- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Waveform degeneracy of binary systems and Lagrange three-body systems

Carlos Jaimel Doctolero* and Ian Vega†

  • *Contact author: cdoctolero@nip.upd.edu.ph
  • †Contact author: ivega@nip.upd.edu.ph
PDF

Phys. Rev. D 114, 044031 – Published 11 August, 2026

DOI: https://doi.org/10.1103/pm77-mfdb

Abstract

A particular solution to the three-body problem is the circular Lagrange three-body system, where the masses move in circular orbits such that they always constitute an equilateral triangle. Such a system has been found to emit gravitational waves with a waveform similar to that of a binary system. In this work, we study the gravitational waveform degeneracy between quasicircular binary systems and Lagrange three-body systems up to 0.5PN order. Assuming we know the parameters of a given binary system, we determine the parameters of the Lagrange triple that produces the same waveform as that of the binary. We show that there exists a mass quadrupole degeneracy in both the plus and cross modes, characterized by two parameters. We also find that there are binary systems and linearly stable Lagrange three-body systems that can have the same mass quadrupole waveform up to the coalescence time. In such cases, the normalized overlap of the waveforms with respect to the power spectral density of the advanced LIGO design remains above 0.97 as long as the binary has nearly symmetric masses. Beyond the mass quadrupole, there is a unique degeneracy at the 0.5PN. However, the Lagrange triple that satisfies this degeneracy is unstable.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (50)

  1. B. Abbott et al. (LIGO Scientific and Virgo Collaborations), Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116, 061102 (2016).
  2. R. Abbott et al. (LIGO Scientific and Virgo Collaborations), GW190412: Observation of a binary-black-hole coalescence with asymmetric masses, Phys. Rev. D 102, 043015 (2020).
  3. R. Abbott et al. (LIGO Scientific and Virgo Collaborations), GW190814: Gravitational waves from the coalescence of a 23 solar mass black hole with a 2.6 solar mass compact object, Astrophys. J. Lett. 896, L44 (2020).
  4. B. Abbott et al. (LIGO Scientific and Virgo Collaborations), GW170814: A three-detector observation of gravitational waves from a binary black hole coalescence, Phys. Rev. Lett. 119, 141101 (2017).
  5. B. Abbott et al. (LIGO Scientific and Virgo Collaborations), GW170817: Observation of gravitational waves from a binary neutron star inspiral, Phys. Rev. Lett. 119, 161101 (2017).
  6. P. Liu, Y.-Y. Yang, J.-W. Zhang, and M. Rah, Simulation of the orbit and spin period evolution of the double pulsars PSR J0737-3039 from their birth to coalescence induced by gravitational wave radiation, Res. Astron. Astrophys. 21, 104 (2021).
  7. I. Romero-Shaw, P. D. Lasky, and E. Thrane, Signs of eccentricity in two gravitational-wave signals may indicate a subpopulation of dynamically assembled binary black holes, Astrophys. J. Lett. 921, L31 (2021).
  8. J. Stegmann, F. Antonini, F. R. Schneider, V. Tiwari, and D. Chattopadhyay, Binary black hole mergers from merged stars in the Galactic field, Phys. Rev. D 106, 023014 (2022).
  9. M. E. Pati and C. M. Will, Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations: Foundations, Phys. Rev. D 62, 124015 (2000).
  10. L. Blanchet, Gravitational Radiation from Post-Newtonian sources and inspiralling compact binaries, Living Rev. Relativity 17, 2 (2014).
  11. L. Blanchet, Post-Newtonian theory for gravitational waves, Living Rev. Relativity 27, 4 (2024).
  12. Z. E. Musielak and B. Quarles, The three-body problem, Rep. Prog. Phys. 77, 065901 (2014).
  13. D. Li, X. Wu, and E. Liang, Figure-eight orbits in three post-Newtonian formulations of triple black holes, Phys. Rev. D 104, 044039 (2021).
  14. C. Moore, Braids in classical dynamics, Phys. Rev. Lett. 70, 3675 (1993).
  15. M. Hénon, A family of periodic solutions of the planar three-body problem, and their stability, Celest. Mech. 13, 267 (1976).
  16. C. Moore and M. Nauenberg, New periodic orbits for the n-body problem, J. Comp. Nonlin. Dyn. 1, 307 (2006).
  17. T. Imai, T. Chiba, and H. Asada, Choreographic solution to the general-relativistic three-body problem, Phys. Rev. Lett. 98, 201102 (2007).
  18. Y. Torigoe, K. Hattori, and H. Asada, Gravitational wave forms for two- and three-body gravitating systems, Phys. Rev. Lett. 102, 251101 (2009).
  19. P. Galaviz and B. Brügmann, Characterization of the gravitational wave emission of three black holes, Phys. Rev. D 83, 084013 (2011).
  20. V. Dmitrašinović, M. Šuvakov, and A. Hudomal, Gravitational waves from periodic three-body systems, Phys. Rev. Lett. 113, 101102 (2014).
  21. H. Essén, On the equilateral triangle solution to the three-body problem, Eur. J. Phys. 21, 579 (2000).
  22. J. E. Mansilla, Stability of Hamiltonian systems with three degrees of freedom and the three body-problem, Celest. Mech. Dyn. Astron. 94, 249 (2006).
  23. G. E. Roberts, Linear Stability of the elliptic Lagrangian triangle solutions in the three-body problem, J. Differ. Equations 182, 191 (2002).
  24. B. Sicardy, Stability of the triangular Lagrange points beyond Gascheau’s value, Celest. Mech. Dyn. Astron. 107, 145 (2010).
  25. P. Gurfil and P. K. Seidelmann, Celestial Mechanics and Astrodynamics: Theory and Practice, Astrophysics and Space Science Library (Springer, Berlin, Heidelberg, 2016), Vol. 436.
  26. A. Morbidelli, H. F. Levison, K. Tsiganis, and R. Gomes, Chaotic capture of Jupiter’s Trojan asteroids in the early solar system, Nature (London) 435, 462 (2005).
  27. J. D. Schnittman, The lagrange equilibrium points L4 and L5 in black hole binary system, Astrophys. J. 724, 39 (2010).
  28. N. Seto and T. Muto, Relativistic astrophysics with resonant multiple inspirals, Phys. Rev. D 81, 103004 (2010).
  29. S. P. Zwart, A. Doelman, and J. Sein, The formation of periodic three-body orbits for Newtonian systems, Astron. Astrophys. 707, A215 (2026).
  30. H. Asada, Gravitational wave forms for a three-body system in lagrange’s orbit: Parameter determinations and a binary source test, Phys. Rev. D 80, 064021 (2009).
  31. T. Ichita, K. Yamada, and H. Asada, Post-Newtonian effects on Lagrange’s equilateral triangular solution for the three-body problem, Phys. Rev. D 83, 084026 (2011).
  32. K. Yamada and H. Asada, Triangular solution to the general relativistic three-body problem for general masses, Phys. Rev. D 86, 124029 (2012).
  33. K. Yamada, T. Tsuchiya, and H. Asada, Post-Newtonian effects on the stability of the triangular solution in the three-body problem for general masses, Phys. Rev. D 91, 124016 (2015).
  34. K. Yamada and H. Asada, Nonchaotic evolution of triangular configuration due to gravitational radiation reaction in the three-body problem, Phys. Rev. D 93, 084027 (2016).
  35. Y. Nakamura and H. Asada, Collinear and triangular solutions to the coplanar and circular three-body problem in the parametrized post-Newtonian formalism, Phys. Rev. D 107, 044005 (2023).
  36. Y. Nakamura and H. Asada, Triangular solution to the planar elliptic three-body problem in the parametrized post-Newtonian formalism, Phys. Rev. D 109, 064067 (2024).
  37. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, San Francisco, 1973).
  38. M. Maggiore, Gravitational Waves: Volume 1: Theory and Experiments (Oxford University Press, Oxford, 2007).
  39. E. Poisson and C. M. Will, Gravity: Newtonian, Post-Newtonian, Relativistic (Cambridge University Press, Cambridge, England, 2014).
  40. J. Creighton and W. Anderson, Gravitational-Wave Physics and Astronomy: An Introduction to Theory, Experiment and Data Analysis (John Wiley & Sons, Ltd, New York, 2011).
  41. M. M. Barandiaran, S. Kuroyanagi, and S. Nesseris, Gravitational waves in the circular restricted three body problem, Classical Quantum Gravity 41, 095002 (2024).
  42. M. Gascheau, Examen d’une classe d’équations différentielles et applicationa un cas particulier du probleme des trois corps, Comptes Rendus 16, 393 (1843).
  43. A. Klein, P. Jetzer, and M. Sereno, Parameter estimation for coalescing massive binary black holes with LISA using the full 2-post-Newtonian gravitational waveform and spin-orbit precession, Phys. Rev. D 80, 064027 (2009).
  44. B. J. Owen, Search templates for gravitational waves from inspiraling binaries: Choice of template spacing, Phys. Rev. D 53, 6749 (1996).
  45. A. Nitz et al., gwastro/pycbc: v2.3.3 release of PyCBC (2024).
  46. LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, LVK Algorithm Library—LALSuite, Free software (GPL) (2018).
  47. B. P. Abbott and others (LIGO Scientific and Virgo Collaborations), LIGO: The laser interferometer gravitational-wave observatory, Rep. Prog. Phys. 72, 076901 (2009).
  48. P. Mahapatra, Octupolar test of general relativity, Phys. Rev. D 109, 024050 (2024).
  49. P. Jamet, A. Barrau, and K. Martineau, Emission and detection of ultrahigh frequency gravitational waves from highly eccentric orbits of compact binary systems, Phys. Rev. D 111, 103032 (2025).
  50. C. J. Doctolero, BinaryLagrangeDegeneracy (2026).

Outline

Information

Phys. Rev. D 114, 044031– Published 11 August, 2026
  • Received 6 April 2026
  • Accepted 21 July 2026
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation