Eccentricity: Definition, Summary & Equations

[Greg Bernhardt]

Definition/Summary

The eccentricity $e$ of a conic section (other than a parabola or a pair of crossed lines) is its focal length divided by its major axis: $e = f/a$

The eccentricity of a conic section (other than a pair of crossed lines) is the distance from any point $P$ on the conic section to a focus $F$ divided by the distance from $P$ to the directrix accompanying $F$.

Eccentricity is a measure of circularity:
e = 0 circle
0 < e < 1 ellipse (other than a circle)
e = 1 parabola
1 < e < $\infty$ hyperbola
e = $\infty$ pair of crossed lines

Equations

For an ellipse or hyperbola with major axis 2a along the x-axis, and focal length 2f:

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

$$\frac{x^2}{a^2}\,+\,\frac{y^2}{a^2 - f^2}\,=\,\frac{x^2}{a^2}\,+\,\frac{y^2}{a^2(1 - e^2)}\,=\,1$$

distance from centre to directrix: $a/e$

Defining $b\,=\,a\sqrt{|1- e^2|}$ gives:

for $e < 1$ (ellipse):

$$f^2\,=\,a^2\,-\,b^2$$

$$\frac{x^2}{a^2}\,+\,\frac{y^2}{b^2}\,=\,1$$ (so the minor axis is 2b)

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

for $e > 1$ (hyperbola):

$$f^2\,=\,a^2\,+\,b^2$$

$$\frac{x^2}{a^2}\,-\,\frac{y^2}{b^2}\,=\,1$$

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

Extended explanation

Orbital eccentricity:

For astronomical orbits or trajectories, an alternative convenient definition is:

$$e = \frac{r_A - r_P}{r_A + r_P}$$

where $r_A = a(1 + e)$ is the apoapse distance

and $r_P = a(1 - e)$ is the periapse distance.

For parabolic trajectories, $r_A$ is taken to be ∞.

For hyperbolic trajectories, $r_A$ is the closest distance if gravity were repulsive.

These formulas of course are valid for any inverse-square-law force.

* 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!

 

[fresh_42]

See also: https://www.physicsforums.com/threads/what-is-a-conic.762859/ and Kepler's laws.