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:

$e=\frac{c}{a}$

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.