A tumble that measures how stiff rubble is

A small asteroid is a pile of rubble held together by its own weak gravity, and it cannot spin faster than a limit set by its density alone, about 2.2 hours. That limit is the clearest evidence that most asteroids larger than a few hundred metres are rubble piles rather than solid rock: thousands of measured spin periods pile up against it, and almost none cross it. A spin limit tests the strength of a body’s interior from the outside. It says the rubble is weak in tension.

There is a second test that looks at the inside from the outside, and it says something different. Most asteroids spin about a single axis — their axis of greatest moment of inertia, the shortest axis of an elongated body — and their light curves repeat with a single period. A minority do not. They tumble: the body turns about one axis while that axis precesses round the direction of the angular momentum, and the light curve beats between two periods that never come back into step. Such motion is called non-principal-axis rotation, and it does not last forever. A tumbling body flexes internally as it turns, the flexing dissipates energy, and a body that loses energy while keeping its angular momentum ends up spinning about its axis of greatest inertia.

How long that takes is set by the body’s interior: how much it flexes for a given stress, which is its rigidity, and how much energy each flex dissipates, which is its quality factor. An asteroid that is still tumbling has not finished settling since whatever disturbed it, and the boundary between the asteroids that tumble and those that do not is, in effect, a measurement of the stiffness and lossiness of rubble.

How long a tumble lasts #

Burns and Safronov worked out the damping time in 1973. A body of radius $R$ and density $\rho$ tumbling at angular rate $\omega$ is stressed by the varying centrifugal forces, which scale as $\rho\,\omega^2 R^2$; it strains by that stress over its rigidity $\mu$; and each cycle of strain loses a fraction $1/Q$ of the stored elastic energy. Balancing the rate of energy loss against the rotational energy that has to be lost gives

$$\tau \approx \frac{\mu Q}{\rho\, K_3^2\, R^2\, \omega^3} ,$$

where $K_3^2$ is a dimensionless factor for the body’s shape. Harris calibrated the expression in 1994 for asteroids, taking the rigidity and quality factor of solid rock, and it reduces to a convenient rule: the damping time in billions of years is the cube of the rotation period in hours, divided by the square of the diameter in kilometres, divided by about five thousand.

The dependence on the period is the striking part. A body’s internal stresses grow as the square of its spin rate and the rate at which it loses energy grows faster still, so a fast rotator settles quickly and a slow one hardly at all: ten times the period means a thousand times the damping time. A kilometre-sized asteroid turning every ten hours would settle in a couple of hundred million years if disturbed. The same asteroid turning every hundred hours would still be tumbling now if it had been disturbed when the solar system formed. Larger bodies settle faster, because their stresses are larger: at a fixed period, a ten-kilometre body damps a hundred times faster than a one-kilometre one.

Where tumblers should be found #

On a chart of rotation period against diameter, the damping time is constant along lines of slope two-thirds.

A body above a given line has not had time to settle since it was last disturbed that long ago. Above the line for the age of the solar system, a tumble started at any time in history would still be going on. Below the ten-million-year line, a tumble would be gone almost as soon as it started, unless something keeps renewing it.

The observed tumblers fall where the formula says they can. Toutatis, a 2.5-kilometre near-Earth asteroid imaged by radar during its close approaches, rotates and precesses with periods of five and seven days; Apophis, the asteroid that will pass within the orbits of geostationary satellites in 2029, tumbles with a period of about thirty hours; ʻOumuamua, the first interstellar object seen passing through the solar system, varied in brightness by a factor of ten on a period of about eight hours in a way no single-axis spin could reproduce. All three lie above the line for the age of the solar system. Surveys of asteroid light curves find tumblers concentrated in the upper left of the chart — small, slow rotators — and nearly absent below the lines for a few hundred million years, which is where the formula says they cannot survive.

A boundary that measures the stiffness of an asteroid #

The calibration used rock: a product of rigidity and quality factor, $\mu Q$, of about $5 \times 10^{11}$ pascals. Rubble is not rock. A pile of boulders bound by gravity, with a quarter or more of its volume empty, should be much less rigid than solid stone, and friction between grains should dissipate much more energy per cycle — both of which make $\mu Q$ smaller and the damping faster.

The boundary moves as the cube root of $\mu Q$. A rubble pile a thousand times softer and lossier than rock would push the boundary for kilometre-sized bodies from about a day to about ten days, and tumblers with periods of a day or two would be rare. They are not: the observed distribution of tumblers fits a boundary near the rock calibration. Taken at face value, the tumblers say that small asteroids dissipate rotational energy about as slowly as solid rock would.

That is a surprise, and it is the most interesting thing the tumblers measure. Several resolutions have been proposed. A rubble pile’s rigidity may be much less than rock’s while its quality factor is much higher — energy is lost mostly at contacts between grains, and a pile with few, large contacts may be surprisingly elastic. The effective $\mu Q$ may depend on the size of the body and the amplitude of the tumble, since a gently flexed pile behaves differently from a strongly flexed one. Or the tumblers may be younger than the formula assumes, disturbed more recently than a pure damping argument allows. The tumblers set a constraint that a model of a rubble pile’s interior has to meet, and it is not obvious how a loose pile meets it.

A rounder body takes longer to settle #

The shape enters through the factor $K_3^2$, which measures how much a body’s own precession flexes it.

A nearly spherical body hardly changes its stress pattern as it tumbles, because every orientation looks alike; an elongated one changes greatly. The damping time is inversely proportional to the shape factor, which spans about a factor of ten between round and strongly elongated bodies. Since the shape of a tumbler can be estimated from the amplitude of its light curve, the shape correction can in principle be applied body by body, and an honest comparison of tumblers with the boundary has to do so.

Measured tumblers against their settling times #

The individual bodies make the arithmetic concrete.

Every one of them could have been set tumbling at any time in the solar system’s history and still be tumbling. ʻOumuamua, small and fast, would need about eleven billion years; Apophis fifty; Toutatis nearly two hundred. Mathilde, fifty-three kilometres across and visited by the NEAR spacecraft in 1997, is the outlier: a large body that should settle quickly, but which rotates once in seventeen days, so slowly that its damping time is longer than the age of the solar system even at its size. Its light curve has been read as evidence of a tumble, and the formula allows it only because of its extraordinary slowness.

The comparison has a second use. For ʻOumuamua, which arrived from interstellar space, the tumble is a clue to its history: a body that has been tumbling for billions of years was set tumbling long ago, perhaps in the event that ejected it from its own planetary system, and has drifted between the stars ever since without anything to settle it.

A light curve that never repeats #

What marks a tumbler observationally is its light curve.

An elongated asteroid spinning about one axis presents its long side and its end alternately, and its brightness rises and falls twice per rotation in a pattern that repeats exactly. The same body tumbling presents a sequence of orientations that does not repeat, because two motions with unrelated periods are combined: the rotation about the body’s own axis and the precession of that axis round the angular momentum. Its light curve beats, and no single period folds it into a clean curve. Identifying a tumbler requires observations over many rotations and a fit with two periods; with too few nights a tumbler looks like a noisy single-period rotator, and the population of known tumblers is therefore a lower limit on the real one. Radar imaging, which sees the shape directly, settles the matter for the bodies that come close enough. The large surveys now measuring light curves for hundreds of thousands of asteroids sample each one sparsely, a few points a night over months, and finding the tumblers among them is a statistical exercise in noticing which light curves refuse every single period offered.

The same settling, from spacecraft to planets #

The physics of the tumble is the physics of any spinning body that can lose energy but not angular momentum, and it appears at every scale. A spacecraft with flexible antennas or sloshing propellant migrates to rotation about its axis of greatest inertia for the same reason an asteroid does, and the early satellites that were spun about their long axes turned end over end within hours because their whip antennas dissipated energy far faster than any rock. A rigid body spun about its intermediate axis does not settle at all but flips end over end periodically, and a tumbling asteroid whose energy lies near that intermediate state wanders between the two families of tumbling motion before it settles.

Comets are tumblers of a special kind, because they are continually disturbed. Gas escaping from their surfaces as they approach the Sun pushes on them off-centre, and the same jets that push a comet also turn it, exciting non-principal rotation faster than their weak, porous interiors can damp it. Comet nuclei are, as a class, far more often found tumbling than asteroids of the same size, and the difference measures how much more strongly they are driven, not how much more slowly they settle.

At the other extreme, the Earth itself has a free wobble — its axis of rotation moving round its axis of greatest inertia by a few metres at the pole, with a period of about 433 days. The formula for an asteroid, applied to the Earth with the Earth’s size and rigidity, gives a damping time of decades, and the wobble should indeed have died away within a human lifetime unless something keeps exciting it. Its persistence is a measure of how strongly the oceans and atmosphere drive it; its damping, inferred from how its amplitude responds, is a measure of the mantle’s dissipation at a period of a year. The tumble of a kilometre-sized asteroid and the wobble of the Earth’s pole are the same phenomenon, read on bodies whose interiors differ by everything except the law that governs them.

What disturbs a settled asteroid #

A tumble needs a cause. The most obvious is a collision: an impact that does not shatter the body still delivers angular momentum off-axis and sets it tumbling. For small asteroids collisions of the right size are rare, every few hundred million years for a kilometre-sized body, which is comparable to the damping times of fast rotators and much shorter than those of slow ones — consistent with tumblers being found mainly among the slow.

The collisions that matter are not the catastrophic ones. An impact that shatters a body resets everything; the tumbles come from the far more numerous impacts too small to break the target but large enough to change its spin, by projectiles a few per cent of its size. How often those occur depends on the size distribution of the small bodies in the main belt and on the strength of the target, and the size at which a body is easiest to break — a few hundred metres, where neither material strength nor gravity holds much — is also roughly the size at which impacts excite tumbles most easily. That puts the most frequent excitation and the slowest damping among the same small, slowly rotating bodies.

A second cause is the YORP effect, the slow torque that sunlight exerts on an irregular body as it absorbs light and re-emits heat — the rotational counterpart of the push that moves an asteroid’s orbit. YORP can spin a small asteroid up to its breakup limit or spin it down towards very slow rotation, and calculations show that a body spun down far enough loses its stable spin altogether and begins to tumble. The slow rotators are thus both the bodies that damp most slowly and the ones that YORP is most likely to have disturbed, and the two effects together make the slow, small corner of the chart the natural home of tumblers.

Close encounters with planets provide a third disturbance. Apophis will pass the Earth in 2029 at about a tenth of the Moon’s distance, and the Earth’s tides will change its rotation measurably — in rate, in the direction of its axis, and in its tumble. Those changes will be watched by radar and by telescopes around the world, and the response of a known tumbler to a known torque is the closest thing to a laboratory experiment on an asteroid’s interior that the solar system offers.

What the formula leaves out #

The damping time comes from treating the body as a uniform elastic solid with a single rigidity and quality factor. A rubble pile is a granular medium, and whether its response to slow, small oscillating stresses can be described by two numbers at all is part of the question. The shape factor is approximate, the density and dissipation are assumed uniform, and the damping is treated as starting from a fully excited tumble; a slightly excited tumble damps at a different rate from a strongly excited one, and the formula is an order-of-magnitude guide over the whole range.

The measured tumblers are taken with single representative periods, where real tumblers have two and the damping depends on both. Their diameters, from brightness and assumed reflectivity, are uncertain by tens of per cent, which moves the damping times by more.

Still open: what is inside a rubble pile #

The spin limit says small asteroids are loose; the tumblers say their interiors dissipate energy slowly, as rock does. Both are inferences from the outside. The spacecraft that have visited rubble-pile asteroids — Itokawa, Ryugu, Bennu, Dimorphos — have found surfaces of boulders and gravel, bulk densities implying a third or more of empty space, and in Bennu’s case a surface so loose that the sampling arm sank into it as if into a ball pit.

The interior remains unseen. A radar sounder carried to a binary asteroid would see the internal structure directly, and the rotation of a moonlet round a rubble pile responds to the pile’s internal dissipation through the tides it raises. Whether the slow damping the tumblers imply comes from a rubble pile that is stiffer than it looks, from large monolithic cores inside loose mantles, or from a dissipation that depends on the size of the tumble, is a question the next missions can answer by doing what the tumblers do from a distance: flexing an asteroid and watching how long it rings.