Astronomy & Celestial Mechanics
the sky, computed
Exoplanets

Planets found by a transit running late

A planet on a fixed orbit transits like a clock. A second planet pulling on it makes the clock run fast and slow by minutes — and fitting that wander weighs a planet that may never cross the star at all.

Assumes Transits and The three-body problem.

16 min read 8 figures Small perturbations, large consequencesThe sky as a clock

A transiting planet on an unperturbed orbit crosses its star at times spaced exactly one period apart. Fitting a straight line through those times — an epoch number against a clock reading — should leave no residual at all.

Where a second planet exists, it does. The residual is a few minutes, it repeats over months, and it contains the mass of the perturber whether or not the perturber ever transits.

A transit that will not keep time. Transit times of a planet of 8 Earth masses on a 3-day orbit, minus the best straight line through them, from a three-body integration in which the second planet at 4.62 days is the only thing perturbing it. The residual swings by ±1.7 minutes and repeats over 58 days — the super-period 1/|3/P₂ − 2/P₁|, which is long precisely because the pair is near the 3:2 resonance and gets longer the nearer it is. The amplitude is proportional to the perturbing planet's mass, so fitting this curve weighs a planet that may never transit at all. The short jagged component is not noise: it is the chopping signal, one kick per conjunction at the 8.6-day synodic period, sampled once every 3 days by the transits themselves and so barely above the rate at which it can be followed. The integration conserved energy to 1.6e-10.
Fig. 1 Transit times of an inner planet minus the best straight line through them, from a three-body integration in which the only perturbation is a second planet just outside the 3:2 resonance. The residual swings by ±1.7 minutes and repeats over 58 days — far longer than either orbital period. The jagged short-period component is not noise; it is one kick per conjunction at the 8.6-day synodic period, sampled once per transit.

The signal is in when, not in whether

Everything else a transit measures is an amplitude: a depth, a duration, a shape. This is a timing measurement, and timing is the thing photometry does best.

A transit’s mid-point can be measured to a small fraction of the ingress duration — tens of seconds for a hot Jupiter with good photometry, and better than ten seconds with space data. Over a four-year baseline that is a clock with a fractional stability of 10−710^{-7}. What a residual of minutes therefore represents is a very large signal on the scale of what can be measured, produced by an interaction that is otherwise entirely invisible.

The magnitude follows from the perturbation being a mass ratio. Two planets of a few tens of Earth masses around a Sun-like star perturb each other at the level of 10−410^{-4} of the central force, so an orbit is displaced by of order 10−410^{-4} of its own size, and a period of days is shifted by of order 10−410^{-4} days — which is about ten seconds. Near a resonance, as below, that accumulates coherently and becomes minutes.

Why the residual repeats so slowly

The most surprising feature of the figure is the timescale. Two planets with periods of 3.00 and 4.62 days produce a residual that repeats every 58 days. Nothing in the system has that period.

It is a beat. Near a first-order resonance j:(j−1)j:(j-1) — here 3:2 — the two planets return to the same relative configuration only slowly, and the rate at which the configuration drifts is

1PTTV=∣jPout−j−1Pin∣.\frac{1}{P_{\text{TTV}}} = \left|\frac{j}{P_{\text{out}}} - \frac{j-1}{P_{\text{in}}}\right|.

For an exact resonance the right-hand side is zero and the super-period is infinite. For a pair 2.7 per cent outside it, the super-period is 58 days; move to 1 per cent outside and it becomes 152 days.

This is the whole reason the method works. A perturbation of ten seconds per orbit that pointed in a random direction each time would average away. One that points the same way for twenty consecutive orbits accumulates, and the accumulated displacement is what is measured. The nearer the pair is to resonance, the longer the coherent accumulation and the larger the amplitude — which is why almost every system with detectable timing variations is near a commensurability, and why that is a selection effect rather than a discovery.

A transit that will not keep time. Transit times of a planet of 8 Earth masses on a 3-day orbit, minus the best straight line through them, from a three-body integration in which the second planet at 4.545 days is the only thing perturbing it. The residual swings by ±5.2 minutes and repeats over 152 days — the super-period 1/|3/P₂ − 2/P₁|, which is long precisely because the pair is near the 3:2 resonance and gets longer the nearer it is. The amplitude is proportional to the perturbing planet's mass, so fitting this curve weighs a planet that may never transit at all. The short jagged component is not noise: it is the chopping signal, one kick per conjunction at the 8.8-day synodic period, sampled once every 3 days by the transits themselves and so barely above the rate at which it can be followed. The integration conserved energy to 1.6e-10.
Fig. 2 The same pair moved from 2.7 per cent outside the 3:2 resonance to 1 per cent outside. The super-period stretches from 58 days to 152 and the amplitude grows from ±1.7 minutes to ±5.2, because the kicks accumulate coherently for longer before the configuration drifts. A factor of 2.7 in the distance from resonance buys a factor of 3.1 in amplitude, which is very nearly the 1/Δ the near-resonant theory asks for. The two effects go together, and both are computed from the integration rather than from a formula.

What is being solved

The figures here come from integrating the equations of motion for three bodies directly — a star and two planets, with every mutual force included, stepped forward with a fourth-order method fine enough that the total energy drifts by two parts in 101010^{10} over the run.

That is not an aesthetic choice. The three-body problem has no closed-form solution, and the analytic TTV expressions in the literature are expansions in the eccentricity and in the distance from resonance. They are excellent where they apply and they do not apply to the interesting cases: high eccentricity, deep resonance, or three or more interacting planets. Every published TTV mass in a compact multi-planet system comes from fitting an N-body integration to the observed times.

A transit that will not keep time. Transit times of a planet of 8 Earth masses on a 3-day orbit, minus the best straight line through them, from a three-body integration in which the second planet at 5.1 days is the only thing perturbing it. The residual swings by ±0.6 minutes and repeats over 17 days — the super-period 1/|2/P₂ − 1/P₁|, which is long precisely because the pair is near the 2:1 resonance and gets longer the nearer it is. The amplitude is proportional to the perturbing planet's mass, so fitting this curve weighs a planet that may never transit at all. The short jagged component is not noise: it is the chopping signal, one kick per conjunction at the 7.3-day synodic period, sampled once every 3 days by the transits themselves and so barely above the rate at which it can be followed. The integration conserved energy to 1.6e-10.
Fig. 3 The same two planets at a period ratio of 1.700, which is near nothing — a third of the way between the 3:2 and the 5:3 — and which is what the signal looks like with the mechanism switched off. A ratio near a small integer one makes the conjunctions recur at nearly the same orbital phase, so the tugs accumulate in one direction for many orbits before reversing and the deviation grows to minutes. Here the conjunctions walk steadily around the orbit instead: the beat is 17 days rather than 58, the accumulation has only a few orbits to work in, and the amplitude falls from ±1.7 minutes to ±0.6. The amplitude of a transit-timing signal is a statement about proximity to resonance before it is a statement about mass, and that is why the method finds the systems it finds.

The two components of the signal

A TTV curve has two parts, and separating them is what breaks the method’s central degeneracy.

The resonant term oscillates at the super-period and carries the product of the perturber’s mass and a function of the eccentricities. On its own it cannot separate the two: a small planet on an eccentric orbit and a larger one on a circular orbit produce nearly the same long-period wave. This is the mass–eccentricity degeneracy, and for a decade it made TTV masses systematically uncertain by factors of several.

The chopping term is the short, jagged component visible in every figure here. It is a discrete kick at each conjunction, so it appears at the synodic period, and its amplitude depends on the perturber’s mass without the eccentricity dependence. Measuring it separately breaks the degeneracy.

Chopping is sampled once per transit, at the inner planet’s period, and the synodic period is only two or three times that — so the signal sits barely above the rate at which it can be followed. It looks like noise, it is not noise, and extracting it is why TTV analyses need dozens of well-measured transits rather than a handful.

The masses come out without a spectrograph

The reason this method matters beyond its novelty is the one it shares with nothing else: it delivers masses for planets around faint stars.

A radial-velocity measurement needs photons — enough of them, at high spectral resolution, to measure a Doppler shift of a metre per second. That requirement scales brutally with stellar brightness, and it excludes almost every star in a transit survey’s catalogue. Most Kepler hosts are twelfth to sixteenth magnitude, where a 1 m/s measurement is out of reach of any existing instrument in any reasonable exposure.

Timing needs only photometry, and the photometry is already there. The transits that established the planets’ radii also establish their times, at no additional observational cost whatever. So for compact multi-planet systems around faint stars — which is the bulk of what Kepler found — the mutual perturbations are the only route to a mass, and therefore the only route to a density.

There is a further asymmetry worth stating: radial velocity measures best what timing measures worst, and the reverse. Velocities favour massive planets close to bright stars on any orbit; timing favours any planets near a resonance around any star bright enough to transit-detect. The two methods overlap on almost nothing, and where they do overlap the masses have generally agreed — which is the cross-check the whole business rests on.

What was actually measured

Kepler-19, 2011. The first planet found by this method alone: a 4.6-minute timing variation on a 9.3-day transiting planet, with a period of about 316 days and no second transit anywhere in the data. The perturber’s existence was established from the timing; its mass was bounded rather than measured, because the degeneracy above could not be broken with the data available.

Kepler-36, 2012. Two planets at 13.8 and 16.2 days — a 7:6 period ratio — whose mutual perturbations produce timing variations of tens of minutes, large enough that the masses come out to a few per cent. The result was the pair of densities, 7.5 and 0.9 g/cm³, that made the system the sharpest evidence for atmospheric stripping in the census. Neither mass could have been obtained by radial velocity: the star is faint and the amplitudes are a few metres per second.

TRAPPIST-1, 2017–2021. Seven planets in a chain of near-resonances, all of them transiting, all perturbing each other. The masses of all seven come from fitting an N-body model to hundreds of transit times, with no spectroscopy of the planets at all — the star is a 0.089 M☉ dwarf far too faint for precision velocities. The resulting densities are uniform to a few per cent, which is a statement about the whole system’s building material that no other technique could have produced. The resonant chain itself is a separate argument about how the system was assembled.

And the classical precedent. In 1846 Urbain Le Verrier fitted the residuals of Uranus’s position against a Keplerian ephemeris, attributed them to an unseen planet, and computed where it would be. Johann Galle found Neptune within a degree of the predicted place on the first night of looking. The structure of the argument — a body’s departure from its own ephemeris, interpreted as the pull of something not yet seen — is exactly what a TTV analysis does, with minutes in place of arcseconds.

What the amplitude is proportional to

The last thing to extract from the figures is the scaling, because it is what turns a curve into a mass.

To first order in the perturbation, the timing amplitude is proportional to the perturbing planet’s mass divided by the star’s, and to the super-period in units of the transiting planet’s period. Written roughly,

Δt∼mpertM⋆ PTTV,\Delta t \sim \frac{m_{\text{pert}}}{M_\star}\, P_{\text{TTV}},

with a coefficient of order unity that depends on which resonance and on the eccentricities. Both factors are measured from the same curve — the amplitude from its height, the super-period from its repetition — so a single well-sampled residual delivers a mass ratio with no external calibration at all.

A transit that will not keep time. Transit times of a planet of 8 Earth masses on a 3-day orbit, minus the best straight line through them, from a three-body integration in which the second planet at 4.62 days is the only thing perturbing it. The residual swings by ±4.3 minutes and repeats over 58 days — the super-period 1/|3/P₂ − 2/P₁|, which is long precisely because the pair is near the 3:2 resonance and gets longer the nearer it is. The amplitude is proportional to the perturbing planet's mass, so fitting this curve weighs a planet that may never transit at all. The short jagged component is not noise: it is the chopping signal, one kick per conjunction at the 8.6-day synodic period, sampled once every 3 days by the transits themselves and so barely above the rate at which it can be followed. The integration conserved energy to 1.0e-10.
Fig. 4 The same configuration with the perturbing planet at 30 Earth masses instead of 12. The super-period is unchanged, because it depends only on the two periods; the amplitude goes from ±1.7 minutes to ±4.3, which is the mass ratio of 2.5 to within a few per cent. That separation — one observable fixed by the geometry, the other by the mass — is what makes the fit well-conditioned once the chopping term has broken the eccentricity degeneracy.

The dependence on the star’s mass is worth noticing too. The signal is a mass ratio, so a TTV analysis gives mp/M⋆m_p/M_\star directly and the planetary mass only through a stellar model — the same dependence a transit depth has on the stellar radius, and the same reason a system’s planetary parameters all move together when the star is re-characterised.

Where the picture stops

The signal requires a near-resonance to be large. Away from a commensurability the coherent accumulation is short and the amplitude drops to seconds. TTV-characterised systems are therefore not a random sample of multi-planet systems; they are the resonant ones, and any statistic drawn from them inherits that.

A timing residual is not unique to a planet. A stellar companion produces a light-travel-time effect: as the transiting system orbits a distant third body, the light arrives early or late by up to the light-crossing time of the orbit. A drifting instrument clock does the same. Both are periodic, and distinguishing them requires either a different signature — light-travel effects are achromatic and strictly sinusoidal — or an independent detection.

The fit is high-dimensional and multi-modal. Two planets have ten free parameters, seven have thirty-five, and the likelihood surface has many local maxima. Published TTV masses are the outcome of long sampling runs, and disagreements between groups analysing the same data have usually been disagreements about which mode the sampler settled in.

Stellar activity moves a transit’s apparent centre. A spot crossed by the planet distorts the light curve asymmetrically, and fitting a symmetric model to an asymmetric event displaces the fitted mid-time by tens of seconds. On an active star that is comparable to the signal, and it varies on the star’s own rotation period — which is a periodicity that can be, and has been, mistaken for a perturbing planet.

And a non-transiting perturber’s inclination is unconstrained. The timing signal depends on the perturber’s mass and orbit, and for a body that never transits, the inclination enters only weakly. TTV masses of unseen planets therefore carry a residual model dependence that TTV masses of transiting ones do not.

Duration variations, which say something different

The transit’s timing is not the only thing a perturbation moves. Its duration changes too, and the two carry different information.

A duration depends on the chord across the star, so it changes when the orbital plane precesses — when a perturber tilts the transiting planet’s orbit, the impact parameter drifts and the transit lengthens or shortens over years. Where a timing variation says a perturber exists and how heavy it is, a duration variation says the orbits are mutually inclined, which is a quantity nothing else in a transit light curve can reach.

A handful of systems show it. Kepler-108’s two giant planets are mutually inclined by more than 20 degrees, measured this way, in a field where nearly every other system is flat to a few degrees. The measurement matters because mutual inclination is the clearest surviving record of whether a system was assembled quietly or violently, and it is exactly the quantity a single edge-on view cannot supply.

A transit of a planet 0.0354 of its star's radius. The star's brightness through one transit, computed by integrating the limb-darkened stellar disc over the region the planet covers. The depth is 0.127%, deeper than (Rp/R⋆)² = 0.001253 because the planet crosses a limb-darkened disc whose centre is brighter than its average. The four contact points are where the two discs are externally and internally tangent, at separations 1 ± 0.0354 stellar radii.
Fig. 5 What a duration variation moves. The same planet at a larger impact parameter crosses a shorter chord, so the event is briefer and its shoulders occupy more of it — and a perturber that tilts the orbit walks the planet slowly along that family from one year to the next. The depth barely changes, which is why a shift of this kind is visible only in a long baseline of well-timed transits.

In the extreme, a precessing orbit stops transiting altogether. Several systems have been watched as the transits shortened, shallowed and disappeared — a planet that has not gone anywhere and is simply no longer crossing the disc from here.

The ephemeris that cannot be extrapolated

There is a practical consequence of everything above that has become the dominant operational problem in the field, and it is about scheduling rather than about physics.

Observing a transiting planet’s atmosphere requires pointing a large telescope at the star during a transit, and the transit lasts a few hours. So the observation has to be scheduled in advance, and the schedule needs a prediction of when the transit will occur — often two or three years after the last time anybody watched.

For a planet with no perturbers that is straightforward: fit a straight line to the observed times, extrapolate, and the uncertainty grows slowly with the number of periods. Even then it grows: a period uncertain by one second accumulates to an hour after three thousand orbits, and several well-known planets have had their predicted transit times drift out of a scheduled window.

For a planet in a resonant multi-planet system it is not straightforward at all, because a straight line is the wrong model. The residual wanders on the super-period, which for the systems that matter is months, so extrapolating linearly from a set of transits taken during one phase of that wander puts the prediction tens of minutes wrong at the wrong phase.

The correct prediction requires the dynamical model — an N-body fit to all the available times, propagated forward — and that fit has the multi-modal likelihood surface described above. Different modes agree about the past, because they were fitted to it, and disagree about the future by amounts that grow with the extrapolation.

The result is that maintaining the ephemerides of the interesting systems is an ongoing observational programme in its own right, carried out largely with small telescopes on the ground, whose whole purpose is to keep the predictions good enough for the large telescope’s schedule.

A perturbation that made the masses measurable also made the clock unreliable, and the same coupling that supplies the science is what has to be modelled before the science can be observed.

The amplitude and the period of the signal depend on quantities that are not the ones being solved for, so it is worth moving two of them and watching what the curve does.

A transit that will not keep time. Transit times of a planet of 3 Earth masses on a 3-day orbit, minus the best straight line through them, from a three-body integration in which the second planet at 4.62 days is the only thing perturbing it. The residual swings by ±1.7 minutes and repeats over 58 days — the super-period 1/|3/P₂ − 2/P₁|, which is long precisely because the pair is near the 3:2 resonance and gets longer the nearer it is. The amplitude is proportional to the perturbing planet's mass, so fitting this curve weighs a planet that may never transit at all. The short jagged component is not noise: it is the chopping signal, one kick per conjunction at the 8.6-day synodic period, sampled once every 3 days by the transits themselves and so barely above the rate at which it can be followed. The integration conserved energy to 9.7e-11.
Fig. 6 The same pair with the transiting planet at 3 Earth masses instead of 8, and the curve is the same curve: ±1.7 minutes over 58 days, unchanged to the precision drawn. A planet’s own mass does not appear in its own timing variation, because what displaces it is the other planet’s pull and what it is displaced from is its own unperturbed orbit — both of which are independent of how heavy it is. That is the property the whole method depends on. A transit-timing fit weighs the perturber, and it does so without needing to know anything about the mass of the object being watched.
A transit that will not keep time. Transit times of a planet of 8 Earth masses on a 3-day orbit, minus the best straight line through them, from a three-body integration in which the second planet at 4.62 days is the only thing perturbing it. The residual swings by ±4.1 minutes and repeats over 58 days — the super-period 1/|3/P₂ − 2/P₁|, which is long precisely because the pair is near the 3:2 resonance and gets longer the nearer it is. The amplitude is proportional to the perturbing planet's mass, so fitting this curve weighs a planet that may never transit at all. The short jagged component is not noise: it is the chopping signal, one kick per conjunction at the 8.6-day synodic period, sampled once every 3 days by the transits themselves and so barely above the rate at which it can be followed. The integration conserved energy to 1.6e-10.
Fig. 7 And around a star of four tenths of a solar mass. The planetary masses are unchanged and their ratio to the star’s has risen by a factor of 2.4, so the amplitude rises from ±1.7 minutes to ±4.1 — which is the reason the transit-timing planets are found around M dwarfs and the radial-velocity ones around anything.

The generalisation

Measuring an unseen body by the departures of a visible one from its expected motion is one of the most productive techniques in the whole subject, and it long predates any of this.

Neptune came out of the residuals of Uranus. Sirius B was inferred from the wobble of Sirius A twenty years before it was seen. The mass of the object at the centre of the Galaxy comes from the orbits of stars around a point where nothing is visible. Pulsar timing found the first planets outside the solar system at all — three bodies around PSR B1257+12 in 1992, detected as millisecond residuals in the arrival times of a neutron star’s pulses, which is this method with a far better clock and a far stranger star.

Resonance is what makes all of these tractable rather than merely present. A perturbation that accumulates coherently is measurable; one that averages away is not. The Kirkwood gaps, the Laplace resonance of Jupiter’s moons and a 58-day beat in a transit ephemeris are the same phenomenon at three scales.

And the same pair at a shorter orbital period, since what a survey can measure is set by how many super-periods fit inside its baseline.

A transit that will not keep time. Transit times of a planet of 8 Earth masses on a 2-day orbit, minus the best straight line through them, from a three-body integration in which the second planet at 3.08 days is the only thing perturbing it. The residual swings by ±1.1 minutes and repeats over 38 days — the super-period 1/|3/P₂ − 2/P₁|, which is long precisely because the pair is near the 3:2 resonance and gets longer the nearer it is. The amplitude is proportional to the perturbing planet's mass, so fitting this curve weighs a planet that may never transit at all. The short jagged component is not noise: it is the chopping signal, one kick per conjunction at the 5.7-day synodic period, sampled once every 2 days by the transits themselves and so barely above the rate at which it can be followed. The integration conserved energy to 1.6e-10.
Fig. 8 The same near-resonant pair with both periods shortened by a third. The super-period shortens with them, so a four-year survey covers several complete cycles rather than one — and a signal whose period is covered is a mass measurement, while one that is not covered is a linear drift indistinguishable from a wrong ephemeris.

Where this goes next

A chain of planets locked into successive resonances is not merely convenient for weighing them. It is a fossil of how the system was assembled, because capture into resonance requires the orbits to have converged slowly — which they cannot do in place.

Later rungs on this anchor: the analytic TTV formalism and its limits. Chopping, and how it breaks the mass–eccentricity degeneracy. Transit duration variations and orbital precession. Light-travel-time effects from a distant companion. Non-transiting perturbers. TTVs in circumbinary systems. Pulsar timing and the first exoplanets. Dynamical stability as a constraint on masses. And Le Verrier’s calculation done properly, which is the ancestor of all of it.

Where this goes next

Three bodies, integrated. Three equal masses integrated forward under mutual gravity: the figure-eight choreography — all three bodies on one closed curve. Every point is a step of the equations of motion, and the total energy is conserved to 2.5e-11 across the run. Three bodies, and what "no solution" actually means The three-body problem is routinely called unsolvable. Trajectories are computed for it every day, exact periodic solutions are known, and both statements are true — the word is doing more work than it looks. The Kirkwood gaps. Asteroid numbers against semi-major axis, with the resonant radii marked. Each gap sits where the orbital period is a simple fraction of Jupiter's, and each of those radii is computed from the harmonic law rather than placed by eye. Resonance clears a gap in one place and locks a moon in another When two orbital periods are in a simple ratio, small tugs stop averaging away and start accumulating. Sometimes that empties a region entirely. Sometimes it holds three moons together for the age of the solar system. A transit of a planet 0.103 of its star's radius. The star's brightness through one transit, computed by integrating the uniform stellar disc over the region the planet covers. The depth is 1.05%, which is exactly (Rp/R⋆)² = 0.01055. The four contact points are where the two discs are externally and internally tangent, at separations 1 ± 0.103 stellar radii. A planet measured by the light it removes A transit gives a depth, and the depth is a ratio of two radii rather than a size. Everything a transit says about a planet is said in units of a star nobody has visited either.

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ChoppingEphemerisMass determinationN-body integrationOrbital periodPerturbationResonanceSuper-periodThree-body problemTransit-timing variation