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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

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  • $\begingroup$ It occurs because Newton's Law of Gravitation is non-linear in $\vec{r}$.(Notice the $r^2$ in the denominator). $\endgroup$ – Sandesh Kalantre Dec 29 '13 at 13:53
  • $\begingroup$ A force is a vector, so to calculate the resulting gravitational force on the asteroid you have to sum the two force vectors exerted by the two stars. $\endgroup$ – fibonatic Dec 29 '13 at 14:17
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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.

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