Specific relative angular momentum

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 $mathbf\{h\},!$, 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:

• $mathbf\{r\},!$ is the orbital position vector of the orbiting body relative to the central body,
• $mathbf\{v\},!$ is the orbital velocity vector of the orbiting body relative to the central body,
• $mathbf\{p\}\; ,$ is the linear momentum of the orbiting body relative to the central body,
• $m\; ,$ is the mass of the orbiting body, and
• $mathbf\{H\}\; ,$ is the relative angular momentum of the orbiting body with respect to the central body.

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 $h,!$:

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

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

$h=\; 2pi\; ab\; /(2pisqrt\{a^3/mu\})\; =\; b\; sqrt\{mu/a\}\; =\; sqrt\{a(1-e^2)mu\}$.

where

• $a$ is semi-major axis
• $b$ is semi-minor axis
• $mu$ is standard gravitational parameter

See also

• Kepler's laws of planetary motion
• Planetary orbit

References