# Neptune dance

Three planets interact with one another through their mutual gravitational forces only; they are isolated in free space and do not interact with any other bodies. They orbit around an axis going through the center-of-mass of the system and perpendicular to the triangle formed by the planets. What should be the angular velocity $$\omega$$ in rad/s so that the shape and size of the triangle remains unchanged during the rotation?

Note: Round this to the nearest nano-radians/sec, so if your answer was $$2.2 \times 10^{-9}\mbox{rad/s}$$ you would round to $$2\times 10^{-9}\mbox{rad/s}$$.

Details and assumptions

• The area of the triangle is $$1~\mbox{A.U.}^2$$.
• The total mass of the planets is $$3 \times 10^{26}~\mbox{kg}$$.
• $$G=6.67\times 10^{-11}~\mbox{m}^3/\mbox{kg s}^2$$.
• This problem was submitted by Italo X.
×