Ways things can rotate

This is an overview of some of the kinds of rotation an object can undergo, with a focus on astronomical objects.

Standard warning: I’m starting to realize that I don’t really know enough about physics to write about this. But I try not to let things like that stop me.

Things to note:

• I’m discussing rotation in 3-dimensional space, and usually assuming that the rotating body is fairly rigid.
• Some kinds of motion are relative, but rotation is not. Rotation is absolute, not relative to anything except maybe the universe as a whole.
• This post is not about long-term effects like precession.
• I’m probably extending and abusing the definition of the word “rotation”.

Uniform rotation – stable

An object with uniform rotation is rotating around a fixed axis (meaning the axis is fixed relative to the object).

The most energetically favorable form of rotational motion is uniform rotation, with the axis chosen so that the moment of inertia is the largest possible. Basically, this is when the average mass is as far away from the axis as possible.

It’s usually easy to determine if an object rotates uniformly or not, assuming you can at least see it clearly with a telescope. If and only if its brightness fluctuates in a periodic pattern, it is rotating uniformly.

A large (say, planet-sized) object will generally find its way into this type of rotation almost immediately.

Any object not in stable uniform rotation, and not under external influence, will eventually find its way to stable uniform rotation, thanks to dampening effects caused by small deformations of the object. There’s a formula for how long this typically takes. For small rigid objects, it can take a very long time — like, trillions of years. By studying which asteroids are and are not rotating uniformly, it’s possible to calculate a rough estimate of the age of the Solar System.

Uniform rotation – metastable

Uniform rotation can happen even if it is not around the most energetically-favorable axis. For example, consider an American football thrown in a “perfect spiral”. In an isolated environment, such rotation would not be stable forever, but it could last a long time.

Uniform rotation – synchronous

Typically, when a massive object like a moon is orbiting a massive object like a planet, tidal acceleration causes its rotation to slow, until its sidereal rotational period is the same as its sidereal orbital period. Thereafter, it always keeps the same hemisphere pointed toward its planet. This is known as synchronous rotation.

Synchronous rotation of a moon becomes more likely the more massive its planet is, and the smaller and more circular the moon’s orbit is. If a moon in the Solar System is notable enough that you recognize its name, it’s probably in synchronous rotation.

The most notable moons that rotate uniformly but not synchronously include Jupiter’s moon Himalia, Saturn’s moon Phoebe, Uranus’s moon Sycorax, and Neptune’s moon Nereid. It seems that Nereid is the largest planetary moon that is not in synchronous rotation.

Moons of dwarf planets and other small bodies stand a better chance of not being in synchronous rotation. Eris’s moon Dysnomia is larger than Nereid, and is not in synchronous rotation, though it’s curiously close to being synchronous.

Some people wonder how Earth’s Moon’s rotation can be so perfectly synchronized that it doesn’t drift at all. Or maybe it does drift, just very very slowly? Well, no, it doesn’t drift. The resonance can be perfect because the Moon’s mass isn’t distributed with perfect spherical symmetry. If you were to give the Moon a little twist, it would most likely return to its present orientation.

Uniform rotation – other resonance

Rotational resonances other than synchronous rotation are possible.

The planet Mercury rotates 3 times for every 2 revolutions around the Sun, and is the only known case of such a spin-orbit resonance in the present Solar System. The eccentricity of Mercury’s orbit is said to be what makes such a resonance possible. But the eccentricity changes over time, leading someone like me to wonder if the 3:2 resonance might be a relatively recent development. Some experts, though, think it has (somehow) been that way since very early in its history.

A different kind of spin resonance may have existed with the Earth, in the distant past. Some scientists propose that a resonance between the Earth’s rotation and the Moon’s orbit held the length of Earth’s day stable at about 21 hours for a billion years or more, with the resonance being broken around 600 million years ago. [Here’s a reference: Analysis of a Precambrian resonance-stabilized day length, by B. Bartlett & D. Stevenson.]

This is one of those that make me do a double-take when I read about it. Why did I not already know this? We’re often told that the Earth’s rotation is gradually slowing down, and has been slowing down since early in its history. But if in fact the Earth spent a good fraction of its lifetime with its rotation not slowing down at all, that’s a misleading claim at best.

Fluid objects

Objects composed largely of liquid, gas, or plasma have to rotate in a way that is pretty much uniform. But it’s not so simple, because they’re, well, fluid. Different latitudes of the Sun’s surface, for example, rotate at different angular velocities, so they have different rotational periods.

Gas giant planets have a rigid core that should have a well-defined rotational period, But it can be difficult to measure the period, and due to atmospheric effects it may not be very steady. Consider that we only know the length of Saturn’s day to within a couple of minutes.

Tumbling (not too chaotic)

Spin a pencil around its long axis, and simultaneously toss it into the air so that it tumbles end over end. That’s a kind of not-too-chaotic “tumbling” motion, with no fixed axis of rotation.

Having skimmed a few documents on the topic, I’m getting the impression that there might not be a universally agreed upon definition of “tumbling”, versus “wobbling”, versus “rotating chaotically”, etc. Sometimes the terms are used interchangeably, sometimes not.

There are different kinds of tumbling/chaotic motion, though I don’t know the extent to which they can be unambiguously classified.

In this category I’ll place the kinds of non-uniform rotation that are not too chaotic, so that the object’s orientation can be predicted significantly in advance. This is the most unruly kind of rotation possible for objects whose symmetry is something like a that of a cylinder.

Tumbling (chaotic)

An object with three substantially different moments of inertia, such as a tennis racket or a pack of cards, is completely unable to rotate uniformly around its “medium” axis. This is known as the tennis racket theorem, the intermediate axis theorem, or the Dzhanibekov Effect.

In extreme cases, such an object can rotate in an almost stable manner for a while, but every now and then it suddenly and dramatically flips over 180 degrees, then resumes rotating in an almost-stable manner in the new orientation.

Saturn’s moon Hyperion rotates chaotically, made possible by its proximity to the large moon Titan, and other factors. Some or all of Pluto’s four smallest moons possibly rotate chaotically, though there seems to be some disagreement about this.

Quantum spin

Almost all fundamental particles, like electrons and photons, have a weird kind of intrinsic angular momentum called spin. (The only known spin-0 fundamental particle is the Higgs boson.)

Quantum spin is definitely not the same thing as classical rotation. The particles are so small that it would be impossible for them to have as much angular momentum as they do, due to limits such as the speed of light.

My understanding is that a single composite particle like an atom can rotate classically, though the amount of angular momentum involved is very small.

Nonrotating

It’s nearly impossible for natural processes to create an object that doesn’t rotate or change its orientation at all. There would have to be some sort of feedback effect that locks the object into a fixed orientation with respect to a distant star, or galaxy, etc.

I suppose it’s possible that a very light object moving toward or away from a star could happen to be shaped enough like a solar sail or parachute to keep it nonrotating for a while.

As for human-made nonrotating objects, I can’t think of very many. Gyroscopes and telescopes come to mind. Though gyroscopes usually have spinning components, they can also have nonrotating components.

An Earth-based telescope can use a so-called equatorial mount that keeps the business end of it nonrotating, so that it can do a long exposure of a distant space scene.

A space telescope might be the best example of a nonrotating object. Though it will surely have spinning components like reaction wheels, as a whole it will spend a lot of time rock-steady, as it observes objects beyond the Solar System.

So, I guess if you want to make collection of nonrotating things, you could start with a telescope and a Higgs boson.