An island a moon could not stay on

Nearly every large moon in the Solar System keeps one face to its planet. The Moon does so with a small rocking that can be measured to fractions of an arcsecond, and that rocking has been used to weigh interiors. The synchronous state is such a general outcome of tidal despinning that a moon’s rotation is rarely even measured before it is assumed. Hyperion, a porous, irregular moon of Saturn some 360 by 266 by 206 kilometres, is the exception, and it became a famous one when Jack Wisdom, Stanton Peale and François Mignard predicted in 1984 that it should be tumbling chaotically.

The account of why usually stops at a single picture, and the essay on the origin of chaos has given it: a moon’s spin-orbit resonances are islands in a phase plane, and for a body as lopsided and as eccentric in its orbit as Hyperion those islands overlap and dissolve into chaos. That is right as far as it goes. But it describes a moon held in the plane of its orbit, spinning about an axis perpendicular to that plane. A real body can tilt. The authors of the 1984 paper knew this and made a second calculation that is less often repeated, and it says something more than the first. An island remains in the plane for Hyperion to sit on, and it cannot sit there.

The story then took a further turn that neither calculation had anticipated. When Voyager and then Cassini looked closely, Hyperion was not tumbling in the slow, wandering way the theory suggested. It was turning at more than four times its orbital rate. So there are three levels of description here, each more honest than the one before, and each changes what is being claimed.

The plane #

Hold the moon’s short axis perpendicular to the orbit, and only one angle remains: that of the long axis, $\theta$. The planet’s tide pulls that axis towards itself with a torque that depends on the asymmetry $\omega_0^2 = 3(B-A)/C$ and on the distance and direction of the planet at that moment:

$$\ddot\theta = -\frac{\omega_0^2}{2}\left(\frac{a}{r}\right)^3\sin 2(\theta - f).$$

The way to see the behaviour is to sample the angle and the spin rate once an orbit, at pericentre, for a number of different starts. Each regular motion appears as a closed loop or a line. Each chaotic one fills an area.

A chaotic region covers nearly the whole area between about half the orbital rate and two and a half. There is a fragmentary line above it, the fast spin that is too rapid for the tide to do anything but nudge it. And there is one substantial island at the centre, and it is worth spending a moment on, because it does not sit where it would be expected to be.

That island is the synchronous state, the one the Moon and every other regular moon occupies. It is centred on a spin rate at pericentre of about 0.46, nowhere near 1. The long axis returns to the same position once every orbit, so the average spin rate over the orbit is exactly the orbital rate. But the moon is lopsided enough, and the orbit eccentric enough, that it swings through a very large forced libration each time round. At pericentre, the moment of this sampling, it happens to be turning at under half the average speed, and it speeds up again later on. So the section contains an island of synchronous rotation, a genuine and stable place to be in the plane. A moon with less asymmetry sits closer to where one would expect.

The tilt #

Now drop the restriction. The full motion has three angles and three rates, and the tide can exert torques about every axis:

$$A\dot\omega_1 - (B - C)\omega_2\omega_3 = \frac{3GM}{r^3}(C-B)\,\hat{r}_2\hat{r}_3,$$

with two more equations of the same form in cyclic order. In the plane the moon’s short axis points along the orbit normal, and all components with subscripts 1 and 2 vanish. This is the same set of equations as those that give the gravity-gradient stabilisation of a satellite. On a circular orbit, for a body with $A$ below $B$ below $C$, the synchronous state is stable to tilt in every direction, the standard engineering result. Nothing says that it must remain stable with an eccentric orbit and a large forced libration.

Test it directly. Start the moon at the centre of the planar island, tilt it by a tenth of a milliradian, and follow what happens.

The result is unambiguous, and it is the essence of what the plane conceals. The planar island is an island only for a moon that cannot tilt. Any tiny departure from the plane, which nothing could have prevented, grows exponentially. After a little over a year, the moon’s spin axis has gone anywhere, and it is tumbling in three dimensions. Wisdom and his colleagues called this attitude instability. In their section, the chaotic zone and the islands inside it were all attitude-unstable, so that a moon entering the chaotic region of the plane would not just wander in spin rate but would very soon be off the plane altogether. So the plane is not a good description of what happens at all. It is a slice through a much larger space, and the slice is itself unstable.

The instability is a kind of parametric resonance. The moon’s forced libration, large in Hyperion’s case, modulates the moments acting on any small out-of-plane motion, and it pumps that motion at a frequency close to its natural frequency. It is a familiar process, seen in a child’s swing being pumped by standing and squatting.

How lopsided a moon has to be #

A sphere has no tide to speak of and no instability either. Stretch it gradually into Hyperion’s shape, and there has to be a point at which the alignment gives out.

The threshold is a real result, and it explains why Hyperion stands alone. The figure says that the synchronous state stays stable for most shapes at this eccentricity, and it is lost only at a high asymmetry. Most regular moons are close to spherical and have asymmetries measured in thousandths, and they move on orbits with eccentricities much lower than a tenth, so they are far from any such threshold. Hyperion got its eccentricity from a 4:3 orbital resonance with Titan, which maintains it against the damping effect of tides, and it got its odd shape from a violent history. Its combination of the two is unusual in the Solar System, and few other moons are candidates for this state. Two small moons of Pluto, Nix and Hydra, were found in 2015 to be in fast non-synchronous rotation, for reasons that seem to involve the binary nature of their primary, and moons of that kind may be more varied than this single case suggests. The Mercury resonance, at three to two, shows another way out of synchrony, and Saturn’s small inner moons show how differently orbits and spin can combine.

An eccentricity that cannot be damped #

An obvious objection is that the orbit should have circularised. Tides raised in a satellite by its planet drain the eccentricity of the orbit, as they have for most regular moons, and on a circular orbit the figure above shows the synchronous state to be safe. The reason that has not happened is Titan. Hyperion’s period is locked at three to four with Titan’s, and the resonance protects it in two ways. It keeps Hyperion from approaching Titan closely, since conjunctions always happen near the apocentre of Hyperion’s orbit, and it transfers angular momentum between the two orbits so that the eccentricity is continually pumped as Titan’s orbit evolves.

Resonant orbits of this kind are common, and a good part of their interest lies in the way an orbit can be held in a state that dissipation would otherwise remove. For Hyperion this holds the eccentricity near a tenth for an indefinite time, so the conditions for attitude instability do not relax. It is a further instance of a rotation that has been forced by something outside the body, the same kind of thing that drives the Moon’s physical libration in longitude, though on a much larger scale. The Moon’s forced swing is a few minutes of arc. Hyperion’s is about as large as a swing can be while still returning.

The chain of cause is then unusually long for a statement about a day. A massive neighbouring moon fixes a resonance, and the resonance fixes an eccentricity. The eccentricity with the body’s shape drives a forced libration, and that forced libration pumps any tilt. Without Titan, Hyperion would probably be an ordinary synchronous moon with one battered face towards Saturn. It would be no less interesting to look at, and its rotation would have had nothing to say about phase space.

It also adds one more reason to wonder how the moon’s shape came about. A larger Hyperion that lost its outer layers in a collision is a common suggestion, since such a body in resonance would keep its orbit while its shape and mass changed around it. If so, then the moon was probably not always in the attitude-unstable part of the diagram, and its chaotic rotation would have begun at a definite moment, with a blow.

What the spacecraft saw #

If that were the end of the argument, the picture would be complete: Hyperion despun by tides into the chaotic zone, off the plane and tumbling slowly and erratically. That was the prediction of 1984. Voyager 2’s images from 1981 were too sparse to test it directly. Ground-based light curves taken in 1987, analysed by Jim Klavetter, did show the aperiodic behaviour expected.

Then the Cassini spacecraft flew past in 2005 and 2006 and joined its images with those from Voyager, giving a rotation state with a clear rate. Hyperion was turning at about seventy degrees per day, about 4.2 times its orbital rate, with its spin axis near its long axis and precessing about it. Twenty-five years apart, the two sets of measurements found much the same state. Matthew Harbison, Peter Thomas and Philip Nicholson’s analysis in 2011, using the full Euler equations, confirmed that this rotation is still chaotic with a short Lyapunov time, while being stable in its rate, in the sense that the rate does not wander far on short timescales.

The figure shows the reason this state can last: at four times the orbital rate, the moon has so much spin that the tide’s torque is a small modulation on it. Its rate has nowhere to go quickly, whereas the synchronous start, with little angular momentum to begin with, is thrown about readily. So there are two kinds of unruly rotation. A slow one is what the theory had described, and a fast one is what the moon was actually doing.

What makes this subtle is the difference between a rate and a direction. A spin rate can stay steady while the orientation is unpredictable.

The figure explains why nobody should simply say “Hyperion’s rotation is chaotic” and leave it there. Spinning about the short axis, the axis of greatest moment and the lowest-energy state for a given angular momentum, a fast Hyperion is regular on these timescales, and its motion could be forecast. Spinning near the middle axis, the moon suffers the same instability as a wingnut in free fall, and the tide makes it chaotic very quickly. Near the long axis, where it was actually observed, the behaviour is in between. The practical upshot is that a measured spin rate says less about predictability than a measured spin axis does, and this moon’s axis has been measured for a very few epochs only.

Why it is still spinning #

Hyperion’s fast spin has a further implication that is worth stating, and it makes the object stranger still. It should not have kept that spin.

The usual tidal calculation predicts that a moon at this distance despins within tens of millions of years at most. It is one reason the 1984 authors expected it to have reached the chaotic zone, as any primordial fast rotation should long since have decayed. So what is sustaining the rotation now? A porous body with a density of about half that of water may dissipate far less than expected. That would extend the time, though perhaps not by the factor needed. A recent impact is possible, and Hyperion’s craters and its sponge-like surface suggest a battered history, though an impact would have to have happened recently. Also possible is a chaotic route by which the moon drifts in and out of fast states, trading energy with the orbit and spending much of its time at high rates, as some integrations have suggested. No one yet has a demonstrated answer. Hyperion’s most basic rotational fact, that it is spinning at about four times the orbital rate, remains unexplained.

What a section leaves out #

The more general moral concerns the kind of picture this subject relies on. A section through phase space is a marvellous device, and the same device shows Kirkwood’s gaps and the edge of order in an asteroid’s motion. It shows just what is on the slice. If the system can leave the slice, then a stable point on it may be unstable in the full space, and the picture will be silent about that. So a fixed point on a two-dimensional section should be read as a candidate for a stable state and not as a proof of one. The same point arises in other settings where a reduced model has been stabilised by construction, including the planar restricted three-body problem and the rigid-body behaviour of a satellite idealised as a point in a plane.

There is a pleasing connection between small and large here. An unstable rotation on a moon the size of a large asteroid has the same mathematical root as the tumbling of a thrown tennis racket and as the flip that surprised the engineers of the first satellites. The tide adds only the time dependence that drives the instability. That a body’s rotation should be sensitive to a tenth of a milliradian of tilt is one of those facts that seems unreasonable, until a calculation is run and shows it directly.

What the picture cannot show #

The body in these figures is a uniform triaxial ellipsoid on a fixed Kepler orbit. Real Hyperion has an irregular shape, so its higher-order gravitational moments may matter, and its orbit is forced by Titan with a variable eccentricity and a precessing pericentre that have been ignored here. Its true moments of inertia cannot be assumed from a uniform-density model, as a porous body is unlikely to be uniform, and the published estimates based on its precession are close to the numbers used but are uncertain.

No tide acts in any of the integrations. The fast-spin runs are started from one or two chosen orientations and can’t represent everything accessible at that spin rate. The divergence figure, in particular, gives one case at each axis, and other starts near the long axis show substantially faster or slower divergence. The despinning estimate assumes a sphere, a uniform body and a constant ratio, and the actual dissipation in a chaotically rotating porous moon is not known. The section uses a finite number of orbits, so that a small island or a narrow chaotic layer could be missed, and the closed loops might be sticky chaotic orbits on a long enough timescale.

Still open: how the spin got there #

Nothing in the rotation dynamics by itself says how Hyperion came to be spinning this way, or for how long it has been doing so. A few more epochs of observation would help, since the spin state’s change between epochs is the cleanest measure of the torque actually acting. Cassini’s mission is over, and no spacecraft is planned for Saturn’s outer moons that will return to look. Telescopes on the ground can measure its light curve, which depends on the orientation but not uniquely. A long, densely sampled set of such light curves, fitted to a dynamical model of the type used in these figures, could constrain the moments of inertia and the orientation well enough to determine whether the moon is near a chaotic or regular region of its fast state. Such constraints are the only way to learn its interior without visiting, and they would say whether the unusually low density comes with an unusually weak tide, which is the most direct proposal for why the moon has not yet finished slowing down.