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Specific relative angular momentum

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In astrodynamics specific relative angular momentum ( $\mathbf{h}\,\!$ ) of orbiting body ( $m_2\,\!$ ) relative to central body ( $m_1\,\!$ ) is the relative angular momentum of $m_2\,\!$ per unit mass. Specific relative angular momentum plays a pivotal role in definition of orbit equations.

Specific relative angular momentum ( $\mathbf{h}\,\!$ )is defined as cross product of position vector and velocity vector of $m_2\,\!$ :

$\mathbf{h}=\mathbf{r}\times \mathbf{v}\,\!$

where:

$\mathbf{r}\,\!$ is orbital position vector of orbiting body relative to central body,
$\mathbf{v}\,\!$ is orbital velocity vector of orbiting body relative to central body.

Under standard assumptions for a 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 $\sqrt{a(1-e^2)\mu}$ .