A kink that measures how deep the lava was

A surface is dated by counting the holes in it. Craters accumulate at a rate known from the lunar samples that calibrate it, so the number per square kilometre larger than some diameter is a clock, and the whole size–frequency distribution — the number larger than every diameter — is a family of curves, one for each age, called isochrons. A count that lies on one isochron has an age.

That reading assumes the surface was made at one moment and left alone. Planetary surfaces are rarely so obliging. Lava floods a basin, then floods it again; ejecta from a distant impact blankets a plain; ice resurfaces a moon’s crust in patches. Each such event buries some craters and spares others, and the craters it spares are chosen by size. The count that results is not on any one isochron. It follows one at large diameters and another at small ones, and the place where it changes is a measurement in its own right.

Which craters a flood spares #

A simple crater — the bowl-shaped kind, less than about fifteen kilometres across on the Moon — has proportions that hardly depend on its size. Its depth is about a fifth of its diameter, and its raised rim stands about 0.036 of its diameter above the surrounding plain, from Pike’s measurements of lunar craters in the 1970s. A crater ten kilometres across has a rim 360 metres high; one a kilometre across, a rim of 36 metres.

A lava flow that spreads across the surface to a depth $h$ fills everything lower than $h$. A crater whose rim is lower than $h$ disappears completely under it. One whose rim stands higher survives, as a ring of rim poking through a smooth floor — a “ghost” crater — or, if the rim is well above the lava, as a recognisable crater with a flattened floor. Since rim height is proportional to diameter, the dividing line is a diameter: every crater smaller than $h/0.036$ is buried and every larger one survives. A flood of 200 metres erases every crater smaller than 5.6 kilometres and spares every larger one.

After the flood the surface is new, and craters begin accumulating on it from zero. So the count on the surface today consists of two populations: every crater made since the flood, of any size, and the pre-flood craters larger than the kink. At large diameters both contribute and the pre-flood population dominates, so the count sits on the old isochron. At small diameters only the post-flood craters remain, and the count sits on the young isochron. The kink between them is a step down from one isochron to the other, and its diameter is the flood’s thickness divided by 0.036.

Fitting an isochron to each branch separately returns both ages exactly — 3.9 billion years above the kink and 3.3 below — because each branch is a pure population. That is the whole payoff: one count of craters on one photograph yields the age of the original surface, the age of the flood that covered it, and the depth of the flood.

Why the rim is a good ruler, and where it stops being one #

The whole method rests on craters having the same shape at every size, and within a range they do. A simple crater forms when the excavation flow opens a bowl whose proportions are set by the ballistics of the ejected material rather than by the target’s strength or gravity, so a one-kilometre crater is a scaled copy of a ten-kilometre one. Above a certain diameter that stops being true. The bowl becomes too deep to support its own walls under the planet’s gravity, the walls slump into terraces, the floor rebounds into a central peak, and the crater becomes complex: shallower for its width, with a depth nearer a tenth of its diameter and a rim proportionally lower.

The transition diameter scales inversely with surface gravity. On the Moon it is about fifteen kilometres, on Mercury and Mars about ten and seven, on the Earth two to four. For the flood-thickness reading that matters in one direction only: a kink above the transition has to be converted with the complex crater’s rim height, which gives a thicker flood for the same kink diameter than the simple-crater ratio would. Most of the kinks measured on the lunar maria lie at a few kilometres, well inside the simple range, where the 0.036 ratio applies; the deepest basin fills, a kilometre or more, put their kinks near or above the transition, where the ruler changes.

How deep, read off a diameter #

The rim criterion is the upper limit on the kink, and the craters that sit near it are informative in their own right. A ghost crater’s rim protrudes by the difference between its original rim height and the lava’s depth at that spot, so measuring the protruding height of many ghost rims — from their shadows, or from altimetry — maps the thickness of the flow across the plain, point by point, without any crater count at all. Maps of mare thickness were built that way from orbital photographs in the 1970s, and they agree with the counts to within the factor the two criteria allow. A flood thinner than the crater’s rim but thicker than its floor leaves the crater visible but altered, and whether a crater counter includes it depends on how they classify ghost craters.

The two criteria differ by a factor of 5.6, the ratio of a crater’s depth to its rim height, and the true burial diameter lies between them. A counter who excludes ghost craters is counting only fully open craters, and the kink appears near the floor criterion; one who includes every rim still visible pushes it towards the rim criterion. For a flood of a hundred metres, the kink sits somewhere between half a kilometre and 2.8 kilometres. That factor is the dominant uncertainty in turning a kink into a thickness, and it is much smaller than the uncertainty in the ages, which is why the thickness of lunar mare flows was known from crater counts before any radar sounder measured it.

Measured this way, individual lunar mare flows are tens to a few hundred metres thick, and the total fill of the large basins is a kilometre or more near their centres. Radar sounding from orbit in the 2000s found buried reflecting layers — old surfaces, weathered before the next flow covered them — at depths of a few hundred metres beneath the maria, consistent with the thicknesses the crater kinks had implied. The kink is one of the few ways to measure a depth beneath a planetary surface using nothing but a photograph of its top.

An age that belongs to neither episode #

The kink also undermines any single age fitted across it.

A crater count is always made over a limited range of diameters: the smallest craters that the images resolve and the largest the counted area contains. If that range lies entirely on one side of the kink, the count returns one of the two true ages. If it straddles the kink, the fit averages the two isochrons, and the result is an intermediate age that corresponds to no event. On the model surface in the figure, a decade from 1.6 to 16 kilometres returns 3.68 billion years, between the surface’s formation and its flooding, and it would be reported with a formal error bar a few hundredths of a billion years wide.

An age quoted without the diameter range it was measured over is not a statement about a surface with a history. The range matters as much as the count, and the range is set by the images, not by the geology. Early counts of the lunar maria, made on photographs that resolved only craters of several kilometres, returned ages weighted towards the older units beneath; later counts on higher-resolution images, reaching craters of a few hundred metres, returned younger ages for the same places. Neither was wrong. They were sampling different parts of one staircase.

A staircase of flows #

The maria were not one flood each. They were built of many flows over hundreds of millions of years, and each leaves its own step in the count.

Two floods give three branches and two kinks. The largest craters, whose rims stand above both layers together, survive from the original surface; those between the two kink diameters survive only from the time between the floods; the smallest date from after the last one. Each branch sits on its own isochron, and each kink diameter is the cumulative thickness of the layers above that generation — here 16 kilometres for the 580 metres of both flows together, and 2.2 kilometres for the last 80 metres alone.

A real mare shows this as a size–frequency distribution with several changes of level, and its interpretation depends on assuming that the production function — the shape of the distribution of new craters — has no wiggles of its own at those diameters. It does have some. The impactor population’s own size distribution is not a pure power law, because the strength of the bodies being broken changes character near a few hundred metres, and the crater production function inherits a wave from it. The standard production functions carry that shape, and a kink is only a flood if it departs from them.

A kink that bends the other way #

There is a second process that puts a kink into crater counts at small diameters, and it mimics a flood unless its direction is read.

On a surface old enough, small craters become so numerous that each new impact destroys, on average, a crater already there: the surface is saturated, and the number of small craters stops growing. The first essay on this subject drew that equilibrium as a diameter below which the count measures the surface’s strength rather than its age. On a count, saturation appears as a kink too: below it the distribution follows the equilibrium line, whose slope is shallower than the production slope.

The two kinks bend in opposite directions relative to the old isochron. Burial removes the small craters and drops the count to a lower isochron, parallel to the old one; saturation holds the small craters at an equilibrium that sits on the old isochron at the kink and becomes shallower below it. A counter who read a flood’s kink as saturation would call the surface uniformly old and heavily cratered; one who read saturation as a flood would invent a resurfacing event that never happened and assign it an age. The discriminant is the slope below the kink: the production slope, displaced downward, for burial; a shallower slope, continuous at the kink, for saturation.

A production rate assumed to be steady #

Every isochron in these figures assumes that craters of every size are made at a rate that changes only as the lunar chronology says — steadily for the last three billion years, faster before. That assumption is tested, and occasionally broken, by what supplies the impactors. The objects that make craters on the Moon and Mars are mostly asteroids delivered from the main belt into planet-crossing orbits, pushed slowly by the recoil of their own thermal emission until they reach a resonance that throws them inward. When a large asteroid breaks up in the belt, it creates a family of fragments whose spread in orbit dates the collision — a family whose membership is itself partly a choice — and some of those fragments are delivered to the inner planets over the following tens to hundreds of millions of years — a temporary increase in the impact rate, concentrated at the sizes the breakup produced.

A spike of that kind would put a feature into the crater count that has nothing to do with the surface: a surface exposed during the spike would carry an excess at those sizes and look older than it is. The size distribution of the delivered fragments differs from the steady background, so the excess would also change the count’s shape. None has been identified unambiguously in the lunar record, but the ages of the youngest large lunar craters cluster in a way that some analyses read as a recent increase in the rate, and a kink produced by a change in the impactors rather than in the surface would be exactly the confusion the flood reading has to rule out.

Where it has mattered #

The technique has been used most heavily not on the Moon, where the samples date the maria directly, but on Mars, where there are no samples and the question of how recently the planet was volcanically active rests entirely on crater counts. The youngest lava plains of Tharsis and Elysium show exactly the staircases drawn here: the largest craters record surfaces billions of years old, the smallest record flows tens to hundreds of millions of years old, and the kinks between them give flow thicknesses consistent with the heights of flow fronts measured from laser altimetry. The conclusion that Mars was volcanically active within the last hundred million years rests on the small-crater branches, and the conclusion that its volcanic provinces are ancient on the large-crater branches. Both are right about different layers.

The same reading applies to the icy moons, where resurfacing is by ice rather than lava — on Europa, a moon kept warm by being flexed by Jupiter, perhaps within the last hundred million years. There the relation between crater proportions and burial is different — craters in ice relax as the ice flows, so old craters flatten even without being buried — and the kink is a combination of burial and relaxation that has to be modelled for each surface. And it applies, in reverse, to the dating of individual impact basins: a basin’s ejecta blanket buries the craters around it, and the count on the blanket dates the basin, provided the counter excludes craters large enough to have survived the blanket.

What the model leaves out #

The figures use a single power-law production function with a cumulative slope of −2, where the real lunar production function has a slope that varies between about −1.8 and −3.8 across the range of diameters counted, with the wave from the impactor population. They treat every flood as uniform in thickness, where real flows thin towards their edges and fill low ground first, so a single flow buries small craters in the valleys and spares them on the ridges, smearing the kink over a factor of two or three in diameter. They ignore the secondary craters — the craters made by debris thrown out of larger ones — which add to the small-diameter counts and have to be removed before either branch can be dated. And they convert counts to ages through the lunar chronology, whose calibration has a gap from 0.8 to 3.1 billion years in which no sample exists; ages in that gap, on the Moon or anywhere the chronology is transferred, are interpolations.

None of these changes the basic reading. The kink’s existence depends only on the scaling of rim height with diameter; its diameter depends on that scaling and on the thickness; the two branches depend on the chronology. The first two are geometry and are well determined. The third carries the chronology’s uncertainty into both ages equally, so their difference — how long the old surface was exposed before the flood — is known better than either.

Still open: the youngest surfaces #

The method works well where both populations are rich enough to count. It works badly on the youngest surfaces, where the post-flood craters are few and the small-crater branch is set by a handful of craters, and badly on surfaces where the burying layer varies in thickness, so that the kink is smeared into a gentle bend. The most consequential young surfaces — the most recent lava flows on Mars, the possibly active regions of Europa’s ice shell, the youngest ejecta on Mercury — are exactly those. Their ages, and therefore the question of whether the planets they belong to are still geologically alive, rest on counts of a few dozen small craters on a surface that may be neither one age nor two, and on images that resolve only the largest of the craters that would separate the cases. The kink is visible when the craters are; below that, the history of a surface is inferred from its count and from what the count’s shape will not allow.