Astronomy & Celestial Mechanics
the sky, computed
Series

Surface chronology — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · orbits
  2. 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.

    part 2 · orbits
  3. 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 calibrating samples 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.

    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.

    part 3 · orbits
  4. One surface, two ages, and a kink between them. The cumulative number of craters per square kilometre larger than a given diameter, both on logarithmic axes, for a surface that formed 3.9 billion years ago and was flooded 3.3 billion years ago by a layer 200 m thick. The dashed lines are the isochrons of the two ages under the lunar chronology. Craters with rims taller than the flood — larger than 5.6 km, since a simple crater's rim stands about 0.036 of its diameter high — survived it, so above the kink the count follows the 3.9-billion-year isochron. Everything smaller was buried, and the craters counted there formed after the flood, so below the kink the count follows the 3.3-billion-year isochron. Fitting the branches separately returns 3.90 and 3.30 billion years. The kink is not noise and not a change in the impactors: its diameter is a measurement of how thick the flood was.

    A kink that measures how deep the lava was

    A surface that was flooded partway through its life carries two crater populations — the old craters too big to bury, and the young ones made since — and its crater count follows one age above a certain diameter and another below it. The diameter of the kink is the thickness of the flood, read through the height of a crater's rim, and a single age fitted across it describes nothing that ever happened.

    part 4 · orbits
  5. 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.

    part 5 · orbits
  6. 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.

    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.

    part 6 · orbits

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