Observable signature of magnetic tidal coupling in hierarchical triple systems

Daniele Pica; Davide Panella; Gianluca Grignani; Marta Cocco; Marta Orselli; Troels Harmark

REVIEW 3 major objections 3 minor 6 cited by

Observable signature of magnetic tidal coupling in hierarchical triple systems

T0 review · 3 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Magnetic tidal fields from a supermassive black hole trigger new orbital resonances in a companion compact binary, boosting eccentricity and altering its gravitational-wave signal.

desk verdict Honest, internally consistent extension of the group's electric-tide resonance work to the 0.5PN magnetic term; the new resonances are real, but Eq. (5.13) has a printed error and App. A's CoM decoupling is asserted, not proved. read the letter →

arxiv 2510.24897 v3 pith:MI2QMM3O submitted 2025-10-28 gr-qc astro-ph.GAhep-th

Marta Cocco , Gianluca Grignani , Troels Harmark , Marta Orselli , Davide Panella , Daniele Pica This is my paper

classification gr-qcastro-ph.GAhep-th

keywords magnetictidalbinarycouplingdynamicseccentricityhierarchicalobservable

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

The paper studies a hierarchical triple system: a small binary of two compact objects (stars or black holes) orbits a supermassive black hole. In Newtonian gravity, the outer black hole exerts familiar 'electric' tidal forces on the inner binary, which can shake its orbit. General relativity also predicts a purely relativistic 'magnetic' tidal force, which has no Newtonian counterpart and depends on the velocities of the bodies. The authors add this magnetic tidal interaction at the leading quadrupole order, a 0.5 post-Newtonian correction.

They show that this magnetic coupling creates new resonances: when the inner binary's periastron precession frequency matches certain combinations of the outer orbit's radial and frame-precession frequencies, the inner orbit's eccentricity jumps. These jumps appear in addition to the previously known electric-tide resonances. The authors integrate the Lagrange Planetary Equations numerically and reproduce the predicted jumps, confirming the analytical resonance conditions. They find the effect is strongest for inclined or aligned orbits and grows with the outer orbit's eccentricity.

The authors argue that these eccentricity jumps would leave phase shifts in the gravitational waves emitted by the inner binary, within LISA's frequency band, and that the extra eccentricity slightly accelerates the merger. However, the analytic resonance model is derived for nearly circular outer orbits, and the LISA detectability is asserted qualitatively rather than through a signal-to-noise calculation.

Extended reading notes

Core claim

The paper's core assertion is that 'magnetic tides introduce new resonances absent at lower order, leading to additional eccentricity excitations and significantly modifying the binary's long-term evolution' (Abstract). Concretely, the 0.5PN magnetic tidal coupling yields precession resonances at ˙γ = −Ω_r + Ω_Ψ, Ω_Ψ, and Ω_r + Ω_Ψ (Eq. 5.12), which are not present for the 0PN electric tides alone, and which produce eccentricity jumps visible in the numerical LPE solutions.

Load-bearing premise

The decoupling of the inner binary's center-of-mass from its relative motion at 0.5PN order. The paper asserts, without solving the system (A.9), that the accelerated-frame corrections to Marck's tetrad are of quadrupolar order and therefore negligible (App. A, Eq. A.4–A.9, text after Eq. A.9). If those corrections contributed at the same order as the magnetic tidal coupling, the Hamiltonian (2.18) would omit 0.5PN terms that could shift or suppress the predicted resonances.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Plainly: this is a good paper, not a flawless one, and the central claim survives scrutiny. The new result is that the 0.5PN magnetic tidal coupling (which vanishes for equal masses and was left out of the group's earlier electric-tide analysis) produces precession resonances at γ̇ = −Ω_r + Ω_Ψ, Ω_Ψ, and Ω_r + Ω_Ψ in the strong-field regime, with q=1 in the resonance condition. The Fourier decomposition of the magnetic tidal tensor is clean, the resonance conditions follow without fitted parameters, and the numerical LPE integrations reproduce the predicted jumps. That is real evidence. The paper is also honest about its regime of validity: the analytic model assumes small outer eccentricity, and the numerics extend it only partially.

The weak spots are three, in increasing order of importance. Minor: the LISA claim is an overreach. Staying in the LISA band is not the same as being detectable; there is no SNR or event-rate estimate, so 'observable signature' in the title is too strong. Moderate: Eq. (5.13) as printed is wrong. The factor cos²I/[2(1−2cosI)] diverges at I=π/3 and is nonzero at I=π, contradicting both the text and Fig. 3. The correct factor is (cosI+cos2I) = (1+cosI)(2cosI−1), which vanishes at both angles. The authors appear to have mis-transcribed their own Eq. (5.6); the misprint does not invalidate the resonance conditions, but it must be fixed.

The load-bearing concern is App. A. The authors need the CoM and relative motion to decouple at 0.5PN order so that X=0, a=0, ω=0 can be imposed in the Hamiltonian. They state, without solving Eq. (A.9), that the accelerated-frame corrections are quadrupolar and therefore negligible. That is a hand-wave. The magnetic resonance is itself a 0.5PN quadrupolar effect, so any omitted correction of the same order could shift the resonance locations. The numerical solutions use the same truncated Hamiltonian, so they cannot validate the truncation. This does not sink the paper: the cited literature (Maeda et al. [22], the gravitoelectromagnetic analysis [95]) suggests the corrections are indeed small, but the paper should either solve the deviation equations at leading order or provide a clean scaling argument. As it stands, App. A is the weakest methodological link.

Who this is for: anyone working on b-EMRIs, resonant three-body dynamics, or strong-field tidal effects. The paper deserves a serious referee. Send it to review, with a request that the authors fix Eq. (5.13), replace the hand-wave in App. A with an actual estimate, and soften the LISA detectability claim.

Referee Report

3 major / 3 minor

Summary. The paper studies hierarchical triples consisting of a compact binary orbiting a supermassive black hole, and extends earlier work on precession resonances from 0PN electric tidal fields to 0.5PN magnetic tidal fields. The main claim is that the magnetic quadrupole tidal coupling, which is linear in the outer orbital angular momentum and has odd parity, produces precession resonances with q=1 in the condition \dot\gamma = k\Omega_{\hat r}+l\Omega_{\hat\Psi}. At leading order in the outer eccentricity, three resonances are predicted: \dot\gamma = -\Omega_{\hat r}+\Omega_{\hat\Psi}, \Omega_{\hat\Psi}, and \Omega_{\hat r}+\Omega_{\hat\Psi}. The authors derive the orbit-averaged 0.5PN magnetic Hamiltonian, compute the relevant Fourier coefficients, and verify the resonance crossings numerically through Lagrange Planetary Equations supplemented by Peters-type radiation reaction. They argue that these resonances produce additional eccentricity jumps, accelerate merger, and leave signatures in the LISA band.

Significance. If the derivation is sound, this is a genuinely new strong-gravity effect: it identifies a concrete observable consequence of purely relativistic magnetic tidal fields in a hierarchical triple, with no Newtonian analogue. The core mechanism is elegant and parameter-free in the technical sense that no parameters are fitted to the phenomenology: the q=1 structure follows from the odd parity of the magnetic tidal tensor, and the resonance locations are fixed by the frequencies of the outer geodesic. The paper also provides explicit analytical amplitudes and numerical confirmation of the resonance crossings, which strengthens the main claim. The main risk is not internal inconsistency of the orbit-averaged calculation itself, but the unproved decoupling of center-of-mass and relative motion at 0.5PN order, on which the central Hamiltonian (2.18) rests. That issue, together with a clear algebraic error in the printed resonant Hamiltonian (5.13), prevents acceptance in the present form.

major comments (3)

[Appendix A, Eqs. (A.4)-(A.9); central Hamiltonian (2.18)] The derivation of the 0.5PN magnetic Hamiltonian (2.18) requires that, after using the accelerated frame defined by Eq. (A.4), the corrections to Marck's tetrad do not contribute at quadrupolar order. The paper states this result is verified 'without explicitly solving the system (A.9)' and under the assumption that \hat r is approximately constant. This is not a derivation. The magnetic resonances in Eq. (5.12) are themselves 0.5PN quadrupolar-order effects; any tetrad correction of the same order can modify the Fourier coefficients (5.11), shift the resonance amplitudes, or introduce new Fourier modes. The numerical LPE calculation uses the same truncated Hamiltonian (4.22), so it cannot validate this truncation. The authors should either solve Eq. (A.9) to the required order, or provide an explicit order-by-order argument that the induced corrections to E_{ij} and B_{ij} in Eq. (2.1)
[Sec. V.B, Eq. (5.13)] Equation (5.13) as printed contains the factor \cos^2 I/[2(1-2\cos I)], which diverges at I=\pi/3, is nonzero at I=\pi, and is negative for I<\pi/3. This contradicts the immediately following text, which states that resonances vanish at both I=\pi/3 and I=\pi, and it is inconsistent with the numerical suppression seen at I=60^\circ in Fig. 3. The correct inclination factor follows from Eq. (5.6) with m_{k,l}=n_{k,l} and vanishing tilde coefficients: it should be proportional to \cos I+\cos 2I=(1+\cos I)(2\cos I-1), which vanishes at both angles. This is not a cosmetic typo: Eq. (5.13) is the principal analytic prediction for the amplitude of the new magnetic resonances, and the printed expression also gives the wrong normalization at aligned inclinations. The numerical LPE solutions do not rely on Eq. (5.13), but the analytical model presented in Sec. V.B needs correction.
[Abstract and Sec. V.B / Table I] The abstract and introduction state that magnetic tides introduce 'new resonances absent at lower order.' This is overstated: Table I shows that the \dot\gamma=\Omega_{\hat\Psi} resonance already appears in the 0PN electric-tidal column. What is new at 0.5PN are the (-1,1) and (1,1) resonances, while the (0,1) resonance receives an additional magnetic contribution. The wording should be adjusted to avoid claiming that all three magnetic resonances are absent at lower order.

minor comments (3)

[Sec. VI] The text describes the numerical computation as 'without performing a Fourier expansion of the Hamiltonian' and as solving the 'full equations of motion.' In practice, the LPE system (6.3) is evolved with the orbit-averaged Hamiltonian (4.22), not with the unaveraged two-body equations. Please rephrase to avoid overstating the independence of the numerical check.
[Abstract, Sec. VII] The title and abstract claim an 'observable signature' and 'potential LISA detectability,' but no waveform or SNR estimate is provided; Sec. VII leaves waveform construction to future work. This is acceptable as a forward-looking statement, but the wording should be qualified, e.g., 'a GW signature that could be observable' rather than a demonstrated detection.
[Throughout] The notation '0.5PN order' is nonstandard; since the term is 1/c beyond Newtonian, a parenthetical definition at first occurrence would help readers not working in the PN convention.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the magnetic-tide resonances are derived from an explicit 0.5PN tidal Hamiltonian and are not fitted to the claimed outputs.

full rationale

The central derivation is self-contained rather than circular. The 0.5PN magnetic tidal coupling enters through the Lagrangian (2.3) and Hamiltonian (2.18), and the resonance condition (5.7) follows from Fourier decomposing B_ij in (5.2)-(5.3) together with the orbit-averaged magnetic Hamiltonian (4.15)-(4.16); the q=1 structure is a consequence of the explicit tensor T_ij in (4.14), not an assumption that the resonances exist. The quasi-circular Fourier coefficients (5.11) are computed from the geodesic solutions (5.8), and the resonance list (5.12) is a direct consequence. No parameter is fitted to the eccentricity jumps, and the numerical LPE solutions use the same Hamiltonian, so they are a consistency check of the averaging analysis rather than a circular derivation. The frequent citations to the authors' earlier papers [20,37,72] supply formalism and background frequencies, but those are standard, independently checkable geodesic/tidal results, and the new magnetic claim does not reduce to any of them. Two genuine non-circular weaknesses should be weighed separately: (i) Appendix A does not solve Eq. (A.9) and says the accelerated-frame tetrad corrections are negligible 'Without explicitly solving the system (A.9)' while assuming r-hat constant; if wrong, the Hamiltonian (2.18) could miss same-order 0.5PN terms, but that is an unproved physical approximation, not a definitional reduction. (ii) Eq. (5.13)'s printed angular factor cos^2 I/[2(1-2cosI)] appears inconsistent with the text and with Eq. (5.6), but this is a typographical/analytical error, not circularity. Neither defect makes the prediction equal to its input.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No parameters are fitted to data; the central result follows from a Hamiltonian with standard physical constants and scenario inputs (masses, semi-major axes, inclinations) chosen by hand. The analytical model introduces no ad hoc constants beyond choosing to truncate the outer-eccentricity expansion at O(ê). However, the paper relies on several domain assumptions (small-tide, geodesic CoM, quasi-circular outer orbit) and on prior self-authored formalism [20,37,72].

assumptions (7)

domain assumption Small-tide approximation: the inner binary's size r is much smaller than the curvature radius R of the SMBH; only quadrupolar tidal terms are kept.
Adopted in Sec. II; ensures the tidal interaction is perturbative and allows truncation at (r/R)^2.
domain assumption The inner binary can be treated as a Newtonian two-body system with small PN corrections; separation much larger than the compact objects' Schwarzschild radii.
Sec. II; needed to use Keplerian orbital elements and Lagrange Planetary Equations.
domain assumption The center-of-mass of the inner binary follows a Schwarzschild geodesic; deviations induced by the accelerated frame are negligible at quadrupolar order.
Sec. II and App. A (Eqs. A.4–A.9); load-bearing for the 0.5PN Hamiltonian (2.18).
domain assumption Resonance crossing is driven mainly by 2.5PN gravitational radiation reaction; environmental effects are neglected or parameterized by α.
Sec. VI (Eq. 6.1) and App. D; assumes T_1PN ≪ T_ZLK and T_RR governs the inspiral.
domain assumption At leading order in outer eccentricity, the outer radial motion and Marck angle are given by the quasi-circular solution (5.8) and the frequencies (5.10).
Sec. V B; the analytical resonance amplitudes and positions rest on this expansion to O(ê).
standard math Standard action-angle formalism for bound geodesics in Schwarzschild spacetime, with fundamental frequencies Ω_r, Ω_ϕ, Ω_Ψ.
Sec. V A; based on Refs. [41,43–45,85].
standard math Lagrange Planetary Equations in the generalized Lagrange gauge are a valid symplectic system for the osculating elements.
Sec. III (Eqs. 3.11–3.17) and Sec. VI (Eq. 6.3); from Refs. [79,83].

how reviews work

0 comments

Cite this review

Pith. "Pith review of Observable signature of magnetic tidal coupling in hierarchical triple systems." pith.science (2026). https://pith.science/paper/MI2QMM3O

@misc{pith2026251024897,
  author       = {Pith},
  title        = {Pith review of: Observable signature of magnetic tidal coupling in hierarchical triple systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MI2QMM3O}},
  note         = {Machine review of arXiv:2510.24897}
}

read the original abstract

We study hierarchical triple systems formed by a compact binary orbiting a supermassive black hole (SMBH), focusing on the role of relativistic magnetic tidal interactions. Extending previous analyses of precession resonances to 0.5 post-Newtonian order, we incorporate quadrupolar magnetic tidal moments, which have no Newtonian counterpart. We find that magnetic tides introduce new resonances absent at lower order, leading to additional eccentricity excitations and significantly modifying the binary's long-term evolution. Numerical solutions of the Lagrange Planetary Equations confirm these analytical predictions and reveal how resonance strength depends on orbital eccentricity and inclination. The resulting dynamics accelerates the binary merger and imprints distinctive signatures on gravitational waves, potentially observable by LISA. Our findings identify magnetic tidal coupling as a novel strong-gravity effect and establish its importance for the resonant dynamics of compact-object binaries near SMBHs.

Figures

Figures reproduced from arXiv: 2510.24897 by the authors.

**Figure 2.** FIG. 2. Evolution of the first precession resonance encountered by the inner binary as a function of the outer semi-major [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** FIG. 3. Evolution of the eccentricity [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Evolution of the eccentricity [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: FIG. 5. Evolution of the eccentricity [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

Discussion (0). Sign in to comment.

Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Periodic line-of-sight velocity-driven modulations to gravitational waves emitted by compact binaries in Keplerian outer orbits
gr-qc 2026-07 conditional novelty 7.0 of 10

New waveform corrections for periodic Doppler shifts from circular and eccentric outer orbits let future gravitational-wave detectors measure the mass and orbit of a third body around a merging binary.
Periodic line-of-sight velocity-driven modulations to gravitational waves emitted by compact binaries in Keplerian outer orbits
gr-qc 2026-07 conditional novelty 6.0 of 10

Periodic non-relativistic line-of-sight velocity of a compact binary’s centre of mass produces 4PN phase and amplitude modulations that improve Fisher forecasts of tertiary mass and outer-orbit size for A+, ET, DECIGO...
Dynamics of Binary System around a Supermassive Black Hole :Binary Scattering and Eccentric vZLK Oscillations
gr-qc 2026-05 unverdicted novelty 6.0 of 10

Binaries around supermassive black holes exhibit four periapsis-dependent scattering regimes in unbound orbits and scattering-type eccentric vZLK oscillations in bound orbits within Kerr spacetime.
Tidal perturbations of an extreme mass ratio inspiral around a Kerr black hole
gr-qc 2026-01 conditional novelty 6.0 of 10

A closed-form Kerr metric under slow quadrupolar tides yields spin-dependent tidal shifts of the ISCO and light ring, with larger shifts for retrograde orbits around fast-spinning holes.
Dynamics of Binary System around a Supermassive Black Hole :Binary Scattering and Eccentric vZLK Oscillations
gr-qc 2026-05 unverdicted novelty 5.0 of 10

Binaries around SMBHs exhibit four scattering regimes in unbound orbits and scattering-type eccentric vZLK oscillations in bound orbits, yielding a unified periapsis-driven tidal dynamics picture in galactic nuclei.
Gravitational waves of extreme-mass-ratio inspirals in a rotating black hole with Dehnen dark matter halo
gr-qc 2026-04 unverdicted novelty 4.0 of 10

EMRI waveforms in a rotating black hole with Dehnen DM halo show amplitude and phase shifts from Kerr, with mismatch rising as DM mass parameter and black hole spin increase.

Reference graph

Works this paper leans on

95 extracted references · 63 linked inside Pith · cited by 4 Pith papers

[22]

Maeda, P

K.-i. Maeda, P. Gupta, and H. Okawa, Phys. Rev. D 107, 124039 (2023), arXiv:2303.16553 [gr-qc]

arXiv 2023
[95]

Chen and Z

X. Chen and Z. Zhang, Phys. Rev. D106, 103040 (2022), arXiv:2206.08104 [astro-ph.HE]

arXiv 2022
[1]

Newton,Philosophiæ Naturalis Principia Mathematica (England, 1687)

I. Newton,Philosophiæ Naturalis Principia Mathematica (England, 1687)
[2]

Euler,Theoria motuum lunae nova methodo pertrac- tata: una cum tabulis astronomicis(1772)

L. Euler,Theoria motuum lunae nova methodo pertrac- tata: una cum tabulis astronomicis(1772)
[3]

Lagrange, M´ emoires de l’Acad´ emie des Sciences de Paris (1783), included in Œuvres de Lagrange, Vol

J.-L. Lagrange, M´ emoires de l’Acad´ emie des Sciences de Paris (1783), included in Œuvres de Lagrange, Vol. IV (Paris: Gauthier-Villars, 1869)
[4]

Delaunay,Th´ eorie du mouvement de la lune, M´ emoires de l’Acad´ emie des sciences de l’Institut de France

C. Delaunay,Th´ eorie du mouvement de la lune, M´ emoires de l’Acad´ emie des sciences de l’Institut de France. S´ erie 2 No. pt. 1 (Didot, 1860)
[5]

Poincar´ e, Bulletin astronomique, Observatoire de Paris8, 12 (1891)

H. Poincar´ e, Bulletin astronomique, Observatoire de Paris8, 12 (1891)
[6]

A. E. H. Love,Some problems of geodynamics : being an essay to which the Adams prize in the University of Cambridge was adjudged in 1911(University Press, Cam- bridge, 1911)

1911

Show all 95 references

[7]

Hinderer, Astrophys

T. Hinderer, Astrophys. J.677, 1216 (2008), [Erratum: Astrophys.J. 697, 964 (2009)], arXiv:0711.2420 [astro- ph]

2008 arXiv
[8]

E. E. Flanagan and T. Hinderer, Phys. Rev. D77, 021502 (2008), arXiv:0709.1915 [astro-ph]

2008 arXiv
[9]

Damour and A

T. Damour and A. Nagar, Phys. Rev. D80, 084035 (2009), arXiv:0906.0096 [gr-qc]

2009 arXiv
[10]

Binnington and E

T. Binnington and E. Poisson, Phys. Rev. D80, 084018 (2009), arXiv:0906.1366 [gr-qc]

2009 arXiv
[11]

Samsing, K

J. Samsing, K. Hendriks, L. Zwick, D. J. D’Orazio, and B. Liu, (2024), arXiv:2403.05625 [astro-ph.HE]

2024 arXiv
[12]

Hendriks, D

K. Hendriks, D. Atallah, M. Martinez, M. Zevin, L. Zwick, A. A. Trani, P. Saini, J. Tak´ atsy, and J. Sam- sing, (2024), arXiv:2411.08572 [astro-ph.HE]

2024
[13]

Saini, L

P. Saini, L. Zwick, J. Tak´ atsy, C. Rowan, K. Hendriks, G. Fabj, D. J. D’Orazio, and J. Samsing, (2025), arXiv:2508.17348 [gr-qc]

2025
[14]

H. V. Zeipel, Astronomische Nachrichten183, 345 (1909)

1909
[15]

Kozai, Astron

Y. Kozai, Astron. J.67, 591 (1962)

1962
[16]

Lidov, Planetary and Space Science9, 719 (1962)

M. Lidov, Planetary and Space Science9, 719 (1962)

1962
[17]

Hu, R.-G

L. Hu, R.-G. Cai, and S.-J. Wang, JCAP08, 010 (2025), arXiv:2411.14047 [gr-qc]. 20

2025 arXiv
[18]

R. S. Chandramouli and N. Yunes, Phys. Rev. D105, 064009 (2022), arXiv:2107.00741 [gr-qc]

2022 arXiv
[19]

Randall and Z.-Z

L. Randall and Z.-Z. Xianyu, Astrophys. J.864, 134 (2018), arXiv:1802.05718 [gr-qc]

2018 arXiv
[20]

Camilloni, T

F. Camilloni, T. Harmark, G. Grignani, M. Orselli, and D. Pica, Mon. Not. Roy. Astron. Soc.531, 1884 (2024), arXiv:2310.06894 [gr-qc]

2024 arXiv
[21]

Antonini and H

F. Antonini and H. B. Perets, Astrophys. J.757, 27 (2012), arXiv:1203.2938 [astro-ph.GA]

2012 arXiv
[23]

Maeda, P

K.-i. Maeda, P. Gupta, and H. Okawa, Phys. Rev. D 108, 123041 (2023), arXiv:2309.12096 [gr-qc]

2023 arXiv
[24]

Maeda and H

K.-i. Maeda and H. Okawa, (2025), arXiv:2504.18934 [gr-qc]

2025 arXiv
[25]

Naoz, Ann

S. Naoz, Ann. Rev. Astron. Astrophys.54, 441 (2016), arXiv:1601.07175 [astro-ph.EP]

2016 arXiv
[26]

Bonga, H

B. Bonga, H. Yang, and S. A. Hughes, Phys. Rev. Lett. 123, 101103 (2019), arXiv:1905.00030 [gr-qc]

2019 arXiv
[27]

Gupta, B

P. Gupta, B. Bonga, A. J. K. Chua, and T. Tanaka, Phys. Rev. D104, 044056 (2021), arXiv:2104.03422 [gr- qc]

2021 arXiv
[28]

´E. E. Flanagan, S. A. Hughes, and U. Ruangsri, Phys. Rev. D89, 084028 (2014), arXiv:1208.3906 [gr-qc]

2014 arXiv
[29]

E. E. Flanagan and T. Hinderer, Phys. Rev. Lett.109, 071102 (2012), arXiv:1009.4923 [gr-qc]

2012 arXiv
[30]

Zwick, K

L. Zwick, K. Hendriks, D. O’Neill, J. Tak´ atsy, P. Kirke- berg, C. Tiede, J. Stegmann, J. Samsing, and D. J. D’Orazio, (2025), arXiv:2506.09140 [astro-ph.HE]

2025 arXiv
[31]

Sharpe, Y

C. Sharpe, Y. B. Ginat, and B. Kocsis, (2025), arXiv:2502.01733 [gr-qc]

2025
[32]

H. G. Bhaskar, G. Li, and D. N. C. Lin, Astrophys. J. 934, 141 (2022), arXiv:2204.07282 [astro-ph.HE]

2022 arXiv
[33]

Stockinger and M

M. Stockinger and M. Shibata, Phys. Rev. D110, 043038 (2024), arXiv:2403.06834 [astro-ph.HE]

2024 arXiv
[34]

Stockinger and M

M. Stockinger and M. Shibata, Phys. Rev. D111, 123044 (2025), arXiv:2502.06998 [astro-ph.HE]

2025 arXiv
[35]

K. S. Thorne and J. B. Hartle, Phys. Rev. D31, 1815 (1984)

1984
[36]

Poisson and I

E. Poisson and I. Vlasov, Phys. Rev. D81, 024029 (2010), arXiv:0910.4311 [gr-qc]

2010 arXiv
[37]

Cocco, G

M. Cocco, G. Grignani, T. Harmark, M. Orselli, and D. Pica, Phys. Rev. D112, 044010 (2025), arXiv:2505.15901 [gr-qc]

2025
[38]

Generozov, N

A. Generozov, N. C. Stone, B. D. Metzger, and J. P. Ostriker, Mon. Not. Roy. Astron. Soc.478, 4030 (2018), arXiv:1804.01543 [astro-ph.HE]

2018 arXiv
[39]

Chen and W.-B

X. Chen and W.-B. Han, Communications Physics1, 53 (2018), arXiv:1801.05780 [astro-ph.HE]

2018 arXiv
[40]

Kuntz, Phys

A. Kuntz, Phys. Rev. D105, 024017 (2022), arXiv:2112.05167 [gr-qc]

2022 arXiv
[41]

van de Meent, Class

M. van de Meent, Class. Quant. Grav.37, 145007 (2020), arXiv:1906.05090 [gr-qc]

2020 arXiv
[42]

D. Bini, A. Geralico, and R. T. Jantzen, Phys. Rev. D 94, 064066 (2016), arXiv:1607.08427 [gr-qc]

2016 arXiv
[43]

Schmidt, Class

W. Schmidt, Class. Quant. Grav.19, 2743 (2002), arXiv:gr-qc/0202090

2002 arXiv
[44]

Fujita and W

R. Fujita and W. Hikida, Class. Quant. Grav.26, 135002 (2009), arXiv:0906.1420 [gr-qc]

2009 arXiv
[45]

Hinderer and E

T. Hinderer and E. E. Flanagan, Phys. Rev. D78, 064028 (2008), arXiv:0805.3337 [gr-qc]

2008 arXiv
[46]

R. M. O’Leary, B. Kocsis, and A. Loeb, Mon. Not. Roy. Astron. Soc.395, 2127 (2009), arXiv:0807.2638 [astro- ph]

2009 arXiv
[47]

Samsing and E

J. Samsing and E. Ramirez-Ruiz, Astrophys. J. Lett. 840, L14 (2017), arXiv:1703.09703 [astro-ph.HE]

2017 arXiv
[48]

Hopman and T

C. Hopman and T. Alexander, Astrophys. J.645, 1152 (2006), arXiv:astro-ph/0601161

2006 arXiv
[49]

J. M. Bellovary, M.-M. Mac Low, B. McKernan, and K. E. S. Ford, Astrophys. J. Lett.819, L17 (2016), arXiv:1511.00005 [astro-ph.GA]

2016 arXiv
[50]

Secunda, J

A. Secunda, J. Bellovary, M.-M. Mac Low, K. E. S. Ford, B. McKernan, N. W. C. Leigh, W. Lyra, Z. San- dor, and J. I. Adorno, Astrophys. J.903, 133 (2020), arXiv:2004.11936 [astro-ph.HE]

2020 arXiv
[51]

Peng and X

P. Peng and X. Chen, Mon. Not. Roy. Astron. Soc.505, 1324 (2021), arXiv:2104.07685 [astro-ph.HE]

2021 arXiv
[52]

Tagawa, Z

H. Tagawa, Z. Haiman, and B. Kocsis, Astrophys. J. 898, 25 (2020), arXiv:1912.08218 [astro-ph.GA]

2020 arXiv
[53]

Cardoso, C

V. Cardoso, C. F. B. Macedo, and R. Vicente, Phys. Rev. D103, 023015 (2021), arXiv:2010.15151 [gr-qc]

2021 arXiv
[54]

Zwick, P

L. Zwick, P. R. Capelo, E. Bortolas, V. Vazquez-Aceves, L. Mayer, and P. Amaro-Seoane, Mon. Not. Roy. Astron. Soc.506, 1007 (2021), arXiv:2102.00015 [astro-ph.GA]

2021 arXiv
[55]

Caneva Santoro, S

G. Caneva Santoro, S. Roy, R. Vicente, M. Haney, O. J. Piccinni, W. Del Pozzo, and M. Martinez, Phys. Rev. Lett.132, 251401 (2024), arXiv:2309.05061 [gr-qc]

2024 arXiv
[56]

Ishibashi and M

W. Ishibashi and M. Gr¨ obner, Mon. Not. Roy. Astron. Soc.529, 883 (2024), arXiv:2412.01925 [astro-ph.HE]

2024 arXiv
[57]

Barausse and L

E. Barausse and L. Rezzolla, Phys. Rev. D77, 104027 (2008), arXiv:0711.4558 [gr-qc]

2008 arXiv
[58]

Barausse, V

E. Barausse, V. Cardoso, and P. Pani, Phys. Rev. D89, 104059 (2014), arXiv:1404.7149 [gr-qc]

2014 arXiv
[59]

Cardoso and C

V. Cardoso and C. F. B. Macedo, Mon. Not. Roy. Astron. Soc.498, 1963 (2020), arXiv:2008.01091 [astro-ph.HE]

1963 arXiv
[60]

P. S. Cole, G. Bertone, A. Coogan, D. Gaggero, T. Karydas, B. J. Kavanagh, T. F. M. Spieksma, and G. M. Tomaselli, Nature Astron.7, 943 (2023), arXiv:2211.01362 [gr-qc]

2023
[61]

R. Chen, R. S. Chandramouli, F. Pozzoli, R. Buscic- chio, and E. Barausse, Phys. Rev. D112, 084053 (2025), arXiv:2507.00694 [gr-qc]

2025
[62]

Abbottet al.(LIGO Scientific, Virgo), Phys

R. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. Lett.125, 101102 (2020), arXiv:2009.01075 [gr-qc]

2020
[63]

Toubianaet al., Phys

A. Toubianaet al., Phys. Rev. Lett.126, 101105 (2021), arXiv:2010.06056 [astro-ph.HE]

2021 arXiv
[64]

M. J. Grahamet al., Phys. Rev. Lett.124, 251102 (2020), arXiv:2006.14122 [astro-ph.HE]

2020 arXiv
[65]

Sbernaet al., Phys

L. Sbernaet al., Phys. Rev. D106, 064056 (2022), arXiv:2205.08550 [gr-qc]

2022 arXiv
[66]

Peißker, M

F. Peißker, M. Zajacek, L. Labadie, E. Bordier, A. Eckart, M. Melamed, and V. Karas, Nature Com- mun.15, 10608 (2024), arXiv:2412.12727 [astro-ph.GA]

2024 arXiv
[67]

J. S. Santos, V. Cardoso, J. Nat´ ario, and M. van de Meent, (2025), arXiv:2506.14868 [gr-qc]

2025 arXiv
[68]

Y. Yin, J. Mathews, A. J. K. Chua, and X. Chen, Phys. Rev. D111, 103007 (2025), arXiv:2410.09796 [gr-qc]

2025 arXiv
[69]

K. Meng, H. Zhang, X.-L. Fan, Y. Yong, and F. Du, (2024), arXiv:2405.07113 [gr-qc]

2024
[70]

Jiang, W.-B

Y. Jiang, W.-B. Han, X.-Y. Zhong, P. Shen, Z.-R. Luo, and Y.-L. Wu, Eur. Phys. J. C84, 478 (2024)

2024
[71]

Amaro-Seoaneet al., arXiv preprint (2017), arXiv:1702.00786 [astro-ph.IM]

P. Amaro-Seoaneet al., arXiv preprint (2017), arXiv:1702.00786 [astro-ph.IM]

2017 arXiv
[72]

Camilloni, G

F. Camilloni, G. Grignani, T. Harmark, R. Oliveri, M. Orselli, and D. Pica, Phys. Rev. D107, 084011 (2023), arXiv:2301.04879 [gr-qc]. 21

2023 arXiv
[73]

Kuntz, F

A. Kuntz, F. Serra, and E. Trincherini, Phys. Rev. D 104, 024016 (2021), arXiv:2104.13387 [hep-th]

2021 arXiv
[74]

Kuntz, F

A. Kuntz, F. Serra, and E. Trincherini, Phys. Rev. D 107, 044011 (2023), arXiv:2210.13493 [gr-qc]

2023 arXiv
[75]

Marck, Proc

J.-A. Marck, Proc. R. Soc. Lond. A385, 431 (1983)

1983
[76]

Goldstein,Classical Mechanics(Addison-Wesley, 1980)

H. Goldstein,Classical Mechanics(Addison-Wesley, 1980)

1980
[77]

Morbidelli, Modern celestial mechanics : aspects of so- lar system dynamics

A. Morbidelli, Modern celestial mechanics : aspects of so- lar system dynamics. London: Taylor and Francis, 2002, ISBN 0415279399-1(2002)

2002
[78]

Valtonen and H

M. Valtonen and H. Karttunen,The Three-Body Problem (Cambridge University Press, Cambridge, 2006)

2006
[79]

Efroimsky and P

M. Efroimsky and P. Goldreich, Astron. Astrophys.415, 1187 (2004), arXiv:astro-ph/0307130

2004 arXiv
[80]

Lagrange, M´ emoires de l’Acad´ emie des Sciences de Paris (1808), included in Œuvres de Lagrange, Vol

J.-L. Lagrange, M´ emoires de l’Acad´ emie des Sciences de Paris (1808), included in Œuvres de Lagrange, Vol. VI, 713 (Paris: Gauthier-Villars, 1877)
[81]

Lagrange, M´ emoires de l’Acad´ emie des Sciences de Paris (1809), included in Œuvres de Lagrange, Vol

J.-L. Lagrange, M´ emoires de l’Acad´ emie des Sciences de Paris (1809), included in Œuvres de Lagrange, Vol. VI, 771 (Paris: Gauthier-Villars, 1877)
[82]

Lagrange, M´ emoires de l’Acad´ emie des Sciences de Paris (1810), included in Œuvres de Lagrange, Vol

J.-L. Lagrange, M´ emoires de l’Acad´ emie des Sciences de Paris (1810), included in Œuvres de Lagrange, Vol. VI, 809 (Paris: Gauthier-Villars, 1877)
[83]

Efroimsky and P

M. Efroimsky and P. Goldreich, J. Math. Phys.44, 5958 (2003), arXiv:astro-ph/0305344

2003 arXiv
[84]

Damour and G

T. Damour and G. Schaefer, Nuovo Cim. B101, 127 (1988)

1988
[85]

Drasco and S

S. Drasco and S. A. Hughes, Phys. Rev. D69, 044015 (2004), arXiv:astro-ph/0308479

2004 arXiv
[86]

Drasco and S

S. Drasco and S. A. Hughes, Phys. Rev. D73, 024027 (2006), arXiv:gr-qc/0509101

2006 arXiv
[87]

P. C. Peters, Physical Review136, 1224 (1964)

1964
[88]

P. C. Peters and J. Mathews, Phys. Rev.131, 435 (1963)

1963
[89]

Maggiore,Gravitational Waves

M. Maggiore,Gravitational Waves. Vol. 1: Theory and Experiments(Oxford University Press, Oxford, 2007)

2007
[90]

Chandrasekhar,The mathematical theory of black holes(Oxford University Press, Oxford, 1983)

S. Chandrasekhar,The mathematical theory of black holes(Oxford University Press, Oxford, 1983)

1983
[91]

Suzuki, Y

H. Suzuki, Y. Nakamura, and S. Yamada, Phys. Rev. D 102, 124063 (2020), arXiv:2009.06999 [gr-qc]

2020 arXiv
[92]

Coleman Miller, M

M. Coleman Miller, M. Freitag, D. P. Hamilton, and V. M. Lauburg, Astrophys. J. Lett.631, L117 (2005), arXiv:astro-ph/0507133

2005 arXiv
[93]

Brink, M

J. Brink, M. Geyer, and T. Hinderer, Phys. Rev. Lett. 114, 081102 (2015), arXiv:1304.0330 [gr-qc]

2015 arXiv
[94]

Wen, Astrophys

L. Wen, Astrophys. J.598, 419 (2003), arXiv:astro- ph/0211492

2003

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [22]

Maeda, P

K.-i. Maeda, P. Gupta, and H. Okawa, Phys. Rev. D 107, 124039 (2023), arXiv:2303.16553 [gr-qc]

arXiv 2023

[2] [95]

Chen and Z

X. Chen and Z. Zhang, Phys. Rev. D106, 103040 (2022), arXiv:2206.08104 [astro-ph.HE]

arXiv 2022

[3] [1]

Newton,Philosophiæ Naturalis Principia Mathematica (England, 1687)

I. Newton,Philosophiæ Naturalis Principia Mathematica (England, 1687)

[4] [2]

Euler,Theoria motuum lunae nova methodo pertrac- tata: una cum tabulis astronomicis(1772)

L. Euler,Theoria motuum lunae nova methodo pertrac- tata: una cum tabulis astronomicis(1772)

[5] [3]

Lagrange, M´ emoires de l’Acad´ emie des Sciences de Paris (1783), included in Œuvres de Lagrange, Vol

J.-L. Lagrange, M´ emoires de l’Acad´ emie des Sciences de Paris (1783), included in Œuvres de Lagrange, Vol. IV (Paris: Gauthier-Villars, 1869)

[6] [4]

Delaunay,Th´ eorie du mouvement de la lune, M´ emoires de l’Acad´ emie des sciences de l’Institut de France

C. Delaunay,Th´ eorie du mouvement de la lune, M´ emoires de l’Acad´ emie des sciences de l’Institut de France. S´ erie 2 No. pt. 1 (Didot, 1860)

[7] [5]

Poincar´ e, Bulletin astronomique, Observatoire de Paris8, 12 (1891)

H. Poincar´ e, Bulletin astronomique, Observatoire de Paris8, 12 (1891)

[8] [6]

A. E. H. Love,Some problems of geodynamics : being an essay to which the Adams prize in the University of Cambridge was adjudged in 1911(University Press, Cam- bridge, 1911)

1911