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.
Besides the observations made on Feb. 18, we've looked at Jupiter and its satellites on three successive classes: Jan. 21, Jan. 28, and Feb. 4. Below are three computer-generated images showing Jupiter and its satellites at 21:00 (9:00 pm) on these dates.
|Positions of Jupiter's satellites at 21:00 on 1/21 (top), 1/28 (middle), and 2/04 (bottom). The satellites are identified by the letters I (Io), E (Europa), G (Ganymede), and C (Callisto).
Looking at the positions of these satellites, you may notice something strange. The three inner satellites, Io, Europa, and Ganymede, appear in nearly the same place each week, while the outermost satellite, Callisto, wanders from one week to the next. This is certainly unexpected. Kepler's third law implies that satellites with smaller orbits move more rapidly. Thus Ganymede, Europa, and Io, which all have smaller orbits than does Callisto, should all move faster than does Callisto. Yet here we see the three inner satellites barely moving, while Callisto seems to be all over the place. What's going on?
The answer involves two facts. One is pure coincidence; the other reveals something very fundamental about Jupiter's satellites.
First, the coincidence: it takes Ganymede exactly 7 days, 3 hours, and 43 minutes to complete one orbit around Jupiter. Thus, if we observe Jupiter's satellites just once a week, we will see Ganymede in almost exactly the same place each time, but while we're not looking, Ganymede travels around Jupiter and almost (but not quite) 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 fundamental fact: Europa's orbital period is half of Ganymede's, and Io's orbital period is half of Europa's! In the time it takes Ganymede to make one orbit, Europa makes two orbits, and Io makes four orbits. So when 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.
|Positions of Jupiter's satellites at 21:00 on every Tuesday from 1/21 through 5/06. The satellites are identified by the letters I (Io), E (Europa), G (Ganymede), and C (Callisto).
The diagram above shows the positions of Jupiter's satellites each Tuesday during the rest of the semester. As you can see, the three inner satellites appear in almost the same positions each week; the slight shifts in their positions from week to week will be explained shortly. The outermost satellite, Callisto, continues to appear all over the place. This occurs because Callisto's orbital period of 16 days, 16 hours, and 32 minutes is a bit more than two weeks; thus observations at weekly intervals find Callisto appearing on more or less opposite sides of Jupiter.
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 3 hours and 43 minutes shorter than Ganymede's orbital period, so Ganymede doesn't quite complete its trip around Jupiter; after one week, it is 2% short of one complete orbit, while Europa is 3% short of two complete orbits, and Io is 4% short of four complete orbits. 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 much the same way, our weekly observations show Io, Europa, and Ganymede all slowly shifting backward.
This slow shift is a good thing for our class, because there'd be little point in observing Jupiter's satellites each week if they were always in the same places! Looking ahead, we can anticipate seeing Europa cross in front of Jupiter in early March, and Io crossing in front of Jupiter in early April; such crossings of a small body in front of a larger one are known as transits. These transit events give us a chance to see the satellites visibly move during a single observing session; we may also be able to see the small shadows Europa and Io cast on Jupiter. Later in March we should be able to see Ganymede moving in and out of Jupiter's shadow; this is called an eclipse. Finally, at the end of the semester we may be able to observe Ganymede passing behind Jupiter; such an event is known as an occultation.
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. Eventually, as the orbits of the three inner satellites continue to drift outward, Ganymede may reach a 2:1 resonance with Callisto, and all four orbits will be locked together.
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 with Jupiter.
We will continue to observe and sketch Jupiter's satellites when it's convenient. You can compare your sketches with the predictions show above to confirm that the satellites appear in their expected positions. The upcoming transits of Europa and Io in front of Jupiter, along with the more spectacular eclipse of Ganymede, will definitely be worth observing.
To see that Io and Europa really do complete four and two orbits, respectively, each time Ganymede completes one orbit, we would have to observe Jupiter much more often than once a week. However, it's often possible to see the satellites with your binoculars. If you make a point of observing Jupiter every night for a while, you will at least be able to see that the satellites move quite rapidly, and that their arrangement around Jupiter changes from night to night - and sometimes even during a single night!
As mentioned above, the gap between the two major rings of Saturn is due to a resonance with one of Saturn's satellites. This gap is fairly easy to see with our 8 inch scopes. Unfortunately, the satellite responsible for creating this gap, Mimas, is too faint to see with our equipment.
Last modified: February 23, 2003