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 ω\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 2.2×109rad/s2.2 \times 10^{-9}\mbox{rad/s} you would round to 2×109rad/s2\times 10^{-9}\mbox{rad/s}.

Details and assumptions

  • The area of the triangle is 1 A.U.21~\mbox{A.U.}^2.
  • The total mass of the planets is 3×1026 kg3 \times 10^{26}~\mbox{kg}.
  • G=6.67×1011 m3/kg s2G=6.67\times 10^{-11}~\mbox{m}^3/\mbox{kg s}^2.
  • This problem was submitted by Italo X.

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