A young surface dated by a handful of holes

Every age quoted for a surface beyond the Earth and the Moon comes from counting the craters on it and converting the count to an age through a chronology calibrated on nine lunar landing sites. For old surfaces the counts are large: a three-billion-year-old plain on Mars carries thousands of craters a kilometre across in an area of a few thousand square kilometres, and the statistical error on their density is a few per cent. The difficulty with old surfaces is the chronology itself, and the history that a single count can fold together.

The youngest surfaces have the opposite problem. A lava flow a few million years old, the floor of a recently active volcanic caldera, a patch of ice on a moon whose shell may still be moving — these are the surfaces that decide whether a world is geologically alive, and they carry so few craters that the count itself is the dominant error. Their ages are statistics of small numbers, and small numbers behave in ways that are worth drawing out before any age is believed.

An age with a Poisson error #

Craters arrive at random, so the number on a surface of a given age and area is a Poisson variable whose mean is the area, times the cratering rate per unit area integrated over the surface’s life, times the fraction of craters larger than the size counted. For anything younger than about three billion years, the rate has been nearly steady, the integral is proportional to the age, and the expected count is simply proportional to area times age. A measured count then gives the age with the Poisson error of the count: about one over the square root of the number counted, as a fraction.

The figure shows what that means in practice. Three craters on a hundred square kilometres date the surface to somewhere between thirty and ninety million years — a factor of nearly three, and the honest answer. Ten narrow it to a factor of 1.8, thirty to 1.4, a hundred to 1.2. Each tenfold increase in the count narrows the range by about the square root of ten. And a count of zero is not an age at all. It says that the surface is younger than an upper limit, here forty-five million years, and nothing about how much younger: the probability is spread across all younger ages, weighted only by the prior.

The numbers depend on two assumptions stated in the figure: a crater population whose cumulative number rises as the inverse square of the diameter, and an impact rate on Mars twice the Moon’s. Real populations of craters tens of metres across are steeper than the inverse square, so every count here is an underestimate and every time a little long, but the shape of the argument — the dependence on the square root of the count, the meaning of zero — is independent of both.

When the holes are not independent #

The Poisson error assumes that each crater is an independent event, and that is the most optimistic assumption in the calculation. Craters arrive in clusters as well as singly. A binary asteroid makes a pair; an impactor broken up in the atmosphere, as small ones on Mars often are, makes a cluster of small craters a few hundred metres across from one event; the debris thrown out by a large impact lands in rays and chains. Each cluster is one event counted as several craters, and a count that includes them has a smaller effective number of independent events than the number of holes suggests.

The effect on a young surface is to make the age’s true uncertainty larger than the Poisson error, sometimes much larger. Ten craters that happen to include a cluster of five are, for the purposes of statistics, six events, not ten, and the age is uncertain by the factor appropriate to six. The careful counts exclude obvious clusters and chains, which reduces the count and widens the range honestly; the less careful ones report a Poisson error on a number that is not Poisson.

There is also a selection running the other way. A surface is chosen for dating because it looks young, which on a Martian plain often means it looks uncratered, and a region that happens by chance to have fewer craters than its age predicts is the one most likely to be picked out and counted. Fluctuations downward are selected, and the published youngest ages are biased slightly young for the same reason the first measurement of any rare thing tends to be an underestimate.

When counting stops being the limit #

The counting error shrinks as more craters are counted. The other error does not.

The chronology for Mars is the Moon’s, scaled by the ratio of the impact rates on the two bodies, and that ratio is computed from models of the asteroid population and its delivery to each planet, not measured. Published values run from about 1.6 to 2.6. Every absolute age on Mars inherits that spread, a factor of 1.6, whatever the count. Below about twenty craters the count dominates; above it, the calibration does, and counting more craters improves nothing about when the surface formed.

The calibration’s own anchor is weaker at the young end than anywhere else. The lunar chronology’s linear branch — the part that applies to every young surface — is pinned by exposure ages of material thrown out by a few young lunar craters, and for ages between about one and three billion years there is no anchor at all. A young age on Mars is therefore a crater count divided by a rate that was measured on the Moon over the last billion years, scaled to Mars by a model, and assumed steady. The count can be made precise; the rate cannot, by any amount of counting.

What an empty surface says #

The most consequential claims about young surfaces are often claims about an absence: a flow with no craters at all larger than some size, read as very young. An absence is an upper limit, and the limit is set by what was searched.

The limit falls in inverse proportion to the area and to the number of craters per unit area above the chosen size, which rises steeply as the size falls. A hundred square kilometres with no crater larger than a hundred metres says only that the surface is younger than about two hundred million years. The same area with no crater larger than ten metres says it is younger than two million — a strong statement, if every ten-metre crater has been found and none has been erased. A small surface with no large craters says almost nothing; a flow of a few square kilometres without a hundred-metre crater could be billions of years old, and the absence would be unremarkable.

So the claims that a surface is very young rest on very small craters, counted in images whose resolution reaches them. At those sizes two other things matter. Small craters are erased fastest — by wind, by frost, by the slow creep of dust — so their absence can mean removal rather than youth. And secondary craters, thrown out by large impacts elsewhere, swarm at exactly these sizes, adding holes that have nothing to do with the steady rain the chronology assumes. An empty surface at ten metres is a strong limit only if neither has happened, and a cratered one is an upper limit if the second has.

How young a count can reach #

The same arithmetic, turned round, gives the youngest age a measurement can resolve at all.

The impact rate on Mars is so low that the youngest flows can only be dated with craters the size of a house, counted over hundreds to thousands of square kilometres. The Cerberus plains in Elysium Planitia, whose lava flows are among the youngest on the planet, have been dated in this way to a few million to a few tens of millions of years, from counts of craters tens of metres across over broad areas. The same region was later found by the seismometer on the InSight lander to be the source of most of the marsquakes it recorded, along the Cerberus Fossae fractures, which is independent evidence that it is still active. The crater ages and the quakes agree in spirit; the crater ages alone, at those sizes and on those areas, would have been a factor-of-a-few statement with a calibration of its own on top.

The same constraint applies with more force further out. Europa’s surface is so sparsely cratered that its average age, from the few dozen craters larger than a kilometre on the whole moon, is estimated at tens of millions of years; but the cratering rate there comes from models of comets, uncertain by a factor of several, and individual regions carry too few craters for an age at all. Whether the ice shell above its ocean is being renewed today in particular places is a question crater counts can only bound, and the bound depends on how much of the surface is searched at what resolution. The heat that would drive such renewal comes from the tides, and how much of it there is depends on the ice’s viscosity in ways that crater counts cannot constrain either; Enceladus, whose south polar region is nearly free of craters and venting water into space today, shows that a surface can be renewed faster than any crater clock can tick.

Venus, where the count is the whole planet #

Venus shows the small-number problem at the scale of a world. Its thick atmosphere stops small impactors altogether, so it has no craters smaller than a few kilometres, and its whole surface carries fewer than a thousand. Spread over an area larger than the Earth’s land, they give the planet an average surface age of a few hundred million years to perhaps three-quarters of a billion, with the calibration’s uncertainty on top.

What the craters cannot give is the age of any one region. A province the size of a large country holds a few dozen at most, and their positions are statistically indistinguishable from random over the whole planet. That randomness was once read as evidence that the entire surface was renewed at once, in a global resurfacing a few hundred million years ago, and since then struck evenly. It is equally consistent with a surface renewed piecemeal, region by region, at a steady rate, provided each renewed patch is small compared with the spacing between craters — because in that case the count in any region is too small to show that its age differs from its neighbours’. The two histories are different planets, and with a thousand craters the Poisson error on any regional comparison is too large to choose.

Venus is the young-surface problem with nowhere to hide: a world whose geological activity cannot be read from its craters, because there are not enough of them anywhere, however carefully they are counted.

A little old ground in a young count #

Young surfaces are rarely clean. Lava flows surround islands of older ground; ice shells have older blocks rafted among younger ones; a mapped unit may include a patch the mapping missed. On an old surface a small admixture of older ground hardly matters. On a young one it can dominate.

Old ground is so densely cratered that it outweighs the young surface in the count long before it outweighs it in area. A three-and-a-half-billion-year-old patch carries several hundred times the crater density of a ten-million-year-old flow, so a fifth of a per cent of the counted area — a single unrecognised island a few hundred metres across in a hundred square kilometres — doubles the age. Above a few per cent, the count is mostly a measurement of the old ground.

The error has a sign. Contamination by older ground can only make a young surface look older, never younger. A young age measured carefully is therefore an upper limit on the true age in a second sense: not only the Poisson spread, but every unrecognised older patch inside the counting area, pushes it upward. The youngest ages in the literature are the ones most carefully cleaned of such patches, and the most sensitive to whatever cleaning was missed.

Which surface came first #

There is one quantity a few dozen craters can measure well, and it is not an age.

On the linear branch of the chronology, the ratio of two surfaces’ ages is the ratio of their crater densities, and the calibration cancels: whatever the impact rate on Mars, it is the same for two flows on the same plain. With few craters on each, the ratio is no better known than either age, because two counts’ errors combine. From about twenty craters each it pulls ahead, and with two hundred it is known to a fifth while the absolute age is still uncertain by two-thirds.

That is how the sequence of events on a planet is reconstructed: not as a list of dates, but as an ordering, with the dates hung on it afterwards. Stratigraphy — which flow overlies which, which fracture cuts which — gives the order directly where units touch; crater densities give it where they do not, and give the intervals between events as ratios. A history in which the Cerberus flows are a tenth as old as the plains they cover, and a tenth again as old as the oldest volcanic shields, is a firmer statement than any of the three ages, and it would survive a revision of the Martian impact rate by a factor of two untouched.

A gap in the calibration, partly filled #

For decades the lunar chronology had no anchor between about one and three billion years, because no sample of known age came from a surface that young. The calibration for every Martian plain dated to that interval was an interpolation between the youngest maria sampled by the Apollo missions and the few young craters whose ejecta had been dated by cosmic-ray exposure.

That changed in 2020, when a robotic mission returned basalt from one of the youngest lava plains on the Moon’s near side, chosen precisely because its crater density placed it in the gap. The rock dated to about two billion years, and the crater density of its plain, measured before the landing, could be compared with an absolute age for the first time in that range. The comparison broadly confirmed the chronology’s shape and moved its middle by a modest amount, the first new point on the curve in half a century.

One point does not make a calibration, and the young end below a billion years is still anchored only by exposure ages of crater ejecta, whose connection to the crater counts on the surrounding ground involves assumptions about which ejecta came from which crater. But the result showed that the chronology is a measurement that can be improved by sampling, not a fixed convention, and every young surface on Mars stands to gain from the next sample returned from a dated plain.

What the counting leaves out #

The figures assume a steady impact rate on the linear branch, a production population with a single slope, and craters that are never erased or confused. Real small-crater populations are steeper and vary with the target’s strength: craters in loose regolith are larger than craters made by the same impact in solid lava, so a count at fixed size on two different materials samples different impactor sizes. Erosion and infilling remove small craters on a time scale that depends on the climate, which on Mars has swung with the planet’s obliquity, so that a young surface’s small-crater population may record its exposure since the last high-obliquity epoch rather than its formation. The prior on age is uniform, which matters only for counts of zero or one. And every figure is drawn for Mars; the Moon’s chronology needs no transfer ratio, and Europa’s has no calibration at all beyond models of the comets that hit it.

Still open: a rate measured where it is used #

The single largest uncertainty in every young age on Mars is the present impact rate there, and that rate is now being measured directly. Repeated imaging of the same terrain by orbiting cameras over more than a decade has found hundreds of new craters, metres to tens of metres across, formed between one image and the next, with the dark blast zones of fresh impacts making them easy to spot against dust. The rate of new crater formation counted that way can be compared with the rate the lunar chronology predicts for Mars, size for size, without a model of asteroid delivery in between. Whether the two agree — and whether the rate measured over a decade is representative of the last few million years — decides whether the youngest crater ages on Mars are a factor of two too old, too young, or about right.