Moon Io orbits Jupiter 4 times as fast as Ganymede. Europa orbits 2 times as fast as Ganymede. This is a kind of "orbital resonance." (Moons not to scale but orbital ratios are exact) Lines connect moon pairs at regular time intervals. Set moon pairs to 1 for just Ganymede to Io. Set to 2 for Ganymede to Europa also. Spacebar erases lines. We make pairs of musical notes play when pairs of moons align (when they have the same angle). The notes are octaves apart. The highest note is for Io and has twice the frequency of the note for Europa and so sounds one octave higher. Europa's note has twice the frequency of Ganymede. This is to bring out the analogy between the doubling of orbital frequencies and musical frequencies (once called the music of the spheres). You can adjust "speed" but the ratios of the angular frequencies of the moons will remain 1:2:4. Moons sizes are not to scale. Distance to Jupiter is not to scale. "p" key for polar view of Jupiter, "s" for side view.
Io goes around Jupiter twice for every time Europa does. Europa circles Jupiter twice for every time Ganymede does. The ratio of their frequencies is 1 to 2 to 4. We are focusing on orbital frequency and their ratios. The simplest way to do that is to focus on angular frequency (or "angular velocity"). We made each moon a separate spinning sprite with the moon a distance (radius, not to scale) from the center of the sprite. Then the code focuses on the speed of the angle change, the frequency with which a 360 degree spin is completed. We set Io to orbit 4 times as fast as Ganymede, and set Europa to turn 2 times as fast as Ganymede. Those moons really have exactly those ratios. For every one orbit Ganymede makes, Europa makes two and Io completes four. Io’s volcanoes are a direct result of this relationship because the gravity tugs of Europa and Ganymede, combined with the tidal force of Jupiter, produce a continuous source of heat for Io’s interior. We give each moon a note. When pairs of moons align we hear pairs of notes. You can hear a musical rhythm (easier to hear at higher speeds). The angular frequency for the moons is not affected by the radius or circumference of the orbits. So the frequencies are not affected by the fact that the moons and their distances from Jupiter are not to scale. Ganymede's angular frequency (in arbitrary units) is set to "speed." Europa's angular velocity is "speed" X 2. Io's angular velocity is "speed" X 4.
