Rings and Resonances

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Ring Systems

All four of the large outer planets have rings. Ring systems obey very strict geometrical laws:

Because all the ringed planets have numerous large satellites, the orbits of ring particles are constantly being disturbed. Thus collisions must still occur. Implications:

Resonances result when two celestial objects interact with each other gravitationally at regular intervals. The regularity of the interaction can do one of two things:

We find examples of both in the Solar System.

Tidal Rotation Lock

A tidal rotation lock is the simplest kind of resonance. Planets and their satellites cause tidal bulges in one another. As the objects rotate, the tidal bulges move through their interiors and encounter friction. The satellites, being smaller, tend to become locked first. Some results:

Mercury

For a long time, astronomers thought Mercury was tidally locked to the Sun as the Moon is to Earth. It actually has a more complex tidal rotation lock; Mercury rotates in 56 days; three times for every two orbits. Some consequences:

Long-Term Effects

As a planet rotates, friction causes its tidal bulges to lead its satellite. The satellite pulls on the near tidal bulge, slowing the planet. The near tidal bulge pulls on the satellite, accelerating it and throwing it into a more distant orbit.

For the Earth-Moon system, the two effects are predicted to balance several billion years in the future, when the Moon will be about 50% farther away and the day and month will both be 40 of our present days long. But that's not the end of the story. The Sun also exerts tidal forces on the Earth, about half as strong as the Moon. The Sun will continue to slow Earth's rotation; the solar tidal bulges will trail the moon, slowing it down and causing it to spiral inward. Whether the Moon will spiral in enough to break up from tidal forces before the Sun goes red giant is unknown.

If the satellite orbits opposite its planet's rotation, the tidal bulges always trail the satellite. They tend to drag the satellite in and may eventually cause it to be destroyed. Neptune's large moon Triton is the only body in the Solar System in this category; it may spiral into Neptune in a few hundred million to several billion years.

The Roche Limit and Tidal Disruption

Tidal forces result when a celestial body exerts a stronger gravitational pull on the near side of another body than the far side. The difference can be great enough to pull fragile objects apart, as Jupiter did to Comet Schumacher-Levy in 1992. Within a certain distance of any planet, called the Roche Limit, tidal forces are greater than the gravitational attraction of small bodies and satellites cannot form.

It is possible for satellites to orbit within the Roche Limit (Our own artificial satellites do; Mars and the Jovian planets have satellites that do) but the satellite is held together by the strength of its materials and not by its gravity.

If a natural satellite is orbiting within its planet's Roche limit, it must have formed elsewhere and perturbed into a close orbit somehow.

There has been speculation that planetary rings may have formed from tidally-disrupted satellites. Possibly Triton and our Moon will be disrupted if they eventually come within the Roche Limit.

Orbital Resonances

Gaps

A asteroid orbiting 3.28 AU from the Sun would circle the Sun in 5.93 years, half the orbital period of Jupiter. Every other revolution, it would receive a tug from Jupiter at the same point in its orbit and eventually have its orbit changed. There are thus no asteroids with periods exactly half that of Jupiter, and almost none with a period 1/3 that of Jupiter. These gaps in the asteroid belt are called Kirkwood Gaps. Principal Kirkwood gaps correspond to orbital periods 1/3, 2/5, 3/7, 1/2 and 3/5 that of Jupiter.

Gaps occur in Saturn's rings due to Saturn's satellites. Particles moving within Cassini's Division would orbit Saturn in periods ranging from 11 hr. 19 min. to 12 hr. 5 min. This is 1/2 the period of the satellite Mimas, 1/3 that of Enceladus, 1/4 that of Tethys and 1/9 that of Rhea. Smaller gaps have been noted in the rings and are due to other resonances.

Shepherd Moons

Consider a moon (it can be tiny) orbiting just outside a ring. Ring particles, being closer to the planet than the satellite, are moving faster. If a particle drifts out of the ring, when it overtakes the satellite, the satellite's gravity will drag the particle back, slow it down, and cause it to fall back into the ring.

Outside Shepherd Moon

Similarly, a moon can orbit just inside a ring. The satellite is moving faster than the ring particles. If a particle drifts inward from the ring, it will be overtaken by the satellite. The satellite's gravity will speed the particle up and throw it back into the ring.

Inside Shpherd Moon

Saturn has a thin ring (the F ring) bounded by two tiny moons that keep it confined. Some of the shepherd moons for the thin rings of Uranus and Neptune are also known.

Preferred Periods

Certain resonances seem to enhance orbital stability by locking bodies in step in such a way they avoid conflict. 3:2 resonances seem to be especially effective. Pluto crosses Neptune's orbit, but its period is 3/2 that of Neptune, so the two objects never approach closely. Of the 40-plus objects discovered orbiting beyond Neptune since 1992, an astounding 40% have periods very close to Pluto's and are also in 3:2 resonance with Neptune. These objects have been dubbed "plutinos".

In contrast to the gaps in the asteroid belt at 1/2 and 1/3 the period of Jupiter, there is a cluster of asteroids orbiting with 3/2 Jupiter's period and a smaller cluster at 4/3.

Lagrangian Points

Johannes Kepler and Isaac Newton solved the equations for orbital motion of one body around another. For centuries after, astronomers struggled to solve the "three-body problem": write an equation for the motion of three orbiting bodies. We now believe such a solution is impossible; in fact, most such systems are chaotic. That is, tiny differences in initial calculations lead to enormous differences later. The best we can do in such a system is what we do with the planets now: achieve reasonable precision over a reasonable time span.

In the late 1700's, the French mathematician Joseph Lagrange solved the three-body problem for certain special cases. In a system consisting of a Primary body (greatest mass), a Secondary (intermediate mass) and a third small object, he showed that there were five points where the third body's motion was predictable.

Points L1 and L2 correspond to orbits around the secondary that match the secondary's period about the primary, for example, a satellite orbiting Earth about 5 times farther than the Moon would have a period of a year. If it orbited in the same direction as Earth in the L2 position or opposite in the L1, it could remain in line with the Earth and Sun. Position L3 orbits opposite the secondary but a bit closer because it feels the combined pull of both the Primary and Secondary.

Lagrangian Points Positions L1, L2 and L3 apply to an ideal system of three bodies and perfectly circular orbits. In the real Solar System they are unstable: any disturbance would cause the third body to drift out of position. Solar observatories have been placed at the L1 position of Earth, but they need a fuel supply to correct for the tendency to drift. In 1985 NASA took advantage of that on-board fuel to divert a probe at the L1 position into a flyby of Comets Giacobini-Zinner and Halley.

Positions L4 and L5 are different. These points form equilateral triangles with the Primary and Secondary. L4 leads the Secondary and L5 trails it. These points are stable. If an object in one of these points is disturbed, it tends to return to its original position. In real life bodies oscillate fairly widely around the stable point

These points were mathematical abstractions until real examples were found. Asteroids occupy the L4 and L5 positions on Jupiter's orbit. The first one discovered was 588 Achilles, so the custom arose of naming these asteroids after figures from the Trojan War. For this reason, objects travelling in an L4 or L5 position are often called Trojan. In the Saturn system, Tethys and Dione have small moons in their L4 or L5 positions. There appear to be faint clouds of dust particles in the L4 and L5 positions of our Moon's orbit. So far no asteroids have been found in the L4 or L5 positions in either Earth's or Saturn's orbit, despite careful searches.

Chaos

Chaos is the opposite of resonance in a way, and subject to a lot of misconceptions.

Chaos does not mean: Chaos does mean:
Events are not governed by laws of nature. Events are governed by the laws of nature but their behavior is too complex for simple description.
Events are random or unpredictable. Events may be predictable in the short term but not over arbitrarily long times. Small errors at the start of a prediction compound later on into great differences.
Chaotic systems are unstable and will eventually fall apart (the Jurassic Park Fallacy) Chaotic systems may remain within fixed bounds but their behavior within those bounds may be hard to predict.

Saturn's moon Hyperion has a thick disk shape that has been likened to a hockey puck. Because of its asymmetry, gravitational interactions with other Saturnian satellites cause it to rotate chaotically. That is, there is no single rotation axis and period that describes Hyperion's rotation for long periods of time.

Three- and larger body systems (and hence the Solar System) are probably chaotic. That is, we could not predict exactly.where the planets will be in exactly a billion years, or say where they were exactly a billion years ago. The only way to predict motions of the planets with extreme accuracy now is to calculate all their gravitational interactions and motions in small steps; tiny errors in our knowledge of masses and distances now would compound into huge errors in billions of years. We can accurately reconstruct celestial events in ancient documents, but on a scale of millions or billions of years our predictions would be increasingly inaccurate. Note that this does not necessarily mean the Solar System is unstable!


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Created 20 May 1997, Last Update 14 December 2009

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