Two isolated spherical planets, X and Y , have uniform mass...

The time period of a satellite orbiting very close to the surface of a planet of mass $M$ and radius $R$ is given by:

$T = 2\pi \sqrt{\dfrac{R^3}{GM}}$

The mass of the planet can be expressed in terms of its uniform mass density $\rho$ as:

$M = \dfrac{4}{3}\pi R^3 \rho$

Substituting this into the expression for the time period, we get:

$T = 2\pi \sqrt{\dfrac{R^3}{G \left(\dfrac{4}{3}\pi R^3 \rho\right)}} = \sqrt{\dfrac{3\pi}{G\rho}}$

This shows that the time period is inversely proportional to the square root of the density of the planet:

$T \propto \dfrac{1}{\sqrt{\rho}}$

Given the ratio of the densities of planets $X$ and $Y$ is $\dfrac{\rho_X}{\rho_Y} = \dfrac{4}{9}$, the ratio of their time periods is:

$\dfrac{T_X}{T_Y} = \sqrt{\dfrac{\rho_Y}{\rho_X}} = \sqrt{\dfrac{9}{4}} = \dfrac{3}{2}$

Answer: $\dfrac{3}{2}$