Orbital eccentricity

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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:

A circle is a "degenerate" ellipse. In a circle, the two foci converge at the center. Therefore

and

.

A parabola is an extreme case of an ellipse and is the first open conic section. For any parabola:

Therefore, for any closed orbit,

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:

or

By the same token, a and e can predict Q and q.

and

Therefore

and

Subtracting the first equation from the second yields

From the above:

and

For , , the orbital radius, as one would expect.