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The Formation of Double Hot Jupiter Systems through von Zeipel-Lidov-Kozai Migration

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Two hot Jupiters can form around both stars of a binary, simulations find

desk verdict A genuinely new channel for double hot Jupiters via mirrored ZLK migration, backed by real N-body work, but the headline ~9% rate is a conditional upper limit that drops to ~1.4% when the second planet's inclination is treated as isotropic. read the letter →

arxiv 2505.04398 v2 pith:UZPWNEJG submitted 2025-05-07 astro-ph.EP

Yurou Liu , Tiger Lu , Malena Rice This is my paper · ORCID
classification astro-ph.EP
keywords hotJupitersvonZeipel-Lidov-Kozaimigrationdoublestellarbinariestidalfrictionhigh-eccentricityexoplanetdynamicsN-bodysimulations
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 a binary star system can produce a hot Jupiter around each star at once, through mirrored von Zeipel-Lidov-Kozai (ZLK) migration: two cold Jupiters, one around each star, are driven by the companion star into high-eccentricity orbits that tidal friction shrinks and circularizes into close-in hot Jupiters. Using N-body simulations with tides and general relativity, the authors show this happens naturally in symmetric binaries and remains robust to modest asymmetries in stellar mass, planet mass, and mutual inclination. They predict that, if most hot Jupiters in binaries form this way and if a cold Jupiter around one star implies one around the other, then up to about 9% of known hot-Jupiter-hosting binaries with stellar separations out to 2000 AU should also host a second hot Jupiter. A curious reader should care because this is a concrete, testable prediction: the companion stars of known hot Jupiter hosts are largely unmonitored, and finding or failing to find these second hot Jupiters would test whether this formation channel operates.

What carries the argument

The mechanism is the von Zeipel-Lidov-Kozai (ZLK) effect with tidal friction, applied in a mirrored four-body configuration: an inclined stellar companion drives coupled oscillations in each planet's eccentricity and inclination, and tides at pericenter drain orbital energy, shrink the semimajor axis, and circularize the orbit into a hot Jupiter. The mirrored setup—two identical planets on orbits whose mutual inclinations with respect to the binary plane sum to 180°—makes the two evolutions identical, so success for one planet implies success for the other. The paper's population-level estimate rests on a Bayesian reduction: for twin binaries, P(both planets become hot Jupiters | one does) equals P(the second planet becomes a hot Jupiter), turning the four-body problem into a single-planet geometry problem whose 15.3%-of-orientations window, multiplied by the 59.9% simulation success rate within that window, yields the ~9% upper limit.

What would settle it

A survey of the secondary stars of known hot-Jupiter-hosting binaries with separations 100-2000 AU that finds zero hot Jupiters around the secondaries in ~50 systems would rule out the ~9% prediction at high confidence; equivalently, measuring the conditional probability P(cold Jupiter around the companion | cold Jupiter around the primary) and finding it close to the field rate would show the predicted rate should be reduced by roughly an order of magnitude.

Watch

Extended reading notes

Core claim

The paper's central claim is that double hot Jupiter systems can arise through simultaneous ZLK migration in stellar binaries, and that this channel is efficient enough to be observable. In a perfectly mirrored four-body configuration—two equal-mass stars, two identical cold Jupiters, with planetary orbital planes inclined by 83° and 97° to the binary plane—both planets undergo the same secular eccentricity-inclination cycles and both circularize into hot Jupiters within a few hundred million years. Relaxing the symmetry, the authors find that inclination asymmetries dominate the outcome while stellar and planetary mass asymmetries mainly shift formation timescales; in equal-mass 'twin' binaries the formation time is minimized. Combining the simulation success rate with a Bayesian argument that the joint formation probability factors into independent per-planet probabilities, they predict that up to ~9% of known hot-Jupiter-hosting binaries with projected separations s ≤ 2000 AU could host a second hot Jupiter, and that the most favorable targets are twin binaries whose orbits bring the stars to pericenter distances of a few hundred AU.

Load-bearing premise

The headline ~9% rate rests on assuming that a cold Jupiter forms around both stars in a binary whenever it forms around one, and that the companion planet's orbit has a favorable inclination; observed cold-Jupiter occurrence around FGK stars is only ~10-15%, and the correlation of planet formation across twin binary components is not yet measured.

Editorial extensions

If this is right

  • A deliberate search of the companion stars of known hot Jupiter hosts in binaries with separations up to 2000 AU should uncover second hot Jupiters at a rate bounded by about 9%.
  • Blind surveys for ZLK-migrated double hot Jupiters should prioritize twin, equal-mass binaries with stellar pericenter approaches of a few hundred AU, where formation is fastest and the success rate is highest.
  • In unequal-mass binaries, if a hot Jupiter formed around the more massive star, the second hot Jupiter is likely to have already formed around the less massive companion, making secondary stars the best targets.
  • For a close, eccentric binary with a*=200 AU and e*=0.7, roughly 46% of isotropically drawn orientations produce double hot Jupiters within 13.8 Gyr, and the rate stays near 33% within 2.3 Gyr.
  • If the mechanism operates, the occurrence of double hot Jupiters should correlate with binary properties—closer pericenters and equal masses—so the predicted systems should be clustered in that region of binary parameter space.

Reading between the lines

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

  • The ~9% figure is best read as an upper limit rather than an expectation: it assumes a cold Jupiter exists around both stars, whereas the field rate of cold Jupiters around FGK stars is only ~10-15%, so the actual yield of a survey could be several times lower if planet formation is not strongly correlated across twin components.
  • The paper's assumption of isotropic orbital orientations may overestimate the rate, because several recent studies find an excess of low mutual inclinations in planet-hosting binaries; folding in that measured inclination distribution would be a direct test of the channel's contribution to the observed population.
  • The same mirrored-migration logic should apply to other close-in planet populations formed by high-eccentricity migration, such as hot Neptunes or super-Earths, so the 'double close-in planet' prediction is not restricted to Jupiter-mass planets.
  • Because the mechanism predicts that second hot Jupiters preferentially orbit the lower-mass star in unequal binaries, measuring the mass ratio of the stellar binary in a survey design directly concentrates the search; this is a testable prioritization strategy the paper gestures toward but does not quantify.
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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. This paper studies the formation of double hot Jupiter (HJ) systems in stellar binaries through simultaneous von Zeipel-Lidov-Kozai (ZLK) migration of two cold Jupiters, one around each star. Using four-body N-body simulations with REBOUND/IAS15 including general relativity and equilibrium tides, the authors first demonstrate a proof of concept in a perfectly mirrored fiducial system, then examine robustness to asymmetries in stellar mass, planetary mass, and initial inclination. They also compare four-body and three-body secular behavior in Appendix A. The paper's main quantitative claim is a population synthesis based on Gaia-resolved HJ-hosting binaries with projected separation s ≤ 2000 AU, from which they derive an upper limit of ~9% for the occurrence of double HJ systems among such binaries, under the stated assumptions that most HJs in binaries form via ZLK migration and that a cold Jupiter forms around both components. The paper recommends that searches for ZLK-migrated double HJs prioritize twin binaries with pericenter approaches of a few hundred AU.

Significance. The core proof of concept is a genuine and useful contribution: the four-body simulations directly demonstrate that mirrored cold Jupiters can both migrate to form double HJs, and they appropriately model the planet-planet interaction that is absent from the standard three-body treatments. The robustness study and the concrete target selection (twin binaries, q ≈ 200 AU) are valuable for future observing programs, and the prediction is falsifiable in principle. The main weakness is that the headline ~9% rate is not the marginal expectation under the assumptions stated in the abstract: it is conditional on a fixed near-orthogonal inclination for the second planet and on a favorable binary pericenter, and it also assumes a cold Jupiter around both stars. These conditionals materially change the rate that a comprehensive, unselected survey would see, and the manuscript should either marginalize over the second planet's orientation or present the number transparently as a conditional upper bound rather than a projected occurrence rate.

major comments (3)
  1. [Section 4 and Abstract] The headline ~9% rate is computed with the second planet's initial inclination fixed at I2 = 93° (Section 4: 'We kept the initial inclination I2 fixed at 93°'), while the introduction states that 'Assuming isotropic initial orbital orientations' the authors find a ~9% rate. For an independent isotropic distribution of I2, the joint probability that both planets fall in the successful band in cos I1 is 0.153^2 ≈ 2.3%, and with the reported 59.9% in-band success fraction the expected rate drops to ~1.4%. Even under perfectly aligned planetary orbits (I2 = I1) the rate is 9%, but that is not one of the two assumptions listed in the abstract. Because the abstract and introduction present 9% as the outcome of a comprehensive search under the stated assumptions, a reader would overestimate the expected occurrence rate by roughly a factor of six. The authors should either run simulations that marginalize over I2 with an isotropic prior, or explicitly and prominently state that the 9% is conditional on the second cold Jupiter's orbit being near-orthogonal (I2 ≈ 90°) and revise the abstract accordingly.
  2. [Section 3.1] The sentence 'For an isotropic set of inclinations for the second planet, the overall success rate of producing double hot Jupiters in the lifetime of the universe is ~46%' is ambiguous and potentially misleading in light of the simulation description that immediately precedes it. The text and Figure 3 state that the simulations are 'initiated uniformly in cos I1', but do not state how I2 was sampled. If I2 was set to the mirror value 180° − I1, then the second planet's orientations are not drawn independently from an isotropic distribution, and the ~46% is a rate for perfectly mirrored systems, not a marginal rate over independent I2. If instead I2 was varied independently, that should be stated explicitly along with the sampling procedure. This ambiguity also affects how the rate connects to Equation (4), where the probability for the second planet is meant to be a function of its own geometry only.
  3. [Section 4, population synthesis setup] The population synthesis assigns every binary a fixed pericenter a*(1−e*) = 200 AU and adopts a* = s (the projected separation), so the resulting ~9% is a simultaneous best-case evaluation over several favorable choices: the second planet's inclination (I2 = 93°), the binary pericenter, and the assumption of a cold Jupiter around both stars. The text does call the result an 'upper limit,' but the abstract and conclusions do not carry the same hedging, and the word 'up to' does not communicate how many separate favorably chosen conditions are being stacked. For the paper's central forecast to be usable, the abstract and the Section 4 discussion should explicitly state that the 9% is conditional on a near-orthogonal second planet, a few-hundred-AU binary pericenter, and the presence of a primordial cold Jupiter around both stars, none of which are guaranteed in an unselected sample of known HJ-hosting binaries.
minor comments (5)
  1. [Abstract] The phrase 'to producedoublehot Jupiter systems' is missing spaces between 'produce', 'double', and 'hot'; this appears to be a LaTeX rendering artifact and should be fixed in the source.
  2. [Figure 4 caption] The caption says 'The shaded region spans a ±1 ZLK cycle error bar as defined in Equation 2,' but Equation (2) is a proportionality relation for a timescale and does not define an error bar; please state how the ±1-cycle uncertainty is computed from the simulation scatter.
  3. [Section 4] The sentence 'The rest of the system parameters, including masses, were set to those shown in Table 1' is confusing because Table 1 includes the fiducial inclinations (I1 = 83°, I2 = 97°) that are explicitly overridden in this section; please state which parameters are retained and which are replaced by the grid or the fixed I2 = 93°.
  4. [Section 5.4] The phrase 'isochrones are so closely separated' is awkward; consider 'isochrones are so closely spaced in age' or similar.
  5. [Throughout] The manuscript would benefit from a data and code availability statement, since the simulations are central to the results and the public packages are named but the initial conditions, analysis scripts, and reproduction details are not deposited.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the ~9% double-hot-Jupiter rate is a forward simulation product, and the fixed-I2 and cold-Jupiter-occurrence caveats are scope limitations rather than circular reductions.

full rationale

The central chain is a forward N-body calculation rather than an inversion or fit. Section 3.1 draws success rates from an isotropic grid in cos I1 for the fiducial binary, and Section 4 extends this to a* <= 2000 AU, reporting: 'This region is 15.3% of the parameter space. Within this region, 59.9% of the systems formed two hot Jupiters, placing an upper limit of ~9%...' The 9% is a simulated success fraction multiplied by a geometric width in cos I1; no equation in the paper reduces the predicted rate to an input by construction. The only analytic overplot, Equation (2), is explicitly 'scaled ... by the ratio of the average simulation timescale to the average analytic value' and is used for trend display, not as a fitted predictor of the headline quantity. Self-citations (Lu et al. 2023 for the tides_spin and BS-integrator codes, Rice et al. 2022 for the q < 0.1 AU hot-Jupiter criterion, Rice et al. 2024 for observed inclination trends, Hand et al. 2025 for the unconstrained correlation caveat) supply tools, thresholds, or background; none is a uniqueness theorem or the source of the central rate. The paper also flags its own load-bearing assumptions: Section 4 says 'it is as-yet unclear the extent to which the outcomes of planet formation are correlated across the two components of twin binary star systems,' and Section 5.1 notes the observed cold-Jupiter occurrence is only '~10-15%'. A separate, non-circularity concern is that Section 4 'kept the initial inclination I2 fixed at 93°' while the abstract describes isotropic orientations; this makes the ~9% an upper limit conditional on a favorable second-planet orientation rather than a marginal expectation. That is a modeling and assumption-statement issue, not a reduction of the output to the input. No circular step was found.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

The 9% prediction is a forward simulation product with explicit population assumptions (ZLK dominance in binaries, correlated cold Jupiter formation in twins), an isotropic inclination prior, and optimistic dynamical choices (fixed 200 AU pericenter, I2=93 deg, Q-rescaling, hot Jupiter survival). No new physical entities are introduced; the free-parameter count is small because the study is not a fit.

free parameters (1)
  • tidal quality factor Q_HJ = 10^3, rescaled to 3x10^5 for reported timescales
    Chosen by hand to speed up simulations; reported formation times are obtained by linear rescaling to Q=3x10^5. This affects absolute timescales and thus success rates within the adopted age limits.
assumptions (8)
  • domain assumption Most hot Jupiters in binary star systems form through ZLK migration of primordially formed cold Jupiters
    Assumption (1) in the Abstract; the population-level interpretation of the 9% rate is conditional on this channel dominating the known hot Jupiter sample.
  • domain assumption If one star in a binary system forms a cold Jupiter, the second does as well
    Assumption (2) in the Abstract; used in Section 4 to posit a cold Jupiter around the companion star. Section 5.1 notes observed cold Jupiter occurrence is ~10-15% and cites Hand et al. (2025) for the unconstrained correlation in twins.
  • domain assumption Isotropic distribution of initial planetary orbital inclinations
    Used for sampling I1 in Sections 3.1 and 4; Section 5.2 discusses observational hints of a coplanarity excess at a<~700 AU and argues the trend is weakest for hot Jupiter hosts.
  • domain assumption Equilibrium tide theory (Eggleton et al. 1998) adequately describes tidal dissipation in high-eccentricity ZLK cycles
    Section 5.3: dynamical tides would dissipate more at pericenter and form hot Jupiters faster; the paper defers this to future work.
  • domain assumption Linear rescaling of tidal quality factor from Q=10^3 to Q=3x10^5 gives realistic formation timescales
    Section 2.2: runs use an artificially dissipative planet; the paper assumes the ZLK cycle period and amplitude are unaffected by tidal dissipation strength, so reported timescales scale linearly with Q.
  • domain assumption Population synthesis binaries have pericenter q=a*(1-e*)=200 AU and a* set equal to projected separation s
    Section 4: binary eccentricities are poorly constrained; fixing q=200 AU is an optimistic configuration for ZLK efficiency, and a*=s overestimates semimajor axis for e* not equal to 0.
  • domain assumption A planet reaching q<0.1 AU counts as a hot Jupiter and survives thereafter
    Sections 2.2 and 5.4: success is scored at q<0.1 AU; the paper notes that post-formation inspiral and engulfment are not modeled, so rates are upper limits especially at high mutual inclination.
  • domain assumption Merging the first formed hot Jupiter into its host star preserves the dynamics of the remaining planet
    Section 2.2 and Appendix A: after HJ1 circularizes it is manually collided with its host star to avoid timestep bottlenecks; total perturbing mass is conserved but the perturber is relocated to the stellar position.

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Pith. "Pith review of The Formation of Double Hot Jupiter Systems through von Zeipel-Lidov-Kozai Migration." pith.science (2026). https://pith.science/paper/UZPWNEJG

@misc{pith2026250504398,
  author       = {Pith},
  title        = {Pith review of: The Formation of Double Hot Jupiter Systems through von Zeipel-Lidov-Kozai Migration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UZPWNEJG}},
  note         = {Machine review of arXiv:2505.04398}
}
abstract

The von Zeipel-Lidov-Kozai (ZLK) mechanism with tidal friction has been demonstrated as a promising avenue to generate hot Jupiters in stellar binary systems. Previous population studies of hot Jupiter formation have largely examined this mechanism in systems comprised of three bodies: two stars and one planet. However, because stars in a binary system form in similar environments with comparable metallicities, the formation of a single hot Jupiter in such a system may imply that the conditions are more likely met for the companion star, as well. We investigate the ZLK mechanism with tidal friction as a potential mechanism to produce double hot Jupiter systems in stellar binaries. Using N-body simulations, we characterize the evolution of two cold Jupiters, each orbiting one star in a binary system, undergoing mirrored ZLK migration. We then examine the robustness of this mechanism to asymmetries in stellar masses, planet masses, and planet orbital inclinations relative to the binary plane. We predict that, under the assumptions that (1) most hot Jupiters in binary star systems form through ZLK migration of primordially formed cold Jupiters and (2) if one star in a binary system forms a cold Jupiter, the second does as well, a comprehensive search could identify double hot Jupiters in up to ~9% of the close- to moderate- separation $a<2000$ AU) binary systems that already host a known hot Jupiter. We also argue that a blind search for ZLK-migrated double hot Jupiters should prioritize twin stellar binaries with pericenter approaches of a few hundred AU.

Figures

Figures reproduced from arXiv: 2505.04398 by the authors.

Figure 1
Figure 1. Schematic of the idealized, symmetric binary system with planetary orbital inclinations I1 = I2. The two planets have the same initial mass, radius, orbital semimajor axis, and orbital eccentricity, and the mutual inclination of the planets with respect to the binary plane adds up to 180◦ . The stars also have the same mass and radius as each other. The symmetric binary system is the most optimal for double hot Jupi… view at source ↗
Figure 2
Figure 2. Semimajor axis, eccentricity, and inclination evo￾lution of the two planetary orbits undergoing ZLK oscilla￾tions. The left and right panels show the evolution of the first and second planet, respectively. The planet 1 evolution is cut off upon manually colliding the planet with its host star after it has become a hot Jupiter. The continued incli￾nation evolution of planet 1 after ZLK cycles conclude is due to prece… view at source ↗
Figure 3
Figure 3. Success rates of double hot Jupiter production as a function of cos I1. The rates are calculated from an isotropic sample of numerical simulations initiated uniformly in cos I1. The provided times are in “real time,” translated from simulation time. The binary semimajor axis, binary eccentricity, and initial planet eccentricities are set to a∗ = 200 AU, e∗ = 0.7, and e1,2 = 0 for this set of simulations. derive the … view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The hot Jupiter formation (HJF) time, in real time, of 4-body double hot Jupiter simulations with asym￾metric masses. System parameters are adopted from [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Left: Hot Jupiter status after simulating for the age of the universe as a function of stellar binary semimajor axis (set such that q = 200 AU) and initial cos I1. The dark blue points are hot Jupiters (with a1(1 − e1) < 0.1 AU), whereas the light blue simulations did …
Figure 6
Figure 6. Figure 6: Maximum planetary orbit peak eccentricity difference between (1) three-body binary star systems with one planet and (2) four-body binary star systems with two planets. Light blue points indicate that at least one simulation in the pair produced an unbound planet. The p…
Figure 7
Figure 7. Figure 7: Comparison of eccentricity evolution between the traditional three-body ZLK setup, with only one planet, and our fiducial 4-body simulation with two planets. System parameters are set to a∗ = 700AU, e∗ = 0.5, I1 = 99◦ . The companion planet produces a phase shift in th…

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Works this paper leans on

151 extracted references · 56 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address archivePrefix author booktitle chapter doi edition editor eprint howpublished institution journal key month note number organization pages publisher school series title misctitle type volume year version url label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.co...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION format.doi doi empty "" "doi:" doi * if FUNCTION format.url url empty "" new.block "" url * "" * if FUNCTION format.eprint eprint empty "" archivePrefix empty "" archivePrefix ":" * if eprint field.or.null * if FUNCTION format.pid eprint empty format.doi format.eprint if FUNCTION n.dashify 't := "" t...

  3. [3]

    EXOFASTv2: Generalized publication-quality exoplanet modeling code

    thebibliography [1] 20pt to REFERENCES 6pt =0pt -12pt 10pt plus 3pt =0pt =0pt =1pt plus 1pt =0pt =0pt -12pt =13pt plus 1pt =20pt =13pt plus 1pt \@M =10000 =-1.0em =0pt =0pt 0pt =0pt =1.0em @enumiv\@empty 10000 10000 `\.\@m \@noitemerr \@latex@warning Empty `thebibliography' environment \@ifnextchar \@reference \@latexerr Missing key on reference command E...

    arXiv 2021

  4. [4]

    2019, Geosciences, 9, 105

    Adibekyan , V. 2019, Geosciences, 9, 105

    2019

  5. [5]

    R., Storch , N

    Anderson , K. R., Storch , N. I., & Lai , D. 2016, , 456, 3671

    2016

  6. [6]

    D., Foreman-Mackey , D., et al

    Angus , R., Morton , T. D., Foreman-Mackey , D., et al. 2019, , 158, 173

    2019

  7. [7]

    Antognini , J. M. O. 2015, , 452, 3610

    2015

  8. [8]

    2020, , 496, 2605

    Bak s , V., Eker , Z., Sar , O., Y \"u cel , G., & Sonba s , E. 2020, , 496, 2605

    2020

Show all 151 references
  1. [9]

    \'A ., Howard , A

    Bakos , G. \'A ., Howard , A. W., Noyes , R. W., et al. 2009, , 707, 446

    2009

  2. [10]

    Barnes , S. A. 2003, , 586, 464

    2003

  3. [11]

    2012, , 751, 119

    Beaug \'e , C., & Nesvorn \'y , D. 2012, , 751, 119

    2012

  4. [12]

    2020, , 160, 254

    Becker , J., Batygin , K., Fabrycky , D., et al. 2020, , 160, 254

    2020

  5. [13]

    \'A ., Hartman , J., et al

    B \'e ky , B., Bakos , G. \'A ., Hartman , J., et al. 2011, , 734, 109

    2011

  6. [14]

    2022, , 516, 75

    Beleznay , M., & Kunimoto , M. 2022, , 516, 75

    2022

  7. [15]

    2012, , 545, A88

    Beust , H., Bonfils , X., Montagnier , G., Delfosse , X., & Forveille , T. 2012, , 545, A88

    2012

  8. [16]

    N., Leconte , J., Hersant , F., & Correia , A

    Bolmont , E., Raymond , S. N., Leconte , J., Hersant , F., & Correia , A. C. M. 2015, , 583, A116

    2015

  9. [17]

    J., Lunine , J., Stevenson , D., et al

    Bolton , S. J., Lunine , J., Stevenson , D., et al. 2017, , 213, 5

    2017

  10. [18]

    B., Matsumura, S., & Rasio, F

    Chatterjee, S., Ford, E. B., Matsumura, S., & Rasio, F. A. 2008, , 686, 580

    2008

  11. [19]

    2023, Proceedings of the National Academy of Sciences, 120, e2304179120

    Chen, D.-C., Xie, J.-W., Zhou, J.-L., et al. 2023, Proceedings of the National Academy of Sciences, 120, e2304179120

    2023

  12. [20]

    2017, , 834, 17

    Chen , J., & Kipping , D. 2017, , 834, 17

    2017

  13. [21]

    2022, , 163, 207

    Christian, S., Vanderburg, A., Becker, J., et al. 2022, , 163, 207

    2022

  14. [22]

    2024, arXiv e-prints, arXiv:2405.10379

    Christian , S., Vanderburg , A., Becker , J., et al. 2024, arXiv e-prints, arXiv:2405.10379

    2024 arXiv

  15. [23]

    L., Angus , R., David , T., et al

    Colman , I. L., Angus , R., David , T., et al. 2024, , 167, 189

    2024

  16. [24]

    P., Marcy, G

    Cumming, A., Butler, R. P., Marcy, G. W., et al. 2008, Publications of the Astronomical Society of the Pacific, 120, 531

    2008

  17. [25]

    I., & Johnson, J

    Dawson, R. I., & Johnson, J. A. 2018, , 56, 175

    2018

  18. [26]

    1991, , 181, 313

    Demircan , O., & Kahraman , G. 1991, , 181, 313

    1991

  19. [27]

    P., & Farr , W

    Denham , P., Naoz , S., Hoang , B.-M., Stephan , A. P., & Farr , W. M. 2019, , 482, 4146

    2019

  20. [28]

    2013, , 51, 269

    Duch \^e ne , G., & Kraus , A. 2013, , 51, 269

    2013

  21. [29]

    J., Kraus, A

    Dupuy, T. J., Kraus, A. L., Kratter, K. M., et al. 2022, , 512, 648

    2022

  22. [30]

    P., Kiseleva , L

    Eggleton , P. P., Kiseleva , L. G., & Hut , P. 1998, , 499, 853

    1998

  23. [31]

    1916, Annalen der Physik, 49, 769

    Einstein, A. 1916, Annalen der Physik, 49, 769

    1916

  24. [32]

    El-Badry, K., Rix, H.-W., & Heintz, T. M. 2021, , 506, 2269

    2021

  25. [33]

    2019, , 489, 5822

    El-Badry , K., Rix , H.-W., Tian , H., Duch \^e ne , G., & Moe , M. 2019, , 489, 5822

    2019

  26. [34]

    2007, , 669, 1298

    Fabrycky, D., & Tremaine, S. 2007, , 669, 1298

    2007

  27. [35]

    H., Naoz , S., Li , G., Rice , M., & Inzunza , N

    Faridani , T. H., Naoz , S., Li , G., Rice , M., & Inzunza , N. 2025, , 978, 18

    2025

  28. [36]

    A., & Valenti, J

    Fischer, D. A., & Valenti, J. 2005, , 622, 1102

    2005

  29. [37]

    J., Rosenthal, L

    Fulton, B. J., Rosenthal, L. J., Hirsch, L. A., et al. 2021, , 255, 14

    2021

  30. [38]

    Gaia Collaboration , Prusti , T., de Bruijne , J. H. J., et al. 2016, , 595, A1

    2016

  31. [39]

    Gaia Collaboration , Brown , A. G. A., Vallenari , A., et al. 2021, , 649, A1

    2021

  32. [40]

    Gaia Collaboration , Vallenari , A., Brown , A. G. A., et al. 2023, , 674, A1

    2023

  33. [41]

    2024, , 967, 74

    Gan , T., Guo , K., Liu , B., et al. 2024, , 967, 74

    2024

  34. [42]

    J., Christian , S., & Vanderburg , A

    Gerbig , K., Rice , M., Zanazzi , J. J., Christian , S., & Vanderburg , A. 2024, , 972, 161

    2024

  35. [43]

    T., & Johnson, J

    Ghezzi, L., Montet, B. T., & Johnson, J. A. 2018, The Astrophysical Journal, 860, 109. http://dx.doi.org/10.3847/1538-4357/aac37c

    2018 doi

  36. [44]

    Habets , G. M. H. J., & Heintze , J. R. W. 1981, , 46, 193

    1981

  37. [45]

    H., & Schlaufman , K

    Hamer , J. H., & Schlaufman , K. C. 2019, , 158, 190

    2019

  38. [46]

    S., Fragione , G., Neunteufel , P., & Kocsis , B

    Hamers , A. S., Fragione , G., Neunteufel , P., & Kocsis , B. 2021, , 506, 5345

    2021

  39. [47]

    S., & Lai , D

    Hamers , A. S., & Lai , D. 2017, , 470, 1657

    2017

  40. [48]

    E., Rice , M., & Gerbig , K

    Hand , J. E., Rice , M., & Gerbig , K. 2025, arXiv e-prints, arXiv:2503.08583

    2025 arXiv

  41. [49]

    R., Millman, K

    Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357

    2020

  42. [50]

    D., Bakos , G

    Hartman , J. D., Bakos , G. \'A ., B \'e ky , B., et al. 2012, , 144, 139

    2012

  43. [51]

    P., & Rauer , H

    Hatzes , A. P., & Rauer , H. 2015, , 810, L25

    2015

  44. [52]

    A., Rosenthal , L., Fulton , B

    Hirsch , L. A., Rosenthal , L., Fulton , B. J., et al. 2021, , 161, 134

    2021

  45. [53]

    1997, Nature, 386, 254

    Holman, M., Touma, J., & Tremaine, S. 1997, Nature, 386, 254

    1997

  46. [54]

    J., & Wiegert , P

    Holman , M. J., & Wiegert , P. A. 1999, , 117, 621

    1999

  47. [55]

    W., Johnson, J

    Howard, A. W., Johnson, J. A., Marcy, G. W., et al. 2010, , 721, 1467

    2010

  48. [56]

    W., Marcy , G

    Howard , A. W., Marcy , G. W., Bryson , S. T., et al. 2012, , 201, 15

    2012

  49. [57]

    Hunter, J. D. 2007, Computing in science & engineering, 9, 90

    2007

  50. [58]

    2022, , 933, L32

    Hwang , H.-C., El-Badry , K., Rix , H.-W., et al. 2022, , 933, L32

    2022

  51. [59]

    2008, , 683, L191

    Jang-Condell , H., Mugrauer , M., & Schmidt , T. 2008, , 683, L191

    2008

  52. [60]

    I., Mahadevan , S., et al

    Kanodia , S., Ca \ n as , C. I., Mahadevan , S., et al. 2024, , 167, 161

    2024

  53. [61]

    R., Parker , R

    King , R. R., Parker , R. J., Patience , J., & Goodwin , S. P. 2012, , 421, 2025

    2012

  54. [62]

    Y., & Katz , B

    Klein , Y. Y., & Katz , B. 2024, , 167, 80

    2024

  55. [63]

    1962, , 67, 591

    Kozai, Y. 1962, , 67, 591

    1962

  56. [64]

    Kratter , K. M. 2011, in Astronomical Society of the Pacific Conference Series, Vol. 447, Evolution of Compact Binaries, ed. L. Schmidtobreick , M. R. Schreiber , & C. Tappert , 47

    2011

  57. [65]

    1997, , 490, 847

    Lai , D. 1997, , 490, 847

    1997

  58. [66]

    2009, , 459, 957

    Lainey , V., Arlot , J.-E., Karatekin , \"O ., & van Hoolst , T. 2009, , 459, 957

    2009

  59. [67]

    A., Tajeddine , R., et al

    Lainey , V., Jacobson , R. A., Tajeddine , R., et al. 2017, , 281, 286

    2017

  60. [68]

    2010, , 516, A64

    Leconte , J., Chabrier , G., Baraffe , I., & Levrard , B. 2010, , 516, A64

    2010

  61. [69]

    V., Howell , S

    Lester , K. V., Howell , S. B., Matson , R. A., et al. 2023, , 166, 166

    2023

  62. [70]

    2009, , 692, L9

    Levrard , B., Winisdoerffer , C., & Chabrier , G. 2009, , 692, L9

    2009

  63. [71]

    2014, , 791, 86

    Li , G., Naoz , S., Holman , M., & Loeb , A. 2014, , 791, 86

    2014

  64. [72]

    Lidov, M. L. 1962, Planetary and Space Science, 9, 719

    1962

  65. [73]

    Lin , D. N. C., & Ida , S. 1997, , 477, 781

    1997

  66. [74]

    J., & Lai , D

    Liu , B., Mu \ n oz , D. J., & Lai , D. 2015, , 447, 747

    2015

  67. [75]

    Love , A. E. H. 1909, Proceedings of the Royal Society of London Series A, 82, 73

    1909

  68. [76]

    2025, , 979, 218

    Lu , T., An , Q., Li , G., et al. 2025, , 979, 218

    2025

  69. [77]

    2023, , 948, 41

    Lu , T., Rein , H., Tamayo , D., et al. 2023, , 948, 41

    2023

  70. [78]

    Mardling , R. A. 1995, , 450, 722

    1995

  71. [79]

    J., & Rasio , F

    Matsumura , S., Peale , S. J., & Rasio , F. A. 2010, , 725, 1995

    2010

  72. [80]

    2010, in Proceedings of the 9th Python in Science Conference, Vol

    McKinney, W. 2010, in Proceedings of the 9th Python in Science Conference, Vol. 445, Austin, TX, 51--56

    2010

  73. [81]

    2014, , 211, 24

    McQuillan , A., Mazeh , T., & Aigrain , S. 2014, , 211, 24

    2014

  74. [82]

    2024, , 527, 3183

    Michel , K.-U., & Mugrauer , M. 2024, , 527, 3183

    2024

  75. [83]

    2019, Nature Astronomy, 3, 424

    Millholland , S., & Laughlin , G. 2019, Nature Astronomy, 3, 424

    2019

  76. [84]

    2023, , 166, 209

    Miyazaki , S., & Masuda , K. 2023, , 166, 209

    2023

  77. [85]

    2017, , 230, 15

    Moe , M., & Di Stefano , R. 2017, , 230, 15

    2017

  78. [86]

    Moe , M., & Kratter , K. M. 2021, , 507, 3593

    2021

  79. [87]

    V., Knutson , H., Line , M., et al

    Morley , C. V., Knutson , H., Line , M., et al. 2017, , 153, 86

    2017

  80. [88]

    2024, , 686, A296

    M \"u ller , S., Baron , J., Helled , R., Bouchy , F., & Parc , L. 2024, , 686, A296

    2024

  81. [89]

    2008, , 678, 498

    Nagasawa , M., Ida , S., & Bessho , T. 2008, , 678, 498

    2008

  82. [90]

    2016, , 54, 441

    Naoz, S. 2016, , 54, 441

    2016

  83. [91]

    M., Lithwick, Y., Rasio, F

    Naoz, S., Farr, W. M., Lithwick, Y., Rasio, F. A., & Teyssandier, J. 2011, Nature, 473, 187

    2011

  84. [92]

    M., Lithwick , Y., Rasio , F

    Naoz , S., Farr , W. M., Lithwick , Y., Rasio , F. A., & Teyssandier , J. 2013 a , , 431, 2155

    2013

  85. [93]

    M., & Rasio, F

    Naoz, S., Farr, W. M., & Rasio, F. A. 2012, , 754, L36

    2012

  86. [94]

    2013 b , , 773, 187

    Naoz , S., Kocsis , B., Loeb , A., & Yunes , N. 2013 b , , 773, 187

    2013

  87. [95]

    2024, Planetary Systems Composite Parameters, vVersion: 2024-06-20, NExScI-Caltech/IPAC, doi:10.26133/NEA13

    NASA Exoplanet Archive . 2024, Planetary Systems Composite Parameters, vVersion: 2024-06-20, NExScI-Caltech/IPAC, doi:10.26133/NEA13. https://catcopy.ipac.caltech.edu/dois/doi.php?id=10.26133/NEA13

    2024 doi

  88. [96]

    R., et al

    Neveu-VanMalle , M., Queloz , D., Anderson , D. R., et al. 2014, , 572, A49

    2014

  89. [97]

    2016, , 586, A93

    ---. 2016, , 586, A93

    2016

  90. [98]

    X., Standish , E

    Newhall , X. X., Standish , E. M., & Williams , J. G. 1983, , 125, 150

    1983

  91. [99]

    A., Hinkley , S., et al

    Ngo , H., Knutson , H. A., Hinkley , S., et al. 2016, , 827, 8

    2016

  92. [100]

    E., Liu , B., & Lai , D

    O'Connor , C. E., Liu , B., & Lai , D. 2021, , 501, 507

    2021

  93. [101]

    S., Dunham, M

    Offner, S. S., Dunham, M. M., Lee, K. I., Arce, H. G., & Fielding, D. B. 2016, The Astrophysical Journal Letters, 827, L11

    2016

  94. [102]

    Ogilvie, G. I. 2014, , 52, 171

    2014

  95. [103]

    Oliphant, T. E. 2006, A guide to NumPy, Vol. 1 (Trelgol Publishing USA)

    2006

  96. [104]

    2019, Monthly Notices of the Royal Astronomical Society, 491, 4481–4487

    Osborn, A., & Bayliss, D. 2019, Monthly Notices of the Royal Astronomical Society, 491, 4481–4487. http://dx.doi.org/10.1093/mnras/stz3207

    2019 doi

  97. [105]

    K., Winters , J

    Pass , E. K., Winters , J. G., Charbonneau , D., et al. 2023, , 166, 11

    2023

  98. [106]

    M., Shappee , B

    Pejcha , O., Antognini , J. M., Shappee , B. J., & Thompson , T. A. 2013, , 435, 943

    2013

  99. [107]

    2015, , 799, 27

    Petrovich , C. 2015, , 799, 27

    2015

  100. [108]

    Pham , D., Rein , H., & Spiegel , D. S. 2024, The Open Journal of Astrophysics, 7, 1

    2024

  101. [109]

    A., Ngo , H., et al

    Piskorz , D., Knutson , H. A., Ngo , H., et al. 2015, , 814, 148

    2015

  102. [110]

    2004, , 351, 487

    Pont , F., & Eyer , L. 2004, , 351, 487

    2004

  103. [111]

    2018, , 155, 157

    Puranam , A., & Batygin , K. 2018, , 155, 157

    2018

  104. [112]

    A., & Ford, E

    Rasio, F. A., & Ford, E. B. 1996, Science, 274, 954

    1996

  105. [113]

    2012, , 537, A128

    Rein, H., & Liu, S.-F. 2012, , 537, A128

    2012

  106. [114]

    Rein , H., & Spiegel , D. S. 2015, , 446, 1424

    2015

  107. [115]

    M., Connelley , M

    Reipurth , B., Guimar \ a es , M. M., Connelley , M. S., & Bally , J. 2007, , 134, 2272

    2007

  108. [116]

    2012, , 492, 221

    Reipurth , B., & Mikkola , S. 2012, , 492, 221

    2012

  109. [117]

    2024, , 167, 126

    Rice , M., Gerbig , K., & Vanderburg , A. 2024, , 167, 126

    2024

  110. [118]

    2022, , 926, L17

    Rice, M., Wang, S., & Laughlin, G. 2022, , 926, L17

    2022

  111. [119]

    I., & Stahler , S

    Sadavoy , S. I., & Stahler , S. W. 2017, , 469, 3881

    2017

  112. [120]

    2024, , 682, L23

    Saffe , C., Miquelarena , P., Alacoria , J., et al. 2024, , 682, L23

    2024

  113. [121]

    Simon , M., & Obbie , R. C. 2009, , 137, 3442

    2009

  114. [122]

    2019, Science, 364, 1046

    Spilker , L. 2019, Science, 364, 1046

    2019

  115. [123]

    G., Oelkers , R

    Stassun , K. G., Oelkers , R. J., Paegert , M., et al. 2019, , 158, 138

    2019

  116. [124]

    Sterne , T. E. 1939, , 99, 451

    1939

  117. [125]

    I., & Lai , D

    Storch , N. I., & Lai , D. 2014, , 438, 1526

    2014

  118. [126]

    Tamayo, D., Rein, H., Shi, P., & Hernandez, D. M. 2019, Monthly Notices of the Royal Astronomical Society, 491, 2885

    2019

  119. [127]

    R., Huber , D., & van Saders , J

    Tayar , J., Claytor , Z. R., Huber , D., & van Saders , J. 2022, , 927, 31

    2022

  120. [128]

    2015, , 456, 2070

    Tokovinin, A., & Kiyaeva, O. 2015, , 456, 2070

    2015

  121. [129]

    2020, , 491, 5158

    Tokovinin , A., & Moe , M. 2020, , 491, 5158

    2020

  122. [130]

    \'A ., Kov \'a cs , G., et al

    Torres , G., Bakos , G. \'A ., Kov \'a cs , G., et al. 2007, , 666, L121

    2007

  123. [131]

    Vick , M., Lai , D., & Anderson , K. R. 2019, , 484, 5645

    2019

  124. [132]

    E., et al

    Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261

    2020

  125. [133]

    2022, , 941, L31

    Vissapragada , S., Chontos , A., Greklek-McKeon , M., et al. 2022, , 941, L31

    2022

  126. [134]

    Vogt , N., Schmidt , T. O. B., Neuh \"a user , R., et al. 2012, , 546, A63

    2012

  127. [135]

    1910, Astronomische Nachrichten, 183, 345

    von Zeipel , H. 1910, Astronomische Nachrichten, 183, 345

    1910

  128. [136]

    Walt, S. v. d., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science & Engineering, 13, 22

    2011

  129. [137]

    Wang , J., & Fischer , D. A. 2015, , 149, 14

    2015

  130. [138]

    Wei, L., Naoz, S., Faridani, T., & Farr, W. M. 2021, The Astrophysical Journal, 923, 118

    2021

  131. [139]

    J., & Marzari , F

    Weidenschilling , S. J., & Marzari , F. 1996, , 384, 619

    1996

  132. [140]

    M., Marcy, G

    Weiss, L. M., Marcy, G. W., Rowe, J. F., et al. 2013, The Astrophysical Journal, 768, 14

    2013

  133. [141]

    C., Naoz , S., & Hansen , B

    Weldon , G. C., Naoz , S., & Hansen , B. M. S. 2024, , 974, 302

    2024

  134. [142]

    2025, , 980, L31

    ---. 2025, , 980, L31

    2025

  135. [143]

    N., Fabrycky, D., Albrecht, S., & Johnson, J

    Winn, J. N., Fabrycky, D., Albrecht, S., & Johnson, J. A. 2010, , 718, L145

    2010

  136. [144]

    A., Wang , S., Horner , J., et al

    Wittenmyer , R. A., Wang , S., Horner , J., et al. 2020, , 492, 377

    2020

  137. [145]

    T., Marcy , G

    Wright , J. T., Marcy , G. W., Howard , A. W., et al. 2012, , 753, 160

    2012

  138. [146]

    2018, , 155, 118

    Wu , Y. 2018, , 155, 118

    2018

  139. [147]

    2003, , 589, 605

    Wu, Y., & Murray, N. 2003, , 589, 605

    2003

  140. [148]

    W., Winn , J

    Yee , S. W., Winn , J. N., & Hartman , J. D. 2021, , 162, 240

    2021

  141. [149]

    W., Winn , J

    Yee , S. W., Winn , J. N., Knutson , H. A., et al. 2020, , 888, L5

    2020

  142. [150]

    2022, , 514, 34

    Y \"u cel , G., & Bak s , V. 2022, , 514, 34

    2022

  143. [151]

    K., & Howard , A

    Zink , J. K., & Howard , A. W. 2023, , 956, L29

    2023

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