One timing curve and five planets that could draw it

A transit that runs late weighs the planet responsible, and in the cases that made the technique famous the planet responsible also transits. Its period is then known to many digits from its own transits, and the timing signal is used for one thing only: the mass. Even then the mass is tangled with an eccentricity.

Most transiting planets whose times wander do not have a second transiting planet to blame. The perturber is inclined a few degrees and passes above or below the star, and what the observer has is one planet’s transit times and a residual with a period and an amplitude. The question is then not only how massive the perturber is but where it is, and the answer turns out to be a list.

A period that names a distance and not a resonance #

The slow part of a timing signal comes from a pair of orbits close to a commensurability. Near a ratio of $j$ to $j-1$ the resonant angle drifts slowly, at the rate

$$\frac{1}{P_{\rm super}} = \left|\frac{j}{P_{\rm out}} - \frac{j-1}{P_{\rm in}}\right|,$$

and the transit times wander with that period. Given the transiting planet’s period and the super-period, the equation has one solution for the perturber’s period for every choice of $j$ and for every choice of which planet is inside. Near 3:2 with the perturber outside is one answer; near 2:1 outside is another; near 3:2 with the perturber inside — the transiting planet then being the outer of the pair — is a third. Each sits at the distance from its own resonance that the super-period measures.

The amplitude then fixes a mass for each guess, and the masses differ. A super-period and an amplitude are two numbers, and a planet has three unknowns — a period, a mass and which side of the orbit it is on. The missing one is supplied by a guess, and each guess returns a consistent planet.

The figure does not solve the analytic expressions for those masses. Each candidate is integrated as a three-body system, its mass is adjusted until the integrated amplitude matches, and the check is that the integration reproduces both the amplitude and the super-period it was meant to. The agreement is better than the analytic theory would give, because the theory is an expansion in the eccentricities and the distance from resonance and the integration is not.

Five planets with masses spanning a factor of almost seven and periods spanning a factor of four draw curves that a timing precision of a minute cannot tell apart. That is not a failure of the fit. It is the content of the data, and a catalogue entry quoting one of the five would be quoting a choice.

Why the masses are not the same #

Two things set how much mass each candidate needs, and neither is obvious from the super-period.

The first is that the candidates are not equally close to their resonances. The fractional distance from exact commensurability, Δ, is related to the super-period by $P_{\rm super} = P_{\rm out}/(j\,|\Delta|)$, so at a fixed super-period Δ grows with the outer period and falls with $j$. For the outside candidates that gives Δ = 0.018 near 4:3, 0.027 near 3:2 and 0.055 near 2:1 — the candidate furthest out sits three times further from its resonance than the closest, and since the near-resonant amplitude is proportional to mass divided by Δ, it needs roughly three times the mass on that account alone. For the inside candidates the transiting planet is the outer body, so Δ is set by its own period: 0.017 near 3:2 and 0.026 near 2:1.

The second is the strength of each resonance’s coupling, a coefficient of order one that depends on $j$ and on which planet is inside. It is not the same for a perturber outside a 2:1 as for one inside it, because the forces a close inner planet exerts include an indirect term — the star itself is displaced by the inner planet and the transiting planet feels the displacement — and for the inner 2:1 case that term nearly cancels the direct one. The result is the outlier in the list: the inside candidate near 2:1 is only modestly far from its resonance and needs 45.8 Earth masses, six times as much as the inside candidate near 3:2.

It is worth pausing on how much each alias is itself a continuum. The mass–eccentricity trade applies within each of them: every candidate mass in the list can be lowered, and the lost amplitude restored by a small free eccentricity with the right orientation.

The list is therefore not five planets but five families of planets, each a curve in the plane of mass and eccentricity. Two degeneracies of different kinds are stacked: a discrete one, which resonance, and a continuous one, how much of the amplitude is mass and how much is eccentricity.

The signal that is not shared #

The candidates were chosen to agree on the super-period and the amplitude of the slow sinusoid. They were not chosen to agree on anything else, and the other component of a timing signal — the chopping, a small kick delivered at each conjunction of the two planets — repeats at the pair’s synodic period, which depends on the perturber’s actual period.

The rows are plainly different, and a timing series good to ten seconds would tell most of them apart without any other measurement. The two outside candidates nearest the transiting planet kick every 11.4 and every 8.6 days, and at 26 and 18 seconds those kicks are the largest signal in the residual. The innermost candidate kicks every 2.85 days, faster than the transiting planet’s own 3-day sampling, and its chopping is folded into a slow pattern by that undersampling and reduced to a second. A transit can only be timed when it happens, so a perturber that passes the transiting planet more often than once per orbit is sampled below the rate its kicks need, and its most distinctive signal is the one the data cannot follow.

Two rows, though, have the same period. The perturber outside near 2:1 and the one inside near 3:2 share a synodic period of 5.70 days, and the agreement is exact rather than a rounding. Writing the candidates’ periods from the super-period relation and taking the synodic frequency with the transiting planet, the outside candidate near $j:(j-1)$ gives

$$\frac{1}{P_{\rm syn}} = \frac{1}{j}\left(\frac{1}{P} + \frac{1}{P_{\rm super}}\right),$$

and the inside candidate near $(j+1):j$ gives precisely the same expression. A second degeneracy is hidden inside the thing that was supposed to break the first, and those two candidates are distinguished by the amplitude and shape of their kicks — 8 seconds against 11 — rather than by their rhythm.

The coincidence is not a curiosity of these particular numbers. It holds for every transiting planet and every super-period, for each pair of an outside candidate and the inside candidate one order up. Any procedure that separates aliases by searching for the chopping frequency alone will find two answers wherever it finds one of those.

Why the lightest candidates kick hardest #

The chopping amplitudes run in the opposite order to the masses. The outside candidate near 4:3 is the lightest of the three outside the orbit, at 7.6 Earth masses, and it delivers the largest kick, 26 seconds; the one near 2:1 is more than three times as heavy and delivers 8. The reason is where the kick comes from. A conjunction is a close approach, and the impulse it delivers is set by the perturber’s mass and by how close the two orbits come, falling steeply with the separation.

The separations follow from the periods. By the harmonic law, orbits near 4:3, 3:2 and 2:1 sit at 1.23, 1.33 and 1.65 times the transiting planet’s orbital radius, so at conjunction the perturbers pass at 0.23, 0.33 and 0.65 of that radius. Taking the kick as roughly the mass over the square of the separation gives relative strengths of 150, 108 and 62 — in the ratio 26 : 19 : 11, against the 26 : 18 : 8 the integrations measured. The rough scaling captures the order and most of the spread; the rest is that a wider pair’s conjunctions are also spread over a longer and gentler encounter, which weakens the kick further than a simple inverse square allows.

So the near-resonant sinusoid and the chopping measure opposite combinations. The sinusoid’s amplitude goes as the mass divided by the distance from resonance, and a distant perturber compensates for its weaker pull by being further from resonance and needing more mass. The chopping goes as the mass divided by the square of the separation, and there the distant perturber’s extra mass does not compensate for its distance. The two components of one timing signal weigh a perturber with two different lever arms, and that is why the chopping breaks the degeneracy the sinusoid creates rather than repeating it.

There is a limit on all of this that has nothing to do with precision. A timing series shorter than one super-period does not measure the super-period at all: it sees part of a swing, which a straight line and a curvature describe as well as a sinusoid does, and the list of candidates cannot even be drawn up. For the pair in the opening figure the super-period is 58 days and a season of transits is enough. For a pair nearer its resonance, with a super-period of years, the first several seasons of a planet’s timing are a slope and a bend with no period in them, and every perturber at every resonance, at almost any mass, is consistent with that.

How the list is actually shortened #

Chopping is one route, and it demands a timing precision that ground-based transits rarely reach. The routes used in practice are four.

Velocities. Each candidate pulls the star with a different velocity semi-amplitude at a different period, because $K$ scales as the mass over the cube root of the period. Across the opening list $K$ runs from 3.1 metres a second to 26.7, a factor of 8.5, at five different periods. A velocity curve gives a period and a minimum mass, and with the timing supplying an amplitude that does not depend on the inclination, one velocity series of modest precision picks out the right line in the list. Kepler-19 is the example: its transiting planet showed a timing wander in 2011 whose non-transiting cause could sit at several periods, and it took years of velocities to place the perturber at 28.7 days and find a third planet besides. Combining the two methods does more than choose: two measurements with different degeneracies return the inclination as well, and therefore a true mass for a planet that has never crossed its star.

The absence of a transit. A perturber on nearly the same plane as the transiting planet would often transit too, and it transits more readily the closer it is to the star, since the chance of a transit goes as the star’s radius over the orbital distance. The innermost candidate at 1.46 days is 2.7 times as likely to transit as the outermost at 6.33, for the same mutual inclination. Not seeing a second transit therefore weighs against the inner candidates more than the outer ones — not decisively, since planetary systems are rarely perfectly flat, but in a direction that can be put into numbers once the distribution of mutual inclinations is assumed.

Dynamical stability. Some candidates cannot exist. An inner perturber of tens of Earth masses at half the transiting planet’s period may sit within a few mutual Hill radii of it, and a pair that close does not keep its spacing for the age of the star. Integrating each candidate for millions of orbits removes such lines from the list, at the cost of an argument from survival rather than from measurement.

A longer series. The super-period is a sinusoid only to first order. Over several cycles the harmonics of the near-resonant term, whose relative sizes also depend on $j$, grow visible, and a series of many super-periods separates candidates the first cycle could not.

The case that established the method went the other way round. KOI-142, a Kepler planet with timing variations of more than ten hours, was modelled in 2013 with a non-transiting perturber near 2:1 outside its orbit, and the model predicted a velocity signal at the perturber’s period and mass. The velocities measured afterwards matched both, which is the kind of test the list makes possible: a timing solution that names a period can be checked against a star that has no stake in the answer.

A different reference, the same shape #

The list is not special to one choice of the reference signal. Moving the reference perturber wide of 2:1 instead of 3:2 changes every number and keeps the structure.

The inner candidate near 2:1 again needs the largest mass by far, and the inside candidate near 3:2 again the smallest. The ordering is set by the resonant coefficients and the distances from resonance rather than by the reference, and it has a practical reading: the aliases that demand the least massive perturber are the ones a velocity survey is least able to see, and so the ones most likely to survive as unresolved possibilities.

The list is also incomplete as drawn, and deliberately. Second-order commensurabilities, near 5:3 or 3:1, produce timing signals too, weaker by a power of the eccentricity, and each adds a candidate that would need a large mass or a large eccentricity to match. They are not drawn, because the first-order list already makes the point, and because a second-order candidate that matches is usually excluded by its stability before anything else is measured.

Resonant systems themselves complicate the picture from the other direction. Pairs that were carried into a commensurability by migration usually settle just wide of it, on the side where capture leaves them, and a population of real systems is therefore not spread evenly across the candidates: the wide side of 3:2 and 2:1 is where observed pairs pile up. A prior built from that population favours those two lines of the list, which is a reasonable use of the census and an assumption that a particular system was assembled the way most were.

What was actually measured #

The first large catalogue of timing variations from space photometry, of some hundreds of Kepler planets, reported wanders with periods and amplitudes, and assigned perturbers only where a second planet in the same system transited near a commensurability whose super-period matched. For the rest, the catalogue was explicitly a list of signals rather than of planets, and the ambiguity described here is why.

Where the perturber was later found by velocities, the agreement has been good enough to trust the method in reverse: when the velocities show a planet at one of the alias periods and the timing amplitude implies a mass consistent with the velocities’ minimum mass, the inclination follows from their ratio. Three bodies have no general solution, and every one of those confirmations was an integration fitted to transit times rather than a formula evaluated.

What the list leaves out #

The candidates are coplanar. The integrations place the perturber in the transiting planet’s orbital plane. A mutual inclination reduces the effective coupling and raises the mass each candidate needs, and a strongly inclined perturber also precesses the transiting planet’s orbit and changes its transit durations, which is a separate observable that no figure here draws.

The eccentricities are zero. Each point in the list is the circular end of its own mass–eccentricity family, not the whole family.

Only one super-period is present. A real residual may contain the slow terms of two perturbers at once, and a list built for one signal can mistake two for one at an intermediate period.

Still open: how many perturbers a single wander can be shown to need #

Every candidate in the list is a single planet. A timing signal can also be reproduced by two or more non-transiting planets whose contributions add, and with enough freedom almost any smooth wander can be fitted that way. What the data actually require — the smallest number of bodies, and which of the many combinations are excluded by stability, by velocities or by the chopping — is the question behind every timing detection of a non-transiting planet, and for most of the wanders in the catalogues the honest answer is still a set of systems rather than a system.