Question #192037

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.


1
Expert's answer
2021-05-14T09:43:10-0400

Consider moons of Jupiter.


1. Metis:     a=128852 km,     T=0.2988 d,2. Io:           a=421700 km,     T=1.7691 d,3. Europa:  a=671034 km,     T=3.5512 d,4. Himalia: a=11497400 km, T=251.86 d,5. Callisto: a=1882709 km,   T=16.689 d.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=T2R3.r=\frac{T^2}{R^3}.

Calculate the Kepler ratio for each moon:


1. r=4.171017 d2/km3.2. r=4.171017 d2/km3.3. r=4.171017 d2/km3.4. r=4.171017 d2/km3.5. r=4.171017 d2/km3.1.\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.


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