# How to identify binary stars in $N$-body simulation?

Binary stars constitute a significant portion of the stars of a globular cluster.

I would like to verify that this is true in my $N$-body simulation, but I don't know how to decide whether a star in the system is a binary.

Visually this is easy to do, as binaries are identified as two stars at very close distance orbiting about their center of mass, but I need a mathematical condition which I can then translate to code.

• @JoshuaBenabou They are bound even in your example. I guess you could start with a fraction of the number $\Sigma^{-1/2}$, where $\Sigma$ is the mass surface density and see how that works out. Measure the number, increase the threshold, measure the number again, increase the threshold again until you reach convergence Oct 12, 2017 at 20:52
• @JoshuaBenabou You're right about the first part. But you can search for neighbors in $O(\log n)$, so the execution time will be reduced by a lot if you take care of distant particles first (which are likely not bound). But again, if your $n$ is small, probably you can just do a $O(n^2)$ search and that's it Oct 12, 2017 at 21:50