Elliptic orbit

2013-12-28T07:17:55Z

2602:306:CF32:1F80:8121:FF3D:FAF3:5580: /* Orbital period */

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

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]].

==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]]

The units of <math>\mathbf{h}\,\!</math> are '''m<sup>2</sup>s<sup>−1</sup>'''.

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]]

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>

==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%27s_laws_of_planetary_motion#Second_law|Kepler's second law of planetary motion]].

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

:<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>.

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.

== See also ==
*[[Areal velocity]]
*[[Kepler's laws of planetary motion]]
*[[Kepler orbit]]
*[[Specific energy]]

{{orbits}}

==References==
{{reflist}}

{{DEFAULTSORT:Specific Relative Angular Momentum}}
[[Category:Orbits]]

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Elliptic orbit