Specific relative angular momentum

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 $\mathbf{h}\,\!$ , is defined as the cross product of the relative position vector $\mathbf{r}\,\!$ and the relative velocity vector $\mathbf{v}\,\!$ .

\mathbf{h} = \mathbf{r}\times \mathbf{v} = { \mathbf{L} \over \mu }

where:

$\mathbf{r}\,\!$ is the relative orbital position vector
$\mathbf{v}\,\!$ is the relative orbital velocity vector
$\mathbf{L} = \mathbf{L_1} + \mathbf{L_2} \,$ is the total angular momentum of the system (i.e. the sum of the angular momenta of each body)
$\mu \,$ is the reduced mass

The units of $\mathbf{h}\,\!$ are m²s⁻¹.

The $\mathbf{h}\,\!$ vector is always perpendicular to the instantaneous osculating orbital plane, which coincides with the instantaneous perturbed orbit. It would not necessarily be perpendicular to an average plane which accounted for many years of perturbations.

As usual in physics, the magnitude of the vector quantity $\mathbf{h}\,\!$ is denoted by $h\,\!$ :

h = \left \| \mathbf{h} \right \|

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 = \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) }

where

$a\,$ is the semi-major axis
$b\,$ is the semi-minor axis
$p\,$ is the semi-latus rectum
$G\,$ is the gravitational constant
$M\,$ , $m\,$ are the two masses.

General	Box Capture Circular Elliptical / Highly elliptical Escape Graveyard Hyperbolic trajectory Inclined / Non-inclined Osculating Parabolic trajectory Parking Synchronous semi sub Transfer orbit

Geocentric	Geosynchronous Geostationary Sun-synchronous Low Earth Medium Earth High Earth Molniya Near-equatorial Orbit of the Moon Polar Tundra Two-line elements

About other points	Areosynchronous Areostationary Halo Lissajous Lunar Heliocentric Heliosynchronous

Parameters

Shape/Size	$e\,\!$ Eccentricity $a\,\!$ Semi-major axis $b\,\!$ Semi-minor axis $Q,q\,\!$ Apsides

Orientation	$i\,\!$ Inclination $\Omega\,\!$ Longitude of the ascending node $\omega\,\!$ Argument of periapsis $\varpi\,\!$ Longitude of the periapsis

Position	$M\,\!$ Mean anomaly $\nu\,\!$ True anomaly $E\,\!$ Eccentric anomaly $L\,\!$ Mean longitude $l\,\!$ True longitude

Variation	$T\,\!$ Orbital period $n\,\!$ Mean motion $v\,\!$ Orbital speed $t_0\,\!$ Epoch

Maneuvers

Collision avoidance (spacecraft) Delta-v Delta-v budget Bi-elliptic transfer Geostationary transfer Gravity assist Gravity turn Hohmann transfer Low energy transfer Oberth effect Inclination change Phasing Rocket equation Rendezvous Transposition, docking, and extraction

Other orbital mechanics topics

Celestial coordinate system Characteristic energy Ephemeris Equatorial coordinate system Ground track Hill sphere Interplanetary Transport Network Kepler's laws of planetary motion Lagrangian point n-body problem Orbit equation Orbital state vectors Perturbation Retrograde motion Specific orbital energy Specific relative angular momentum

List of orbits

References

↑ Pandian, Jagadheep D. "Eclipse" (PDF). Curious about Astronomy?. Cornell University.

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