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:''See also: [[Classical central-force problem#Specific angular momentum|Classical central-force problem]]''


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In [[celestial mechanics]], the '''specific relative angular momentum (h)''' of two [[orbiting body|orbiting bodies]] is the [[vector product]] of the relative position and the relative velocity. Equivalently, it is the total [[angular momentum]] divided by the [[reduced mass]].<ref>{{cite web| url = http://curious.astro.cornell.edu/pdf-files/eclipse.pdf|first=Jagadheep D.|last= Pandian|work=Curious about Astronomy?|publisher=Cornell University|title=Eclipse}}</ref> Specific relative angular momentum plays a pivotal role in the analysis of the [[two-body problem]].
 
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==Definition==
Specific relative angular momentum, represented by the symbol <math>\mathbf{h}\,\!</math>, is defined as the [[cross product]] of the relative [[orbital position vector|position vector]] <math>\mathbf{r}\,\!</math> and the relative [[orbital velocity vector|velocity vector]] <math>\mathbf{v}\,\!</math>.
:<math>\mathbf{h} = \mathbf{r}\times \mathbf{v} = { \mathbf{L} \over \mu } </math>
where:
*<math>\mathbf{r}\,\!</math> is the relative [[orbital position vector]]
*<math>\mathbf{v}\,\!</math> is the relative [[orbital velocity vector]]
*<math> \mathbf{L} = \mathbf{L_{M}} + \mathbf{L_{n}} \, </math> is the total [[angular momentum]] of the system
*<math> \mu \, </math> is the [[reduced mass]]


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The units of <math>\mathbf{h}\,\!</math> are '''m<sup>2</sup>s<sup>−1</sup>'''.
 
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For unperturbed orbits the <math>\mathbf{h}\,\!</math> vector is always perpendicular to the fixed [[orbital plane (astronomy)|orbital plane]]. However, for perturbed orbits the <math>\mathbf{h}\,\!</math> vector is generally not perpendicular to the [[osculating orbit|osculating orbital plane]]


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As usual in physics, the [[Magnitude (mathematics)|magnitude]] of the vector quantity <math>\mathbf{h}\,\!</math> is denoted by <math>h\,\!</math>:
 
:<math>h = \left \| \mathbf{h} \right \| </math>
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In an [[elliptical orbit]], the specific relative angular momentum is twice the area per unit time swept out by a chord from the primary to the secondary: this area is referred to by [[Kepler%27s_laws_of_planetary_motion#Second_law|Kepler's second law of planetary motion]].
 
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Since the area of the entire orbital ellipse is swept out in one [[orbital period]], <math>h\,\!</math> is equal to twice the area of the ellipse divided by the orbital period, as represented by the equation


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:<math> h = \frac{ 2\pi ab }{2\pi \sqrt{ \frac{a^3}{ G(M\!+\!m) }}} = b \sqrt{\frac{ G(M\!+\!m) }{a} } = \sqrt{a(1-e^2) G(M\!+\!m) } = \sqrt{ p G(M\!+\!m) }</math>.
 
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where
*<math>a\,</math> is the [[semi-major axis]]
*<math>b\,</math> is the [[semi-minor axis]]
*<math>p\,</math> is the [[semi-latus rectum]]
*<math>G\,</math> is the [[gravitational constant]]
*<math>M\,</math>, <math>m\,</math> are the two masses.


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== See also ==
 
*[[Areal velocity]]
  <li>[http://cgi.www5c.biglobe.ne.jp/~kk_aoi/bbs/apeboard_plus.cgi/ http://cgi.www5c.biglobe.ne.jp/~kk_aoi/bbs/apeboard_plus.cgi/]</li>
*[[Kepler's laws of planetary motion]]
 
*[[Kepler orbit]]
  <li>[http://www.hw5699.com:81/forum.php?mod=viewthread&tid=1600632&fromuid=54907 http://www.hw5699.com:81/forum.php?mod=viewthread&tid=1600632&fromuid=54907]</li>
*[[Specific energy]]
 
  <li>[http://site.vhostapp.com/news/html/?1911431.html http://site.vhostapp.com/news/html/?1911431.html]</li>
 
  <li>[http://matchsouls.com/tweet/?module=rijcgrad http://matchsouls.com/tweet/?module=rijcgrad]</li>
 
  <li>[http://math-preview.wmflabs.org/wiki/User:Dylhnrep#Air_Max_2014_Nz_Editor.27s_Choice http://math-preview.wmflabs.org/wiki/User:Dylhnrep#Air_Max_2014_Nz_Editor.27s_Choice]</li>
 
</ul>


== Longchamp Le Pliage Nz In New South Wales ==
{{orbits}}


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==References==
 
{{reflist}}
  <li>[http://advertsbook.net/forum/read.php?28,212495 http://advertsbook.net/forum/read.php?28,212495]</li>
 
 
{{DEFAULTSORT:Specific Relative Angular Momentum}}
  <li>[http://enseignement-lsf.com/spip.php?article64#forum18432291 http://enseignement-lsf.com/spip.php?article64#forum18432291]</li>
[[Category:Orbits]]
 
  <li>[http://enseignement-lsf.com/spip.php?article172#forum18417004 http://enseignement-lsf.com/spip.php?article172#forum18417004]</li>
 
  <li>[http://gxjd.aliapp.com/read.php?tid=126788&displayMode=1&page=e#a] http://gxjd.aliapp.com/read.php?tid=126788&displayMode=1&page=e#a]]</li>
 
  <li>[http://www.748yx.com/forum.php?mod=viewthread&tid=3611 http://www.748yx.com/forum.php?mod=viewthread&tid=3611]</li>
 
</ul>

Revision as of 08:17, 28 December 2013

See also: Classical central-force problem

In celestial mechanics, the specific relative angular momentum (h) of two orbiting bodies is the vector product of the relative position and the relative velocity. Equivalently, it is the total angular momentum divided by the reduced mass.[1] Specific relative angular momentum plays a pivotal role in the analysis of the two-body problem.

Definition

Specific relative angular momentum, represented by the symbol h, is defined as the cross product of the relative position vector r and the relative velocity vector v.

h=r×v=Lμ

where:

The units of h are m2s−1.

For unperturbed orbits the h vector is always perpendicular to the fixed orbital plane. However, for perturbed orbits the h vector is generally not perpendicular to the osculating orbital plane

As usual in physics, the magnitude of the vector quantity h is denoted by h:

h=h

Elliptical orbit

In an elliptical orbit, the specific relative angular momentum is twice the area per unit time swept out by a chord from the primary to the secondary: this area is referred to by Kepler's second law of planetary motion.

Since the area of the entire orbital ellipse is swept out in one orbital period, h is equal to twice the area of the ellipse divided by the orbital period, as represented by the equation

h=2πab2πa3G(M+m)=bG(M+m)a=a(1e2)G(M+m)=pG(M+m).

where

See also

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