How often the belt breaks, counted from its own collisions

An asteroid family is a clump of orbits — fragments of one parent body, shattered in a single collision and still travelling on nearly the same path round the Sun. The families have been dated by the spread of their fragments, which drift apart at a rate set by their size; checked by their colours, which the fragments of one body share; and, for the youngest, timed to within thousands of years by turning their orbits back until they meet. What the catalogue of families has not yet been made to answer is the plainest question about them: how often the belt breaks.

That question can be asked from the other end. The asteroids collide with one another at a rate that can be computed — from how often two orbits bring their bodies to the same place, from how many bodies there are of each size, and from how much energy it takes to shatter a body of each size. Each of those ingredients is measured or calibrated independently of the families. Put together, they predict how many catastrophic disruptions the belt should have had of parents of each size, over any interval, and the families found can then be set against the prediction.

This essay does that arithmetic. It finds that today’s belt accounts for the large families without needing an earlier, busier one, that the agreement rests on a strength law no experiment can test at the relevant size, and that the youngest large family in the catalogue is an unlikely event.

The targets and the bullets #

The belt is a population of a little over a million bodies larger than a kilometre, eight thousand larger than ten and about two hundred and thirty larger than a hundred. Its size distribution is not a single power law: it has a slope that steepens and flattens across the range, with a well-known bump near a hundred kilometres where the primordial survivors sit and another near a few kilometres where the collisional cascade has carved it. The model used here passes through a set of anchor points that span the range and interpolates between them with power laws.

Every target has a set of bullets. A body is catastrophically disrupted — shattered, with at least half its mass dispersed — when a projectile delivers enough kinetic energy per unit mass of target, the quantity called $Q^*_D$. At the belt’s mean impact speed of 5.3 kilometres a second, that fixes a smallest shattering projectile for each target, and the number of potential shatterers is the number of bodies larger than that. A hundred-kilometre asteroid is shattered by anything larger than about twenty kilometres, and there are about two and a half thousand of those. A ten-kilometre asteroid is shattered by anything larger than seven hundred metres, of which there are more than two million.

How small a stone can break each asteroid #

The ratio of bullet to target is the strength law seen from a different side. Small bodies are held together by the strength of their rock, which weakens with size because a larger body contains larger cracks; large bodies are held together by their own gravity, because fragments that are broken apart must also be thrown faster than escape speed or they fall back together. The weakest bodies sit between the two regimes, at a few hundred metres, where a projectile two per cent of the target’s diameter — a ten-metre boulder against a half-kilometre asteroid — suffices.

Above that size, the ratio climbs steeply. A hundred-kilometre asteroid needs a projectile a fifth of its size, and a five-hundred-kilometre one needs a projectile two-fifths of its size: Vesta, at 525 kilometres, could be shattered only by a body larger than two hundred kilometres, of which the belt now holds about thirty. The steepness is the reason large asteroids survive for so long, and it is set by the gravitational term in the strength law — a term calibrated on computer simulations of impacts into self-gravitating rubble, because no laboratory can hit a body a hundred kilometres across.

How long an asteroid lasts #

The rate at which a given target is shattered is the intrinsic collision probability — the chance per year that two particular asteroids in typical belt orbits come within a kilometre of each other, $2.86 \times 10^{-18}$ per square kilometre per year, computed from the distribution of orbits — times the target’s collisional cross-section, times the number of bullets. Its inverse is the collisional lifetime.

A one-kilometre asteroid lasts a little under a billion years before something shatters it: the belt’s kilometre-sized bodies have been replaced several times over the age of the solar system, made from the fragments of larger ones. A ten-kilometre asteroid lasts about five billion years, comparable to the age of the solar system. A hundred-kilometre asteroid lasts nearly forty. So the largest bodies are mostly primordial, survivors from the belt’s formation, and only a fraction of them have been shattered since. When one is, its fragments are a family — and because a large parent makes fragments of many sizes, including large ones that outlive the event by billions of years, its family persists.

How many families four billion years should make #

Summing the disruption rate over every target larger than a given size gives the number of family-forming events of parents that large per unit time. Over four billion years, with today’s belt and the standard strength law, the prediction is about twenty-three disruptions of parents larger than a hundred kilometres, seventy larger than fifty, and six hundred larger than twenty.

The catalogue of families, built from the clustering of proper orbits and from the spread of their fragments, contains about twenty whose parents are estimated to have been larger than a hundred kilometres. The agreement is close enough to be informative, and its meaning is that the belt as it is now could have produced its large families on its own. That was not guaranteed. The belt is thought to have been much more massive early on, depleted by a factor of a hundred or more as Jupiter’s resonances and the giant planets’ migration swept through it; a belt that was ten times busier for its first few hundred million years would have made ten times as many families in that interval, and the old large families would outnumber the prediction. They do not, by much, which says that most of the large families now visible formed after the early depletion, or that the early ones have been erased.

The agreement also depends on the strength law. Halve the strength and the count of large-parent disruptions rises to about thirty; double it and the count falls below twenty. The gravitational part of the strength at a hundred kilometres is a simulation result, and simulations of different internal structures — solid rock, rubble, porous material — differ by factors of two or more. So the twenty families are as much a measurement of how hard it is to shatter a hundred-kilometre asteroid as they are a test of the rate: given the collision rate, the family count calibrates the strength law at the size no experiment can reach.

Why the small families are missing #

The prediction for small parents is a different matter. Six hundred disruptions of parents larger than twenty kilometres over four billion years is far more than the number of families with parents that size in the catalogue. The difference is not a failure of the rate. It is the fading of small families.

A family is recognised because its fragments cluster in orbit more tightly than the background — how tightly is itself a choice of cutoff — and that clustering decays. The fragments drift apart through the Yarkovsky effect, the slow push of re-emitted sunlight, at rates inversely proportional to their size, and they are themselves broken up by collisions on their own lifetimes — under a billion years for kilometre-sized pieces. A family from a twenty-kilometre parent has no fragment larger than a few kilometres; within a billion years most of its members have drifted out of the cluster or been ground down, and it merges into the background. A family from a hundred-kilometre parent has fragments of tens of kilometres that outlast the event by billions of years, and its core stays recognisable.

So the observed census is the formation rate multiplied by a detectability that falls steeply with decreasing parent size and increasing age. The large families compare cleanly with the rate because their detectability is close to one for their whole history; the small families compare only through a model of their fading, which is why the young small families, still compact, are the other clean test.

What the last ten million years should hold #

In a window short enough that nothing has faded, the rate can be compared with the count directly. Over the last ten million years, today’s belt should have shattered about ten bodies larger than ten kilometres, one or two larger than twenty, and a hundred-kilometre body with a probability of about six per cent.

The young families found so far fit that picture at the small end. The Karin family, dated to 5.8 million years by the convergence of its members’ orbital angles, came from a parent of about thirty-three kilometres, and the expected number of such events in its window is about one-half — an ordinary event. Several other young families, a few hundred thousand to a few million years old, came from parents of a few to ten kilometres, and their number is in line with the ten or so expected.

The Veritas family is the outlier. Its age, 8.3 million years, is measured precisely, and its parent was larger than Karin’s — estimates range up to about a hundred kilometres or more, depending on how much of the family’s mass is hidden in fragments too small to detect. If the parent was near a hundred kilometres, its disruption in the last ten million years had about one chance in eighteen in a steady belt. That is unlikely but not absurd, and there is only one such family; the reasonable reading is that the belt was lucky, or that the parent was smaller than the larger estimates. Either way it matters beyond the belt: the Veritas and Karin collisions threw out dust that spreads round the Sun and is seen as bands in the infrared sky, and a share of the dust that falls on the Earth — including the extraterrestrial helium-3 spike found in ocean sediments about eight million years old — has been traced to them.

Dust that reaches the Earth from a breakup #

A catastrophic disruption makes far more than the fragments a survey can count. The same size distribution that gives a family its few dozen kilometre-sized members gives it trillions of grains, and the grains do not stay in the family’s orbit. Sunlight acting on them — the radiation pressure that blows the smallest out of the solar system and drags the larger inward over tens of thousands of years — spreads them into a torus round the Sun at the family’s inclination, and then draws them slowly towards the inner solar system. The infrared survey satellites of the 1980s found such tori as bands of thermal emission above and below the ecliptic, and two of the brightest were traced to the Karin and Veritas families.

That dust reaches the Earth. Interplanetary dust carries helium-3 implanted by the solar wind, and the concentration of extraterrestrial helium-3 in deep-sea sediments is a record of how much dust has fallen over time. The record shows a spike beginning about eight million years ago that rose to several times the background and decayed over about a million and a half years — the time it would take dust from a fresh breakup to spiral in and be used up. Its timing matches the Veritas family’s age, measured independently from the convergence of its orbits.

The comparison works in the other direction too. If the rate here is right, the sediments should hold a spike of that kind every few hundred million years from parents of fifty kilometres or more, and more often from smaller ones. The record is long enough to test that, and the spikes identified in it so far are few. The rate of large disruptions is therefore written twice in the solar system: in the clumps of orbits in the belt, and in the dust layers on the floor of the Earth’s oceans — two records made by different physics that ought to agree, and so far do.

What a hundred million years should hold #

Stretching the window to a hundred million years tests the census of compact families rather than individual events. A steady belt should have produced about a hundred disruptions of parents larger than ten kilometres, fifteen larger than twenty, one or two larger than fifty, and has a fair chance — forty-four per cent — of one larger than a hundred. Families from that interval are still compact enough to be found, and the census of young families from the current generation of surveys, which reach asteroids of a few hundred metres, is approaching completeness for parents above about twenty kilometres.

If that census finds substantially more young families than these numbers, the present rate is higher than the long-term average — perhaps because one recent large disruption has flooded part of the belt with projectiles, raising the local collision rate. If it finds fewer, either the strength law is too weak at the relevant sizes or the size distribution of small projectiles is shallower than modelled. Either result recalibrates the arithmetic used for the families’ ages and for the flux of fragments delivered towards the Earth through the resonances, the same resonances that emptied the gaps in the belt.

What the arithmetic leaves out #

The size distribution is a smooth model through anchor points, not the measured distribution with its waves, and below half a kilometre it is extrapolated. The intrinsic collision probability and mean impact speed are belt-wide averages; the inner belt, where speeds are higher, and the outer belt, where bodies are darker and more porous, differ. The strength law is the basalt fit from impact simulations, applied at all sizes and at a single speed, and it ignores the porosity and rubble-pile structure that most bodies above a few hundred metres are thought to have. The rate is today’s, held constant over four billion years, which ignores the belt’s early depletion and the slow decline of its population since. The family counts quoted are rounded summaries from the literature, and the parent sizes of large families depend on corrections for fragments too small to see.

Still open: whether the belt broke more often in its youth #

The large families agree with today’s rate, and that agreement has two readings. Either the belt has been near its present state for most of its history, its early massive phase short and its families from that time erased, or the strength of large asteroids is higher than the standard law and the early busier belt made more families that the higher strength then prevented from forming later. The two are distinguished by the ages of the large families: a steady rate gives ages spread uniformly over four billion years, an early burst gives a pile-up of old ones.

The ages of the old families are read from the V-shaped spread their fragments acquire, and that spread depends on the Yarkovsky drift rate, which depends on the fragments’ sizes, densities, spins and thermal properties — all of which have to be modelled — and on how much the initial ejection velocities contributed. The ages are uncertain by tens of per cent for the oldest families. A census of large families with ages good to ten per cent would show whether their formation was steady. Until it exists, the arithmetic says only that the belt now breaks about as often as its large families say it has been breaking, and that it could not have broken much more often for long without leaving more wreckage than there is.