Question

The star Pax has four planets orbiting it. The planet Morana lies 17 AU away from Pax and takes 32 Earth years to orbit Pax. The planet Noveria lies 3.5 AU away from Pax. How long does it take Noveria to orbit Pax?
You do not need to convert AU to meters or years to seconds to solve this problem.
(a) 0.72 years
(b) 1.35 years
(c) 1.74 years
(d) 2.99 years

Answer 1

The third Kepler’s law of planetary motion states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis (mean distance) of its orbit:

$$\dfrac{P_M^2}{a_M^3}=\dfrac{P_N^2}{a_N^3},$$

here, $P_M=32\ years$ is the orbital period of Morana, $P_N$ is the orbital period of Noveria, $a_M=17\ AU$ is the mean distance of Morana from Pax and $a_N=3.5\ AU$ is the mean distance of Noveria from Pax.

Then, from this equation we can find the length of the Noveria year (or its orbital period):

$$P_N=\sqrt{\dfrac{a_N^3P_M^2}{a_M^3}},$$ $$P_N=\sqrt{\dfrac{(3.5\ AU)^3\cdot (32\ years)^2}{(17\ AU)^3}}=2.99\ years.$$

Answer:

(d) $P_N=2.99\ years.$