What is the ratio T1 : T2 of the orbital periods of the two planets in the following case? Consider two planets of mass m and 2m, respectively, orbiting the same star in circular orbits. The more massive planet is 4.5 times as far from the star as the less massive one. What is the ratio T1:T2 of the orbital periods of the two planets?

Expert Answers

An illustration of the letter 'A' in a speech bubbles

The two planets P1 and P2 have a mass of m and 2m respectively. P2 is 4.5 times as far from the star as P1. The orbital period T of a planet that is moving in a circular orbit around a star is given by T = 2*pi*sqrt(a^3/u) where a is the radius of the path followed by the planet and u is the standard gravitational parameter that is the product of the gravitational constant G and the mass of the planet.

If the radius of the orbit of P1 is r, the radius of the orbit of P2 is 4.5*r. This gives T1 = 2*pi*sqrt(r^3/G*m), T2 = 2*pi*sqrt((4.5*r)^3/G*2m)

The ratio T1:T2 = [2*pi*sqrt(r^3/G*m)]/[2*pi*sqrt((4.5*r)^3/G*2m)]

=> T1:T2 = (2*pi/2*pi)*[sqrt(r^3/G*m)/sqrt((4.5*r)^3/G*2m)]

=> T1:T2 = sqrt[(r^3/G*m)/((4.5*r)^3/G*2m)]

=> T1:T2 = sqrt[(1/((4.5)^3/2)]

=> T1:T2 = sqrt[(2/(4.5)^3]

=> T1:T2 = sqrt(2/91.125)

=> T1:T2 = 0.148

The ratio T1: T2 of the orbital periods of the two planets is approximately 0.148

Approved by eNotes Editorial Team