Astronomy & Celestial Mechanics
the sky, computed
Gravitation

A moon the Earth does not hold

Seen from the Earth, the small asteroid Kamoʻoalewa circles the planet once a year, backwards, between forty and a hundred times as far away as the Moon. It is not a moon. It orbits the Sun, with the Earth's period and a slightly different eccentricity, and the loop is the difference between two ellipses — one the Earth would trace even with no mass at all. What the Earth's gravity does is subtler: it keeps the loop from drifting away.

Assumes Lagrange points, Hill sphere and The three-body problem.

15 min read 6 figures Two bodies, and no moreSmall perturbations, large consequences

In April 2016 a survey telescope in Hawaii found a small asteroid, perhaps forty to a hundred metres across, that has since been followed every spring. Its distance from the Earth never falls below about forty times the Moon’s, nor rises above about a hundred, and over a year it goes once round the Earth — in the opposite sense to the Moon. It was named Kamoʻoalewa, a Hawaiian word for an oscillating celestial fragment, and it is often described as the Earth’s second moon. Its orbit has been traced backward for a century and forward for several more, and it has been circling the Earth in this way since about 1900 and will continue until about 2300.

It is not a moon in any sense the word carries. The Earth’s gravity at its distance is a few hundred times weaker than the difference between the Sun’s pull on it and on the Earth; it is not bound to the Earth and is not orbiting it. It orbits the Sun, on an ellipse with exactly the Earth’s period, and the loop it seems to make round the Earth is an artefact of watching one ellipse from a point moving on another. The interesting question is not why it circles the Earth — that needs no force at all — but why it keeps doing so for centuries rather than drifting off, and the answer is a piece of the same physics that holds the bodies that keep station at the Lagrange points.

A loop that needs no force

Take a body on an orbit round the Sun with exactly the Earth’s semi-major axis, and so exactly the Earth’s period, but with an eccentricity of a tenth where the Earth’s is nearly zero. Start it at the same mean longitude as the Earth. Over a year the two return to their starting positions together. In between, the eccentric body runs ahead of the Earth near its perihelion, where it moves faster, and falls behind near its aphelion, where it moves slower; it is closer to the Sun than the Earth for half the year and farther for the other half.

A loop round the Earth that the Earth does not make. The path over one year, in the frame that turns with the Earth (Earth at the centre, the Sun far to the left, the Earth's motion upward), of bodies on heliocentric orbits with the Earth's period and eccentricities of 0.05, 0.1, 0.15, started at the Earth's mean longitude. Each traces a retrograde loop round the Earth, about 2ae long in the direction of motion and ae wide towards the Sun: for e = 0.1, 0.395 by 0.199 AU. The small circle is the Earth's Hill sphere, 0.010 AU in radius, inside which the Earth's gravity dominates the Sun's tide. Every loop lies far outside it. The loop is a difference between two ellipses with one period, and would be traced the same way if the Earth had no mass at all.
Fig. 1 One year’s path, in the frame turning with the Earth, of bodies with the Earth’s period and eccentricities 0.05, 0.1 and 0.15. Each traces a retrograde loop round the Earth about 2ae long along the orbit and ae wide towards the Sun — 0.4 by 0.2 AU for e = 0.1. The small disc at the centre is the Earth’s Hill sphere, 0.01 AU in radius.

In a frame that turns with the Earth, so that the Earth stays fixed, those two oscillations — ahead and behind, inward and outward, a quarter of a year out of step — combine into a closed loop round the Earth’s position. It is twice as long along the direction of motion as it is towards the Sun, and its size is set by the eccentricity: a tenth gives a loop four tenths of an astronomical unit long. The loop is traced backwards relative to the Earth’s own orbital motion, which is what makes it retrograde.

Why backwards is worth a sentence, because it is the opposite of what a moon does. At perihelion the eccentric body is inside the Earth’s orbit, nearer the Sun, and moving faster than the Earth; so in the turning frame it is on the sunward side and moving forward. At aphelion it is outside and moving slower, so it is on the far side and moving backward. Sunward and forward, then outward and backward: in a frame where the Sun is to the left and the Earth’s motion is up, that is a clockwise circuit, while the Earth’s own motion round the Sun, and the Moon’s round the Earth, are anticlockwise seen from the same side. The sense of the loop is fixed by the geometry of the two ellipses and cannot be otherwise. A body on a prograde loop round the Earth at these distances would need a different period from the Earth’s, and would drift away within a year.

None of this involves the Earth’s mass. The loop is pure kinematics, the epicycle that any slightly eccentric orbit traces round the circular orbit of the same period, and it would be the same if the Earth were a point with no gravity at all. The ancient epicycles that described the planets’ apparent motions were the same geometry seen from the other end, and the curve that a body cannot cross in the rotating frame says nothing against it because the loop is far from any such boundary.

Outside the region where the Earth wins

The Earth’s gravitational reach in this problem is the Hill sphere: the region inside which the Earth’s pull on a body exceeds the Sun’s tidal pull, the difference between the Sun’s attraction on the body and on the Earth. For the Earth it is a hundredth of an astronomical unit in radius, about four times the Moon’s distance. Inside it a body can orbit the Earth; outside it, it orbits the Sun.

At a quasi-satellite's distance the Earth hardly pulls. The Earth's gravitational acceleration and the Sun's tidal acceleration — the difference between the Sun's pull on the Earth and on a body near it — in units of the Sun's pull at one AU, against distance from the Earth, both logarithmic. They are equal at the Hill radius, 0.0100 AU. Across the shaded band, the 0.05–0.3 AU at which quasi-satellites circle the Earth, the tide is 125 to 26,969 times the Earth's pull. The Moon, at 0.0026 AU, sits well inside the radius where the Earth wins.
Fig. 2 The Earth’s pull and the Sun’s tidal acceleration against distance from the Earth. They cross at the Hill radius, 0.01 AU. Across the shaded band, 0.05–0.3 AU, where quasi-satellites loop round the Earth, the tide is a hundred to tens of thousands of times stronger. The Moon sits well inside the Hill radius.

The Earth’s pull falls as the inverse square of the distance and the tide grows in proportion to the distance, so the gap between them widens as the cube. At the distances where Kamoʻoalewa loops, between a tenth and a quarter of an astronomical unit, the Sun’s tide is several hundred to several thousand times the Earth’s attraction. A body there is governed by the Sun. The Moon, at a quarter of the Hill radius, is governed by the Earth, which is what makes it a moon. The region a planet may keep a moon in is a fraction of the Hill sphere — about half for prograde moons, and somewhat more for retrograde ones — and a quasi-satellite lies ten to thirty Hill radii outside it.

So the right description of Kamoʻoalewa is the one astronomers use: a quasi-satellite. It is a co-orbital asteroid, sharing the Earth’s orbit in the sense of sharing its period, that happens to sit at the same mean longitude as the Earth and so appears to circle it.

What the Earth’s pull is for

If the loop needs no force, the Earth’s gravity might seem irrelevant. It is not, because no body has exactly the Earth’s period, and a small difference is fatal over time.

What the Earth's pull does: it holds the loop in place. The longitude of a body relative to the Earth, sampled once a year at the same point of its loop, over 150 years, for an orbit of eccentricity 0.1 and semi-major axis 0.004 AU larger than the Earth's: with the Earth's gravity (solid) and with the Earth's mass set to zero (dashed). Without the Earth the slightly longer period carries the loop steadily backwards, 2.16° a year, and it leaves the Earth's neighbourhood within a few years. With it, each passage nudges the orbit's period across the Earth's, the drift reverses, and the loop's centre swings within ±8° of the Earth on a cycle of about 20 years. The Earth does not hold the body; it holds the body's period.
Fig. 3 The longitude of the loop’s centre relative to the Earth, averaged over each year, for e = 0.1 and a semi-major axis 0.004 AU larger than the Earth’s: with the Earth’s gravity (solid) and with the Earth’s mass set to zero (dashed). Without the Earth the loop drifts backwards at 2.2° a year; with it, the centre swings within ±8° of the Earth on a cycle of about twenty years.

A semi-major axis larger than the Earth’s by four parts in a thousand gives a period longer by six parts in a thousand, and the loop’s centre drifts backward by about two degrees a year. Without the Earth’s gravity the body would move off along the orbit; within a few years it would be well away from the Earth, looping round an empty point that drifts steadily round the Sun, and within eighty it would be on the far side of the Sun. With the Earth’s gravity the drift is stopped and reversed. As the body passes the Earth each year its orbit receives a small kick, and the kicks have a systematic sign: when the loop’s centre has drifted to one side of the Earth, the kicks change the body’s semi-major axis so that its period moves to the other side of the Earth’s, and the drift reverses. The centre swings back and forth through the Earth’s position, a few degrees either way, on a cycle of decades.

That is the whole role of the Earth’s mass. It does not hold the body; it holds the body’s period, nudging it across the Earth’s own whenever the two have drifted apart. The force involved is tiny — a few hundredths of a per cent of the Sun’s tide — but it acts with the same sign for years at a time, and a small force applied consistently changes an orbital period by the small amount needed. This is the same mechanism that shows a planet’s companion on the same orbit in its own transit timing round other stars: a body sharing a planet’s period trades period with it, and each trade shows up in the planet’s clock.

Drawn in the turning frame over three decades, the trading becomes visible as a change in the loops themselves.

The loop walks back and forth along the Earth's orbit. The path over 30 years, in the frame that turns with the Earth (Earth at the centre, the Sun to the left), of a body with eccentricity 0.1 and a semi-major axis 0.004 AU larger than the Earth's. Each year it traces one retrograde loop; the loops do not close on themselves, because the slightly longer period carries each one a little behind the last, and the Earth's yearly kicks then carry them forward again. Over 30 years the loops sweep a band 0.68 AU long along the orbit while staying 0.22 AU wide — the guiding centre's slow swing drawn out in full.
Fig. 4 Thirty years of the same body’s path in the frame turning with the Earth. Each year’s loop is displaced a little from the last as the period runs slow and is then corrected; together they sweep a band about 0.68 AU long along the orbit and 0.22 AU wide, always enclosing the Earth.

No single year’s loop is the orbit; the orbit is the whole band, and every loop in it encloses the Earth. The loops shrink and swell along the direction of motion as the guiding centre swings, and cross over each other on either side of the Earth where the swing reverses. This is the picture that makes the name quasi-satellite fair: from the Earth the body is always somewhere round it, never far ahead or far behind, even though nothing binds it.

The correction has a limit, and it is not generous.

A mismatch too large for the Earth to correct. The longitude of a body relative to the Earth, sampled once a year at the same point of its loop, over 150 years, for an orbit of eccentricity 0.1 and semi-major axis 0.012 AU larger than the Earth's: with the Earth's gravity (solid) and with the Earth's mass set to zero (dashed). Without the Earth the slightly longer period carries the loop steadily backwards, 6.48° a year, and it leaves the Earth's neighbourhood within a few years. With it, the yearly kicks are too weak to reverse a drift this fast: the loop's centre is slowed, then carried away from the Earth, and the body leaves the quasi-satellite state for a horseshoe-like wandering round the orbit. There is a largest mismatch of period the Earth can correct, and it is small. The Earth does not hold the body; it holds the body's period.
Fig. 5 The same body with a semi-major axis 0.012 AU larger than the Earth’s. The drift, 6.5° a year, is too fast for the yearly kicks to reverse: the centre is slowed, then carried away, and the body wanders round the orbit with brief returns. The Earth can only correct a small mismatch of period.

With a semi-major axis larger than the Earth’s by a little over one per cent, the drift is three times faster and the kicks cannot keep up. The loop’s centre is slowed near the Earth, then carried off, and the body circulates round the orbit with brief returns, sometimes turning back, sometimes passing through. A quasi-satellite is possible only for bodies whose periods already match the Earth’s to well under a per cent, which is why the known ones are so few and why each was found by accident.

Three ways to share an orbit

The same integration, started in other places, produces the two other ways a body can share a planet’s orbit.

Three ways to share the Earth's orbit. The longitude relative to the Earth, sampled once a year, of bodies with the Earth's period — quasi-satellite, e = 0.1 (−1° to 1°); horseshoe (24° to 336°); tadpole round L4 (49° to 74°), integrated over 800 years in the Sun–Earth restricted problem; 180° is the far side of the Sun, and 0° and 360° are both the Earth. The tadpole librates round the point 60° ahead of the Earth; the horseshoe swings from one side of the Earth round the far side of the Sun to the other, turning back each time it approaches; the quasi-satellite's loop stays centred on the Earth itself. All three are held by the same mechanism — the Earth's pull trading the body's period back and forth across its own — and differ only in where the body started.
Fig. 6 Longitude relative to the Earth, once a year for 800 years, of three bodies with the Earth’s period in the Sun–Earth restricted problem: a quasi-satellite (within a degree of the Earth), a horseshoe (23° to 337°, passing behind the Sun at 180°) and a tadpole round L4 (49° to 74°). One mechanism, three starting points.

A body started near the point sixty degrees ahead of the Earth, the Lagrange point L4, librates round it — the tadpole, named for the shape of its path in the rotating frame. Jupiter’s Trojan asteroids, thousands of them, are on tadpoles round its L4 and L5, and the Earth has two known Trojans of its own. A body started behind the Sun with a period very slightly longer than the Earth’s drifts round towards the Earth, is turned back when it approaches within twenty or so degrees, and drifts all the way round to the other side, where it is turned back again — the horseshoe, which encloses L3, L4 and L5 together. Several near-Earth asteroids follow horseshoes, among them Cruithne, which takes about seven and a half centuries to traverse its horseshoe once. And a body started at the Earth’s own longitude with some eccentricity is a quasi-satellite.

In each case the Earth’s pull does the same thing: each close-ish approach exchanges a little orbital period, and the exchange always reverses the drift. What differs is where the reversal happens. Tadpoles reverse round the stable Lagrange points; horseshoes reverse near the Earth and pass behind the Sun; quasi-satellites reverse round the Earth itself. The three are not separate phenomena but three topologies of one motion, and a body can switch between them. Kamoʻoalewa is a quasi-satellite now, was on a horseshoe before about 1900, and will return to one after about 2300; its eccentricity and inclination slowly change under the other planets’ influence, and with them the shape of its relative motion.

That switching is the strongest evidence that the classification is about the observer’s frame and not about the body’s nature. A horseshoe and a quasi-satellite differ in whether the body’s guiding centre happens to cross the Earth’s longitude, and the boundary between them is a separatrix in a slowly varying system, crossed when the eccentricity drifts.

The real temporary moons

The contrast with the Earth’s genuine temporary moons sharpens the point. Every few years a small asteroid, a few metres across, drifts into the Earth’s Hill sphere through one of the necks near the inner and outer Lagrange points and is captured for a few months to a couple of years — the minimoons, of which 2006 RH120 and 2020 CD3 are the best observed. These are bound: inside the Hill sphere the Earth’s pull wins, and for a while the body orbits the Earth on a looping, irregular path that the Sun’s tide continually distorts. It leaves the same way it came in, through a neck, because the tube that leads out of a neck leads in as well, and capture of this kind is the transient half of an arrival the Sun hands over to the Earth for a short time.

A minimoon is a body the Earth holds for months. A quasi-satellite is a body the Earth does not hold at all, for centuries. The difference in duration is the opposite of what the words suggest, and it comes from the same arithmetic: a body inside the Hill sphere is strongly perturbed by the Sun on every orbit and soon thrown out, while a body far outside it, matching the Earth’s period on an orbit of its own, needs only the gentlest correction to stay. Retrograde orbits are in general the more durable ones near a planet — the stability limit for retrograde moons lies further out than for prograde ones — and the quasi-satellite is the extreme case, a retrograde relative motion so far out that it is no longer an orbit about the planet at all.

None of this has a closed formula. The restricted problem of three bodies has no general solution, and what “unsolvable” means there is exactly that motions like these — tadpoles, horseshoes, quasi-satellites and the transitions between them — have to be found by following them step by step, as the figures here do.

Why such bodies are temporary

The Earth’s quasi-satellites and horseshoes are not primordial. A co-orbital configuration with a planet as small as the Earth is fragile: the other planets’ perturbations change the body’s eccentricity and inclination over thousands of years, and the region of the co-orbital zone that is stable for the age of the solar system is narrow or absent for the inner planets. The bodies found in these states are near-Earth asteroids that were scattered into the Earth’s orbital neighbourhood and are passing through the co-orbital zone for a few centuries to a few tens of thousands of years, before being scattered out again. An invariant that is only almost one governs how such encounters reshape their orbits.

Jupiter’s Trojans are the contrast. Jupiter’s mass is three hundred times the Earth’s, its co-orbital zone is correspondingly wider and its tadpoles are stable for billions of years, so its Trojans are a permanent population, some of them perhaps survivors from the solar system’s first few hundred million years. The Earth holds its co-orbitals the way a shallow dish holds marbles on a moving table: for a while.

Where a piece of the Moon might come from

Kamoʻoalewa’s spectrum, measured in 2021, is unusually red and resembles the space-weathered silicates of lunar samples more than those of typical near-Earth asteroids. One proposal is that it is a fragment of the Moon, thrown off by an impact into an orbit round the Sun close to the Earth’s, and the co-orbital geometry makes that plausible: ejecta from the Moon leaving the Earth–Moon system at low relative speed start on heliocentric orbits with almost the Earth’s period, which is exactly what co-orbital motion requires. Simulations of lunar ejecta find a small fraction entering quasi-satellite and horseshoe states, for times comparable to Kamoʻoalewa’s. Whether this one is lunar is being tested directly: a sample-return mission launched in 2025 is to visit it and bring material back.

If it is, the quasi-satellite is a sample of the Moon’s surface delivered to a place where it is easy to visit, held there for a few centuries by the period-trading that this essay has described — the most accessible piece of another world, kept in place by a force too weak to bind it.

A flat, circular problem standing in for a tilted one

The figures use the planar circular restricted problem: the Earth on a circular orbit, everything in one plane, no Moon, no other planets. Kamoʻoalewa’s real orbit is inclined by about eight degrees, so its loop round the Earth is tilted and its distance oscillates up and down as well as along and across; the Earth’s real eccentricity shifts the loop’s shape from year to year; the Moon perturbs the close passages; and Venus and Jupiter change the body’s eccentricity and inclination over centuries, which is what drives the transitions between quasi-satellite and horseshoe. The once-a-year sampling also hides the loop itself in the longitude plots, showing only where its centre is. None of these changes the two claims the figures make — that the loop is kinematic and far outside the Hill sphere, and that the Earth’s pull acts on the guiding centre by trading period — but every specific number for Kamoʻoalewa needs the full problem.

Still open: how many the Earth has, and for how long

Kamoʻoalewa is the most stable of a handful of known Earth quasi-satellites, and the others are smaller and more transient. How many there are at any time depends on the supply of near-Earth asteroids with orbits close enough to the Earth’s to enter the co-orbital zone, and on how long each stays — numbers that surveys of small near-Earth objects are now starting to constrain, because quasi-satellites are found only when they happen to be bright enough near their closest approach. If the population turns out to be larger than scattering from the asteroid belt can supply, the excess points at the Moon as a source, and the co-orbital zone becomes a record of lunar impacts over the last few million years. If it matches what the asteroid supply predicts, Kamoʻoalewa’s lunar spectrum is an individual coincidence, and the sample it is about to give will say which.

Where this goes next

The five Lagrange points at mass fraction 0.12. The five points at which a small body can keep station with two larger ones. The three on the line of centres are roots of a quintic and are unstable; the two forming equilateral triangles are stable for a sufficiently lopsided mass ratio. Five places that keep station, in a problem with no solution Three bodies under gravity cannot be solved. Restrict the problem slightly and five exact answers fall out anyway — three of them roots of a quintic, two of them perfect equilateral triangles. A 10 Earth-mass Trojan started 10° from its point, seen in the planet's transits. A Jupiter-mass planet on a 4-day orbit about a 1 solar-mass star, with a 10 Earth-mass companion started 10 degrees beyond the leading Lagrange point of the same orbit, integrated for 160 days. Above, the angle between the companion and the planet as seen from the star: it swings about 60 degrees, between 51.1 and 70.3, with a period of 48.5 days measured off the curve, against 49.8 from the small-amplitude formula P/√(27μ/4). Below, the planet's own transit times minus a straight line: they swing by ±300.0 seconds at the same period, because the planet and the companion orbit their common centre of mass and the companion's libration moves that centre along the orbit. The companion shares the planet's period exactly, so it produces no timing signal at any orbital period of its own and would not transit on a schedule distinct from the planet's; the libration period is the only clock it has. A companion on the same orbit, seen in the planet's clock A body sharing a planet's orbit at one of its Lagrange points has the planet's period exactly, so no search for periodic dips or wobbles can find it at a period of its own. It shows instead in two ways the planet's own signals carry — a slow swing of the transit times at the companion's libration period, and a fixed offset between the planet's transit and its star's velocity curve. The Hill radius against the mass ratio, and where four planets keep their outermost moon. The Hill radius R_H = a(μ/3)^⅓ and the exact L₁ distance from the same quintic, both in units of the planet's own semi-major axis, against the mass ratio μ = m/(M+m) from 10⁻⁸ to 0.12. The two are the same line to the width of the stroke across the whole planetary range — the cube-root formula runs 0.33% short of the exact L₁ distance at the Earth's mass ratio and 10.0% short at the right-hand edge — so the departure between them is drawn on its own percentage scale at the right, which is the only place it can be seen. Below the curves are the numerically determined stability limits, 0.5 R_H for a prograde satellite and 0.7 R_H for a retrograde one, and the outermost known satellite of each of four planets, each at a Hill fraction computed from that planet's own mass and orbit: Moon 0.257, Sinope 0.451, Phoebe 0.198, Neso 0.424. Of the four, three are retrograde — Sinope, Phoebe, Neso — and they reach 0.451 of their planet's Hill radius against 0.257 for the one prograde satellite: 1.8 times as far out, in a diagram whose two stability limits stand in the ratio 0.7 to 0.5. No equilibrium argument predicts that, and it is what the drawing makes plain. The region a planet may keep a moon in A satellite is not held by orbiting but by orbiting inside the radius at which the Sun's tidal field would take it away. That radius is the same distance the innermost Lagrange point sits at, asked a different question — and the boundary the sky actually respects is smaller, by a factor that depends on which way the moon is going round.

About the same objects

Not linked from either essay — found by the objects both name.

The objects this essay names

Each one links to every other essay that touches it.

Co-orbitalHill sphereHorseshoe orbitLagrange pointsQuasi satelliteRestricted three-body problemRotating frameTadpole orbit