This spring, I've sketched Jupiter and its moons most nights I've been observing. I thought it would be an interesting exercise to figure out what the orbital period of one of the moons based on my logs.
I picked Ganymede since it usually seemed to be farther out than the other moons, so would have a longer period. Other moons would be possible, but it would be better to observe on consecutive nights to estimate shorter periods more easily. I don't think I had two nights in a row of clear skies.
I made the following observations. All were made in Bellevue, WA, with an Omni XLT 150 reflector.
I cheated a little and used Stellarium to figure out which moon was which. From here, I have the following data:
Days since 4/22 | Location of G | "Side"
0 | G west of J | 1
32 | G east of J | -1
35 | G near J | 0
38 | G west of J | 1
40 | G east of J | -1
45 | G west of J | 1
50 | G west of J | 1
I converted east/west location to just a -1, 0, 1 scale ("Side") since I didn't have much confidence in how consistent the sizes of things were in my sketches. (Since planets are so small, I find it's easy to draw them bigger than they actually appear.)
On 5/27 (day 35), since Ganymede was quite close to Jupiter, I approximated this as halfway through its orbit, so its motion could be described by the equation
x(t) = sin(2 pi (t - 35) / P)
where P is the period in days.
From here, I experimented graphing the function above for various values of P. I found that 7.0 < P < 7.4; values outside this range meant that the predicted location of Ganymede was inconsistent with my observations (that is, it would have been on the wrong side). Taking the average, I estimated P = 7.2 days (plus or minus 0.2 days). Here's the estimated location using P = 7.2
Checking on Wikipedia, the actual value is 7.155 days, surprisingly close!