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Concept
Specific angular momentum
Summary
In celestial mechanics, the specific relative angular momentum (often denoted or ) of a body is the angular momentum of that body divided by its mass. 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 and the relative velocity vector .
where is the angular momentum vector, defined as .
The vector is always perpendicular to the instantaneous osculating orbital plane, which coincides with the instantaneous perturbed orbit. It is not necessarily be 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 mass of one object is much greater than the mass of the other one. ()
The coordinate system is inertial.
Each object can be treated as a spherically symmetrical point mass.
No other forces act on the system other than the gravitational force that connects the two bodies.
The proof starts with the two body equation of motion, derived from Newton's law of universal gravitation:
where:
is the position vector from to with scalar magnitude .
is the second time derivative of . (the acceleration)
is the Gravitational constant.
The cross product of the position vector with the equation of motion is:
Because the second term vanishes:
It can also be derived that:
Combining these two equations gives:
Since the time derivative is equal to zero, the quantity is constant. Using the velocity vector in place of the rate of change of position, and for the specific angular momentum:
is constant.
Official source
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