Specific angular momentum

In celestial mechanics, the specific relative angular momentum (often denoted

[1] In the case of two orbiting bodies it is the vector product of their relative position and relative linear momentum, divided by the mass of the body in question.

Specific relative angular momentum plays a pivotal role in the analysis of the two-body problem, as it remains constant for a given orbit under ideal conditions.

"Specific" in this context indicates angular momentum per unit mass.

The SI unit for specific relative angular momentum is square meter per second.

The specific relative angular momentum is defined as the cross product of the relative position vector

is the angular momentum vector, defined as

vector is always perpendicular to the instantaneous osculating orbital plane, which coincides with the instantaneous perturbed orbit.

It is not necessarily perpendicular to the average orbital plane over time.

Under certain conditions, it can be proven that the specific angular momentum is constant.

The conditions for this proof include: The proof starts with the two body equation of motion, derived from Newton's law of universal gravitation:

where: The cross product of the position vector with the equation of motion is:

Since the time derivative is equal to zero, the quantity

in place of the rate of change of position, and

, because it does not include the mass of the object in question.

Kepler's laws of planetary motion can be proved almost directly with the above relationships.

The proof starts again with the equation of the two-body problem.

This time the cross product is multiplied with the specific relative angular momentum

The left hand side is equal to the derivative

After some steps (which includes using the vector triple product and defining the scalar

to be the radial velocity, as opposed to the norm of the vector

Now this equation is multiplied (dot product) with

which is the equation of a conic section in polar coordinates with semi-latus rectum

The second law follows instantly from the second of the three equations to calculate the absolute value of the specific relative angular momentum.

for the area of a sector with an infinitesimal small angle

(triangle with one very small side), the equation

Kepler's third is a direct consequence of the second law.

Integrating over one revolution gives the orbital period[1]

and the specific relative angular momentum with

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.