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...

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?

Asked on by cchsiao787

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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

Sources:
tomson12's profile pic

tomson12 | Student | eNotes Newbie

Posted on

I did the same technique you did but for a planet that is 6.0 times farther from the star and got .096225 but when entering into my computer software for homework, it says its wrong. Rounding it to two sig figs is .096

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