Ratio of Time Periods of Planetary Revolutions Around Different Stars

To find the ratio of the time periods of revolution of planet $P_2$ and planet $P_1$, we use the expression for the orbital time period of a planet revolving around a central star.

The time period $T$ of a planet in a circular orbit of radius $r$ around a star of mass $M_s$ is given by: $T = 2\pi \sqrt{\frac{r^3}{G M_s}}$

where $G$ is the universal gravitational constant.

From this relation, the time period $T$ is proportional to: $T \propto \sqrt{\frac{r^3}{M_s}}$

For planet $P_1$:

• Mass of the star, $M_1 = 2M$
• Radius of the orbit, $r_1 = R$
• Time period $T_1 = 2\pi \sqrt{\frac{R^3}{G(2M)}}$

For planet $P_2$:

• Mass of the star, $M_2 = 4M$
• Radius of the orbit, $r_2 = 2R$
• Time period $T_2 = 2\pi \sqrt{\frac{(2R)^3}{G(4M)}}$

Taking the ratio of $T_2$ to $T_1$: $\frac{T_2}{T_1} = \sqrt{\left(\frac{r_2}{r_1}\right)^3 \times \left(\frac{M_1}{M_2}\right)}$

Substitute the given values into the equation: $\frac{T_2}{T_1} = \sqrt{\left(\frac{2R}{R}\right)^3 \times \left(\frac{2M}{4M}\right)}$

$\frac{T_2}{T_1} = \sqrt{2^3 \times \frac{1}{2}} = \sqrt{8 \times \frac{1}{2}} = \sqrt{4} = 2$

Thus, the ratio of the time periods of revolution of $P_2$ and $P_1$ is $2$.