Specific relative angular momentum

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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.

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:

$\mathbf{h}=\mathbf{r}\times \mathbf{v} = { \mathbf{r} \times \mathbf{p} \over m } = { \mathbf{H} \over m}$

where:

$\mathbf{r}\,\!$ is the orbital position vector of the orbiting body relative to the central body,
$\mathbf{v}\,\!$ is the orbital velocity vector of the orbiting body relative to the central body,
$\mathbf{p} \,$ is the linear momentum of the orbiting body relative to the central body,
$m \,$ is the mass of the orbiting body, and
$\mathbf{H} \,$ is the relative angular momentum of the orbiting body with respect to the central body.

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

The magnitude of $\mathbf{h}\,\!$ is denoted as $h\,\!$ :

$h=\left|\mathbf{h}\right|\,\!$

For an elliptical orbit, it is twice the area per unit time swept out, hence twice the area of the ellipse divided by the orbital period, hence $2\pi ab /(2\pi\sqrt{a^3/\mu}) = b \sqrt{\mu/a}$ , which is $\sqrt{a(1-e^2)\mu}$ .