"Gravitational Radiation from Post-Newtonian Sources
and Inspiralling Compact Binaries"
Luc Blanchet 
1 Introduction
1.1 Analytic approximations and wave generation formalism
1.2 The quadrupole moment formalism
1.3 Problem posed by compact binary systems
1.4 Post-Newtonian equations of motion
1.5 Post-Newtonian gravitational radiation
A Post-Newtonian Sources
2 Non-linear Iteration of the Vacuum Field Equations
2.1 Einstein’s field equations
2.2 Linearized vacuum equations
2.3 The multipolar post-Minkowskian solution
2.4 Generality of the MPM solution
2.5 Near-zone and far-zone structures
3 Asymptotic Gravitational Waveform
3.1 The radiative multipole moments
3.2 Gravitational-wave tails and tails-of-tails
3.3 Radiative versus source moments
4 Matching to a Post-Newtonian Source
4.1 The matching equation
4.2 General expression of the multipole expansion
4.3 Equivalence with the Will–Wiseman formalism
4.4 The source multipole moments
5 Interior Field of a Post-Newtonian Source
5.1 Post-Newtonian iteration in the near zone
5.2 Post-Newtonian metric and radiation reaction effects
5.3 The 3.5PN metric for general matter systems
5.4 Radiation reaction potentials to 4PN order
B Compact Binary Systems
6 Regularization of the Field of Point Particles
6.1 Hadamard self-field regularization
6.2 Hadamard regularization ambiguities
6.3 Dimensional regularization of the equations of motion
6.4 Dimensional regularization of the radiation field
7 Newtonian-like Equations of Motion
7.1 The 3PN acceleration and energy for particles
7.2 Lagrangian and Hamiltonian formulations
7.3 Equations of motion in the center-of-mass frame
7.4 Equations of motion and energy for quasi-circular orbits
7.5 The 2.5PN metric in the near zone
8 Conservative Dynamics of Compact Binaries
8.1 Concept of innermost circular orbit
8.2 Dynamical stability of circular orbits
8.3 The first law of binary point-particle mechanics
8.4 Post-Newtonian approximation versus gravitational self-force
9 Gravitational Waves from Compact Binaries
9.1 The binary’s multipole moments
9.2 Gravitational wave energy flux
9.3 Orbital phase evolution
9.4 Polarization waveforms for data analysis
9.5 Spherical harmonic modes for numerical relativity
10 Eccentric Compact Binaries
10.1 Doubly periodic structure of the motion of eccentric binaries
10.2 Quasi-Keplerian representation of the motion
10.3 Averaged energy and angular momentum fluxes
11 Spinning Compact Binaries
11.1 Lagrangian formalism for spinning point particles
11.2 Equations of motion and precession for spin-orbit effects
11.3 Spin-orbit effects in the gravitational wave flux and orbital phase

10 Eccentric Compact Binaries

Inspiralling compact binaries are usually modelled as moving in quasi-circular orbits since gravitational radiation reaction circularizes the orbit towards the late stages of inspiral [340*, 339*], as we discussed in Section 1.2. Nevertheless, there is an increased interest in inspiralling binaries moving in quasi-eccentric orbits. Astrophysical scenarios currently exist which lead to binaries with non-zero eccentricity in the gravitational-wave detector bandwidth, both terrestrial and space-based. For instance, inner binaries of hierarchical triplets undergoing Kozai oscillations [283, 300] could not only merge due to gravitational radiation reaction but a fraction of them should have non negligible eccentricities when they enter the sensitivity band of advanced ground based interferometers [419]. On the other hand the population of stellar mass binaries in globular clusters is expected to have a thermal distribution of eccentricities [32]. In a study of the growth of intermediate black holes [235*] in globular clusters it was found that the binaries have eccentricities between 0.1 and 0.2 in the eLISA bandwidth. Though, supermassive black hole binaries are powerful gravitational wave sources for eLISA, it is not known if they would be in quasi-circular or quasi-eccentric orbits. If a Kozai mechanism is at work, these supermassive black hole binaries could be in highly eccentric orbits and merge within the Hubble time [40]. Sources of the kind discussed above provide the prime motivation for investigating higher post-Newtonian order modelling for quasi-eccentric binaries.

10.1 Doubly periodic structure of the motion of eccentric binaries

In Section 7.3 we have given the equations of motion of non-spinning compact binary systems in the frame of the center-of-mass for general orbits at the 3PN and even 3.5PN order. We shall now investigate (in this section and the next one) the explicit solution to those equations. In particular, let us discuss the general “doubly-periodic” structure of the post-Newtonian solution, closely following Refs. [142, 143*, 149*].

The 3PN equations of motion admit, when neglecting the radiation reaction terms at 2.5PN order, ten first integrals of the motion corresponding to the conservation of energy, angular momentum, linear momentum, and center of mass position. When restricted to the frame of the center of mass, the equations admit four first integrals associated with the energy E and the angular momentum vector J, given in harmonic coordinates at 3PN order by Eqs. (4.8) – (4.9) of Ref. [79].

The motion takes place in the plane orthogonal to J. Denoting by r = |x| the binary’s orbital separation in that plane, and by v = v1 − v2 the relative velocity, we find that E and J are functions of r, r˙2, v2 and x × v. We adopt polar coordinates (r,ϕ) in the orbital plane, and express E and the norm J = |J |, thanks to 2 2 2 ˙2 v = ˙r + r ϕ, as some explicit functions of r, 2 r˙ and ˙ϕ. The latter functions can be inverted by means of a straightforward post-Newtonian iteration to give ˙r2 and ϕ˙ in terms of r and the constants of motion E and J. Hence,


where ℛ and 𝒮 denote certain polynomials in 1∕r, the degree of which depends on the post-Newtonian approximation in question; for instance it is seventh degree for both ℛ and 𝒮 at 3PN order [312*]. The various coefficients of the powers of 1∕r are themselves polynomials in E and J, and also, of course, depend on the total mass m and symmetric mass ratio ν. In the case of bounded elliptic-like motion, one can prove [143] that the function ℛ admits two real roots, say rp and ra such that rp ≤ ra, which admit some non-zero finite Newtonian limits when c → ∞, and represent respectively the radii of the orbit’s periastron (p) and apastron (a). The other roots are complex and tend to zero when c → ∞.

Let us consider a given binary’s orbital configuration, fully specified by some values of the integrals of motion E and J corresponding to quasi-elliptic motion.70 The binary’s orbital period, or time of return to the periastron, is obtained by integrating the radial motion as

∫ ra dr P = 2 ∘-----. (332 ) rp ℛ [r]
We introduce the fractional angle (i.e., the angle divided by 2 π) of the advance of the periastron per orbital revolution,
∫ ra K = -1 dr∘-𝒮[r]-, (333 ) π rp ℛ [r]
which is such that the precession of the periastron per period is given by Δ Φ = 2π(K − 1). As K tends to one in the limit c → ∞ (as is easily checked from the usual Newtonian solution), it is often convenient to pose k ≡ K − 1, which will then entirely describe the relativistic precession.

Let us then define the mean anomaly ℓ and the mean motion n by


Here tp denotes the instant of passage to the periastron. For a given value of the mean anomaly ℓ, the orbital separation r is obtained by inversion of the integral equation

∫ r --ds--- ℓ = n ∘ -----. (335 ) rp ℛ [s]
This defines the function r(ℓ) which is a periodic function in ℓ with period 2π. The orbital phase ϕ is then obtained in terms of the mean anomaly ℓ by integrating the angular motion as
∫ ℓ 1- ϕ = ϕp + n 0 dl𝒮 [r(l)], (336 )
where ϕp denotes the value of the phase at the instant tp. We may define the origin of the orbital phase at the ascending node 𝒩 with respect to some observer. In the particular case of a circular orbit, r = const, the phase evolves linearly with time, ϕ˙= 𝒮 [r] = Ω, where Ω is the orbital frequency of the circular orbit given by
Ω = Kn = (1 + k)n. (337 )
In the general case of a non-circular orbit it is convenient to keep that definition Ω = Kn and to explicitly introduce the linearly growing part of the orbital phase (336*) by writing it in the form

Here W (ℓ) denotes a certain function of the mean anomaly which is periodic in ℓ with period 2π, hence periodic in time with period P. According to Eq. (336*) this function is given in terms of the mean anomaly ℓ by

1 ∫ ℓ W (ℓ) = -- dl(𝒮 [r(l)] − Ω ). (339 ) n 0
Finally, the decomposition (338) exhibits clearly the nature of the compact binary motion, which may be called doubly periodic in that the mean anomaly ℓ is periodic with period 2π, and the periastron advance K ℓ is periodic with period 2πK. Notice however that, though standard, the term “doubly periodic” is misleading since the motion in physical space is not periodic in general. The radial motion r(t) is periodic with period P while the angular motion ϕ(t) is periodic [modulo 2π] with period P∕k where k = K − 1. Only when the two periods are commensurable, i.e., when k = 1∕N where N ∈ ℕ, is the motion periodic in physical space (with period N P).

10.2 Quasi-Keplerian representation of the motion

The quasi-Keplerian (QK) representation of the motion of compact binaries is an elegant formulation of the solution of the 1PN equations of motion parametrized by the eccentric anomaly u (entering a specific generalization of Kepler’s equation) and depending on various orbital elements, such as three types of eccentricities. It was introduced by Damour & Deruelle [149*, 150] to study the problem of binary pulsar timing data including relativistic corrections at the 1PN order, where the relativistic periastron precession complicates the simpler Keplerian solution.

In the QK representation the radial motion is given in standard parametric form as

r = ar (1 − er cos u), (340 )
where u is the eccentric anomaly, with ar and er denoting two constants representing the semi-major axis of the orbit and its eccentricity. However, these constants are labelled after the radial coordinate r to remember that they enter (by definition) into the radial equation; in particular er will differ from other kinds of eccentricities et and eϕ. The “time” eccentricity et enters the Kepler equation which at the 1PN order takes the usual form
( 1) ℓ = u − etsin u + 𝒪 -4 , (341 ) c
where the mean anomaly is proportional to the time elapsed since the instant tp of passage at the periastron, ℓ = n (t − tp) where n = 2π∕P is the mean motion and P is the orbital period; see Eqs. (334). The “angular” eccentricity eϕ enters the equation for the angular motion at 1PN order which is written in the form
( ) ϕ − ϕp 1 -------= v + 𝒪 -4 , (342 ) K c
where the true anomaly v is defined by71
[ ( 1 + e )1∕2 u ] v ≡ 2arctan ----ϕ-- tan -- . (343 ) 1 − eϕ 2
The constant K is the advance of periastron per orbital revolution defined by Eq. (333*); it may be written as K = Φ- 2π where Φ is the angle of return to the periastron.

Crucial to the formalism are the explicit expressions for the orbital elements n, K, ar, er, et and eϕ in terms of the conserved energy E and angular momentum J of the orbit. For convenience we introduce two dimensionless parameters directly linked to E and J by


where μ = m ν is the reduced mass with m the total mass (recall that E < 0 for bound orbits) and we have used the intermediate standard notation h ≡ GJm-. The equations to follow will then appear as expansions in powers of the small post-Newtonian parameter 𝜀 = 𝒪 (1∕c2),72 with coefficients depending on j and the dimensionless reduced mass ratio ν; notice that the parameter j is at Newtonian order, 0 j = 𝒪 (1∕c ). We have [149*]


The dependence of such relations on the coordinate system in use will be discussed later. Notice the interesting point that there is no dependence of the mean motion n and the radial semi-major axis ar on the angular momentum J up to the 1PN order; such dependence will start only at 2PN order, see e.g., Eq. (347a).

The above QK representation of the compact binary motion at 1PN order has been generalized at the 2PN order in Refs. [170*, 379*, 420*], and at the 3PN order by Memmesheimer, Gopakumar & Schäfer [312*]. The construction of a generalized QK representation at 3PN order exploits the fact that the radial equation given by Eq. (331a) is a polynomial in 1∕r (of seventh degree at 3PN order). However, this is true only in coordinate systems avoiding the appearance of terms with the logarithm ln r; the presence of logarithms in the standard harmonic (SH) coordinates at the 3PN order will obstruct the construction of the QK parametrization. Therefore Ref. [312*] obtained it in the ADM coordinate system and also in the modified harmonic (MH) coordinates, obtained by applying the gauge transformation given in Eq. (204*) on the SH coordinates. The equations of motion in the center-of-mass frame in MH coordinates have been given in Eqs. (222); see also Ref. [9*] for details about the transformation between SH and MH coordinates.

At the 3PN order the radial equation in ADM or MH coordinates is still given by Eq. (340*). However, the Kepler equation (341*) and angular equation (342*) acquire extra contributions and now become


in which the true anomaly v is still given by Eq. (343*). The new orbital elements ft, fϕ, gt, gϕ, it, iϕ, ht and hϕ parametrize the 2PN and 3PN relativistic corrections.73 All the orbital elements are now to be related, similarly to Eqs. (345), to the constants 𝜀 and j with 3PN accuracy in a given coordinate system. Let us make clear that in different coordinate systems such as MH and ADM coordinates, the QK representation takes exactly the same form as given by Eqs. (340*) and (346). But, the relations linking the various orbital elements ar, er, et, eϕ, ft, fϕ, ... to E and J or 𝜀 and j, are different, with the notable exceptions of n and K.

Indeed, an important point related to the use of gauge invariant variables in the elliptical orbit case is that the functional forms of the mean motion n and periastron advance K in terms of the gauge invariant variables 𝜀 and j are identical in different coordinate systems like the MH and ADM coordinates [170*]. Their explicit expressions at 3PN order read


Because of their gauge invariant meaning, it is natural to use n and K as two independent gauge-invariant variables in the general orbit case. Actually, instead of working with the mean motion n it is often preferable to use the orbital frequency Ω which has been defined for general quasi-elliptic orbits in Eq. (337*). Moreover we can pose

( ) Gm Ω 2∕3 x = ---3-- (with Ω = Kn ), (348 ) c
which constitutes the obvious generalization of the gauge invariant variable x used in the circular orbit case. The use of x as an independent parameter will thus facilitate the straightforward reading out and check of the circular orbit limit. The parameter x is related to the energy and angular momentum variables 𝜀 and j up to 3PN order by

Besides the very useful gauge-invariant quantities n, K and x, the other orbital elements ar, er, et, eϕ, ft, gt, it, ht, fϕ, gϕ, iϕ, hϕ parametrizing Eqs. (340*) and (346) are not gauge invariant; their expressions in terms of 𝜀 and j depend on the coordinate system in use. We refer to Refs. [312*, 9*] for the full expressions of all the orbital elements at 3PN order in both MH and ADM coordinate systems. Here, for future use, we only give the expression of the time eccentricity et (squared) in MH coordinates:


Again, with our notation (344), this appears as a post-Newtonian expansion in the small parameter 𝜀 → 0 with fixed “Newtonian” parameter j.

In the case of a circular orbit, the angular momentum variable, say jcirc, is related to the constant of energy 𝜀 by the 3PN gauge-invariant expansion

( ) ( 2) ( [ ] 2 3) ( ) j = 1 + 9-+ ν- 𝜀+ 81-− 2ν + ν-- 𝜀2+ 945- + − 7699-+ 41π2 ν + ν--+ ν-- 𝜀3+ 𝒪 1- . circ 4 4 16 16 64 192 32 2 64 c8

This permits to reduce various quantities to circular orbits, for instance, the periastron advance is found to be well defined in the limiting case of a circular orbit, and is given at 3PN order in terms of the PN parameter (230*) [or (348*)] by

( 27 ) ( 135 [ 649 123 ] ) ( 1 ) Kcirc = 1 + 3x + ---− 7 ν x2 + ----+ − ----+ ---π2 ν + 7ν2 x3 + 𝒪 -8 . 2 2 4 32 c

See Ref. [291] for a comparison between the PN prediction for the periastron advance of circular orbits and numerical calculations based on self-force theory in the small mass ratio limit.

10.3 Averaged energy and angular momentum fluxes

The gravitational wave energy and angular momentum fluxes from a system of two point masses in elliptic motion was first computed by Peters & Mathews [340*, 339*] at Newtonian level. The 1PN and 1.5PN corrections to the fluxes were provided in Refs. [416, 86*, 267*, 87*, 366*] and used to study the associated secular evolution of orbital elements under gravitational radiation reaction using the QK representation of the binary’s orbit at 1PN order [149]. These results were extended to 2PN order in Refs. [224*, 225] for the instantaneous terms (leaving aside the tails) using the generalized QK representation [170, 379, 420]; the energy flux and waveform were in agreement with those of Ref. [424] obtained using a different method. Arun et al. [10*, 9*, 12*] have fully generalized the results at 3PN order, including all tails and related hereditary contributions, by computing the averaged energy and angular momentum fluxes for quasi-elliptical orbits using the QK representation at 3PN order [312], and deriving the secular evolution of the orbital elements under 3PN gravitational radiation reaction.74

The secular evolution of orbital elements under gravitational radiation reaction is in principle only the starting point for constructing templates for eccentric binary orbits. To go beyond the secular evolution one needs to include in the evolution of the orbital elements, besides the averaged contributions in the fluxes, the terms rapidly oscillating at the orbital period. An analytic approach, based on an improved method of variation of constants, has been discussed in Ref. [153*] for dealing with this issue at the leading 2.5PN radiation reaction order.

The generalized QK representation of the motion discussed in Section 10.2 plays a crucial role in the procedure of averaging the energy and angular momentum fluxes ℱ and 𝒢i over one orbit.75 Actually the averaging procedure applies to the “instantaneous” parts of the fluxes, while the “hereditary” parts are treated separately for technical reasons [10*, 9*, 12*]. Following the decomposition (308*) we pose ℱ = ℱinst + ℱhered where the hereditary part of the energy flux is composed of tails and tail-of-tails. For the angular momentum flux one needs also to include a contribution from the memory effect [12*]. We thus have to compute for the instantaneous part

∫ P ∫ 2π ⟨ℱinst⟩ = -1 dtℱinst =-1- du dℓ-ℱinst, (351 ) P 0 2 π 0 du
and similarly for the instantaneous part of the angular momentum flux 𝒢i.

Thanks to the QK representation, we can express ℱinst, which is initially a function of the natural variables r, ˙r and v2, as a function of the varying eccentric anomaly u, and depending on two constants: The frequency-related parameter x defined by (348*), and the “time” eccentricity e t given by (350). To do so one must select a particular coordinate system – the MH coordinates for instance. The choice of et rather than er (say) is a matter of convenience; since et appears in the Kepler-like equation (346a) at leading order, it will directly be dealt with when averaging over one orbit. We note that in the expression of the energy flux at the 3PN order there are some logarithmic terms of the type ln (r ∕r0) even in MH coordinates. Indeed, as we have seen in Section 7.3, the MH coordinates permit the removal of the logarithms ′ ln(r∕r0) in the equations of motion, where ′ r0 is the UV scale associated with Hadamard’s self-field regularization [see Eq. (221*)]; however there are still some logarithms ln(r∕r0) which involve the IR constant r0 entering the definition of the multipole moments for general sources, see Theorem 6 where the finite part ℱ 𝒫 contains the regularization factor (42*). As a result we find that the general structure of ℱ inst (and similarly for 𝒢inst, the norm of the angular momentum flux) consists of a finite sum of terms of the type

du-∑ αl(x,et) +-βl(x,-et)-sin-u +-γl(x,-et)-ln(1-−-etcos-u) ℱinst = dℓ (1 − e cos u)l+1 . (352 ) l t
The factor du ∕dℓ has been inserted to prepare for the orbital average (351*). The coefficients αl, βl and γl are straightforwardly computed using the QK parametrization as functions of x and the time eccentricity et. The βl’s correspond to 2.5PN radiation-reaction terms and will play no role, while the γ l’s correspond to the logarithmic terms ln (r∕r ) 0 arising at the 3PN order. For convenience the dependence on the constant lnr0 has been included into the coefficients αl’s. To compute the average we dispose of the following integration formulas (l ∈ ℕ)76

In the right-hand sides of Eqs. (353b) and (353c) we have to differentiate l times with respect to the intermediate variable z before applying z = 1. The equation (353c), necessary for dealing with the logarithmic terms, contains the not so trivial function

[ ∘1--−-e2 + 1] [ ∘1--−-e2 − 1 ] Z (z,et) = ln -------t----- + 2 ln 1 + ----∘--t----- . (354 ) 2 z + z2 − e2t
From Eq. (353a) we see that there will be no radiation-reaction terms at 2.5PN order in the final result; the 2.5PN contribution is proportional to r˙ and vanishes after averaging since it involves only odd functions of u.

Finally, after implementing all the above integrations, the averaged instantaneous energy flux in MH coordinates at the 3PN order is obtained in the form [9]

32c5 ( ) ⟨ℱinst⟩ = -----ν2x5 ℐ0 + xℐ1 + x2 ℐ2 + x3 ℐ3 , (355 ) 5G
where we recall that the post-Newtonian parameter x is defined by (348*). The various instantaneous post-Newtonian pieces depend on the symmetric mass ratio ν and the time eccentricity et in MH coordinates as

The Newtonian coefficient ℐ0 is nothing but the Peters & Mathews [340] enhancement function of eccentricity that enters in the orbital gravitational radiation decay of the binary pulsar; see Eq. (11*). For ease of presentation we did not add a label on et to indicate that it is the time eccentricity in MH coordinates; such MH-coordinates et is given by Eq. (350). Recall that on the contrary x is gauge invariant, so no such label is required on it.

The last term in the 3PN coefficient is proportional to some logarithm which directly arises from the integration formula (353c). Inside the logarithm we have posed

Gm x0 ≡ -2--, (357 ) c r0
exhibiting an explicit dependence upon the arbitrary length scale r0; we recall that r0 was introduced in the formalism through Eq. (42*). Only after adding the hereditary contribution to the 3PN energy flux can we check the required cancellation of the constant x0. The hereditary part is made of tails and tails-of-tails, and is of the form
( ) 32c5-2 5 3∕2 5∕2 3 ⟨ℱhered⟩ = 5G ν x x 𝒦3∕2 + x 𝒦5∕2 + x 𝒦3 , (358 )
where the post-Newtonian pieces, only at the 1.5PN, 2.5PN and 3PN orders, read [10*]

where φ (et), ψ(et), ζ(et), κ (et) and F (et) are certain “enhancement” functions of the eccentricity.

Among them the four functions φ (e ) t, ψ(e ) t, ζ(e ) t and κ(e ) t appearing in Eqs. (359) do not admit analytic closed-form expressions. They have been obtained in Refs. [10*] (extending Ref. [87*]) in the form of infinite series made out of quadratic products of Bessel functions. Numerical plots of these four enhancement factors as functions of eccentricity et have been provided in Ref. [10*]; we give in Figure 3* the graph of the function φ(et) which enters the dominant 1.5PN tail term in Eq. (358*).

View Image
Figure 3: Variation of the enhancement factor φ (e) with the eccentricity e. This function agrees with the numerical calculation of Ref. [87*] modulo a trivial rescaling with the Peters–Mathews function (356a). The inset graph is a zoom of the function at a smaller scale. The dots represent the numerical computation and the solid line is a fit to the numerical points. In the circular orbit limit we have φ(0) = 1.

Furthermore their leading correction term 2 et in the limit of small eccentricity et ≪ 1 can be obtained analytically as [10]


On the other hand the function F (et) in factor of the logarithm in the 3PN piece does admit some closed analytic form:

[ ] F(et) = -----1----- 1 + 85e2t + 5171-e4t + 1751-e6t + 297-e8t . (361 ) (1 − e2t)13∕2 6 192 192 1024

The latter analytical result is very important for checking that the arbitrary constant x0 disappears from the final result. Indeed we immediately verify from comparing the last term in Eq. (356d) with Eq. (359c) that x0 cancels out from the sum of the instantaneous and hereditary contributions in the 3PN energy flux. This fact was already observed for the circular orbit case in Ref. [81]; see also the discussions around Eqs. (93*) – (94*) and at the end of Section 4.2.

Finally we can check that the correct circular orbit limit, which is given by Eq. (314), is recovered from the sum ⟨ℱinst⟩ + ⟨ℱhered⟩. The next correction of order e2t when et → 0 can be deduced from Eqs. (360) – (361*) in analytic form; having the flux in analytic form may be useful for studying the gravitational waves from binary black hole systems with moderately high eccentricities, such as those formed in globular clusters [235].

Previously the averaged energy flux was represented using x – the gauge invariant variable (348*) – and the time eccentricity e t which however is gauge dependent. Of course it is possible to provide a fully gauge invariant formulation of the energy flux. The most natural choice is to express the result in terms of the conserved energy E and angular momentum J, or, rather, in terms of the pair of rescaled variables (𝜀, j) defined by Eqs. (344). To this end it suffices to replace et by its MH-coordinate expression (350) and to use Eq. (349) to re-express x in terms of 𝜀 and j. However, there are other possible choices for a couple of gauge invariant quantities. As we have seen the mean motion n and the periastron precession K are separately gauge invariant so we may define the pair of variables (x, ι), where x is given by (348*) and we pose

-3x--- ι ≡ K − 1. (362 )
Such choice would be motivated by the fact that ι reduces to the angular-momentum related variable j in the limit 𝜀 → 0. Note however that with the latter choices (𝜀, j) or (x, ι) of gauge-invariant variables, the circular-orbit limit is not directly readable from the result; this is why we have preferred to present it in terms of the gauge dependent couple of variables (x, e t).

As we are interested in the phasing of binaries moving in quasi-eccentric orbits in the adiabatic approximation, we require the orbital averages not only of the energy flux ℱ but also of the angular momentum flux 𝒢i. Since the quasi-Keplerian orbit is planar, we only need to average the magnitude 𝒢 of the angular momentum flux. The complete computation thus becomes a generalisation of the previous computation of the averaged energy flux requiring similar steps (see Ref. [12*]): The angular momentum flux is split into instantaneous 𝒢inst and hereditary 𝒢hered contributions; the instantaneous part is averaged using the QK representation in either MH or ADM coordinates; the hereditary part is evaluated separately and defined by means of several types of enhancement functions of the time eccentricity et; finally these are obtained numerically as well as analytically to next-to-leading order e2 t. At this stage we dispose of both the averaged energy and angular momentum fluxes ⟨ℱ⟩ and ⟨𝒢 ⟩.

The procedure to compute the secular evolution of the orbital elements under gravitational radiation-reaction is straightforward. Differentiating the orbital elements with respect to time, and using the heuristic balance equations, we equate the decreases of energy and angular momentum to the corresponding averaged fluxes ⟨ℱ ⟩ and ⟨𝒢⟩ at 3PN order [12]. This extends earlier analyses at previous orders: Newtonian [339] as we have reviewed in Section 1.2; 1PN [86, 267]; 1.5PN [87, 366] and 2PN [224, 153*]. Let us take the example of the mean motion n. From Eq. (347a) together with the definitions (344) we know the function n(E, J) at 3PN order, where E and J are the orbit’s constant energy and angular momentum. Thus,

dn- ∂n--dE-- ∂n-dJ- dt = ∂E dt + ∂J dt . (363 )
The usual balance equations for energy and angular momentum

have already been used at Newtonian order in Eqs. (9). Although heuristically assumed at 3PN order, they have been proved through 1.5PN order in Section 5.4. With the averaged fluxes known through 3PN order, we obtain the 3PN averaged evolution equation as

dn- ∂n-- ∂n- ⟨dt ⟩ = − ∂E ⟨ℱ ⟩ − ∂J⟨𝒢 ⟩. (365 )
We recall that this gives only the slow secular evolution under gravitational radiation reaction for eccentric orbits. The complete evolution includes also, superimposed on the averaged adiabatic evolution, some fast but smaller post-adiabatic oscillations at the orbital time scale [153, 279].
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