Algebraic function: Difference between revisions

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In [[celestial mechanics]], the '''eccentricity vector''' of a [[Kepler orbit]] is the [[vector (geometric)|vector]] that points towards the [[periapsis]] and has a magnitude equal to the orbit's scalar [[orbital eccentricity|eccentricity]]. The magnitude is [[unitless]]. For Kepler orbits the '''eccentricity vector''' is a constant of motion. Its main use is in the analysis of almost circular orbits, as perturbing (non-Keplerian) forces on an actual orbit will cause the [[Osculating orbit|osculating]] eccentricity vector to change continuously. For the eccentricity and [[argument of periapsis]] parameters, eccentricity zero (circular orbit) corresponds to a singularity.
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The '''eccentricity vector''' <math> \mathbf{e} \,</math> can be calculated<ref>{{cite book |title=The Kepler Problem |first=Bruno |last=Cordani |page=22 |location= |publisher=Birkhaeuser |year=2003 |isbn=3-7643-6902-7 }}</ref>
 
 
  <li>[http://www.365wango.com/forum.php?mod=viewthread&tid=68457 http://www.365wango.com/forum.php?mod=viewthread&tid=68457]</li>
:<math> \mathbf{e} = {\mathbf{v}\times\mathbf{h}\over{\mu}} - {\mathbf{r}\over{\left|\mathbf{r}\right|}}</math>
 
where:
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*<math> \mathbf{v}\,\!</math> is [[velocity vector]]
 
*<math>\mathbf{h}\,\!</math> is [[Specific relative angular momentum|specific angular momentum vector]] (equal to <math>\mathbf{r}\times\mathbf{v}</math>)
  <li>[http://www.nokia4bh.hyperphp.com/vb/showthread.php?p=3365#post3365 http://www.nokia4bh.hyperphp.com/vb/showthread.php?p=3365#post3365]</li>
*<math>\mathbf{r}\,\!</math> is [[position vector]]
 
*<math>\mu\,\!</math> is [[standard gravitational parameter]]
  <li>[http://dirtyhotshot.com/activity/p/27150/ http://dirtyhotshot.com/activity/p/27150/]</li>
 
 
or equivalently:
  <li>[http://www.hngggj.com/news/html/?53800.html http://www.hngggj.com/news/html/?53800.html]</li>
:<math> \mathbf{e} = {\mathbf{\left |v \right |}^2 \mathbf{r} \over {\mu}} - {(\mathbf{r} \cdot \mathbf{v} ) \mathbf{v} \over{\mu}} - {\mathbf{r}\over{\left|\mathbf{r}\right|}}</math>
 
 
</ul>
==See also==
*[[Kepler orbit]]
*[[Orbit]]
*[[Eccentricity (orbit)|Eccentricity]]
*[[Laplace–Runge–Lenz vector]]
 
==References==
{{Reflist}}
 
{{DEFAULTSORT:Eccentricity Vector}}
[[Category:Orbits]]
[[Category:Vectors]]

Latest revision as of 20:31, 16 September 2014

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