A past that can only be given as odds

A planet’s obliquity — the tilt of its spin axis to its orbit — is not a constant. The Sun pulls on the planet’s equatorial bulge, and the resulting torque makes the axis precess, sweeping a cone around the orbit’s pole with a period of tens of thousands to hundreds of thousands of years. At the same time the orbit’s plane itself wobbles, pulled by the other planets, at a handful of slow frequencies. When the axis’s precession rate matches one of those frequencies, the two lock into resonance and the tilt can change by many degrees. The Earth’s precession is fast enough, because the Moon triples it, to sit above all of those frequencies, and its tilt varies by only a degree and a half.

Mars has no large moon. Its precession constant is about eight arcseconds a year, a sixth of the Earth’s, and at its present tilt that puts its precession rate right among the frequencies of its orbital plane’s wobble. Several resonances lie within a few tens of degrees of its present tilt, and they are wide enough to overlap. An axis in overlapping resonances does not settle into any of them; it wanders between them, and the wandering is chaotic. The word is meant precisely: the equations contain no randomness at all, and the same starting axis always gives the same history. What they lack is stability — any difference in the start, however small, grows until it is as large as the tilt itself.

Why slow precession invites resonance #

The torque that drives precession is set by three things: the planet’s flattening, which gives the Sun something to pull on; the planet’s spin, which gives the axis its gyroscopic stiffness; and the strength of the tidal pull, which falls with distance from the Sun. They combine into the precession constant, and the actual precession rate is that constant times the cosine of the obliquity. For Mars, a smaller and more distant body than the Earth, with no Moon adding its own torque, the constant is about 8.3 arcseconds a year, and the precession period at its present tilt is about 170,000 years.

The orbital plane’s wobble is described by the secular theory of the planetary system: each planet’s orbit responds to a set of normal modes shared among all the planets, each with its own frequency, and the plane’s orientation is a sum of them. Those that matter for Mars have periods of about 50,000 to 250,000 years, frequencies of roughly 5 to 26 arcseconds a year. A precession rate of 8.3 times the cosine of the tilt matches the mode near 7 arcseconds a year at a tilt of about 31 degrees, and the mode near 5.6 at about 47 degrees. Around each exact match there is a band of tilts over which the resonance can capture the axis, and for Mars the bands are wide enough to touch.

That is the classic recipe for chaos: two resonances whose widths exceed their separation, which is the same criterion that makes the general three-body problem unpredictable and that Boris Chirikov stated for any system driven at more than one frequency. Between them, the axis has no single frequency to follow, and it is passed back and forth between the two, driven by the others, in a way that is deterministic and unpredictable.

A forecast with a horizon #

The unpredictability can be measured directly. Two integrations of the same planet, identical except that their starting axes differ by a tiny angle, stay together for a while and then separate.

That exponential growth is the defining property of chaos, and its rate — the Lyapunov exponent — sets how far ahead or behind a calculation can see. In the model drawn here the e-folding time is about a million years; the careful integrations of Mars with the full solution for the planets’ orbits find about five. Either way the consequence is the same. A prediction in a chaotic system carries an expiry date, and for Mars’s obliquity it is a few tens of millions of years, beyond which even a perfect knowledge of the present axis, to any attainable precision, leaves the past undetermined.

The integrations run forward in time, but the equations are reversible, and the same fan opens into the past. The last ten million years of Mars’s tilt can be computed and are known: the axis was between about fifteen and forty-five degrees, cycling on a 120,000-year period modulated by a 1.2-million-year one, and it spent a few million years at higher average tilt before a shift about five million years ago. Earlier than about twenty million years, individual histories diverge, and the question “what was Mars’s obliquity three billion years ago?” has no answer. What has an answer is a different question.

The distribution is the prediction #

A chaotic system that cannot be followed can still be characterised statistically. Over a long enough time, the axis visits every tilt within its accessible range, and the fraction of time it spends at each is a stable property of the dynamics — the same for any starting point within the chaotic zone and for any of the diverging histories.

The integration drawn uses illustrative orbital perturbations of the size Mars’s own have, not the full planetary solution, and its details should not be over-read. The published calculation by Jacques Laskar and colleagues in 2004, using the full solution and an ensemble of integrations spanning four billion years, found a similar picture: a most probable obliquity of about 42 degrees over the planet’s history, a range from nearly zero to more than 80 degrees, and a present tilt that is typical of the last few million years but low by the standard of the last few billion. Both calculations agree on the qualitative result that matters for Mars’s climate: for most of its history, Mars has been tilted far more than it is now. The present moment is not special in the dynamics; it is special only in being the moment at which the tilt is measured, and there is no reason to expect the measured value to be the typical one. A visitor arriving at a random time in Mars’s history would more often find it tilted by forty degrees or more than by the twenty-five it has today.

That is a statement about the past that can be made with confidence precisely because it is statistical. It is not a statement that Mars was at 42 degrees at any particular date. It is a statement about how Mars’s climate record should look if it faithfully registered the tilt: long stretches at high obliquity, shorter ones at low, and transitions between them at intervals of a few million years.

The claim that the distribution does not depend on where the history starts can be tested directly: run it again from a different tilt.

That agreement is the practical content of chaos theory for a problem like this one. An individual trajectory is lost; the statistics of trajectories are not, and they are what a long record averages over. It is the same distinction that governs the accumulation of numerical error in a long integration: a computed position becomes wrong, while the computed rate at which things happen stays right. For Mars’s axis, “how much time at each tilt” is the robust question and “which tilt, when” is the lost one, and the history of the planet’s climate has to be argued from the first.

Where the sunlight goes #

Obliquity matters to a climate because it decides how sunlight is shared between latitudes over the year. The annual total reaching a planet does not depend on it, but its distribution does, as it does on the Earth, where the tilt alone draws the climate zones. At zero tilt the poles get almost nothing and the equator everything; as the tilt grows, the summer poles face the Sun more directly and the annual-mean sunlight at the poles grows as the sine of the obliquity.

For Mars that swing is the difference between polar ice caps and equatorial glaciers. At the present tilt, water ice is stable at the poles, where the annual sunlight is least, and it is found there in the polar layered deposits — kilometres-thick stacks of ice and dust. At high tilt, the poles become warmer on average than the equator, the polar ice sublimates, and the water is redeposited at lower latitudes, where the cold, shaded ground can hold it. The threshold matters because water ice is only stable where the surface stays cold enough all year, and the coldest places move with the tilt: at low obliquity the poles, at high obliquity the mid-latitudes and even the tropics, whose annual sunlight then falls below the poles’. There is geological evidence for exactly that: flow features at mid-latitudes resembling debris-covered glaciers, and ice-rich mantles that appear to have been laid down in the last few million years.

The layers as a record #

The polar layered deposits are the record the distribution is tested against. Each layer is a slightly different mixture of ice and dust, and the sequence of layers, exposed in the walls of troughs cut into the caps and imaged from orbit, carries a pattern of thicknesses and brightnesses. If the layers were laid down under the control of the insolation, the pattern should contain the periods of the obliquity and precession cycles — 120,000 years, 1.2 million years, 51,000 years for the precession of the perihelion — in the same way that ice cores and ocean sediments on the Earth contain the Milankovitch periods.

Analyses of the brightness profiles of the north polar deposits have found periodicities consistent with the obliquity cycles, at a deposition rate of about half a millimetre a year, which would make the upper few hundred metres of the northern cap a record of the last few million years — within the forecast horizon, where the obliquity history can be computed and compared layer by layer. The deeper layers, and the south polar deposits, which are older, reach back into the chaotic past, where only the distribution applies. There the test is statistical: the relative thicknesses of the deposits laid down in high- and low-tilt epochs should reflect the time the axis spent in each.

Seasons that are not just warmer #

A high tilt does not only move the annual mean. It sharpens the seasons: at 45 degrees a Martian polar summer would bring the Sun within 45 degrees of the zenith at the pole for weeks, delivering more sunlight per day than the equator ever receives, and a polar winter would be longer and darker. The carbon dioxide that freezes onto the winter pole each year, a quarter of Mars’s thin atmosphere, would condense and sublimate in larger amounts, and the atmospheric pressure would swing more over the year. At low tilt, by contrast, the poles would be cold all year, carbon dioxide could freeze permanently, and the atmosphere could partly collapse onto the caps.

So the tilt history is also a history of Mars’s atmosphere. A reservoir of buried carbon dioxide ice detected by radar beneath the south polar cap, enough to double the present atmosphere if released, is interpreted as frozen out during a recent epoch of lower tilt and not yet returned. The obliquity is one of the few processes on Mars that could change its surface pressure by a factor of two within a few hundred thousand years, and it does so repeatedly.

What the model leaves out #

The spin-axis equation is exact for a rigid planet whose orbit plane moves; what is illustrative is the orbital wobble, represented by five modes with amplitudes of the right size rather than the full planetary solution, which contains many more terms and some chaos of its own. That makes the model’s Lyapunov time shorter than the careful calculations’ and its exact distribution different in detail. It is chosen to show the mechanism — overlapping resonances between a slow precession and an orbital wobble — and to compute the consequences of that mechanism, not to reproduce Mars’s history.

The precession constant is also not quite constant. It depends on the planet’s flattening, which responds to the redistribution of ice between poles and equator and to the formation of the great Tharsis volcanic rise early in Mars’s history; changes of a few per cent move the resonances by several degrees. And over billions of years the spin rate itself may have changed. The published ensembles vary these within their uncertainties, which is part of why their predictions are distributions rather than tracks.

Finally, the model contains no climate. The insolation curve says how sunlight is distributed, not where ice ends up, which depends on the atmosphere’s thickness, on dust, and on the ice’s own insulating lag deposits. The step from obliquity to layers passes through all of that.

A moon as a stabiliser, and why the claim is weaker than it sounds #

The comparison drawn in the first figure — Mars as it is, and Mars precessing as fast as the Earth — is the basis of a widely repeated claim: that a large moon stabilises a planet’s tilt and so its climate, and that the Earth’s habitability owes something to the Moon. The figure supports the first half. A precession rate raised well above the orbital frequencies does keep the tilt steady, and without the Moon the Earth’s precession would be slow enough to enter a chaotic zone at tilts between about 60 and 90 degrees, and would reach tilts of a few tens of degrees from its present 23 only over hundreds of millions of years.

The second half needs care. The Earth’s present tilt, without the Moon, would lie below the chaotic zone for Earth’s own orbital frequencies, and later work found that a moonless Earth’s tilt would vary by perhaps ten to twenty degrees over billions of years rather than wandering over the whole range as Mars’s does. A moon is one way to avoid chaotic obliquity; being in the right place in the frequency spectrum is another. Mars happens to be in the wrong place, and that — not its lack of a moon alone — is what made its past unknowable.

The same analysis now has to be done for planets around other stars, and there the frequencies are different. In compact systems of several planets packed close to their star, the secular frequencies of the orbital planes are much higher than in the Solar System, and a rocky planet’s precession, unless it is very fast, falls among them. Some calculations find that a planet in the habitable zone of such a system could have a tilt that swings by tens of degrees over a few hundred thousand years — far faster than anything Mars experiences — while others in the same system are stable. Whether a given exoplanet’s tilt is steady is therefore a question about its whole system’s architecture, and the one known planet whose tilt is chaotic, Mars, is the calibration for how such a planet’s climate might respond.

Still open: whether the layers read the tilt #

The link between the obliquity distribution and the ice record rests on the assumption that the layers are a climate record in the way terrestrial ice cores are: that each layer’s composition registers the insolation when it formed and that no layers are missing. Both are uncertain. Periods of erosion can remove layers and leave unconformities that break the sequence, and dust storms add variation with no orbital cause. Radar sounding from orbit, which images the layers inside the caps in three dimensions, has found unconformities that may mark epochs of high tilt when the ice retreated. Whether the full sequence can be matched to computed insolation over the last few million years, and so extended as a tested chronology into the older layers, is the question on which the use of Mars’s poles as a climate archive depends — and the answer will say whether the chaotic history computed from celestial mechanics was actually recorded in the ice.