# Specific relative angular momentum

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Specific relative angular momentum

In astrodynamics, the specific relative angular momentum of an orbiting body with respect to a central body is the relative angular momentum of the first body per unit mass. Specific relative angular momentum plays a pivotal role in definition of orbit equations.

Definition

Specific relative angular momentum, represented by the symbol $mathbf\left\{h\right\},!$, is defined as the cross product of the position vector $mathbf\left\{r\right\},!$ and velocity vector $mathbf\left\{v\right\},!$ of the orbiting body relative to the central body::$mathbf\left\{h\right\}=mathbf\left\{r\right\} imes mathbf\left\{v\right\} = \left\{ mathbf\left\{r\right\} imes mathbf\left\{p\right\} over m \right\} = \left\{ mathbf\left\{H\right\} over m\right\}$where:
*$mathbf\left\{r\right\},!$ is the orbital position vector of the orbiting body relative to the central body,
*$mathbf\left\{v\right\},!$ is the orbital velocity vector of the orbiting body relative to the central body,
*$mathbf\left\{p\right\} ,$ is the linear momentum of the orbiting body relative to the central body,
* $m ,$ is the mass of the orbiting body, and
*$mathbf\left\{H\right\} ,$ is the relative angular momentum of the orbiting body with respect to the central body.

The units of $mathbf\left\{h\right\},!$ are m2s-1.

Under standard assumptions for an orbiting body in a trajectory around central body at any given time the $mathbf\left\{h\right\},!$ vector is perpendicular to the osculating orbital plane defined by orbital position and velocity vectors.

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

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

:$h= 2pi ab /\left(2pisqrt\left\{a^3/mu\right\}\right) = b sqrt\left\{mu/a\right\} = sqrt\left\{a\left(1-e^2\right)mu\right\}$.

where
*$a$ is semi-major axis
*$b$ is semi-minor axis
*$mu$ is standard gravitational parameter

*Kepler's laws of planetary motion
*Planetary orbit

References

Wikimedia Foundation. 2010.

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