# Total angular momentum in system of binary brown dwarfs on wide orbit from primary star

I want to calculate the total angular momentum of a system in which two brown dwarfs are orbiting one another (P ~ 10 years), and this system is stably orbiting a third primary ZAMS star (P ~ $10^4$ years). I was thinking of approaching this question from a Hamiltonian mechanics perspective, but the problem doesn't become easier in the rotating reference frame.

Let A denote the primary star, and B,C denote the two components of the binary. I was thinking the angular momentum would simply be the sum of rotational (S) and orbital angular (L) momentums: $$L_{tot} = S_{A} + S_{B} + S_{C} + L_{BC} + L_{A+BC}$$

However, I am both worried that I am partially "double counting" with the last term and I don't really know how to treat the last term. Do I treat BC like a single mass at their barycenter?

If there is a Hamiltonian method to approach this problem, I am all ears.

The question I am actually concerned with is what would the rotation rate of the primary star be if the total angular momentum of the system was in the primary star? But I would really just like an accurate description of the angular momentum of the system to do this.

• I think you might want to start counting the angular momentum in the center-of-mass system. We know the CMS system to be non-accelerated thanks to one of the Noether's theorems, so it is also non-accelerated for a 3-body problem and thus a viable starting point for your problem. – AtmosphericPrisonEscape Aug 9 at 11:07