It's called specific angular momentum because it's not the actual angular momentum, but the angular momentum per mass. Thus, the word "specific" in this term is short for "mass-specific" or divided-by-mass:
The vector is always perpendicular to the instantaneous osculatingorbital plane, which coincides with the instantaneous perturbed orbit. It would not necessarily be perpendicular to an average plane which accounted for many years of perturbations.
As usual in physics, the magnitude of the vector quantity is denoted by :
Proof that the specific relative angular momentum is constant under ideal conditions
One looks at two point masses and , at the distance from one another and with the gravitational force acting between them. This force acts instantly, over any distance and is the only force present. The coordinate system is inertial.
The further simplification is assumed in the following. Thus is the central body in the origin of the coordinate system and is the satellite orbiting around it. Now the reduced mass is also equal to and the equation of the two-body problem is
It is important not to confound the gravitational parameter with the reduced mass, which is sometimes also denoted by the same letter .
Distance vector , velocity vector , true anomaly and flight path angle of in orbit around . The most important measures of the ellipse are also depicted (among which, note that the true anomaly is labeled as ).
One obtains the specific relative angular momentum by multiplying (cross product) the equation of the two-body problem with the distance vector
The cross product of a vector with itself (right hand side) is 0. The left hand side simplifies to
This vector is perpendicular to the orbit plane, the orbit remains in this plane because the angular momentum is constant.
One can obtain further insight into the two-body problem with the definitions of the flight path angle and the transversal and radial component of the velocity vector (see illustration on the right). The next three formulas are all equivalent possibilities to calculate the absolute value of the specific relative angular momentum vector
Kepler's third is a direct consequence of the second law. Integrating over one revolution gives the orbital period
for the area of an ellipse. Replacing the semi-minor axis with and the specific relative angular momentum with one gets [References 4]
There is thus a relationship between the semi-major axis and the orbital period of a satellite that can be reduced to a constant of the central body. This is the same as the famous formulation of the law:
The square of the period of a planet is proportional to the cube of its mean distance to the Sun.