# 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

• It occurs because Newton's Law of Gravitation is non-linear in $\vec{r}$.(Notice the $r^2$ in the denominator). – Sandesh Kalantre Dec 29 '13 at 13:53
• 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. – fibonatic Dec 29 '13 at 14:17