The hilltop that holds the Trojans

The orbit that cannot be made smaller reduces a planet’s orbit to a marble rolling in a one-dimensional well — the effective potential, gravity plus the centrifugal barrier — and reads stability off its shape: a circular orbit is stable where the well has a minimum and unstable where it has a maximum. The rule is the one everybody learns first about equilibrium. A ball rests stably in a valley and not on a hilltop.

In 1772 Joseph-Louis Lagrange found five places where a small body can keep station with two larger ones orbiting each other. Three lie on the line through the two bodies and are unstable, as their shape suggests. The other two, L4 and L5, sit at the third corners of the two equilateral triangles whose base is the line between the large bodies — sixty degrees ahead of the smaller body in its orbit and sixty degrees behind. In 1906 Max Wolf found an asteroid near Jupiter’s L4, named it Achilles, and began a tradition of naming these bodies after the heroes of the Trojan war; more than twelve thousand are now known. They are there in their thousands, and they should not be, if the first rule of equilibrium were the whole story. L4 and L5 are hilltops.

Five places to rest, two of them hilltops #

In the frame turning with the two large bodies they are stationary, and a small body feels two things besides their gravity: the forces that are not there. One of them, the centrifugal force, depends only on position, and it can be folded into the gravity to make an effective potential. The drawing contours it for the Earth and the Moon. Deep wells surround each body; far away the centrifugal term dominates and the potential falls without limit. Between them are five flat spots. Three are saddles on the axis. The other two are the summits of long, low ridges curving along the orbit, and the contours round them close as they would round the top of a hill.

A small body put exactly at L4, at rest in the turning frame, stays there: the forces balance. Displace it a little, and if the effective potential were the whole story, it would roll downhill and away. The second fictitious force — Coriolis — is what the potential leaves out. It depends on velocity, not position, so it cannot be written into any potential, and it does no work: it only turns a moving body sideways. The deflection that closes on itself finds what that does to a body moving freely in a turning frame: it bends its path into a circle. A body sliding off the hill at L4 starts to pick up speed downhill, and the Coriolis force turns that motion sideways — along the contour, round the hill instead of down it.

Why sixty degrees #

That the balance points sit at exactly sixty degrees, whatever the masses, is a small piece of geometry worth seeing. Put a small body at the third corner of an equilateral triangle with the two large ones. It is the same distance from each, so each pulls on it in proportion to its mass alone, and the two pulls add to a force pointing at the point between the large bodies that divides the line in the ratio of their masses — the centre of mass, about which the whole system turns. The small body therefore feels a pull directed at the centre of its circle, and the strength of that pull, set by the common distance, turns out to be exactly what circular motion at the system’s angular speed requires. The configuration rotates rigidly, and it would do so even if all three bodies were massive: Lagrange’s equilateral solution is one of the very few exact solutions of the three-body problem, and it holds for any three masses.

The two points on the line through the bodies, by contrast, move with the mass ratio: L1, between the Earth and the Sun, sits where the Sun’s pull minus the Earth’s matches the centripetal need of a body orbiting closer in than the Earth but at the Earth’s rate, and its distance depends on how heavy the Earth is. The equilateral points need no such adjustment. Their position is set by symmetry, and their stability by the Coriolis force and the mass ratio — two separate questions, with separate answers, which is part of why the result that they are hilltops surprised the astronomers who first computed it.

Tadpoles and a horseshoe #

The drawing integrates the full equations of motion in the turning frame, starting bodies at rest near L4. With Jupiter’s mass ratio they do not leave. A body started ten degrees past L4 swings back and forth along the orbit between about fifty and seventy degrees from Jupiter, and drifts slightly inward and outward as it does so, tracing a closed, lopsided loop round the hilltop — a tadpole, named for its shape, fat on the side away from Jupiter and tapered towards it. A body started twenty-five degrees past L4 traces a larger one. Superimposed on each is a faster wiggle, the body’s epicycle, once per orbit. The orbit is stable, and the check on the integration is that the one quantity conserved in the turning frame, the Jacobi constant, stays fixed to better than a part in a million.

Start a body far enough from L4 and its tadpole grows until it reaches round the far side of the orbit and joins its twin at L5: the body sweeps from near L4, the long way round past L3, to near L5 and back, never approaching the planet more closely than about twenty-four degrees. That is a horseshoe orbit. Saturn’s small moons Janus and Epimetheus are the best-known pair on such orbits relative to each other: they share nearly the same orbit, and every four years the inner one catches up with the outer, they exchange a little momentum and swap orbits without ever coming closer than about ten thousand kilometres. Several near-Earth asteroids follow horseshoe paths relative to the Earth.

The mass ratio above which the hilltop stops holding #

The Coriolis rescue has a limit, and it was found by Edward Routh in 1875. Linearise the motion about L4 and there are two kinds of small oscillation, one fast and one slow, with frequencies $s$ given by

$$s^4 - s^2 + \tfrac{27}{4}\mu(1-\mu) = 0,$$

where $\mu$ is the smaller body’s fraction of the total mass. For a light secondary the fast frequency is close to the orbital frequency — the epicycle every orbiting body makes — and the slow one is close to $\sqrt{27\mu/4}$ times it, the libration round the hill. As $\mu$ grows the two approach each other. At $\mu = (1 - \sqrt{23/27})/2 \approx 0.0385$ they meet, and beyond that the equation’s roots are complex: the frequencies acquire imaginary parts, oscillations turn into growth, and any small displacement from L4 grows exponentially.

The number means that the larger body must outweigh the smaller by at least about twenty-five to one. The Sun outweighs Jupiter by a thousand to one, and the Earth outweighs the Moon by eighty-one to one, so both have stable Trojan points; the Moon’s L4 and L5 hold faint clouds of dust, reported in 1961 by Kazimierz Kordylewski and still argued about. Pluto is only about eight times heavier than Charon, and no Trojans can hold there. The ratio also governs planets forming in a disc: two bodies of comparable mass sharing an orbit cannot sit sixty degrees apart for long, which is one of the arguments about whether the Moon formed from debris of a Mars-sized body that had grown at the Earth’s L4 or L5 until it passed Routh’s limit and was knocked out.

Either side of Routh’s ratio #

The integration makes the threshold concrete. Four mass ratios straddling Routh’s value, four bodies started a degree from L4: the two below the threshold swing about the point indefinitely, and the two above it depart, the one just above after thirty orbits, the one well above after seven. Nothing dramatic distinguishes the four systems in the effective potential; the hills at L4 are nearly the same height. What differs is a race between two rates: how fast the hill pushes a displaced body outward, which grows with the mass of the smaller body, and how fast the Coriolis force turns it, which is fixed by the rotation. When the push wins, the turning cannot keep up and the body spirals off the summit.

The same race decides a device used in every physics laboratory that measures atomic masses. Nothing can be held still by a static field proves that no arrangement of electric charges can hold another charge in stable equilibrium, and lists the escapes. One is the Penning trap: an electric field that pushes an ion towards the centre along one axis and outward in the plane at right angles — a hill in two directions — combined with a magnetic field along the axis. The magnetic force, like the Coriolis force, is velocity-dependent and does no work, and it turns the ion’s outward drift into a slow circling round the hilltop called the magnetron motion. The trap is stable only if the magnetic field is strong enough compared with the electric one — a condition of exactly the same form as Routh’s. The Trojan asteroids and the ions in a Penning trap are held on hilltops by the same kind of force.

A spinning top on the same hilltop #

The Penning trap is not the only relative. A spinning top standing on its point is balanced at the highest point of its potential energy — tip it and gravity pulls it further over — and it stays up only while it spins fast enough. The top that nods before it settles follows what happens: gravity’s torque on a spinning body does not topple it but turns its axis sideways, into precession, which is the gyroscopic cousin of the Coriolis turning. A top spinning faster than a threshold set by its weight and its moments of inertia sleeps upright; slow it below that threshold and it falls, exactly as a Trojan falls off L4 once the smaller body is too heavy. Routh, who found the Trojan limit, was also one of the founders of the theory of spinning tops, and the two problems were solved with the same mathematics.

The contrast with the axis that will not hold makes the point sharper. A body spinning about the axis of its intermediate moment of inertia is unstable even though no potential is involved at all — the gyroscopic coupling there makes displacements grow rather than circle. Whether a velocity-dependent coupling stabilises or destabilises depends on the arrangement, and in each case the answer comes from the same kind of characteristic equation, with Routh’s condition marking the boundary.

A third relative uses a different mechanism to the same end. A pendulum can be made to stand upside down by shaking its pivot up and down rapidly, and the force that averages to nothing explains the upright pendulum by an effective potential produced by the averaging of a rapid oscillation — which turns the hilltop into a valley. The Trojans and the Penning trap do something subtler. Their hilltops stay hilltops; nothing is averaged. The body is held by being turned, continuously, along the contour rather than down it.

Where spacecraft park #

The five points are working locations. L1 and L2 of the Sun–Earth system, a million and a half kilometres sunward and anti-sunward, are saddles, unstable in the plane, and the observatories stationed near them — the solar observatory SOHO at L1, the James Webb Space Telescope at L2 — must fire their thrusters every few weeks to stay on orbits round the points. The instability is gentle, with an e-folding time of about three weeks, and it is also useful: a small push sends a spacecraft away from a saddle along any of a family of low-energy paths, which mission designers use to move between the Lagrange points of the Sun–Earth and Earth–Moon systems with very little fuel.

L4 and L5 need no station-keeping at all, for the reason the drawings give, and they offer something the saddles cannot: a view of the Sun from the side. A spacecraft at the Sun–Earth L5, sixty degrees behind the Earth, sees the solar surface that will rotate to face the Earth days before it does, and sees eruptions heading towards the Earth side-on, where their speed can be measured. The European Space Agency’s Vigil mission, planned for the early 2030s, is designed to be the first to be stationed there for space-weather forecasting. It will librate round the hilltop, slowly, like an asteroid.

How slowly a Trojan swings #

The libration is slow, and slower the lighter the secondary. A Jupiter Trojan takes a century and a half to swing once round its hilltop; a body at the Sun–Earth L4 would take more than two centuries. The slowness is a measure of how gentle the hill is — the restoring effect of the Coriolis turning is proportional to the square root of the mass ratio — and it controls how easily a Trojan is lost. A slow libration is easily disturbed by the pull of other planets, whose periodic tugs can resonate with it, and the Trojan regions of the lighter planets are correspondingly sparse and short-lived. Two asteroids have been found at the Earth’s L4 point, and both are on temporary orbits that will take them away within a few thousand years. Jupiter’s, heavy and isolated, holds thousands.

Saturn’s moons show the scheme at the smallest scale. Tethys has two small moons, Telesto and Calypso, at its L4 and L5 points, and Dione has two more, Helene and Polydeuces. Nothing in the system is special except that the mass ratios are far below Routh’s limit and the other perturbations are small.

Where the flat, circular problem stops #

Circular orbits. The drawings assume the two large bodies move on circles. Jupiter’s orbit has an eccentricity of 0.05, which makes the effective potential pulse once per orbit and shrinks the stable region; the calculation for Jupiter’s actual orbit, and with Saturn’s perturbations added, is done numerically, and it finds a stable region that survives for the age of the Solar System but is narrower than the circular problem’s.

A massless third body. The small body’s own gravity is ignored. For asteroids that is exact enough. For the co-orbital moons of Saturn it is not, and Janus and Epimetheus, whose masses differ by a factor of four, both move.

Two dimensions. The orbits are drawn in the plane of the large bodies’ orbit. Real Trojans have inclinations of up to thirty degrees or more, and their motion out of the plane adds a third frequency to the libration.

What the pictures cannot show #

The drawings show orbits in the turning frame, where the Trojans loop round fixed points. Seen from outside, in the frame of the stars, there are no loops: each Trojan moves on an ordinary, slightly eccentric, slightly tilted ellipse round the Sun, held in its orientation by the inverse-square law’s extra conserved vector, with the same period as Jupiter’s, and the tadpole is only the slow difference between its ellipse and Jupiter’s. The hill, the loop and the Coriolis force are all properties of the turning frame, chosen because in it the problem is steady. That is the most useful thing a change of frame can do, and it is easy to forget that the hill is not a place anyone could stand on.

Still open: where the Trojans came from #

The Jupiter Trojans are not what a simple story predicts. If they were captured from nearby material as Jupiter formed, they should resemble the asteroids of the outer main belt; instead many are dark, reddish objects more like the bodies beyond Neptune, and their orbits are more tilted than capture in a calm disc would give. The leading explanation is that they were captured later, when the giant planets’ orbits shifted and scattered the outer Solar System’s small bodies inward, and that some of that population was trapped in Jupiter’s Trojan regions as the resonances swept past. NASA’s Lucy spacecraft, launched in 2021, is visiting several Trojans at L4 and L5 through the late 2020s and early 2030s to compare them directly, and the answer bears on how far the giant planets moved.

The habit worth carrying away is to ask whether a force that does no work is present before reading stability off a potential. In a turning frame, or in a magnetic field, a velocity-dependent force can hold a body on a hilltop by turning every slide into a circle — provided the turning is quick enough compared with the push off the summit, which for two orbiting bodies means the lighter one must weigh less than about a twenty-sixth of the pair.