SOURCE FOR CONTENT: An Introduction to Modern Astrophysics, Carroll & Ostlie, Cambridge University Press. Ch.2 Celestial Mechanics
Here is my solution to one of the problems in the aforementioned text. I derive the total energy of a binary system making use of center-of-mass coordinates. In order to conceptualize it I have used the binary Alpha Centauri A and Alpha Centauri B. While writing this I stumbled upon the Kepler problem, the two-body problem, and the N-body problem. Leave a comment if you would like me to consider that in another post.
Derivation of the Total Energy of a Binary Orbit:
Setup: Consider the nearest binary star system to our solar system: Alpha Centauri A and Alpha Centauri B. These two stars orbit each other about a common center of mass; a point called a barycenter. The orbital radius vector of Alpha Centauri A is and the orbital radius vector of Alpha Centauri B is . The masses of Alpha Centauri A and B are , and , respectively. The total mass of the binary orbit is the sum of the individual masses of each component. In the context of this system, we encounter what is called the two-body problem of which there exists a special case known as the Kepler Problem (by the way let me know if that would be something that you guys would want to see…). We can simplify this two-body problem by making use of center-of-mass coordinates wherein we define the reduced mass . Therefore, the derivation of the total energy of the binary system of Alpha Centauri A and B will be carried out in such a coordinate system.
To derive this energy equation, one would typically make use of center-of-mass coordinates in which
where represents the reduced mass given by
Recall from conservation of energy that
where represents the separation distance between the two components. Let us take the derivative of Eqs.(0.1) and (0.2) to get
Upon making use of the definition of the reduced mass (Eq. (0.3)) we arrive at
If we solve for in Eq.(0.3) we get the total energy of the binary Alpha Centauri A and B. This is true for any binary system assuming center-of-mass coordinates.