Jupiter has been on my mind recently. First of all, I've been watching all the action happening on Ganymede in The Expanse on Amazon Prime 😁. And then, Saturn came into its closest conjunction with Jupiter in 800 years. Over the past few weeks I've been listening to Prof. Richard Pogge's Astronomy 161 lectures and I was particularly fascinated by his lecture on Jupiter in which he explained how the Galilean moons of Jupiter are tidally locked in orbital ratios of 4:2:1.
Here's what I set out to do:
1. Measure the closest conjunction of Saturn.
How close in the sky will Saturn be to Jupiter?
NASA published 0.1 degrees or 6 arc-minutes or 360 arc-minutes. Not many significant figures here.
Astronomy Now published 6 arc-minutes or 360 arc-seconds. Not many significant figures here, either.
When the Curves Line Up published 377 arc-seconds. More significant figures doesn't equal more accuracy, but I'll go with this number.
Several sources published 6 arc-seconds, but that seems like a misprint.
On the day of conjunction, December 21st, 2020, I was initially plagued with fog and then high clouds. They did eventually clear enough for me to photograph the two planets.
For the measurement, I used a Canon SL1 camera with a 300 mm f5.6 lens. Exposure was 1/60th of a second and ISO setting was 3200. I tried manual focus, but it was just too fussy. Fortunately Jupiter is bright enough that auto-focus worked. To minimize vibration, I put the camera on a tripod, and put the camera in "live" mode to lock up the mirror. I then used a remote to trigger the camera.
I took photos with these and other settings in raw mode. I then selected the best candidate for measurement and opened it in UFRaw and exported it into .ppm format. I could then use GIMP to mark the coordinates in pixels on the image by moving the cursor over each planet, and reading the coordinates in the lower left. To calculate the distance in pixels I used the following formula:
Now we have the distance in pixels. To convert the distance to an angle, we need to get the resolution of of the camera in terms of an angle, rather than pixels. The trick is to take the arc-tangent of the distance between pixels over the focal length, then do some unit conversions to get arc-seconds.
With a distance of 126.6 pixels between Jupiter and Saturn, and a camera resolution of 2.9 arc seconds, we get a distance of 366 arc-seconds, which is within about 3% of the published value. Not bad.
2. Show Orbital Ratios of the Moons of Jupiter.
About a month before the conjunction I started taking practice photographs of Jupiter. I noticed that if I made the exposure long enough, I could see the Galilean moons of Jupiter. Sometimes I also saw interesting things like spent boosters. On the top is Jupiter with Europa to the left. Ganymede and Callisto are to the right. At the bottom is a spent Zenit-2 rocket body which just happened to pass by as I snapped the photo.
I started taking photos every other day, starting on November 23rd, but my efforts were hampered by bad weather, so I wasn't able to get as many shots as I would have liked. However, I was able to make six measurements.
This table uses the methods shown in section one to show the distance from Jupiter to its Galilean moons as a function of the number of days since my first observation.
When plotted, they should look like sine waves, but I really don't have enough data points. Since Io orbits about as often as I was taking measurements, it almost looks like it wasn't moving. This is called aliasing. It's what makes wagon wheels in movies look like they're moving backwards. I think there may be have enough data points to curve fit Ganymede, though. Then I can see if sine waves of 2x and 4x Ganymede's frequency overlay Europa and Io, thus confirming tidal lock. Callisto is not in tidal lock, but it does look that I got half an orbit.
I was going to do the sinusoidal curve fit in a spreadsheet, but I decided it would be more fun and more useful to do it in Python. That will be the subject of the next post.
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