A surface that slides before the body sheds

A rubble pile cannot spin faster than a period set by its density alone: at that spin the material at its equator weighs nothing, and anything faster flings it off. The barrier is visible in the rotation periods of thousands of small asteroids, which crowd up against it and do not cross. What happens when a body reaches it was the subject of the essay on fission: material is shed from the equator, or the body splits in two, and which of the two happens depends on how the body’s strength is distributed.

Both of those are endings. This essay is about what happens on the way, in the interval between a body spinning comfortably and one at its limit, and the answer is that a great deal happens. The surface of a rubble pile is loose material held in place by gravity and a little friction, and friction gives way at a definite slope. On a spinning body the slopes grow with the spin, and they grow fastest not at the equator, where the material eventually leaves, but at the latitudes on either side of it. The surface starts to flow long before anything is shed — and it flows towards the equator, which is where the ridges of small asteroids are.

Where a spinning sphere is steep #

On a sphere that is not spinning, the pull of gravity points straight at the centre and is perpendicular to the surface everywhere. Nothing slides, because nothing is a slope. Spin adds a centrifugal acceleration directed away from the axis, and the sum of the two is no longer perpendicular to the surface except at two kinds of place: the poles, where the centrifugal term vanishes, and the equator, where it points straight outward along the same line as gravity. Everywhere else the combined pull has a component along the surface, directed towards the equator, and the surface is effectively tilted.

The tilt on a sphere is

$$\tan\theta = \frac{s\cos\varphi\sin\varphi}{1 - s\cos^2\varphi},$$

where $\varphi$ is the latitude and $s$ is the spin measured as the square of its ratio to the critical spin. At slow spins the slope is greatest at mid-latitudes and small everywhere: at thirty per cent of critical, ten degrees at latitude forty. As the spin rises the steepest band moves towards the equator and grows rapidly, because the denominator — the part of gravity that the centrifugal term has not yet cancelled — shrinks there. At ninety-seven per cent the slope reaches seventy degrees at latitude ten, ten degrees from an equator where it is still, by symmetry, zero.

The figure’s shaded band is the angle of repose, the steepest slope a heap of loose rock can hold: thirty to forty degrees, nearly independent of the grain size. It is crossed by the steepest part of the surface at about eighty per cent of the critical spin.

A sixth slower than the limit #

In rotation periods the interval is easy to state. For the density of the two carbonaceous asteroids that spacecraft have sampled — Bennu and Ryugu, both about 1.19 grams per cubic centimetre — the equator would lift off at a period of 3.03 hours, and the steepest slope passes thirty-five degrees at 3.5 hours. For a denser stony body of two grams per cubic centimetre the two periods are 2.33 and 2.7 hours. In both cases the surface starts to fail at a spin seventeen per cent slower than the one that would shed it, and since a period scales as the inverse square root of the density, that ratio is the same for every density.

Spin-up by the thermal torque of sunlight — the YORP effect, whose clock grows as the square of the size — takes a body through that interval slowly, and for all of that time the surface is flowing. A spin barrier crowded with bodies near the limit is therefore also a population of bodies whose surfaces have been reshaping themselves for a large fraction of their recent history.

Bennu and Ryugu are both outside the interval now. Bennu, at 4.3 hours, has a steepest slope of nineteen degrees on the sphere drawn here — its real surface is rougher, but not systematically steeper — and Ryugu, at 7.6 hours, has slopes of a few degrees. Neither surface is failing. Both have pronounced equatorial ridges.

How long a body spends in the interval sets how much reshaping it does, and the arithmetic is short. The thermal torque changes a body’s spin rate at a roughly constant rate for a given shape and orbit, so the time to cross the last seventeen per cent of period before the limit is about a sixth of the time to double the spin. For an asteroid a few hundred metres across at the distance of the inner asteroid belt, that doubling time is a few million years, and the crossing takes several hundred thousand — long compared with the minutes a landslide takes, and long enough for the failing belt to be reworked many times. The same torque spreads the members of an asteroid family by drifting their orbits, so the small members of an old family have been through this interval, some of them more than once, and the population of top-shaped asteroids among them is a record of how often.

Numbers make the interval common rather than rare. The small asteroids that the thermal torque governs are most of the asteroids by count, and the spin barrier is crowded: in surveys of rotation periods, a substantial fraction of bodies a few hundred metres across spin within a quarter of their limit. Each of them is a surface that is flowing now or has flowed recently.

How much of the surface moves #

The onset is abrupt. Below about three-quarters of the critical spin nothing is past thirty-five degrees; a few per cent above it, half the surface is. The failing region is a belt that opens at the mid-latitudes and widens both ways — towards the equator, where the slopes grow steepest, and towards the poles — while a narrow strip along the equator itself and caps around the poles stay below repose.

The angle of repose matters, and it is not perfectly known for material in a gravity a hundred thousand times weaker than the Earth’s. Laboratory measurements, and the slopes of boulder fields on Bennu and Ryugu, suggest that it is about the same as on the Earth: the geometry of interlocking grains, not the strength of gravity, sets it. A repose angle of thirty degrees moves the onset to two-thirds of critical; forty, to four-fifths. The conclusion that the surface fails well before the equator sheds survives either way.

Downhill is towards the equator #

Where the surface fails, the material moves down the local slope, and on a spinning body down means towards the equator.

The cleanest way to see this is through the effective potential — gravity’s potential plus the rotation’s. On a sphere, gravity’s potential is the same at every point of the surface; the rotation’s falls away from the axis as the square of the distance from it. So the equator is the lowest point of the surface in the only sense that matters for loose material, and the poles are its hilltops. At ninety per cent of the critical spin, the pole of a spherical rubble pile stands, in effect, nearly half its radius above its equator. Material released anywhere in the failing belt slides towards the equator and piles up there.

That is how a ridge is built. The equator collects what the belt sheds, the body becomes a spinning top — fatter at the equator, flatter at the poles — and the process continues until the new shape is level enough at the current spin for the slopes to fall back below repose. The equator itself is the last place to fail: its slope is zero by symmetry until the moment the material there weighs nothing at all.

The analogy with a different object in this collection is close. A star in a close binary fills the equipotential surface that bounds it, and the gas leaves through the saddle point between the two stars; a rubble pile at its limit fills its own rotating equipotential and loses material from the equator, the saddle of its own effective potential. The asteroid’s surface flow before shedding is the part of that story with no stellar counterpart, because a star’s gas is a fluid and cannot hold a slope at all.

A fluid levels itself; a heap does not have to #

There is a body for which the whole of this is familiar, and the comparison shows what is special about a rubble pile. The Earth spins once a day, and its surface — the oceans, and over geological time the rock — has flowed until the effective potential is the same everywhere on it. The result is the equatorial bulge, twenty-one kilometres of extra radius at the equator, which a satellite’s slowly turning orbit reads off as the planet’s flattening. A fluid body has no slopes in the sense used here: it takes whatever shape makes every slope zero, and it takes it at any spin.

A rubble pile is not obliged to. Friction lets it hold any shape whose slopes stay below the angle of repose, so a slowly spinning asteroid can be a potato, a peanut or a sphere with no preference among them. Only as the spin rises do the slopes of its present shape grow past repose, and only then is it forced towards the fluid’s answer — a body flattened by its rotation, with the material moved to the equator. The top shape is the rubble pile’s approximation to a fluid’s equilibrium, reached only where the spin leaves it no choice, and frozen there when the spin later slows.

A ridge lowers the spin that sheds it #

The ridge changes the body that built it.

A flatter body has more of its mass near the equatorial plane and less along the axis, and at its equator the pull of gravity is weaker relative to the equatorial radius than on a sphere of the same density. So it reaches the shedding condition at a slower spin. The effect is modest — a flattening to nine-tenths, about the axis ratio of Bennu and Ryugu, moves the shedding period from 3.03 to 3.09 hours — but it has a definite direction: reshaping by surface flow brings the body closer to the next stage. A rubble pile spun up by sunlight first flows into a top and then sheds from the top’s ridge, at a spin a few per cent slower than a sphere would have needed.

Shedding from a ridge is also different from shedding from a sphere’s equator in what it leaves behind. Material lifted from a ridge carries more angular momentum per unit mass than material from the same spin at a sphere’s equator, because the ridge sits further from the axis, and it leaves with a lower relative speed. That favours the formation of a debris ring and, from it, a small satellite, rather than escape — one reason binary asteroids, roughly one in six among the near-Earth asteroids larger than a few hundred metres, so often have top-shaped primaries.

A cohesion of less than a pascal decides it #

Everything so far has assumed a surface of loose, cohesionless grains, held by friction alone. Real regolith has some cohesion — van der Waals forces between fine grains, electrostatic forces, the interlocking of irregular fragments — and on a body whose gravity is a hundred-thousandth of the Earth’s, a cohesion that would be imperceptible on the Earth is not small.

The figure asks how thick a layer has to be before its weight on the slope can break the bonds that hold it. A thin layer has too little weight; a thick one has enough. For a body two hundred and fifty metres across spinning at ninety per cent of its limit, a cohesion of a tenth of a pascal holds layers thinner than about eight metres: the surface can still flow, in slabs of a few metres. A cohesion of one pascal holds layers up to seventy-seven metres thick: the surface can fail only in deep landslides of a substantial part of the body. A cohesion of twenty-five pascals — the value at which the spin barrier acquires its corner — holds layers thicker than the body itself: nothing slides, and the body keeps its shape until it breaks.

So whether a small asteroid spun towards its limit reshapes itself into a top or keeps its shape and then fissions depends on the cohesion of its surface in a range from a tenth of a pascal to a few. That is the range the spacecraft have been probing. When OSIRIS-REx touched Bennu’s surface to collect its sample, the sampling head sank in as though into a pit of plastic balls, and the surface’s response implied almost no cohesion — so weak that the outer layers of Bennu are held together by little more than their own feeble weight. A surface like that flows readily, and Bennu’s ridge, and the landslides and mass movement seen in its images, are what the slopes here predict for a body that has been spun close to its limit in the past.

Shapes as a record of past spins #

Bennu and Ryugu both spin too slowly for their surfaces to be failing now, and both wear the shape that failure builds. The obvious reading is that both were spinning faster in the past and have since slowed — spin-down is as available to the thermal torque as spin-up, depending on the body’s shape and orientation — and that their ridges are fossils of the interval they spent near their limits. For Ryugu, models of its shape find its slopes most nearly level at a period of about three and a half hours, half its present period, which is where the onset figure puts the failure of its surface.

A third body sits where the others once were. Didymos, the primary of the binary pair a spacecraft was flown into in 2022, spins in 2.26 hours with a density close to 2.8 grams per cubic centimetre, which puts its spin at about three-quarters of its critical rate — right at the onset of surface failure in the figures here. It has a ridge, and it has a moon, and its moonlet’s material is likely to have come from its equator. On the arithmetic of this essay it is a body in the middle of the process: surface slopes at the edge of repose, flow towards the equator under way or recently paused, and the next stage — shedding from the ridge, feeding its satellite — a few per cent more spin away.

What the sphere leaves out #

The calculation is for a homogeneous sphere or spheroid with a smooth surface. Real rubble piles have irregular shapes whose local slopes differ from the sphere’s by tens of degrees — boulders, craters, the ridge itself — and their gravity fields are not those of a homogeneous body; a denser core or a more porous outer layer changes the balance between gravity and rotation at the surface. The sphere’s slopes are the large-scale trend on which the local roughness rides, and the onset of failure in the real body is spread over a range of spins rather than occurring at one.

The angle of repose is treated as a single number. Granular materials have two — a static angle at which a heap starts to fail and a lower dynamic angle at which flowing material stops — and the difference, a few degrees, means that a surface once set flowing continues to flow at slopes the static angle would have held. That makes the onset of flow more abrupt than the figures show, and its end more gradual.

And the dynamics of the flow are not computed at all. Material moving towards the equator changes the body’s moment of inertia and so its spin, which conserves angular momentum by slowing slightly; it also changes the body’s response to sunlight, and so the torque driving the spin. Simulations of granular rubble piles spun up slowly find exactly the sequence described — mid-latitude failure, equatorward flow, ridge growth, then shedding from the ridge — but also find that the outcome depends on details of the internal structure that no observation yet constrains.

Still open: how deep the reshaping goes #

The essay on fission ended on the question of whether shedding or splitting dominates when a rubble pile reaches its limit. The slopes add a prior step and sharpen the question. A body with a weak surface reshapes itself before it reaches its limit, and arrives there as a top with a ridge from which it sheds; a body with a strong surface or a strong interior keeps its shape and arrives as whatever it was, from which it splits. The cohesion that separates the two cases is of order a pascal, and it may differ between the surface and the interior of the same body.

What would decide it is the depth of the reshaping. If only the outer few metres flow, a top-shaped asteroid has the same interior as before, wrapped in a reworked skin; if the failure reaches deep, the whole body has been rearranged. Measurements of the interior — radar sounding, seismic experiments on a surface, or the gravity field mapped closely enough to see how density varies between the ridge and the core — would tell which, and with it whether the size at which a body is easiest to break is also the size at which it is easiest to reshape.