An age read from angles that meet

The asteroids that belong to a collisional family are fragments of one body, shattered by an impact and thrown onto neighbouring orbits at a few metres a second. They are recognised by what they still share: proper elements, the parts of their orbits that the planets’ perturbations leave constant, which cluster tightly when everything else about the orbits has been scrambled. And they are dated, most of the time, by how far sunlight has since pushed them apart: small fragments drift faster than large ones, so the family spreads into a V in a plot of semi-major axis against size, and the V’s opening angle is an elapsed time. It is the only way to date most families, and it is a clock with a calibration uncertain by a quarter and a membership that is itself a choice.

For a handful of very young families there is a second clock, and it is of a different kind. It does not infer a time from a rate. It finds an instant.

Orbits that turn at slightly different rates #

Every asteroid’s orbit precesses. The long axis of the ellipse turns slowly forward, and the line where the orbit crosses the plane of the planets turns slowly backward, both driven mostly by Jupiter; in the outer belt the rates are around seventy arcseconds a year each way, a full turn in about eighteen thousand years. The rate depends on the asteroid’s distance from the Sun — it rises steeply towards Jupiter — and the linear theory that describes it gives it as a function of the semi-major axis alone.

The fragments of a collision are thrown onto orbits whose semi-major axes differ by a few thousandths of an astronomical unit, and their precession rates therefore differ by a fraction of an arcsecond a year. At the distance of the family drawn here, the rate changes by about seventy arcseconds a year for each astronomical unit of semi-major axis, so two fragments a thousandth of an astronomical unit apart precess at rates differing by seven hundredths of an arcsecond a year — twenty degrees per million years. That is nothing at first. Immediately after the breakup the fragments’ orbits are nearly identical in orientation: their perihelia and nodes differ by a few degrees at most, set by the direction each fragment was thrown. But a fraction of an arcsecond a year for a million years is several degrees, and for five million years it is tens of degrees or, for the members at the edges of the family, several whole turns. A family a few million years old has had its members’ orientations spread round the whole circle, and a snapshot of their present perihelia looks as random as a snapshot of any unrelated asteroids.

Run backwards, they meet #

The spreading is deterministic, and it can be undone. Each member’s present orientation is its initial one plus its precession rate times the elapsed time; running the calculation backwards subtracts the same amount; and at the instant the fragments were made, every member’s angle returns to within a few degrees of every other’s.

The spread of the angles against time before the present is a flat line at its random value almost everywhere and a sharp dip at one time. The dip’s bottom is set by the few degrees the collision itself spread the orientations; its sharpness by how different the members’ rates are. For a family like the one drawn — members spread over seven thousandths of an astronomical unit, their rates spread by about seventy degrees per million years — the angles stay within twice their minimum spread for only about fifty thousand years either side of it. An age of 5.75 million years with an uncertainty of fifty thousand is a measurement of an instant, not an inference from a drift, and it is closer in kind to timing an eclipse than to reading an isotopic clock.

The nodes give a second convergence, from the same members and the same backward calculation, because the nodes regress while the perihelia advance. In the linear theory used for the figures the two rates are exactly equal and opposite, so the two convergences are mirror images and add nothing; in the real family the rates come from a full theory in which they differ, and two convergences at the same instant from two different sets of angles are a genuine check that the instant is the collision and not a coincidence of rates.

The small ones need the sunlight put back #

A backward calculation with fixed semi-major axes works for the large members and fails for the small ones, and the reason is the same force that draws the V-shape.

A small asteroid absorbs sunlight on its day side and re-emits it as heat a few hours later, towards its evening side, and the recoil of the delayed emission pushes its orbit outward if it spins in the same sense as it orbits and inward if it spins the other way. The push is inversely proportional to the body’s size: a few ten-thousandths of an astronomical unit per million years for a body a kilometre across, a tenth of that for ten kilometres. Over five million years a kilometre-sized fragment’s semi-major axis has changed by about a thousandth of an astronomical unit — comparable to the spread the collision gave it — and its precession rate has changed with it.

Run such a member back with its present semi-major axis and its rate is wrong throughout, by an amount that grows the further back the calculation goes. The error in its angle grows as the square of the time. In the figure the members smaller than four kilometres never come closer together than seventeen degrees when their drift is ignored, while the larger members converge to a few degrees either way. Put each member’s drift back — let its semi-major axis change linearly with time in the backward calculation — and the small members converge as sharply as the large ones.

The drift rates that make them converge are not assumed; they are fitted, one per member, as the rates that bring its angles to the common value at the common instant. That makes each converging family a measurement of the thermal drift of kilometre-sized bodies — of their spin directions and of how well their surfaces conduct heat — for objects no telescope can resolve. The drift rates recovered for the young families are consistent with surfaces of fine regolith on the larger members and bare, more conductive rock on the smallest, what the drift rate says about the surface read off a family’s backward convergence.

A family found inside a family #

The first family dated this way was found by looking more closely at an old one. The Koronis family in the outer belt is about two billion years old, spread by drift into a broad V, and its proper elements had been computed for a few hundred members. When the catalogue of asteroids with well-determined orbits grew into the tens of thousands at the turn of the century, and proper elements were computed for all of them, a tight sub-cluster appeared inside Koronis: a few dozen small bodies clumped far more closely than the old family’s members, around a seventeen-kilometre asteroid named Karin. A clump that tight in proper elements had to be young, because drift and chaotic diffusion would have spread an old one.

The test of youth was the backward integration. The members’ orbits were integrated backwards through the full gravitational field of the planets — not the linear theory used for the figures here, but a direct numerical integration of every orbit — and their perihelia and nodes, spread round the circle today, came together at about five million eight hundred thousand years ago. Including the thermal drift of the smaller members later sharpened that to 5.75 million years with an uncertainty of about fifty thousand. The collision that made them happened at around the time the human and chimpanzee lineages were separating, and its date is known far more precisely than that one.

Pairs that parted a few thousand years ago #

The same backward calculation works at the smallest scale there is: two asteroids on orbits so similar that they were one body within human prehistory. Such pairs were first noticed in 2008, by searching for asteroids whose osculating orbits — the instantaneous ones, not the proper — were nearly identical. For a pair only a few thousand years apart, even the fast short-period oscillations have not had time to separate them, and the backward integration converges not just in the secular angles but in the positions of the two bodies themselves, which can be brought to within a few times their own radii of each other at a specific date.

The pairs are not collisional. The larger member of almost every pair is a fast rotator, spinning close to the rate at which a loosely bound body would fly apart, and the smaller is a few per cent of its mass or less. They are the products of rotational fission: a rubble pile spun up by the same asymmetric re-emission of sunlight that drives the drift — acting on its spin rather than its orbit — until a piece came away at the spin barrier. The backward convergence dates the fission, and the pairs’ separation speeds, a few centimetres a second, are the escape speeds of kilometre-sized bodies. It is the same clock, pointed at a different kind of breakup.

A horizon set by what is not known #

The method has a range, and the range is set by the thing that had to be put back.

Two kinds of error accumulate in a backward calculation. An error in a member’s present semi-major axis — orbits of well-observed asteroids are known to a hundred-thousandth of an astronomical unit — makes its rate wrong by a fixed amount, and the error in its angle grows in proportion to the time; at that precision the angle would still be known to a quarter of a turn after four hundred million years. An error in its drift rate grows faster. A member whose spin axis and surface are unknown has a drift that can be anything between its two extremes, and the resulting angle error grows as the square of the time: a quarter of a turn after five million years for a kilometre-sized body, after sixteen million for one ten kilometres across.

So the horizon is the drift, not the orbit. A family older than about fifteen or twenty million years has members whose angles cannot be run back reliably unless their drifts are known independently, and the drifts are what the convergence itself was measuring. Weak resonances with the planets add chaotic diffusion on top — small, random changes in the proper elements that grow with time and that no backward calculation can remove. The families dated this way are therefore few and all young: one of about five and three-quarter million years in the outer belt, one of about eight million at the belt’s outer edge, and several of a few hundred thousand years or less, down to pairs of asteroids that split apart only a few thousand years ago.

Two clocks that barely overlap #

The V-shape clock, which dates the old families, has the opposite limitation: it cannot date young ones at all.

A V-shape is read as drift, but the collision itself spread the fragments before any drifting began. Fragments of five kilometres thrown at fifteen metres a second land on semi-major axes that differ by a few thousandths of an astronomical unit, and that initial spread draws the same V that seventy million years of drift would. A family younger than that reads as seventy million years old by the V alone, and even an old family carries the quarter uncertainty of the drift’s calibration. The converging angles have the reverse profile: their error is a fixed interval of tens of thousands of years, so their fractional error falls as the family ages, until the drifts scramble the angles.

The two clocks’ ranges barely meet. Below ten million years the angles give ages to a per cent; above a hundred million the V gives them to a quarter; between, an age requires fitting the collision’s own ejection velocities together with the drift, which ties the answer to a model of how the parent body broke. The ten to a hundred million years in between is where the evidence about the belt’s collisional history is thinnest, and where the rate of family-forming collisions in the recent past is least well measured.

Dust on the Earth from a collision in the belt #

The young families have a connection to the Earth that no old one could.

A collision that makes a family also makes dust, far more of it by surface area than the fragments though far less by mass, and the dust is ground down further by collisions among its own grains. It spreads in the same way the fragments do: its orbits precess at rates set by their semi-major axes, so within a million years or so a dust cloud from one collision has been spread into a band round the Sun, confined in inclination by the family’s shared orbital plane. The infrared sky surveys of the 1980s found such bands in the zodiacal light, at inclinations matching specific families — and the brightest of them match the youngest families, those dated by converging angles, because the dust from older collisions has had time to be ground down and swept inward.

Dust from the asteroid belt spirals inward under the drag of absorbing and re-emitting sunlight, and some of it reaches the Earth. Deep-sea sediments record the rate at which it arrived, through a rare isotope of helium that interplanetary dust carries implanted from the solar wind and that terrestrial rock lacks. The sediments show a spike in that helium beginning about eight million years ago and decaying over the next million or two — a surge of extraterrestrial dust arriving at the Earth. The collision that made the eight-million-year-old family, dated by its members’ converging angles to within a hundred thousand years, was followed by a dust surge dated in ocean-floor mud to the same time. Two clocks that share nothing, one run on asteroid orbits and one on marine sediments, read the same instant.

What the synthetic family does not include #

The figures use a synthetic family, placed at the position of the five-and-three-quarter-million-year-old family in the outer belt and run through the linear secular theory, not the actual members fitted with a full theory. The rates in the linear theory depend on the semi-major axis alone, which makes the node and perihelion convergences exact mirrors and removes the check that their agreement provides. The members’ drifts are drawn from a uniform range and are known exactly in the backward calculation that includes them; in a real analysis they are fitted and carry uncertainties, and the convergence they produce is correspondingly broader. The collision’s initial spread in angles is drawn as a few degrees for every member regardless of size, whereas real fragments of different sizes were thrown at different speeds. And the secular theory used here ignores the weak resonances that make the real integrations chaotic over long times; the horizon drawn is the drift’s alone.

Still open: how often the belt breaks #

The converging families are few, but they are the only families whose ages are measurements, and they are young enough to say something about the present. A handful of families formed in the last ten million years from parents tens of kilometres across implies a rate of large collisions in the belt today that can be compared with the rate the belt’s size distribution and the number of old families imply — and with the rate at which impacts have cratered the Moon and the Earth recently. Those rates do not yet agree cleanly, and the gap between the two clocks is where the disagreement could be resolved: families of twenty to eighty million years, too old for the angles to be run back and too young for the V to be read, dated only by modelling how their parents broke. How often the belt breaks, measured rather than inferred, depends on closing that gap.