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.

Definition

Specific relative angular momentum, represented by the symbol $\backslash mathbf\{h\}\backslash ,\backslash !$, is defined as the cross product of the position vector $\backslash mathbf\{r\}\backslash ,\backslash !$ and velocity vector $\backslash mathbf\{v\}\backslash ,\backslash !$ of the orbiting body relative to the central body:

where:

• $\backslash mathbf\{r\}\backslash ,\backslash !$ is the orbital position vector of the orbiting body relative to the central body,

• $\backslash mathbf\{v\}\backslash ,\backslash !$ is the orbital velocity vector of the orbiting body relative to the central body,

• $\backslash mathbf\{p\}\backslash ,$ is the linear momentum of the orbiting body relative to the central body,

• $m\; \backslash ,$ is the mass of the orbiting body, and

• $\backslash mathbf\{H\}\; \backslash ,$ is the relative angular momentum of the orbiting body with respect to the central body.

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

Under standard assumptions for an orbiting body in a trajectory around central body at any given time the $\backslash mathbf\{h\}\backslash ,\backslash !$ 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 $\backslash mathbf\{h\}\backslash ,\backslash !$ is denoted by $h\backslash ,\backslash !$:

Elliptical orbit

In an elliptical orbit, the specific relative angular momentum is twice the area per unit time swept out by a chord from the central mass to the orbiting body: this area is 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, $h\backslash ,\backslash !$ is equal to twice the area of the ellipse divided by the orbital period, giving the equation

where

• $a$ is semi-major axis

• $b$ is semi-minor axis

• $\backslash mu$ is standard gravitational parameter

See also

• Kepler's laws of planetary motion

• Planetary orbit