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.