pith. machine review for the scientific record. sign in

arxiv: 2604.08383 · v1 · submitted 2026-04-09 · 🌌 astro-ph.EP

Recognition: 2 theorem links

· Lean Theorem

Planetesimal-Driven Instabilities in Resonant Chains of Cold Neptunes and Their Dynamical Outcomes

Ryan LoRusso , Cristobal Petrovich , Hareesh Gautham Bhaskar

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:52 UTC · model grok-4.3

classification 🌌 astro-ph.EP
keywords resonant chainsplanetesimal diskscold Neptunesdynamical instabilityN-body simulationshot Neptune formationtidal disruptionexoplanet dynamics
0
0 comments X

The pith

Planetesimal disks with 1-4% of planetary mass disrupt resonant chains of cold Neptunes and trigger instabilities 1 Myr to 1 Gyr after gas dispersal.

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

The paper uses N-body simulations to demonstrate that remnant planetesimal disks around systems of cold Neptunes can break their resonant orbital chains long after the gas disk has vanished. These disruptions initiate chaotic global instabilities that rearrange orbits, cause planet collisions and ejections, and scatter planets inward to roughly 0.1 AU where tidal forces often capture or destroy them. A sympathetic reader would care because the process supplies a delayed mechanism for the observed scarcity of resonant configurations among close-in planets and creates a pathway to forming hot Neptunes from originally distant ones. The model also forecasts specific populations of cold, wide-orbit, and free-floating planets whose abundances can be checked with microlensing data.

Core claim

Planetesimal disks containing approximately 1-4 percent of the total planetary mass efficiently disrupt initially resonant chains of cold Neptunes, triggering global dynamical instabilities on timescales of 1 Myr to 1 Gyr. The ensuing instability produces large-scale orbital rearrangement together with planet loss through collisions, tidal disruption, and ejections; in most cases at least one planet reaches approximately 0.1 AU on 10-100 Myr timescales, where tidal capture or disruption can occur.

What carries the argument

N-body simulations tracking gravitational perturbations from remnant planetesimal disks acting on multi-Neptune systems locked in resonant chains during the gas-disk phase.

If this is right

  • Compact inner resonant chains are destroyed by cold sub-Neptunes on roughly 100 Myr timescales.
  • Tidal capture of scattered planets supplies a formation channel for hot Neptunes.
  • The instability produces mass-segregated planets whose relative numbers of cold, wide-orbit, and free-floating planets can be tested directly.
  • Instability timescales vary strongly with planetesimal-disk mass, allowing diverse outcomes depending on disk properties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The strong dependence on disk mass implies that modest differences in planetesimal budgets can produce the full range of observed exoplanet architectures.
  • If the mechanism operates widely, resonant fractions should decline steadily with system age on 100 Myr timescales.
  • Detection of the predicted free-floating and wide-orbit populations would also constrain the typical mass of remnant planetesimal disks.

Load-bearing premise

Resonant chains must assemble during the gas-disk phase and the planetesimal disks must keep realistic masses and radial distributions once the gas has gone.

What would settle it

Microlensing surveys that measure abundances of wide-orbit and free-floating Neptune-mass planets differing substantially from the model's predicted mass-segregated ratios would falsify the disruption pathway.

Figures

Figures reproduced from arXiv: 2604.08383 by Cristobal Petrovich, Hareesh Gautham Bhaskar, Ryan LoRusso.

Figure 1
Figure 1. Figure 1: Representative orbital evolution from a simulation in the fiducial f5.6n2m suite. From left to right, panels show: i) capture of cold Neptunes into resonance via disk-driven migration; ii) gradual removal of damping representing disk dispersal; iii) introduction of planetesimals and scattering of cold Neptunes until the first close encounter; iv) onset of global instability, during which one or more planet… view at source ↗
Figure 2
Figure 2. Figure 2: Angular momentum deficit (AMD) evolution for the same simulation as in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Destabilization times for five-planet systems, defined as the time of the first close encounter between any pair of planets. Colors and symbols distinguish between simulation suites. Time is measured from the introduction of planetesimals. Each small symbol represents an individual system, while large symbols indicate the median destabilization time. Distributions are shown as lines: thin lines trace the f… view at source ↗
Figure 5
Figure 5. Figure 5: Cumulative distribution functions of the mini￾mum pericenter distance reached by any planet in a given system at selected times. Times are measured in years since system destabilization and are ordered from earliest to latest by increasing opacity. Panel a) corresponds to the fiducial f5.6n2m suite, and panel b) corresponds to the sub-Neptune mm(-3)f5.3n4m suite (1/3 Neptune mass). The innermost boundary m… view at source ↗
Figure 4
Figure 4. Figure 4: Average number of planets per system in each outcome as a function of time across the simulation suite. Colors denote the class of state, while the color bar on the right indicates the average number of planets per system in the corresponding end state. Panels show suites of thirty planetesimals with different planetesimal masses (values of ϵ), increasing from top to bottom [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 6
Figure 6. Figure 6: Times of first approach within 0.1 au for five-planet systems. Colors and symbols distinguish between simulation suites, with the f5 series and other series shown in separate panels. Only systems that both destabilize and reach a first approach within 1.1 Gyr are included; the fraction of such systems in each suite is indicated by the ratios on the left. Time is measured from destabilization. Each small sy… view at source ↗
Figure 7
Figure 7. Figure 7: Final eccentricities (upper panel) and inclinations (lower panel) as a function of semimajor axis for Neptunes in the f5 series that remain bound after 1.146 Gyr across all simulation suites. Colors indicate the system multiplicity, symbol size scales with planetary mass, and stars denote the innermost planets. Only destabilized systems are shown; the initial semimajor axes are indicated by vertical dotted… view at source ↗
Figure 8
Figure 8. Figure 8: Fraction of Neptunes classified as wide-orbiting above a given threshold in average separation ⟨r⟩ as a func￾tion of time for the fiducial f5.6n2m simulations. Different thresholds are indicated by curves with varying opacity. arises from the tendency for collisions to occur on av￾erage before scatterings, which thus produces an inner population of massive (≥ 2 MNep) collided Neptunes and an outer, scatter… view at source ↗
Figure 9
Figure 9. Figure 9: Proof-of-concept evolution of an inner resonant chain of super-Earths coupled to a destabilizing outer reso￾nant chain of cold Neptunes. The super-Earths are initialized with the masses and resonances of the TOI-1136 system, ex￾cept for the 7:5 resonance (planets e-f) that is replaced by a 4:3 resonance to enhance its stability. Colored lines trace the orbital evolution of each planet, with warm colors den… view at source ↗
Figure 10
Figure 10. Figure 10: As [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
read the original abstract

Cold Neptunes and sub-Neptunes are among the most common products of planet formation and likely dominate the angular-momentum budgets in most planetary systems, yet their dynamical impact on planetary architectures remains poorly understood. Using N-body simulations, we investigate the evolution of multi-Neptune systems assembled into resonant chains during the gas-disk phase and later coupled to remnant planetesimal disks. We show that planetesimal disks containing $\simeq 1$-$4\%$ of the planetary mass efficiently disrupt resonant chains and trigger global dynamical instabilities on timescales of $1~\mathrm{Myr}$-$1~\mathrm{Gyr}$, providing a pathway for delayed instability long after gas-disk dispersal, albeit with instability timescales that are highly sensitive to disk mass. The ensuing instability drives large-scale orbital rearrangement and loss of planets through collisions, tidal disruption, and ejections. Notably, in most systems at least one planet is scattered inward to $\sim 0.1~\mathrm{au}$ on $\sim 10$-$100$ Myr timescales (for $\sim 5$-$50\; M_\oplus$ planets) following instability onset, with a substantial fraction undergoing tidal capture or disruption. This tidal capture can provide a natural pathway to hot Neptune formation, while compact inner chains, if present, would be destroyed on $\sim 100~\mathrm{Myr}$ timescales by cold sub-Neptunes, naturally explaining the observed decline in the resonant fraction. We argue that the predictions of our model, which yields mass-segregated planets and corresponding relative abundances of cold, wide-orbit, and free-floating planets, can be tested by ongoing and upcoming microlensing surveys.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper uses N-body simulations to investigate the post-gas-disk evolution of resonant chains of cold Neptunes coupled to remnant planetesimal disks. It claims that disks containing ≃1-4% of the total planetary mass efficiently disrupt these chains, triggering global instabilities on timescales of 1 Myr to 1 Gyr. The resulting dynamical rearrangement leads to planet losses via collisions, ejections, and tidal disruptions, with frequent inward scattering of planets to ~0.1 au that can produce hot Neptunes via tidal capture; the model also yields mass-segregated architectures and relative abundances of cold, wide-orbit, and free-floating planets testable by microlensing surveys.

Significance. If the central results hold, the work identifies a viable pathway for delayed instabilities in multi-Neptune systems long after gas dispersal, offering a natural explanation for the observed scarcity of resonant chains and the existence of hot Neptunes. The forward N-body approach generates concrete, observationally testable predictions for planet abundances and orbital distributions. The sensitivity of outcomes to disk mass is explicitly noted, which strengthens the paper's transparency about parameter dependence.

major comments (3)
  1. [Abstract] Abstract: The central claim that planetesimal disks of ≃1-4% planetary mass disrupt resonant chains on 1 Myr–1 Gyr timescales is load-bearing, yet the text provides no justification from formation models or observations for why this specific mass fraction (and associated radial profile) is realistic after gas dispersal; the abstract itself states that timescales are highly sensitive to disk mass, so the efficiency and window would change substantially under plausible variations.
  2. [Numerical methods] Numerical methods section: The abstract reports outcomes from N-body simulations but supplies no information on the integrator, timestep, number of realizations per initial condition, or convergence/sensitivity tests with respect to numerical parameters or small perturbations in the initial resonant chains. Given the chaotic nature of the reported instabilities, these details are required to assess robustness of the quoted timescales and scattering statistics.
  3. [Initial conditions] Initial conditions and setup: The assumption that resonant chains assembled in the gas phase remain intact until the planetesimal disk is introduced after dispersal is taken as given; no tests are described for how the chains respond to small perturbations expected during the final stages of gas-disk dispersal, which could alter the subsequent disruption efficiency.
minor comments (2)
  1. [Abstract] The abstract uses ≃1-4% without defining the exact total planetary mass reference or showing how this fraction maps onto absolute disk masses for the simulated systems.
  2. [Figures] Figure captions and text should explicitly state the number of realizations and the range of initial conditions explored for each disk-mass case to allow readers to gauge statistical significance of the reported fractions (e.g., “at least one planet scattered inward”).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their positive assessment of the significance of our work and for the constructive major comments. We address each comment in detail below and have made revisions to the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that planetesimal disks of ≃1-4% planetary mass disrupt resonant chains on 1 Myr–1 Gyr timescales is load-bearing, yet the text provides no justification from formation models or observations for why this specific mass fraction (and associated radial profile) is realistic after gas dispersal; the abstract itself states that timescales are highly sensitive to disk mass, so the efficiency and window would change substantially under plausible variations.

    Authors: The referee raises a valid point regarding the motivation for the chosen planetesimal disk mass range. Our simulations demonstrate that disks with 1-4% of the planetary mass lead to the reported outcomes, and we explicitly note the high sensitivity to this parameter. To provide better context, we will revise the manuscript to include a discussion in the introduction section referencing relevant planet formation models that predict remnant planetesimal disk masses in the range of a few percent of the total planetary mass after gas dispersal. We will also update the abstract to better reflect that this mass fraction represents a plausible and effective range for triggering the instabilities. revision: yes

  2. Referee: [Numerical methods] Numerical methods section: The abstract reports outcomes from N-body simulations but supplies no information on the integrator, timestep, number of realizations per initial condition, or convergence/sensitivity tests with respect to numerical parameters or small perturbations in the initial resonant chains. Given the chaotic nature of the reported instabilities, these details are required to assess robustness of the quoted timescales and scattering statistics.

    Authors: We agree that the numerical methods section lacks essential details for reproducibility and robustness assessment. This information was inadvertently omitted. In the revised manuscript, we will expand the Numerical Methods section to specify the N-body integrator employed, the adopted timestep, the number of realizations performed for each initial condition, and the results of convergence and sensitivity tests. These additions will confirm that the instability timescales and dynamical outcomes are not sensitive to small numerical variations or initial perturbations within the explored parameter space. revision: yes

  3. Referee: [Initial conditions] Initial conditions and setup: The assumption that resonant chains assembled in the gas phase remain intact until the planetesimal disk is introduced after dispersal is taken as given; no tests are described for how the chains respond to small perturbations expected during the final stages of gas-disk dispersal, which could alter the subsequent disruption efficiency.

    Authors: The referee correctly identifies that we did not perform or report tests on the response of the resonant chains to perturbations during the gas-disk dispersal phase. Our study assumes that the chains are stable at the start of the planetesimal disk phase, focusing on the subsequent evolution. To strengthen this, we will add a brief analysis or note in the revised manuscript discussing the expected perturbations and include a small set of test simulations showing that moderate perturbations do not prematurely disrupt the chains before the planetesimal disk takes effect. This will better justify the initial setup. revision: yes

Circularity Check

0 steps flagged

No circularity: outcomes are direct numerical consequences of input initial conditions

full rationale

The paper reports results exclusively from forward N-body integrations of resonant chains interacting with planetesimal disks of given mass (1-4% of planetary mass) and radial profile. These are exploratory simulations whose outputs (instability timescales, scattering, tidal captures) follow from the chosen physics and initial conditions rather than any algebraic reduction, fitted parameter renamed as prediction, or self-referential definition. The disk mass and resonant-chain setup are explicitly treated as inputs whose variation affects outcomes, with no claim that the results are independent of or derive those inputs. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior author work are invoked to force the central claims. The derivation chain is therefore self-contained as a set of numerical experiments.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard gravitational N-body dynamics plus domain assumptions about the survival and mass fraction of planetesimal disks after gas dispersal; no new entities are postulated.

free parameters (1)
  • planetesimal disk mass fraction
    The 1-4% range is the interval reported to produce efficient disruption; it is the key control parameter explored in the simulations.
axioms (2)
  • standard math Newtonian gravitational dynamics govern planet-planetesimal interactions
    Invoked throughout the N-body simulations described in the abstract.
  • domain assumption Resonant chains form during the gas-disk phase and persist until planetesimal interactions begin
    Stated as the starting condition for the post-gas evolution.

pith-pipeline@v0.9.0 · 5614 in / 1493 out tokens · 28488 ms · 2026-05-10T17:52:05.812216+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

86 extracted references · 83 canonical work pages · 1 internal anchor

  1. [1]

    J., et al., 2006, @doi [ ] 10.1111/j.1365-2966.2005.09675.x , http://adsabs.harvard.edu/abs/2006MNRAS.365...11C 365, 11

    Agol, E., Steffen, J., Sari, R., & Clarkson, W. 2005, MNRAS, 359, 567, doi: 10.1111/j.1365-2966.2005.08922.x Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068 Astropy Collaboration, Price-Whelan, A. M., Sip˝ ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Astropy Co...

    work page doi:10.1111/j.1365-2966.2005.08922.x 2005

  2. [2]

    2026, arXiv e-prints, arXiv:2603.27093

    Batygin, K., & Morbidelli, A. 2026, arXiv e-prints, arXiv:2603.27093. https://arxiv.org/abs/2603.27093 Beaug´ e, C., & Nesvorn´ y, D. 2012, ApJ, 751, 119, doi: 10.1088/0004-637X/751/2/119

    work page doi:10.1088/0004-637x/751/2/119 2026

  3. [3]

    A., Liquet, M., Signell, J., et al

    Bednar, J. A., Liquet, M., Signell, J., et al. 2024, holoviz/colorcet: Version 3.1.0, v3.1.0, Zenodo, doi: 10.5281/zenodo.10729196

    work page doi:10.5281/zenodo.10729196 2024

  4. [4]

    G., & Perets, H

    Bhaskar, H. G., & Perets, H. B. 2025, ApJ, 991, 132, doi: 10.3847/1538-4357/adf4e2

    work page doi:10.3847/1538-4357/adf4e2 2025

  5. [5]

    2023, A&A, 669, A63, doi: 10.1051/0004-6361/202245004

    Bourrier, V., Attia, O., Mallonn, M., et al. 2023, A&A, 669, A63, doi: 10.1051/0004-6361/202245004

    work page doi:10.1051/0004-6361/202245004 2023

  6. [6]

    2022, Matplotlib label lines, v0.5.1, Zenodo, doi: 10.5281/zenodo.7428071 Castro-Gonz´ alez, A., Bourrier, V., Lillo-Box, J., et al

    Cadiou, C. 2022, Matplotlib label lines, v0.5.1, Zenodo, doi: 10.5281/zenodo.7428071 Castro-Gonz´ alez, A., Bourrier, V., Lillo-Box, J., et al. 2024, A&A, 689, A250, doi: 10.1051/0004-6361/202450957

    work page doi:10.5281/zenodo.7428071 2022

  7. [7]

    Chatterjee, S., & Ford, E. B. 2015, ApJ, 803, 33, doi: 10.1088/0004-637X/803/1/33

    work page doi:10.1088/0004-637x/803/1/33 2015

  8. [8]

    , keywords =

    Chatterjee, S., Ford, E. B., Matsumura, S., & Rasio, F. A. 2008, ApJ, 686, 580, doi: 10.1086/590227

    work page doi:10.1086/590227 2008

  9. [9]

    Two-stage disruption of resonant chains

    Choksi, N., Lithwick, Y., Chiang, E., & Li, R. 2026, arXiv e-prints, arXiv:2604.05035. https://arxiv.org/abs/2604.05035

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  10. [10]

    Dai, F., & Winn, J. N. 2017, AJ, 153, 205, doi: 10.3847/1538-3881/aa65d1

    work page doi:10.3847/1538-3881/aa65d1 2017

  11. [11]

    2023, AJ, 165, 33, doi: 10.3847/1538-3881/aca327

    Dai, F., Masuda, K., Beard, C., et al. 2023, AJ, 165, 33, doi: 10.3847/1538-3881/aca327

    work page doi:10.3847/1538-3881/aca327 2023

  12. [12]

    2024, AJ, 168, 239, doi: 10.3847/1538-3881/ad83a6

    Dai, F., Goldberg, M., Batygin, K., et al. 2024, AJ, 168, 239, doi: 10.3847/1538-3881/ad83a6

    work page doi:10.3847/1538-3881/ad83a6 2024

  13. [13]

    Origins of Hot Jupiters

    Dawson, R. I., & Johnson, J. A. 2018, ARA&A, 56, 175, doi: 10.1146/annurev-astro-081817-051853

    work page Pith review doi:10.1146/annurev-astro-081817-051853 2018

  14. [14]

    M., & Batygin, K

    Deck, K. M., & Batygin, K. 2015, ApJ, 810, 119, doi: 10.1088/0004-637X/810/2/119

    work page doi:10.1088/0004-637x/810/2/119 2015

  15. [15]

    2025, ApJL, 994, L23, doi: 10.3847/2041-8213/ae18c7

    Dugan, E., Wang, X.-Y., Heron, A., et al. 2025, ApJL, 994, L23, doi: 10.3847/2041-8213/ae18c7

    work page doi:10.3847/2041-8213/ae18c7 2025

  16. [16]

    1995, ApJ, 455, 640, doi: 10.1086/176611

    Eggleton, P., & Kiseleva, L. 1995, ApJ, 455, 640, doi: 10.1086/176611

    work page doi:10.1086/176611 1995

  17. [17]

    I., Stef´ ansson, G., Petrovich, C., et al

    Espinoza-Retamal, J. I., Stef´ ansson, G., Petrovich, C., et al. 2024, AJ, 168, 185, doi: 10.3847/1538-3881/ad70b8

    work page doi:10.3847/1538-3881/ad70b8 2024

  18. [18]

    A., & Ip, W.-H

    Fernandez, J. A., & Ip, W.-H. 1984, Icarus, 58, 109, doi: 10.1016/0019-1035(84)90101-5 20 Figure 10.As Figure 5, but across allf5suites of simulations. From left to right, top to bottom, these correspond to the f5.3n1m,f5.3n2m,f5.3n4m,f5.6n1m,f5.6n2m,f5.6n4m,f5.12n1m,f5.12n2m, andf5.12n4msimulations. In other words; total mass of planetesimals increases f...

    work page doi:10.1016/0019-1035(84)90101-5 1984

  19. [19]

    B., & Rasio, F

    Ford, E. B., & Rasio, F. A. 2008, ApJ, 686, 621, doi: 10.1086/590926

    work page doi:10.1086/590926 2008

  20. [20]

    Meyer Marcy, G.W., Rubenzahl, R.A., Weiss, L.M., Wright, J.T.: California Legacy Survey

    Fulton, B. J., Rosenthal, L. J., Hirsch, L. A., et al. 2021, ApJS, 255, 14, doi: 10.3847/1538-4365/abfcc1

    work page doi:10.3847/1538-4365/abfcc1 2021

  21. [21]

    Gaudi, B. S. 2010, arXiv e-prints, arXiv:1002.0332, doi: 10.48550/arXiv.1002.0332

    work page doi:10.48550/arxiv.1002.0332 2010

  22. [22]

    2023, ApJ, 943, 8, doi: 10.3847/1538-4357/aca58e

    Ghosh, T., & Chatterjee, S. 2023, ApJ, 943, 8, doi: 10.3847/1538-4357/aca58e

    work page doi:10.3847/1538-4357/aca58e 2023

  23. [23]

    A., Morais, M

    Giuppone, C. A., Morais, M. H. M., & Correia, A. C. M. 2013, MNRAS, 436, 3547, doi: 10.1093/mnras/stt1831

    work page doi:10.1093/mnras/stt1831 2013

  24. [24]

    2023, ApJ, 948, 12, doi: 10.3847/1538-4357/acc9ae

    Goldberg, M., & Batygin, K. 2023, ApJ, 948, 12, doi: 10.3847/1538-4357/acc9ae

    work page doi:10.3847/1538-4357/acc9ae 2023

  25. [25]

    2022, Icarus, 388, 115206, doi: 10.1016/j.icarus.2022.115206

    Goldberg, M., Batygin, K., & Morbidelli, A. 2022, Icarus, 388, 115206, doi: 10.1016/j.icarus.2022.115206

    work page doi:10.1016/j.icarus.2022.115206 2022

  26. [26]

    Goldberg, M., & Petit, A. C. 2025, arXiv e-prints, arXiv:2511.11329, doi: 10.48550/arXiv.2511.11329

    work page doi:10.48550/arxiv.2511.11329 2025

  27. [27]

    2025, arXiv e-prints, arXiv:2507.08968, doi: 10.48550/arXiv.2507.08968 —

    Hadden, S., & Wu, Y. 2025, arXiv e-prints, arXiv:2507.08968, doi: 10.48550/arXiv.2507.08968 —. 2026, arXiv e-prints, arXiv:2602.21349, doi: 10.48550/arXiv.2602.21349

    work page doi:10.48550/arxiv.2507.08968 2025

  28. [28]

    M., & Malhotra, R

    Hahn, J. M., & Malhotra, R. 1999, AJ, 117, 3041, doi: 10.1086/300891

    work page doi:10.1086/300891 1999

  29. [29]

    1993, Solving Orginary Differential Equations I (Springer), doi: 10.1007/978-3-540-78862-1

    Hairer, E., Nørsett, S., & Wanner, G. 1993, Solving Orginary Differential Equations I (Springer), doi: 10.1007/978-3-540-78862-1

    work page doi:10.1007/978-3-540-78862-1 1993

  30. [30]

    S., An, J

    Han, C., Gaudi, B. S., An, J. H., & Gould, A. 2005, ApJ, 618, 962, doi: 10.1086/426115

    work page doi:10.1086/426115 2005

  31. [31]

    B., & Batygin, K

    Handley, L. B., & Batygin, K. 2026, arXiv e-prints, arXiv:2601.04140, doi: 10.48550/arXiv.2601.04140

    work page doi:10.48550/arxiv.2601.04140 2026

  32. [32]

    R., Millman, K

    Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2

    work page doi:10.1038/s41586-020-2649-2 2020

  33. [33]

    Progress of Theoretical Physics Supplement , year = 1981, month = jan, volume =

    Hayashi, C. 1981, Progress of Theoretical Physics Supplement, 70, 35, doi: 10.1143/PTPS.70.35

    work page doi:10.1143/ptps.70.35 1981

  34. [34]

    1983, Celestial Mechanics, 30, 197, doi: 10.1007/BF01234306

    Henrard, J., & Lemaitre, A. 1983, Celestial Mechanics, 30, 197, doi: 10.1007/BF01234306

    work page doi:10.1007/bf01234306 1983

  35. [35]

    2023, ApJL, 949, L35, doi: 10.3847/2041-8213/acd6f5

    Hixenbaugh, K., Wang, X.-Y., Rice, M., & Wang, S. 2023, ApJL, 949, L35, doi: 10.3847/2041-8213/acd6f5

    work page doi:10.3847/2041-8213/acd6f5 2023

  36. [36]

    S., Matthews, B

    Holland, W. S., Matthews, B. C., Kennedy, G. M., et al. 2017, MNRAS, 470, 3606, doi: 10.1093/mnras/stx1378

    work page doi:10.1093/mnras/stx1378 2017

  37. [37]

    C., Ford, E

    Hsu, D. C., Ford, E. B., Ragozzine, D., & Ashby, K. 2019, AJ, 158, 109, doi: 10.3847/1538-3881/ab31ab 21

    work page doi:10.3847/1538-3881/ab31ab 2019

  38. [38]

    2026, ApJ, 998, 245, doi: 10.3847/1538-4357/ae394f

    Huang, X., & Lai, D. 2026, ApJ, 998, 245, doi: 10.3847/1538-4357/ae394f

    work page doi:10.3847/1538-4357/ae394f 2026

  39. [39]

    Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55

    work page doi:10.1109/mcse.2007.55 2007

  40. [40]

    N., et al

    Izidoro, A., Bitsch, B., Raymond, S. N., et al. 2021, A&A, 650, A152, doi: 10.1051/0004-6361/201935336

    work page doi:10.1051/0004-6361/201935336 2021

  41. [41]

    N., et al

    Izidoro, A., Ogihara, M., Raymond, S. N., et al. 2017, MNRAS, 470, 1750, doi: 10.1093/mnras/stx1232 Juri´ c, M., & Tremaine, S. 2008, ApJ, 686, 603, doi: 10.1086/590047

    work page doi:10.1093/mnras/stx1232 2017

  42. [42]

    2015, arXiv e-prints, arXiv:1509.03700, doi: 10.48550/arXiv.1509.03700

    Kovesi, P. 2015, arXiv e-prints, arXiv:1509.03700, doi: 10.48550/arXiv.1509.03700

    work page doi:10.48550/arxiv.1509.03700 2015

  43. [43]

    V., & Wyatt, M

    Krivov, A. V., & Wyatt, M. C. 2021, MNRAS, 500, 718, doi: 10.1093/mnras/staa2385

    work page doi:10.1093/mnras/staa2385 2021

  44. [44]

    1997, A&A, 317, L75

    Laskar, J. 1997, A&A, 317, L75

    1997

  45. [45]

    Laskar, J., & Petit, A. C. 2017, A&A, 605, A72, doi: 10.1051/0004-6361/201630022

    work page doi:10.1051/0004-6361/201630022 2017

  46. [46]

    H., Fabrycky, D., & Lin, D

    Lee, M. H., Fabrycky, D., & Lin, D. N. C. 2013, ApJ, 774, 52, doi: 10.1088/0004-637X/774/1/52

    work page doi:10.1088/0004-637x/774/1/52 2013

  47. [47]

    H., & Peale, S

    Lee, M. H., & Peale, S. J. 2002, ApJ, 567, 596, doi: 10.1086/338504

    work page doi:10.1086/338504 2002

  48. [48]

    E., & Rasio, F

    Li, J., O’Connor, C. E., & Rasio, F. A. 2026, ApJL, 998, L5, doi: 10.3847/2041-8213/ae3a9a

    work page doi:10.3847/2041-8213/ae3a9a 2026

  49. [49]

    2025, AJ, 169, 323, doi: 10.3847/1538-3881/adce0c

    Li, R., Chiang, E., Choksi, N., & Dai, F. 2025, AJ, 169, 323, doi: 10.3847/1538-3881/adce0c

    work page doi:10.3847/1538-3881/adce0c 2025

  50. [50]

    J., Ragozzine, D., Fabrycky, D

    Lissauer, J. J., Ragozzine, D., Fabrycky, D. C., et al. 2011, ApJS, 197, 8, doi: 10.1088/0067-0049/197/1/8

    work page doi:10.1088/0067-0049/197/1/8 2011

  51. [51]

    2012, ApJL, 756, L11, doi: 10.1088/2041-8205/756/1/L11 Lopez Murillo, A

    Lithwick, Y., & Wu, Y. 2012, ApJL, 756, L11, doi: 10.1088/2041-8205/756/1/L11 Lopez Murillo, A. I., Mann, A. W., Barber, M. G., et al. 2026, AJ, 171, 63, doi: 10.3847/1538-3881/ae231a

    work page doi:10.1088/2041-8205/756/1/l11 2012

  52. [52]

    1993, Nature, 365, 819, doi: 10.1038/365819a0 —

    Malhotra, R. 1993, Nature, 365, 819, doi: 10.1038/365819a0 —. 1995, AJ, 110, 420, doi: 10.1086/117532

    work page doi:10.1038/365819a0 1993

  53. [53]

    1982, Celestial Mechanics, 26, 311, doi: 10.1007/BF01230725 Matr` a, L., Marino, S., Wilner, D

    Marchal, C., & Bozis, G. 1982, Celestial Mechanics, 26, 311, doi: 10.1007/BF01230725 Matr` a, L., Marino, S., Wilner, D. J., et al. 2025, A&A, 693, A151, doi: 10.1051/0004-6361/202451397

    work page doi:10.1007/bf01230725 1982

  54. [54]

    2013, , 208, 16, 10.1088/0067-0049/208/2/16

    Mazeh, T., Nachmani, G., Holczer, T., et al. 2013, ApJS, 208, 16, doi: 10.1088/0067-0049/208/2/16

    work page doi:10.1088/0067-0049/208/2/16 2013

  55. [55]

    2025, A&A, 700, A146, doi: 10.1051/0004-6361/202451142

    Mignon, L., Delfosse, X., Meunier, N., et al. 2025, A&A, 700, A146, doi: 10.1051/0004-6361/202451142

    work page doi:10.1051/0004-6361/202451142 2025

  56. [56]

    V., et al

    Montesinos, B., Eiroa, C., Krivov, A. V., et al. 2016, A&A, 593, A51, doi: 10.1051/0004-6361/201628329 Mo´ or, A., K´ osp´ al,´A., ´Abrah´ am, P., et al. 2016, ApJ, 826, 123, doi: 10.3847/0004-637X/826/2/123 M¨ uller, S., Baron, J., Helled, R., Bouchy, F., & Parc, L. 2024, A&A, 686, A296, doi: 10.1051/0004-6361/202348690

    work page doi:10.1051/0004-6361/201628329 2016

  57. [57]

    M., & Armitage, P

    Murphy, M. M., & Armitage, P. J. 2022, MNRAS, 512, 2750, doi: 10.1093/mnras/stac750

    work page doi:10.1093/mnras/stac750 2022

  58. [58]

    2002, ApJ, 565, 608, doi: 10.1086/324536

    Murray, N., Paskowitz, M., & Holman, M. 2002, ApJ, 565, 608, doi: 10.1086/324536

    work page doi:10.1086/324536 2002

  59. [59]

    2011, ApJ, 742, 72, doi: 10.1088/0004-637X/742/2/72

    Nagasawa, M., & Ida, S. 2011, ApJ, 742, 72, doi: 10.1088/0004-637X/742/2/72

    work page doi:10.1088/0004-637x/742/2/72 2011

  60. [60]

    2008, ApJ, 678, 498, doi: 10.1086/529369

    Nagasawa, M., Ida, S., & Bessho, T. 2008, ApJ, 678, 498, doi: 10.1086/529369

    work page doi:10.1086/529369 2008

  61. [61]

    2017, Icarus, 293, 52, doi: 10.1016/j.icarus.2017.04.010

    Obertas, A., Van Laerhoven, C., & Tamayo, D. 2017, Icarus, 293, 52, doi: 10.1016/j.icarus.2017.04.010

    work page doi:10.1016/j.icarus.2017.04.010 2017

  62. [62]

    2026, ApJ, 996, 91, doi: 10.3847/1538-4357/ae1d78

    Ogihara, M., & Kunitomo, M. 2026, ApJ, 996, 91, doi: 10.3847/1538-4357/ae1d78

    work page doi:10.3847/1538-4357/ae1d78 2026

  63. [63]

    T., Gaudi, B

    Penny, M. T., Gaudi, B. S., Kerins, E., et al. 2019, ApJS, 241, 3, doi: 10.3847/1538-4365/aafb69

    work page doi:10.3847/1538-4365/aafb69 2019

  64. [64]

    2015, ApJ, 808, 120, doi: 10.1088/0004-637X/808/2/120

    Petrovich, C. 2015, ApJ, 808, 120, doi: 10.1088/0004-637X/808/2/120

    work page doi:10.1088/0004-637x/808/2/120 2015

  65. [65]

    2024, in IAU Symposium, Vol

    Petrovich, C. 2024, in IAU Symposium, Vol. 382, Complex Planetary Systems II: Latest Methods for an Interdisciplinary Approach, ed. A. Lemaitre & A.-S. Libert, 30–40, doi: 10.1017/S1743921323004775

    work page doi:10.1017/s1743921323004775 2024

  66. [66]

    2014, ApJ, 786, 101, doi: 10.1088/0004-637X/786/2/101

    Petrovich, C., Tremaine, S., & Rafikov, R. 2014, ApJ, 786, 101, doi: 10.1088/0004-637X/786/2/101

    work page doi:10.1088/0004-637x/786/2/101 2014

  67. [67]

    2020, MNRAS, 494, 4950, doi: 10.1093/mnras/staa1102

    Pichierri, G., & Morbidelli, A. 2020, MNRAS, 494, 4950, doi: 10.1093/mnras/staa1102

    work page doi:10.1093/mnras/staa1102 2020

  68. [68]

    2015, ApJ, 807, 44, doi: 10.1088/0004-637X/807/1/44

    Pu, B., & Wu, Y. 2015, ApJ, 807, 44, doi: 10.1088/0004-637X/807/1/44

    work page doi:10.1088/0004-637x/807/1/44 2015

  69. [69]

    S., Charalambous, C., Ben´ ıtez-Llambay, P., & Beaug´ e, C

    Ramos, X. S., Charalambous, C., Ben´ ıtez-Llambay, P., & Beaug´ e, C. 2017, A&A, 602, A101, doi: 10.1051/0004-6361/201629642

    work page doi:10.1051/0004-6361/201629642 2017

  70. [70]

    Rein, H., & Liu, S. F. 2012, A&A, 537, A128, doi: 10.1051/0004-6361/201118085

    work page doi:10.1051/0004-6361/201118085 2012

  71. [71]

    I., Jenkins, J

    Rubenstein, R. I., Jenkins, J. S., Pe˜ na R., P. A., et al. 2025, A&A, 702, A139, doi: 10.1051/0004-6361/202449563

    work page doi:10.1051/0004-6361/202449563 2025

  72. [72]

    2015, MNRAS, 453, 4089, doi: 10.1093/mnras/stv1924

    Silburt, A., & Rein, H. 2015, MNRAS, 453, 4089, doi: 10.1093/mnras/stv1924

    work page doi:10.1093/mnras/stv1924 2015

  73. [73]

    , keywords =

    Sumi, T., Koshimoto, N., Bennett, D. P., et al. 2023, AJ, 166, 108, doi: 10.3847/1538-3881/ace688

    work page doi:10.3847/1538-3881/ace688 2023

  74. [74]

    2017, , 840, L19, 10.3847/2041-8213/aa70ea

    Tamayo, D., Rein, H., Petrovich, C., & Murray, N. 2017, ApJL, 840, L19, doi: 10.3847/2041-8213/aa70ea

    work page doi:10.3847/2041-8213/aa70ea 2017

  75. [75]

    Tamayo, D., Rein, H., Shi, P., & Hernandez, D. M. 2020, MNRAS, 491, 2885, doi: 10.1093/mnras/stz2870

    work page doi:10.1093/mnras/stz2870 2020

  76. [76]

    Tanaka, H., Takeuchi, T., & Ward, W. R. 2002, ApJ, 565, 1257, doi: 10.1086/324713

    work page doi:10.1086/324713 2002

  77. [77]

    Tanaka, H., & Ward, W. R. 2004, ApJ, 602, 388, doi: 10.1086/380992

    work page doi:10.1086/380992 2004

  78. [78]

    1993, in Astronomical Society of the Pacific Conference Series, Vol

    Tremaine, S. 1993, in Astronomical Society of the Pacific Conference Series, Vol. 36, Planets Around Pulsars, ed. J. A. Phillips, S. E. Thorsett, & S. R. Kulkarni, 335–344

    1993

  79. [79]

    2023, Dynamics of Planetary Systems (Princeton University Press)

    Tremaine, S. 2023, Dynamics of Planetary Systems (Princeton University Press)

    2023

  80. [80]

    Tsiganis, K., Gomes, R., Morbidelli, A., & Levison, H. F. 2005, Nature, 435, 459, doi: 10.1038/nature03539 22

    work page doi:10.1038/nature03539 2005

Showing first 80 references.