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,

$$0 \le e < 1$$

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.

Excentricité orbitale Excentricidade orbital