Comparison of Escape Velocities for Spherical Planets of Equal Density

To determine the correct options, we analyze the relationship between the surface area, mass, radius, density, and escape velocity for each spherical planet.

1. Determination of Radii

The surface area of a spherical planet of radius $R$ is given by: $A = 4\pi R^2$

For planets $P$ and $Q$:

• Surface area of planet $P$: $A_P = A \implies 4\pi R_P^2 = A \implies R_P = \sqrt{\frac{A}{4\pi}}$
• Surface area of planet $Q$: $A_Q = 4A \implies 4\pi R_Q^2 = 4A \implies R_Q = 2 R_P$

Thus, the relation between the radii of planets $P$ and $Q$ is: $R_Q = 2 R_P$

---

2. Determination of Masses

The mass $M$ of a uniform sphere of radius $R$ and density $\rho$ is given by: $M = \frac{4}{3}\pi R^3 \rho$

Since all planets have the same uniform density $\rho$, the mass of each planet is proportional to the cube of its radius ($M \propto R^3$):

• Mass of planet $P$: $M_P = \frac{4}{3}\pi R_P^3 \rho$
• Mass of planet $Q$: $M_Q = \frac{4}{3}\pi R_Q^3 \rho = \frac{4}{3}\pi (2 R_P)^3 \rho = 8 M_P$

For planet $R$, its mass is given as: $M_R = M_P + M_Q = M_P + 8 M_P = 9 M_P$

Since $M_R = \frac{4}{3}\pi R_R^3 \rho = 9 M_P$, we find the radius of planet $R$: $\frac{4}{3}\pi R_R^3 \rho = 9 \left( \frac{4}{3}\pi R_P^3 \rho \right) \implies R_R^3 = 9 R_P^3 \implies R_R = 9^{1/3} R_P$

---

3. Escape Velocity Derivation

The escape velocity $V$ from the surface of a spherical planet of mass $M$, radius $R$, and density $\rho$ is: $V = \sqrt{\frac{2 GM}{R}} = \sqrt{\frac{2 G \left(\frac{4}{3}\pi R^3 \rho\right)}{R}} = \sqrt{\frac{8\pi G \rho}{3}} \cdot R$

Since $\rho$ and $G$ are constant for all planets, the escape velocity is directly proportional to the radius of the planet ($V \propto R$): $V_P = \sqrt{\frac{8\pi G \rho}{3}} R_P$ $V_Q = \sqrt{\frac{8\pi G \rho}{3}} R_Q = 2 V_P$ $V_R = \sqrt{\frac{8\pi G \rho}{3}} R_R = 9^{1/3} V_P$

---

4. Comparison of Escape Velocities

• Ratio of $V_P$ to $V_Q$: $\frac{V_P}{V_Q} = \frac{R_P}{R_Q} = \frac{R_P}{2 R_P} = \frac{1}{2}$ Therefore, Option (D) is correct.

• Ratio of $V_R$ to $V_P$: $\frac{V_R}{V_P} = \frac{R_R}{R_P} = 9^{1/3} \approx 2.08 \neq 3$ Therefore, Option (C) is incorrect.

• Ordering of Escape Velocities: Since $1 < 2 < 9^{1/3}$ (because $2^3 = 8 < 9$), we have: $R_P < R_Q < R_R \implies V_P < V_Q < V_R \implies V_R > V_Q > V_P$ Therefore, Option (B) is correct.

---

Conclusion

The correct options are B and D.