Third post-Newtonian accurate generalized quasi-Keplerian parametrization for compact binaries in eccentric orbits

Achamveedu Gopakumar; Gerhard Sch\"afer; Raoul-Martin Memmesheimer

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Third post-Newtonian accurate generalized quasi-Keplerian parametrization for compact binaries in eccentric orbits

Raoul-Martin Memmesheimer, Achamveedu Gopakumar, Gerhard Schäfer

General Relativity and Quantum Cosmology · Submitted 13 Jul 2004 · 33 references

Pith's verdict: A careful, fully explicit 3PN quasi-Keplerian parametrization in both ADM and harmonic coordinates; the algebra is long but the cross-gauge agreement on n and Φ is a real check, not just a formality.

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CitationAchamveedu Gopakumar, Gerhard Sch\"afer, Raoul-Martin Memmesheimer. "Third post-Newtonian accurate generalized quasi-Keplerian parametrization for compact binaries in eccentric orbits." arXiv:gr-qc/0407049 (2004). https://pith.science/paper/45KO6JJF

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Bottom line: this is the 3PN extension of the Damour–Schäfer–Wex generalized quasi-Keplerian parametrization, done in both ADM-type and harmonic coordinates, with every orbital element and correction function written out in terms of E, h, and η. The harmonic version is the first beyond 1PN. If you work on eccentric-binary templates, PN/NR comparison, or pulsar timing, this is the reference you will pull the formulas from. It is a tool paper, and it does the tool job properly. The 2PN limit reproduces [14,15], and the structure (r = a_r(1−e_r cos u), Kepler equation with (v−u), sin v, sin 2v, sin 3v, angle equation through sin 5v) is exactly what the 2PN pattern predicts. Nothing here is conceptually surprising, and the authors do not pretend otherwise. On the stress-test worry: it is half right. The ṙ² reconstruction check does reuse the same polynomial and roots, so it validates the trigonometric conversion and the e_φ determination but not the coefficients A1/A3 themselves. But the n and Φ agreement across gauges is stronger than the note allows. The two derivations start from genuinely different inputs — the ADM Hamiltonian versus harmonic E and J — so a shared algebraic slip in the 3PN coefficients would generically break that identity. It is an expected result, but confirming it at this order with this much algebra is real evidence. I would also push back on the claim that degree-7 in s is merely asserted: A3a–A3i display it explicitly. The genuine soft spots are smaller. The intermediate κ_i and λ_i are computed and then withheld (Sec. III, after Eq. 13), so a reader cannot retrace the trig step. The consistency checks are reported, not shown. And the harmonic parametrization is tied to one specific 3PN harmonic coordinate system — the log-removal of [28] plus the regularization fixing of [23,30,31] — so 'harmonic gauge' here means that choice, which the authors should flag more loudly than they do. None of this threatens the central claim. Who should read it: anyone building 3.5PN eccentric templates, anyone setting up PN diagnostics for NR runs, and pulsar-timing people thinking about higher-order timing-formula corrections. Worth a serious referee, mainly to have someone redo a slice of the algebra independently.

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Major objections (3)

[Sec. III, paragraph following Eq. (21b); Sec. IV, Eqs. (A3)-(A4)] The validation of the 3PN coefficients rests on reconstructing rdot^2 and phidot^2 from the parametrization and comparing to the equations of motion. As the authors note, a_r and e_r come from the roots s_pm of the same rdot^2 polynomial (Eqs. 9-11) and the Kepler equation from integrating that same polynomial (Eqs. 12-13), so this check tests the trigonometric conversion and the e_phi determination but not the correctness of the polynomial coefficients A1/A3 or the root extraction. A common algebraic error in, e.g., the 3PN energy coefficient of Eq. (23d) would propagate identically into derivation and check. The paper would be strengthened by at least one check that does not route through the same polynomial, for example a numerical integration of the Hamiltonian equations of motion (ADM) or of the energy/angular-momentum equations (harmonic) at sample (E, h, eta) compared against the parametrized r(t), phi(t) over several radial periods. This is a correctness-risk concern, not a claim that an error exists.
[Sec. IV, paragraphs preceding Eqs. (22)-(23); Eq. (A3)] The harmonic-coordinate derivation depends on two external inputs: removal of the logarithmic terms from the 3PN E and J via the coordinate transformation of [28], and the fixing of the regularization ambiguity via [23,30,31]. Both are cited published results and are properly supported as such. What the manuscript asserts rather than shows is that the resulting rdot^2 is a degree-seven polynomial in s = 1/r (Eq. A3), which is the property that makes the turning-point factorization and root extraction possible. A short explicit demonstration (or a pointer to where in [28] the log-free form is exhibited in this variable) would make the harmonic construction self-contained and would let a reader verify that no residual log or non-polynomial term survives.
[Sec. IV, Eqs. (25)-(26) versus Sec. III, Eqs. (20)-(21)] The paper states that the structure of the parametrization is identical in the two gauges and that n and Phi agree to 3PN, which is presented as a powerful check. That check is meaningful for n and Phi, but the remaining elements (a_r, e_r, e_t, e_phi and the correction functions) differ between gauges, and the paper gives no cross-gauge relation for them beyond Eq. (27) for e_r^2. Since the two derivations share the same overall procedure and much of the algebra, a brief statement of which intermediate quantities were computed independently in the two gauges (and which were reused) would help the reader judge how much independent confirmation the gauge comparison actually provides.

Minor objections (8)

[Title and abstract] The abstract says the elements are given in terms of the conserved energy and angular momentum; it would be worth stating explicitly that the mass ratio eta also enters, as it does throughout Eqs. (20)-(21) and (25)-(26).
[Eqs. (20d), (25d)] In the expression for e_t^2 the terms -24 sqrt(-2E h^2)(-5+2eta) and +24 sqrt(-2E h^2)(-5+2eta) appear with opposite signs in the same bracket; please confirm this cancellation is intended and not a transcription slip.
[Eqs. (20r), (25r)] The e_phi^2 expressions are split across a bracket boundary in a way that makes the c^4 and c^6 groupings hard to parse; a consistent factorization by powers of (-2E) would help the reader.
[Eq. (21a) and Eq. (26a)] The relations between e_t and e_r are given to 3PN but the corresponding relation between e_phi and e_r (Eqs. 21b, 26b) uses a different normalization of the 1/(-2E h^2) factors; please check that the two are stated in a uniform convention.
[Sec. II, Eqs. (3a)-(3c)] The Newtonian elements are written with factors of G M; since the rest of the paper uses units with G M = 1 (r = R/(GM)), a one-line statement of the unit convention at the start of Sec. II would prevent confusion.
[Sec. III, Eq. (13) and Appendix B] The temporary coefficients kappa_i are said to be omitted because they are only needed temporarily; since the final expressions are fully explicit this is acceptable, but a note on how the kappa_i were checked (e.g. against the 2PN limit of [15]) would be useful.
[Fig. 1] The figure caption refers to the auxiliary circle and the focus but the figure itself is not described in enough detail to be read without the text; a sentence identifying the labeled points and the angle definitions would improve clarity.
[References] Reference [16] is a diploma thesis cited as containing an incomplete earlier ADM representation; since the present paper supersedes it, it would help to state in the text which specific results of [16] are confirmed and which are corrected.

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The paper

The paper claims

The paper derives a Kepler-style description of eccentric binary orbits accurate to third post-Newtonian order, with every orbital element written explicitly in terms of the orbit's conserved energy and angular momentum.

The reading

Two compact objects in an eccentric orbit, described by general relativity expanded to third post-Newtonian (3PN) order in $v^2/c^2$, do not move on ellipses: the orbit precesses, the period shifts, and the radial and angular motions fall out of step. The paper's aim is to show that this motion can still be written in the old Kepler form — a radius $r = a_r(1 - e_r cos u)$, a Kepler equation relating time to the eccentric anomaly $u$, and an angle equation relating $v$ to $u$ — provided one allows a few extra sine harmonics and lets the three eccentricities $e_r$, $e_t$, $e_φ$ differ from one another by PN corrections. The authors carry this out completely, giving every element and every correction function explicitly in terms of the conserved energy $E$, angular momentum $h$, and mass ratio $η$, and doing it twice: once in ADM-type coordinates and once in harmonic coordinates. The payoff, if it holds, is a ready-made analytic orbit that can be plugged into gravitational-wave search templates for inspiralling eccentric binaries, into post-Newtonian diagnostics of numerical-relativity simulations, and into higher-order corrections to binary-pulsar timing formulas.

What carries the argument

The load-bearing device is the turning-point factorization. Because $ṙ^2$ and $dφ/ds$ (with $s = 1/r$) both come out as polynomials of degree seven in $s$ at 3PN order, the two positive roots $s_-$ and $s_+$ that survive as $1/c → 0$ can be read as the periastron and apastron. Writing $a_r = (s_- + s_+)/(2 s_- s_+)$ and $e_r = (s_- - s_+)/(s_- + s_+)$ turns the radial integral into the standard Kepler-like form and yields the radial period $P$, hence the mean motion $n = 2π/P$. The angular motion is handled by evaluating the $dφ/ds$ integral between the same two roots to get $Φ$, then introducing a third eccentricity $e_φ$ and choosing its PN corrections so that no $sin v$ term survives in the angle equation — the requirement that fixes $e_φ$ uniquely and reproduces the Keplerian structure at first PN order. Two consistency checks carry the weight of the algebra: $ṙ^2$ and $φ̇^2$ recomputed from the parametrization are compared against the same quantities computed from the Hamiltonian equations of motion (ADM) or from the conserved $E$ and $J$ (harmonic), and both agree to 3PN order.

Extended reading notes

Core claim

The central claim is that the 3PN-accurate conservative equations of motion for a non-spinning compact binary in an eccentric orbit admit a generalized quasi-Keplerian representation of exactly the same structural shape as the 2PN one: $r = a_r(1 - e_r cos u)$; a Kepler equation $l = n(t-t_0) = u - e_t sin u + (g_{4t}/c^4 + g_{6t}/c^6)(v-u) + (f_{4t}/c^4 + f_{6t}/c^6) sin v + (i_{6t}/c^6) sin 2v + (h_{6t}/c^6) sin 3v$; and an angle equation $2π(φ-φ_0)/Φ = v + (f_{4φ}/c^4 + f_{6φ}/c^6) sin 2v + (g_{4φ}/c^4 + g_{6φ}/c^6) sin 3v + (i_{6φ}/c^6) sin 4v + (h_{6φ}/c^6) sin 5v$, with $v = 2 arctan[((1+e_φ)/(1-e_φ))^{1/2} tan(u/2)]$. All elements — $a_r$, $e_r^2$, $n$, $e_t^2$, $Φ$, $e_φ^2$ — and all twelve orbital functions are written out explicitly in terms of $E$, $h$ and $η$ (Eqs. 20 and 25), together with the PN relations linking the three eccentricities (Eqs. 21 and 26). A second, independent claim is that the mean motion $n$ and the periastron advance per revolution $Φ$ come out identical in ADM-type and harmonic coordinates to 3PN order, which the authors present as a nontrivial confirmation of the expectation that these two quantities, expressed through gauge-invariant $E$ and $h$, should be coordinate-independent. From that invariance they extract gauge-invariant expressions for the dimensionless energy $E/c^2$ and squared angular momentum $(hc)^2$ in terms of $x = (GMn/c^3)^{2/3}$ and $k' = (Φ-2π)/(6π)$, generalizing the circular-orbit energy–frequency relation to eccentric orbits.

Load-bearing premise

The whole construction rests on $ṙ^2$ and $dφ/ds$ remaining polynomials of degree seven in $s = 1/r$ at 3PN order, which in harmonic coordinates holds only after adopting a specific coordinate transformation to remove logarithmic terms and a specific value for the regularization ambiguity — both taken from earlier work rather than derived here.

Editorial extensions

If this is right

Search templates for gravitational waves from inspiralling eccentric compact binaries can be extended from the existing 2.5PN treatment to a fully 3.5PN-accurate orbital description.
The explicit 3PN orbit supplies the input needed to compute $O(1/c^{11})$ corrections to orbit-averaged far-zone fluxes for eccentric binaries, once those fluxes are available.
Because $n$ and $Φ$ are gauge invariant, eccentric orbits can be characterized by quantities that do not depend on the coordinate system, giving a cleaner post-Newtonian diagnostic for comparing numerical-relativity simulations against PN predictions.
The parametrization provides the analytic solution needed to extend the relativistic binary-pulsar timing formula beyond its current PN order, which matters for long-baseline timing of systems such as the double pulsar.
The gauge-invariant relations in Eq. (28) generalize the circular-orbit energy–frequency relation to eccentric orbits, opening the way to 3PN expressions for $dx/dt$ and $dk'/dt$ in terms of $x$ and $k'$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The identical form of $n(E,h)$ and $Φ(E,h)$ in two coordinate systems is the paper's strongest internal evidence; a reader can treat it as a numerical cross-check that a bug in either derivation would almost certainly break.
Because the parametrization is purely conservative, the elements are constants here; the authors note that with radiation reaction switched on they become slowly evolving variables, which is what would make the construction usable inside an adiabatic inspiral template.
The degree-seven polynomial structure in $s = 1/r$ is what makes the whole approach work; if a future 4PN treatment introduces non-polynomial terms (as logarithms already do at 3PN in harmonic coordinates before the coordinate transformation), the turning-point method would need to be rethought rather than merely extended.
The eccentricity relations in Eqs. (21) and (26) mean that a single chosen eccentricity (say $e_r$) fixes the other two; a reader comparing PN eccentricity definitions across the literature should expect offsets of PN order, not agreement.

Original abstract

We present Keplerian-type parametrization for the solution of third post-Newtonian (3PN) accurate equations of motion for two non-spinning compact objects moving in an eccentric orbit. The orbital elements of the parametrization are explicitly given in terms of the 3PN accurate conserved orbital energy and angular momentum in both Arnowitt, Deser, and Misner-type and harmonic coordinates. Our representation will be required to construct post-Newtonian accurate `ready to use' search templates for the detection of gravitational waves from compact binaries in inspiralling eccentric orbits. Due to the presence of certain 3PN accurate gauge invariant orbital elements, the parametrization should be useful to analyze the compatibility of general relativistic numerical simulations involving compact binaries with the corresponding post-Newtonian descriptions. If required, the present parametrization will also be needed to compute post-Newtonian corrections to the currently employed `timing formula' for the radio observations of relativistic binary pulsars.

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What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.18 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.12 · mentions code or data availability 0.03.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We present Keplerian-type parametrization for the solution of third post-Newtonian (3PN) accurate equations of motion for two non-spinning compact objects moving in an eccentric orbit. The orbital elements of the parametrization are explicitly given in terms of the 3PN accurate conserved orbital energy and angular momentum in both Arnowitt, Deser, and Misner-type and harmonic coordinates. Our representation will be required to construct post-Newtonian accurate `ready to use' search templates for the detection of gravitational waves from compact binaries in inspiralling eccentric orbits. Due to the presence of certain 3PN accurate gauge invariant orbital elements, the parametrization should be useful to analyze the compatibility of general relativistic numerical simulations involving compact binaries with the corresponding post-Newtonian descriptions. If required, the present parametrization will also be needed to compute post-Newtonian corrections to the currently employed `timing formula' for the radio observations of relativistic binary pulsars.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present Keplerian-type parametrization for the solution of third post-Newtonian (3PN) accurate equations of motion for two non-spinning compact objects moving in an eccentric orbit.	contribution 1.00	0.95	0.56	asserted 1.00
s1	The orbital elements of the parametrization are explicitly given in terms of the 3PN accurate conserved orbital energy and angular momentum in both Arnowitt, Deser, and Misner-type and harmonic coordinates.	contribution 0.49	0.92	0.17	asserted 1.00
s2	Our representation will be required to construct post-Newtonian accurate `ready to use' search templates for the detection of gravitational waves from compact binaries in inspiralling eccentric orbits.	future work 0.55	0.21	0.21	asserted 0.73
s3	Due to the presence of certain 3PN accurate gauge invariant orbital elements, the parametrization should be useful to analyze the compatibility of general relativistic numerical simulations involving compact binaries with the corresponding post-Newtonian descriptions.	interpretation 0.84	0.24	0.08	hedged possibility 0.93
s4	If required, the present parametrization will also be needed to compute post-Newtonian corrections to the currently employed `timing formula' for the radio observations of relativistic binary pulsars.	future work 0.67	0.13	0.06	hedged possibility 0.96

What kind of paper the classifier reports this is: theoretical result 0.68 · evidence from proof or derivation 1.00.

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Figures

**Figure 1.** FIG. 1: The geometrical interpretation of the eccentric and

Figures reproduced from arXiv:gr-qc/0407049 by the authors.

References

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gr-qc 2015-03 accept novelty 7.0 of 10

A 3PN analytic formula for the orbit-averaged redshift of an eccentric binary matches numerical self-force results to about 1% at moderate separations, and the numerical data constrain two previously unknown 4PN coefficients.
Next-to-next-to-leading gravitational radiation-reaction corrections to quasi-elliptic and quasi-hyperbolic compact binary orbits
gr-qc 2026-11 accept novelty 6.0 of 10

Complete radiation-reaction corrections up to 4.5PN order are derived for quasi-elliptic and quasi-hyperbolic compact binary orbits using the Lagrange method of variation of constants, including non-local tail and rec...
Post-adiabatic effective-one-body waveforms for coalescing compact binaries
gr-qc 2026-10 conditional novelty 6.0 of 10

A post-adiabatic equinoctial-element formulation speeds up eccentric effective-one-body waveform generation by a median of roughly 2.6x for neutron star binaries and 1.5x for black holes, with mismatches below 5e-7 fo...
Periastron advance from the perturbed Binet equation
gr-qc 2026-09 conditional novelty 6.0 of 10

A Poincaré-Lindstedt treatment of the perturbed Binet equation derives the periastron advance from the first Fourier coefficient of the perturbation and yields new multipolar formulas.
Preferential accretion onto eccentric and unequal binary black holes
astro-ph.HE 2026-09 accept novelty 6.0 of 10

The secondary black hole in a supermassive binary tends to out-accrete the primary, with this preference strongest for low-eccentricity, low-mass-ratio binaries and becoming increasingly time-variable at high eccentricity.
Analytical Solution of Spinning, Eccentric Binary Black Hole Dynamics at the Second Post-Newtonian Order
gr-qc 2026-03 unverdicted novelty 6.0 of 10

An analytical 2PN solution is constructed for the orbital and spin dynamics of eccentric, arbitrarily spinning binary black holes, with spin oscillations retained only at 1.5PN accuracy.
Radiated momentum and radiation-reaction in gravitational two-body scattering including time-asymmetric effects
gr-qc 2022-10 conditional novelty 6.0 of 10

Including time-asymmetric tail and radiation-reaction effects is necessary for the post-Minkowskian expansion of the radiated four-momentum to be polynomial in the two masses, and that polynomiality then fixes the 4PM...
Radiation-reaction force and multipolar waveforms for eccentric, spin-aligned binaries in the effective-one-body formalism
gr-qc 2021-04 accept novelty 6.0 of 10

The paper derives eccentric-orbit radiation-reaction forces and gravitational-wave modes through 2PN order, including tail effects and aligned spins, in a factorized form that plugs into the SEOBNRv4HM waveform model.
Combining Post-Circular and Pad\'e approximations to compute Fourier domain templates for eccentric inspirals
gr-qc 2020-09 conditional novelty 6.0 of 10

Padé-resummed post-circular expressions yield fully analytic frequency-domain eccentric inspiral templates that stay faithful to a semi-analytic benchmark up to e0 = 0.6.
Fourier domain gravitational waveforms for precessing eccentric binaries
gr-qc 2018-01 conditional novelty 6.0 of 10

Two new frequency-domain waveform families reproduce an equivalent time-domain eccentric precessing model with median faithfulness above 0.993 up to initial eccentricity about 0.3, and circular templates can bias high...
An eccentric binary black hole inspiral-merger-ringdown gravitational waveform model from numerical relativity and post-Newtonian theory
gr-qc 2017-09 conditional novelty 6.0 of 10

An eccentric non-spinning black hole binary waveform model, built from a post-Newtonian inspiral plus an interpolated circular numerical-relativity merger and calibrated to 23 simulations, reaches better than 97% fait...
Complete waveform model for compact binaries on eccentric orbits
gr-qc 2016-09 conditional novelty 6.0 of 10

The ax-model reproduces eccentric numerical-relativity waveforms through merger for equal-mass and 2:1 binaries, and implies GW150914 and GW151226 could have had eccentricities up to 0.15 and 0.1 at 14 Hz and still be...
Gravitational-wave phasing for low-eccentricity inspiralling compact binaries to 3PN order
gr-qc 2016-05 accept novelty 6.0 of 10

Explicit 3PN-order, leading-order-in-eccentricity phase corrections are derived for the TaylorT1, T2, T3, T4 and F2 post-Newtonian waveform approximants, with the analytic formulas valid up to initial eccentricity abo...
Frequency and time domain inspiral templates for comparable mass compact binaries in eccentric orbits
gr-qc 2016-02 conditional novelty 6.0 of 10

Two post-Newtonian inspiral template prescriptions for eccentric compact binaries are derived: a 2PN Fourier-domain waveform and an exact-in-eccentricity time-domain TaylorT4 extension.
Third post-Newtonian gravitational waveforms for compact binary systems in general orbits: Instantaneous terms
gr-qc 2015-01 conditional novelty 6.0 of 10

The authors derive the instantaneous 3PN spherical-harmonic modes of gravitational waves from nonspinning compact binaries in general and quasi-elliptical orbits.
Reduced Hamiltonian for next-to-leading order Spin-Squared Dynamics of General Compact Binaries
gr-qc 2010-02 conditional novelty 6.0 of 10

A spin-supplementary-condition-reduced Hamiltonian at next-to-leading order in spin-squared is derived for general compact binaries and shown to agree with an independent effective-field-theory result up to a canonica...
Third post-Newtonian angular momentum flux and the secular evolution of orbital elements for inspiralling compact binaries in quasi-elliptical orbits
gr-qc 2009-08 accept novelty 6.0 of 10

The 3PN angular momentum flux and the resulting secular evolution of orbital elements are derived for inspiralling compact binaries in quasi-elliptical orbits, completing the eccentric-orbit phasing input at 3PN.
Inspiralling compact binaries in quasi-elliptical orbits: The complete third post-Newtonian energy flux
gr-qc 2007-11 conditional novelty 6.0 of 10

The authors derive the complete third post-Newtonian energy flux for inspiralling compact binaries in quasi-elliptical orbits, combining new instantaneous multipole contributions with previously computed hereditary ta...
Tail effects in the third post-Newtonian gravitational wave energy flux of compact binaries in quasi-elliptical orbits
gr-qc 2007-11 conditional novelty 6.0 of 10

The hereditary tail contribution to the 3PN gravitational-wave energy flux of eccentric compact binaries is computed as a set of eccentricity-dependent enhancement functions multiplying the known circular-orbit coefficients.
Dynamics of Relativistic Binaries in Structured and Stochastic Environments: A Lagrange-Fourier-Hansen Framework
gr-qc 2026-06 unverdicted novelty 5.0 of 10

A new framework projects perturbations onto resonant frequencies via Hansen coefficients to produce efficient coupled ODEs for orbital elements in GW-driven relativistic binaries, demonstrated on tidal fields and accr...
Eccentricity Effects on Modeling Dynamic Quantities and Their Correlations in Binary Black Hole Mergers
gr-qc 2025-01 conditional novelty 5.0 of 10

Varying the initial orbital phase of an eccentric binary black hole at fixed eccentricity produces an envelope of radiated energy, momentum, and spin, so eccentric mergers span broad domains of remnant properties rela...
Defining eccentricity for gravitational wave astronomy
gr-qc 2023-02 conditional novelty 5.0 of 10

The authors adopt a waveform-only definition of eccentricity and mean anomaly for gravitational wave binaries, implement it in an open-source Python package, and show it works across post-Newtonian, effective-one-body...
A 3PN Fourier Domain Waveform for Non-Spinning Binaries with Moderate Eccentricity
gr-qc 2019-03 conditional novelty 5.0 of 10

A 3PN Fourier-domain waveform for eccentric non-spinning binaries, built from the stationary phase approximation and a two-frequency harmonic expansion, matches a numerical time-domain PN model at 97-99% for moderate ...
Foundations of an effective-one-body model for coalescing binaries on eccentric orbits
gr-qc 2017-07 conditional novelty 5.0 of 10

The authors recast effective-one-body binary dynamics in orbital-geometry variables and fundamental frequencies, deriving eccentric evolution equations and 1.5PN fluxes that reproduce known post-Newtonian results in t...
First Law of Mechanics for Compact Binaries on Eccentric Orbits
gr-qc 2015-06 conditional novelty 5.0 of 10

The circular-orbit first law of binary mechanics is extended to eccentric orbits as delta M = omega delta L + n delta R + sum <z_a> delta m_a, verified to 3PN order and used to extract effective-one-body potentials fr...
Next-to-next-to-leading order post-Newtonian linear-in-spin binary Hamiltonians
gr-qc 2013-02 conditional novelty 5.0 of 10

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The Overlap of Numerical Relativity, Perturbation Theory and Post-Newtonian Theory in the Binary Black Hole Problem
gr-qc 2014-08 accept novelty 3.0 of 10

Coordinate-invariant comparisons across post-Newtonian theory, black hole perturbation theory, and numerical relativity show that perturbation theory, rescaled by the symmetric mass ratio, tracks comparable-mass binar...
Hamiltonian formulation of general relativity and post-Newtonian dynamics of compact binaries
gr-qc 2018-05 accept novelty 2.0 of 10

A Living Reviews synthesis of Hamiltonian post-Newtonian dynamics of compact binaries, covering spinless and spinning bodies, tidal interactions, and radiation reaction through 5PN and beyond.

How later papers used it

108 later papers cite it

Evidence: substantial. 97 of the citing papers share no author with it, and 35 of those carry a confident label from Pith's citation classifier.

Pith read 29 of the 223 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

4 later papers by other authors are listed below: 4 used its method or tool and 1 compared its own results with it. 1 of them appears under more than one label. 2 more papers by the same authors are listed: 1 built on it and 1 used its method or tool.

Builds on it 1 paper · 1 by the same authors

By the same authors

Computed 4PN order corrections to this paper's 3PN-accurate generalized quasi-Keplerian parametric solution in Arnowitt, Deser, and Misner coordinates.

Generalized quasi-Keplerian solution for eccentric, non-spinning compact binaries at 4PN order and the associated IMR waveform (2021) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

Uses its method or tool 5 of 36 papers with this label · 1 by the same authors

Followed this paper's well-explained method to obtain the quasi-Keplerian parametrization, which it briefly summarized.

Spin effects in gravitational waveforms and fluxes for binaries on eccentric orbits to the third post-Newtonian order (2023) · 1.00 · paraphrased: the licence does not allow quoting
Following another work, used this paper's formalism to obtain a PN estimate for the eccentricity.

Anomalies in the gravitational recoil of eccentric black-hole mergers with unequal mass ratios (2021) · 1.00 · paraphrased: the licence does not allow quoting
Utilized this paper's 3PN accurate quasi-Keplerian orbital parametrization.

Gravitational Waves from the Inspiral of Supermassive Black Holes in Galactic-scale Simulations (2019) · 1.00 · paraphrased: the licence does not allow quoting
We employ similar techniques which allowed to obtain a simple parametrization for the solution of Post Newtonian accurate equations of motion for compact binaries in eccentric orbits [ref] .

Testing modified gravity via Yukawa potential in two body problem: Analytical solution and observational constraints (2022) · Introduction · 0.93

By the same authors

We let BH binaries move in 3PN-accurate precessing eccentric orbits with the help of generalized quasi-Keplerian parametrization [ref] , where the 3PN-accurate description incorporates $(v/c)^6$-order general relativistic corrections to Newtonian motion.

Pulsar timing array signals induced by black hole binaries in relativistic eccentric orbits (2020) (same authors) · Introduction · 0.99

Compares with it 1 of 5 papers with this label

Recovered this paper's results to 2PN in the nonspinning case.

Spin effects in gravitational waveforms and fluxes for binaries on eccentric orbits to the third post-Newtonian order (2023) · 0.97 · paraphrased: the licence does not allow quoting

How the literature uses it

83 arXiv papers cite this in 177 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 98 sentences without a confident label

Builds on it 1 paper

This paper computes 4PN order corrections to the 3PN-accurate generalized quasi-Keplerian parametric solution of Ref. [ref] in the Arnowitt, Deser, and Misner (ADM) coordinates.

Generalized quasi-Keplerian solution for eccentric, non-spinning compact binaries at 4PN order and the associated IMR waveform · Introduction · 1.00

Uses its method or tool 24 papers

We use the 3 PN conservative quasi-Keplerian orbit equations [ref] , combined with the 2 PN evolution of the orbital elements [ref] to construct adiabatic PN waveforms, determined by four independent initial parameters.

Comparisons of eccentric binary black hole simulations with post-Newtonian models · Introduction · 1.00
Anyway, as the spins do not precess, the quasi-Keplerian parameterisation [ref] can be employed to obtain a parametric solution to the dynamics.

Aligned Spins: Orbital Elements, Decaying Orbits, and Last Stable Circular Orbit to high post-Newtonian Orders · Introduction · 0.96
In this section, we follow closely the paper [ref] [ref] on the eccentric PN model since only the early-stage inspiral phase is considered in our case.

The GWs from the S-stars revolving around the SMBH at Sgr A* · The GWs from eccentric BBHs system · 0.99

Uses its data or values 5 papers

The explicit expressions of $E$ and $h$ up to the 2PN order can be found in Eqs. (23a) – (23h) in Ref. [ref] .

Analytical coordinate time at the second post-Newtonian order · Preliminaries · 0.89
+ 1/c^2H_1(r,p̂) +1/c^4 H_2(r,p̂) +1/c^6 H_3(r,p̂) +1/c^8 H_4(r,p̂) , where the explicit expressions for the 1PN, 2PN and 3PN contributions are given in Eq. (7) of Ref. [ref] .

Generalized quasi-Keplerian solution for eccentric, non-spinning compact binaries at 4PN order and the associated IMR waveform · Incorporating local-in-time 4PN Hamiltonian: generalized quasi-Keplerian solution · 0.90
In ADM coordinates, the 3PN-accurate expressions for the orbital elements read [ref]

Comparison Between Self-Force and Post-Newtonian Dynamics: Beyond Circular Orbits · The generalized quasi-Keplerian representation · 0.83

Compares with it 5 papers

It should be obvious that the resulting 4PN order $Φ$ expression depends on $E, h$ and $η$ and we have verified that our expression is consistent with Eq. (20-k) in Ref. [ref] .

Generalized quasi-Keplerian solution for eccentric, non-spinning compact binaries at 4PN order and the associated IMR waveform · Incorporating local-in-time 4PN Hamiltonian: generalized quasi-Keplerian solution · 1.00
In the nonspinning case, we recover the results in Ref. [ref] to 2PN, at which the structure of the 3PN spin contribution is similar to the 2PN nonspinning contribution.

Spin effects in gravitational waveforms and fluxes for binaries on eccentric orbits to the third post-Newtonian order · Quasi-Keplerian parametrization · 0.97
We have checked our expressions for $r$ and $ϕ̇$, as well as the 3 PN Kepler equation, by deriving them from the orbital elements in terms of $E$ and $h$ [ref] , and comparing with the explicit expressions in terms of $n$ and $e_t$ [ref] .

Comparisons of eccentric binary black hole simulations with post-Newtonian models · Eccentric post-Newtonian model · 0.95

Cites it as background 33 papers

(Reference [ref] also includes the corresponding 3PN QK representation of [ref] for nonspinning systems in harmonic coordinates.) It may be worth noting that the QK representation presented in [ref] is in harmonic coordinates or more specifically in modified harmonic (MH) coordinates that were introduced to remove the presence of logarithmic terms in expressions for mass quadrupole moment at 3PN order [ref] and have been extensively used in numerous efforts focusing on modeling compact binary inspirals in elliptical orbits [ref] . 5pt The QK representation involves connecting variables describing compact binaries in general orbits to those characterizing the orbit one wishes to specialize to.

Spin effects in Spherical Harmonic Modes of Gravitational Waves from Eccentric Compact Binary Inspirals · quasi-keplerian representation · 0.83
These extensions, detailed in Refs. [ref] , may be referred to as the `generalized quasi-Keplerian' parametric solution for describing PN-accurate orbital dynamics of compact binaries in eccentric orbits.

Generalized quasi-Keplerian solution for eccentric, non-spinning compact binaries at 4PN order and the associated IMR waveform · Keplerian type solution at 4PN order · 0.83
Indeed, the quasielliptic motion was studied using an elegant generalization of the Kepler solution to first post-Newtonian (1PN) order, dubbed the quasi-Keplerian (QK) parametrization, which was introduced by Damour and Deruelle [ref] , and generalized up to 3PN order [ref] and partially to 4PN order [ref] .

Quasi-Keplerian parametrization for compact binaries on hyperbolic orbits in scalar-tensor theories at second post-Newtonian order · Introduction · 0.88

Discussion

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