Jupiter and its satellites are sometimes called a `miniature Solar System', but the truth is more complicated. Three of Jupiter's four satellites are locked in resonant orbits. This has interesting consequences for our weekly observations, and for the history and fate of the Jovian system.
Jupiter has four bright satellites which are easily seen with a telescope. These bodies travel around the planet in nearly-circular orbits; in order of increasing distance from Jupiter, they are called Io, Europa, Ganymede, and Callisto. Kepler's third law implies that satellites with smaller orbits move more rapidly. Thus Ganymede, Europa, and Io, which are all closer to Jupiter than Callisto, should all move faster than Callisto.
If we try to test this prediction by making observations every Monday night, we will find the satellites at the positions shown in Fig. 1. Looking at this figure, you may notice something strange. The three inner satellites, Io, Europa, and Ganymede, seem to barely move from one week to the next, while Callisto jumps all over the place. That's not what Kepler's third law led us to expect! What's going on?
The answer involves two facts. One is pure coincidence; the other is a remarkable fact about Jupiter's satellites.
First, the pure coincidence: it takes Ganymede 7.155 days to orbit Jupiter. This is slightly more than one week, so if we observe Jupiter's satellites on the same day every week, we'll see Ganymede in almost exactly the same place each time. While we're not looking, Ganymede goes around Jupiter and almost returns to where it was the week before. This is a matter of luck; it just so happens that the 7-day week we use on Earth is almost exactly equal to Ganymede's orbital period.
Second, the remarkable fact: Europa's orbital period (3.578 days) is half of Ganymede's, and Io's orbital period (1.789 days) is half of Europa's! So, in the time it takes Ganymede to make one orbit, Europa makes two orbits, and Io makes four orbits. If we observe once a week, we see all three of these satellites almost exactly where they were the week before. The relationship between Ganymede, Europa, and Io's orbital periods is not a coincidence; the odds of such celestial clockwork occuring by chance are very small.
If the time between our observations exactly matched Ganymede's orbital period, we would see the three inner satellites in the same places each week. But the amount of time between our observations is 0.155 days (or 3 hours, 43 minutes) shorter than Ganymede's orbital period, so Ganymede doesn't quite complete its trip around Jupiter. The result is a bit like photographing a clock once every 59 minutes; a series of such photographs shows the minute hand slowly moving backward because it doesn't quite finish its trip around the dial between photographs. In the same way, our weekly observations show Io, Europa, and Ganymede all slowly shifting backwards.
Fig. 1 shows the outermost satellite, Callisto, appearing all over the place. This happens because Callisto's orbital period of 16.689 days is somewhat more than two weeks; thus observations at weekly intervals find Callisto appearing on more or less opposite sides of Jupiter.
Fig. 1. Jupiter's satellites at 20:30 (HST) on lab nights during Fall 2011. The satellites are identified by their initials.
The relationship between the orbital periods of Io, Europa, and Ganymede is an example of a resonance. More generally, we say that two orbits are resonant when the ratio of their periods is a ratio of whole numbers. For example, Pluto's orbital period is 247.7 years, while Neptune's orbital period is 164.8 years. The ratio 247.7:164.8 is equal to 3:2, so Pluto completes two orbits around the Sun in the same time it takes Neptune to complete exactly three orbits. This resonance explains how Pluto and Neptune can cross orbits without colliding: Pluto only comes within Neptune's orbit when Neptune is on the other side of the Solar System. It's also possible to have resonances between orbital motion and rotation; for example, the Moon's orbital period and rotation period are both 27.3 days, so their ratio is 1:1 exactly.
In the case of Jupiter's satellites, it's likely that Io, Europa, and Ganymede developed their resonance as a result of gravitational attraction. One possible scenario starts with Io, Europa, and Ganymede all orbiting closer to Jupiter than they do today. As a result of the tides Io created on Jupiter, Io's orbit slowly drifted outward, and as it did so it would eventually approach a 2:1 resonance with Europa. Once that happened, the orbits of the two satellites would be `locked' by gravity, and both would drift outward together. Eventually, as Europa's orbit grew larger, it would have reached a 2:1 resonance with Ganymede, and the orbits of all three satellites would lock into their present relationship.
Resonances play an important role throughout the Solar System. For example, some of the gaps in Saturn's rings occur as a result of resonances between particles in the rings and Saturn's satellites. Likewise, there are gaps in the asteroid belt as a result of resonances between asteroids and Jupiter.
To see that Io and Europa really do complete four and two orbits, respectively, in the time it takes Ganymede complete one orbit, we would have to observe Jupiter in between our weekly lab meetings. However, Stellarium and similar planetarium programs can be used to display Jupiter's satellites and speed up their motion; we may try this out on a cloudy night.
We will observe and sketch Jupiter's satellites when it's convenient. You can compare your sketches with the predictions shown in Fig. 1 to check that the satellites appear in their expected positions.
As Jupiter's satellites orbit they periodically pass in front of or behind the planet; they also pass through Jupiter's shadow or cast their own shadows on Jupiter's disk. A passage in front of the planet is called a transit, while a passage behind is called an occultation. An eclipse occurs when a satellite passes through Jupiter's shadow, while a shadow transit occurs when satellite's shadow falls on Jupiter.
The table below lists various events involving Jupiter's satellites which we may be able to observe during lab nights. All dates and times are given in HST. Notice that the same events involving Io recur week after week, shifting about two hours later each week; in between one week and the next, Io makes four orbits around Jupiter!
|04-Oct||19:18||Io shadow transit ends|
|19:54||Io transit ends|
|11-Oct||19:02||Io shadow transit begins|
|19:30||Io transit begins|
|21:12||Io shadow transit ends|
|21:38||Io transit ends|
|18-Oct||20:56||Io shadow transit begins|
|21:14||Io transit begins|
We may also be able to see Jupiter's `Great Red Spot', a giant storm in the planet's atmosphere. Listed below are the dates and times when the GRS will facing the Earth; the best views are within half an hour of these times.
|18-Oct||19:46||GRS on meridian|
|25-Oct||20:31||GRS on meridian|
|01-Nov||21:16||GRS on meridian|
|08-Nov||22:01||GRS on meridian|
Joshua E. Barnes
(barnes at ifa.hawaii.edu)
21 August 2011