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:
{\displaystyle e={\frac {c}{a}}}
A circle is a "degenerate" ellipse. In a circle, the two foci converge at the center. Therefore
{\displaystyle {\mathit {c}}=0\!}
and
{\displaystyle {\mathit {e}}=0\!}.
A parabola is an extreme case of an ellipse and is the first open conic section. For any parabola:
{\displaystyle {\mathit {e}}=1\!}
Therefore, for any closed orbit,
{\displaystyle 0\leq 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:
{\displaystyle a={\frac {Q+q}{2}}}
{\displaystyle e={\frac {Q-q}{Q+q}}}
or
{\displaystyle e=1-{\frac {2}{(Q/q)+1}}}
By the same token, a and e can predict Q and q.
{\displaystyle {\frac {Q}{q}}={\frac {1+e}{1-e}}}
and
{\displaystyle {\mathit {Q}}+{\mathit {q}}={\mathit {2a}}\!}
Therefore
{\displaystyle Q-q{\frac {1+e}{1-e}}=0}
and
{\displaystyle {\mathit {Q}}+{\mathit {q}}={\mathit {2a}}\!}
Subtracting the first equation from the second yields
{\displaystyle q\left(1+{\frac {1+e}{1-e}}\right)=2a}
From the above:
{\displaystyle q=a(1-e)\!}
and
{\displaystyle Q=a(1+e)\!}
For {\displaystyle {\mathit {e}}=0\!}, {\displaystyle {\mathit {Q}}={\mathit {q}}={\mathit {a}}={\mathit {r}}\!}, the orbital radius, as one would expect.
