- Specific relative angular momentum
In

astrodynamics , the**specific relative angular momentum**of anorbiting body with respect to acentral body is therelative angular momentum of the first body per unitmass . Specific relative angular momentum plays a pivotal role in definition oforbit equation s.**Definition**Specific relative angular momentum, represented by the symbol $mathbf\{h\},!$, is defined as the

cross product of the position vector $mathbf\{r\},!$ and velocity vector $mathbf\{v\},!$ of the orbiting body relative to the central body::$mathbf\{h\}=mathbf\{r\}\; imes\; mathbf\{v\}\; =\; \{\; mathbf\{r\}\; imes\; mathbf\{p\}\; over\; m\; \}\; =\; \{\; mathbf\{H\}\; over\; m\}$where:

*$mathbf\{r\},!$ is theorbital position vector of the orbiting body relative to the central body,

*$mathbf\{v\},!$ is theorbital velocity vector of the orbiting body relative to the central body,

*$mathbf\{p\}\; ,$ is thelinear momentum of the orbiting body relative to the central body,

* $m\; ,$ is themass of the orbiting body, and

*$mathbf\{H\}\; ,$ is therelative angular momentum of the orbiting body with respect to the central body.The units of $mathbf\{h\},!$ are

**m**.^{2}s^{-1}Under standard assumptions for an

orbiting body in a trajectory aroundcentral body at any given time the $mathbf\{h\},!$ vector is perpendicular to theosculating orbital plane defined by orbital position and velocity vectors.As usual in physics, the magnitude of the vector quantity $mathbf\{h\},!$ is denoted by $h,!$::$h=left|mathbf\{h\}\; ight|,!$

**Elliptical orbit**In an

elliptical orbit , the specific relative angular momentum is twice the area per unit time swept out by a chord from from the central mass to the orbiting body: this area is that referred to by Kepler's second law of planetary motion.Since the area of the entire ellipse of the orbit is swept out in one

orbital period , $h,!$ is equal to twice the area of the ellipse divided by the orbital period, giving the equation:$h=\; 2pi\; ab\; /(2pisqrt\{a^3/mu\})\; =\; b\; sqrt\{mu/a\}\; =\; sqrt\{a(1-e^2)mu\}$.

where

*$a$ issemi-major axis

*$b$ issemi-minor axis

*$mu$ isstandard gravitational parameter **See also***

Kepler's laws of planetary motion

*Planetary orbit **References**

*Wikimedia Foundation.
2010.*

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