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.

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[edit] 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}\times \mathbf{v} = { \mathbf{r} \times \mathbf{p} \over m } = { \mathbf{H} \over m}

where:

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

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.

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

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

[edit] 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= 2\pi ab /(2\pi\sqrt{a^3/\mu}) = b \sqrt{\mu/a} = \sqrt{a(1-e^2)\mu}.

where

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