit{2a}\!

Subtracting the first equation from the second yields

q\left (1 + \frac{1+e}{1-e}\right ) = 2a

From the above:

q = a(1-e)\!

and

Q = a(1+e)\!

For \mathit{e} = 0\!, \mathit{Q} = \mathit{q} = \mathit{a} = \mathit{r}\!, the orbital radius, as one would expect.

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