Astronomy & Celestial Mechanics
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Nine dates for every surface in the solar system

Every absolute age quoted for a planetary surface — Martian volcanism, Mercury's plains, the resurfacing of Europa — descends from radiometric dates on rocks returned from nine landing sites on one body. Six of them fall between 3.15 and 3.92 billion years, and there is nothing at all between 0.8 and 3.1.

Assumes Surface chronology, Asteroid families and Non-gravitational forces.

16 min read 8 figures Scales that defeat intuitionMeasuring the unreachable

The first rung of this anchor established what a crater count is and what it is not: a number per unit area, converted into an age by a curve, with a saturation diameter below which the count stops meaning anything. The second removed the craters that were made by ejecta rather than by impactors.

This rung is about the curve. It is one curve, it was fitted to nine points, and every absolute age given to any surface in the solar system is a reading off it.

Nine dates, and a gap from 0.8 to 3.1 billion years. The lunar crater chronology and the samples that fix it: crater density per square kilometre against age, the density axis logarithmic. Every point is a laboratory measurement on returned rock — six mare and highland units dated by crystallisation, four young craters dated by how long their ejecta has been exposed to cosmic rays. The curve is a fit of N(T) = A(e^(λT) − 1) + BT with the exponential rate swept over 6 to 7.9 per billion years and the two amplitudes refitted at each; the best fit lands at λ = 6.95, against the 6.93 the published chronology uses, which is a check on the fitting rather than an input to it. What the figure exists to show is where the anchors fall in time. Six of them lie between 3.15 and 3.92 billion years, four between 0.026 and 0.80, and there is nothing whatever between 0.8 and 3.1 — a gap covering half the age of the solar system. The consequence is drawn as the spread of the acceptable curves: a surface whose crater density says 3.5 billion years is dated to ±0.03 billion by this family, and a surface at 2 billion to ±0.13. The chronology is a measurement where the astronauts landed and an interpolation everywhere else.
Fig. 1 The lunar crater chronology and the samples that fix it. Every point is a laboratory measurement on returned rock — six mare and highland units dated by crystallisation, four young craters dated by how long their ejecta has been exposed to cosmic rays. The curve is a fit of N(T) = A(e^(λT) − 1) + BT with the exponential rate swept and the amplitudes refitted at each; the best fit lands at λ = 6.95 per billion years against the 6.93 the published chronology uses, which is a check on the fitting rather than an input to it. The shaded band from 0.8 to 3.1 billion years holds no sample of any age. Curves that fit the anchors equally well date a 3.5-billion-year surface to ±0.03 billion and a 2-billion-year one to ±0.13.

The gap is the subject. It covers rather more than half the age of the solar system, and it is exactly the interval in which most of the interesting questions about Mars lie.

It is worth saying at once what this essay is not arguing. Crater chronology works. The surfaces it dates are ordered correctly, its ages are consistent with every independent check that has been possible, and no alternative exists for any body nobody has visited. The point is narrower and sharper: an absolute planetary age is a reading off a curve whose shape between two clusters of anchors is assumed rather than measured, and the quoted uncertainties describe the count rather than the curve.

Where the nine came from

Six of the anchors are basalts and breccias returned by Apollo and Luna, dated by potassium–argon and rubidium–strontium decay. Those are absolute ages in the strongest sense available: a closed system, a known decay constant, and a measured daughter-to-parent ratio. They cluster between 3.15 and 3.92 billion years because that is when the maria were flooded, and the maria are where it is safe to land.

Four are young, and they are a different kind of measurement. A crater’s ejecta blanket has been sitting exposed to cosmic rays since the crater was made, accumulating spallation products at a rate that can be computed. Measuring those gives an exposure age: 26 million years for Cone crater, 50 for North Ray, 109 for Tycho, and 800 million for Copernicus, that last from ray material sampled at the Apollo 12 site rather than at the crater.

Both kinds of anchor are good measurements. The trouble is entirely in where they fall.

There is a subtlety in each kind that is worth knowing, because it affects how tightly the points constrain the curve. A crystallisation age dates when a lava solidified; the crater count is made on the surface of the flow, which is the same event, so those two agree by construction. An exposure age dates when material was excavated, which is when the crater formed — but the crater count that goes with it is made on the ejecta blanket, a surface younger than the terrain around it, and the blanket is small, so the count has few craters in it and a large Poisson error. The four young anchors that fix the whole linear branch of the chronology rest on counts of a few tens of craters each.

That asymmetry is why the young anchors carry much larger uncertainties in the figure above than the mare basalts, and why the family of admissible curves fans out toward the recent end as well as in the middle.

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.
Fig. 2 The chronology drawn with the sampled sites on it and a steeper production slope than the standard. Almost all of the curve’s shape is in two places: an exponential rise before about 3.5 billion years, fixed by the highland samples, and a straight line since, fixed by the four young craters. Between them the curve is the sum of the two branches and is determined by neither. Nothing about the functional form is derived — it is a two-branch fit chosen because the data have two branches — and any other function passing through the same points is equally admissible.

It is worth being explicit about what “returned rock” means as a constraint, because it is the strongest thing in planetary science and the rarest. Six Apollo landings and three Luna sample returns brought back 382 kilograms of material between 1969 and 1976, and three Chang’e missions have added to it since — the last of those, in 2020, returning basalt from a young mare unit that dated to about two billion years and landed in the middle of the gap this essay is about. That single sample is the most important addition to the chronology in forty years, and it moved the curve: the crater density on the unit it came from implies an older age than the standard curve gives, by a few hundred million years.

That is what a point in the gap does. It is also, so far, one point.

What the gap costs

The consequence is not that intermediate ages are wrong. It is that their uncertainties are not what they look like.

A crater count on a Martian lava flow produces a density with a formal error from Poisson statistics — count a thousand craters and the density is good to three per cent. Passing that through the chronology produces an age with a formal error of a few per cent, and that is the number usually quoted. The chronology’s own uncertainty in the interpolated region is several times larger and is not usually propagated, because there is no accepted way to express it: the curve has no error band, only a functional form and nine points.

Nine dates, and a gap from 0.8 to 3.1 billion years. The lunar crater chronology and the samples that fix it: crater density per square kilometre against age, the density axis logarithmic. Every point is a laboratory measurement on returned rock — six mare and highland units dated by crystallisation, four young craters dated by how long their ejecta has been exposed to cosmic rays. The curve is a fit of N(T) = A(e^(λT) − 1) + BT with the exponential rate swept over 5.5 to 8.5 per billion years and the two amplitudes refitted at each; the best fit lands at λ = 7.00, against the 6.93 the published chronology uses, which is a check on the fitting rather than an input to it. What the figure exists to show is where the anchors fall in time. Six of them lie between 3.15 and 3.92 billion years, four between 0.026 and 0.80, and there is nothing whatever between 0.8 and 3.1 — a gap covering half the age of the solar system. The consequence is drawn as the spread of the acceptable curves: a surface whose crater density says 3.5 billion years is dated to ±0.05 billion by this family, and a surface at 2 billion to ±0.14. The chronology is a measurement where the astronauts landed and an interpolation everywhere else.
Fig. 3 The same fit with the crater densities allowed 30 per cent uncertainty and the exponential rate swept over a wider range. The family of acceptable curves widens, and it widens most in the unsampled interval — which is what it means for the anchors to constrain the ends and not the middle. The spread is still an underestimate of the real ignorance, because every curve drawn has the same two-branch functional form and the true production history need not.

There is a further difficulty in the oldest part of the curve, and it is the reason the exponential branch is contentious rather than merely uncertain. The steep rise before 3.8 billion years is read by some as the tail of the declining bombardment left over from accretion, and by others as a distinct spike — a cataclysm around 3.9 billion years in which the giant planets rearranged and the belt was emptied into the inner solar system. The two readings imply different curves before 3.9 and are almost indistinguishable after, because the highland samples are all from the narrow interval when either would have been happening.

There is one more thing the gap hides, and it concerns the shape of the linear branch rather than its position. A constant impact rate for the last three billion years is an assumption, not a result: the four young anchors are all within the last billion years and the six old ones are all before 3.1, so nothing in the calibration would notice if the rate had been, say, half its present value between 3 and 1 billion years ago and higher since. Several lines of argument suggest the rate has been roughly constant — the ages of lunar and terrestrial impact glasses, the crater retention of ancient terrestrial cratons — and none of them is a direct measurement of the rate in that interval.

Below 410 metres the oldest surface stops counting. Cumulative crater counts — the number of craters at least as large as D, per square kilometre — for four lunar surfaces of different ages, with both axes logarithmic. The production curves are parallel straight lines of slope −2.00, one above the other in proportion to how long each surface has been exposed: age enters as a vertical shift and nothing else. The heavy line crossing them is the empirical saturation density, of slope −1.83, which is where craters are packed so tightly that each new one destroys about one old one and the count stops rising. Because production is the steeper of the two, saturation is reached first at the small diameters, and it is reached at all only on ancient ground: the 3.9 Gyr surface saturates below 410 metres while the 0.2 Gyr surface does not saturate anywhere on this plot. So a count that includes small craters on old terrain is measuring the strength of the regolith rather than the age of the ground, and the diameter at which the measured distribution bends away from the production slope is where a count has to stop being believed.
Fig. 4 The size–frequency distributions of four surfaces spanning the gap and its ends, with the saturation line that every count eventually meets — 410 metres for the oldest surface drawn, below which every new crater destroys an old one and the count stops being a clock. The curves are parallel, because the production function’s shape is assumed independent of time — another statement that the calibration cannot test, since the two sampled epochs are the only ones whose production function has ever been measured. If the impactor population’s size distribution changed between them, every intermediate age computed by extrapolating a count to one kilometre inherits the error.

Taking it to another planet

Everything so far concerns the Moon, which is the only body outside Earth whose rocks have been dated in a laboratory. Every other surface is dated by taking the lunar curve and multiplying.

The multiplier is the ratio of the impact rate at the target to the impact rate at the Moon, and it is not measured. It is computed: from a model of the impactor population, its orbital distribution, the encounter speeds at each planet, and the conversion from impactor size to crater size at each planet’s gravity — with an atmosphere to correct for at Mars and Venus and Titan.

One crater count, three Martian ages 0.3 billion years apart. Age against crater density, for the Moon and for Mars under two assumptions about how much faster Mars is cratered. The lunar curve is the calibrated one. The Martian curves are the same curve with the density divided by an impact-rate ratio, because a surface cratered R times faster reaches a given count in less time — and that ratio is not measured. It is computed from a model of the impactor population: the number of bodies on Mars-crossing orbits against the number on Earth-crossing ones, their relative speeds, and how a given impactor size converts into a crater at each planet's gravity and atmosphere. Published values range from about 1.5 to about 2.6, and the two drawn here bracket most of them. A Martian surface carrying 8.0e-4 craters of a kilometre or more per square kilometre is 0.68 billion years old on the lowest ratio drawn and 0.37 on the highest. That difference, 0.31 billion years, is larger than the formal uncertainty on almost any published Martian age, and it is a systematic shared by every one of them — so two surfaces dated with the same chronology can be compared and a Martian age cannot be compared with a lunar one without carrying it.
Fig. 5 Age against crater density for the Moon and for Mars under three assumptions about the impact-rate ratio. A young Martian surface carrying 8·10⁻⁴ craters of a kilometre or more per square kilometre is 0.68 billion years old at a ratio of 1.4 and 0.37 at 2.6 — the same count, 310 million years apart, and the choice between them is a choice between published models rather than between measurements.

The impactor population itself is supplied by the asteroid families, whose membership and ages are themselves uncertain by tens of per cent, and it reaches the inner solar system through the resonances that a family’s fragments drift into as Yarkovsky pushes them. Neither of those steps has ever been checked against a count of anything. So the Martian chronology rests on the lunar chronology, which rests on nine samples; and the ratio between them rests on a model of a population whose calibration rests on a clustering cutoff.

One crater count, and a flux ratio that barely matters. Age against crater density, for the Moon and for Mars under two assumptions about how much faster Mars is cratered. The lunar curve is the calibrated one. The Martian curves are the same curve with the density divided by an impact-rate ratio, because a surface cratered R times faster reaches a given count in less time — and that ratio is not measured. It is computed from a model of the impactor population: the number of bodies on Mars-crossing orbits against the number on Earth-crossing ones, their relative speeds, and how a given impactor size converts into a crater at each planet's gravity and atmosphere. Published values range from about 1.5 to about 2.6, and the two drawn here bracket most of them. A Martian surface carrying 2.0e-2 craters of a kilometre or more per square kilometre is 3.73 billion years old on the lowest ratio drawn and 3.63 on the highest. The difference is only 101 million years, and that is the other half of the story: on the exponential branch, where an ancient surface sits, the chronology rises so steeply that dividing the count by a factor of 1.6 moves the age hardly at all. The flux ratio is a crippling uncertainty for a young lava flow and almost none for the southern highlands.
Fig. 6 And the same comparison on an ancient surface. At a density of 2·10⁻² per square kilometre the two ratios give 3.73 and 3.63 billion years — a hundred million years apart, on a surface nearly four billion years old. The chronology’s exponential branch is so steep there that dividing the count by a factor of 1.6 barely moves the age. The flux ratio is a crippling uncertainty for a young lava flow and almost none for the southern highlands, which is the opposite of the usual expectation that older things are dated worse.

That inversion is worth holding on to, and it is the same shape as the one a mismeasured inner radius produces for a black hole’s spin: where the calibration curve is steep, an error in the observable is a small error in the answer.

What is actually counted

Underneath the chronology sits the count, and the count has its own difficulties which the earlier rungs of this anchor are about.

The crater clock is sharpest where the surface is oldest. What an uncertainty of a factor of 3 in the crater count is worth in years, plotted against the age it is worth it at. A count is never better than about this: craters have to be recognised, degraded ones argued over, and secondaries thrown out. On a young surface the chronology is linear, so a factor of 3 in the count is a factor of 3 in the age — at 0.2 billion years that is 267 million years of uncertainty, and the worst case on this curve is 1450 million years at 1.5. On the oldest ground the exponential is so steep that the same factor of 3 moves the answer by only 161 million years, at 4.2 billion. That inversion is worth sitting with, because it is the opposite of every other dating method: the crater clock is at its most precise where the surface is most ancient, and at its vaguest on the young lava flows a geologist would most like to sequence. It is also why the Apollo 16 highland breccias pinned the early bombardment so firmly while the ages of young Martian flows are quoted with error bars comparable to the ages themselves.
Fig. 7 The effect of resolution. A count made at three times worse resolution loses the small craters first, and the small craters are the numerous ones — so the density measured depends on the image as well as on the surface. Comparing counts made with different instruments requires extrapolating each to a common diameter through an assumed production slope, and the slope is itself fitted to counts.
A branch that adds nothing above 6 km and everything below it. Cumulative crater counts on a 2.5-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 6 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 12506 to one. A count taken at 100 metres and read through the primary production curve returns an age of 4.60 billion years for ground that is 2.5, and it returns it with a small formal error, because the counting statistics are excellent. The error is not in the counting.
Fig. 8 And the contamination that the second rung of this anchor is about. Craters made by ejecta from other craters add nothing above about six kilometres and can dominate below it, so a count taken at small diameters on a young surface can be mostly secondaries and can read as an age several times too old. The two effects — resolution pushing counts to small diameters and secondaries dominating there — pull in the same direction, and both are worse for the young surfaces whose ages the flux ratio has already made uncertain.

What is actually measured

The honest summary of a published planetary age is a chain, and it is worth setting out in full because no single step of it is weak.

An image is taken; craters are identified and measured; a size–frequency distribution is built; a diameter range free of saturation and of secondaries is chosen; the count in that range is extrapolated to one kilometre through a production function fitted on the Moon; the result is divided by an impact-rate ratio computed from a dynamical model; and the quotient is passed through a curve fitted to nine radiometric dates on one body.

Two of those steps deserve a closer look because they are where the largest quiet corrections live. Choosing the diameter range is a judgement, made by looking at the size–frequency distribution for the departure from a straight line that marks where secondaries take over — and different workers reading the same distribution choose different ranges. And the extrapolation to one kilometre magnifies whatever the slope error is: a surface counted at 200 metres and extrapolated over two thirds of a decade with a slope wrong by 0.15 has its density wrong by twenty per cent, which on the linear branch is twenty per cent in the age.

Six of the eight steps involve a model calibrated elsewhere. The two that do not — counting the craters and measuring their diameters — are the only ones the quoted error bar usually reflects.

The comparison worth making is with the one place a crater chronology has been checked against an independent absolute date. The Earth has both craters and radiometrically dated rocks, and the terrestrial cratering rate over the last hundred million years — from the small number of large craters whose ages are known — agrees with the lunar rate scaled by the appropriate ratio to within a factor of about 1.5. That is a genuine test and it passes, and it tests only the most recent one part in forty of the curve.

None of this makes planetary ages worthless. The relative ordering of surfaces on one body, dated with one chronology, is robust: a surface with ten times the crater density is older, and by roughly the right factor. What is fragile is any absolute number, and any comparison between bodies.

What the ages are used for

It is easy to treat the numbers as a technicality until the arguments that rest on them are set out.

The duration of Martian volcanism — whether the planet was geologically active a hundred million years ago or stopped a billion years ago — is a crater-count result and nothing else. The history of water on Mars, the sequence of the valley networks against the outflow channels against the gullies, is a stratigraphy whose absolute dates all come from this curve. The age of Europa’s surface, and therefore the resurfacing rate that any model of its ocean has to produce, is a crater count divided by an impact rate that is a factor of several uncertain. The timing of the late heavy bombardment, and therefore the window in which life on Earth could have begun, is the exponential branch above.

Each of those is stated in the literature with an uncertainty, and each of those uncertainties is dominated by something that does not appear in it.

Why nobody has fixed it

The obvious remedy has been obvious for fifty years and it is worth saying why it has not happened.

A sample return from a lunar surface of intermediate age is a small mission by modern standards: land, scoop, return a few hundred grams. What makes it hard is not the engineering but the requirement that the sample’s provenance be unambiguous — the rock has to come from the unit whose craters were counted, not from ejecta thrown in from somewhere else, which on a surface that has been bombarded for two billion years is a demanding condition. The Chang’e 5 sample succeeded on exactly that point: it came from a young mare unit thick enough and flat enough that the surface material is local.

The second reason is that the payoff is diffuse. A date in the gap improves every Martian age by a factor of two, and no single result by more than that. Missions are proposed for results, and “the uncertainty on a great many other people’s numbers would halve” is a weak proposal even when it is true.

The generalisation

The structure worth extracting is that a calibration curve is only as good as the distribution of its calibration points, not their number or their individual quality.

Nine excellent dates clustered at two epochs constrain a two-parameter model well and a three-parameter one badly, and they say almost nothing about the interval between them however precise each of them is. Adding a tenth date at 3.5 billion years would improve nothing. Adding one at 2 billion — a sample returned from a mare unit of intermediate age, which is a mission that has been proposed repeatedly — would halve the uncertainty on every intermediate age in the solar system.

There is a second, sharper reading. A model with more parameters than the data can distinguish will produce a confident interpolation and a meaningless one. The two-branch functional form has three free numbers and nine points; it fits, and the fit is good, and the goodness of the fit says nothing about the region where no point lies. A reader shown the curve and the points naturally reads the curve as the measurement, because that is what a curve through points usually is. Here it is the assumption, and the points are the measurement.

The same shape governs every ladder in this collection. Four distances to the same galaxy agree because the rungs overlap; a ladder whose rungs did not overlap would have the same problem as this curve, and for the same reason. And a helium abundance read as a count of particle species — an entirely different subject with the same structure — survives its own weakest input precisely because the quantity read off it is a slope rather than a value in an unsampled interval.

There is a third reading, about how a field’s confidence and its evidence can drift apart. Crater counting became routine and then became infrastructure: a Martian age is quoted in an abstract with no more ceremony than a coordinate. Nothing in that usage carries the information that the curve behind it has a hole in it, and the people who put the hole there said so plainly at the time. Confidence accumulates through citation in a way uncertainty does not, and a number that has been used a thousand times acquires a solidity that its derivation never had.

The corollary is a question to ask of any fitted relation. Where are the points, and where is the answer being read? If the second is not inside the first, the quoted uncertainty is describing the fit rather than the reading.

Where the ladder goes next

The next rung takes the one thing that would settle the interpolated region: a returned sample of known provenance from a surface of intermediate age. The candidate units are known, the crater counts on them have been made, and the mission has been designed several times. What the rung is about is what such a sample would and would not fix — it would pin the curve at one point in the gap, which halves the uncertainty on Martian volcanism, and it would say nothing at all about the impact-rate ratio.

Further rungs on this anchor: the cataclysm hypothesis and what a sample of an unambiguously pre-Imbrium surface would decide; the chronologies of the outer solar system, where the impactors are comets rather than asteroids and the ratio to the Moon is uncertain by an order of magnitude rather than a factor; the youngest surfaces, where crater counts run out of craters entirely and ages come from morphology; and the crater counts on asteroids themselves, where the target’s own gravity is so low that most of the ejecta escapes and the production function is a different function.

Where this goes next

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. A family of 276, or of 582, depending on one number. Membership of a synthetic family against the cutoff velocity used to define it, for a family of 260 genuine fragments ejected at 15 metres a second sitting in a background of 417 unrelated bodies. The clustering is single linkage in the standard proper-element metric, started from one object and grown until nothing more is within the cutoff of anything already absorbed. Three curves: the number of objects claimed, scaled to its largest value; the fraction of the real family recovered; and the fraction of the claim that is background. The shape is the whole difficulty. At low cutoff the family is fragmented and only its core is found. There is then a plateau — near 58 metres a second here, giving 276 members of which 6 per cent are background — and that plateau is what every published family list is chosen at. Past it the count runs away, because single linkage absorbs an object and then searches from the object it has just absorbed: one chance interloper bridges the family to the belt and the algorithm returns half the main belt. Completeness and contamination rise together, so there is no cutoff at which both are good and the plateau is a compromise rather than a discovery. A family whose size is a choice Nothing observable distinguishes a collisional fragment from an asteroid that happens to be nearby. Membership is assigned by clustering at a cutoff velocity, and the same family has 276 members or 582 according to which cutoff is used — with completeness and contamination rising together, so no cutoff is good.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

CalibrationCrater chronologyCrater countingCratering rateExposure ageImpact fluxLate heavy bombardmentPlanetary surfacesProduction functionRadiometric dating