The dance of the planets

Consider three planets of masses $$m_{1}= m_{*}, m_{2}=m_{*}/2$$ and $$m_{3}=m_{*}/3$$ located at the vertices of an equilateral triangle with sides of length $$d$$. It turns out that the planets, under the action of the mutual forces of gravitation, can move in such a way that they always form an equilateral triangle of variable side $$d(t)$$. Assume that maximum and minimum distances between the planets are $$d_{max}=2\times 10^{6}~\text{km}$$ and $$d_{min}=5\times 10^{5}~\text{km}$$ and that after $$T=3\times 10^{5}~\text{s}$$ the planets return to their original configuration shown in the figure. That is, $$T$$ is the period of the orbital motion of the planets. Using the information provided, determine the maximum speed in kilometers per second of planet $$m_{1}$$.

Details and assumptions

The universal gravitational constant is $G=6.67\times 10^{-11}~\text{m}^{3}\text{kg}^{-1} \text{s}^{-2}$

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