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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