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:

$e=c/a$

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 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.