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.

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