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