Q:

Center of mass and gravitational force

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. 
https://www.dropbox.com/s/ft5ipvknczqsr0a/Photo%2029-12-13%207%2028%2013%20pm.jpg

A:

The other answers given are correct, but I want to elaborate - We can use a simplifying assumption of "gravity acts as if all mass is at the center of the object" ONLY if the object is a sphere or uniform mass density. Because really, portions of star #1 are closer than others (and thus exert more force), yet we consider all the mass of star #1 to be concentrated at its center. We can't apply this assumption to systems of multiple spheres.

