# 7.13 Resonance and Harmonics

Why do the ring systems of giant planets have such amazing structure? We might expect material orbiting a planet to be in a smooth, uniform disk shape. Instead, we see a complex pattern of concentric rings with widely varying widths and opacity. Sometimes we see gaps in the pattern. The closer we look at Saturn’s rings, the more intricate their structure appears. What causes this complexity? You may be astonished to learn that the answer is related to how a guitar produces a rich, full sound!

Natural-color mosaic of?Cassini?narrow-angle camera images of the unilluminated side of Saturn's D, C, B, A and F rings (left to right) taken on May 9, 2007. Click here for original source URL

Every object has a natural frequency of vibration or oscillation — whether it’s a star, a planet, a bridge, a wineglass, or a guitar string. We will use a string as our example, because the sound that results from its vibration is so familiar. When one string is vibrating, it can set another string vibrating at the same frequency (this happens with bells, too). This is called resonance.

Pushing a person in a?swing?is a common example of resonance. The loaded swing, pendulum, has a natural frequency of oscillation, its resonant frequency, and resists being pushed at a faster or slower rate. Click here for original source URL.

You’re probably familiar with resonance if you’ve ever been on a swing at the playground. If you push on the swing at random times, you don't "pump up" the swing's motion, because the pushes tend to cancel each other out. But if you push at exactly the interval that matches the swing's natural period, the swing gets higher and higher. Similarly, soldiers have to break step when crossing a bridge, because their marching rhythm might match the natural frequency of the bridge, sending it into violent oscillation and possible disruption. Most mechanical objects have a natural resonant frequency. Perhaps there is an engine rotation frequency at which your car vibrates, or a wheel rotation frequency at which your steering shakes?

A string actually has many modes of vibration. In addition to the fundamental frequency f, the string can vibrate through more complete cycles each second, corresponding to higher frequencies. These harmonics have integer multiples of the fundamental frequency: f, 2f, 3f, 4f, and so on.

Harmonic frequency = n f,

where n is an integer. The period P of a vibration (time in seconds) is just the inverse of the frequency (number of vibrations per second), so we have:

Harmonic period = P / n

Octaves displayed on a grand staff with corresponding names of common naming systems. Click here for original source URL.

In the musical scale, if the fundamental frequency was middle C, 2f would be an octave higher, 3f would be another fifth higher, and 4f would be two octaves above middle C.

Animation of orbital resonances of three of Jupiter's moons. Click here for original source URL.

The idea of harmonics works for orbits, too! But resonances between orbits are caused by gravity, instead of sound waves. Suppose there’s a satellite orbiting outside a ring system. Let’s say the satellite has an orbital frequency f. Kepler’s laws state that the ring particles will move faster than the satellite, because they are orbiting at a smaller radius. There is one place in the ring system where the orbital frequency is exactly twice that of the satellite, 2f. At this radius, a ring particle will pass the satellite every other orbit, and be slightly tugged out of position by the satellite’s gravity. Each tug is tiny, but the effect is cumulative. Over many orbits, the particles will move away from that radius. The particles are like the swing, getting pushed resonantly at just the right interval to "pump up" their velocity, and thus change their orbits. This creates a gap.

The same thing occurs when the ring particle has an orbital frequency three times that of the satellite, or four times, or any other integer multiple. So this orbital resonance should create gaps wherever the particles have 2 or 3 or 4 (or more) times the orbital frequency of the satellite. Equivalently, a gap is created where the ring particles have periods of 1/2 or 1/3 or 1/4 (or less) of the satellite’s period.

A number of features in Saturn's rings are related to?resonances?with Mimas. Mimas is responsible for clearing the material from the?Cassini Division, the gap between Saturn's two widest rings, the?A Ring?and?B Ring. Particles in the?Huygens Gap?at the inner edge of the Cassini division are in a 2:1?resonance?with Mimas. They orbit twice for each orbit of Mimas. The repeated pulls by Mimas on the Cassini division particles, always in the same direction in space, force them into new orbits outside the gap. The boundary between the C and B ring is in a 3:1 resonance with Mimas. Click here for original source URL.

The first big satellite beyond Saturn’s rings is Mimas, at a radius of 185,000 kilometers from Saturn. Kepler’s third law relates period and semi-major axis: P^{2} ∝ a^{3}. We can manipulate this to give a ∝ P^{2/3}. For a higher harmonic, the period is P / n, and the distance goes down by:

a ∝ (P / n)^{2/3}

So for the n = 2 harmonic, the distance where gravitational resonance will clear out the orbit is (1/2)^{2/3} = 0.63 times the satellite distance, or 117,000 kilometers. This is exactly the radius of the largest gap in Saturn’s rings, the Cassini Division. Giovanni Cassini discovered this gap between the A and the B rings in 1675. The next harmonic, at n = 3, is close to the gap between the B and the C rings. Resonances like this have been located in all the ring systems. The resonance when the ratio (Ring period / Satellite period) is equal to 1/2 or 1/3 or 1/4 is just one set of harmonics. Resonances also occur whenever the ratio (Ring period / Satellite period) is equal to a ratio of whole numbers, like 2/3 or 3/7 or 4/9. Just as the large number of harmonics leads to the richness of musical sounds, the large number of resonances leads to the rich complexity of planetary ring systems.

More than two thousand years ago, Pythagoras believed that the universe was governed by simple relationships between numbers. He even speculated on the "harmony of the spheres," a celestial music that mirrored the harmonics produced by a plucked string. Kepler was strongly influenced by these ideas. Shakespeare gave a nod to Pythagoras in The Merchant of Venice:

"Sit, Jessica. Look how the floor of heaven

Is thick inlaid with patens of pure gold.

There’s not the smallest orb which thou behold’st

But in his motion like an angel sings.

Such harmony is in immortal souls,

But whilest this muddy vesture of decay

Doth grossly close it in, we cannot hear it."

The planets, satellites and rings produce no sounds to propagate through the vacuum of space. However, the astral harmonics that Pythagoras and Kepler sought are present in the dance of gravity.