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} imes mathbf{v} = { mathbf{r} imes 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} ight|,!

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