Astronomy & Celestial Mechanics
the sky, computed
Orbits

A clock checked by the craters it is making now

Every young age on Mars is a crater count divided by a rate borrowed from the Moon. Since 2006 the orbiting cameras have been watching that rate happen: hundreds of fresh craters, metres across, that were not there in the previous image. They form several times less often than the borrowed rate predicts at the smallest sizes, and more often than it at a few tens of metres — and the youngest surfaces on Mars are dated from exactly the sizes where the two disagree most.

Assumes Surface chronology and Surface chronology.

14 min read 6 figures The sky as a clockMeasuring the unreachable

An age measured by counting craters is a ratio. The numerator is the number of craters per square kilometre on a surface, which an image provides; the denominator is the rate at which craters form, per square kilometre per year, which no image of an old surface can provide. On the Moon, the denominator was measured by bringing rocks home: the Apollo and Luna samples date particular surfaces, the crater densities of those surfaces are known, and the two together calibrate the clock. Every other body is dated from the same nine dated landing sites, by moving that calibration across the solar system — scaled for the target’s gravity, its distance from the asteroid belt, the speed at which impactors arrive, and for Mars its atmosphere.

That transfer is the weakest link in the dating of the youngest surfaces on Mars, the lava flows and ice deposits a few million to a few hundred million years old that decide whether the planet is still volcanically alive. The scaling factors are each uncertain by tens of per cent, and they multiply. A young age on Mars might be wrong by a factor of two or three for reasons unrelated to how carefully its craters were counted.

Since the mid-2000s, the denominator has been measurable directly. Cameras in orbit around Mars have imaged much of the planet repeatedly at resolutions of metres, and comparison of images years apart finds craters that formed in between — fresh holes a few metres to a few tens of metres across, surrounded by dark blast zones where the impact stripped away the bright surface dust. Each is an impact that happened while someone was watching. Counted over the area and time the images cover, they give the present impact rate on Mars, size by size, with no Moon in between.

Craters forming on Mars now, against the rate the clock assumes. The rate at which craters larger than a given diameter form on Mars, per square kilometre per year, against that diameter, both on logarithmic scales. The black curve is the production function scaled from the Moon, the rate every crater-count age on Mars assumes. The red curve and its points are the craters found forming between successive orbital images — 248 above 3.9 metres over the survey's coverage, 1.65 × 10⁻⁶ per square kilometre per year, with a cumulative slope of −2.2, shallower than the scaled function's −3. At 4 metres the scaled function predicts 6.4 times what is seen; the two cross near 42 metres; above that the survey has found too few craters to say. Bars are the counts' Poisson errors.
Fig. 1 The rate at which craters larger than a given diameter form on Mars, per square kilometre per year. The black curve is the production function scaled from the Moon; the red curve and points are the craters seen forming between orbital images — 248 above 3.9 metres, 1.65×10−61.65 \times 10^{-6} per square kilometre per year, with a cumulative slope of −2.2 against the scaled function’s −3. At 4 metres the scaled function predicts 6.4 times what is seen; the two cross near 42 metres.

Holes that were not there before

The first new craters were found by accident. In 2006 a team reviewing images from a camera that had been orbiting Mars since 1997 noticed dark spots on dusty terrain that had not appeared in earlier images of the same places, and higher-resolution follow-up showed each spot to be a fresh crater, or a cluster of them, with a blast zone of disturbed dust around it. Twenty such impacts had happened within the camera’s own lifetime. By the mid-2010s systematic searches through the overlapping coverage of two cameras on a later orbiter had found several hundred, each dated to the interval between the last image without it and the first image with it, often a span of a few years and sometimes of a few days.

The catalogue is a strange kind of data. Each entry is an event rather than a feature, and the rate it gives depends on the area and time over which a new crater would have been noticed — which in turn depends on dust cover, lighting, image resolution, and how often each place was imaged. Turning a list of events into a rate per square kilometre per year means estimating that effective area-time product, and most of the uncertainty in the measured rate lies there rather than in the count itself.

Two rates that cross

The figure sets the two rates side by side. At the smallest sizes the survey reaches, four metres, the new craters form at a sixth of the rate the scaled function predicts. The measured rate falls off more slowly with size, though — its cumulative slope is about −2.2, against the scaled function’s −3 — so the gap closes toward larger craters, and the two cross at a few tens of metres. Above that the survey’s points run out, because craters that large form only a few times in the area and time the images cover.

The comparison as drawn is a simplification of the published one, which uses the full shape of the lunar production function rather than a single power law, and whose details depend on how the scaled function is normalised to Mars. What survives in every version is the pattern: a deficit of new craters at the smallest sizes by a factor of several, a size distribution shallower than the borrowed one, and agreement only over a narrow range of sizes in the tens of metres.

Neither curve is the truth. The scaled function inherits every uncertainty of the transfer from the Moon. The measured rate is a decade’s sample of a process that varies on much longer timescales, from a survey whose detection efficiency falls off for the smallest craters and on terrains without bright dust to darken. But the measured rate has one property the scaled function cannot have. It is a measurement of Mars.

How far each size’s clock runs fast

If the measured rate is right and representative of the time a surface has existed, every age computed with the scaled function from craters of a given size is off by the ratio of the two rates.

How far each crater size's clock runs fast. The ratio of the rate the scaled production function assumes to the rate measured from new craters, against crater diameter, on logarithmic scales. A crater-count age is the count divided by the assumed rate, so this ratio is the factor by which an age from craters of that size would have to be multiplied if the measured rate is right and representative: 6.4 at 4 metres, 3.1 at 10, 1.8 at 20, and about one near 42 metres. The youngest surfaces on Mars are dated from exactly the smallest craters, because those are the only ones numerous enough on them to count — so the correction is largest where it is most used.
Fig. 2 The ratio of the assumed to the measured rate, against crater diameter. Ages from craters of that size would have to be multiplied by it: 6.4 at 4 metres, 3.1 at 10, 1.8 at 20, about one near 42 metres. Above 50 metres the survey has too few new craters to test.

The correction is largest exactly where it is most used. A surface a few million years old has accumulated only the smallest craters in numbers large enough to count — a lava flow of a few square kilometres with an age of two million years holds a few dozen craters above ten metres and almost none above fifty — so its age is a count of craters of ten or twenty metres, divided by the rate at those sizes, and the rate at those sizes is where the scaled function exceeds the measurement by a factor of two or three. The youngest ages on Mars, the ones that bear on present-day volcanism, would grow by that factor.

There is an irony in this. The young-surface problem was already that a few dozen craters give an age with a large Poisson error, and that a little contamination by older ground can double it. The new-crater comparison adds a systematic error of the same size, in a consistent direction, to every age in the range where the Poisson errors are worst.

A rate measured where the craters are small

The measured rate covers the sizes it covers because of arithmetic.

A rate measured where the craters are small. The number of new craters larger than a given diameter that a survey of the reach the published catalogues have — an area-time product of 1.5 × 10⁸ square-kilometre-years — would find, against that diameter, on logarithmic scales. It holds 248 above 3.9 metres, 31 above 10, 3 above 30 and 0.2 above 100. The rate is measured to 6% at the smallest sizes and not at all above a few tens of metres, while the young lava plains and ice deposits of Mars are dated from craters of tens to hundreds of metres. The measurement and its use overlap in a narrow window, and outside it the comparison is an extrapolation along an assumed slope.
Fig. 3 The number of new craters larger than a given diameter that a survey with the reach of the published catalogues — an area-time product of 1.5×1081.5 \times 10^8 square-kilometre-years — would find: 248 above 3.9 metres, 31 above 10, 3 above 30, 0.2 above 100. The shaded range is where young surfaces are dated.

The rate of crater formation falls steeply with size, as the inverse square or steeper, so a survey that finds two hundred and fifty craters above four metres finds thirty above ten, a few above thirty, and almost certainly none above a hundred. Its precision follows: six per cent at the smallest size, twenty per cent at ten metres, and nothing at all at the sizes from tens to hundreds of metres from which most young Martian surfaces are dated. The measurement and its use overlap in a window from about ten to about forty metres, and outside that window the comparison is an extrapolation of the measured slope.

The extrapolation is the uncomfortable part. A slope of −2.2 measured between four and thirty metres, carried to a hundred, predicts a rate above the scaled function’s — and a surface dated from hundred-metre craters would then be younger, not older, than its quoted age. Whether the slope continues is not something the survey can say, because the survey has no hundred-metre craters.

How long the cameras must keep looking

The precision of a counted rate improves as the square root of the number of events, and the number of events grows with time.

How long the cameras must keep looking. The fractional uncertainty of the new-crater rate above 10, 30, 100 metres against the number of years of repeated imaging at the coverage rate of the present surveys, on logarithmic scales. The error falls as the inverse square root of the time. After a decade the rate above 10 metres is known to 20% and above 30 metres to 66%; above 100 metres a decade holds less than one new crater, and the uncertainty exceeds the rate. Reaching 10% takes 40 years, more than a century, more than a century. The sizes that date the youngest plains will not be measured directly by any survey with the present coverage within the lifetime of a spacecraft.
Fig. 4 The fractional uncertainty of the new-crater rate above 10, 30 and 100 metres against years of repeated imaging at present coverage. After a decade the rate above 10 metres is known to 20% and above 30 metres to 66%; above 100 metres a decade holds less than one crater. Reaching 10% takes about forty years at 10 metres and more than a century at 30.

The rate above ten metres reaches ten per cent precision after about forty years of imaging at the coverage the present cameras achieve; above thirty metres, after well over a century. No spacecraft lasts that long, and the cameras now in orbit around Mars are nearing the ends of their lives. The survey that measured the present rate is not going to be extended to the sizes that date the youngest plains, unless a future orbiter images far more of the planet far more often — which a wide-field camera with a few metres’ resolution, imaging the whole planet every few months, could do.

There is a second limit on what time can buy. A decade’s rate is a decade’s rate. The flux of impactors at Mars varies as asteroids are fed out of the main belt by resonances, sometimes in surges when a large body breaks up — the collision that makes an asteroid family — and its fragments drift into the resonances over a few million years, so that the inner solar system receives a pulse of impactors whose timing is set by one event in the belt, and the present rate may be above or below the average over the last few hundred million. The scaled function averages over billions of years of the Moon’s history; the new-crater rate is a snapshot. They differ in what they measure as well as in how well they measure it.

There is a way to test whether the snapshot is typical without waiting for it to lengthen. Small craters on Mars are erased over millions of years by wind, dust and frost, and the size distribution of the small craters on a young lava flow reflects both their production and their erasure. A flow whose count at ten metres matches the new-crater rate times its age, where the age is fixed by larger craters, says the present rate is representative over that flow’s lifetime; one whose count falls short says either that the rate was lower or that erasure is faster than assumed. The comparison has been made for a few flows and is not yet decisive, because the erasure rate is itself known only roughly.

An atmosphere that edits the record

One reason for the deficit at the smallest sizes has nothing to do with the impact rate. Mars has an atmosphere, the Moon does not, and a small body entering the Martian air is decelerated and often broken up before it reaches the ground.

An atmosphere that edits the smallest craters out. The share of impactors able to make a crater of a given diameter that reach the ground intact, against that diameter, for surface pressures of 1, 6, 60 millibars — the second is Mars today, the third a guess at an earlier, thicker atmosphere. Small bodies are slowed or broken by the air, making smaller craters or clusters of small ones, and the screen's edge scales with the column of air the body must cross: half get through at 2 metres, 3.3 metres, 13 metres. The Moon has no such screen, so a production function scaled from it overpredicts the smallest Martian craters for a reason unrelated to the impact rate — and the same screen, thicker in the past, would have made the ancient small-crater record thinner still.
Fig. 5 The share of impactors able to make a crater of a given size that reach the ground intact, for surface pressures of 1, 6 and 60 millibars — the second is Mars today. Half get through at 2, 3.3 and 13 metres. The Moon has no screen at all.

The atmosphere’s effect is a screen whose edge is set by the column of air a body must push through: a body whose mass is small compared with the air in front of it is slowed to a fraction of its speed and makes a smaller crater, or breaks into fragments that make a cluster of small craters instead of one larger one. About half of the new craters on Mars are in fact clusters — groups of small craters from one impactor that fragmented on the way down — which is direct evidence of the screen at work. At the present pressure, the screen removes impactors that would have made craters of a few metres; it is the reason the deficit is largest at four metres and smaller at ten.

The screen also has a history. The Martian atmosphere was probably much thicker billions of years ago, when water flowed on the surface. A thicker atmosphere screens larger impactors, and ancient surfaces dated from small craters would then have recorded fewer of them than the present screen allows — making their crater-count ages too young by an amount that depends on how fast the atmosphere thinned. That is a correction the new-crater survey cannot calibrate, because it can only see today’s atmosphere at work.

Three young ages on two clocks

What the correction would mean for actual surfaces depends on which crater sizes each age rests on.

Three young ages, read on two clocks. Model-age estimates for three young Martian surfaces, on a logarithmic scale in millions of years, as dated with the lunar-scaled production function (open) and as they would read if the rate measured from new craters held at each size and over each surface's lifetime (filled). A lava flow counted from 10 m craters: 2 million years becomes 6.2. A lava flow counted from 30 m craters: 10 million years becomes 13. A lava plain counted from 100 m craters: 100 million years becomes 49. The youngest ages move most, and older, because they rest on the smallest craters, where the measured rate is lowest relative to the assumed one. The third moves the other way, younger, because above about forty metres the measured slope passes above the assumed one — but it is dated from craters the new-crater survey cannot reach, so its correction is the measured slope carried beyond its data. The figure shows what that extrapolation would mean, not that it is right.
Fig. 6 Model ages for three young surfaces, read with the lunar-scaled function (open) and with the rate measured from new craters (filled). A lava flow counted from 10 m craters goes from 2 to 6.2 million years; one counted from 30 m craters from 10 to 13; a lava plain counted from 100 m craters from 100 to 49.

The youngest age moves most: a lava flow dated to two million years from ten-metre craters would be six million years old on the measured rate. That changes the reading of the youngest volcanic activity on Mars from “within the last few million years” to “within the last ten million”, which is still recent geologically and still implies a planet that is not volcanically dead, but it moves the most recent eruptions further from the present than the published ages say.

A surface dated from thirty-metre craters barely moves, because thirty metres is close to where the two rates cross. And a surface dated from hundred-metre craters moves the other way, younger by a factor of two — if the measured slope continues that far, which is the assumption the survey cannot test. The ages of the young plains are not uniformly too old or too young. They are wrong in a size-dependent way whose sign changes in the middle of the range they are measured in.

The Moon’s clock checked the same way

The same method has been turned on the Moon, and it reads the source of the calibration rather than its transfer. A camera in lunar orbit since 2009 has reimaged much of the Moon at about a metre’s resolution, and comparison of image pairs found two hundred new craters by 2016, with rates for craters above ten metres about a third higher than the lunar production function predicts. The lunar function was built from crater counts on surfaces billions of years old, so a mismatch with the present rate is either a change in the impact rate over time or an error in how the old counts were converted into a present-day rate; at the level of a third, either is plausible.

That comparison is the cleanest of all, because no transfer between bodies is involved: the same function and the same kind of surface, with only the time between them. It suggests that even the Moon’s clock, at the smallest sizes, is uncertain by tens of per cent in its present rate. On Mars, where the transfer adds its own factors and an atmosphere edits the smallest impacts, the factors of several in the comparison above are not surprising.

The impactors themselves are the other half of the question. Bodies reach the inner solar system from the main asteroid belt by drifting into resonances with Jupiter, and the drift is driven by the heat an asteroid radiates unevenly as it rotates — a push that acts faster on smaller bodies. The size distribution of what arrives is therefore not the size distribution of the belt, and the delivery of metre-sized bodies is a different process from the delivery of kilometre-sized ones. A production function that is a single power law across that range assumes away a change in the physics of delivery.

What the comparison leaves out

The new craters have their own biases. They are found most easily on dusty terrain, where a fresh impact leaves a dark scar visible from orbit at lower resolution than the crater itself; on dust-free ground, many new craters are missed. The survey corrects for this by restricting the rate to areas it has reimaged under suitable conditions, and the correction is itself uncertain by tens of per cent.

The scaled function, for its part, is contaminated at small sizes by craters that were not primary — secondaries made by debris thrown out of larger impacts, which on the Moon dominate the count below a few hundred metres. A production function built from lunar counts includes secondaries at small sizes, and a Martian surface’s count includes Martian secondaries; whether the two contaminations are similar is not known. The new craters are primary by construction, since each is an isolated fresh impact, so the comparison of the two rates partly measures the secondary contamination of the scaled function rather than any difference in the impact rate.

Still open: whether a decade speaks for a hundred million years

The new-crater survey converted a question that had been answered only by scaling into one answered by watching, and the answer it gave is uncomfortable: the clock borrowed from the Moon runs several times fast at the smallest sizes, about right at a few tens of metres, and perhaps slow above. The ages of the youngest surfaces on Mars need correcting by factors of a few, in directions that depend on which craters were counted.

The correction rests on two assumptions the survey cannot check. One is that the measured slope continues beyond the sizes it measures; the other is that the present decade is representative of the past few hundred million years. The first can be tested by the next generation of orbiting cameras, if any is built to image the whole planet repeatedly. The second can only be tested by an absolute date — a sample of a young Martian lava, returned to Earth and dated in a laboratory, from a surface whose crater count is already known. That is exactly how the Moon’s clock was calibrated half a century ago, and how a lava plain’s depth was read from a kink in its count only once a sampled surface fixed the clock’s shape, and it remains the only way to calibrate a clock: by reading it once against a known time.

Where this goes next

The age a handful of craters allows. The probability of each age for a 100 square-kilometre surface on Mars, given a count of 0, 3, 10, 30, 100 craters 50 metres across or larger, per unit of the logarithm of age, with a uniform prior on age. No craters: younger than 45 Myr at 95 per cent, and nothing more; 3: 55 Myr, from 31 Myr to 88 Myr; 10: 158 Myr, from 116 Myr to 212 Myr; 30: 457 Myr, from 382 Myr to 547 Myr; 100: 1,502 Myr, from 1,359 Myr to 1,660 Myr. Each tenfold increase in the count narrows the range by about the square root of ten, from a factor of 2.8 with three craters to 1.22 with a hundred. The counting error is only part of the story: the chronology converting a crater density into an age is uncertain by a factor of about 1.6 on Mars before any crater is counted. A young surface dated by a handful of holes The surfaces that decide whether a planet is still geologically alive — the freshest lava on Mars, the smoothest ice on Europa — are the ones with fewest craters to count. Their ages are Poisson estimates from a few dozen holes, they cannot be younger than the area and crater size allow, and a fraction of a per cent of older ground in the counting area can double them. What survives best is not when a surface formed but which of two surfaces formed first. A clock that is a straight line for three billion years and then is not. The lunar chronology function: craters of a kilometre or more per square kilometre against the age of the surface, with the count logarithmic and time not. The dashed line is the present impact rate extrapolated backwards, and it accounts for the whole curve up to 3.1 billion years — over that entire range, dating a surface is dividing a crater count by a constant. The rate itself, read off the slope of the drawn curve at the present day, is 8.4·10⁻⁴ craters per square kilometre per billion years, which over the whole Moon is about 32 new craters of a kilometre or more per million years. Past three and a half billion years the exponential term takes over and the curve turns almost vertical: ground that is 4.1 billion years old carries 26 times the crater density of ground 3.5 billion years old, for a difference in age of six hundred million years. Most of the craters on the Moon were made in a small fraction of its life, and nothing that happened after them is recorded anything like as densely. The six marked ages are laboratory measurements on returned rock, and they are what makes the curve a chronology rather than a shape — the Moon is the only body whose crater counts and whose radiometric ages have ever been measured on the same square kilometre. A surface dated by counting holes in it Every age quoted for a surface in the solar system outside the Earth — a Martian lava flow, a crater on Mercury, the ice of Europa — comes from counting craters and passing the count through one curve. That curve was calibrated on nine square kilometres of the Moon, and it is nearly a straight line for three billion years and then is not. A branch that adds nothing above 1 km and everything below it. Cumulative crater counts on a 3.5-billion-year-old surface, with the population split into the craters made by objects arriving from outside and the craters made by blocks thrown out of larger ones on the same surface. The two are indistinguishable in a photograph and completely different as a statistic. Secondaries stop at about 1 kilometre, because that is the largest crater a block leaving at a few hundred metres a second can excavate, so the upper half of the plot is unaffected. Below it they are steeper — slope -3.2 against the primaries' -2 — and by the smallest diameter drawn they outnumber the primaries 292 to one. A count taken at 100 metres and read through the primary production curve returns an age of 4.34 billion years for ground that is 3.5, and it returns it with a small formal error, because the counting statistics are excellent. The error is not in the counting. The craters that were not primary Counting craters dates a surface, and the method works because impacts from space arrive at a known rate. Some of the holes were not made from space. They were made by rock thrown out of the larger holes on the same surface, and they are far more numerous than anything that arrived.

The objects this essay names

Each one links to every other essay that touches it.

Atmospheric entryChronology functionCrater countingImpact fluxMarsPoisson statisticsProduction functionSecondary cratersSurface chronology