- 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

Analytical solution of spinning, eccentric binary black hole dynamics at the second post-Newtonian order

Tom Colin1,*, Sashwat Tanay1,2,†, and Laura Bernard1,‡

  • *Contact author: tom.colin@obspm.fr
  • †Contact author: stanay@utec.edu.pe
  • ‡Contact author: laura.bernard@obspm.fr
PDF

Phys. Rev. D 114, 064051 – Published 10 September, 2026

DOI: https://doi.org/10.1103/tr9z-qky6

Abstract

Recent gravitational wave (GW) detections showing signatures of eccentricity and spin precession underscore the need to model binary black holes (BBHs) possessing these features simultaneously. Most efforts over the past fifteen years to model spinning BBHs and their corresponding GWs have relied on heuristically twisting waveforms from nonprecessing systems. This approach is based on empirical observations rather than first principles. This article aims to model the dynamics of spinning and eccentric BBHs from a first-principles approach using the post-Newtonian (PN) approximation within general relativity. Building on the already-existing 1.5PN solution, we construct an analytical solution for the time evolution of the relative separation vector, the individual black hole spin vectors, and the orbital angular momentum vector at 2PN order for BBHs with arbitrary spins and eccentricity. Such a solution is not fully 2PN accurate in that the tiny orbital timescale fluctuations in the solutions for the spins are only leading 1.5PN order accurate, instead of 2PN. However, it is shown that our new 2PN solution is still an order of magnitude improvement over the earlier 1.5PN solution, underlining the subdominant nature of the neglected next-to-leading-order oscillations in the spin solutions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (62)

  1. M. Hannam, C. Hoy, J. E. Thompson, S. Fairhurst, V. Raymond et al., General-relativistic precession in a black-hole binary, Nature (London) 610, 652 (2022).
  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. I. M. Romero-Shaw, P. D. Lasky, E. Thrane, and J. C. Bustillo, GW190521: Orbital eccentricity and signatures of dynamical formation in a binary black hole merger signal, Astrophys. J. Lett. 903, L5 (2020).
  4. D. Gerosa and M. Fishbach, Hierarchical mergers of stellar-mass black holes and their gravitational-wave signatures, Nat. Astron. 5, 749 (2021).
  5. I. Mandel and A. Farmer, Merging stellar-mass binary black holes, Phys. Rep. 955, 1 (2022).
  6. I. Romero-Shaw, J. Stegmann, H. Tagawa, D. Gerosa, J. Samsing, N. Gupte, and S. R. Green, GW200208_222617 as an eccentric black-hole binary merger: Properties and astrophysical implications, Phys. Rev. D 112, 063052 (2025).
  7. I. M. Romero-Shaw, D. Gerosa, and N. Loutrel, Eccentricity or spin precession? Distinguishing subdominant effects in gravitational-wave data, Mon. Not. R. Astron. Soc. 519, 5352 (2023).
  8. S. Tibrewal, A. Zimmerman, J. Lange, and D. Shoemaker, Misinterpreting spin precession as orbital eccentricity in gravitational-wave signals, arXiv:2601.02260.
  9. J. Stegmann, D. Gerosa, I. Romero-Shaw, G. Fumagalli, H. Tagawa, and L. Zwick, Distinguishing the origin of eccentric black hole mergers with gravitational-wave spin measurements, Astrophys. J. Lett. 994, L47 (2025).
  10. V. Baibhav, Inferring eccentricity of binary black holes from spin-orbit misalignment, arXiv:2512.22044.
  11. L. Lehner, Numerical relativity: A review, Classical Quantum Gravity 18, R25 (2001).
  12. U. Sperhake, The numerical relativity breakthrough for binary black holes, Classical Quantum Gravity 32, 124011 (2015).
  13. K. D. Kokkotas and B. G. Schmidt, Quasi-normal modes of stars and black holes, Living Rev. Relativity 2, 2 (1999).
  14. E. Berti, V. Cardoso, and A. O. Starinets, Quasinormal modes of black holes and black branes, Classical Quantum Gravity 26, 163001 (2009).
  15. L. Blanchet, Gravitational radiation from post-Newtonian sources and inspiralling compact binaries, Living Rev. Relativity 17, 2 (2014).
  16. A. Buonanno and T. Damour, Effective one-body approach to general relativistic two-body dynamics, Phys. Rev. D 59, 084006 (1999).
  17. S. Khan, S. Husa, M. Hannam, F. Ohme, M. Pürrer, X. J. Forteza, and A. Bohé, Frequency-domain gravitational waves from nonprecessing black-hole binaries. II. A phenomenological model for the advanced detector era, Phys. Rev. D 93, 044007 (2016).
  18. P. Schmidt, M. Hannam, S. Husa, and P. Ajith, Tracking the precession of compact binaries from their gravitational-wave signal, Phys. Rev. D 84, 024046 (2011).
  19. P. Schmidt, M. Hannam, and S. Husa, Towards models of gravitational waveforms from generic binaries: A simple approximate mapping between precessing and nonprecessing inspiral signals, Phys. Rev. D 86, 104063 (2012).
  20. M. Boyle, R. Owen, and H. P. Pfeiffer, A geometric approach to the precession of compact binaries, Phys. Rev. D 84, 124011 (2011).
  21. K. Chatziioannou, A. Klein, N. Yunes, and N. Cornish, Constructing gravitational waves from generic spin-precessing compact binary inspirals, Phys. Rev. D 95, 104004 (2017).
  22. J. N. Arredondo, A. Klein, and N. Yunes, Efficient gravitational-wave model for fully-precessing and moderately eccentric, compact binary inspirals, Phys. Rev. D 110, 044044 (2024).
  23. G. Morras, G. Pratten, and P. Schmidt, Improved post-Newtonian waveform model for inspiralling precessing-eccentric compact binaries, Phys. Rev. D 111, 084052 (2025).
  24. T. Damour and N. Deruelle, General relativistic celestial mechanics of binary systems. I. The post-Newtonian motion, Ann. l’I.H.P. Phys. Théor. 43, 107 (1985).
  25. T. Damour, P. Jaranowski, and G. Schäfer, Nonlocal-in-time action for the fourth post-Newtonian conservative dynamics of two-body systems, Phys. Rev. D 89, 064058 (2014).
  26. G. Cho, S. Tanay, A. Gopakumar, and H. M. Lee, Generalized quasi-Keplerian solution for eccentric, nonspinning compact binaries at 4pn order and the associated inspiral-merger-ringdown waveform, Phys. Rev. D 105, 064010 (2022).
  27. M. Tessmer, J. Hartung, and G. Schafer, Motion and gravitational wave forms of eccentric compact binaries with orbital-angular-momentum-aligned spins under next-to-leading order in spin-orbit and leading order in spin(1)-spin(2) and spin-squared couplings, Classical Quantum Gravity 27, 165005 (2010).
  28. M. Tessmer, J. Hartung, and G. Schäfer, Aligned spins: Orbital elements, decaying orbits, and last stable circular orbit to high post-Newtonian orders, Classical Quantum Gravity 30, 015007 (2013).
  29. G. Cho and H. M. Lee, Analytic Keplerian-type parametrization for general spinning compact binaries with leading order spin-orbit interactions, Phys. Rev. D 100, 044046 (2019).
  30. R. V. Wagoner and C. M. Will, Post-Newtonian gravitational radiation from orbiting point masses., Astrophys. J. 210, 764 (1976).
  31. G. Schäfer and N. Wex, Second post-Newtonian motion of compact binaries, Phys. Lett. A 174, 196 (1993).
  32. R.-M. Memmesheimer, A. Gopakumar, and G. Schaefer, Third post-Newtonian accurate generalized quasi-Keplerian parametrization for compact binaries in eccentric orbits, Phys. Rev. D 70, 104011 (2004).
  33. R. Arnowitt, S. Deser, and C. W. Misner, Republication of: The dynamics of general relativity, Gen. Relativ. Gravit. 40, 1997 (2008).
  34. R. Samanta, S. Tanay, and L. C. Stein, Closed-form solutions of spinning, eccentric binary black holes at 1.5 post-Newtonian order, Phys. Rev. D 108, 124039 (2023).
  35. J. D. Schnittman, Spin-orbit resonance and the evolution of compact binary systems, Phys. Rev. D 70, 124020 (2004).
  36. D. Gerosa, M. Kesden, R. O’Shaughnessy, A. Klein, E. Berti, U. Sperhake, and D. Trifirò, Precessional instability in binary black holes with aligned spins, Phys. Rev. Lett. 115, 141102 (2015).
  37. D. Gerosa, M. Kesden, U. Sperhake, E. Berti, and R. O’Shaughnessy, Multi-timescale analysis of phase transitions in precessing black-hole binaries, Phys. Rev. D 92, 064016 (2015).
  38. S. Tanay, L. C. Stein, and J. T. Gálvez Ghersi, Integrability of eccentric, spinning black hole binaries up to second post-Newtonian order, Phys. Rev. D 103, 064066 (2021).
  39. E. Racine, Analysis of spin precession in binary black hole systems including quadrupole-monopole interaction, Phys. Rev. D 78, 044021 (2008).
  40. L. A. Gergely, Spin spin effects in radiating compact binaries, Phys. Rev. D 61, 024035 (2000).
  41. A. Klein and P. Jetzer, Spin effects in the phasing of gravitational waves from binaries on eccentric orbits, Phys. Rev. D 81, 124001 (2010).
  42. L. E. Kidder, Coalescing binary systems of compact objects to (post) 5/2-Newtonian order. V. Spin effects, Phys. Rev. D 52, 821 (1995).
  43. B. M. Barker and R. F. O’Connell, Nongeodesic motion in general relativity, Gen. Relativ. Gravit. 5, 539 (1974).
  44. M. D. Hartl and A. Buonanno, The dynamics of precessing binary black holes using the post-Newtonian approximation, Phys. Rev. D 71, 024027 (2005).
  45. A. Gopakumar and C. Konigsdorffer, The Deterministic nature of conservative post-Newtonian accurate dynamics of compact binaries with leading order spin-orbit interaction, Phys. Rev. D 72, 121501 (2005).
  46. T. Colin and S. Tanay, mathematica notebook for the analytical 2PN spinning, eccentric binary black hole solution, https://github.com/sashwattanay/BBH-PN-Toolkit (2026), directory: 2PN_BBH_solution. Main file: 2PN_BBH_solution. nb (also provided in .m, .pdf, and .txt formats).
  47. M. H. L. Pryce, The mass-centre in the restricted theory of relativity and its connexion with the quantum theory of elementary particles, Proc. R. Soc. A 195, 62 (1948).
  48. T. D. Newton and E. P. Wigner, Localized states for elementary systems, Rev. Mod. Phys. 21, 400 (1949).
  49. T. Damour, Coalescence of two spinning black holes: An effective one-body approach, Phys. Rev. D 64, 124013 (2001).
  50. B. M. Barker, S. N. Gupta, and R. D. Haracz, One-graviton exchange interaction of elementary particles, Phys. Rev. 149, 1027 (1966).
  51. B. M. Barker and R. F. O’Connell, Gravitational two-body problem with arbitrary masses, spins, and quadrupole moments, Phys. Rev. D 12, 329 (1975).
  52. J. Steinhoff, Canonical formulation of spin in general relativity, Ann. Phys. (Berlin) 523, 296 (2011).
  53. E. Poisson and C. M. Will, Gravity: Newtonian, Post-Newtonian, Relativistic (Cambridge University Press, Cambridge, England, 2014).
  54. J. Spitale and R. Greenberg, Numerical evaluation of the general Yarkovsky effect: Effects on eccentricity and longitude of periapse, Icarus 156, 211 (2002).
  55. D. J. Scheeres and W. Hu, Secular motion in a 2nd degree and order-gravity field with no rotation, Celest. Mech. Dyn. Astron. 79, 183 (2001).
  56. S. Tanay, L. C. Stein, and G. Cho, Action-angle variables of a binary black hole with arbitrary eccentricity, spins, and masses at 1.5 post-Newtonian order, Phys. Rev. D 107, 103040 (2023).
  57. J. José and E. Saletan, Classical Dynamics: A Contemporary Approach (Cambridge University Press, Cambridge, England, 1998).
  58. A. da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics No. 1764 (Springer, New York, 2001).
  59. A. Fasano and S. Marmi, Analytical Mechanics: An Introduction, Oxford Graduate Texts (OUP, Oxford, 2006).
  60. Q. Henry and M. Khalil, Spin effects in gravitational waveforms and fluxes for binaries on eccentric orbits to the third post-Newtonian order, Phys. Rev. D 108, 104016 (2023).
  61. L. V. Drummond and S. A. Hughes, Precisely computing bound orbits of spinning bodies around black holes. I. General framework and results for nearly equatorial orbits, Phys. Rev. D 105, 124040 (2022).
  62. L. V. Drummond and S. A. Hughes, Precisely computing bound orbits of spinning bodies around black holes. II. Generic orbits, Phys. Rev. D 105, 124041 (2022).

Outline

Information

Phys. Rev. D 114, 064051– Published 10 September, 2026
  • Received 3 April 2026
  • Accepted 28 July 2026
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation