Question

The mean distance of Mars from the Earth is 0.5 AU and its orbital period is 687 days. Calculate the orbital period of Jupiter given that its mean distance from the Earth is 4 AU.

Answer 1

Apply Kepler's third law:

$$\frac{a^3_M}{a^3_J}=\frac{T^2_M}{T^2_J},\\ \space\\ T_J=T_M\sqrt{\bigg(\frac{a_J}{a_M}\bigg)^3}.$$

That simple, just substitute the values and get the answer? Uh no, the distances are from the Earth. Meanwhile the distance between the Earth and the Sun is 1 AU. So add 1 AU to each distance:

$$T_J=687\sqrt{\bigg(\frac{4+1}{0.5+1}\bigg)^3}=4180.9^\text{d}.$$

Also, this is 11.45 years. NASA says it's 11.86 years, just 3.5% error! Very good.