Orbital eccentricity

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:

<math>e=\frac{c}{a}</math>

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

<math>\mathit{c} = 0\!</math>

and

<math>\mathit{e} = 0\!</math>.

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

<math>\mathit{e}=1\!</math>

Therefore, for any closed orbit,

<math>0 \le e \le 1</math>

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:

<math>a=\frac{Q + q}{2}</math>

<math>e=\frac{Q - q}{Q + q}</math>

or

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

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

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

and

<math>\mathit{Q} + \mathit{q} = \mathit{2a}\!</math>

Therefore

<math>Q - q\frac{1+e}{1-e} = 0</math>

and

<math>\mathit{Q} + \mathit{q} = \mathit{2a}\!</math>

Subtracting the first equation from the second yields

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

From the above:

<math>q = a(1-e)\!</math>

and

<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.