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# 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, $0\leq e<1$

## 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. ${\frac {Q}{q}}={\frac {1+e}{1-e}}$

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. Browse