Say we have a two star system with both stars of equal mass $M$. The center of mass of this system is by definition in the center of the two stars.
There is a small asteroid with mass m in orbit around the center of mass of the two stars. (What is known as a halo orbit apparently) The orbital distance is x.
The question is why the gravitational force on the asteroid cannot be calculated using the center of mass. ie $F=Gm(2M)/x^2$?
I can only get the force on the asteroid if I resolve the centripetal component of force on the asteroid due to each star individually and sum them.