A collision dated by a scatter plot

Almost nothing in the solar system carries a date. A crater can be counted against other craters, a meteorite can be dated in a laboratory once it has arrived, and everything else is inferred from a model of how things must have gone. The bodies themselves are silent: an asteroid seen tonight looks exactly as it would have looked a hundred million years ago, and the six numbers that fix its orbit contain no memory of when it acquired them.

A collisional family is the exception, and the reason is a force so weak that measuring it on a single body took a decade of radar and two spacecraft apparitions. Integrated over a hundred million years across a few thousand fragments, that force turns a shapeless cloud into a shape — and the shape is a clock.

What is being dated #

The object being dated is not an asteroid. It is an event: the catastrophic disruption of one parent body, which happened once, at a definite moment, and left several thousand fragments on orbits that were nearly the parent’s own.

Those fragments are recognisable because the orbital elements a catalogue reports are not the ones to look in. An orbit fitted to a few weeks of astrometry gives osculating elements, which contain the body’s own eccentricity plus the part that belongs to Jupiter, and the second part swings by far more than the whole width of a family. The change of basis that removes it — averaging away the short-period terms and subtracting the forced ones — leaves quantities that hold still for a hundred million years, and in those the belt is not smooth.

The force, and why it sorts by size #

Sunlight absorbed by a rotating body is not re-radiated straight back at the Sun. A rotating surface carries its heat round with it, so the warmest hemisphere is the afternoon one, and the recoil from the thermal photons has a component along the direction of motion. That is the Yarkovsky effect, and its size is set by a ratio the whole of this essay depends on: the absorbed power goes as the cross-section, the mass goes as the volume, so the acceleration goes as one over the diameter.

Two consequences follow immediately, and they are what make a family into a V rather than a blur.

The first is that the drift is ordered by size: fragments are sorted along the semi-major-axis axis in inverse proportion to their diameters, with no scatter of their own. The second is that the drift has a sign, and the sign is set by which way the fragment spins.

The spin does not hold still #

The account so far treats each fragment’s obliquity as a constant handed to it by the collision. It is not, and the reason is the same radiation that produces the drift.

The thermal photons leaving an irregular body carry away angular momentum as well as linear momentum, and for a shape that is not symmetric the net torque does not average to zero over a rotation. The effect spins a small asteroid up or down, and — more consequentially here — it drives the obliquity towards one of two attractors, near zero and near a hundred and eighty degrees.

That works for the method rather than against it. A fragment driven to an obliquity of zero or of a hundred and eighty degrees is drifting at the full rate rather than at some intermediate one, so the population migrates towards the two edges of the V and away from its middle. An old family should therefore be hollow — dense along the two boundaries, sparse between them — and several are observed to be exactly that.

The timescale is what makes it a correction rather than a rewriting. Driving a kilometre-sized body to an attractor takes tens of millions of years, comparable with the ages being measured, so a family carries fragments in every stage of the process at once. The largest members, which the torque reaches most slowly, keep something close to their original spins; the smallest have been reoriented several times over.

The consequence for a fitted age is a systematic one and it runs in a known direction. A fragment that spent part of its life at an inefficient obliquity has drifted less than the full-rate assumption credits it with, so the envelope it defines is narrower than a steady-drift calculation would produce, and an age fitted to it comes out too low. A fragment that has been reoriented from inward to outward drift has doubled back, which widens the cloud without extending the envelope. Modern determinations handle this by simulating the spin evolution rather than by correcting the slope, which is why a published family age now carries a model of thermal torques inside it and why two groups fitting the same points can differ by twenty per cent.

Reading the age off the picture #

Put the two together. After a time $t$ a fragment of diameter $D$ and obliquity $\gamma$ has moved

$$\Delta a \;=\; \frac{C}{D}\,t\,\cos\gamma,$$

with $C$ a constant of the body’s density, albedo and distance from the Sun. Plotted against $1/D$ the extreme fragments — those with $\cos\gamma = \pm 1$ — lie on two straight lines through the family’s centre, of slopes $\pm Ct$. The slope is the age.

That is a stronger statement than it looks, because it does not require knowing where any individual fragment started. The family’s original spread does not enter; only the envelope does. Nor does it require the drift of any one body to be measurable — the drift of a five-kilometre asteroid over a human lifetime is a few kilometres, which is nothing, and over a hundred million years it is a hundredth of an astronomical unit, which is the whole width of the picture.

The size dependence is what makes the method work at all. If every fragment drifted at the same rate the cloud would spread into a rectangle, and a rectangle has no slope to fit. A rate that goes as $1/D$ converts a duration into a shape, and a shape survives having no clock in it — much as an age is read off the bend in a cluster’s main sequence rather than off any one star in it.

What was actually measured #

The Erigone family drawn at the top of this essay sits near 2.37 AU, is composed of dark carbonaceous material, and has a V whose fitted slope gives an age of about 130 million years. Its members span diameters from roughly 1.5 to 30 kilometres; the smallest have moved 0.05 AU and the largest a tenth of that.

None of that is an observation of an age. What is observed is a list of proper elements, a list of absolute magnitudes, and an assumed albedo that turns magnitudes into diameters. The chain from those to a date has three assumptions in it and each is worth naming.

The albedo. Diameter enters as $1/D$, so a systematic error in the albedo scales the fitted age directly. Families are typically homogeneous in composition — that is one of the things that identifies them — so the assumption is defensible, but a ten per cent error in albedo is a five per cent error in the age.

The thermal efficiency. The constant $C$ contains the conductivity of the surface and the spin period, neither known for any individual member. It is calibrated against the handful of near-Earth asteroids whose drift has actually been measured, and transplanted to the belt with a correction for distance — the same efficiency four measured near-Earth objects agree on.

And the assumption that the drift has been steady. A fragment’s spin state is not permanent: collisions reorient it and the thermal torques that accompany the same radiation spin bodies up and down over tens of millions of years. Over a family’s lifetime a small member changes obliquity several times, so its drift is a random walk with a bias rather than a straight line. That widens the envelope and biases the age high, and the correction is the largest single term in a modern family-age error bar.

The family that was dated twice #

One family has been dated by a method that shares nothing with the one above, and the agreement is the best evidence the technique has.

About 466 million years ago a body roughly 150 kilometres across broke up in the inner belt. Its fragments are the L-chondrite parent population, and a large share of them reached the Earth: Ordovician limestone quarried in southern Sweden contains fossil meteorites, altered but recognisable, at a concentration a hundred times the background rate, over a stratigraphic interval a couple of million years thick.

Two independent clocks agree on when that happened. The meteorites themselves carry cosmic-ray exposure ages — the accumulated damage from galactic cosmic rays, which begins the moment a fragment becomes small enough for the rays to reach its interior — and those ages are short, a hundred thousand years to a million, meaning the fragments arrived quickly after being liberated. Their shock ages, measured by argon retention in the laboratory, cluster at 470 million years. And the limestone they are embedded in is dated by its own biostratigraphy to the same interval.

The dynamical clock is the one this essay is about: the family left in the belt has a V, and the V gives an age consistent with the rock.

The rock and the scatter plot are measuring the same event through entirely different physics. One is a fitted envelope in a plot of orbital elements, sensitive to thermal conductivity and to an assumed albedo. The other is a layer of Baltic seafloor, sensitive to sedimentation rates and to the survival of a chondrule through 466 million years of diagenesis. There is no shared assumption to make them agree, and they do.

Where the picture stops #

The V is not a clean object, and four things spoil it.

Interlopers. A family is identified as a clustering, and the belt has a background. Any body that happens to lie inside the V’s boundary is counted, whether or not it came from the parent, and the interlopers are preferentially the large ones — a big background asteroid is common and a big family member is rare. Since the fit is driven by the envelope, a single large interloper near the outer edge can add tens of per cent to an age.

Resonances. A family that spreads far enough reaches a mean-motion resonance, and the resonance does not merely bound it — it removes the members that arrive. So the oldest families are truncated, their V cut off at the resonance rather than continued, and their ages are lower limits. Chaotic diffusion. Proper elements are quasi-integrals rather than integrals. In the outer belt, where secular resonances overlap, they wander slowly, and after a billion years the wandering is comparable with the family’s own width. And the ejection velocity. The fragments did not start in one place. A catastrophic disruption gives them relative speeds of tens of metres a second, which through the variational equation for the semi-major axis is a spread of a few thousandths of an astronomical unit — the rounding at the bottom of the V. For a young family that spread is larger than the drift, the V has no slope to fit, and the age has to come from somewhere else: for families a few million years old the fragments’ orbits can be integrated backwards until their nodes and perihelia converge, which dates the Karin family to 5.8 million years and works for nothing older than about ten.

The same construction at a much older age and in a different part of the belt shows what the method loses with time.

Finding a family by its shape rather than by its clustering #

The method has an application that inverts its usual direction, and it recovers objects that clustering cannot.

A family is normally identified first and dated second: a clustering algorithm finds an overdensity in proper elements, and the V is then fitted to its members. That order fails for the oldest families. After a billion years the drift has carried the small members so far that the group is no longer an overdensity — it has spread across a substantial fraction of the belt, its members are outnumbered by the background at every point, and no clustering algorithm will return it.

What survives is the shape. However far the family has spread, its members still satisfy the relation between semi-major axis and inverse diameter, and the background does not: a background asteroid’s position in that plane is uncorrelated with its size. So a search that looks for V-shaped correlations rather than for density peaks can find a family that has stopped being a cluster.

Searches of that kind have recovered several ancient groupings in the inner belt, with fitted ages approaching four billion years — old enough that they are candidates for the primordial planetesimals rather than for later collisions among them. The claim is contested in exactly the way it should be, because the signal is a weak correlation against a large background, and the statistical question of how often a random subset of the belt happens to fall inside a V is not trivial.

Two things make the result worth pursuing anyway. The first is that the recovered groups have coherent spectral properties, which a chance alignment in one plane has no reason to produce. The second is that the number of large asteroids not assigned to any family is itself a measurement: if most of the belt’s big bodies turn out to belong to a handful of ancient families, then the belt was assembled from a much smaller number of parent bodies than its present population suggests, which is a statement about the earliest solar system rather than about collisions in it.

The method also has a boundary worth naming. Everything here requires a population whose members share an origin and are sorted by a size-dependent force acting steadily for a long time. Outside the main belt the sorting is spoiled — a near-Earth object’s orbit is rewritten by planetary encounters far faster than the thermal drift can order it — so the V exists only where nothing else is stirring.

The generalisation #

The pattern here recurs wherever a population is acted on by a force that depends on a property the population happens to spread over.

A cluster of stars dissolving into a galaxy loses its lowest-mass members first, because two-body relaxation drives them to the highest speeds, so the surviving mass function is a clock on the same principle. Interstellar dust is sorted by radiation pressure in proportion to its cross-section over its mass, which is again one over the size. A debris disc’s grains spiral in under Poynting–Robertson drag at a rate that goes as one over the grain radius, so the observed size distribution of the dust dates its production.

In each case the ingredient that makes it work is the same: a size-dependent rate, acting for a time nobody watched, converting a duration into a shape that can be measured now.

And the belt itself drawn with a denser background, which is the condition every family identification has to work against.

Where this ladder goes next #

Later rungs on this anchor: the size–frequency distribution of a family, whose slope is a signature of how the parent broke and whose knee records the smallest fragments the collision made; the identification problem itself, which is a clustering algorithm run in a three-dimensional space with a metric that has to be chosen and whose choice sets how many families exist; halo and satellite families, produced when a family member is itself disrupted; the very young families, dated by orbital convergence rather than by drift; and the connection to meteorites, where the spectral class of a family and the composition of a meteorite fall are matched to say which rock in a museum came from which event in the belt.