- #1
- 19,449
- 10,038
Definition/Summary
The eccentricity [itex]e[/itex] of a conic section (other than a parabola or a pair of crossed lines) is its focal length divided by its major axis: [itex]e = f/a[/itex]
The eccentricity of a conic section (other than a pair of crossed lines) is the distance from any point [itex]P[/itex] on the conic section to a focus [itex]F[/itex] divided by the distance from [itex]P[/itex] to the directrix accompanying [itex]F[/itex].
Eccentricity is a measure of circularity:
e = 0 circle
0 < e < 1 ellipse (other than a circle)
e = 1 parabola
1 < e < [itex]\infty[/itex] hyperbola
e = [itex]\infty[/itex] pair of crossed lines
Equations
For an ellipse or hyperbola with major axis 2a along the x-axis, and focal length 2f:
[tex]e = \frac{f}{a}[/tex]
[tex]\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[/tex]
distance from centre to directrix: [itex]a/e[/itex]
Defining [itex]b\,=\,a\sqrt{|1- e^2|}[/itex] gives:
for [itex]e < 1[/itex] (ellipse):
[tex]f^2\,=\,a^2\,-\,b^2[/tex]
[tex]\frac{x^2}{a^2}\,+\,\frac{y^2}{b^2}\,=\,1[/tex] (so the minor axis is 2b)
[tex]e\,=\,\frac{f}{a}\,=\,\sqrt{1 - \left (\frac{b}{a} \right)^2}[/tex]
for [itex]e > 1[/itex] (hyperbola):
[tex]f^2\,=\,a^2\,+\,b^2[/tex]
[tex]\frac{x^2}{a^2}\,-\,\frac{y^2}{b^2}\,=\,1[/tex]
[tex]e\,=\,\frac{f}{a}\,=\,\sqrt{1 + \left (\frac{b}{a} \right)^2}[/tex]
Extended explanation
Orbital eccentricity:
For astronomical orbits or trajectories, an alternative convenient definition is:
[tex]e = \frac{r_A - r_P}{r_A + r_P}[/tex]
where [itex]r_A = a(1 + e)[/itex] is the apoapse distance
and [itex]r_P = a(1 - e)[/itex] is the periapse distance.
For parabolic trajectories, [itex]r_A[/itex] is taken to be ∞.
For hyperbolic trajectories, [itex]r_A[/itex] 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!
The eccentricity [itex]e[/itex] of a conic section (other than a parabola or a pair of crossed lines) is its focal length divided by its major axis: [itex]e = f/a[/itex]
The eccentricity of a conic section (other than a pair of crossed lines) is the distance from any point [itex]P[/itex] on the conic section to a focus [itex]F[/itex] divided by the distance from [itex]P[/itex] to the directrix accompanying [itex]F[/itex].
Eccentricity is a measure of circularity:
e = 0 circle
0 < e < 1 ellipse (other than a circle)
e = 1 parabola
1 < e < [itex]\infty[/itex] hyperbola
e = [itex]\infty[/itex] pair of crossed lines
Equations
For an ellipse or hyperbola with major axis 2a along the x-axis, and focal length 2f:
[tex]e = \frac{f}{a}[/tex]
[tex]\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[/tex]
distance from centre to directrix: [itex]a/e[/itex]
Defining [itex]b\,=\,a\sqrt{|1- e^2|}[/itex] gives:
for [itex]e < 1[/itex] (ellipse):
[tex]f^2\,=\,a^2\,-\,b^2[/tex]
[tex]\frac{x^2}{a^2}\,+\,\frac{y^2}{b^2}\,=\,1[/tex] (so the minor axis is 2b)
[tex]e\,=\,\frac{f}{a}\,=\,\sqrt{1 - \left (\frac{b}{a} \right)^2}[/tex]
for [itex]e > 1[/itex] (hyperbola):
[tex]f^2\,=\,a^2\,+\,b^2[/tex]
[tex]\frac{x^2}{a^2}\,-\,\frac{y^2}{b^2}\,=\,1[/tex]
[tex]e\,=\,\frac{f}{a}\,=\,\sqrt{1 + \left (\frac{b}{a} \right)^2}[/tex]
Extended explanation
Orbital eccentricity:
For astronomical orbits or trajectories, an alternative convenient definition is:
[tex]e = \frac{r_A - r_P}{r_A + r_P}[/tex]
where [itex]r_A = a(1 + e)[/itex] is the apoapse distance
and [itex]r_P = a(1 - e)[/itex] is the periapse distance.
For parabolic trajectories, [itex]r_A[/itex] is taken to be ∞.
For hyperbolic trajectories, [itex]r_A[/itex] 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!