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# Relative angular momentum

In celestial mechanics, the relative angular momentum ( \( {\displaystyle \mathbf {H_{{2}/{1}}} \,\!} ) of an orbiting body ( \( {\displaystyle m_{2}\,\!} \) ) relative to a central body ( \( {\displaystyle m_{1}\,\!}\) ) is the moment of ( \( {\displaystyle m_{2}\,\!}\) )'s relative linear momentum:

\( {\displaystyle \mathbf {H_{{2}/{1}}} =\mathbf {r} \times m_{2}\mathbf {v} \,\!} \)

where:

\( {\displaystyle \mathbf {r} \,\!}\) is the orbital position vector of the orbiting body relative to the central body,

\( {\displaystyle \mathbf {v} \,\!} \) is the orbital velocity vector of the orbiting body relative to the central body,

\( {\displaystyle m_{2}\,\!}\) is mass of the orbiting body.

For a body in an unperturbed orbit about a central body, the orbital plane is stationary, and the relative angular momentum ( \( {\displaystyle \mathbf {H_{{2}/{1}}} \,\!}\) ) is perpendicular to the orbital plane.

For perturbed orbits where the orbital plane is in motion, the relative angular momentum vector is perpendicular to the (osculating) orbital plane at only two points in the orbit.

Uses

In astrodynamics relative angular momentum is usually used to derive specific relative angular momentum ( h {\displaystyle \mathbf {h} \,\!} \) ):

\( {\displaystyle \mathbf {h} =\mathbf {H_{{2}/{1}}} /m_{2}\,\!} \)

where:

\( {\displaystyle m_{2}\,\!}\) is mass of the orbiting body.

See also

Angular momentum

Specific relative angular momentum

Linear momentum

Momentum

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