Calculate the period T of a planet whose orbit has a semimajor axis of 3.6 AU? What is Y?

Question

Calculate the period T of a planet whose orbit has a semimajor axis of 3.6 AU? What is Y?

Answer 1

This problem is based on Kepler's Third Law: The squares of the orbital periods of the planets are directly proportional to the cubes of the semi-major axes of their orbits. This is often phrased into an equation, T² = (4π²/GM)a³, using metric units, the mass of the sun M, Newton's gravitational constant, etc. In this problem, when comparing a planet's orbit to Earth's, it's much easier to just rephrase the Law as a proportionality: T²/T_Earth² = a³/a_Earth³

For Earth, the orbital period is one year (I assume this is the 'Y' that you've referred to above) and the semi-major axis is one astronomical unit, or 1 AU.

Substituting the values for Earth, we see that T² = a³ - pretty straightforward solution from here!

T² = 3.6³ = 46.656 -> T = 6.83 years