The Exoplanet Edge: Planets Don't Induce Observable TTVs with a Dominant TTV Period Faster than Half their Orbital Period

TL;DR

Transit timing variations (TTVs) encode gravitational perturbations in exoplanetary systems but are often degenerate. This work identifies the exoplanet edge—a lower bound on the dominant TTV period for distant perturbers—via TTVFast simulations and Lomb-Scargle analysis, showing that aliasing and tidal effects yield dominant periods at $P_{ ext{pert}}$ or $P_{ ext{pert}}/2$. The authors connect these signals to observable patterns in Kepler data (Holczer2016), uncovering 13 two-planet outliers and 695 single-planet TTVs that likely imply distant companions or moons. They illustrate the edge with Kepler-16, Kepler-1513 b, and Solar System analogs, and discuss how measurements can guide future distant-planet searches using RV and Gaia astrometry, enhancing our ability to detect elusive exomoons and wide companions.

Abstract

Transit timing variations (TTVs) are observed for exoplanets at a range of amplitudes and periods, yielding an ostensibly degenerate forest of possible explanations. We offer some clarity in this forest, showing that systems with a distant perturbing planet preferentially show TTVs with a dominant period equal to either the perturbing planet's period or half the perturbing planet's period. We demonstrate that planet induced TTVs are not expected with dominant TTV periods below this exoplanet edge (lower period limit) and that systems with TTVs that fall below this limit likely contain additional mass in the system. We present an explanation for both of these periods, showing that both aliasing of the conjunction induced synodic period and the near $1:2$ resonance super-period and tidal effects induce TTVs at periods equal to either the perturber's orbit or half-orbit. We provide three examples of known systems for which the recovered TTV period induced by a distant perturbing planet is equal to the perturber's orbital period or half its orbital period. We then investigate \textit{Kepler} two-planet systems with TTVs and identify 13 two-planet systems with TTVs below this TTV period lower limit -- thus potentially uncovering the gravitational influence of new planets and/or moons. We conclude by discussing how the exoplanet edge effects can be used to predict the presence of distance companion planets, in situations where TTVs are detected and where nearby companions can be ruled out by additional observations, such as radial velocity data.

Figures (5)

• Figure 1: 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. [Top:]TTVFast systems initialized with zero eccentricity [Middle:]TTVFast systems initialized with eccentricities between 0 and 0.4. [Bottom:]TTVFast systems initialized with any physical eccentricity. For circular planets, TTVs with a period equal to the perturber's period dominates for distant perturbers. When moderately eccentric systems are sampled, TTV periods commensurate with the perturber's orbit and half orbit are the dominant signals. For highly eccentric systems, the parameter space becomes more degenerate, but the exoplanet edge provides a lower limit for TTV period.
• Figure 2: Schematic demonstrating how conjunction induced TTVs produce an observable TTV period equal to $P_\mathrm{pert}$ when $P_\mathrm{pert}/P_\mathrm{trans} > 2$. For an observer to the right of the diagram, as shown above, during the first half of the perturber's orbit, it induces early TTVs (pulling planet in same direction as orbit) and for the second half of its orbit it induces late TTVs (pulling planet in opposite direction as orbit). Thus the observed TTV will have a period equal to $P_\mathrm{pert}$.
• Figure 3: Schematic showing the rotating-tidal distortion induced lower exoplanet edge with an observed TTV at period $P_\mathrm{pert}/2$. The gravitational influence of the distant outer perturbing planet tidal distorts the orbit of the transiting planet, with a rotational period equal to $P_\mathrm{pert}$
• Figure 4: Two-planet KOIs from the Holczer2016 catalog that exhibit TTVs with $\Delta$BIC $>$ 6 overplot on top of TTVFast simulated TTVs. Marker size is indicative of TTV amplitude. TTVs split into three populations: (i) TTV periods that are greater than the period of the pertuber and thus likely near MMR induced TTVs, (ii) TTV periods consistent with exoplanet edges and thus likely driven by either aliased conjunction induced TTVs or tidal distortions, and (iii) TTV periods that are anomolously fast thus indicating the likely presence of additional mass in the system.
• Figure 5: Hypothetical perturbing planet periods, in single-planet systems from Holczer2016 with TTVs ($\Delta$BIC $>$ 6), if the observed TTVs are [left] upper and [right] lower exoplanet edge TTVs. Marker size is indicative of TTV amplitude.