Image:Eccentricity.svg
All types of conic sections, arranged with increasing eccentricity. Note that curvature decreases with eccentricity, and that none of these curves intersect.
In mathematics, the eccentricity, denoted e or Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon , is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.
In particular,
- The eccentricity of a circle is zero.
- The eccentricity of an (non-circle) ellipse is greater than zero but less than 1.
- The eccentricity of a parabola is 1.
- The eccentricity of a hyperbola is greater than 1 but less than infinity.
Furthermore, two conic sections are similar if and only if they have the same eccentricity.
Definitions
For every conic section, there exist a fixed point F, a fixed line L and a non-negative number e such that the conic section consists of all points whose distance to F equals e times their distance to L. e is called the eccentricity of the conic section.
The linear eccentricity of a conic section, denoted c or e, is the distance between its center and its focus (or one of its two foci).
Alternate Names
The eccentricity is sometimes called first eccentricity to distinguish it from the second eccentricity and third eccentricity defined for ellipses (see below). The eccentricity is also sometimes called numerical eccentricity.
In the case of ellipses and hyperbolas the linear eccentricity is sometimes called half-focal separation.
Notation
Two notational conventions are in common use:
- e for the eccentricity and c for the linear eccentricity.
- Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon
for the eccentricity and e for the linear eccentricity.
We will use the first one in this article.
Values
| conic section |
equation |
eccentricity (e) |
linear eccentricity (c) |
| circle |
Failed to parse (Missing texvc executable; please see math/README to configure.): x^2+y^2=r^2 |
Failed to parse (Missing texvc executable; please see math/README to configure.): 0 |
Failed to parse (Missing texvc executable; please see math/README to configure.): 0 |
| ellipse |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\sqrt{a^2-b^2}}{a} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{a^2-b^2} |
| parabola |
Failed to parse (Missing texvc executable; please see math/README to configure.): y^2=4ax |
Failed to parse (Missing texvc executable; please see math/README to configure.): 1 |
Failed to parse (Missing texvc executable; please see math/README to configure.): a |
| hyperbola |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\sqrt{a^2+b^2}}{a} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{a^2+b^2} |
Ellipses
For any ellipse, let a be the length of its semi-major axis and b be the length of its semi-minor axis.
We define a number of related additional concepts (only for ellipses):
| name |
symbol |
value in term of a and b |
value in term of Failed to parse (Missing texvc executable; please see math/README to configure.): o\!\varepsilon |
| first eccentricity |
Failed to parse (Missing texvc executable; please see math/README to configure.): e |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\sqrt{a^2-b^2}}{a} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \sin(o\!\varepsilon) |
| second eccentricity |
Failed to parse (Missing texvc executable; please see math/README to configure.): e' |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\sqrt{a^2-b^2}}{b} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \tan(o\!\varepsilon) |
| third eccentricity |
Failed to parse (Missing texvc executable; please see math/README to configure.): e'' |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\sqrt{a^2-b^2}}{\sqrt{a^2+b^2}} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\sin(o\!\varepsilon)}{\sqrt{2-\sin(o\!\varepsilon)^2}} |
| angular eccentricity |
Failed to parse (Missing texvc executable; please see math/README to configure.): o\!\varepsilon |
Failed to parse (Missing texvc executable; please see math/README to configure.): \arccos\left(\frac{b}{a}\right) |
Failed to parse (Missing texvc executable; please see math/README to configure.): o\!\varepsilon |
Quadrics
The eccentricity of a three-dimensional quadric is the eccentricity of a designated section of it. For example, on a triaxial ellipsoid, the meridional eccentricity is that of the ellipse formed by a section containing both the longest and the shortest axes (one of which will be the polar axis), and the equatorial eccentricity is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in the equatorial plane).
Celestial Mechanics
In celestial mechanics, for bound orbits in a spherical potential, the definition above is informally generalized. When the apocentre distance is close to pericentre distance, the orbit is said to have low eccentricity; when they are very different, the orbit is said be eccentric or having eccentricity near unity. This definition coincides with the mathematical definition of eccentricity for ellipse, in Keplerian, i.e., Failed to parse (Missing texvc executable; please see math/README to configure.): 1/r
potentials.
See also
External links
|
Orbits |
| Orbital parameters |
|
Classical orbital elements
Failed to parse (Missing texvc executable; please see math/README to configure.): i\,\!
Inclination
Failed to parse (Missing texvc executable; please see math/README to configure.): \Omega\,\!
Longitude of the ascending node
Failed to parse (Missing texvc executable; please see math/README to configure.): e\,\!
Eccentricity
|
Failed to parse (Missing texvc executable; please see math/README to configure.): \omega\,\!
Argument of periapsis
Failed to parse (Missing texvc executable; please see math/README to configure.): a\,\!
Semi-major axis
Failed to parse (Missing texvc executable; please see math/README to configure.): M_o\,\!
Mean anomaly at epoch
|
Other parameters
Failed to parse (Missing texvc executable; please see math/README to configure.): \nu\,
True anomaly
Failed to parse (Missing texvc executable; please see math/README to configure.): b\,
Semi-minor axis
Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon\,
Linear eccentricity
Failed to parse (Missing texvc executable; please see math/README to configure.): E\,
Eccentric anomaly
|
Failed to parse (Missing texvc executable; please see math/README to configure.): L\,
Mean longitude
Failed to parse (Missing texvc executable; please see math/README to configure.): l\,
True longitude
Failed to parse (Missing texvc executable; please see math/README to configure.): T\,
Orbital period
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| Types |
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| Maneuvers |
Bi-elliptic transfer · Geostationary transfer · Gravity assist · Hohmann transfer · Inclination change · Phasing · Rendezvous |
| Related topics |
Apsis · Celestial coordinate system · Delta-v budget · Epoch · Ephemeris · Equatorial coordinate system · Gravity turn · Ground track · Interplanetary Transport Network · Kepler's laws of planetary motion · Lagrangian point · n-body problem · Oberth effect · Orbit equation · Orbital state vectors · Perturbation · Retrograde and direct motion · Specific orbital energy · Specific relative angular momentum |
| List of orbits |
af:Eksentrisiteit (wiskunde)
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