Elliptic orbit - Revision history

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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]]~~.

	~~==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]~~]

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

2013-12-28T07:17:55Z

Orbital period

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199.209.144.62: /* Radial elliptic trajectory */

2012-07-10T19:47:29Z

Radial elliptic trajectory

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