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Constants of motion and fundamental frequencies for elliptic orbits at fourth post-Newtonian order

David Trestini

TL;DR

This work delivers the complete conservative map between energy $E$ and angular momentum $J$ and the fundamental orbital frequencies $(n,\omega)$ for nonspinning binaries on elliptic orbits at 4PN order, including both instantaneous and tail contributions. The author deploys an action-angle (Delaunay) framework, localizes the nonlocal tail via a contact transformation, and performs orbit averaging to obtain a local Hamiltonian $H(i_{r\phi},i_\phi)$ whose derivatives yield $(n,\omega)$ and the orbit-averaged redshift $\langle z_1\rangle$, with perfect agreement against self-force results. A carefully resummed tail enhancement function $\Lambda_0(e)$ controls eccentricity effects across all $e$, enabling accurate expressions for the 4PN redshift and for recasting 3PN fluxes in terms of the gauge-invariant frequencies. The circular-orbit limits reproduce known results and the approach lays the groundwork for 4PN eccentric phasing, while also outlining necessary steps to extend to spins and higher PN orders. Overall, the paper advances precision modeling for eccentric binary dynamics and provides robust, gauge-invariant connections between orbital constants and observables relevant for gravitational-wave data analysis.

Abstract

In the case of nonspinning compact binary systems on quasi-elliptic orbits, I obtain the conservative map between the constants of motion (energy and angular momentum) and the fundamental (radial and azimuthal) frequencies at the fourth post-Newtonian order, including both instantaneous and tail contributions. This map is expressed in terms of an enhancement function of the eccentricity, which is appropriately resummed to ensure accuracy for any eccentricity; in particular, I recover known results for circular orbits. In order to obtain this map, the local dynamics are expressed using an action-angle formulation. The tail term is treated as a perturbation, which is first localized in time, then Delaunay-averaged. Both operations require a contact transformation of the phase-space variables, which I explicitly control. Using the first law of binary black hole mechanics, I then obtain the orbit-averaged redshift invariant for eccentric orbits at fourth post-Newtonian order; when properly accounting for the tail contributions, it perfectly agrees with analytical self-force at postgeodesic order [arXiv:2203.13832]. Finally, I use these results to re-express the fluxes of energy and angular momentum obtained at third post-Newtonian order in [arXiv:0711.0302] and [arXiv:0908.3854] in terms of fundamental frequencies.

Constants of motion and fundamental frequencies for elliptic orbits at fourth post-Newtonian order

TL;DR

This work delivers the complete conservative map between energy and angular momentum and the fundamental orbital frequencies for nonspinning binaries on elliptic orbits at 4PN order, including both instantaneous and tail contributions. The author deploys an action-angle (Delaunay) framework, localizes the nonlocal tail via a contact transformation, and performs orbit averaging to obtain a local Hamiltonian whose derivatives yield and the orbit-averaged redshift , with perfect agreement against self-force results. A carefully resummed tail enhancement function controls eccentricity effects across all , enabling accurate expressions for the 4PN redshift and for recasting 3PN fluxes in terms of the gauge-invariant frequencies. The circular-orbit limits reproduce known results and the approach lays the groundwork for 4PN eccentric phasing, while also outlining necessary steps to extend to spins and higher PN orders. Overall, the paper advances precision modeling for eccentric binary dynamics and provides robust, gauge-invariant connections between orbital constants and observables relevant for gravitational-wave data analysis.

Abstract

In the case of nonspinning compact binary systems on quasi-elliptic orbits, I obtain the conservative map between the constants of motion (energy and angular momentum) and the fundamental (radial and azimuthal) frequencies at the fourth post-Newtonian order, including both instantaneous and tail contributions. This map is expressed in terms of an enhancement function of the eccentricity, which is appropriately resummed to ensure accuracy for any eccentricity; in particular, I recover known results for circular orbits. In order to obtain this map, the local dynamics are expressed using an action-angle formulation. The tail term is treated as a perturbation, which is first localized in time, then Delaunay-averaged. Both operations require a contact transformation of the phase-space variables, which I explicitly control. Using the first law of binary black hole mechanics, I then obtain the orbit-averaged redshift invariant for eccentric orbits at fourth post-Newtonian order; when properly accounting for the tail contributions, it perfectly agrees with analytical self-force at postgeodesic order [arXiv:2203.13832]. Finally, I use these results to re-express the fluxes of energy and angular momentum obtained at third post-Newtonian order in [arXiv:0711.0302] and [arXiv:0908.3854] in terms of fundamental frequencies.

Paper Structure

This paper contains 37 sections, 172 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Notations
  3. Summary of results
  4. Hamiltonian formulation for the 4PN dynamics of compact binaries
  5. The 4PN equations of motion and their derivation from an action principle
  6. Formalism for constructing the 4PN Hamiltonian and treatment of the hereditary piece
  7. Obtaining the local equations of motion from the local Hamiltonian
  8. Action-angle formulation of the local dynamics
  9. Action variables in the local sector
  10. Local Hamiltonian in terms of action variables
  11. Angle variables in the local problem
  12. Tail corrections to the local Hamiltonian in action-angle variables
  13. Splitting the tail term into a logarithmic and hereditary contribution
  14. Localizing the hereditary Hamiltonian
  15. Delaunay averaging of the tail Hamiltonian
  16. ...and 22 more sections

Figures (1)

  • Figure 1: (Top panel) Comparison of various estimates for $\Lambda_0(e)$, normalized by the divergent factor $(1-e^2)^{-7/2}$: (i) the black dots are the data points for the reference numerical estimate; (ii) the blue curve corresponds to the small-eccentricity expansion \ref{['eq:Lambda_0_small_e_expansion']}, including terms up to $\mathcal{O}(e^{10})$; (iii) the red curve corresponds to the naive resummation \ref{['eq:Lambda_0_resummed_naive_truncated_N']} with $N = 4$; and (iv) the green curve corresponds to the proposed resummation \ref{['eq:Lambda_0_resummed_truncated_N']} with $N=4$. The small eccentricity expansion becomes fully inadequate for $e \ge 0.5$; whereas both resummations provide a reasonable estimate for larger $e$. (Inset plot in top panel) Same as the top panel, but zooming into the large eccentricity region, namely $0.8 \le e \le 1$. The 'naive' resummation is not accurate in this regime, whereas the proposed resummation is extremely accurate. (Bottom panel) Relative error of the proposed resummation $\Lambda_0^{[N]}(e)$ for $N=4$ [see Eq. \ref{['eq:Lambda_0_resummed_truncated_N']}], with respect to the numerical estimate for $\Lambda_0(e)$. For any value of the eccentricity, the relative error always remains smaller than $4\cdot 10^{-6}$. For $e \le 0.15$, the error is dominated by numerical noise, due to the division by $|\Lambda_0(e)|$ which vanishes in the $e\rightarrow 0$ limit. For $e>0.15$, the error is dominated by the inaccuracy of the resummation; this required controlling the numerical estimate very precisely by summing over many modes. For $e=0.2$, $p_\mathrm{max}=20$ modes are enough, whereas for $e=0.99$, $p_\mathrm{max}=15\,000$ modes were required!