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\,\!], is defined as the cross product of the position vector [\mathbf\,\!] and velocity vector [\mathbf\,\!] of the orbiting body relative to the central body:

[\mathbf=\mathbf\times \mathbf = \times \mathbf \over m } = \over m} ]

where:

[\mathbf\,\!] is the orbital position vector of the orbiting body relative to the central body,
[\mathbf\,\!] is the orbital velocity vector of the orbiting body relative to the central body,
[ \mathbf \, ] is the linear momentum of the orbiting body relative to the central body,
[ m \, ] is the mass of the orbiting body, and
[ \mathbf \, ] 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\,\!] vector is perpendicular to the osculating orbital plane defined by orbital position and velocity vectors.

The magnitude of [\mathbf\,\!] is denoted as [h\,\!]:

[h=\left|\mathbf\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) = b \sqrt], which is [\sqrt].

The units of [\mathbf\,\!] are km²s^-1.

Specific relative angular momentum

See also