Answer to Question #192037 in Astronomy | Astrophysics for Pranali Patil

Question

There are four planets in our solar system with a large number of moons; Jupiter, Saturn, Uranus and Neptune.  Choose one of these planets and for that planet, randomly select five moons.  For each of these moons, record (or calculate) the mean orbital radius and orbital period (in units of your choosing).  Then use this information to calculate the Kepler ratio for each of these moons orbiting its planet.  Compare the ratios and determine whether or not Kepler’s third law applies for moons around a planet other than Earth.

Answer 1

Consider moons of Jupiter.

$1.\text{ Metis: }\space\space\space\space a=128852\text{ km,}\space\space\space\space\space T=0.2988 \text{ d},\\ 2.\text{ Io: }\space\space\space\space\space\space\space\space\space\space a=421700\text{ km, }\space\space\space\space T=1.7691 \text{ d},\\ 3.\text{ Europa: }\space a=671034\text{ km, }\space\space\space\space T=3.5512\text{ d},\\ 4.\text{ Himalia: }a=11497400\text{ km, }T=251.86 \text{ d},\\ 5. \text{ Callisto: }a=1882709\text{ km, }\space\space T=16.689 \text{ d}.$

The Kepler ratio is

$$r=\frac{T^2}{R^3}.$$

Calculate the Kepler ratio for each moon:

$1.\space r=4.17·10^{-17} \text{ d}^2/\text{km}^3.\\ 2.\space r=4.17·10^{-17} \text{ d}^2/\text{km}^3.\\ 3.\space r=4.17·10^{-17} \text{ d}^2/\text{km}^3.\\ 4.\space r=4.17·10^{-17} \text{ d}^2/\text{km}^3.\\ 5.\space r=4.17·10^{-17} \text{ d}^2/\text{km}^3.\\$

Thus, Kepler’s third law applies for moons around a planet other than Earth.