14. Johannes Kepler (1571–1630AD) developed 3 revolutionary laws (pun intended) about planetary motion. Those laws are given below, although we don’t need the second one.

(1) The orbits of planets are ellipses with the Sun at one of the foci.

(2) The line joining the planet to the Sun sweeps out equal areas in equal times.

(3) The square of a planet’s orbital period, P (the time it takes to complete an orbit of the Sun), is proportional to the cube of the semi-major axis of its orbit, a. That is, in the solar system, a3/P2 has the same value for each planet.

Later work by Isaac Newton (1643–1727AD) expanded on these laws to show they are true not just for planets around the Sun, but also for moons around each planet, such as Jupiter.

(a) Jupiter has a very large number of moons (79 at current count, 53 of which are named). Look up the so-called Galilean moons and list them, including one feature of each that you think is interesting and also find the eccentricities of their orbits. Comment on the shape of their orbit based on their eccentricities.

(b) Verify Kepler’s 3rd law by looking up the orbital period (the time it takes to complete one orbit) and semi-major axis for each of these moons and showing that the constant of proportionality is roughly equal for all of them.

(c) Considering the orbital periods of the Galilean satellites, find some ratios of the orbital periods of these satellites. You should notice some interesting answers. Comment on what you discover.

(d) Find one named moon of Jupiter that has a much more elliptical orbit than the Galilean satellites — your moon should have an orbit with eccentricity greater than 0.2. Check that the semi-major axis and orbital period still satisfy Kepler’s third law.