Layers that keep the rhythm and lose the dates

Mars’s axis wanders far more than the Earth’s, and beyond a few million years its tilt can only be given as a distribution of possible histories. Within those few million years, though, the history can be computed, and it ought to be written in the ice. Both of Mars’s poles carry layered deposits kilometres thick — stacks of ice with varying amounts of dust, exposed in the walls of spiral troughs and imaged from orbit, and sounded from above by radar that sees layers inside the caps. The layers are taken to record the climate as the summer sunshine at the poles rose and fell with the planet’s tilt and orbit.

The difficulty is that a stack of layers is a record in depth, and the forcing is a record in time. Converting one to the other needs the rate at which ice accumulated, and that rate is not known, varied, and may have depended on the very sunshine the layers are supposed to record. What a stack laid down under such conditions can still say, and what it cannot, is a question that can be answered by building one.

What the sunshine does #

At the pole on the day of the summer solstice the Sun is above the horizon all day, at a height set by the tilt, and the energy arriving depends on the tilt and on how close to the Sun the planet is at that moment. A larger tilt raises the Sun higher; a solstice that falls near perihelion brings the planet closer. The first effect varies with the tilt’s cycles, dominated on Mars by a period of about 120,000 years. The second depends on where the perihelion lies relative to the seasons, which goes round once in about 51,000 years, and on how eccentric the orbit is, which varies over 95,000 years and 2.4 million.

The forcing drawn here is built from those periods rather than integrated, so that the argument is about how such a forcing is recorded and not about which layer formed when. It swings by about a factor of two between its extremes: a slow swell from the tilt, and a faster ripple from the precession whose strength rises and falls with the eccentricity. On the Earth the corresponding swings are smaller, and they have been read in ocean sediments and ice cores for half a century — sometimes with surprises about which beat the ice obeyed. On Mars they are larger, the ice is exposed at the surface, and no core has been drilled.

Why Mars’s swings are so large #

The Earth’s tilt varies by little more than a degree either side of its mean, because the Moon’s pull on the equatorial bulge makes the axis precess too fast to resonate with the slow wobbles of the orbital plane. Mars has no such stabiliser, and its tilt swings by several degrees over a hundred thousand years and by tens of degrees over millions. Its orbit is also more eccentric, so the distance at the solstice matters more. The two together make the polar summer sunshine on Mars swing by factors of two over a few hundred thousand years, where the Earth’s swings by tens of per cent.

The annual total of sunshine reaching a planet is untouched by the precession, and the tilt alone draws the climate zones by deciding how that total is shared between latitudes. On Mars, a tilt above about 54 degrees gives the poles more annual sunshine than the equator, and the polar ice then migrates to lower latitudes; at the present tilt the poles are the coldest places and the ice is where it is. The layered deposits record the last few million years in which the tilt has stayed in the range that keeps ice at the poles, and their base marks a time when it did not.

So Mars’s caps record a stronger forcing than any terrestrial ice, in a climate simpler than the Earth’s — no oceans, little weather, an atmosphere a hundredth as thick. If orbital forcing can be read cleanly anywhere, it should be there.

A stack built two ways #

The simplest model of a deposit lays down ice at the same rate every year and mixes into it dust in proportion to the sunshine, so that dusty bands mark sunny epochs. A less simple one lets the rate fall when the summer sunshine is strong — less ice is left behind when summers are warm — by an amount controlled by a single factor, k.

The two columns are the same history, and they look recognisably alike: dusty and clean bands in the same sequence. The difference is in the spacing. In the second column the sunny epochs, laid down slowly, are squeezed into thin bands, and the shady epochs, laid down quickly, are stretched into thick ones. A layer’s thickness has become a measurement of the climate as well as of the time, and the two cannot be separated from the column alone.

At k = 2.5 the rate falls to near zero in the sunniest summers without becoming negative, so no ice is lost and the total is exactly that of the even case. Beyond that, the strongest summers remove ice, and parts of the history are not merely compressed but gone.

The rhythm survives #

The standard way to read a stack is to look for regularity in depth: to take the spectrum of the brightness or dust profile and find the wavelengths at which it varies most.

The two strongest peaks fall at the wavelengths the two main periods would occupy at the average rate — 120,000 years becomes sixty metres, 51,000 years becomes twenty-five — and their ratio is 2.35, the ratio of the periods. That ratio is what makes a spectral identification convincing. A single peak could be anything; two peaks in the proportion of the obliquity and precession periods, without the rate having been assumed, are hard to explain any other way. Analyses of the brightness profiles of Mars’s northern deposits have found layering at a wavelength of a few tens of metres, consistent with the tilt’s cycle at a rate of about half a millimetre a year, by exactly this logic.

The uneven deposit has the same peaks in the same places, which is the surprising part. Its precession peak is weaker and its spectrum has some spurious structure — combination tones, where the stretching of one cycle by the other creates wavelengths the forcing never had — but the main wavelengths, and their ratio, are where they were.

The reason is that stretching and squeezing average out over a cycle. Within one obliquity cycle, the shady half is thicker and the sunny half thinner than an even rate would make them, but the whole cycle occupies the average rate times its period, because the average rate is by definition what the cycles add up to. Both cycles are sampled by the same deposit with the same average, so their wavelengths stay in the ratio of their periods. Even when erosion removes the sunniest spells entirely, the cycles that remain still span the right thicknesses on average. The rhythm of a forcing survives a recording that distorts it unevenly, as long as the distortion is itself a function of the forcing and not of something else.

A record long enough to resolve its own peaks #

A spectrum can only separate two periods if the record spans enough cycles of their difference. The tilt’s two main terms, at 120,000 and 99,000 years, beat against each other with a period of more than half a million years, and separating them needs a record of a million years or more; the eccentricity’s 2.4-million-year term needs several million. A spectrum is also prone to peaks that the sampling makes rather than the signal, and a short, unevenly sampled record finds them readily.

The youngest few hundred metres of the northern cap, the part laid down since the tilt last swung high, span a few hundred thousand years at the inferred rate: two or three obliquity cycles and a handful of precession cycles. That is enough to find the main peaks and their ratio, not enough to resolve the obliquity’s fine structure, and far too little for the eccentricity’s long cycles. Deeper layers lengthen the record and bring in the erosion counted further on, so the trade between a long record and a complete one is set by the planet’s own climate history.

The dates do not #

What does not survive is the correspondence between a particular layer and a particular moment.

Reading the deposit at its average rate assigns each depth an age, and in an uneven deposit that age is wrong in a way that oscillates: too old in the stretched shady phases, too young in the squeezed sunny ones, returning to the right value at the end of each cycle. The errors do not accumulate, which is why the wavelengths are right, but within a cycle they reach a substantial fraction of the precession period. At k = 3 a layer can be misdated by more than twenty thousand years, which is close to half a precession cycle — enough to place a dusty layer at the opposite phase of the forcing from the one it actually recorded.

That matters for the most ambitious reading of the deposits, which is not spectral but layer by layer: tying a particular bright band to a particular high-tilt epoch, and so dating it. Such a tuning assumes a relation between the forcing and the accumulation, adjusts the depth–age map until the record matches the computed sunshine, and reports the result as a chronology. The figure says that the match can be good in rhythm and wrong in phase by close to half a cycle, and that nothing in the stack’s own profile reveals which.

Time that is not there #

A strong enough dependence on sunshine does more than distort. It removes.

Once the sunniest summers erode ice, each such spell removes not only its own record but some of the record below it, and the history acquires gaps. They fall in the sunniest phases of every cycle, and they are marked in the stack at most by an unconformity — a surface where younger ice rests on older with the layers between missing. Radar soundings of Mars’s northern deposits have found exactly such a surface a few hundred metres below the top of the cap, extending across much of it, and above it a sequence interpreted as the ice that has accumulated since the planet’s tilt last swung high, a few hundred thousand years ago. The unconformity is read as the erosion of an epoch whose summers were sunny enough to remove ice; the stack’s own record of that epoch is gone.

Gaps change the spectrum less than might be expected, as the ratio figure showed, because the remaining cycles still average correctly. They change the chronology a great deal. A count of cycles down from the top undercounts the time by whatever was eroded, and an age assigned to the base of a sequence from its thickness is a lower limit.

What a bright layer is #

Every argument so far treats the dust content of a layer as a faithful copy of the sunshine when it formed. The quantity actually measured is less direct. From orbit, a layer exposed in a trough wall is seen as a band of a certain brightness, and its brightness depends on how much dust it holds, on the size of the ice grains, which coarsen with age and temperature, on any frost that has settled on the exposure, and on the angle at which the wall faces the Sun and the camera. Radar sees layers as reflections where the electrical properties change abruptly, which happens at boundaries between dusty and clean ice but also at changes in density.

So the profile analysed for its spectrum is the sunshine filtered twice: once through the accumulation, which the figures model, and once through whatever turns a layer’s composition into a measured brightness, which they do not. The second filter can add its own periodicities — a trough wall’s slope varies with depth, and so does the illumination of the exposed layers — and it can weaken the record of the faster cycle relative to the slower, if thin layers are blurred together by the camera’s resolution. A twenty-five-metre precession band, exposed on a gentle slope, is spread across a wide strip of the wall and easily resolved; on a steep wall it is a few pixels across.

The independent measurements matter for this reason. Where a layer is seen both in an image of a trough wall and as a radar reflector inside the cap, and where both agree on its depth and continuity, the composition change it records is real and not an artefact of the exposure.

The Earth’s records, and the same trap #

None of this is peculiar to Mars. The Earth’s ocean sediments, ice cores and lake beds are read for the same orbital periods, and their chronologies are often built by the same tuning: adjusting the depth–age relation until a record matches the computed sunshine, on the assumption that the climate responded to the sunshine with a known lag. The figures here give that practice its two faces. The rhythm of the orbital periods, found in a record’s spectrum with the right ratios, is strong evidence that the record is orbitally paced, and it survives distortions of the rate that depend on the forcing. The phase — which peak of the record answers which peak of the forcing — depends on the assumed relation between them, and the record cannot test that assumption against itself.

On the Earth there are independent clocks to break the circularity: radiometric dates on volcanic ash layers, reversals of the magnetic field recorded in the sediments, and the decay of isotopes trapped in ice. Each pins a few depths to absolute ages and lets the tuning be tested between them. On Mars there are none yet. The polar deposits contain no datable ash and no dated magnetic reversals, and the only absolute anchor is the age of the surface, from the craters on top of the caps, which are few.

What a returned core would add #

The decisive measurement would be a sample of the layers with their own clock in them. Ice cores on the Earth are dated in their upper parts by counting annual layers and in their lower parts by the decay of trapped isotopes and by the flow of the ice; a Martian core could be dated by the accumulation of cosmic-ray-produced isotopes in its dust, by the decay of trapped gases, or by counting annual layers if the seasons left any. A few absolute ages along a core would fix the depth–age relation where the spectrum cannot, and the comparison of the dated layers with the computed sunshine would measure the relation between climate and accumulation directly — the k that the figures here have had to assume.

Short of a core, the radar soundings are the strongest constraint. They trace individual reflecting layers across the whole cap, and the variation of a layer’s depth from place to place records where ice accumulated faster or slower at the same moment. A layer that is thicker on one side of the cap than the other is a direct measurement that accumulation was uneven, and the pattern of unevenness is a map of the climate at that moment — something no single profile, however well its spectrum is measured, can provide.

What the model leaves out #

The forcing is a sum of the main periodic terms, not an integration of Mars’s orbit; the real history has more terms and, beyond a few million years, cannot be computed at all. The sunshine is the solstice value at the pole, when the quantity that controls accumulation may be the summer’s total or the sunshine at the edge of the cap. The accumulation rate depends linearly on the sunshine with one factor, and real accumulation depends on temperature, the supply of water vapour from elsewhere on the planet, the dust load of the atmosphere and the albedo of the ice itself, which feeds back on the sunshine absorbed. The dust content is taken to follow the sunshine directly. And the stack is a single column, with no flow of the ice, no compaction and no lateral variation. Each of these changes what the record looks like; none of them changes the result that a forcing-dependent rate preserves the rhythm and scrambles the phase.

Still open: whether the accumulation followed the sunshine or the water supply #

The figures treat accumulation as a function of the local sunshine. It may instead be set mainly by how much water vapour reaches the pole, which depends on the sunshine at the other pole, on the dust storms that warm the atmosphere, and on the reservoirs of ice at middle latitudes that grow and shrink as the tilt changes. If so, the rate is a function of the forcing, but a different function, with its own lags, and the phases of the layers relative to the polar sunshine would be shifted by amounts the spectrum cannot reveal. Separating those possibilities needs either dated layers or a climate model that follows the water round the planet through a whole tilt cycle — and the comparison of the two poles’ records, which should respond to the same forcing in opposite phase if the sunshine at each pole is what matters.