The Rings of Saturn and the Harmony of the Spheres

“And there, in the middle of the light, they saw stretching from the heavens the ends of its bonds—for this light is what binds the heavens, like the cables underneath a trireme, thus holding the entire revolving thing together. From those ends hangs the spindle of Necessity, by means of which all the revolving things are turned. [….] The spindle as a whole revolved at the same speed, but within the revolving whole the seven inner circles gently revolved in the opposite direction to the whole. [….] The spindle revolved on the lap of Necessity. On top of each of its circles stood a Siren, who was carried around by its rotation, emitting a single sound, one single note. And from all eight in concord, a single harmony was produced.” (Republic, X.616b-617b)

Legend has it that a sign hung over the entrance to Plato’s Academy, declaring “Let none but geometers enter here.” The story is most likely apocryphal, but it does capture something significant about Plato’s project—thus the anecdote’s enduring appeal. Now, Plato himself wasn’t much of a mathematician. At least, there’s no evidence of such talent in his surviving works. The few calculations we find are riddled with errors, and his account of the cosmos, based on the spindles used for weaving fabric, is a feat of literature, not science. Nonetheless, Plato called on his students to seek rational order in the chaotic motions of the planets. In doing so, he directly inspired the development of a Greek science of astronomy. There is a direct line from Plato to the complete Geocentric system of Ptolemy, which ruled the western mind for over a thousand years, and then on through Copernicus.

Plato’s musical metaphor also proved influential. Drawing upon his Pythagorean roots, he linked the complex order of the planetary motions to the musical harmony of distinct notes. Most astronomers after him sought the same harmony in the skies. Witness history’s greatest astronomer, Johannes Kepler:

“The heavenly motions are nothing but a continuous song for several voices, to be perceived by the intellect, not by the ear; a music which […] sets landmarks in the immeasurable flow of time.” (Harmony of the Worlds, 1619)

Kepler-HarmonyStrictly speaking, this harmony stuff tended to get Kepler into trouble. His commitment to celestial purity may have motivated his work, but it held back his theorizing, delaying his crucial breakthrough: the idea that planetary orbits followed ellipses, rather than perfect circles. Once he was willing to bend the notes a little, he was quickly able to work out his three laws of planetary motion.

Kepler’s circophilia is but one example of a recurring problem in the history of science: the projection of one’s own theoretical commitments onto the nature we observe. The rings of Saturn provide a good illustration of the point.

Here’s Kant in the Critique of Pure Reason:

“The predicates of appearance can be attributed to the object in itself, in relation to our sense, […] but the illusion can never be attributed to the object as predicate, precisely because that would be to attribute to the object for itself what pertains to it only in relation to the senses or in general to the subject, e.g., the two handles that were originally attributed to Saturn.” (Aesthetic, B69-70)

If you’ve read much Kant, you’ll know that he rarely gives examples to support his points, and he never develops them in much detail. In this case, he doesn’t really explain what he means by the “handles” of Saturn. When I first read this, I probably thought this was just a bad translation of “rings.” But Kant has something quite definite in mind.

Galileo’s discovery of the moons of Jupiter was crucial for winning support for the new Heliocentric model. They challenged the defining principle of Geocentrism: that every celestial body rotated around the Earth. So astronomers began looking for moons around other planets. As their telescopes improved, they began to notice something peculiar going on with Saturn. The sketches of their observations varied wildly:

SaturnEarly on, they saw satellites, which were fresh on their mind. When that clearly didn’t work, they turned to a concept from everyday life: handles. (It was only much later that the now-familiar ring account triumphed, thanks to the work of Huygens.) But here’s what I find fascinating about this episode. We know what kind of telescopes and lenses these astronomers were using, so we can recreate their experience of viewing Saturn. And when we do this, it’s just plainly obvious that Saturn has rings. In fact, it seems inexplicable that they couldn’t see what we see. Here, the specter of theory-laden observation rears its head.

Now, of course, we can photograph the rings in all their spectacular glory. Here’s a recent photo from NASA’s Cassini spacecraft:

Saturn Cassini

Saturn Cassini

This is a shot looking back toward the center of the solar system, so we’re in the photo–at least the full-resolution version. (Clicking on the photo links to a detail, showing the Earth–a little dot below the right side of the ring.) By the way, this is a natural light photo. The odd colors come from Saturn being backlit by the sun, which is directly behind it.

Notice the gaps in the rings. Have you ever wondered why they’re there? I can recall reading as a kid that the gaps were cleared out by Saturn’s moons. That’s not entirely wrong, but it doesn’t really capture what’s going on. I pictured a satellite careening along, blasting the ice and rocks out of its orbital path. But the actual explanation is more interesting and much more beautiful. (Thanks to Keith Johnson at the Rowan Planetarium for sharing this with me.)

Basically, there are a lot of objects orbiting around Saturn. Their speed varies directly with their distance from the planet. (When you’re in close, you have to go very fast to keep from falling into Saturn; when you’re farther away, it takes less speed to maintain your position. This is Kepler’s third law.) So, picture the moon Mimas, traveling slightly outside the rings, orbiting Saturn roughly once per day. Within the rings, there will be an area where objects take exactly half the time to orbit Saturn. Imagine a lonely chunk of ice in this location. It spins around Saturn twice for every orbit Mimas completes. But because their orbital periods have an exact 2:1 ratio, the ice always laps Mimas at the very same spot. This creates an orbital resonance.

Here’s an example of what I have in mind, based on a similar orbital resonance between various moons of Jupiter:

orbital-resonance(Thanks: http://scientificgamer.com/resonating-resonances/. You might have to click on the image to trigger the animation.)

Since they always meet at the same point, the occasional gravitational tugs of Mimas keep adding up, gradually pulling the ice out of its orbital path. Basically, any objects that make their way into this part of the ring are inevitably pulled out of position by Mimas, leaving the gap. (Other satellites have a similar effect, producing their own smaller gaps in Saturn’s rings.)

I hope I’ve done some justice to the sheer beauty of this explanation. But what I like most about it is that it brings us back to Plato. We find these orbital resonances everywhere in the solar system, and thank goodness for that. They’re essential to keeping objects lined up and maintaining the long-term stability of their orbits. And so, we’ve found what we were looking for all along: the harmony of the spheres.

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2 Responses to The Rings of Saturn and the Harmony of the Spheres

  1. Josh says:

    Nates –

    How do Kepler’s notes on a staff relate to the underlying phenomena? Are they representative of melodies that someone fall out of the apparent orbits of those planets from earth? I get that his better work was more empirical; I was just wondering how he came up with those notes and what significance they held for him.

  2. Nates says:

    To be honest, Josh, I don’t fully understand how this works. But the basics are straightforward. The range of notes corresponds to the range of variation in a planet’s angular velocity. So, a planet like Venus, which has an almost perfectly circular orbit, always sings the same note. Mercury and Mars, which have very elliptical orbits (moving quickly near the sun, slowly on the other end), play a wider range of notes. Earth is closer to Venus, with a fairly limited range. As Kepler quips: “The Earth sings Mi, Fa, Mi: you may infer even from the syllables that in this our home misery and famine hold sway.” (If there’s a reason why it’s Mi and Fa in particular–other than setting up the joke–I don’t know what it is.)

    Kepler also thought he was spotting harmonic proportions in these ranges of velocities, but I don’t know enough about either astronomy or music theory to make sense of it. If you want to take a stab at it, there’s a brief overview here: http://en.wikipedia.org/wiki/Harmonice_Mundi

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