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What is eccentricity

  1. Jul 23, 2014 #1
    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!
     
  2. jcsd
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