Specific angular momentum - Definition

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

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

<math>\mathbf{h}=\mathbf{r}\times \mathbf{v}\,\!<math>

where:

• <math>\mathbf{r}\,\!<math> is orbital position vector of orbiting body relative to central body,
• <math>\mathbf{v}\,\!<math> 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 <math>\mathbf{h}\,\!<math> vector is perpendicular to the osculating orbital plane defined by orbital position and velocity vectors.

The magnitude of <math>\mathbf{h}\,\!<math> is denoted as <math>h\,\!<math>:

<math>h=\left|\mathbf{h}\right|\,\!<math>

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 <math>2\pi ab /(2\pi\sqrt{a^3/\mu}) = b \sqrt{\mu/a}<math>, which is <math>\sqrt{a(1-e^2)\mu}<math>. The units of <math>\mathbf{h}\,\!<math> are km²s^-1.