Question

  Suppose a new planet is discovered in circular orbit about the Sun with a period of 512 years. Find the radius of the orbit.  

Answer 1

Use Kepler's third law. Take the Earth as the second planet in the equation: the period is 1 year and the radius is 1 AU. Then

$$\frac{T^2}{T_E^2}=\frac{a^3}{a_E^3},\\\space\\ a=a_E\sqrt[3]{\frac{T^2}{T_E^2}}=1\cdot\sqrt[3]{\frac{512^2}{1^2}}=8\text{ AU}.$$