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Orbital eccentricity

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Examples of orbits characterized by different eccentricities.

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.

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