A wall with no size in it

There is a way to find out whether a body is a single rock or a heap of loose fragments without going anywhere near it. Watch it turn.

A body held together by its own gravity and nothing else cannot spin arbitrarily fast. Set the centrifugal acceleration at the equator equal to the surface gravity, $\omega^{2}R = GM/R^{2}$, put in $M = \tfrac{4}{3}\pi\rho R^{3}$, and everything about the body cancels except its density:

$$P_{\text{crit}} = \sqrt{\frac{3\pi}{G\rho}}.$$

Three and a third hours for water ice. About 2.3 hours for the two grams per cubic centimetre that a typical asteroid’s bulk density comes out at. And — this is the part that makes the criterion useful — no size anywhere in it. A boulder and a mountain of the same density have the same critical period.

The derivation above is the crudest possible version and it is worth saying what it leaves out, because the leaving-out is where the interesting corrections live. It treats the body as a sphere, which no small body is; it compares accelerations at a single point rather than solving for the stress throughout; and it assumes the material has no strength whatever, neither tensile nor frictional. A proper treatment for a cohesionless pile with internal friction — the same continuum mechanics a civil engineer uses for a heap of sand — gives a critical period that depends on the body’s elongation and on the friction angle, and for realistic values it lands within twenty per cent of the naive answer. The naive answer is the right one to think with.

The same arithmetic as a Roche limit, turned inward #

This is not a new criterion. It is the Roche limit with the roles rearranged. There, a satellite is pulled apart because the tide across it beats its own gravity; here, a body is pulled apart because the rotation at its equator beats its own gravity. Both are a comparison between a disruptive acceleration and $GM/R^{2}$, and both come out as a statement about density with the size divided out.

There is a satisfying corollary. A body orbiting at the Roche distance and rotating synchronously is at both limits at once, which is not a coincidence: synchronous rotation at the Roche distance means the centrifugal term at the equator and the tidal term are the same size, because both are set by the same orbital frequency and the orbital frequency at the Roche distance is set by the primary’s density, which equals the satellite’s.

There is a third member of the family, and it is the one that made the connection historically. A liquid mass held together by surface tension and spun up passes through a sequence of equilibrium figures — an oblate spheroid, then a triaxial one, then a pear, and then it separates. The sequence was worked out in the nineteenth century for a rotating fluid, revived for rotating self-gravitating bodies, and applied to asteroids only when it became clear that asteroids behave more like fluids than like rocks. The observed shapes of large rubble piles sit on that sequence, which is a stronger claim than the barrier alone: not merely that they cannot spin faster, but that they take the shapes a strengthless body would take.

What the wall is a measurement of #

If asteroids were monolithic rocks, the barrier would not exist. Granite has a tensile strength of some tens of megapascals, which is enormous compared with the stresses in a small body: a ten-kilometre rock could be spun in minutes without cracking. The fact that nothing above a couple of hundred metres is found spinning faster than the cohesionless limit is therefore a statement that nothing above a couple of hundred metres has any meaningful tensile strength.

That is the modern picture: most asteroids above that size are rubble piles, gravitational aggregates of fragments produced in collisions that shattered a parent body and then reaccumulated. They have porosities of twenty to fifty per cent, they have no through-going strength, and they respond to a push by rearranging rather than by cracking.

The reasoning that gets there is worth laying out, because it is an inference from an absence and those are easy to overstate. The argument is not that a fast-spinning large asteroid would be impossible; it is that a large asteroid with strength would have no reason to avoid fast spins, and that spins are being changed all the time by the torque described below. A population under a slow random walk in spin rate, with no barrier, would fill the space above the line. It is empty. The emptiness is the measurement. The bodies below the barrier’s reach — the fast rotators, some turning in minutes — are all small. Nothing about gravity changes at two hundred metres. What changes is that a body that small can plausibly be a single coherent fragment, with real tensile strength, and can therefore be spun as fast as anything has spun it. The barrier is a size threshold for a change in what a body is.

One further reading of the same figure is worth having. The barrier’s position depends on density, so a population with a range of densities should show a range of barriers — and it does, faintly. The fastest large rotators cluster around 2.2 hours, which corresponds to a bulk density near 2.2 grams per cubic centimetre; the few that go a little faster are of a spectral class associated with metal-rich material, and a metal-rich rubble pile is a denser one. The wall is therefore not one wall but several, sorted by composition, and where a body sits against its own class’s wall is more informative than where it sits against the average one.

Not quite zero cohesion #

The clean statement has an interesting complication. Fine regolith is not perfectly frictionless: van der Waals forces between micron-scale grains give a small cohesive strength, of order tens of pascals — negligible for a mountain and not negligible for a body whose self-gravity is measured in micro-gravities.

A cohesion of even a few pascals lets a small rubble pile spin somewhat faster than the barrier, and the excess grows as the body shrinks, because cohesive stress is independent of size while gravitational stress scales as $\rho^{2}R^{2}$. So the barrier is not a step but a corner, sharp at large sizes and softening below a kilometre. Several bodies a few hundred metres across are found just inside it, and the standard reading of those is not that they are monoliths but that they have a little cohesion.

The most direct test came from a spacecraft. When a probe touched a rubble-pile asteroid to collect a sample, the surface gave way far more than expected — the material behaved like a fluid of near-zero strength, and the arm sank in tens of centimetres before firing. A body whose surface offers essentially no resistance is one whose cohesion is at the low end of the estimates, and it makes the spin barrier’s sharpness easier to understand.

The same visit produced a second piece of evidence of a kind nobody had planned for. The spacecraft’s own gravity measurements, made by tracking it in orbit for months, gave a bulk density around one and a quarter grams per cubic centimetre against a meteorite density of about two and a half — a porosity of fifty per cent, half of the body’s volume being empty space. A body that is half empty is not a rock with cracks in it. It is a pile.

What spins them up #

A barrier is only interesting if bodies are being pushed against it, and something is. The same thermal recoil that drifts an asteroid’s orbit also applies a torque, because an irregular body re-radiates asymmetrically about its spin axis. The torque is tiny and it is secular: over a few million years it can double or halve a small asteroid’s spin rate.

That gives the population a dynamic rather than a static explanation. Bodies are driven toward the limit, reach it, and then something has to give — mass is shed from the equator, the body reshapes, its moment of inertia changes, and the spin drops back. Several observed asteroids have equatorial ridges that look exactly like the result: a bulge of material that migrated to the equator and stayed there, in some cases with a small satellite in orbit that appears to have been shed from it.

This is now the leading explanation for why a substantial fraction of near-Earth asteroids are binaries. A rubble pile driven past its spin limit does not explode; it sheds, and the shed material can end up in orbit.

What the barrier is used for #

Bulk density from a rotation. Turn the criterion round — the same manoeuvre that turns a line width into a distance, where a dynamical quantity that was a nuisance becomes the measurement. A body observed rotating just under the barrier must have at least the density that puts the barrier there, so a light-curve period gives a lower bound on bulk density with no mass measurement at all, which is worth having for an object whose orbit is known to nine figures and whose mass is not known at all. For the fastest large rotators this is the only density estimate available.

Interpreting the size distribution. The steady-state cascade that grinds a population toward a power law assumes a strength law, and the strength law changes character at exactly the size where the barrier stops applying. Below it, bodies are strength-dominated and hard to break; above it, gravity-dominated and easy. The minimum in fragmentation energy sits near a few hundred metres, which is the same scale, and for the same reason.

Deflection. If a hazardous object has to be pushed, it matters enormously whether it is a rock or a heap. A rock can be given an impulse; a heap absorbs it, redistributes it internally, and may simply reshape. The kinetic-impact test flown to a small binary asteroid produced a momentum transfer several times the incoming momentum, because the impact excavated and ejected a great deal of material — behaviour characteristic of a weak, porous target and not of a monolith.

Impact history. A porous, strengthless body absorbs an impact very differently from a monolith: the shock is damped within a few crater radii rather than propagating, so a rubble pile can survive impacts that would shatter a rock of the same size. That is part of why the population is dominated by piles — being a pile is, in the collisional environment of the belt, a survival strategy — and it also means the crater counts used to date surfaces mean something different on a pile than on a solid body.

What was actually measured #

Nothing in the hero figure is a direct observation of a spin. What is measured is a light curve: an asteroid is an irregular object reflecting sunlight, so its brightness varies as it turns, and the period of that variation is half the rotation period for a body with two long sides. Tens of thousands of asteroid light curves have been collected, mostly by small telescopes, and the barrier emerged from the accumulation rather than from any single observation.

A period from a light curve is also an ambiguous quantity in a specific way. A body with two long sides gives two brightness maxima per rotation, so the light-curve period is half the spin period; a body with an odd shape may give three. Resolving the ambiguity needs either a well-sampled curve with unequal maxima or a shape model, and the early asteroid catalogues contain a scattering of periods that are wrong by exactly a factor of two. That matters here more than it usually would, because a factor of two is the difference between sitting at the barrier and sitting well clear of it.

Two selection effects have to be kept in view. Slow rotators are under-represented, because a survey that observes a field for a few hours cannot measure a period of days. And very fast rotators among large bodies would be easy to find and are not found, which is the part of the claim that carries the weight — the barrier is an absence, and an absence is only evidence if the search would have seen the thing.

The densities that go into the barrier’s position come from elsewhere entirely: from the orbits of small satellites where a body has one, from the deflection of a passing spacecraft, and in a few cases from the mutual perturbations of two asteroids that pass close to each other. Those give masses; a shape model from radar or from light curves gives a volume; and the quotient is a bulk density that is systematically lower than the density of the meteorites believed to come from the same bodies. That difference is the porosity, and it is a second, independent piece of evidence that these objects are piles.

The other end of the size range #

The barrier applies to bodies held together by gravity and nothing else, and there is a size above which gravity stops merely holding a body together and starts deciding its shape.

Below a few hundred kilometres a body’s own gravity is too weak to overcome the strength of its material, so it keeps whatever shape a collision left it with — irregular, lumpy, in some cases two lobes touching. Above that size the material yields under its own weight and the body relaxes towards an equilibrium figure, which for a non-rotating body is a sphere and for a rotating one is a flattened spheroid.

The threshold is not sharp and it depends on composition: icy bodies relax at smaller sizes than rocky ones, because ice is weaker, so the smallest round objects in the solar system are icy and the largest irregular ones are rocky. The transition sits near two hundred kilometres for ice and four hundred for rock.

That gives a second, independent inference about interiors from a shape. A body large enough to be round has relaxed, which means it was at some point warm enough or weak enough to flow — so roundness is a statement about a body’s thermal history rather than only about its size. And a body that is round but not in the equilibrium figure its rotation demands is a body that relaxed at some earlier rotation rate and has since been spun down, which is a fossil of a spin state.

Between the two thresholds is the whole population this essay is about: too large to have strength and too small to have relaxed, held in a shape a collision gave them by a gravity that is only just sufficient. That is a narrow window in the abstract and it contains almost every small body in the solar system.

The window is narrow in size and wide in number, which is the usual arrangement for a steep size distribution.

The barrier is a density comparison, so it is worth reading at a wider spread of densities and around a different primary.

Where the ladder goes #

The barrier is the simplest of a family of statements about strength inferred from dynamics. The next rungs are about what happens on either side of it: how a body that is a pile responds to being spun past the limit, what shapes it takes on the way, and why so many small asteroids share the same top-shaped figure with an equatorial ridge.

There is also a thread that runs the other way, into how these bodies formed. A rubble pile is a record of a collision, and the collisional history of the belt is written in the families and in the crater counts on the surfaces the fragments made. The spin barrier is the present-day cross-section of that history: whatever happened, it left almost nothing large that is still in one piece.