Reference: Carroll, Bradley W. & Ostlie, Dale A. (2007), An Introduction to Modern Astrophysics, 2nd Edition; Pearson Education – Chapter 2, Problems 2.4-2.5.
For two masses and interacting via gravity, it’s easiest to do calculations in the centre of mass frame. The position of the centre of mass is defined as
where is the position of in a coordinate system where the origin could be anywhere.
In the centre of mass frame, we take the centre of mass to be at the origin, so that , giving
where now (without the prime) indicates the position relative to the centre of mass. If we introduce the relative position
In terms of the reduced mass
The velocities of the two masses are just the derivatives of the positions, so that
and the rate of change of the relative position is
The total energy of the two mass system is the sum of the two kinetic energy terms and the gravitational potential energy term, so
The gravitational potential energy can be written in terms of the reduced mass and the total mass as
Thus the total energy is
which is the energy of a mass moving around another mass , the latter of which is fixed at the origin.
The orbital angular momentum (that is, the angular momentum due to the masses orbiting about each other, not including any angular momentum due to each mass rotating on its own axis) is
Thus the angular momentum is due to the reduced mass alone, which is consistent with this mass orbiting about a fixed mass at the origin.