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

Daniel A. Yahalomi, David Kipping

TL;DR

This work tackles the multi-modal interpretation of transit timing variations in single-planet systems by mapping the perturbing-planet solution space with extensive $N$-body simulations using $\texttt{TTVFast}$ and by deriving analytic predictions for dominant near-MMR super-periods and synodic periods in the low-eccentricity regime. It then prescribes a practical Bayesian workflow that partitions the orbital-period space into uniform priors aligned with resonant windows, enabling efficient sampling and model averaging across multiple local fits to recover the global posterior. A model-comparison framework between perturbing planets and moons is presented, usinglocal evidences and demographics-informed priors, to distinguish between these scenarios. A key finding is the exoplanet edge: TTVs with dominant periods faster than half the perturber–planet period are not observed, implying a minimum detectable TTV regime and providing a diagnostic for unseen companions. These results yield a concrete, scalable methodology to infer unseen perturbers and to constrain planetary architecture from TTV data in current and future surveys.

Abstract

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 $\texttt{TTVFast}$ 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. 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.

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

TL;DR

This work tackles the multi-modal interpretation of transit timing variations in single-planet systems by mapping the perturbing-planet solution space with extensive -body simulations using and by deriving analytic predictions for dominant near-MMR super-periods and synodic periods in the low-eccentricity regime. It then prescribes a practical Bayesian workflow that partitions the orbital-period space into uniform priors aligned with resonant windows, enabling efficient sampling and model averaging across multiple local fits to recover the global posterior. A model-comparison framework between perturbing planets and moons is presented, usinglocal evidences and demographics-informed priors, to distinguish between these scenarios. A key finding is the exoplanet edge: TTVs with dominant periods faster than half the perturber–planet period are not observed, implying a minimum detectable TTV regime and providing a diagnostic for unseen companions. These results yield a concrete, scalable methodology to infer unseen perturbers and to constrain planetary architecture from TTV data in current and future surveys.

Abstract

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 -body simulations with 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. 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.

Paper Structure

This paper contains 16 sections, 16 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Numerical Simulations
  3. TTVFast
  4. Lomb-Scargle Periodogram
  5. Removing Non-Observable Systems
  6. Analytic TTV Periods
  7. External Perturber
  8. Internal Perturber
  9. Chaotic Region
  10. Discussion
  11. Combining the Analytic and Numerical Solutions
  12. The Exoplanet Edge
  13. Parameter Estimation and Model Selection
  14. Sampling the Global Posterior
  15. Planets vs. Moons
  16. ...and 1 more sections

Figures (5)

  • Figure 1: Peak TTV periods recovered via Lomb-Scargle (LS) periodograms fit to transit times simulated with TTVFast. Each column shows a different set of planetary masses. Each row shows a different range of eccentricities. TTVs split into three groups: (i) systems that fail the hill and/or chaos stability criteria (red x markers), (ii) systems with unreliable TTV amplitudes less than 1 minute (orange x markers), (iii) and those that pass both tests and are thus observable TTVs (blue circle markers).
  • Figure 2: Peak TTV periods recovered via Lomb-Scargle (LS) periodograms fit to TTVFast simulated systems that pass stability and amplitude tests for all combined masses simulated in Figure \ref{['fig: orbital_landscape_mass_sep']}. [Top:]TTVFast systems initialized with zero eccentricity [Bottom:]TTVFast systems initialized with eccentricities between 0 and 0.2. We also show the chaotic boundaries as defined by Deck2013 for two Earth mass and two Jupiter mass planets, respectively, around a Solar mass star.
  • Figure 3: Analytic TTVs from super-period equation and their aliases for some $j:k$ near MMR TTVs for external perturbers. Also analytic TTVs from synodic period and its alias. [Top] $1:k$ super-periods shown. [Bottom] Aliases of first-order super-periods shown. Here we show how first-order super-periods can induce all the same TTV periods as higher order super-periods expected only for large eccentricities.
  • Figure 4: Analytic TTVs from super-period equation and their aliases for some $j:k$ near MMR TTVs for internal perturbers. [Top] $1:k$ super-periods shown. [Bottom] Aliases of first-order super-periods shown. Here we show how first-order super-periods can induce all the same TTV periods as higher order super-periods expected only for large eccentricities.
  • Figure 5: Analytic and numerical results for TTV period vs. orbital period ratio space of planet-planet TTVs. The analytic model, explained solely by first-order mean-motion resonant super periods and the synodic period (and their aliases), fits the numerical results very well. Numerical simulations combine all observable TTVs that passed tests explained in Figure \ref{['fig: orbital_landscape_mass_sep']}. We also show the chaotic boundaries as defined by Deck2013 for two Earth mass and two Jupiter mass planets, respectively, around a Solar mass star. Also recommended orbital period ratio prior boundaries for efficient and accurate modeling.