Orbits of the Galilean Moons

In the previous post I described my measurements of Jupiter's moons - four of which were discovered by Galileo. One fascinating thing about the inner three moons is that they are tidally locked into a [4:2:1] orbital resonance. That means that Io orbits twice as fast as Europa, which orbits twice as fast as Ganymede. I though it would be neat to take some measurements of these moons and try to verify that resonance. Observing an orbiting object side-on, it appears to oscillate back and forth in a sinusoidal pattern; that is, as long is the orbit isn't too elliptical. Fortunately these moons have a small eccentricity. I was able to get five measurements over a week, and then one more, three weeks later. That's not many. Ganymede, the outer-most of the three tidally locked moons orbits Jupiter once per week, so I figured I had enough samples to get it's orbital period. The other two moons, having periods of half and one quarter of Ganymede were too under-sampled to make the calculation, but I figured if I plotted their predicted orbits based on the Ganymede data I could claim success if the few sample points I had lined up on the predicted curve. 

 To fit a sine wave to my data points I used the curve_fit function in the SciPi Python library. You can find the source code to the Python script I wrote here. Plugging the data in, I got this:

The dots are the measured distances and the line is the best-fit curve. The calculated orbital period was 7.22 days. According to Wikipedia, the actual orbital period is 7.15 days. Looks like I'm off by about 1%. Not bad at all. 

Now to overlay the predicted and measured orbits of Europa and Io. When I first did this, the measured points were nowhere near the predicted curve. It turns out that each moon is 180 out of phase with its neighbor. I hadn't seen that mentioned in any of the literature. I did, however, notice it in an animated simulation of the orbits of the three moons. When I added the phase shift, the dots stayed pretty close to the line, although not perfectly. I'm sure there was plenty of measurement error in my technique!

Now that Jupiter is starting to be obscured by the setting sun, I don't think I'll be able to take more measurements until next year when Jupiter is once more visible.