The size at which a body is easiest to break

A population that grinds itself down settles into a power law, and if every collision looks the same at every scale — if a collision between two ten-kilometre bodies is a scaled-up copy of a collision between two ten-metre ones — the power law has a fixed exponent, 3.5 for the differential number of bodies per unit diameter. The argument that gives it is elegant, general and wrong in one respect that matters: collisions do not look the same at every scale. What holds a body together changes with its size, and the change leaves a mark on the distribution that can be read from the asteroid belt.

The mark is a preferred size. Somewhere between pebbles and moons there is a size at which a body is easier to destroy than at any other, and the population is thinned there more than elsewhere.

Two things that hold a body together #

A body survives a collision if the energy the impactor delivers is too small to both break it and throw the pieces apart. The standard measure is the specific energy of disruption: the kinetic energy of the impactor per unit mass of the target at which the largest surviving fragment carries exactly half the original mass. Below it the target is cratered or cracked; above it the target is destroyed.

For a small body, the energy is spent breaking chemical bonds, and the difficulty is set by the material’s strength. Strength is not a single number for a rock. A rock breaks along its flaws, and a larger rock contains larger flaws, because the probability of a big crack somewhere in a big volume is higher. Laboratory experiments and the theory of brittle fracture agree that the energy needed to shatter a rock per unit mass falls with its size — roughly as the diameter to the power minus a third, a little more steeply for ice. A pebble is proportionately much harder to break than a boulder.

For a large body, breaking it is not enough. The pieces must also be given enough speed to escape each other’s gravity, or they fall back together as a rubble pile — a body that has been shattered and has reassembled. The energy needed to disperse the pieces is set by the body’s own gravitational binding energy per unit mass, which grows as the square of its size, and the fits to impact simulations give a dependence slightly steeper still, because a larger body also compresses and absorbs more of the impact’s energy in its interior.

The two contributions add. The first falls with size, the second rises, and their sum has a minimum. For rock struck at a few kilometres a second the minimum is at about 235 metres in diameter; for ice, about 200. At that size a body needs only about 130 joules per kilogram of impact energy to be destroyed — the kinetic energy per kilogram of motion at sixteen metres a second. A body of that size can be shattered and dispersed by an impactor a hundred thousand times less massive than itself, moving at the five kilometres a second typical of collisions in the asteroid belt.

The same transition between material and gravitational control appears in how asteroids spin. Bodies smaller than a few hundred metres can spin faster than the rate at which a rubble pile would fly apart, because they hold together by cohesion; bodies larger than that cannot, and the spin barrier has a corner at about the same size. Two independent measurements — how fast asteroids spin and how hard they are to break — locate the change of regime at the same few hundred metres.

A slope set by how strength scales #

The self-similar cascade assumes that the energy needed to disrupt a body is the same per unit mass at every size. If it is not, the equilibrium changes, and the change can be worked out exactly.

The steady state of a cascade balances the rate at which bodies of each size are destroyed against the rate at which they are supplied as fragments of larger ones. The destruction rate for a body of diameter D depends on how many projectiles are large enough to destroy it, and the smallest such projectile scales with the disruption energy: a body that is harder to break needs a relatively larger projectile. If the disruption energy scales as the diameter to a power s, carrying that through the balance gives an equilibrium index

$$p = \frac{7 + s/3}{2 + s/3},$$

a result published in 2003 that reduces to 3.5 when s is zero. In the strength regime, where s is about −0.38 for rock, the larger bodies of each size range are relatively easy to destroy and are depleted: the distribution steepens to 3.67. In the gravity regime, where s is about 1.36, the larger bodies are relatively hard to destroy and survive longer: the distribution flattens to 3.04.

The direction of each change is intuitive once it is stated. A cascade is a flow of mass from large bodies to small ones, and in steady state the flow must be the same through every size. Where bodies are easy to destroy, they pass the mass on quickly and fewer of them are needed at any moment to carry the flow; where they are hard to destroy, they hold on to it and more of them accumulate. The index measures how the number needed to carry a fixed flow changes with size, and the disruption energy’s scaling is what sets it.

The effect is not small. A few tenths in the index compounds over the many decades of size that a cascade spans. Over five decades of diameter, a difference of 0.46 in the index is a factor of two hundred in the relative number of the smallest and largest bodies.

The knee in the distribution #

A population that spans the minimum in strength — as any real one does, from dust to bodies hundreds of kilometres across — therefore has two slopes, and the break between them sits near the weakest size.

Starting from the largest bodies and working down, the gravity regime’s shallow slope accumulates fewer bodies than a 3.5 cascade would, so by the time the knee is reached the population is down to a few per cent of the self-similar count. Below the knee the strength regime’s steeper slope climbs faster than 3.5 and recovers part of the deficit, but it never recovers it all over the range where the distribution is observed. The net shape, on a logarithmic plot, is a distribution that is shallow for large bodies, bends at a few hundred metres, and is steep for small ones.

That is what the asteroid belt looks like. The size distribution of main-belt asteroids, measured by surveys down to a few hundred metres and extrapolated below that from the craters they leave on other asteroids, changes slope at roughly the size the strength argument predicts — shallower above a kilometre or so, steeper below. The asteroid belt is not a pure equilibrium cascade; its largest bodies are survivors of its formation rather than products of collisions, and Jupiter’s resonances remove bodies of all sizes continuously. But the knee is where the strength curve puts it, and it is the one feature of the distribution that the self-similar theory cannot produce.

The Kuiper belt is harder to count than the asteroid belt, because its members are faint and distant, but the size distribution inferred from surveys and from the craters on Pluto and Charon has a feature of the same kind: a change of slope at a few kilometres to tens of kilometres, and a marked deficit of small craters on Charon that implies a shallow distribution of small impactors. Whether that feature is the strength knee, a relic of how the bodies formed, or both, is argued. The ice curve says only that a knee is expected somewhere near a few hundred metres for a population ground by collisions, and that the Kuiper belt’s much slower collisions may not have had time to grind it into place.

A cascade caught in a single collision #

The equilibrium picture describes a population ground down over billions of years by countless collisions. The asteroid belt also contains the products of individual ones, and they show the same strength physics from the other side.

An asteroid family is the debris of one catastrophic disruption: dozens to thousands of bodies sharing nearly the same orbit, because they left the parent with speeds of a few tens of metres a second — small compared with the orbital speed, and comparable to the parent’s own escape speed, which is exactly what the gravity regime says the dispersal must supply. How large a family is taken to be depends on where its membership is cut off, and its age can be read from how far its small members have drifted since, but the size distribution of its members is a direct record of how one body broke. Family size distributions are much steeper than the background’s — indices of four and above are common — because a single disruption produces a burst of fragments that the cascade has not yet had time to grind into equilibrium.

The ratio of the largest fragment to the parent is the other record. A parent struck just above its disruption threshold leaves a largest remnant of about half its mass, by the definition of the threshold; one struck far above it is pulverised. Reconstructing the parent from the family and comparing the largest remnant with it gives the energy of the event relative to the threshold, and families spanning a range of parent sizes trace out the disruption curve’s gravity-regime slope from observation rather than simulation. The agreement with the simulated curves is good, within the uncertainties of the reconstruction.

A wave that follows a cutoff #

There is a second departure from the self-similar power law, and it comes from the other end of the distribution.

A real cascade has a smallest size. In the asteroid belt and in debris discs, grains smaller than about a micron are blown out of the system by radiation pressure as soon as they are made. That removes the projectiles that would otherwise destroy grains just above the cutoff, so those grains are over-abundant; the excess of them destroys more of the grains a little larger, which are then under-abundant; and the alternation propagates up the distribution as a wave, decaying slowly with size. The same thing happens at the strength knee, where the change in slope perturbs the balance on either side.

The waves are real features of detailed cascade models and are seen in some observed distributions, but their amplitude and phase depend on the details of how fragments are distributed in each collision, which is the least certain input to any cascade calculation. The broken power law drawn here is the envelope around which those waves oscillate, not a replacement for them.

What the knee does to a debris disc’s mass #

The most consequential use of all this is in debris discs, the belts of dust around other stars that are the visible products of cascades among unseen planetesimals.

A debris disc is seen through its dust: the grains a few microns across that absorb starlight and reradiate it in the infrared, and that carry almost all of the cross-section. Its mass lies in its largest bodies, which are invisible. The mass and the light are in different bodies, and the only way to get from the observed brightness to the mass of the reservoir that supplies it is to assume a size distribution and integrate it from the dust to the largest planetesimals.

The result is a warning about how much the conversion depends on the assumed slope. A population described by the gravity regime’s index all the way down would have almost four orders of magnitude less emitting area per unit mass than a 3.5 cascade — so a given observed brightness would imply four orders of magnitude more mass. The two-regime distribution happens to come back close to the self-similar answer, within a factor of one and a half, because the steep strength-regime slope below the knee restores what the shallow slope above it removed. That near-agreement is a coincidence of the particular exponents, not a law, and a small change in either regime’s slope — from a different material, a different impact speed, or porosity — moves it by factors of several.

The practical consequence is that masses quoted for debris discs, converted from their infrared brightness, carry an uncertainty of an order of magnitude or more that comes from the size distribution alone. The largest bodies, which hold the mass, are never observed; the knee, which shapes the conversion, is set by the strength of materials nobody has sampled; and the dust that is observed is the product of collisions at sizes a factor of a hundred million smaller than the bodies that dominate the mass.

What the calculation assumes #

The disruption energies drawn here are fits to simulations of impacts into homogeneous targets at a single speed. Real asteroids are not homogeneous: many of those larger than a few hundred metres are rubble piles, already shattered and reassembled — the same aggregates whose response to a tide decides whether a satellite can hold together — whose strength is closer to that of a gravel heap than of solid rock, and porous bodies absorb impact energy by compaction rather than transmitting it as a shock. Both effects move the curve, and the porous case can move the minimum by a factor of several in size. The impact speed matters too: the strength regime depends only weakly on it, while the gravity regime strengthens at higher speed because more of the energy is lost to heating.

The speed matters for where each population sits on the curves. Collisions in the main asteroid belt happen at about five kilometres a second, because Jupiter keeps the orbits eccentric and inclined. In the Kuiper belt, where orbits are cold and the orbital speed itself is only a few kilometres a second, typical collision speeds are about one kilometre a second — twenty-five times less energy per kilogram of impactor — so a body there must be struck by a relatively much larger projectile to be destroyed, and the cascade grinds far more slowly. In a debris disc the collision speed depends on how strongly the planetesimals have been stirred, by planets or by their own largest members, and a disc that has not been stirred barely grinds at all. The strength curve says where a population is weakest; the stirring says whether that weakness is ever tested.

The equilibrium slope formula assumes a cascade in steady state, with the same impact speed at every size and fragments distributed by a fixed power law. The asteroid belt is not in steady state at its large end, where the bodies are primordial, and its impact speeds are set by Jupiter’s perturbations rather than by the cascade itself. The formula gives the right sense and roughly the right size of the effect; it does not give a distribution to be fitted to the belt point by point.

Still open: the strength of bodies nobody has touched #

The knee is set by the material strength of bodies a few hundred metres across, and that strength has now been measured directly for only a handful of objects. Spacecraft that have sampled small asteroids found them to be rubble piles of astonishing weakness, their surfaces behaving more like a fluid than a solid when disturbed — and an impactor deliberately crashed into one changed its orbit by more than models of a coherent target predicted, because it ejected far more material and reshaped the body it struck. If the bodies at the weakest size are weaker still than the curves drawn here, the knee moves and deepens, the distribution below it steepens, and every conversion of a debris disc’s glow into a mass changes with it. The strength of the smallest bodies is the input on which the dynamics of the largest populations depends, and it is measured one asteroid at a time.