Two hypothetical planets A and B have the same average mass density, but the radius of planet A is twice that of planet B. Two satellites revolve very close to the surfaces of their respective planets. The ratio of the time periods of the satellites of planet A to planet B is

Step-by-step solution

For a satellite orbiting very close to the surface of a planet, the orbital radius r is approximately equal to the radius of the planet R ( r R ). The time period of a satellite is given by: T = 2 r^3 GM Substituting r = R and the mass of the planet M = Volume Density = 4 3 R^3 : T = 2 R^3 G ( 4 3 R^3 ) T = 3 G This expression shows that the time period of a satellite orbiting close to the surface of a planet depends only on the density of the planet and is independent of its radius R . Since planets A and B have t