Two isolated spherical planets, X and Y , have uniform mass densities in the ratio 4 : 9 . A satellite orbits very close to the surface of planet X with a time period T_X . Another satellite orbits very close to the surface of planet Y with a time period T_Y . What is the ratio T_X T_Y ?

Step-by-step solution

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 R^3 GM The mass of the planet can be expressed in terms of its uniform mass density as: M = 4 3 R^3 Substituting this into the expression for the time period, we get: T = 2 R^3 G ( 4 3 R^3 ) = 3 G This shows that the time period is inversely proportional to the square root of the density of the planet: T 1 Given the ratio of the densities of planets X and Y is _X _Y = 4 9 , the ratio of their time