Mapping the Orbital Landscape of Perturbing Planet Solutions for Single-Planet Systems with TTVs

There are now thousands of single-planet systems observed to exhibit transit timing variations (TTVs), yet we largely lack any interpretation of the implied masses responsible for these perturbations. Even when assuming these TTVs are driven by perturbing planets, the solution space is notoriously multi-modal with respect to the perturber’s orbital period and there exists no standardized procedure to pinpoint these modes, besides from blind brute force numerical efforts. Using $N$-body simulations with TTVFast and focusing on the dominant periodic signal in the TTVs, we chart out the landscape of these modes and provide analytic predictions for their locations and widths, providing the community with a map for the first time: the TTV circus tent diagram. We then introduce an approach for modeling single-planet TTVs in the low-eccentricity regime, by splitting the orbital period space into a number of uniform prior bins over which there aren’t these degeneracies. We show how one can define appropriate orbital period priors for the perturbing planet in order to sufficiently sample the complete parameter space. We demonstrate, analytically, how one can explain the numerical simulations using first-order near mean-motion resonance super-periods, the synodic period, and their aliases – the expected dominant TTV periods in the low-eccentricity regime. Using a Bayesian framework, we then present a method for determining the optimal solution between TTVs induced by a perturbing planet and TTVs induced by a moon.

💡 Research Summary

The paper addresses the long‑standing problem that thousands of single‑transiting exoplanet systems exhibit measurable transit‑timing variations (TTVs) but lack a systematic interpretation of the unseen perturbers that must be responsible. Assuming the TTVs arise from gravitational interactions with an unseen planet, the authors note that the solution space is highly multimodal in the perturber’s orbital period because many near‑mean‑motion resonances (MMRs) and their aliases can produce the same dominant TTV frequency. Traditional approaches rely on blind, computationally expensive brute‑force searches that often miss important modes.

To overcome this, the authors combine large‑scale N‑body integrations (using TTVFast) with analytic theory to map the entire “TTV circus‑tent diagram”: a two‑dimensional landscape of TTV period versus perturber‑to‑transiting‑planet period ratio. They first derive the dominant low‑eccentricity TTV periods analytically. For a pair of planets with periods P₁ (transiting) and P₂ (perturber) near a j : k resonance, the super‑period is

 P_sup = 1 / |j/P₁ − k/P₂|,

and the synodic (or “chopping”) period is

 P_syn = 1 / |1/P₁ − 1/P₂|.

Because real observations sample the TTV signal only at integer multiples of P₁, the measured frequency ν is aliased according to

 ν = |1/P_TTV + m/P₁|, m ∈ ℤ,

or equivalently the observed aliased period

 P̄_TTV = 1 / |1/P_TTV + m/P₁|.

The authors show that for low eccentricities, first‑order resonances (|j − k| = 1) dominate, and that their aliases (with m = −1, −2, …) generate the same apparent TTV periods as higher‑order resonances that would only be visible for large eccentricities. They illustrate this for external perturbers (P₂/P₁ ≈ 1.5–10) and internal perturbers (P₂/P₁ ≈ 0.1–0.8), plotting the locations of 1 : k and k‑1 : k super‑periods and their aliases, as well as the synodic period and its alias. A key result is that the alias of the 1 : 2 super‑period becomes equal to half the perturber’s orbital period for P₂/P₁ > 4, and the alias of the synodic period equals the perturber’s period for P₂/P₁ > 2.

The numerical component consists of thousands of two‑planet N‑body integrations spanning the same period‑ratio ranges. For each simulation the dominant TTV frequency is extracted via Fourier analysis and plotted on the same diagram. The simulated points fall precisely on the analytically predicted resonance lines and alias curves, confirming the theory. The authors also discuss the chaotic boundary identified by Deck et al. (2013): when the period ratio satisfies

 P₂/P₁ ≲ 1 + 2.2 ε_p^{2/7},

with ε_p the total planet‑to‑star mass ratio, first‑order resonances become chaotic and the TTV signal becomes unstable. This defines a “chaotic region” that must be avoided when constructing priors.

Armed with the circus‑tent map, the authors propose a practical prior‑construction scheme. The map’s mode centers and widths are used to divide the perturber period axis into uniform bins that each contain a single, non‑degenerate mode. Separate nested‑sampling runs are then performed within each bin, guaranteeing that all plausible solutions are explored without wasting computational effort on empty regions.

Finally, the paper tackles the distinction between planet‑planet TTVs and planet‑moon (exomoon) TTVs. In a Bayesian framework, two models are defined: (i) a perturbing planet model with the period‑bin priors described above, and (ii) an exomoon model that predicts a short‑period “chopping” signal superimposed on a small amplitude. By computing the Bayesian evidence Z for each model, the authors show that a log‑evidence difference Δln Z > 5 decisively favors one scenario over the other. The method is demonstrated on synthetic data, where the correct model is recovered with high confidence.

In summary, this work delivers the first comprehensive analytic‑numeric map of the multimodal TTV solution space for single‑transiting systems, introduces a rigorously justified prior‑binning strategy to fully sample that space, and provides a Bayesian model‑comparison tool to separate planetary from lunar origins of TTVs. These contributions substantially advance the ability to infer unseen companions from TTVs, turning a previously intractable problem into a tractable, systematic procedure.