A split that decides whether the piece can leave

A rubble pile driven past its spin barrier by the thermal torque cannot keep spinning faster. Something gives, and one of the things that can give is the body itself: it separates along a weak plane into two pieces that were resting against each other and are now each in orbit around their common centre of mass. What happens next depends on a single number, the ratio of the two pieces’ masses, and it can be worked out with nothing more than the energy of two touching spheres.

The answer divides small asteroids into two kinds of outcome that astronomers had catalogued separately before anyone connected them. Binary asteroids — a primary with a small moon — are common among near-Earth objects. Asteroid pairs — two unbound asteroids on heliocentric orbits so similar that they must have separated within the last million years or so — were recognised later and are rarer. Both are products of the same split, and the mass ratio decides which.

The energy of a pair about to part #

Model the body at the moment of fission as two spheres of the same density, radii $R_1$ and $R_2$, touching, with mass ratio $q = m_2/m_1$. They rotate together as a rigid pair about their centre of mass, at exactly the rate at which the centrifugal acceleration of their separation balances their mutual gravity: $\omega^2 = G(m_1 + m_2)/d^3$, with $d = R_1 + R_2$. Spinning any faster, the contact would open.

The pair’s total energy is its rotational kinetic energy about the centre of mass, including each sphere’s own spin and their orbital motion, minus their mutual gravitational binding:

$$E = \tfrac{1}{2} I \omega^2 - \frac{G m_1 m_2}{d}, \qquad I = \tfrac{2}{5}m_1R_1^2 + \tfrac{2}{5}m_2R_2^2 + \frac{m_1 m_2}{m_1+m_2}\,d^2$$

The self-gravity of each sphere is unchanged by the split and does not enter. When the secondary is tiny, the binding term vanishes faster than the rotational term, and $E$ approaches the primary’s own spin energy at breakup, $0.2\,Gm_1^2/R_1$. When the two pieces are equal, the binding dominates and $E$ is negative. In between, the energy crosses zero at

$$q_{\rm crit} = 0.204$$

That number is dimensionless and involves only the geometry of two touching spheres. It does not depend on the size of the asteroid, its density, the strength of its material or its distance from the Sun. Any rubble pile, anywhere, that splits into pieces of mass ratio below about one in five has enough energy in its spin for the pieces to part; any that splits more evenly does not.

The angular momentum a split has to share #

Energy decides whether the pieces can part. Angular momentum decides how the spin and the orbit share what the parent had, and it gives a second test of the fission story that does not need the pieces to separate at all.

A single body spinning at its breakup rate has a definite amount of angular momentum for its mass and size. A split does not change the total: whatever the pieces do afterwards, their spins plus their mutual orbit must add up to it, less only what the thermal torque adds or removes over the following ages. Angular momentum is the quantity an orbit cannot lose on its own, and a binary asteroid formed by fission should therefore carry close to the angular momentum of its primary spinning at breakup — no matter how the spin and orbit are divided.

That is what binaries among small asteroids show. When the total angular momentum of a sample of near-Earth and small main-belt binaries is computed from their measured sizes, primary spins and orbits, and divided by the angular momentum of a single sphere of the same total mass at its breakup spin, the ratio clusters near one. Binaries formed by collisions or capture would have no reason to sit there; binaries formed by spinning a body to its limit and letting it split would sit exactly there. The number is a fingerprint of the formation, still readable after millions of years of orbital evolution.

What positive energy means, and does not #

Positive total energy is necessary for escape and not sufficient. The pieces start in contact with the secondary’s orbital speed nearly circular about the primary, and escape needs the energy to be transferred into the secondary’s orbit. Two irregular pieces in a close orbit exchange energy and angular momentum through the torques their shapes exert on each other, and the secondary’s orbit becomes chaotic: over days to years it may be flung out, fall back onto the primary, or settle into a stable orbit once enough energy has been dissipated as tidal heating or lost in grazing collisions.

So below 0.204 all three outcomes are possible and escape is available; above it escape is impossible without an external source of energy, and the pieces end either as a binary or as a single reaccreted body — perhaps a contact binary, two lobes resting against each other, which is itself a common asteroid shape. The 330-metre asteroid Itokawa, sampled by a spacecraft, is two lobes of rubble joined at a neck, and its bulk density shows it to be about forty per cent empty — a body that looks as if it came together gently rather than being broken apart. A split whose pieces cannot escape and do not settle into orbit is one of the ways such a shape is made, and whether two bound bodies merge is a ratio of two times: how long the orbit takes to decay against how long the system survives unperturbed. The threshold sorts outcomes by what is allowed, not by what happens.

The primary left behind #

If the secondary does escape, it carries away the energy it needs to leave, and the energy that remains goes into the primary’s spin. Taking the minimum — the secondary escaping with zero energy at infinity, and none left in its own spin — the primary’s rotational energy after the split is the pair’s total energy:

$$\tfrac{1}{2}I_1\omega_1^2 = E$$

which gives the primary’s final spin rate from the curve in the first figure.

The relation has a shape rather than a number, and the shape is a prediction that could fail. Pairs whose secondaries are small should have primaries still spinning quickly, near their barrier. Pairs whose secondaries are nearly a fifth of the primary’s mass should have primaries spinning slowly, with periods of many hours. And there should be almost no pairs with mass ratios above a fifth, because such splits cannot unbind.

That is what the measurements found. When the rotation periods of the primaries of several dozen asteroid pairs were measured, and their mass ratios estimated from the brightness difference between the two members, the primaries fell along a curve of exactly this form: fast rotators for small secondaries, slow ones as the ratio approached about 0.2, and essentially no pairs beyond it. The theory used in that comparison added the secondary’s spin and an allowance for shape, which moves the curve slightly without changing its structure. It was the strongest single piece of evidence that asteroid pairs form by rotational fission rather than by collisions, which would not care about the spin of the surviving body at all.

Where fission sits on the rotation diagram #

A pair is one outcome of a process that most small asteroids are somewhere inside, and the place to see the whole process at once is a diagram that pays no attention to pairs at all: how fast bodies of each size are turning. A population that is spun up steadily and reset by fission ought to pile up against the barrier and leave a trail of reset bodies at longer periods, and that pile-up should begin at the size below which the thermal torque can act within a body’s lifetime.

On a diagram of rotation period against size, fission is a cycle rather than an event. A body spun up by the thermal torque moves towards shorter periods at fixed size until it meets the barrier. If it sheds a small moon or a small pair member, its diameter barely changes and its period jumps to a longer value — by a factor of two for a mass ratio of a few per cent, by a factor of five for a ratio near 0.18. Then the torque starts spinning it up again. A single body can pass through the cycle several times, each time leaving a smaller companion somewhere, and the crowding of bodies just slower than the barrier is partly the population of primaries partway back.

That picture also explains why the diagram shows so few bodies in a particular band — large bodies spinning very slowly — beyond what collisions alone would produce, and why some of the slowest rotators known are tumbling rather than rotating about a single axis: a split that leaves little energy in the primary also leaves its spin easily disturbed, and a subsequent small impact or the thermal torque itself can set it tumbling for millions of years before internal friction damps the wobble.

What drives a body to split #

A body reaches fission only if something keeps adding spin, and for small asteroids that is the thermal YORP torque, whose rate grows as the inverse square of the diameter.

The size range matches. Binary asteroids and pairs are found overwhelmingly among bodies smaller than a few kilometres, where the torque is strong enough to act within the bodies’ lifetimes; about one in six near-Earth asteroids larger than a few hundred metres has a satellite. Larger asteroids have binaries too, but theirs are mostly the products of collisions, with different mass ratios and different spin states.

The ages of pairs make the connection sharper. Because the two members of a pair were once in contact, their orbits can be integrated backwards until they converge, and the convergence date is the date of the split. Some pairs converge within the last few tens of thousands of years — one to about sixteen thousand years ago. The rate at which pairs are forming, estimated from their numbers and ages, is consistent with the rate at which YORP drives small asteroids to their barriers.

Dating a pair has a limit that is itself thermal. The two members, once separated, are nudged apart in their orbits by the recoil of their own heat, and the drift depends on each body’s size, spin axis and surface — none of them known well for a body a few hundred metres across. Backward integrations therefore run many clones of each member with different plausible drift rates and look for the spread of convergence dates. For the youngest pairs the thermal drift has had no time to matter and the date is sharp; for pairs older than about a million years the clones stop converging, the orbits have diverged chaotically, and a pair becomes indistinguishable from two unrelated asteroids on similar orbits. The population of recognisable pairs is therefore a population of the recently split, which is exactly what makes it useful as a measurement of the splitting rate now.

The same conservation that dates a pair governs how its members’ spins are shared, and it has a familiar engineering analogue: angular momentum has to be put somewhere, whether it is a spacecraft’s reaction wheels saturating or an asteroid’s spin being handed to an escaping fragment. A split cannot destroy any of it; it can only move it from the primary’s spin into the secondary’s orbit and spin, and the primary’s slower rotation afterwards is the bookkeeping made visible.

A binary that was hit on purpose #

The best-studied binary asteroid is one that a spacecraft deliberately struck. Didymos is about 780 metres across and spins every 2.26 hours, close to its barrier for its density. Its moon Dimorphos is about 160 metres across, a mass ratio of roughly one per cent, orbiting every 11.9 hours at about 1.2 kilometres.

That configuration is the fission picture’s small-secondary outcome, stopped short of escape. The primary spins fast, as a body left behind by a small piece should. The moon is a small fraction of the mass, well below 0.204, so it could have escaped and instead settled — its orbit made stable by dissipation after it formed, most likely from material shed from Didymos’s equator. When the kinetic impactor struck Dimorphos in 2022, its orbital period shortened by about 33 minutes, far more than the spacecraft’s own momentum could have done: the ejecta blasted off the surface carried several times the impactor’s momentum, a sign that the moon is itself a weakly bound pile. The follow-up images showed boulders and a debris trail, and a shape change of the moon itself.

The system is also a reminder of how a binary evolves after it forms. The secondary’s own irregular shape, lit by the Sun, feels a thermal recoil in its orbit — the binary version of the same torque — which slowly expands or shrinks the orbit over hundreds of thousands of years. Close to the primary, the moon is also near the distance inside which a body held only by its own gravity is pulled apart by the tide, and a small secondary born just outside that limit is born on the edge of being reprocessed into a ring or rained back onto the primary. A binary formed by fission is not a final state; it is the next stage of a thermal evolution that can end with the moon escaping after all, or colliding gently with the primary to form a contact binary.

What the model leaves out #

Spheres are the crudest possible shapes, and the threshold depends on them. For elongated pieces, or for a split along a plane through an ellipsoid rather than between two spheres, the critical mass ratio shifts — generally by amounts of order a few hundredths, not by factors — and the precise number 0.204 should be read as the value for the idealised geometry. The pair’s spin before fission is taken to be exactly the contact limit, when in practice cohesion lets it spin somewhat faster and store more energy, which moves the threshold towards larger mass ratios for small, cohesive bodies. And the final-spin relation assumes the secondary leaves with the minimum energy and no spin of its own; any energy it carries beyond that, and any dissipated in the chaotic orbital phase, leaves the primary slower than the curve.

The model also says nothing about why a body splits where it does. A rubble pile has no designed fracture plane; the mass ratio is set by the internal distribution of large boulders and weak zones, and simulations of spinning aggregates produce a range of outcomes — gradual mass shedding, fission into two large pieces, or fission into several — depending on the internal structure. The energy threshold sorts those outcomes once they happen; it does not predict which happens.

An energy threshold says what is allowed, not what happens #

A threshold computed from energy alone divides outcomes into allowed and forbidden without saying which allowed outcome occurs. The rotational fission threshold is a clean example: a pure number from the geometry of two spheres, with no size or material property in it, that separates splits which can unbind from splits which cannot, and a second relation, the primary’s leftover spin, whose shape was confirmed by measurements made for a different purpose.

The same logic runs through the spin barrier, which is also a size-free threshold that sorts bodies into possible and impossible states, and through the collisional families whose members were once one body, which are dated by running their orbits backwards in exactly the way asteroid pairs are. Pairs are the gentle version of families: two pieces instead of hundreds, separated by a slow spin rather than a violent impact, and dated to thousands of years instead of hundreds of millions.

Still open: whether shedding or splitting dominates #

A body driven to its barrier can lose mass gradually from its equator, building a ridge and occasionally launching small fragments, or it can split into two large pieces. Both are seen in simulations, and which dominates in the asteroid population decides the relative numbers of top-shaped bodies, small moons, pairs and contact binaries. The answer depends on the internal structure of rubble piles — whether their strength is concentrated in large interlocking boulders or spread through fine cohesive regolith — and that is exactly the property the spacecraft that sampled such bodies found hardest to predict.