A companion on the same orbit, seen in the planet's clock

Every timing signal from a second planet is read through a period the second planet has and the first does not. A super-period is a beat between two periods; a chopping kick comes once per conjunction, which requires the two planets to pass one another. A companion with no period of its own leaves neither.

Such companions are not hypothetical. Jupiter shares its orbit with thousands of asteroids clustered sixty degrees ahead of it and sixty behind, at two of the five places where a small body can keep station with a planet. Neptune has Trojans, Mars has a few, and the Earth has two known. They orbit the Sun with exactly the planet’s period, and the question here is what a planet with such a companion looks like from another star, when the companion is too small to see and never transits at any period its planet does not.

Why a point sixty degrees ahead holds anything #

The leading and trailing Lagrange points are strange places to be stable. In the frame that turns with the planet they sit at the tops of hills of the effective potential, not at the bottoms of valleys, and a body placed near the top of a hill would ordinarily roll off. What holds it is the Coriolis force. A body displaced from the point starts to fall away, picks up speed in the rotating frame, and the Coriolis force turns that speed sideways, so that instead of rolling downhill it circles the hilltop.

The arrangement only works if the planet is small enough. Edward Routh showed in 1875 that the triangular points are stable only while $27\mu(1-\mu) < 1$, with μ the planet’s share of the total mass — a limit of about 3.85 per cent. Jupiter, at a thousandth of the Sun, is far inside it, and so is every planet around a Sun-like star. Two stars of comparable mass have no stable triangular points at all, which is why the idea of a Trojan belongs to planetary systems and not to binaries.

The same geometry is what the zero-velocity curves draw: a body with a little less energy than the triangular points’ potential is confined to a narrow region elongated along the orbit, and its path inside that region, seen in the turning frame, is a tadpole. A body with more energy wanders round a horseshoe that encloses both triangular points and the far collinear one, and one with more still escapes into an ordinary orbit of its own. The unstable collinear points on either side of the planet are the gateways between these regimes, and the tubes that lead out through them are how a spacecraft moves between them cheaply.

A period of its own after all #

A Trojan does not sit still at its Lagrange point. Every real one is displaced, and a displaced one oscillates about the point along its tadpole, the companion’s angle from the planet rising and falling. The oscillation has a period, and it is the one clock the companion has. For a small libration it is

$$P_{\rm lib} = \frac{P}{\sqrt{27\mu/4}},$$

where $P$ is the planet’s orbital period. For a Jupiter about a Sun-like star μ is about a thousandth, and the libration takes some fifty orbits. That is the period measured off the opening figure, 48.5 days, within three per cent of the formula’s 49.8; the difference is the libration’s amplitude, since the formula is the limit of a vanishing swing and a real one of ten degrees samples a less quadratic part of the potential.

The libration period depends only on the planet’s mass and its orbital period, not on the companion. A timing signal at a period near fifty orbits, with no second planet anywhere else to explain it, therefore points to a co-orbital body with a measured period that already constrains the planet it orbits with.

It constrains it well. Inverting the formula gives $\mu = (4/27)(P/P_{\rm lib})^2$, and the 48.5-day period measured off the opening figure returns a planet of 1.008 thousandths of the star’s mass, against the 0.955 thousandths the integration was given — high by under six per cent, the part of the swing’s amplitude the small-oscillation formula leaves out. That is a mass for the planet obtained from the timing of a body too small to see, with no velocity measurement and no inclination in it. A velocity curve returns only a minimum mass, because it sees the star’s motion projected on the line of sight; a libration is an oscillation within the orbital plane, and its period depends on the planet’s true mass whatever angle the plane is viewed from. For a transiting planet the inclination is known anyway, but for a planet that does not transit, a companion’s libration would be one of very few routes to a mass free of the projection.

The two triangular points do not reveal the same things. A companion at the leading point shifts the planet-companion barycentre ahead of the planet, and one at the trailing point shifts it behind, so the static offset described below changes sign between them while the timing swing has the same form at either. The solar system does not populate the two equally: Jupiter’s leading swarm outnumbers its trailing one by about half again, an asymmetry for which several capture histories have been proposed and none established. Around another star a timing swing alone would not say which point was occupied, and the velocity offset would.

Why the planet’s clock swings #

The planet and its companion orbit their common centre of mass as it goes round the star, and the companion’s libration moves that centre along the orbit. The planet responds in the opposite direction by the mass ratio: a companion swinging ten degrees ahead of and behind its mean position drags the planet by ten degrees times the ratio of their masses. In time, that is a displacement of the transits by

$$\delta t \approx \frac{m_{\rm T}}{m_{\rm p}}\,\theta_{\rm lib}\,\frac{P}{2\pi},$$

with $\theta_{\rm lib}$ the libration amplitude in radians. For a 10 Earth-mass companion of a Jupiter, a ten-degree swing and a four-day orbit, that is 300 seconds, which is what the integration returns.

The mechanism is the reason neither body in a pair is still, applied to a pair that shares one orbit instead of orbiting each other. It has a close relation in the solar system that is worth naming. Saturn’s small moons Janus and Epimetheus share an orbit on horseshoe paths rather than tadpoles, and every four years they approach each other and swap orbits, the inner one moving out and the outer one moving in by some tens of kilometres. The swap is exactly the exchange of angular momentum that makes a Trojan’s planet swing, taken to the extreme where the two bodies are nearly equal in mass and the swing is not a small correction but a trade of places.

The proportionality is nearly exact while the companion is small, and falls off as it grows, because a hundred Earth masses is a third of the planet and the two bodies’ libration becomes a mutual one. The more consequential feature is the second family of points. A companion’s mass and its libration amplitude enter the timing signal as a product, so a massive companion swinging a little and a light one swinging a lot are indistinguishable in the planet’s transits. That is the same shape of degeneracy as a timing mass tangled with an eccentricity, met this time with no resonance between different periods involved at all.

How small a companion can be seen #

The figure’s precision lines settle the practical question. A single transit of a hot Jupiter, timed from space, is good to about ten seconds; the same transit from the ground to about a minute. A companion of an Earth mass, swinging twenty degrees, moves its Jupiter by about a minute.

The libration period sets a second requirement, and it is not about precision. A swing can only be recognised once the series spans a cycle of it, and the period grows as the planet gets lighter: fifty orbits for a Jupiter, but for a Neptune, a twentieth of the mass, about 214 days on a four-day orbit and 536 days on a ten-day one. The two methods in this essay therefore favour opposite planets. Timing reaches small companions only around massive planets on short orbits, where the libration is quick and the planet’s transits are deep and sharply timed; the velocity offset described below is largest around light planets, whose barycentre a companion displaces further. A survey for co-orbitals that used one method alone would find them, if at all, in one corner of the population and conclude something about all of it.

That places Earth-mass Trojans of hot Jupiters within reach of transit timing, and nothing has been found. A companion on an orbit inclined by a degree or two to its planet’s would also transit, shallowly, at orbital phases sixty degrees before and after the planet’s own transit, and a transit of an Earth-sized body across a Sun-like star removes about eighty parts per million of the light — invisible in a single orbit and within reach of the same photometry that measured the planet once hundreds of orbits are folded together at those phases. Stacking the Kepler light curves of every planet at its Lagrange-point phases returned nothing significant. The dust apparently collected at one Lagrange point of the young planet PDS 70 b, seen in submillimetre images, is the nearest thing to a detection anywhere, and it is not a body.

At larger libration the swing grows faster than the formula predicts.

The widest stable tadpoles reach from about twenty-four degrees from the planet round to the far side of the orbit, and beyond them lie horseshoes and then escape. A timing swing therefore has an upper limit as well as a lower one for a given companion mass, and a measured swing larger than a stable libration allows points to a different explanation.

Where co-orbitals would come from #

Whether anything should be found depends on how co-orbitals form and what happens to them afterwards, and the two routes that have been worked out predict different populations.

One route is formation in place. The Lagrange points of a growing giant planet collect gas and solid material from the disc around it, and models of that accumulation produce co-orbital bodies of up to several Earth masses. The other is capture. Jupiter’s Trojans are thought to have been caught during the rearrangement of the giant planets early in the solar system’s history, when scattered planetesimals moved the planets’ orbits and briefly unlocked and relocked the co-orbital regions; the Trojans are what was trapped when the doors closed.

For a hot Jupiter the history that matters most is the journey inwards. A planet migrating slowly through its gas disc can carry its co-orbital material with it, since the libration is an adiabatic invariant that survives a gradual change. A planet delivered by high eccentricity — thrown inwards by a distant perturber and then circularised by tides — passes through orbits where its triangular points are violently disturbed and almost certainly loses whatever shared its orbit. The null results are therefore not only a limit on companions; they are a weak test of which way hot Jupiters arrived.

A clock and a velocity that disagree #

The second signature needs no libration and no long series, and it is the cleverer of the two.

Without a companion, a transiting planet’s star moves directly away from the planet, so its radial velocity crosses zero at exactly the moment of transit: the star is moving across the line of sight while the planet is in front of it. With a companion at the leading Lagrange point, the star answers not to the planet but to the centre of mass of planet and companion, which lies a little ahead of the planet along the orbit. The velocity curve is shifted earlier than the transit by the angle between the two directions.

The offset is static: it is there on the first night, it needs no libration and no baseline of months, and its sign says which Lagrange point is occupied. It is also largest exactly where a timing swing is hardest to measure. A companion of a light planet shifts the velocity curve by hours, because the centre of mass is displaced by a larger fraction of the orbit. The two signatures together measure the product of mass and libration once and the mass on its own once, and so break the degeneracy the timing left.

The method was proposed by Eric Ford and Scott Gaudi in 2006, and its weakness is that it sets a velocity zero crossing against a transit time, two measurements made by different instruments with different systematics. Two effects in particular shift a velocity curve’s apparent phase by minutes. The star’s rotation distorts the lines during transit — a shadow crossing a rotating line — and if those velocities are not removed they drag the fitted zero crossing. And a small eccentricity moves the zero crossing away from conjunction on its own: an eccentricity of a thousandth with the wrong orientation shifts a four-day orbit’s zero crossing by several minutes, and fits of nearly circular orbits are biased towards reporting a small eccentricity even when there is none. A Trojan offset of tens of minutes survives both; one of a few minutes does not, unless the eccentricity is constrained independently, for instance by the timing of the planet’s secondary eclipse.

A longer orbit makes the offset larger in minutes and the measurement no easier, because the velocity curve is shallower and its zero crossing less sharply defined. At four days the offset is three times its uncertainty; at ten days a little over twice. The method’s reach is set mainly by the planet’s mass and the velocity precision, not by where in the system the planet is.

What was actually measured #

Searches combining archival velocities with transit ephemerides have tested this offset for dozens of transiting planets. None shows a significant shift in the direction a Trojan would cause, and the upper limits reach tens of Earth masses for the best-measured hot Jupiters — well above what the timing method could see from space, and far below the mass at which a co-orbital body would itself have been noticed in the light curve. What the null results constrain is the population of large co-orbitals, and they constrain it more tightly than theory predicted it.

The solar system’s co-orbitals were found by their positions, not by their planets’ motions. The Earth’s known Trojans shift the Earth’s orbital phase by amounts far too small to measure, and Jupiter’s thousands of Trojans, together far less massive than the Moon, displace Jupiter by metres at most. The technique in this essay is a method for other systems only; in the solar system the companions are bright enough to be seen directly long before their planets’ clocks would notice them.

What the integrations leave out #

The system is coplanar. A real co-orbital would be inclined to its planet’s orbit, and an inclined libration adds a vertical oscillation that the planar integrations do not contain and that changes transit durations slightly.

The star is a point. Over 160 days the tides of a close-in planet and its star, and the star’s own flattening, precess the orbits by amounts that could imitate a very slow timing trend; at the libration period they are negligible.

The companion is started by hand. Its libration amplitude is an initial condition chosen for each figure. How large a libration a real co-orbital would have after forming or being captured, and migrating with its planet, is set by a history the figures do not model.

Still open: whether hot Jupiters lose their companions on the way in #

Migration through a gas disc can carry a co-orbital body inwards with its planet or strip it away, depending on the disc’s structure and on how fast the planet moves; migration by scattering and tidal circularisation almost certainly destroys one. The null results of the timing and velocity searches are therefore a measurement about how hot Jupiters arrived, as well as about what accompanies them now. Which fraction of hot Jupiters should still have a co-orbital, under each route, has been predicted in outline and not in numbers that the surveys’ upper limits could yet reject.