Skip to main content
SpringerLink
Log in

Regular dynamics of canonical post-Newtonian Hamiltonian for spinning compact binaries with next-to-leading order spin-orbit interactions

  • Research Article
  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

Abstract

This paper shows that a conservative canonical post-Newtonian Hamiltonian formulation of spinning compact binaries with a pure orbital part up to third post-Newtonian order and spin-orbit contributions at the next-to-leading post-Newtonian order is explicitly integrable and regular because there are 5 independent exact isolating integrals in the 10-dimensional phase space. With the help of symplectic integrators and the fast Lyapunov indicators of two nearby trajectories, numerical investigations also support the absence of chaos.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Damour T., Jaranowski P., Schäfer G.: Equivalence between the ADM-Hamiltonian and the harmonic-coordinates approaches to the third post-Newtonian dynamics of compact binaries. Phys. Rev. D 63, 044021 (2001)

    Article  ADS  Google Scholar 

  2. Blanchet L., Iyer B.R.: Third post-Newtonian dynamics of compact binaries: equations of motion in the centre-of-mass frame. Class. Quantum Gravity 20, 755 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Königsdörffer C., Gopakumar A.: Post-Newtonian accurate parametric solution to the dynamics of spinning compact binaries in eccentric orbits: The leading order spin-orbit interaction. Phys. Rev. D 71, 024039 (2005)

    Article  Google Scholar 

  4. Gopakumar A., Königsdörffer C.: Deterministic nature of conservative post-Newtonian accurate dynamics of compact binaries with leading order spin-orbit interaction. Phys. Rev. D 72, 121501(R) (2005)

    ADS  Google Scholar 

  5. Levin J.: Fate of chaotic binaries. Phys. Rev. D 67, 044013 (2003)

    Article  ADS  Google Scholar 

  6. Levin J.: Chaos and order in models of black hole pairs. Phys. Rev. D 74, 124027 (2006)

    Article  ADS  Google Scholar 

  7. Cornish N.J., Levin J.: Chaos and damping in the post-Newtonian description of spinning compact binaries. Phys. Rev. D 68, 024004 (2003)

    Article  ADS  Google Scholar 

  8. Levin J., Grossman R.: Dynamics of black hole pairs. I. Periodic tables. Phys. Rev. D 79, 043016 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  9. Grossman R., Levin J.: Dynamics of black hole pairs. II. Spherical orbits and the homoclinic limit of zoom-whirliness. Phys. Rev. D 79, 043017 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  10. Wu X., Xie Y.: Symplectic structure of post-Newtonian Hamiltonian for spinning compact binaries. Phys. Rev. D 81, 084045 (2010)

    Article  ADS  Google Scholar 

  11. Levin J.: Gravity waves, chaos, and spinning compact binaries. Phys. Rev. Lett 84, 3515 (2000)

    Article  ADS  Google Scholar 

  12. Cornish N.J.: Comment on “Gravity Waves, Chaos, and Spinning Compact Binaries”. Phys. Rev. Lett. 85, 3980 (2000)

    Article  ADS  Google Scholar 

  13. Cornish N.J.: Chaos and gravitational waves. Phys. Rev. D 64, 084011 (2001)

    Article  ADS  Google Scholar 

  14. Cornish NJ., Levin J.: Comment on “Ruling Out Chaos in Compact Binary Systems”. Phys. Rev. Lett 89, 179001 (2002)

    Article  ADS  Google Scholar 

  15. Xie Y., Huang T.Y.: Chaos in compact binaries with frequency map analysis. Chinese. J. Astron. Astrophys. 6, 705 (2006)

    ADS  Google Scholar 

  16. Hartl M.D., Buonanno A.: Dynamics of precessing binary black holes using the post-Newtonian approximation. Phys. Rev. D 71, 024027 (2005)

    Article  ADS  Google Scholar 

  17. Wu X., Xie Y.: Revisit on “Ruling out chaos in compact binary systems”. Phys. Rev. D 76, 124004 (2007)

    Article  ADS  Google Scholar 

  18. Wu X., Xie Y.: Resurvey of order and chaos in spinning compact binaries. Phys. Rev. D 77, 103012 (2008)

    Article  ADS  Google Scholar 

  19. Zhong S.Y., Wu X.: Manifold corrections on spinning compact binaries. Phys. Rev. D 81, 104037 (2010)

    Article  ADS  Google Scholar 

  20. Damour T., Jaranowski P., Schäfer G.: Hamiltonian of two spinning compact bodies with next- to-leading order gravitational spin-orbit coupling. Phys. Rev. D 77, 064032 (2008)

    Article  ADS  Google Scholar 

  21. Damour T., Jaranowski P., Schäfer G.: Effective one body approach to the dynamics of two spinning black holes with next-to-leading order spin-orbit coupling. Phys. Rev. D 78, 024009 (2008)

    Article  ADS  Google Scholar 

  22. Damour T., Jaranowski P., Schäfer G.: ADM canonical formalism for gravitating spinning objects. Phys. Rev. D 77, 104018 (2008)

    Article  MathSciNet  Google Scholar 

  23. Damour T., Jaranowski P., Schäfer G.: Poincaré invariance in the ADM Hamiltonian approach to the general relativistic two-body problem. Phys. Rev. D 62, 021501 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  24. Damour T., Jaranowski P., Schäfer G.: Dimensional regularization of the gravitational interaction of point masses. Phys. Lett. B 513, 147 (2001)

    Article  ADS  MATH  Google Scholar 

  25. Arnold V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1978)

    MATH  Google Scholar 

  26. Lichtenberg A.J., Lieberman M.A.: Regular and Chaotic Dynamics. Springer, New York (1983)

    Google Scholar 

  27. Binney J., Tremaine S.: Galactic Dynamics. 2nd edn. Princeton University Press, Princeton (2008)

    MATH  Google Scholar 

  28. Lubich C., Walther B., Brügmann B.: Symplectic integration of post-Newtonian equations of motion with spin. Phys. Rev. D 81, 104025 (2010)

    Article  ADS  Google Scholar 

  29. Zhong S.Y., Wu X., Liu S.Q., Deng X.F.: Global symplectic structure-preserving integrators for spinning compact binaries. Phys. Rev. D 82, 124040 (2010)

    Article  ADS  Google Scholar 

  30. Liao X.H.: Symplectic integrator for general near-integrable Hamiltonian system. Celest. Mech. Dyn. Astron. 66, 243 (1997)

    Article  ADS  MATH  Google Scholar 

  31. Preto M., Saha P.: On post-Newtonian orbits and the galactic-center stars. Astrophys. J. 703, 1743 (2009)

    Article  ADS  Google Scholar 

  32. Wisdom J., Holman M.: Symplectic maps for the n-body problem. Astron. J 102, 1528 (1991)

    Article  ADS  Google Scholar 

  33. Wu X., Huang T.Y., Zhang H.: Lyapunov indices with two nearby trajectories in a curved spacetime. Phys. Rev. D 74, 083001 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  34. Froeschlé C., Lega E., Gonczi R.: Fast Lyapunov indicators. Appl. Asteroidal Motion. Celest. Mech. Dyn. Astron 67, 41 (1997)

    Article  ADS  MATH  Google Scholar 

  35. Wang Y., Wu X.: Next-order spin-orbit contribution to chaos in compact binaries. Class. Quantum Gravity 28, 025010 (2011)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Astronomy and Department of Physics, Nanchang University, 330031, Nanchang, China

    Xin Wu & Shuang-Ying Zhong

Authors
  1. Xin Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shuang-Ying Zhong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xin Wu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wu, X., Zhong, SY. Regular dynamics of canonical post-Newtonian Hamiltonian for spinning compact binaries with next-to-leading order spin-orbit interactions. Gen Relativ Gravit 43, 2185–2198 (2011). https://doi.org/10.1007/s10714-011-1171-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10714-011-1171-0

Keywords

Search

Navigation