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Dance of the Moons

Elliot

I traveled about 50 feet for my field trip. But I looked over 300 million miles.

elliot-jupiter

The planet Jupiter


To the naked eye the night sky doesn't change quickly. The constellations move slowly around the sky over the span of a year. Planets drift against the stellar background over weeks or months, and can take many years to complete a trip around the Sun. Trying to actually measure the orbits of the planets in the solar system, from the inside, where we sit, would be far beyond the skills or time available to a beginner like me.

But there is one place in the solar system that puts on a show over a time span of days or even hours—Jupiter and its satellites. First discovered by Galileo in 1610, the four innermost moons of Jupiter seem to shuttle back and forth, across and behind the giant planet. At certain times a change is visible over an hour or less. It's like watching from the outside a small solar system running on a fast clock, and since it runs fast I could measure it in a relatively short period of time.

My goal was, if possible, to observe and record the positions of the moons so I could determine their relative distances from Jupiter and their patterns moving around it, using the data to construct an accurate model of the inner Jovian system.

  When to Look, How to Look
Jupiter was in a favorable position for viewing during the spring of 2004, a bright dot slowly arcing across the southern sky. Even at its closest, Jupiter is too far away to show a disk to the unaided eye, though a small pair of binoculars will show Jupiter and the inner moons very well. But just looking is one thing, quantifying is quite another. To make measurements I needed a bigger image and a way to record those images: a telescope and a camera.

The telescope was a 10" Schmidt-Cassegrain. For a neophyte it had the wonderful capability of automatically finding a known object in the sky and then tracking it through the evening with minimal adjustments. For a camera I used the family digital camera. Unlike film cameras, digital cameras have the advantage that images can be examined immediately after exposure and additional shots taken if the first ones didn't come out well. The images can also be manipulated with standard computer photo software, which made the subsequent measurements much easier.

Since we have no permanent mounting for the telescope, this project meant setting it up each evening in the driveway in front of the house (the 50 feet mentioned earlier). After turning it on and getting it tracking, I took pictures of Jupiter and its moons each available night from when it first got dark enough (about 7:30 p.m.) until as late as midnight. Observing sessions were usually ended by either the ground fog common in our area in the spring, or by falling temperatures, which caused dew on the optical surface of the telescope. Actually, most sessions never happened at all due to the famous Oregon cool-season cloud cover.

Jupiter and the Three Inner Moons, 3 April 2004 (Click to enlarge)


Taking astrophotos is  not  easy. Just centering the image in the camera field can be a hair-tearing experience. To get decent-sized images, I used an eyepiece that gave a magnification of about 250 times (and operated the camera at a 3x zoom setting); at such magnifications the telescope could be fairly described as one of the better vibration detectors commonly available. Even a moderate breeze could move the scope enough to make the image wander out of view, and by just walking too hard near the tripod I could turn it into a blur. And focusing the image using the electronic camera screen was a project in itself. (Take a picture, enlarge it, tweak the focus, take another, and so on until the image was sharp.) Even very elementary astrophotography can be painful. But it's also rewarding, as the sequence in the illustration shows.

The picture shows images of Jupiter with Io, Europa, and Ganymede, taken on April 3, 2004. You can almost feel them swirling in towards Jupiter. The fuzziness in the topmost image comes from shooting in late twilight; when I increased the image brightness, the faint sky glow became visible. Some smearing of the moons' images, due to the unstable atmosphere, is also evident.

 

Seeing Is Believing—When You Can See It All
While the moons of Jupiter may be predictable, the sky in between us isn't.

Look at the problem. First, no viewing in the daytime. Second, even if the sky is dark, Jupiter has to be well above the horizon (lights and hills). Third, clouds or thick haze mean no images. And even if the sky appears clear, there is the question of  seeing , the term for the stability of the atmosphere. The air can act like a distorting lens as one looks up through layers of different winds and temperatures. On poor nights, looking at Jupiter was like looking through a heat mirage on a summer highway, and getting good pictures was impossible. At higher magnifications the image of the planet would pulse and stretch like a water drop on a hot plate.

Favorable viewing combinations are not the norm in western Oregon in the spring. In practice I was able to collect pictures on nine nights over a two-week period in the first half of April 2004. Then the clouds rolled back in.

 

Recording the Observations—Endless Pickiness
Even with clear skies, recording the necessary data was no picnic. I had to use  exactly  the same setup for each observing session. Since I was going to be making quantitative measurements of the photographs for the distances of the moons from Jupiter, I needed to settle on a particular combination of telescope eyepiece and camera settings and use that  every  time. I couldn't vary the magnification or the size of the images would vary, making my measurements incompatible. I also had to attach the camera to the telescope so that I would have the same view each time, and have extreme steadiness, but also have the option to move the camera around a bit if necessary. This was complicated by the fact that the camera wasn't really designed for this kind of photography in the first place.

The problem was solved by gluing a camera filter adapter into some PVC plumbing pipe, which was then cut and sanded to slip snugly over a smaller piece of pipe on the telescope eyepiece holder. When the adapter was screwed onto the camera, the camera/pipe assembly could be snugged over the scope eyepiece. The camera was focused to infinity, and the image was focused in the camera using the telescope.

I collected data on the three innermost Galilean satellites: Io, Europa, and Ganymede. This focus was due to the final eyepiece/camera configuration, which gave just enough of a view to see Ganymede and Jupiter at their greatest separation in the same frame.

 

Analyzing the Pictures
One of the first things I found was that the moons are faint when compared to Jupiter itself. In the computer I had to drastically increase the brightness to even see  the moons, resulting in a total washout of the image of Jupiter itself. (Since I wasn't examining Jupiter, this wasn't a problem.) Then I rotated the images so that the plane of the orbits was horizontal. I drew two crossing lines across the disk of the planet in each image and took the crossing point as the center. Then I drew a line from the center of Jupiter to the center of the moon's image and noted the number of pixels apart they were. These distances were then matched to the time at which the picture was taken (another great thing about digital cameras is that they record date and time for each picture).

Jupiter and the Three Inner Moons (Click to enlarge)


I converted the date/time data to minutes, starting with time zero for the first frame shot on April 1. The illustration shows a sample of these data. On this evening the moons were on opposite sides of Jupiter, and I had to take two exposures for each time point. Also, all four moons were visible: Callisto was wandering through the frames (just to make figuring out which moon was which a bit harder).

A good question: How can you tell which moon is which? When they are near their maximum distance from Jupiter it is obvious (Ganymede is farther out than Europa, which in turn is farther out than Io), but when they are all diving in close to the planet, telling them apart can be difficult, especially Europa and Io. Sometimes their identity could be deduced by examining the previous or next night's data, or by noting how fast each one moved, but sometimes I just had to look at the fit of the data to the orbit and swap the readings from one column to another.

This data set is also a good illustration of the finite size of Jupiter itself. The planet blocks (or washes out) almost a 90,000-square-mile swath of sky. Both Io and Europa disappeared in the glare of Jupiter after 9 p.m.

 

Raw Data in Time vs. Distance Plot (Click to enlarge)


Final Data and Calculations
A graph of time-distance data is shown below. Positive and negative values (in pixels) refer to which side of Jupiter the moon was on. Time (minutes) is the time elapsed since the first frame was taken, at 7:56 p.m. on April 1. The clearest and sharpest frames, out of about 300, were used. It's interesting to note that there are fewer points for Io. With its smaller orbit, it spends more time obscured by Jupiter than the other two moons.

When I first plotted the raw data from the nine nights as distance vs. time, the result was pretty but a bit ... confusing, a follow-the-dots drawing with most of the dots left out. My advantage was that I knew (hoped) that there was a regular pattern in those dots.

 

Marbles from the Side
Put a marble on a turntable and watch it from the top; it follows a circle at a constant speed. But get down and watch it from the side, and it will appear to go back and forth in a line, faster at the middle, slower at the ends. Same object, but different views need different descriptions. My view of Jupiter's moons was like watching marbles from the side. I had to take the back-and-forth motion I could see and use it to figure out the circular orbits.

One way to figure out the period would have been to use the observations of the moons just before and after they passed in front of (or behind) Jupiter. By drawing a line through the points I could find where they crossed the x-axis, then measure the distance between crossings. Three crossings (corresponding to in front, behind, then in front again) would give me the period of the moon.

A problem with this approach is that it depends on the combination of night, seeing, and the moon's position being available at just the right times. If you think about it, the moons spend most of their time  away  from the x-axis. It also discards data taken at other times, data that are just as valid in describing the overall shape of the curve.

The pattern a marble follows from the side view of a turntable is described by a sine function. Just as circular motion from the top view is described by the circle's radius and its rotational speed, the side view is described by the sine wave's amplitude (height) and period (the time it takes for the curve to get back to an identical starting position).

The approach I used was to combine a personal computer and a program capable of performing regression analysis on periodic data (I'll bet Kepler would have loved one of these). By entering the time-distance data into the program, I could calculate the sine curves that best fit the overall data. The resulting equations gave the values for the orbital period and radius of each moon's orbit. (I made the simplifying assumptions that the moons are observed in an exact horizontal plane and that the orbits are perfectly circular, neither of which is exactly correct.)

The following charts show the best-fit equation (blue lines) and the experimental points (red points) for each moon. The two parallel lines across each graph represent the portion of the orbits blocked by Jupiter.

Best-fit equation and experimental points: Io (Click to enlarge)


Best-fit equation and experimental points: Europa (Click to enlarge)


Best-fit equation and experimental points: Ganymede (Click to enlarge)


 

So how did I do? Below are the values I obtained for the periods and relative orbital radii of the inner three Jovian moons compared to accepted values.

One obvious result is that the ratios of the periods of the moons Io: Europa: Ganymede are almost  exactly  1:2:4. The odds of this being due to chance struck me as being in the winning-the-lottery range. I subsequently learned that it has nothing to do with chance and everything to do with the gravitational interactions of the moons over billions of years, akin to the Earth's synchronizing of the Moon's rotation with its revolution.

 

Values obtained for the periods and relative orbital radii of the inner three Jovian moons compared to accepted values:

  Io Europa Ganymede
Orbital Period (experimental) 2,551 min 5,116 min 10,198 min
Orbital Period (literature) 2,547 min 5,113 min 10,303 min
Difference % 0.15% 0.06% 1.1%
Orbital Period Ratios (Io = 1) 1 2.005 3.998
Orbital Radius (Io = 1) (experimental) 1 1.54 2.55
Orbital Radius (literature) 1 1.59 2.53
Difference %   3.1% 0.8%
 

This chart shows the superimposed equations for the three moons, making the almost exact multiples of the periods unmistakable. This type of chart makes a nice complement to the looking-down-from-the-top picture, giving a feel for the patterns of motion of the moons over time. It also allows for predictions of the moons' positions in the future.

Chart of superimposed equations for the three moons (Click to enlarge)


 

Finally, a scale-model drawing of the Jovian system, using the amplitude of the sine equations I found as the mean orbital radii. The diameter of Jupiter is to scale, but the moons are not. The red arrows indicate relative orbital velocities. In the time Io makes one half an orbit, Europa makes one quarter and Ganymede one eighth.

Scale-model drawing of the Jovian system (Click to enlarge)


 

Afterword
I didn't expect to make any new factual discoveries from my observations, but I did learn a lot about the processes involved. Collecting astronomical data requires planning, some ingenuity, and lots of patience (not to mention a bit of luck). No matter how involved the work in the field, the work at the desk afterward is at least as difficult and time-consuming. Collected data don't just sit up and speak to you; they must be sorted out, digested, interpreted. A  model must be created, then tested against observations.

On a personal level I was pleased that my results came so close to the accepted values. This is tempered by the realization I had many advantages (even for a beginner) in modern observing and recording equipment, ready-made computer tools for analysis, and, not least, a near-perfect perch to observe from. I wondered what it would be like to try the same project from inside Jupiter's miniature solar system, say from the surface of Io, and shuddered at the difficulties.
But imagine the view ...

 

Notes

I used my father's telescope and camera, and he trained me in their operation.

The software used for image manipulation was Photoimpact SE Version 3.01 (Ulead Systems, Inc.).

Data were analyzed using Microsoft Excel 97 (Microsoft Corp.) and SlideWrite Plus Version 4.0 (Advanced Graphics Software).

 

References

Astronomy Lab: Moons of Jupiter. Retrieved from the World Wide Web on 8 December 2004. http://www.missioncollege.org/depts/astro/Astrolab/Jupiter.pdf

Burnham, Robert, et al. A Guide to Backyard Astronomy. San Francisco: Fog City Press, 1997.

Cayless, Alan. Astrophotography with a Digital Camera. Retrieved from the World Wide Web on 8 December 2004. http://www.cloudynights.com/astrophotography/afocal1.htm

Ferris, Timothy, Seeing in the Dark. New York: Simon and Schuster, 2002.

Miller, Doug. Moondark for July: Easy Afocal Astrophotography. Retrieved from the World Wide Web on 8 December 2004. http://home.dmv.com/dcmiller/moondark/july02.html

Page, Thornton, ed. Telescopes: How to Make Them and Use Them. New York: Macmillan Company, 1966.

 

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