Transit Timing Variation

The transit clock is not a clock

A single transit is a beautiful observation. Each time a planet crosses in front of its host star, the integrated stellar flux drops by an amount proportional to the area ratio (R_p/R_s)², the dip lasts for a chord-traversal time, and the centre of the dip occurs at a well-defined moment we call the mid-transit time. Stack many transits, fit a straight line through the times, and you have a linear ephemeris: T_n = T_0 + n·P, where n is the transit number and P the orbital period. The textbook picture is that subsequent transits arrive on this grid forever — that is what "periodic orbit" means in the two-body problem.

Almost no real planetary system contains only two bodies. The moment you place a second planet around the same star, the orbits couple. The transiting planet still completes roughly one revolution per period, but the period itself, and the precise moment of the transit, jitter as the perturber pulls the planet forward or backward, raises or lowers its eccentricity, or precesses its argument of pericentre. Plot the observed minus computed (O−C) transit times against transit number and you get an oscillating residual rather than a flat line. That residual is the transit timing variation.

Two parallel predictions in 2005

The effect had been noted in the pulsar timing literature for years — the pulsar PSR B1257+12 planets were discovered in 1992 entirely from pulse-arrival-time residuals — but its application to optical transits had to wait for the photometric precision delivered by CoRoT and Kepler. Two papers appeared independently in early 2005:

• Holman & Murray (2005), "The use of transit timing to detect terrestrial-mass extrasolar planets". They worked out that an Earth-mass perturber near a 2:1 mean-motion resonance with a hot Jupiter could produce TTVs of order one minute, well within the reach of the upcoming Kepler mission.
• Agol, Steffen, Sari & Clarkson (2005), "On detecting terrestrial planets with timing of giant planet transits". A parallel analytic treatment with worked-out scaling laws for the resonant amplitude and for non-resonant chopping.

Both papers made the same critical observation: TTVs are most sensitive precisely where radial velocity is least sensitive — to small planets in resonant configurations. RV scales with m_p/M_*; TTV scales with m_pert/M_* divided by the fractional resonance offset Δ, with the Δ in the denominator acting as an amplification factor.

Why mean-motion resonance is the megaphone

Consider two transiting planets, inner with period P_1 and outer with period P_2 > P_1. The synodic period — the interval between conjunctions, when they are aligned with the star — is

P_syn = 1 / (1/P_1 − 1/P_2)

At every conjunction the planets exchange a small impulse. Far from resonance, successive conjunctions happen at uncorrelated orbital phases of each planet, so the impulses average to nearly zero. Near a j:k mean-motion resonance — where P_2/P_1 ≈ j/k for small integers j, k — conjunctions happen at very nearly the same phase each time, and the impulses add coherently. The TTV amplitude scales as

A_TTV ≈ P · (m_pert / M_star) / |Δ|     (first-order resonance)

where Δ = (P_2/P_1)·(k/j) − 1 is the fractional distance from exact commensurability. A planet 1% off resonance amplifies the signal by a factor of 100 compared with the bare m_pert/M_* prediction.

The waveform is also distinctive. Near a first-order resonance the TTVs of the two planets are anti-correlated on a long timescale (the "super-period" P_sup = 1/|Δ|·P_1) and modulated by a faster "chopping" signal at the conjunction cadence. Fitting the super-period gives Δ; fitting the chopping amplitude separates the two masses unambiguously. Lithwick, Xie & Wu (2012) wrote down the analytic formulae that turn the four observable amplitudes and phases into m_1, m_2, and the two free combinations of eccentricity vectors near the resonance.

Kepler-9: the first confirmed TTV pair

Kepler-9 b and c, reported by Holman and 39 co-authors in Science on 26 August 2010, were the first system in which TTVs were detected and used to weigh transiting planets. The host is a Sun-like G2V star at 619 parsecs. The two planets have periods of approximately 19.2 and 38.9 days respectively — sitting just outside 2:1 — and radii of about 9.4 and 9.2 Earth radii (Saturn-class). The decisive observation: when Kepler-9 b transited early, Kepler-9 c transited late, and vice versa. Plotted side by side, the two O−C curves were mirror images of each other.

The amplitude was striking: about 4 minutes peak-to-peak for b, and 39 minutes for c. The ratio of amplitudes gave the mass ratio directly — Kepler's third law plus N-body integration delivered m_b ≈ 0.25 M_J and m_c ≈ 0.17 M_J without a single radial-velocity measurement. Subsequent HIRES RV data confirmed the masses to within the TTV uncertainties.

The detection settled the debate. The Holman-Murray and Agol et al. predictions were not theoretical curiosities; they were the basis for a viable mass-measurement technique that Kepler had just realised in its first year of operation.

TRAPPIST-1: seven masses from one method

The most spectacular application of TTV inversion to date is the TRAPPIST-1 system, an M8V ultracool dwarf at 12.5 parsecs hosting seven roughly Earth-sized transiting planets. The seven periods are in a chain of three-body Laplace resonances; consecutive pairs are near 8:5, 5:3, 3:2, 3:2, 4:3 and 3:2. Each planet's transit times are perturbed by its two nearest neighbours and (weakly) by the further chain.

Grimm et al. (2018) ran a joint N-body fit to 284 transit times collected by Spitzer, K2, and the ground-based TRAPPIST and SPECULOOS facilities. They recovered masses for all seven planets:

Planet	Period (d)	Mass (M⊕)	Radius (R⊕)	Density (g/cc)
TRAPPIST-1 b	1.510	1.017 ± 0.154	1.121	3.97
TRAPPIST-1 c	2.421	1.156 ± 0.142	1.095	4.87
TRAPPIST-1 d	4.050	0.297 ± 0.039	0.784	3.39
TRAPPIST-1 e	6.099	0.772 ± 0.075	0.910	5.65
TRAPPIST-1 f	9.207	0.934 ± 0.078	1.046	4.51
TRAPPIST-1 g	12.353	1.148 ± 0.095	1.148	4.20
TRAPPIST-1 h	18.767	0.331 ± 0.049	0.773	3.97

The precision on the masses is 5-10 percent — better than what radial velocity could deliver for a system this faint. The resulting bulk densities place all seven planets in the rocky regime, with subtle differences (b, d, h slightly less dense; c, e, f, g closer to Earth) consistent with either ice or volatile envelopes for the lower-density worlds. None of this would have been possible without the resonant chain doing the amplifying work.

Detecting planets that never cross the disk

Because the perturbation does not care whether the perturber happens to cross the star, TTVs can flag a planet that never transits. The classical case is Kepler-46b (KOI-872), where periodic timing residuals on a transiting hot-Saturn could only be reproduced by a second, non-transiting planet on a slightly inclined 57-day orbit. Several other Kepler systems have similar evidence for hidden companions inferred purely from TTV morphology.

The hidden-planet inference is degenerate in two important ways. First, mass and eccentricity correlate near resonance: a more massive low-eccentricity planet can mimic a less massive eccentric one. Second, the orbital inclination of the non-transiting planet is unconstrained beyond "not in our line of sight". Both degeneracies can be broken by combining TTVs with TDVs (which constrain inclination via the impact-parameter drift) or with high-precision radial velocity, but for the faintest Kepler hosts neither follow-up is feasible.

Worked example: predicting the TTV amplitude

Take a hot-Jupiter system. Inner planet: hot Jupiter with P_1 = 3.0 days, mass M_J. Suspected companion: Earth-mass terrestrial planet near 2:1 outer resonance, period P_2 = 6.05 days, so Δ = (P_2/P_1)/2 − 1 = (6.05/3.0)/2 − 1 = 0.00833 (0.83% off resonance). Host star mass 1 M_☉.

m_pert / M_star = (1 M⊕) / (1 M☉) = 3.0 × 10⁻⁶
A_TTV  ≈ P_1 · (m_pert/M_star) / |Δ|
       = 3.0 d × 3.0×10⁻⁶ / 0.00833
       = 1.08 × 10⁻³ d
       ≈ 1.6 minutes

For a system that Kepler observed for 4 years (~485 transits at P_1 = 3 d), a coherent 1.6-minute signal is well above the photon-limited timing precision of about 10-30 seconds for a bright (V = 11) host. The same Earth-mass planet at twice the resonance offset (Δ = 1.67%) would produce only 48 seconds of TTV; at ten times the offset, just 9.6 seconds — below the detection floor for that host. The 1/Δ scaling is what makes resonant amplification so important: it is the difference between detection and silence.

Other causes of timing residuals

Not every wiggle in an O−C plot is a planet-planet interaction. Real surveys must rule out:

• Light-travel-time (LTT, Rømer) effect. If the transiting planet's host star is itself orbiting a distant companion, the finite speed of light makes transits arrive earlier when the star is closer and later when it is farther. The signal is strictly sinusoidal with the outer orbital period and amplitude (a_*·sin i)/c. A long-period brown-dwarf or stellar companion is the classic source.
• Apsidal precession. For an eccentric transiting planet, the line of pericentre rotates due to general relativity, stellar oblateness J_2, or a perturber. Precession shifts the time between primary and secondary transits and creates a slow drift in mid-transit times.
• Stellar activity. A starspot crossed during transit shortens the apparent dip on one side of the transit centre and lengthens it on the other, biasing the inferred mid-time by tens of seconds. Repeated spot crossings produce a quasi-periodic signal at the stellar rotation period.
• Tidal decay. Hot Jupiters lose orbital angular momentum to their host stars via tidal dissipation; the period shrinks secularly. WASP-12 b shows a parabolic O−C with d P/dt ≈ −29 ms/yr, the cleanest case yet of orbital decay.
• Systematic timing offsets. Detrending choices, transit-model parametrisation, and clock-corrections between observatories all add 10-100 second systematics. A real planet-planet signal must survive a complete reanalysis with independent pipelines.

The inverse problem and its degeneracies

Given a list of measured transit times {t_n} with uncertainties {σ_n}, the forward problem integrates an assumed N-body system and predicts the times. The inverse problem — recovering masses and elements from the data — is high-dimensional and non-convex. Standard practice now is to:

1. Use an analytic approximation (Lithwick-Xie-Wu near first-order resonance, or the chopping formulae) to get a starting estimate for the masses and the eccentricity vector free parameters.
2. Switch to a full N-body integration with REBOUND or TTVFast (Deck et al. 2014) for the likelihood, propagating the system through the full data baseline at adaptive timestep.
3. Sample the posterior with MCMC (emcee) or nested sampling (dynesty, MultiNest), typically with weakly informative priors on planet density to break the mass-eccentricity degeneracy when independent radius measurements are available.

The principal degeneracy is between the perturber mass and its eccentricity at the resonance: both increase the TTV amplitude. Far from resonance the chopping signal partially breaks the degeneracy (its amplitude depends only on mass), but the cleanest break comes from a longer time baseline that captures the libration period directly. This is why the precision on TRAPPIST-1 masses kept improving every year Spitzer monitored the system, even after the resonant phases were nominally pinned.

TTV versus the other clocks

Method	Measures	Best at	Limitation
Transit timing (TTV)	Perturber mass, e	Near MMR, faint hosts	Needs multi-transiting + resonance
Transit duration (TDV)	Plane evolution, i drift	Out-of-plane perturbers	Smaller amplitude than TTV
Radial velocity (RV)	m sin i	Bright stars, all geometries	Photon-limited for faint M-dwarfs
Astrometry (Gaia)	Mass, full inclination	Long periods, bright stars	Limited time baseline so far
Pulsar timing	Mass, e, ω of NS companions	Sub-Earth-mass on neutron stars	Only applies to pulsars
Direct imaging	Photometry, separation	Wide, young, self-luminous	Cannot weigh the planet alone

The methods are largely complementary. TTV and RV are most powerful in different parts of parameter space: RV needs Doppler precision the star photons can support; TTV needs photometric precision the timing precision can resolve. A bright sun-like star with a transiting hot-Neptune is well served by either. A faint M-dwarf with a chain of resonant terrestrial planets is uniquely well served by TTV.

From Kepler to TESS and PLATO

Kepler's prime mission (2009-2013) discovered roughly 100 systems with significant TTVs out of about 2,300 confirmed planets. K2 (2014-2018) and TESS (2018-) extended the catalogue toward brighter hosts, though TESS's 27-day-per-sector baseline limits its sensitivity to long-period perturbers. CHEOPS provides exquisite individual transit timings for known multi-planet systems, used to refine TTV solutions year by year. PLATO, scheduled for launch in 2026, will deliver a Kepler-class baseline on much brighter stars — many of the best TTV systems are expected to come from PLATO's nominal 2-year long-stare fields.

The community's running list of confirmed TTV systems is maintained at the Visual TTV catalogue and at the NASA Exoplanet Archive. The Kepler TTV catalogue (Holczer et al. 2016) systematically searched all 2,599 Kepler Objects of Interest for periodic O−C signals and tabulated more than 130 with detections above 5σ.

Common pitfalls

• Calling a flat O−C "no perturber". Absence of TTV evidence is not evidence of absence: a perturber far from resonance, or one whose super-period exceeds the observing baseline, may simply not yet have produced a detectable signal. The right statement is an upper limit on m_pert at each resonance offset.
• Confusing first- with second-order resonance. The 2:1 and 3:2 are first-order; the 3:1 and 5:3 are second-order with amplitude scaling as Δ⁻² and a different angular dependence. Applying first-order analytic formulae to a 3:1 system gives nonsense.
• Ignoring the eccentricity-mass degeneracy. Near a first-order MMR, only the combination z_free = e·exp(iω) − e'·exp(iω') is constrained well. A mass quoted without acknowledging this is misleading; full N-body MCMC posteriors are mandatory.
• Treating starspots as random noise. Quasi-periodic spot crossings at the rotation period can mimic a TTV near rotation/orbit commensurability. Joint modelling of the in-transit residual is required.
• Linear-ephemeris bias. Fitting a straight line to all transits and then computing O−C against that line absorbs the long-term trend (e.g. tidal decay) into the fit. Quadratic or higher-order ephemerides should be tested; the AIC/BIC tells you whether the residuals truly demand a periodic explanation.