Question

if a planet existed that had a period twice that of earth, what would be the length of the semi - major axis as it orbits the sun? (1 point) 1.59 au 2 au 4 au 2.83 au

Explanation:

Step1: Use Kepler's third law

Kepler's third law is \(T^{2}=a^{3}\) (where \(T\) is the orbital period in years and \(a\) is the semi - major axis in AU). For Earth, \(T_{E} = 1\) year and \(a_{E}=1\) AU. Let the period of the other planet be \(T_{P}=2T_{E}\).

Step2: Substitute into Kepler's third law

Substitute \(T_{P}\) into \(T^{2}=a^{3}\). We have \((2T_{E})^{2}=a_{P}^{3}\). Since \(T_{E} = 1\), then \(4=a_{P}^{3}\).

Step3: Solve for \(a_{P}\)

Take the cube - root of both sides: \(a_{P}=\sqrt[3]{4}\approx1.59\) AU.

Answer:

1.59 AU