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TOI-4504: Exceptionally large Transit Timing Variations induced by two resonant warm gas giants in a three planet system

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read TOI-4504 c shows the largest transit timing variations ever observed—about two days of wobble—and the wobble is caused by a hidden Jupiter-mass planet in a 2:1 resonance.

desk verdict A genuine TTV record with a secure perturber, but the resonant architecture rests on a 41 d signal that the RVs alone barely support. read the letter →

arxiv 2412.05609 v1 pith:EYAWMZLH submitted 2024-12-07 astro-ph.EP

Michaela Vítková , Rafael Brahm , Trifon Trifonov , Petr Kabáth , Andrés Jordán , Thomas Henning , Melissa J. Hobson , Jan Eberhardt
show 19 more authors
Marcelo Tala Pinto Felipe I. Rojas Nestor Espinoza Martin Schlecker Matías I. Jones Maximiliano Moyano Susana Eyheramendy Carl Ziegler Jack J. Lissauer Andrew Vanderburg Karen A. Collins Bill Wohler David Watanabe George R. Ricker Roland Vanderspek Sara Seager Joshua N. Winn Jon M. Jenkins Marek Skarka
This is my paper · ORCID
classification astro-ph.EP
keywords transittimingvariationsmean-motionresonancewarmJupiternon-transitingplanetexoplanetdynamicsTOI-4504radialvelocity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the transiting planet TOI-4504 c is the most extreme known case of transit timing variations, with its transits arriving up to roughly two days early or late in a regular ~930-day cycle. Using those timing variations together with radial-velocity measurements, the authors uncover a previously unknown non-transiting planet, TOI-4504 d, that is causing the wobble. The system contains three planets: a hot sub-Neptune and two warm Jupiter-mass gas giants locked in a 2:1 mean-motion resonance. The result matters because strongly resonant giant-planet pairs are direct evidence that these planets migrated smoothly inward through the protoplanetary disk.

What carries the argument

The central object is the transit timing variation signal of TOI-4504 c: a roughly sinusoidal modulation of transit arrival times with ~930-day super-period and ~2-day peak-to-node amplitude, produced by the gravitational tug of the inner non-transiting planet d. The argument is carried by a Jacobi-coordinate N-body dynamical model that simultaneously fits space-based transit times and ground-based radial velocities, using a fast analytic transit-timing prescription for the TTV computation and a resonance-angle diagnostic ($\theta_1 = \lambda_c - 2\lambda_d + \omega_c$); the libration of $\theta_1$ around $0^\circ$ is the key signature that the two Jupiters are genuinely trapped in a 2:1 mean-motion resonance.

What would settle it

A decisive test is to obtain high-precision radial-velocity series around the predicted 40.56-day signal of TOI-4504 d: if the 41.2-day periodogram peak does not persist above the 1% false-alarm level once the ~103 m/s jitter is modeled together with stellar activity indicators, the planetary interpretation of the TTV signal is weakened; conversely, a clean detection at ~91 m/s semi-amplitude would confirm the hidden planet.

Watch

Extended reading notes

Core claim

The paper claims that the warm Jupiter TOI-4504 c, with osculating period $P = 82.54 \pm 0.02$ days and dynamical mass $m = 3.77 \pm 0.18\,M_{\rm J}$, shows transit timing variations with peak-to-node semi-amplitude of about 2 days and a super-period of about 930 days—the largest TTV amplitude ever recorded. The joint TTV plus radial-velocity dynamical model identifies the perturber as an interior non-transiting planet TOI-4504 d, with period $P = 40.56 \pm 0.04$ days and dynamical mass $m = 1.42^{+0.07}_{-0.06}\,M_{\rm J}$, placing the pair in a first-order 2:1 mean-motion resonance. The paper also reports a transiting sub-Neptune TOI-4504 b with period $2.42614$ days and radius $2.69 \pm 0.19\,R_{\oplus}$, whose mass is unconstrained by the data. Long-term N-body integrations of the giant pair show stability over 10 million years, with the first-order resonance angle $\theta_1$ librating around $0^\circ$, which the authors take as evidence of an active 2:1 resonance rather than a mere near-commensurability.

Load-bearing premise

The 41.2-day radial-velocity signal attributed to TOI-4504 d is only marginally significant, crossing just the 10% false-alarm threshold, and the fitted radial-velocity jitter of 103 m/s is as large as the 91 m/s semi-amplitude of planet d itself; the paper's masses and periods for d therefore depend on the assumption that this unexplained jitter does not bias the two-planet solution.

Editorial extensions

If this is right

  • If the dynamical solution is correct, TOI-4504 becomes the benchmark for the largest observed transit-timing amplitude, roughly twice the previous record holder Kepler-30 b.
  • The pair of warm Jupiters in a 2:1 resonance adds to the small sample of resonant giant-planet systems whose growing occurrence supports smooth, disk-driven migration into resonance.
  • Because the perturber d is non-transiting, an observer whose line of sight made d transit would see TTVs more than 50% larger than those seen for c, making the resonance an even stronger timing signal from other viewing geometries.
  • The predicted transit times over the next decade give a direct observational target: future photometry near those windows can test the model's forecasts.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the large libration amplitude (~65–73°) persists over many super-periods, the TOI-4504 pair could become a probe of resonant libration damping, which is expected to slowly shrink the libration amplitude over time.
  • The unmodeled ~100 m/s radial-velocity jitter is comparable in size to planet d's own signal, so a plausible extension is that additional planets or persistent stellar activity hide in the data; higher-precision velocities would either sharpen d's mass or reveal a fourth body.
  • Because the model implies planet d is more likely to transit than c, a dedicated transit search of the inner companion could detect it directly, and its TTV signal would be more than half again as large as the one observed for c.
  • Applying the same joint timing-and-velocity analysis to other 2:1 resonant warm-Jupiter pairs could turn the growing sample into a population-level test of disk-driven migration rates.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper reports a joint TTV and radial-velocity analysis of the TOI-4504 system, confirming a transiting warm Jupiter (TOI-4504 c, P ≈ 82.54 d, m ≈ 3.77 MJup) and an inner non-transiting Jovian planet (TOI-4504 d, P ≈ 40.56 d, m ≈ 1.42 MJup) claimed to be in a 2:1 mean-motion resonance. The TTVs of TOI-4504 c have a peak-to-node amplitude of about 2 days and a super-period near 930–950 d, making this the largest absolute TTV signal reported to date. The paper also characterizes an inner sub-Neptune (TOI-4504 b) with P ≈ 2.426 d and Rp ≈ 2.69 R⊕. The detection of the large-amplitude TTV is secure, but the specific period and mass of planet d rest on a multimodal joint fit and on an RV signal that only crosses the 10% false-alarm-probability threshold, with an adopted RV jitter comparable to the claimed RV semi-amplitude of d.

Significance. If the inferred architecture is correct, TOI-4504 provides a new benchmark for TTV amplitudes and a further example of a warm-Jupiter pair in a first-order mean-motion resonance, which is relevant to disk-migration theories. The paper's strengths include the secure detection of very large TTVs, the joint TTV+RV N-body modeling with publicly available tools, the explicit presentation of predicted future transit times, and a careful stellar characterization. However, the central claim about the period, mass, and resonant state of TOI-4504 d is not yet uniquely established, and the paper's own text acknowledges that the large unexplained RV scatter could not be fully modeled.

major comments (3)
  1. [§4.3 and Fig. A1] The global nested-sampling search for the period of TOI-4504 d is reported as multimodal, but the paper does not provide a quantitative comparison between the preferred 40.56 d, 2:1 solution and the other modes. The text says the 41-day solution leads to "significantly better fits," but no Δln Z, ΔBIC, or equivalent evidence is given. Because the claimed mass and period of d, and hence the resonance interpretation, depend on the uniqueness of this solution, please report either the Bayes factor or BIC for the 40.56 d solution relative to the other modes in Fig. A1. In addition, a fit using only the TTV data (without RVs) should be shown to demonstrate whether the 40.56 d period is identified by the TTVs alone; otherwise the RV data, which are marginal, may be driving the period selection.
  2. [§4.2 and Table 4] The 41.2 d RV peak crosses only the 10% FAP threshold, and the fitted RV jitter is 103.4 m/s, which is comparable to K_d = 90.8 m/s. The paper itself states that the large RV scatter "cannot be fully resolved" and that the source of the jitter remains unclear. This makes the RV contribution to the d mass and period uncertain. Please quantify how the derived K_d, mass, and period of d change if the RVs are excluded from the fit or if the jitter is modeled more flexibly (e.g., with a Gaussian process or an activity-correlated component). If the mass is not robust to these choices, the text should temper the claims about the mass and period of planet d.
  3. [§1 and §5] The abstract and summary describe TOI-4504 c as having "the largest TTV amplitude ever observed." The paper's own Fig. 10 shows that this statement holds for absolute peak-to-node amplitude, but not for the relative TTV amplitude (TTV amplitude divided by orbital period). Please clarify this distinction explicitly in the abstract and main text so that the record claim is not overstated.
minor comments (5)
  1. [Abstract, Fig. 6 caption, §5] The super-period of the TTVs is quoted inconsistently: the abstract says ~930 d, the Fig. 6 caption says 946.5 d, and §5 says ~2.9 years (about 1060 d). Please harmonize these values.
  2. [§4.2] The text first states that after subtracting the 84 d signal "no other significant signals were detected," but then refers to the 41.2 d peak as "prominent" despite it crossing only the 10% FAP threshold. This wording is contradictory; please specify that 10% FAP is not a significant detection threshold in the usual sense.
  3. [Table A1] The prior for the inclination of planet c is listed as U(89.7, 0.1), which appears to be a typo (likely N(89.7, 0.1), as in Table 4, or a misprinted uniform range). Please correct it.
  4. [Table 4] The derived eccentricity uncertainties (e.g., e_d = 0.0445 ± 0.0010) are quoted to three decimal places; given the 103 m/s jitter and the marginal RV detection of d, please verify that these uncertainties are not underestimated by the adopted white-noise jitter model.
  5. [§2.4] The eight non-detections of predicted transits of c are attributed to large TTVs. It would be helpful to explicitly show that these non-detections are consistent with the best-fit TTV model, for example by plotting the expected transit times against the observed windows; otherwise the non-detections could be interpreted as tension.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reported masses, periods, and eccentricities are fitted parameters from independent TTV and RV data, and the Table A2 future transit times are extrapolations of the fitted model that the paper does not use as evidence.

full rationale

The derivation chain in TOI-4504 is a standard TTV+RV N-body fit. Transit times are measured independently from TESS photometry (Sect. 4.1.2), and RV data are independent FEROS measurements (Sect. 4.2); the joint model then fits K, P, e, omega, and inclination for planets c and d (Sect. 4.3, Table 4). No fitted parameter is relabeled as a prediction: the masses (3.77 MJup, 1.42 MJup) and periods (82.54 d, 40.56 d) are reported as fit results, and the 'predicted' future transit times in Table A2 are explicitly extrapolations from the same best-fit model, not used to validate the model. The super-period relation (Eq. 1) is a standard TTV approximation used to motivate the search range for the perturber period, and the joint fit with broad priors (Table A1) is multimodal but selects the 41-day solution on likelihood grounds (Fig. A1), so the conclusion is not forced by definition. Citations to Trifonov et al. (2021, 2023) concern the public Exo-Striker fitting/stability protocol and are code-reproduced, not load-bearing external evidence. The acknowledged limitations (41.2 d GLS peak crossing only the 10% FAP; RV jitter 103.4 m/s comparable to K_d; residual RV scatter not fully modeled) are statistical robustness concerns, not circularity: the TTV data independently establish a strong perturbing companion, and the marginal RV peak only weakens, but does not definitionally force, the period assignment. I therefore find no step where an output equals an input by construction.

Assumptions & free parameters 13 free parameters · 5 assumptions · 1 invented entities

The central model is a Newtonian N-body fit with 13 fitted parameters (amplitudes, periods, eccentricities, angles, inclinations, and RV offset/jitter). The system relies on standard N-body dynamics and on assumptions that planet b and additional bodies do not affect the c-d pair, and that the unmodeled RV jitter is not biasing the solution. The 2:1 resonance claim rests on libration of θ1 in numerical integrations using the fitted parameters.

free parameters (13)
  • RV semi-amplitude K_c = 190.9 m/s
    Fitted in the N-body model; directly sets the mass of TOI-4504 c.
  • RV semi-amplitude K_d = 90.8 m/s
    Fitted in the N-body model; directly sets the mass of TOI-4504 d.
  • Osculating period P_c = 82.54 d
    Fitted osculating period of TOI-4504 c at epoch BJD 2458400.0.
  • Osculating period P_d = 40.56 d
    Fitted osculating period of TOI-4504 d at epoch BJD 2458400.0.
  • Eccentricity e_c = 0.0320
    Derived from fitted h = e sin(ω) and k = e cos(ω) for planet c.
  • Eccentricity e_d = 0.0445
    Derived from fitted h = e sin(ω) and k = e cos(ω) for planet d.
  • Mean anomaly λ_c = 83.97 deg
    Fitted mean longitude parameter for planet c.
  • Mean anomaly λ_d = 9.89 deg
    Fitted mean longitude parameter for planet d.
  • Inclination i_d = 85.0 deg
    Fitted orbital inclination of non-transiting planet d.
  • Inclination i_c = 89.69 deg
    Fitted orbital inclination of transiting planet c, consistent with its transits.
  • Longitude of ascending node difference ΔΩ = 0.0 deg
    Fitted difference between ascending nodes of c and d; sets mutual inclination.
  • RV offset (FEROS) = 2067 m/s
    Fitted constant velocity offset for the FEROS dataset.
  • RV jitter (FEROS) = 103 m/s
    Fitted additional white noise to absorb stellar activity and unmodeled signals; this jitter is comparable to the planet d signal amplitude.
assumptions (5)
  • standard math Newtonian gravitational dynamics for the N-body system
    The TTV and RV model integrates Newtonian equations of motion via Exo-Striker and TTVfast; this is standard physics.
  • domain assumption TTVs are caused solely by the gravitational interaction between planets c and d
    The analysis excludes planet b from the N-body fit, arguing its Hill distance makes its effect negligible. Additional unknown planets are also ignored.
  • domain assumption The unmodeled 103 m/s RV jitter does not bias the two-planet solution
    The paper states the jitter source is unclear and could be stellar activity or unresolved planets; the derived masses assume this jitter is not systematically corrupting the signals.
  • ad hoc to paper Stability criteria used in the 10 Myr integrations (20% semi-major axis deviation and orbit-crossing thresholds) are sufficient to assess resonance
    The stability analysis follows the criteria from Trifonov et al. (2021), which are adopted for this system without independent validation.
  • standard math Libration of θ1 implies a 2:1 mean-motion resonance
    The identification of MMR follows the standard criterion of Lee (2004), applied to the fitted orbital parameters.
invented entities (1)
  • Planet TOI-4504 d independent evidence
    purpose: To explain the observed 2-day TTVs of TOI-4504 c and the 41.2 d radial velocity signal.
    The planet is inferred from the TTV and RV data. It has a falsifiable handle outside the current data: Table A2 lists predicted future transit times of c that would shift in a specific way if the model is correct, and future RV measurements can test the 41.2 d signal directly.

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Cite this review

Pith. "Pith review of TOI-4504: Exceptionally large Transit Timing Variations induced by two resonant warm gas giants in a three planet system." pith.science (2026). https://pith.science/paper/EYAWMZLH

@misc{pith2026241205609,
  author       = {Pith},
  title        = {Pith review of: TOI-4504: Exceptionally large Transit Timing Variations induced by two resonant warm gas giants in a three planet system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EYAWMZLH}},
  note         = {Machine review of arXiv:2412.05609}
}
abstract

We present a joint analysis of TTVs and Doppler data for the transiting exoplanet system TOI-4504. TOI-4504 c is a warm Jupiter-mass planet that exhibits the largest known transit timing variations (TTVs), with a peak-to-node amplitude of $\sim$ 2 days, the largest value ever observed, and a super-period of $\sim$ 930 d. TOI-4504 b and c were identified in public TESS data, while the TTVs observed in TOI-4504 c, together with radial velocity (RV) data collected with FEROS, allowed us to uncover a third, non-transiting planet in this system, TOI-4504 d. We were able to detect transits of TOI-4504 b in the TESS data with a period of 2.4261$\pm 0.0001$ days and derive a radius of 2.69$\pm 0.19$ R$_{\oplus}$. The RV scatter of TOI-4504 was too large to constrain the mass of TOI-4504 b, but the RV signals of TOI-4504 c \& d were sufficiently large to measure their masses. The TTV+RV dynamical model we apply confirms TOI-4504 c as a warm Jupiter planet with an osculating period of 82.54$\pm 0.02$ d, mass of 3.77$\pm 0.18$ M$_{\rm J}$ and a radius of 0.99$\pm 0.05$ R$_{\rm J}$, while the non-transiting planet TOI-4504 d, has an orbital period of 40.56$\pm 0.04$ days and mass of 1.42$_{-0.06}^{+0.07}$ M$_{\rm J}$. We present the discovery of a system with three exoplanets: a hot sub-Neptune and two warm Jupiter planets. The gas giant pair is stable and likely locked in a first-order 2:1 mean-motion resonance (MMR). The TOI-4504 system is an important addition to MMR pairs, whose increasing occurrence supports a smooth migration into a resonant configuration during the protoplanetary disk phase.

Figures

Figures reproduced from arXiv: 2412.05609 by the authors.

Figure 2
Figure 2. High-resolution imaging from SOAR for TOI￾4504. The inside image shows a speckle auto-correlation function. The 5σ contrast curve is shown as the black points with the linear fit as the black solid line. PTTV = [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Phase plot for TOI-4504 b transit. Light curve was binned into one-hour bins. For the transit analysis of TOI-4504 b, we used 2-min PDCSAP data. Before the analysis, we deleted transits of TOI-4504 c from the time series. We used broad un￾informative priors (see [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. GLS power spectrum of FEROS spectroscopic products of TOI-4504. From top to bottom panels, as la￾beled, RVs used in this work, RV residuals after subtract￾ing the dominant signal of TOI-4504 c at 84 d, the final the best-fit TTV+RV model residuals, BIS, Hα, He I Na II, and log(R ′ HK) activity indicators, respectively. False alarm prob￾ability levels of 10%, 1%, and 0.1% are marked with dashed lines, respectively. T… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 6
Figure 6. Figure 6: TESS TTV time series of TOI-4504 c and a model consistent with two Jovian-mass planets with periods close to the 2:1 MMR commensurability, with the non-transiting planet being interior (top panel). The TTV signal is expressed as the deviation of the TESS transit events…
Figure 7
Figure 7. Figure 7: Phased RV signals for the planets TOI-4504 c and d. The top two panels display the planetary signals along with the osculating N-body model, phased to the best-fit periods from [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Orbital evolution of the best TTV+RV N-body model of the TOI-4504 system for a short extent of 20 yr long N-body simulation using the Exo-Striker. The top row from left to right panels shows the evolution of the planetary period ratio (Pc/Pd) (magenta) and the evolutio…
Figure 9
Figure 9. Figure 9: Position of TOI-4504 c in the age-radius diagram (black dot). Three different models with an isodensity core (ρ = 15 [g cm3 ]) with different masses, and surrounded by an H/He envelope are overplotted (solid lines). with the heavy elements condensed in an inert isoden￾…
Figure 10
Figure 10. Figure 10: Position of TOI-4504 c, other planets with signif￾icant TTVs and planets from Holczer et al. (2016) in period￾TTV amplitude and period-TTV amplitude and period ra￾tio diagram. TTV amplitude is a peak-to-node amplitude of cosinus fit, for Kepler-90 g it is the maximum …

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Pith tools

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