Use the equations in Exercise 102 to plot the paths of the following moons in our solar system.

a. Each year our moon revolves around Earth about n = 13.4 times, and the distance from the Sun to Earth is approximately a = 389.2 times the distance from Earth to our moon.

b. Plot a graph of the path of Callisto (one of Jupiter’s moons) that corresponds to values of a = 727.5 and n = 259.6. Plot a small portion of the graph to see the detailed behavior of the orbit.

c. Plot a graph of the path of Io (another of Jupiter’s moons) that corresponds to values of a = 1846.2 and n = 2448.8. Plot a small portion of the path of Io to see the loops in its orbit. 

Data from Exercise 102

An idealized model of the path of a moon (relative to the Sun) moving with constant speed in a circular orbit around a planet, where the planet in turn revolves around the Sun, is given by the parametric equations

x(θ) = a cos θ + cos nθ, y(θ) = a sin θ + sin nθ. 

The distance from the moon to the planet is taken to be 1, the distance from the planet to the Sun is a, and n is the number of times the moon orbits the planet for every 1 revolution of the planet around the Sun. Plot the graph of the path of a moon for the given constants; then conjecture which values of n produce loops for a fixed value of a.