A tilt built from a thousand near misses

The Moon’s orbit is tilted by five degrees and a sixth to the plane of the Earth’s orbit, and that tilt is a puzzle. A moon born from debris in its planet’s equatorial plane, just outside the distance at which a moon stops holding together, as the Moon is thought to have been, orbits close in about a plane set by the planet’s bulge — the equator — and far out about one set by the Sun — the ecliptic. Between the two lies a smooth transition, and a moon receding slowly through it, as the tides carry the Moon outward, follows the transition plane faithfully: the angle it keeps to that plane is an adiabatic invariant. The Moon should have arrived at its present distance lying in the ecliptic. It has five degrees instead.

So the tilt is a fossil of something fast — fast compared with the precession of the Moon’s orbit, which the slow recession was not. Three fast things have been proposed: resonances between the Moon’s orbit and the Sun that it passed through on the way out — the strongest of them is the term that survives today as the evection — an Earth left by the giant impact spinning on a very tilted axis, and the subject of this essay, encounters with the bodies left over from planet formation. A body passing the Earth–Moon pair at a few kilometres a second does its work in a few hours, and a few hours is instantaneous to an orbit whose month was then a week or two.

The question is quantitative. How big is the kick one passage gives, how many passages does five degrees need, and is there any reason to think the young solar system supplied that many?

A walk with no preferred direction #

Each passage is a kick, and the kicks come from every direction. A body moving past on a straight line gives every mass it passes an impulse toward the line, of size $2Gm/(v d)$ for a passer of mass $m$, speed $v$ and distance of closest approach $d$. The Earth and the Moon both receive one. What changes the Moon’s orbit is the difference — the extra impulse the Moon receives because it sits at a different place — and its component out of the orbital plane tilts the orbit.

Because the passers arrive from random directions, the out-of-plane impulses are as often up as down, and the inclination performs a random walk: the typical tilt after $N$ kicks is the typical single kick times $\sqrt{N}$. The figure shows four such walks with identical statistics. They wander. After a thousand passages one has reached nearly four degrees and another has come almost back to zero, and that spread is intrinsic. A random walk does not say what tilt a particular history produced; it says what tilt a typical history produces, and how likely the observed one is.

That matters for reading the answer. The observed five degrees is one draw. A tilt has two components — which way the orbit leans, as well as how far — and a random walk in two components gives a tilt distributed as a Rayleigh distribution: the chance of exceeding an angle $x$ is $e^{-x^2/i_{\mathrm{rms}}^2}$, where $i_{\mathrm{rms}}$ is the typical tilt. A population whose typical outcome is three degrees produces five or more in about one history in sixteen; one whose typical outcome is five, in about a third; one whose typical outcome is ten, in four out of five. So an encounter model is comfortably consistent with the Moon if its typical tilt is five degrees or more, strained if it is three, and effectively excluded if it is one. The distribution also says that an encounter history which overshot is as natural as one that fell short, which is worth remembering when a model gives twelve degrees and is judged a failure.

Why a kick is a difference #

The impulse a passing body gives is the simplest calculation in gravitational dynamics, and the reason it is so simple is speed. A body passing at five kilometres a second at a distance of a hundred thousand kilometres spends about five hours near closest approach, while the Moon at thirty Earth radii takes nine days to go round. During the passage the Moon hardly moves, so the passer’s pull can be added up along a straight line with everything else held still, and the result is an impulse of $2Gm/vd$ toward the line. That is the impulse approximation, the same one used for a star cluster crossing the Galaxy’s disc, and it is accurate whenever the encounter is much faster than the orbits it disturbs.

The Earth receives an impulse too, and an impulse that the Earth and the Moon share equally moves the pair without changing the orbit at all. What changes the orbit is the part they do not share: the difference between two pulls at two places, which is exactly what a tide is. For a distant passer the difference is a small fraction of either impulse — proportional to the Moon’s distance from the Earth divided by the passer’s — and this is why the kick grows with the Moon’s distance and falls steeply with the passer’s.

The kick changes the Moon’s velocity by a few metres a second against an orbital speed of a kilometre and a half. The orbit’s plane is the plane containing the Moon’s position and velocity, and turning the velocity out of that plane by a few metres a second in fifteen hundred tilts it by a fraction of a degree. That fraction, times the square root of the number of kicks, is the whole of the result.

Near misses do the work #

The size of a single kick depends steeply on how close the passer comes.

A body passing just outside the Moon’s orbit gives a kick comparable to the one it gives the Earth, but in a different direction, and the difference tilts the Moon’s orbit by about a fifth of a degree. Further out, the impulses on the Earth and the Moon become nearly parallel and the difference falls off — the kick becomes tidal, proportional to the Moon’s distance from the Earth divided by the square of the passer’s — and the tilt falls nearly as the inverse square of the impact parameter.

Passers are spread evenly over the area they cross, so the number passing at each distance grows in proportion to the distance. But the square of the tilt, which is what a random walk adds, falls as the fourth power, and the sum is dominated by the closest passages. The figures count passers within ten lunar distances; widening the count to a hundred would add many passages and almost no tilt. A random walk of tilts is, in practice, a count of near misses.

The later the better #

The tilt a given passage produces depends on where the Moon was at the time, because the Moon receded from a few Earth radii to sixty over the first billion years and more.

Two effects compete. A body passing at a fixed distance from the Earth tilts a close-in Moon less, because the difference between its pulls on the Moon and the Earth is small when the two are close together, and because a close Moon moves fast in its orbit and is harder to turn. A more distant Moon, on the other hand, is passed closely by more of the bodies streaming past — its orbit is a bigger target. Together the typical tilt grows a little faster than the square root of the Moon’s distance, and the number of passages needed for five degrees falls from seven thousand when the Moon was at ten Earth radii to under a thousand at sixty.

The figure says when the encounters would have to happen to be efficient: late in the Moon’s recession, which works against them, because the leftover population was largest early and dwindled as it was swept up. There is a second constraint the figure does not show. A tilt acquired while the Moon was close to the Earth, inside the transition between the equatorial and the ecliptic planes, is measured relative to the equator, and the slow passage through the transition then carries it outward adiabatically — preserved as an angle to the local plane rather than to the ecliptic. Kicks received inside about twenty Earth radii survive the transition; kicks received outside it are tilts relative to the ecliptic directly. Either way they count, but the ones received early are smaller.

A few big ones or many small ones #

The kick is proportional to the passer’s mass, and in a random walk the number of kicks needed goes as the inverse square of each kick.

So the question of how many passages is really a question about the mass distribution of what was left over. Two thousand passages of Moon-sized bodies, a couple of hundred of bodies three times the Moon’s mass, or a few dozen of Mars’s mass all give the same typical tilt. A population dominated by small bodies is nearly useless: two hundred thousand bodies a tenth of the Moon’s mass are needed. The kick budget favours a leftover population with most of its mass in a few large bodies — which is also what models of the end of planet formation produce, since the final stage is a phase of giant impacts between a few dozen planetary embryos, and the embryos that did not join a planet were the leftovers.

A population already counted #

Whether the young solar system supplied that many passages is not a free assumption, because the same population left a measurable record in the Earth itself.

The Earth’s mantle contains platinum, iridium, gold and other metals that dissolve in iron, in amounts several hundred times larger than they would be if the mantle had been in equilibrium with the iron core. The core took those metals when it formed. They must have arrived afterwards, delivered by impacts of bodies whose metals stayed in the mantle — a late veneer, which on the measured abundances amounts to about half a per cent of an Earth mass. The impacts that delivered it are a sample of the leftover population, and the number of near passages per impact is fixed by geometry.

Both impacts and passages are a flux of bodies times a cross-section. The Earth’s cross-section for impact is its physical size enlarged by its gravity, which bends slow trajectories inward; at five kilometres a second the focusing enlarges it sixfold. A circle ten lunar distances across is enlarged hardly at all — gravity is weak out there — but its area is tens of thousands of times the Earth’s. So for every body that struck the Earth, fifteen thousand passed within ten lunar distances when the Moon was at thirty Earth radii.

If the late veneer was delivered by bodies of lunar mass, half a per cent of an Earth mass is less than one impact — so the expected number of near passages is less than fifteen thousand by the same factor, around six thousand. That is three times the two thousand passages a five-degree tilt needs at thirty Earth radii. If the veneer arrived in a few bodies the size of Mars, the passages are fewer but each is heavier, and the kick budget is met just as well. The arithmetic does not prove that encounters tilted the Moon. It says that the population which must have existed to deliver the Earth’s precious metals was large enough to do it.

The eccentricity that tides took back #

There is an obvious objection. A kick that tilts an orbit also changes its shape.

The kicks have in-plane components as large as their out-of-plane ones, and the in-plane components change the orbit’s eccentricity. The two grow together as random walks, and a history that tilted the Moon by five degrees would also have given it an eccentricity of a tenth or more — twice the present value. That would be a problem for the encounter account if eccentricity and inclination lasted equally long. They do not. Tides raised in the Moon by the Earth flex the Moon twice an orbit when the orbit is eccentric and dissipate energy, which circularises the orbit on a timescale of tens to hundreds of millions of years for a young, warm Moon. The same tides act on inclination far more weakly, because a tilted circular orbit raises an almost steady tide in the Moon. So a bombardment would leave an orbit that was eccentric and tilted, and the tides would then return the eccentricity toward the value the Sun forces while leaving most of the tilt. The present eccentricity of 0.055 is set by those later processes and says nothing about what the encounters gave.

What the kicks leave out #

The model treats each passage as a straight line at constant speed, which is accurate for distant passages and less so for the closest, where the passer’s path bends around the Earth. Bent paths spend longer near the pair and give larger kicks, which would lower the numbers needed. It keeps the Moon on a circle between kicks, which ignores the eccentricity the kicks themselves build up; an eccentric orbit is turned slightly more easily at apogee and less at perigee, and on average the difference is small. And it ignores the Earth’s oblateness and the Sun’s torque, which make the orbit’s plane precess between kicks. Precession scrambles the direction of each kick relative to the orbit and is part of why the walk is random; it does not change the walk’s statistics.

What the model cannot supply is the leftover population’s history: its mass distribution, its approach speeds, and when it was swept up relative to the Moon’s recession. Models of the late stage of planet formation put most of the leftover mass in a few large bodies with approach speeds of a few kilometres a second, cleared over the first hundred million years.

The approach speed matters twice. Slower bodies give larger kicks, because the impulse goes as the inverse of the speed, and they are focused more strongly toward the Earth, so more of them pass close. Both effects favour encounters in the earliest era, when the leftovers moved on orbits similar to the Earth’s and approached slowly, over later ones, when the survivors had been stirred onto more eccentric orbits by repeated encounters with the planets. The Moon was then at twenty to forty Earth radii, depending on how fast the early Earth’s tides acted on a planet whose day was a few hours long — which is the same unknown that decides when the Moon passed through its Cassini transition.

Still open: which record to trust #

The encounter account turns the Moon’s tilt into a second measurement of the leftover population, beside the metals in the mantle. The two need not agree, and that is what makes it a test. The metals measure the total mass that struck the Earth; the tilt measures the sum of the squares of the passers’ masses that came close. A population of many small bodies can deliver the veneer and fail to tilt the Moon; a population of a few large ones can do both, and could also deliver its metals unevenly, as a few large impacts would.

The metals have their own uncertainty, because a large impactor’s iron core may have sunk through the mantle and taken part of its metals with it, and then the veneer underestimates the mass that arrived. And the other explanations of the tilt — the resonances on the way out, the strongly tilted early Earth — remain, each needing conditions the encounters do not. What the figures establish is that the encounter account is not merely possible but arithmetically comfortable: a few thousand near misses of Moon-sized bodies, about what the delivery of the Earth’s gold implies, give a tilt of about five degrees. Whether the tilt was made that way is a question about the giant impacts’ last survivors, and the place their record is most complete is the Moon’s oldest, most heavily cratered highlands, where counting the holes gives a rate for the bodies that struck — and through the same focusing arithmetic, for the bodies that only passed.

There is a satisfying symmetry in the account, whichever explanation turns out to be right. Kepler’s equal areas are angular momentum conserved, and the Moon’s tilt is the one property of its orbit that the long, slow exchange of angular momentum with the Earth’s tides could not change. Whatever set it acted quickly, and the record it left is a direction — the one thing a slow process keeps.