A tilt the Moon should not have

Kepler’s second law is angular momentum in disguise, and the disguise hides half of it. Equal areas in equal times says that the size of the angular momentum is fixed. It says nothing about its direction, because for two bodies alone the direction is fixed too, and the orbit stays in one plane forever. Add a third influence that is not a point mass — an equatorial bulge, a distant Sun — and the size can stay constant while the direction does not. The orbit’s plane turns. Its angular momentum precesses.

For a moon the turning is governed by two torques that pull towards two different planes, and the competition between them has a geometric answer: a plane, intermediate between the planet’s equator and its orbit around the Sun, about which the moon’s orbit precesses at a steady angle. Laplace found it in 1805. The Moon is the case where that plane has a history, because the Moon has not always been where it is, and its present orbit poses a question that the plane makes sharp: why is the Moon tilted by five degrees to it?

Two torques and the plane between them #

The Earth is not a sphere. It spins once a day, and the spin flattens it by a part in three hundred, leaving a ring of extra mass around the equator. A moon on an orbit inclined to that equator feels the ring pull its orbit towards the equatorial plane, and since the orbit carries angular momentum, the pull does not flatten it but turns it: the orbit’s pole precesses around the Earth’s spin axis, at a rate proportional to the Earth’s flattening and falling steeply with distance, as $J_2 (R/a)^2$ times the orbital frequency.

The Sun does the same thing from outside. Seen from the Earth, the Sun circles once a year in the ecliptic, and averaged over the year its pull on the moon is the pull of a ring in the ecliptic plane. That ring also torques the orbit, and the orbit’s pole precesses around the ecliptic pole, at a rate that grows with distance — slowly for a close satellite, which orbits fast enough that the Sun barely changes during one orbit, and quickly for a distant one.

So there are two axes, and the orbit’s pole is being turned about both at once. The averaged equation of motion for the unit vector along the orbit’s angular momentum, $\mathbf n$, is

$$\frac{d\mathbf n}{dt} = -A\,(\mathbf n\cdot\mathbf s)\,(\mathbf s\times\mathbf n) \;-\; B\,(\mathbf n\cdot\mathbf k)\,(\mathbf k\times\mathbf n),$$

with $\mathbf s$ the Earth’s spin axis, $\mathbf k$ the ecliptic pole, and $A$ and $B$ the two rates. It has a fixed point: a direction in the plane of $\mathbf s$ and $\mathbf k$ at which the two turnings cancel. That direction is the Laplace pole, and its plane is the Laplace plane. An orbit whose pole sits exactly on it does not precess at all; an orbit tilted from it precesses about it, at a steady angle, like a top about the vertical.

The ratio of the two rates goes as the fifth power of the distance, so the Laplace pole does not drift gradually from one axis to the other. It sits on the spin axis until a particular distance, the Laplace radius, swings across within a factor of two of it, and sits on the ecliptic pole beyond. The radius is the fifth root of $J_2 R^2 a_\oplus^3 M_\oplus/M_\odot$, and for today’s Earth it is 8.4 Earth radii — about 54,000 kilometres. The same transition sorts the satellites of the giant planets into regular ones that orbit in their planet’s equator and irregular ones that do not.

The Earth spun faster then #

The Earth’s bulge is not fixed. It is a consequence of the spin, and its size goes as the square of the spin rate, while the spin has been slowing for four and a half billion years as the tides pay the Earth’s rotation into the Moon’s orbit. The two are tied by a conservation law: the angular momentum of the Earth’s spin plus the Moon’s orbit is nearly constant, the Sun’s tides on the Earth being a small correction. A Moon at five Earth radii — near where the giant impact that made it would have left it — orbited an Earth whose day was 5.3 hours long. Its bulge was twenty times today’s.

A bigger bulge pushes the transition outward, as the fifth root of the bulge, and for the early Earth the Laplace radius was near fourteen Earth radii rather than eight. That is the solid curve in the first figure, and it is the one that matters, because it describes the Earth the Moon was actually orbiting when it crossed.

Turning fast, moving slowly #

The Moon has moved since it formed — from a few Earth radii to sixty, pushed outward by the torque of the tidal bulge it raises. That movement was slow, and the precession was fast. How different the two are is the whole of the argument.

The precession period near the transition is a century or two. The recession, at rates scaled back from what the laser ranging measures today, takes tens of thousands of years to move the Moon by its own distance there. The plane the orbit precesses about changes direction as the Moon moves, but it changes over thousands of turns of the orbit around it.

There is a theorem about systems like that. When a periodic motion’s parameters change slowly compared with its period, certain combinations of its variables — the adiabatic invariants — stay fixed even though the motion itself changes completely. A pendulum whose string is slowly shortened keeps the ratio of its energy to its frequency. A charged particle in a slowly strengthening magnetic field keeps the flux through its orbit. And an orbit precessing about a slowly moving axis keeps the angle it precesses at. The axis can swing through twenty-three degrees; the orbit follows it, at the same tilt it started with.

Integrated: the Moon arrives in the ecliptic #

That is a claim the equation of motion can be asked about directly.

The Moon in the figure starts in the equator, which is where the giant-impact debris disc it accreted from would have been: the disc was made of material flung off in the impact and spun up with the Earth, and a disc that settles in a spinning planet’s gravity settles into its equatorial plane. At four Earth radii the equator is the Laplace plane, so the Moon starts with no free inclination at all. As it recedes, the Laplace plane swings from the equator to the ecliptic, and the Moon’s orbit swings with it. By the time it has passed the transition its free inclination is still a hundredth of a degree, and from there out to sixty Earth radii it orbits in the ecliptic.

The passage in the figure is made deliberately fast — thirty precession periods to cross the transition, a hundred times faster than tidal recession could have moved it — and the orbit still follows. The real crossing was gentler still. The conclusion is not sensitive to how the tides behaved, how fast the early Moon receded or how much the Earth’s dissipation has varied: as long as the motion is slow, the free inclination at the end equals the free inclination at the start.

The Moon’s free inclination today is 5.15°. Its Laplace plane is the ecliptic to a few thousandths of a degree, so all of that is free — it is a tilt nothing is forcing, precessing around the ecliptic pole once every 18.6 years, the period that runs through the Earth’s nutation and the cycle of eclipse seasons. On the invariance argument it has been 5.15° to the local Laplace plane, or more, since before the Moon reached fourteen Earth radii. It cannot have been acquired on the way out.

What exactly is kept, and how a calculation can lose it #

The invariant is worth stating precisely, because the precise statement is what makes it trustworthy. An orbit’s pole precessing about the Laplace pole traces a closed loop on the sky. Far from the transition the loop is a circle around one axis or the other; near it, where both torques matter, it is an oval. What slow change preserves is not the angle at every instant but the area the loop encloses — the solid angle swept out once per precession. A circle that starts with a given area ends as a circle with the same area, so a free inclination that starts at ten degrees ends at ten degrees, even though in between the pole wanders on an oval whose distance from the Laplace pole varies around the loop.

That makes the claim a delicate thing to compute. The quantity being asked about is a small, slowly preserved property of a fast oscillation, carried through tens of thousands of oscillations. A numerical method that loses a tiny fraction of the loop’s area every turn — as any method that is merely accurate per step, and not built to preserve the geometry, may do — erodes the free inclination steadily, and it erodes it in the direction of the answer the Moon contradicts: towards zero. A first version of the integration in this essay, taking too few steps per precession, turned a ten-degree start into one and a half degrees by sixty Earth radii; it looked like a physical result and was a numerical one, and the same flaw would have made the Moon’s arrival in the ecliptic look more certain than it is. Taken with enough steps, the ten degrees are kept to a hundredth of a degree, and the conclusion is the same for the right reason.

What the invariance does with a tilt the Moon started with #

An inclination present from the start would survive.

So the problem can be restated more carefully. Slow recession carries a free inclination through unchanged, but it cannot create one. The Moon needs either to have been born tilted, or to have been tilted by something fast. Both are harder than they sound.

Born tilted is difficult because of where the Moon was born. A debris disc around a rapidly spinning, strongly flattened Earth settles into the equator in a few orbits — the precession at three Earth radii is fast and the disc’s own collisions damp out any inclination — and the Moon assembled from it in months to years would inherit the disc’s plane. The giant impact that made the disc could leave the Earth’s spin axis tilted, but the disc forms around whatever spin the impact leaves, and its plane is that spin’s equator.

And there is a further loss. The Moon is locked with one face to the Earth, in a Cassini state that keeps its own spin axis tilted from its orbit by a degree and a half. The tides the Earth raises in a tilted Moon dissipate energy, and that dissipation slowly reduces the orbit’s inclination. Chen and Nimmo estimated in 2016 that the Moon’s inclination has been damped this way by a factor of two or more over its history, if the Moon was partly molten early on. The 5.15° seen now may be the remnant of ten or more.

How fast would have been fast enough #

The invariance holds only if the change is slow. A passage through the transition fast enough to break it would leave a free inclination behind — and the question of how fast that has to be has a numerical answer.

The curve is the adiabatic theorem in reverse. If the Laplace plane had swung across in much less than one precession period, the orbit would have been left behind, tilted from the new plane by nearly the whole obliquity; if it swung across in several periods, the orbit kept up. The Moon’s present inclination would need a crossing of about seven-tenths of a period — the Moon would have had to move from eleven to eighteen Earth radii in about a century. Tidal recession took tens of thousands of years over that distance. No variation in the tides plausible for the early Earth closes a factor of ten thousand.

The fast things that have been proposed #

Since the Moon’s tilt cannot be the residue of a slow process, it has to be the record of a fast one, and every proposed explanation is a way of breaking the invariance.

Resonances on the way out. As the Moon receded, the period of the precession of its orbit’s long axis passed through simple ratios with the year, and at those resonances the Sun’s pull on the Moon’s orbit was coherent rather than averaging away. The evection, the largest of the Moon’s solar inequalities, is the present-day trace of one such term. Touma and Wisdom showed in the 1990s that passage through the evection resonance could pump the Moon’s eccentricity, and through a related resonance its inclination, by amounts that depend on the tidal parameters of the time. A resonance acts quickly compared with recession, which is what breaking the invariance needs.

Encounters with what was left over. In the first hundred million years the inner solar system still held bodies of lunar mass and larger, the leftovers of planet formation. Pahlevan and Morbidelli showed in 2015 that a few close passages of such bodies by the Earth–Moon system, each a sudden kick to the Moon’s orbit lasting days, could have tilted it by several degrees while it was still close to the Earth. A passing body is the archetype of a non-adiabatic event, and the leftovers are needed anyway: the same bombardment delivered the late veneer of metals to the Earth’s mantle.

A very tilted Earth. Ćuk and collaborators proposed in 2016 that the impact left the Earth spinning with an obliquity of seventy degrees or more. At obliquities above about sixty-nine degrees the Laplace plane itself becomes unstable through part of the transition: the equilibrium orbit stops being a stable fixed point, and the Moon’s eccentricity and inclination are excited as it passes. That would give the Moon its tilt, and the subsequent evolution through resonances would reduce the Earth’s obliquity towards its present value — tying the Moon’s tilt and the Earth’s seasons to one event.

Each mechanism needs conditions that are not independently known: the early Earth’s tidal response, the population of leftover bodies, the obliquity the impact left. What the invariance does is narrow the search to fast events, and say that the Moon’s inclination is a fossil of whichever one happened.

A test the other planets set #

The invariance makes predictions elsewhere, and one is a check on the reasoning. Uranus’s spin axis is tilted by ninety-eight degrees, and its five large moons orbit in its equatorial plane, all well inside its Laplace radius. If Uranus had been tipped over quickly after its moons formed, they would have been left orbiting in the old equator, tilted by up to ninety-eight degrees to the new one. They are not. Either the moons formed after the tilt, from a disc that settled into the new equator, or the tilt happened slowly — over many precession periods of the moons’ orbits — and the moons followed the equator over, keeping their zero free inclination as the Moon would have kept its own. Both are actively proposed, and the second is the adiabatic argument applied to a planet rather than a moon.

Saturn’s Iapetus sits in the transition region of its planet, where its Laplace plane is tilted about fifteen degrees from Saturn’s equator, and it is inclined by about eight degrees to that local plane — a free inclination the regular satellites inside it do not have. It too was presumably formed in Saturn’s equator, and it too has a tilt slow evolution cannot have given it.

What the figures assume #

The integration holds the Earth’s spin axis fixed in space, and it is not. The Moon and the Sun make the Earth’s axis precess — today once in twenty-six thousand years, and much faster when the Moon was close — and in the full problem the Earth’s axis and the Moon’s orbit precess about the total angular momentum of the system together. That coupling changes the Laplace plane’s location in detail, and near the transition it introduces resonances of its own, which are part of what the high-obliquity scenario uses. It does not change the invariance: the coupled system still changes slowly compared with its own precession, and the free inclination is still kept.

The bulge is computed from the spin by today’s relation between the two, which assumes the early Earth was in hydrostatic equilibrium. The orbits are circular, and the eccentricity resonances that break the invariance are not in the figures at all. And the recession is a power law scaled from today’s rate, which is known to be several times faster than the average over the Moon’s history; that changes the date at which the Moon crossed the transition, not the fact that it crossed slowly.

Still open: which fast event tilted the Moon #

The inclination is one number, five degrees and a sixth, and it has to be explained by an event that left no other mark visible today — or by one that left marks only in quantities as poorly known as the early Earth’s obliquity. The tests are indirect. The resonance explanations constrain the early Earth’s tidal response and should leave a correlated signature in the Moon’s orbital eccentricity history; the encounter explanation needs a population of leftover bodies that also has to account for the metals in the Earth’s mantle and the cratering record of the Moon; the high-obliquity explanation needs an impact geometry that the isotopic similarity of the Earth and the Moon also constrains. Each is a reading of a different record, and the angle the Moon makes with its orbital plane is the one place they all have to agree.