A spin barrier with a corner in it

The spin barrier has no size in it. A body held together by nothing but its own gravity flies apart when the centrifugal acceleration at its equator exceeds its surface gravity, and both scale with the radius in the same way, so the critical period is $\sqrt{3\pi/G\rho}$ — 2.33 hours for a bulk density of two grams per cubic centimetre, whether the body is two hundred metres across or twenty kilometres. The asteroid population respects it. Above a few hundred metres, almost nothing spins faster.

“Almost nothing” and “a few hundred metres” are both soft, and the softness has a cause that can be written down. A rubble pile is not quite strengthless. Fine grains cling to each other by van der Waals forces, and the resulting cohesion — a few to a few hundred pascals — is utterly negligible in a mountain and not negligible at all in a body whose gravity is a millionth of the Earth’s. Adding it to the balance turns the size-free wall into a curve with a corner, and the position of the corner is a measurement of how strong the pile is.

A stress that does not care about size #

Inside a spinning body the centrifugal stress that pulls it apart and the gravitational stress that holds it together both grow with the radius. The centrifugal acceleration at the surface is $\omega^2 R$ and the pressure it produces over a column of the body’s own material is of order $\rho\,\omega^2 R^2$. The gravitational pressure is of order $G\rho^2 R^2$. Both go as $R^2$, which is why the balance between them contains only the density.

Cohesion is different. It is a strength of the material — a stress the grains can support without separating — and a stress does not depend on the size of the body it is in. Add it to the side holding the body together, and the balance becomes

$$\omega^2_{\rm crit} = \frac{4\pi}{3}G\rho + \kappa\,\frac{C}{\rho R^2}$$

with κ a number of order one that depends on the internal friction and the shape. The first term is the size-free barrier. The second grows without limit as the body shrinks. Below the size at which the two terms are equal,

$$R_* = \frac{1}{\rho}\sqrt{\frac{3\kappa C}{4\pi G}}$$

cohesion dominates, $\omega_{\rm crit} \propto 1/R$, and the fastest possible period is proportional to the diameter. A body a tenth the size of the corner can spin ten times faster than the gravity-only limit.

The corner scales as the square root of the cohesion. A hundredfold increase in strength moves it only tenfold, which is why the three nonzero curves in the first figure are evenly spaced a factor of three apart in diameter for factors of ten in cohesion. It also scales as the inverse of the density, which makes the corner a stronger function of porosity than of material.

What a few tens of pascals looks like #

A cohesion of ten pascals is a stress equal to the weight of a one-millimetre layer of water. A body of loose regolith that could not support a coin’s weight on its edge on the Earth can have that much. Lunar soil, compressed under a gravity a sixth of the Earth’s for billions of years, has a cohesion of hundreds to a thousand pascals; the surface of a rubble pile that has been shaken by impacts and thermal cycling in near-zero gravity is expected to be weaker.

The observed population fits a picture in which cohesions of order ten to a hundred pascals are common. The barrier is sharp for bodies of a kilometre and larger and fades between about a kilometre and a couple of hundred metres, which is the range the corners of those cohesions occupy. Below that, the fast rotators appear — bodies of tens of metres spinning in minutes — and for those the question of whether they are rubble piles with modest cohesion or single coherent rocks becomes hard to answer from the spin alone, because both are permitted.

The population the corner is read from #

What the catalogues of rotation periods actually show is a distribution rather than a line. Light curves from wide-field surveys give periods for tens of thousands of asteroids, and when they are plotted against diameter the region faster than about 2.2 hours is nearly empty for bodies larger than a few hundred metres and increasingly populated below. The transition is not at one size. It is a band, a factor of a few wide, in which some bodies spin faster than the barrier and most do not.

Reading that band as cohesion is an inversion: for each fast-spinning body of known size, the curve through its size and period gives the minimum cohesion it needs. Bodies near the corner need only a few pascals; bodies far below the barrier at a few hundred metres need tens; a kilometre-sized body spinning faster than the barrier needs more. The inversion depends on the density, which is known for only a handful of asteroids from spacecraft or binary orbits, and on the shape coefficient κ, which is not known for any. What it gives robustly is the order of magnitude, and every analysis that has done it lands in the tens of pascals — weaker than dry sand held together by damp, and far stronger than nothing.

There is also a selection effect in the band that works in the barrier’s favour. A fast rotator’s light curve repeats in a couple of hours, and surveys that sample each asteroid a few times a night alias short periods into long ones; very fast rotators among small bodies were missed until dedicated rapid-cadence observations were made. The emptiness above the barrier for large bodies is not a product of that selection — large bodies are bright and well sampled — but the counts of fast small rotators have risen as observations have become faster, and the band’s lower edge is still being mapped.

A kilometre-sized body spinning too fast #

One asteroid makes the argument almost directly. (29075) 1950 DA is about 1.3 kilometres across, rotates once every 2.12 hours, and has a bulk density near 1.7 grams per cubic centimetre, measured from the thermal and radar observations that fixed its size and shape. At that density the gravity-only barrier is 2.5 hours. The asteroid spins faster than its own gravity can hold loose material on its equator.

It is not a monolith: its thermal inertia is that of a regolith-covered rubble pile. The resolution offered when the measurement was published was cohesion — tens of pascals, about sixty-four in the published estimate, enough to move the corner of the barrier to a size larger than the asteroid, so that the body sits on the steep, cohesion-held part of the curve. That is a strength about equal to the van der Waals attraction expected between fine grains of the kind a small asteroid’s surface is made of. One fast-spinning kilometre-sized body is not a population, but it is a direct lower bound on cohesion in an object where no other explanation fits.

What pushes bodies to the barrier #

A barrier is only interesting if bodies are driven into it, and the thermal recoil that moves asteroids’ orbits also changes their spins. An irregular body absorbs sunlight and re-emits it as heat, and if its shape is asymmetric the recoil from that emission has a torque about the spin axis. The torque’s average over a rotation and an orbit is not zero, and it changes the spin rate steadily for as long as the shape and the spin axis stay the same. This is the YORP effect, after the four people whose work described it.

It has been measured on individual asteroids by comparing rotation phases over years: a body whose period is shortening accumulates a phase drift that grows as the square of the time. For (1862) Apollo, a near-Earth asteroid about 1.4 kilometres across, the measured acceleration is $5.5 \times 10^{-8}$ radians per day squared.

The scaling is what matters. The torque goes as the absorbed sunlight, so as $1/a^2$ with distance from the Sun. It is applied at a lever arm proportional to the body’s size, to a surface area proportional to its size squared, and acts on a moment of inertia proportional to its size to the fifth power: the angular acceleration goes as $1/D^2$. A body a tenth the size is spun up a hundred times faster.

That is the same direction in which cohesion makes the barrier permissive. Small bodies are pushed hardest towards fast rotation and are also the bodies that can survive it. Large bodies are pushed slowly, often more slowly than their spins are reset by collisions, and they meet a sharp barrier when they arrive. The softening of the barrier below a kilometre and the dominance of YORP below a few kilometres are two faces of one size scale.

What happens at the barrier #

A body pushed past its limit does not explode. Material near the equator, where the centrifugal acceleration is largest and the gravity smallest, is the first to lose contact, and it moves outward and either settles into an equatorial ridge or leaves the surface. A body losing mass from its equator reduces its moment of inertia at the equator more than elsewhere, and the loss carries angular momentum with it; the spin slows and the barrier is respected again, until the torque drives it back.

The resulting shape is a spinning top: a nearly circular equator with a raised ridge, and flattened poles. Radar images of many near-Earth asteroids and the spacecraft images of Bennu and Ryugu show exactly that shape, and the leading explanation is repeated episodes of spin-up and equatorial reshaping. Bennu’s current spin of 4.3 hours is well below its barrier, and its own measured YORP acceleration, obtained from years of rotation measurements before and during the spacecraft’s visit, is spinning it up now.

The alternative outcome of spin-up is fission: the body splits into two pieces, which either stay in orbit about each other as a binary or separate entirely. Whether a split piece can leave is a question of energy, and the energy depends on the mass ratio of the pieces — which is what makes that outcome a measurement too.

How much of the belt this touches #

The size range where cohesion and the thermal torque both matter is not a corner of the asteroid population. It is most of it by number. Asteroid size distributions are steep — the mass is in the few large bodies and the numbers are in the many small ones — so for every body ten kilometres across there are hundreds a kilometre across and tens of thousands a hundred metres across. The bodies whose spins are governed by a sunlight-driven torque, and whose shapes and survival are governed by a few tens of pascals of grain-to-grain attraction, are the majority of all asteroids.

That has consequences for things that look unrelated. A small asteroid’s spin sets how its surface is heated and how its orbit drifts, since the thermal recoil that moves the orbit depends on the rotation. It sets whether loose material stays on the surface or is shed, and therefore what a spacecraft touching down finds. And it sets how often small asteroids are converted into binaries or pairs, which changes the counts that the size of an asteroid family is estimated from. Roughly one in six near-Earth asteroids larger than a few hundred metres has a satellite — a fraction far too large to have been produced by capture, and one that spin-up driven to the barrier produces naturally.

Why the torque is not constant forever #

The YORP torque depends on the shape, and the shape changes. A small crater, a boulder moved by a surface landslide, or a shift in the regolith can change the torque’s sign. Simulations of spin evolution therefore describe a random walk overlying the steady torque: a body spins up for a few hundred thousand years, reaches the barrier or reshapes, and may find its torque reversed afterwards. The doubling times drawn here are the steady rates; over millions of years the spin distribution they produce is a statistical one, with an excess of very slow and very fast rotators, which is what surveys of small asteroids find.

The torque also turns the spin axis. Its component perpendicular to the spin drives the obliquity towards values near 0° or 180° — spin axes perpendicular to the orbit — and that, combined with the orbit’s own thermal drift, explains a pattern that looked like coincidence: in families of asteroids produced by a single collision, the members’ spin axes are aligned in a way that depends on which side of the family’s centre they now lie. Prograde rotators drift outwards and retrograde ones inwards, and both have had their axes pushed to the extremes by the same torque.

What the figures leave out #

The cohesion model is a scaling with one coefficient, and the real limit depends on the internal friction angle of the grains, on the body’s shape, and on whether it fails at the surface or through its interior. A body that is elongated fails at a longer period than a sphere of the same density, because its tips are further from the centre; a body with a denser core and a looser surface fails from the surface first. The corner sizes in the figures carry an uncertainty of a factor of a few.

The YORP normalisation is one asteroid’s. The torque coefficient depends so strongly on shape — and on thermal inertia, which delays the emission and shifts the recoil’s direction — that bodies of the same size and distance can differ by an order of magnitude or have opposite signs. Apollo’s measured value is a representative rather than a typical one, and the $1/D^2$ scaling applies to the ensemble, not to each body. A surface dated by counting its craters on a small asteroid is also a surface whose regolith has been shaken and resettled by spin changes, which is one of the complications for that method on bodies below a few kilometres.

A third force with a different scaling puts a size into a limit #

A limit derived from a balance of two effects that scale the same way has no size in it, and adding a third effect with a different scaling introduces one. The size-free spin barrier and the corner that cohesion adds are one example; the Roche limit of a satellite, which contains no size for a fluid body and acquires one when the body has strength, is another. In each case the size at which the new effect takes over is where the two stresses match, and measuring where a population’s behaviour changes is measuring the strength.

The coincidence of scales between the corner and the thermal torque’s reach is not a coincidence of physics but of the solar system’s asteroids: both are set by gravity being weak in small bodies, one through the stress it supports and the other through the moment of inertia it has to overcome. The smallest asteroids are therefore both the most easily spun up and the most able to survive it, which is why the fastest rotators known are all tiny.

Still open: what a split leaves behind #

A rubble pile driven past its barrier can shed material gradually or split in two. When it splits, the question is whether the smaller piece can escape. The energy of two touching pieces spinning together at their shared limit is positive when the smaller piece is a small fraction of the whole and negative when it is large, and the sign changes at a mass ratio that is a pure number of the geometry. Which side a split falls on decides whether it becomes a binary asteroid or a pair of asteroids on nearly identical orbits — and how fast the larger piece is left spinning.