A planet \((P_1)\) is moving around the star of mass \(2M\) in the orbit of radius \(R\). Another planet \((P_2)\) is moving around another star of mass \(4M\) in a orbit of radius \(2R\). Ratio of time periods of revolution of \(P_2\) and \(P_1\) is _______.

Using Kepler's third law, the time period of revolution of a planet is given by \(T = 2\pi \sqrt{\dfrac{r^3}{GM_s}}\).

For planet \(P_1\), the orbital radius is \(r = R\) and the mass of the star is \(M_s = 2M\):
\(T_1 = 2\pi \sqrt{\dfrac{R^3}{G(2M)}}\)

For planet \(P_2\), the orbital radius is \(r = 2R\) and the mass of the star is \(M_s = 4M\):
\(T_2 = 2\pi \sqrt{\dfrac{(2R)^3}{G(4M)}} = 2\pi \sqrt{\dfrac{8R^3}{4GM}} = 2\pi \sqrt{\dfrac{2R^3}{GM}}\)

Taking the ratio of \(T_2\) and \(T_1\):
\(\dfrac{T_2}{T_1} = \dfrac{\sqrt{\dfrac{2R^3}{GM}}}{\sqrt{\dfrac{R^3}{2GM}}} = \sqrt{2 \times 2} = 2\)

Answer: \(2\)