What is the Mathematical Definition of Orbital Eccentricity?

[Greg Bernhardt]

Definition/Summary

Eccentricity is the measure of the 'roundness' of the orbit.
For circular orbits: e=0
For elliptical orbits: 0<e<1
For parabolic trajectories: e=1
For hyperbolic Trajectories: e>1

Equations

$$e= \frac{c}{a}$$

For ellipses:
$$e = \frac{r_A-r_P}{r_a+r_P}$$

$$e = \sqrt{ 1- \left ( \frac{b}{a} \right )^2}$$

For hyperbolas:
$$e = \sqrt{ 1 + \left( \frac{b}{a} \right)^2}$$

Extended explanation

a is the semi-major axis of the orbit. For an elliptical orbit, this is equal to one half the longest length of the ellipse. For an hyperbolic path, it is equal to the distance of periapsis to to point where the asymptote lines cross (the center of the hyperbola).

b is the semi-minor axis of the orbit. For an elliptical orbit, this is equal to one half the width of the ellipse. See attached image for a hyperbolic orbit.

c is the distance between the center and the focus of the orbit.


r_A is the apoapsis distance as measured from the focus.

r_P is the periapsis distance as measured from the focus.

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[Aaron Paulo]

Thanks for the overview on orbital eccentricity!