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

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.

- Suppose you photograph a clock once every 62 minutes. If you put your photographs together in the order they were taken, which way will the minute hand appear to move? Which will appear to move faster, the minute hand or the hour hand?
- Mercury rotates once every 58.646 days and orbits around the Sun once every 87.969 days. Is this some kind of a resonance, and if so, what's the ratio of orbital and rotation periods?
- Here is a table of names and orbital periods (in hours) for
some of Saturn's larger satellites. Are any of these satellites
in resonances with each other?
Mimas Enceladus Tethys Dione Rhea Titan 22.62 32.89 45.31 65.69 108.42 382.69 - Saturn's satellites Tethys, Dione, Rhea, and Titan can all be seen with our 8 inch scopes. Would you expect any of these satellites to appear in roughly the same place if observed once a week?

Joshua E. Barnes (barnes@ifa.hawaii.edu)

Last modified: February 23, 2003

`http://www.ifa.hawaii.edu/~barnes/ASTR110L_S03/jupitersat.html`