A tilt held still by being too fast to resonate

The first rung of this anchor established the obliquity as a dynamical variable with equilibria of its own, and noted that Mars’s has swung between about thirteen and forty-five degrees while the Earth’s has stayed within a degree and a half of twenty-three and a half. The usual explanation is the Moon, and the usual explanation is usually stated in a way that is misleading.

The Moon does not hold the axis. Nothing holds an axis; a torque makes it precess, and precession is a circulation rather than a restraint. What the Moon does is change a rate, and the rate matters because of what else in the solar system happens at similar rates.

It is also worth being clear about what “chaotic obliquity” means, since the word does a lot of work. It does not mean the axis tumbles, or that the tilt changes on human timescales. It means that the obliquity’s trajectory is exponentially sensitive to its starting value, so that the sequence of values it takes over the next hundred million years is not computable from present measurements — while the range it explores, and how long it takes to explore it, are perfectly well determined. A chaotic obliquity is a variable with a known distribution and an unknown history, which is the same object a prediction with an expiry date describes in the orbital case.

The two rates, and why they can meet #

The spin axis precesses about the orbit normal at α cos ε, where α is a constant of the planet — its oblateness, its moment of inertia, its rotation rate and the strength of the torques on it — and ε is the obliquity. That is the twenty-six-thousand-year circle the pole traces, and by itself it is perfectly regular.

The orbit normal is not fixed. The Earth’s orbit plane is pulled by the other planets and its node regresses, not at one rate but as a superposition of the linear secular theory’s eigenmodes — a set of discrete frequencies belonging to the solar system as a whole rather than to any planet. Their magnitudes run from 0.69 to 26.35 arcseconds a year.

So the axis is precessing about a plane that is itself turning. When the two rates are unrelated, the obliquity is very nearly constant and the whole thing is bookkeeping. When they match, the resonant angle stops circulating and starts librating, and the obliquity swings with it.

The condition for a resonance is $\alpha\cos\varepsilon = |s_i|$, which for a given planet is an equation in the obliquity alone. Every crossing in the figure is a solution of it.

What happens at a crossing is worth spelling out, because “resonance” is doing a lot of work in that sentence. In the frame that turns with the orbit’s node, the axis feels a slowly varying torque instead of a rapidly averaging one, and the obliquity acquires an equilibrium. Nearby, it librates about that equilibrium; the amplitude of the libration is set by how strongly the relevant secular mode is excited in the planet’s orbit, which for the Earth means how inclined the orbit plane is relative to the invariable plane. The Earth’s is inclined by about 1.6 degrees, which is small, so even a resonance would produce a modest libration — a few degrees rather than tens.

That is the reason a single resonance is not the danger. The danger is several overlapping, because then the obliquity can wander from one to the next without limit, and the excursion is bounded only by where the resonances stop.

It is worth being concrete about the magnitudes, because they are the reason the question has an answer at all. The secular frequencies are of order ten arcseconds a year, which is a period of a hundred and thirty thousand years — comparable with the slower Milankovitch cycles, and utterly negligible on any human scale. The spin precession is fifty arcseconds a year, a period of twenty-six thousand. Those two are within a factor of a few of each other, which is why they can meet. Had the Earth’s precession been a hundred times faster or a hundred times slower, no arrangement of the planets could have produced a resonance and the whole subject would not exist.

The near-coincidence is not designed and it is not surprising either. Both rates come from the same kind of quantity — a small mass ratio times an orbital frequency — so they land within an order of magnitude of one another for the same reason that most secular timescales in the solar system are tens to hundreds of thousands of years.

What the Moon actually contributes #

The Moon supplies about two thirds of the torque that drives the precession and the Sun the remaining third. Removing it therefore does not remove the precession; it divides α by about three.

That is enough to move the whole curve down through the forest. With α = 54.8 arcseconds a year, α cos ε exceeds every secular frequency until the obliquity passes sixty degrees. With α = 17.5 — a Moonless Earth still turning once a day — the maximum value of α cos ε is 17.5, which is below three of the seven frequencies and above four, so the crossings occur at obliquities the planet can plausibly occupy.

A useful way to see the size of the effect is to ask how much of the Moon could be removed before it mattered. The precession constant scales with the satellite’s mass and the inverse cube of its distance, so halving the Moon’s mass would take α from 54.8 to about 37, which is still above the fastest secular mode but no longer clear of it by a factor of two — and the crossings would move from sixty degrees down towards forty-five. The Moon is not comfortably sufficient; it is sufficient with a margin of about a factor of two, and that margin is being eroded as the Moon recedes.

That last point deserves stating because it is a real prediction rather than a curiosity. The Moon is receding at 3.8 centimetres a year, so α is falling, so the crossings are moving down towards the Earth’s obliquity. Extrapolating the recession is unsafe — the present rate is anomalously high — but on any plausible history the Earth’s obliquity enters the resonance forest at some point in the next few billion years, well before the Sun leaves the main sequence.

The other half of the mechanism is overlap. An isolated resonance produces a libration — the obliquity oscillates between two values and returns, which is orderly. Chaos requires two or more resonances close enough that their libration zones touch, so that a trajectory leaving one is captured by the next and there is no invariant surface between them. That is where chaos comes from generally, and it is why the argument is about the density of crossings rather than about the existence of one.

The number nobody has #

The published widths of the Moonless chaotic zone have changed, and the reason is a single unmeasured input.

The dependence is not even of a settled sign, which is the part worth dwelling on.

Neither scaling is a refinement of the other and there is no third one waiting to arbitrate. The choice is a statement about how completely a planet’s interior has relaxed to its own rotation, and that is not a thing the counterfactual specifies.

The 14.6-hour boundary in the last three figures deserves one more sentence, because it is where the whole counterfactual turns. A Moonless Earth spinning faster than that is safe under the fluid scaling and unsafe under the rigid one; slower than that, the reverse. Nothing about a planet says which scaling to use — it depends on how completely the interior has relaxed to its own rotation, which for a body with a fluid outer core and a mantle that creeps over ten thousand years is neither limit exactly.

The first careful calculation, in 1993, integrated the spin axis of a Moonless Earth through the secular system and found chaotic obliquity over essentially the whole range from zero to eighty-five degrees for prograde rotation. A later and more thorough one, in 2012, found the chaotic zone narrower — large but not universal, and strongly dependent on the initial spin period and on whether the rotation was prograde or retrograde. Neither result is wrong. They are answers to slightly different questions, and the difference between the questions is a number that cannot be observed.

What the Cassini states are doing underneath #

The resonances above are not exotic objects. They are the same equilibria the first rung of this anchor drew.

What decides how many of those equilibria exist is a ratio no planet holds fixed. Tides lengthen the day and so lower the precession rate, and the secular frequencies themselves drift as the other planets exchange angular momentum — so an axis can be delivered into a libration zone that did not exist when the planet formed.

What this does and does not license #

The argument is often carried further than it goes, and it is worth marking where it stops.

It supports: an Earth without a large satellite would have had a substantially more variable obliquity, over tens of millions of years, with excursions of tens of degrees.

It does not support: that such a planet would be uninhabitable. Obliquity excursions of that size change the seasonal distribution of sunlight enormously and change the annual mean at each latitude by a real but bounded amount — and the climate consequences of a forty-five-degree obliquity have been modelled and are dramatic rather than sterilising. A planet with an ocean and a carbon cycle is a robust object.

It also does not support: that the Moon is rare and therefore habitable planets are. The Moon’s origin in a giant impact is one of a class of events that planet-formation models make common rather than exceptional, and the frequency with which such impacts leave a satellite of the required mass is not known to better than an order of magnitude.

What is actually observed #

Almost none of the above has been observed, and the honest accounting is short.

The precession constant itself is measured beautifully. It follows from the Earth’s dynamical ellipticity, which is measured from satellite orbits to eight significant figures, and from the observed general precession in longitude, which is measured against quasars. The two agree, and the agreement is a test of the whole Earth-model chain rather than of anything in this essay.

The secular frequencies are computed, not measured. They come from the linear secular theory or from a long numerical integration, and the two agree to a few parts in a thousand over the range of frequencies that matters here. They are not observable individually — nothing in the sky oscillates at s₃ in a way anybody can point at — but their superposition is what the Earth’s orbital elements do, and that superposition is checked against a numerical integration of the whole solar system.

The Earth’s obliquity is measured directly and its variation over the last few million years is recorded in the sedimentary record’s forty-one-thousand-year term, which agrees with the computed history. That is a real check on the theory, over one per cent of the Earth’s age and with the Moon present throughout.

The obliquity of Mars is the case the whole argument is really about, and it is instructive that it does not need the Moon’s absence to be chaotic. Mars has two small satellites that contribute essentially nothing, so its precession constant is set by the Sun alone and by its own flattening: 8.3 arcseconds a year, which puts α cos ε at about 7.6 — right in the middle of the forest, between s₂ at 7.06 and s₁ at 5.61. Mars is not chaotic because it lacks a moon. It is chaotic because it is slightly less oblate, slightly further from the Sun, and turning at very nearly the Earth’s rate.

Mars’s obliquity variation is not observed at all. It is computed, from the same secular theory applied to a planet with a precession constant of about 8.3 arcseconds a year, which puts it squarely in the forest. The evidence that the computation is right is circumstantial and good: layered deposits at the Martian poles with a spacing consistent with the computed cycles, and mid-latitude ice whose presence requires the obliquity to have been much larger recently than it is now.

What would settle the Moonless case #

It is worth asking what evidence could decide between the 1993 and the 2012 answers, since both are integrations of the same equations.

Nothing observational can, because the object does not exist. What could narrow it is a better theory of what a giant impact leaves behind: the impact that made the Moon set the Earth’s rotation rate as well, and any Moonless counterfactual has to specify a different impact history. Planet-formation simulations do produce distributions of final spin rates for terrestrial planets, and those distributions are broad — peaking somewhere in the range of a few hours to half a day, with a long tail. Feeding that distribution through the calculation would give a probability that a given Moonless Earth is chaotic, which is the right form for the answer and is not the form the question is usually asked in.

The other thing that would help is more planets. A statistical sample of terrestrial planets with measured obliquities would test the whole framework directly, since the framework predicts which of them should be variable. Measuring an exoplanet’s obliquity is at the very edge of what is conceivable — it requires resolving a seasonal signal in a thermal light curve — and it has not been done.

The generalisation #

The structure worth extracting is that stability against a resonance is usually a statement about a frequency being out of range, not about a restoring force being strong.

Nothing in this essay restrains the Earth’s axis. The axis precesses freely, exactly as a Moonless one would; what differs is that its precession is too fast to be caught. That is the same reason a driven oscillator far above its resonance responds hardly at all — not because it is stiff but because it is out of tune.

The same shape governs a great deal of solar-system dynamics. A resonance clears a gap by capturing bodies whose period matches, and leaves alone those whose period does not; the stability of a planetary system is a question about whether its secular frequencies overlap; and the bound that holds only in the linear theory fails precisely where two of those frequencies approach each other.

There is a second reading, about how a mechanism is best described. Saying “the Moon stabilises the Earth’s axis” is true and conveys nothing about why, and it invites the wrong picture — a heavy anchor holding something down. Saying “the Moon triples the precession rate and lifts it out of the secular frequencies” is a longer sentence and contains the whole mechanism, including its limits: it says immediately that a smaller moon would do less, that a more tilted planet would be less protected, and that a planet in a system with faster secular frequencies would need a bigger one. A mechanism stated as a rate comparison carries its own boundary conditions; a mechanism stated as an agency does not.

The corollary is a warning about counterfactuals. A calculation that removes something removes its consequences too, and the consequences may include the initial conditions the calculation needs. The Moonless Earth’s rotation rate is unknowable for exactly the reason the question is being asked, and no amount of computing power fixes that.

Where the ladder goes next #

The next rung takes the same machinery to Mars, where the precession constant is small, the resonances are dense, and the obliquity history is chaotic over a few tens of millions of years — so that the distribution of past obliquities can be computed while any particular past value cannot. That distribution is a real, checkable prediction, and it is what the polar layered deposits are being read against.

Further rungs on this anchor: the obliquity of a tidally locked planet, where the spin rate is fixed by the orbit and α is not free; the capture of Saturn’s axis into a Cassini state with Neptune’s node, which is the one large obliquity in the solar system with a specific proposed explanation; the obliquity histories of planets in compact multi-planet systems, where the secular frequencies are far larger and the forest is far denser; and whether a large satellite is a requirement for a habitable planet, which is the claim this essay’s argument is usually enlisted for and which it does not quite support.