A surface dated by counting holes in it

Somebody standing on a lava flow in the Elysium region of Mars, asked how long ago it erupted, would have no way of finding out. Nothing on the surface carries a date. The rock has minerals in it whose radioactive clocks could be read in a laboratory, and there is no laboratory; the flow overlies older terrain and underlies younger, which fixes an order and not an age; and no photograph of it distinguishes a hundred million years from a billion.

It is nevertheless dated, and the number has an error bar, and the method is to count the craters on it. The situation is the one this collection keeps arriving at from different directions — nothing in the sky is weighed in kilograms and nothing in it is dated by being asked — and the response is the same: find a process whose rate is known, and count what it has left behind. That method is the only chronometer the solar system possesses that works from orbit, and everything from the resurfacing of Europa to the volcanic history of Venus rests on it.

Why a count is an age at all #

The premise is almost embarrassingly simple. Impacts arrive at some rate; each leaves a crater; craters accumulate; so the number per unit area increases with the time the surface has been exposed. Reset the surface — flood it with lava, resurface it with ice — and the count starts again from zero.

Two things have to be true for that to be a measurement rather than a slogan.

The rate has to be known, which is what the calibration supplies. And the craters have to accumulate rather than saturate, which is not always true and is the subject of the next section.

There is a third condition that is usually left unstated because it is usually met: the surface has to be old enough to have collected a statistically useful number of craters and young enough not to have collected too many. A hundred craters on a counting area gives a ten per cent Poisson error, which is the same error bar that comes from counting anything and which is rarely the dominant one. Ten craters gives thirty per cent, and on a small young flow ten is often all there are.

The slope of minus two is not arbitrary. It follows from the size distribution of the impactors, which is itself a collisional equilibrium: a population being ground down by mutual collisions settles towards a power law of a particular index, and the fragments of a single disruption join that population and are worn into it. The impactor distribution and the crater distribution are related by an impact-scaling law — how large a hole a given projectile at a given speed makes in a given material — which is calibrated by explosion experiments and hypervelocity gun ranges, and which is one of the softer joints in the chain. The impactor sizes themselves are inferred rather than measured, since the bodies are unresolved points: what is observed is a brightness, and turning a brightness into a diameter requires an assumed albedo, while an occultation gives a true size for the handful of objects that happen to pass in front of a star.

The count that stops counting #

A surface cannot accumulate craters indefinitely. At some density every new crater lands on top of an old one and destroys it, and the number stops rising.

This is the single most common way a crater age goes wrong, and it goes wrong in a specific direction: counting small craters on old ground underestimates the age, because the small craters stopped accumulating long ago and the count is reporting the strength of the regolith rather than the exposure of the surface. A count that includes them is a measurement of something real and it is not a measurement of time.

The diagnostic is visible in the data itself. A measured size–frequency distribution that follows the production slope down to the smallest craters counted is a surface below saturation everywhere; one that bends away from the production slope at some diameter has saturated below that diameter, and the bend is where the counting has to stop.

There is a second and nastier contaminant at small sizes. A large impact throws out debris at above the local escape speed, and that debris comes down as secondary craters — thousands of them, clustered and often in chains, all made in one instant. A field of secondaries looks like an old surface and is a single event, and distinguishing them from primaries below a kilometre is genuinely difficult on any body with a low escape speed.

The two contaminants pull the same way and for different reasons, which is unhelpful. Saturation removes small craters that should be there; secondaries add small craters that should not be counted; and both are confined to the small end, which is exactly where the counting statistics are best and the temptation to count is strongest. The discipline the method requires is to throw away the part of the data that has the most of it in.

The calibration, and how narrow it is #

The curve in the first figure has a shape from the crater counts and an absolute scale from somewhere else entirely: rocks brought back from the Moon and dated in laboratories by argon and rubidium–strontium methods.

The whole chronology of the solar system rests on nine landing sites — six Apollo and three Luna — and, more precisely, on the requirement that the samples returned from each be genuinely representative of the surface whose craters were counted. That requirement is not automatic. A sample from a crater rim may be older than the plain it sits on; a breccia is an average over whatever the impact assembled; and the Apollo 14 material is thought to date the Imbrium basin rather than the Fra Mauro plains it was collected from.

Nine points, calibrating a curve that spans eight orders of magnitude in crater density and is then applied to every solid surface in the solar system. That is worth holding in mind whenever a Martian age is quoted to two significant figures.

The transfer to another body adds its own factors. The impact rate at Mars is not the rate at the Moon: Mars sits nearer the belt, its encounter speeds are lower, and its larger mass focuses more of what passes. Each of those is a multiplicative correction, each is uncertain by tens of per cent, and they do not cancel.

Where the impactors come from #

The projectiles are not a fixed background. They are the small end of a population that is continuously resupplied, and the resupply chain is one this collection has already built. The delivery is efficient enough that a fragment’s dynamical lifetime once it leaves the belt is a few million years — short compared with everything else in this essay — so the population of Earth-crossing bodies at any moment is a steady state between supply and removal rather than a reservoir being drained. The quantity that survives a planetary encounter is what makes that population traceable at all: an object’s orbit changes at every pass, and one combination of its elements does not, so a body can be recognised as belonging to a source region long after its orbit has stopped resembling one.

The chain has a consequence the crater clock has to assume away. A large collision in the belt raises the supply of small bodies, and the supply then decays as the fragments are removed. The impact flux is therefore not constant on timescales of tens of millions of years, and the chronology function’s linear term is an average over exactly the fluctuations it cannot see.

The clock is sharpest where the surface is oldest #

The most counter-intuitive property of the method falls straight out of the shape of the curve.

The practical consequence is that the early solar system is much better dated than the recent one. The heavy bombardment is pinned to within a hundred million years or so; the eruption ages of young Martian flows are quoted with error bars comparable to the ages themselves, and two studies of the same flow can differ by a factor of three without either being wrong.

The flux, measured directly #

The chronology’s linear term is an impact rate averaged over billions of years, and there is now a direct measurement of the present rate to compare it against.

The method is repeat imaging. A spacecraft in orbit photographs the same terrain twice, years apart, at a resolution fine enough to see a crater a few metres across, and the difference between the two images contains any crater that formed in between. Hundreds of new impacts have been found on Mars this way, and dozens on the Moon.

Each detection is unambiguous in a way a count is not: there is a before and an after, so nothing has to be assumed about saturation, about secondaries, or about what counts as a crater. What has to be assumed instead is the effective search area and the search efficiency, which depend on how much terrain was imaged twice and on how well the automated comparison finds a small dark spot.

The rates that come out are broadly consistent with the chronology’s linear term, and “broadly” is doing some work: the direct measurement is a rate for craters of metres and the chronology is anchored on craters of kilometres, so comparing them requires extrapolating a size distribution across three decades. The comparison tests the shape as much as the normalisation.

There is a further complication that the direct measurement uncovered. A substantial fraction of the new Martian craters are clusters rather than single holes, because a small projectile fragments in the atmosphere before arrival. That is a target-specific effect with no lunar analogue, and it means the Martian small-crater population is not the lunar one scaled — which is exactly the assumption a transfer function makes.

A direct rate is a check on an average and not a substitute for it, since the present flux is a snapshot of a population that fluctuates, and the chronology needs the integral.

What the steep part means #

The exponential term is the reason the curve turns vertical, and its interpretation is contested in a way the linear term is not.

Read literally, it says the impact flux four billion years ago was hundreds of times the present one and fell with an e-folding time of about a hundred and forty million years. Read as a spike rather than a decline — the late heavy bombardment hypothesis — it says the flux was low, rose sharply around 3.9 billion years ago, and fell again, which is a different claim about a different mechanism: a rearrangement of the giant planets’ orbits destabilising the belt and the outer disc at once. That rearrangement is not a free invention — the outer solar system contains several structures that are hard to make without it — but it is a hypothesis being supported by the very record it is invoked to explain, and the long-term integrations that would settle it run past their own predictability horizon long before four billion years.

The distinction matters because a monotonic decline is what a leftover population of planetesimals does naturally, and a spike is an event that has to be caused. The evidence is a cluster of impact-melt ages near 3.9 billion years in the returned samples, and the counter-argument is that those samples are dominated by the Imbrium event and would cluster whether or not anything special happened. It remains open, and it is one of the few questions in the subject whose resolution is a matter of going somewhere: a returned sample from a single well-chosen basin, dated in a laboratory, would settle in an afternoon what four decades of argument has not.

The clock’s two failure modes are worth drawing together, because they act at opposite ends of the size range and for entirely unrelated reasons.

Where there is no calibration at all #

Everything above rests on returned samples, and samples have been returned from two bodies. Every age quoted for anything else is an extrapolation, and the extrapolation gets worse the further out it goes.

For Mars the transfer is defensible: the impactor population is largely the same, the encounter speeds differ by a computable amount, and there are Martian meteorites whose ejection ages provide a weak independent check. The published Martian chronologies differ from one another by factors approaching two at young ages, which is an honest statement of how well the transfer is known.

For the outer solar system it is much worse. The impactors at Jupiter are not asteroids but Kuiper belt objects and their fragments, a population whose size distribution is measured poorly and whose flux at each moon depends on the planet’s own gravitational focusing — enormous for Jupiter, and different for each satellite depending on its orbit. The chronology functions in use are constructed from dynamical models of that population rather than calibrated against anything.

The consequence is that the ages quoted for Europa’s surface — commonly a few tens of millions of years, which is the basis for calling it geologically active — rest on a crater count divided by a modelled flux, with no laboratory measurement anywhere in the chain. Different models of the impactor population give ages differing by an order of magnitude, and the disagreement is not narrowing.

That is worth stating plainly because the number is used. Whether Europa’s ice shell is resurfaced on a timescale of tens of millions of years or hundreds is the difference between an ocean vigorously exchanging with the surface and one largely sealed off, and the case for the first is a crater count passed through a function nobody can calibrate.

A chronometer calibrated in one place and applied in another is only as good as the transfer, and in the outer solar system the transfer is the whole measurement.

Where the picture stops #

The scaling law from projectile to crater is calibrated in the wrong regime. Laboratory impacts are at centimetre scale into sand and metal; the craters being dated are kilometres across in fractured rock and ice under different gravity. The extrapolation is many orders of magnitude and it enters every age as a multiplicative factor.

And the target matters. Ice is not rock. A crater in Europa’s ice relaxes viscously over geological time, so an old surface can look sparsely cratered because its craters have flowed away rather than because they were never made — which converts a chronology into an argument about rheology.

Counting is not automatic. Two experienced people counting the same image differ by tens of per cent, systematically, in ways that depend on how degraded a feature has to be before it stops being a crater. Automated counting removes the disagreement and replaces it with a different one, about what the algorithm was trained on.

And a count is a measurement of one number where an orbit determination is a measurement of six. The comparison is worth making because it is unflattering. An asteroid’s orbit is fitted to astrometry and its uncertainty is an ellipsoid whose long axis lies along the track, which is a rich and honest description of what is and is not known. A crater age is a single number with an error bar that hides the impact-scaling law, the transfer function, the saturation cutoff and the counter, and there is no established way to propagate any of them.

And the chronology function itself over the full history, since it is the one part of the method that is calibrated rather than counted.

Where this ladder goes next #

Later rungs on this anchor: the size–frequency distribution as a signature of the impactor population rather than of the surface, and what a change in its slope at a given diameter is evidence for; resurfacing, where a partly buried crater population leaves a kink whose position dates the flooding; secondary craters and the statistical tests that separate them; the transfer functions between bodies, which is where a lunar chronology becomes a Martian one and where most of the error is; and the direct measurement of the present-day flux, from repeat imaging that has now caught hundreds of new craters forming.