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,

$0\leq e\leq 1$

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.