Transit Timing Variations

The clock that runs early, then late

A single planet on a fixed orbit is one of the most precise clocks in nature. Each time it passes in front of its star it carves the same dip in the light curve, and those dips arrive at strictly even intervals — its orbital period. Measure a handful of transits, fit a straight line through their mid-times, and you can predict the next one to within seconds, years in advance. That linear ephemeris,

T(n) = T0 + n · P

where T0 is a reference mid-transit, P the period, and n the integer transit number, is the bedrock of all transit work. A transit timing variation is simply what happens when that clock refuses to keep time. The planet shows up a minute early on one orbit, a couple of minutes late several orbits later, then drifts back. The deviations are tiny — seconds to tens of minutes against periods of days — but utterly diagnostic, because a lone planet in an otherwise empty system cannot do this. Something is pulling on it. In nearly every confirmed case, that something is another planet, and the rhythm of the lateness encodes the puller's mass and orbit. The transit clock has become a gravimeter.

The mechanism: gravity shows up in the timing

Picture two planets orbiting the same star. They are not isolated two-body systems; each feels the other's gravity. Over an orbit, the inner planet is alternately pulled ahead (speeding it up, shortening that lap) and pulled back (slowing it, lengthening the next). Because the transit happens at a fixed geometric point — when the planet crosses our line of sight to the star — any change in how fast the planet sweeps through its orbit shifts the moment it arrives there. The accumulated shift is the TTV.

To extract it, you don't model the dip; you model the schedule. Fit the linear ephemeris, then plot the residual of each observed mid-transit against the prediction:

(O − C)_n = T_observed(n) − [T0 + n · P]

This observed-minus-calculated, or O−C, curve is the central diagnostic of the field. A perfect single planet gives a flat line scattered around zero. A perturbed planet gives a structured, oscillating curve — sinusoidal in the simplest cases, sawtooth-like near resonance. The amplitude scales with the perturber's mass relative to the star; the period of the oscillation is set by how close the two planets sit to a resonance. Reading both off the O−C curve, then running an N-body integration that reproduces it, returns the masses and eccentricities of the interacting planets. No spectrograph touches the star: the same Newtonian gravity that drives radial velocities is read out in time rather than velocity.

Why resonance turns a whisper into a shout

Most planet pairs are too widely separated, or too poorly aligned in their orbital phases, for their mutual tugs to accumulate. The kicks come at random phases and largely cancel. But when the orbital periods sit near a low-integer ratio — a mean-motion resonance such as 2:1, 3:2, or 4:3 — the geometry repeats. The two planets return to nearly the same relative configuration each cycle, so the same kick is delivered at the same phase over and over and the effect builds coherently. A perturbation that would be a one-second wobble far from resonance becomes a tens-of-minutes swing near one.

The variations no longer oscillate at the orbital period; they oscillate at the much longer resonant super-period:

P_TTV = 1 / | j / P_out − (j − 1) / P_in |

for a j : (j−1) resonance. As a pair sits closer to exact commensurability the denominator shrinks and P_TTV grows — often to months or years, far longer than either planet's day-scale orbit. The other hallmark is anti-correlation: because the two planets exchange orbital angular momentum, when the inner one runs early the outer one runs late, and vice versa. Two transiting planets that show timing signals at the same super-period but in opposite phase are gravitationally bound to each other — a smoking gun that no instrumental drift or starspot can fake.

The numbers: amplitudes, periods, and what they imply

TTV amplitudes span an enormous range. The smallest secure detections are at the few-second level; the largest, in strongly resonant systems, reach tens of minutes to over an hour. The order-of-magnitude scaling near a first-order resonance is that the timing amplitude is roughly the orbital period times the perturber-to-star mass ratio divided by the fractional distance from exact resonance — so a planet a few hundredths from resonance with an Earth-to-Sun-class mass ratio (≈3 × 10⁻⁶) on a 10-day orbit produces variations of minutes, comfortably within Kepler's ~tens-of-seconds-per-transit timing precision for a bright target.

Quantity	Typical value	Why it matters
Single-transit timing precision (Kepler, bright star)	~10 s – 1 min	Sets the smallest detectable variation
TTV amplitude, off-resonance pair	seconds	Usually below the noise floor
TTV amplitude, near 2:1 or 3:2 resonance	minutes – >1 hour	Coherent build-up makes it detectable
Super-period P_TTV	months – years	Needs a long, continuous baseline
Mass ratio probed	~10⁻⁶ – 10⁻⁴ (planet/star)	Sub-Neptunes to gas giants
WASP-12b orbital decay (a different timing effect)	−29 ms/yr	Long-term drift, not periodic TTV

The mass sensitivity is what makes the method valuable. For the small, abundant sub-Neptunes that dominate Kepler's catalogue, the reflex radial-velocity signal is often well under 1 m/s — beyond the reach of ground-based spectrographs on the faint (V ≈ 12–16) Kepler host stars. TTVs sidestep the brightness problem entirely: the timing precision depends on the transit depth and cadence, not on resolving a Doppler shift in the star's spectrum.

How it's measured: from light curve to mass

The pipeline is conceptually clean. (1) Detect a transiting planet and assemble every transit you can. (2) Measure each individual mid-transit time, typically by fitting a transit model (Mandel-Agol) to one transit at a time and recording the centre. (3) Fit a linear ephemeris and form the O−C residuals. (4) If the residuals are structured rather than flat noise, model them. The early literature used analytic perturbation formulae (Agol, Steffen, Sari & Clarkson 2005; Holman & Murray 2005); modern analyses run a full N-body integration whose initial masses, periods, eccentricities and phases are varied — usually inside a Markov chain Monte Carlo — until the integrated mid-transit times reproduce the data.

The state of the art is the photodynamical model, which abandons the two-step "measure times, then fit times" approach and instead fits the raw photometry directly with an N-body simulation, so that the transit shapes, the durations, and the timings are all explained by one self-consistent dynamical solution. This is essential in compact systems where transits of different planets can overlap, or where transit-duration variations (TDVs, changes in how long the crossing lasts) and impact-parameter changes carry extra information about mutual inclinations. KOI-126 and Kepler-16 (the first circumbinary planet) were landmark photodynamical fits.

Worked example: spotting an unseen planet in the O−C

Suppose a transiting planet b has a clean ephemeris, P_b = 9.29 d, established over the first season. As more transits accumulate, the O−C residuals stop scattering around zero and instead trace a clear sinusoid with amplitude A ≈ 5 minutes and a period of roughly 300 days. What can you say?

First, the variation is real and periodic, not noise — so a perturber exists. Second, its 300-day super-period, combined with P_b, pins down where the perturber's period must lie to produce that beat through the resonant-super-period relation; you solve for P_c. Third, the 5-minute amplitude, fed into an N-body model, fixes the perturber's mass to within a factor set by the mass-eccentricity degeneracy — for a near-resonant sub-Neptune perturber this lands the mass in the few-to-tens-of-Earth-mass range. And critically: none of this required planet c to transit. If c happens not to cross the stellar disk from our viewpoint, it leaves no light-curve dip — yet its full gravitational fingerprint is written into b's schedule.

This is exactly the story of Kepler-19. In 2011, Sarah Ballard and collaborators reported that the transiting Kepler-19b (period 9.29 d) showed TTVs of about 5 minutes with a roughly 300-day super-period, betraying a non-transiting companion they named Kepler-19c. It was the first planet discovered solely through the transit-timing perturbations it imposed on a transiting neighbour — the exoplanet analogue of predicting Neptune from Uranus's orbital irregularities.

Discovery and the missions that made it routine

The idea long predates exoplanets. In 1846 Urbain Le Verrier and John Couch Adams independently predicted Neptune's position from the timing and positional anomalies of Uranus's orbit; Johann Galle then found Neptune within a degree of the prediction. The same logic — infer an unseen mass from the perturbation it imposes on a body you can track — is the conceptual root of TTV.

For exoplanets, the technique was set out theoretically in 2005 in two simultaneous papers: Eric Agol, Jason Steffen, Re'em Sari and Will Clarkson, and independently Matthew Holman and Norman Murray, who showed that a transiting hot Jupiter's mid-times could be shifted by minutes by an Earth-mass companion near resonance — flagging TTV as a route to terrestrial-mass detection years before such planets were transit-detectable. Early ground-based attempts produced false alarms, but NASA's Kepler mission (launched 2009) was transformative: a continuous, ~30-minute-cadence stare at one field for four years delivered the long baselines and uniform photometry the method demands. Kepler-9 (Holman et al. 2010) was the first system with clearly anti-correlated TTVs and a dynamical mass; Kepler-19c (Ballard et al. 2011) the first non-transiting discovery; and large catalogues followed (e.g. Mazeh et al. 2013, Hadden & Lithwick 2014/2017) that delivered masses and densities for dozens of small planets. The TRAPPIST-1 system — seven Earth-sized planets in a resonant chain around an M dwarf — is the modern showcase: its rich TTVs, refined with Spitzer and now JWST timing, pin down all seven masses to a few percent.

TTV among the detection methods

TTV is best understood beside the other ways we find and weigh planets. It is the only method on this list whose strength grows for the small, low-mass, faint-star systems where radial velocity fails, and the only one that routinely detects bodies that never cross the stellar disk.

Method	Measures	Best for	Detects non-transiting planet?
Transit photometry	Radius (depth), period	Aligned, short-period planets	No
Radial velocity	m·sin i, period, eccentricity	Bright stars, massive/close planets	Yes (any orbit)
Transit timing variations	Mass, eccentricity (dynamical)	Resonant multis around faint stars	Yes — its signature feature
Microlensing	Mass ratio, projected separation	Cold, wide planets toward bulge	n/a (one-off event)
Direct imaging	Luminosity, orbit, spectrum	Young, wide, self-luminous giants	n/a
Astrometry	Mass, full 3D orbit	Nearby stars, wide planets	Yes

Its closest cousin is radial velocity: both read out Newtonian gravity and both deliver mass. The difference is the channel. RV needs a high-resolution spectrum and so needs a bright star; TTV needs only precise transit photometry and a long baseline, so it thrives on faint Kepler targets where RV is impossible. The two are complementary — and when both are available, a joint fit is the gold standard.

Related timing phenomena and variants

• Transit duration variations (TDVs). Mutual gravity can also change how long a transit lasts and the chord across the star, by altering the planet's sky-projected velocity or the orbit's inclination. TDVs carry information about mutual inclinations and, in principle, about exomoons (Kipping 2009).
• Eclipse timing variations (ETVs). The same logic applied to eclipsing binary stars: a third body (a circumbinary planet) periodically shifts the eclipse times. This is how some circumbinary worlds and post-common-envelope planet candidates are flagged.
• Light-travel-time (Rømer) effect. If the transiting system orbits a distant stellar companion, the changing path length to Earth shifts all transit times together with the wide binary's period — a timing signal that is not internal planetary perturbation.
• Orbital decay. A slow, secular (non-periodic) shortening of the period as a hot Jupiter spirals inward by tidal dissipation. The textbook case is WASP-12b, whose transit times have been observed to advance, implying a period decreasing by about 29 milliseconds per year — a quadratic trend in the O−C diagram, distinct from a periodic TTV.
• Apsidal precession. Slow rotation of an eccentric orbit's line of apsides produces a long-period timing trend and an anti-phased pattern between primary transit and secondary eclipse.

Common misconceptions and subtleties

• "A TTV means the planet's period is changing." Not in the periodic case. The orbit is being perturbed back and forth; the long-term mean period is essentially constant. A genuinely changing period (orbital decay) shows up as a parabolic, not oscillatory, O−C curve.
• "More transits always give a better mass." Only if the baseline spans the super-period. Sampling a small arc of a multi-year TTV cycle can mimic a linear trend or alias to the wrong amplitude; the constraint tightens dramatically once you cover at least one full P_TTV.
• "TTV mass equals the true mass." The amplitude depends on a degenerate combination of mass and free eccentricity. Studies have found TTV masses running systematically a little low versus radial-velocity masses for overlapping samples — a reminder that breaking the mass-eccentricity degeneracy needs either resonant-pair structure, a long baseline, or complementary RV.
• "Any timing wobble is a planet." Starspots crossed during transit, a blended eclipsing binary, the Rømer delay from a stellar companion, and red noise can all distort mid-times. The planetary signature is the coherent, anti-correlated, resonant-super-period pattern shared between two transiting planets — that is what cannot be faked.
• "It only works near exact resonance." Resonance amplifies the signal, but any sufficiently massive, sufficiently close perturber leaves a TTV. The method simply favours near-resonant pairs because that is where small planets become detectable.