# Linear momentum flux from inspiralling compact binaries in quasi-elliptical
orbits at 2.5 Post-Newtonian order

Shilpa Kastha<EMAIL_ADDRESS>Albert-Einstein-Institut, Max-Planck-
Institut für Gravitationsphysik, Callinstraße 38, 30167 Hannover, Germany
Leibniz Universität Hannover, 30167 Hannover, Germany

###### Abstract

Emission of anisotropic gravitational radiation from compact binary system
leads to a flux of linear momentum. This results in recoil of the system. We
investigate the rate of loss of Linear momentum flux in the far zone of the
source using various mass type and current type multipole moments for
inspiralling compact binary merger in quasi-elliptical orbits at 2.5 Post
Newtonian order. We compute the linear momentum flux accurate upto
$\mathcal{O}(e_{t})$ in harmonic coordinate. A 2.5 Post Newtonian Quasi-
Keplarian representation of the parametric solution to the Post Newtonian
equation of motion for the compact binary system has been adopted here. We
also provide a closed form expression for the accumulated linear momentum from
remote past through the binary evolution.

##  I Introduction

Anisotropic gravitational radiation leads to a flux of linear momentum from
the compact binary system Peres (1962); Papapetrou (1962). To conserve the
total linear momentum, the system recoils. The direction of recoil changes
continuously over an orbit. As a result, for a perfectly circular trajectory,
no net recoil builds up over an orbit. On the contrary, for inspiralling
compact binaries, the recoil accumulates over the inspiralling orbits and
imparts a kick to the remnant at the merger. A reasonably high kick imparted
to the binary black hole (BBH) merger could be of great importance in
understanding the structure formation of globular clusters. If the kick is
greater than the escape velocity of the host galaxy, the remnant black
hole(BH) may even be ejected Komossa et al. (2008) from the galaxy. Even if
the kick is not high enough to eject the remnant BH, it might cause
significant dynamical changes at the core of the galaxy. A detailed discussion
on various astrophysical aspects of BH kicks can be found in ref Merritt et
al. (2004).

Though a compact binary merger may have significant eccentricity at the birth,
due to the gravitational radiation, it gets circularized Peters and Mathews
(1963a); Peters (1964). By the time their gravitational wave (GW) frequency
enters the sensitivity bands of the ground based interferometric GW detectors,
they may have negligible eccentricity. Yet there may be astrophysical
processes which may retain their eccentricity even in the late stages of their
dynamical evolution. For example, in dense stellar clusters, interactions
between pairs of BBH systems may eject one of the BHs leading to the formation
of a stable hierarchical triple system. If the two orbital planes are tilted
with respect to each other, the third body can increase the eccentricity of
the inner binary via Kozai mechanismKozai (1962). Binaries in such
hierarchical triple systems may have non-zero eccentricities even towards the
late stages of the inspiral. Further, binary neutron star systems in globular
clusters may have a thermal distribution of eccentricities Benacquista (2002)
if formed by exchange interactions as opposed to the formation scenario
through the common envelope. Similarly, there have been mechanisms proposed
for binaries consisting of supermassive BHs in the LISA band Gultekin et al.
(2004); Hoang et al. (2019, 2018) which may have detectable eccentricities.
The recent detection Abbott et al. (2020a, b) of the heaviest BBH merger event
to date, GW190521, has been argued to have high eccentricity Bustillo et al.
(2021); Romero-Shaw et al. (2020); Gayathri et al. (2020) at merger. Motivated
by these scenarios, we study the emission of linear momentum flux (LMF) in
case of non-spinning compact binary systems in quasi-elliptical orbits.

The first formal treatment of gravitational recoil for a general self-
gravitating system in linearized gravity is explored in refs. Peres (1962);
Bekenstein (1973). It is valid for any kind of motion (rotational, vibrational
or any other kind) given the source is localized within a finite volume.
Later, within the Post-Newtonian (PN) framework, the leading order
contribution (Newtonian) to the LMF and recoil of a compact binary system is
discussed in refs. Fitchett (1983); Fitchett and Detweiler (1984). The first
PN correction to this was computed by Wiseman Wiseman (1992) and the quasi-
circular orbit scenario was discussed as a special case there. Much later, a
closed form expression for the recoil in case of compact binary in quasi-
circular orbit is quoted in Blanchet et al. (2005) at the second post-
Newtonian (PN) order. Its extension to 2.5PN order, accounting for the
radiation reaction effects, is discussed in ref.Mishra et al. (2012).
According to these studies, the BH recoil for nonspinning systems could be in
the range 74-250 km $\rm{s}^{-1}$. Using the effective one-body(EOB) approach,
the recoil estimates for BBH is obtained considering the contributions from
inspiral, plunge and ringdown phases in ref. Damour and Gopakumar (2006). The
typical estimate obtained here, lies in the range $49$-$172$ km $\rm{s}^{-1}$.
In ref. Favata et al. (2004), BH perturbation theory is used to compute the
accumulated recoil up to the innermost stable circular orbit (ISCO)
($10$-$100$ km $\rm{s}^{-1}$) for a system where a test particle inspirals
into a BH. In principle, these estimates are valid for extreme mass ratio
inspirals but they have extrapolated the results till mass ratio of about
$\sim 0.4$.

Along with the various analytical and semi-analytical studies, the recent
progress in numerical relativity techniques has led to more accurate estimates
for the recoil of the remnant BH. As quoted in refs. Campanelli (2005); Baker
et al. (2006); Herrmann et al. (2007a); Gonzalez et al. (2009), the recoil
velocity can reach up to a few hundreds of km $\rm{s}^{-1}$ while the
component masses are nonspinning for compact binaries in quasi-circular orbit.
But for the spinning case Herrmann et al. (2007b); Koppitz et al. (2007);
Gonzalez et al. (2007), the recoil velocity can be of the order of few
thousands km $\rm{s}^{-1}$. In case of maximally spinning BHs, it could be as
high as 4000 km $\rm{s}^{-1}$ Campanelli et al. (2007). Such a large recoil
velocity may lead to ejection of the merged binary from its host galaxy. A
detailed study on multipolar analysis of the gravitational recoil is also
discussed in ref. Schnittman et al. (2008). They have explored the build up of
the recoil through the different phases of the binary evolution,
(inspiral+merger+ringdown) due to the relative amplitude and the phases of
various modes of the GWs.

Using a formal approach within the Multipolar Post-Minkowskian Post-Newtonian
(MPM-PN) formalism, the leading order Newtonian contribution to the LMF and
the associated recoil of a compact binary system is explored by Fitchett in
refs. Fitchett (1983); Fitchett and Detweiler (1984). The authors have assumed
the inspiralling binary to be composed of two point particles moving in a
Keplerian orbit. A rough estimate of maximum BH recoil quoted in these studies
are $\sim 1500\rm{km/s}$. Assuming the periastron advance to be small, a crude
estimate of the recoil at 1PN is also quoted. The first extension of these
estimates at 1PN for binaries moving in generic orbits is explored by Wiseman
Wiseman (1992) and concluded that higher order correction reduces the net
momentum ejection. As a special case, they also studied BNS systems moving in
quasi-circular orbits. They found the upper bound on the velocity of the
center of mass very near to the coalescence to be 1 km $\rm{s}^{-1}$. In
another study Sopuerta et al. (2007), the authors showed a 10% increase in the
recoil estimate compared to the quasi-circular case for small eccentricities
$(e<0.1)$ using close limit approximation. They also claimed that the maximum
recoil takes place for the value of the symmetric mass ratio of around
$\eta\sim 0.19$ and the magnitude could be as high as $216-242$ km/s.

In order to obtain a correct recoil estimates one needs to compute the linear
momentum from the system. Here we investigate the linear momentum flux at 2.5
post-Newtonian order for compact binaries moving in quasi-elliptical orbits
with non-spinning component masses.

The paper is organized as follows. In section II we discuss the multipolar
decomposition of LMF in terms of the various source type multipole moments and
their non-linear interactions. In section III, we discuss the orbital dynamics
and in section IV we quote the instantaneous contribution to the LMF in terms
of the orbital parameters. In section V and VI we summarize the keplerian and
generalized quasi-keplerian representations (QKR) of the orbital dynamics and
re-express the instanteneous contribution to the LMF in terms of the QK
parameters. Section VII and VIII consists of the computation of hereditary and
the post-adiabatic contributions to the LMF. We quote the complete LMF at
2.5PN in section IX. Finally in section X we compute the accumulated linear
momentum through the inspiralling orbits from remote past.

##  II Multipole decomposition of Linear momentum flux

For an isolated source, gravitational wave generation is well studied under
the framework of multipolar decomposition Thorne (1980a). Following Thorne
(1980a), we explicitly write down the multipolar decomposition of far-zone
linear momentum flux (LMF) for an isolated source at the 2.5 post-Newtonian
order in terms symmetric trace-free (STF) radiative mass type and current type
multipole moments [see Eq.(2.1) of ref. Mishra et al. (2012)].

$\displaystyle{\mathcal{F}_{i}}$ $\displaystyle=$
$\displaystyle\frac{G}{c^{7}}\,\biggl{\\{}\left[\frac{2}{63}\,U^{(1)}_{ijk}\,U^{(1)}_{jk}+\frac{16}{45}\,\varepsilon_{ijk}U^{(1)}_{ja}\,V^{(1)}_{ka}\right]+{1\over
c^{2}}\left[\frac{1}{1134}\,U^{(1)}_{ijkl}\,U^{(1)}_{jkl}+\frac{1}{126}\,\varepsilon_{ijk}U^{(1)}_{jab}\,V^{(1)}_{kab}+\frac{4}{63}\,V^{(1)}_{ijk}\,V^{(1)}_{jk}\right]$
(1) $\displaystyle+{1\over
c^{4}}\left[\frac{1}{59400}\,U^{(1)}_{ijklm}\,U^{(1)}_{jklm}+\frac{2}{14175}\,\varepsilon_{ijk}U^{(1)}_{jabc}\,V^{(1)}_{kabc}+\frac{2}{945}\,V^{(1)}_{ijkl}\,V^{(1)}_{jkl}\right]+{\cal
O}\left({1\over c^{6}}\right)\biggr{\\}}.$

Here $U_{K}^{(p)}$ and $V_{K}^{(p)}$ ( $K=i_{1}i_{2}\cdots i_{k}$ represents
the multi-index structure of the tensors of order $k$ in three dimension) are
the $p^{th}$-time derivative of mass-type and current-type radiative multipole
moments respectively. $\varepsilon_{ijk}$ is the usual three dimensional Levi-
Civita tensor, with a value $+1$ in case of all even permutations and $-1$ for
all the odd ones. The multipole moments in the formula, are functions of
retarded time $(t-({\it r}/c))$ in radiative coordinates, where $\it r$ and
$t$ denote the distance of the source from the observer and the time of
observation respectively.

Using the multipolar post Minkowskian (MPM) formalism Thorne (1980b); Blanchet
and Damour (1984, 1986); Blanchet (1987); Blanchet and Damour (1988, 1989,
1992); Blanchet (1995); Blanchet et al. (1995, 2002); Damour et al. (2001);
Blanchet et al. (2004) the two types of radiative moments, $(U_{L},V_{L})$ can
be expressed in terms of two canonical moments $(M_{L},S_{L})$ and eventually
as a function of all the source multipole moments
$(I_{L},J_{L},X_{L},W_{L},Y_{L},Z_{L})$ Blanchet et al. (2008) at the 2.5PN
order. Every radiative moment has two types of contributions. One of them is
only a function of retarded time and hence called the instantaneous part. The
other one depends on the dynamical behavior of the system throughout its
entire past and referred to as the hereditary contributions. These
contributions contain information about various multipolar interactions the
gravitational wave undergoes, as it propagates from the source to the
detector.

For the reader’s convenience, here we explicitly quote all the radiative
moments in terms of the source moments, accurate up to the order necessary for
the present calculation(see Eqs. (3.1)-(3.18b) of refs. Mishra et al. (2015)).
Since the leading order term in the LMF expression (see Eq. 1) consists of the
mass quadrupole moment $(U_{ij})$, the desired accuracy of $U_{ij}$ is 2.5PN.
Furthermore, the decomposition of mass quadrupole moment into instantaneous
and hereditary parts is as follows,

$U_{ij}=U_{ij}^{\rm inst}+U_{ij}^{\rm hered},$ (2)

where the instantaneous and the hereditary parts explicitly read

$\displaystyle U_{ij}^{\rm inst}(U)$ $\displaystyle=I^{(2)}_{ij}(U)+{G\over
c^{5}}\left\\{{1\over 7}I^{(5)}_{a\langle i}I_{j\rangle a}-{5\over
7}I^{(4)}_{a\langle i}I^{(1)}_{j\rangle a}-{2\over 7}I^{(3)}_{a\langle
i}I^{(2)}_{j\rangle a}+{1\over 3}\varepsilon_{ab\langle i}I^{(4)}_{j\rangle
a}J_{b}+4\left[W^{(4)}I_{ij}+W^{(3)}I_{ij}^{(1)}\right.\right.$
$\displaystyle\left.\left.-W^{(2)}I_{ij}^{(2)}-W^{(1)}I_{ij}^{(3)}\right]\right\\}+\,\,\mathcal{O}\left(\frac{1}{c^{7}}\right),$
(3a) $\displaystyle U_{ij}^{\rm hered}(U)$ $\displaystyle={2GM\over
c^{3}}\int_{-\infty}^{U}d\tau\left[\ln\left({U-\tau\over
2\tau_{0}}\right)+{11\over 12}\right]I^{(4)}_{ij}(\tau)+{G\over
c^{5}}\left\\{-{2\over 7}\int_{-\infty}^{U}d\tau I^{(3)}_{a\langle
i}(\tau)I^{(3)}_{j\rangle a}(\tau)\right\\}$ $\displaystyle+2\left({GM\over
c^{3}}\right)^{2}\int_{-\infty}^{U}d\tau\left[\ln^{2}\left({U-\tau\over
2\tau_{0}}\right)+{57\over 70}\ln\left({U-\tau\over
2\tau_{0}}\right)+{124627\over
44100}\right]I^{(5)}_{ij}(\tau)+\,\,\mathcal{O}\left(\frac{1}{c^{7}}\right)\,.$
(3b)

In the above expression, $M$ represents the Arnowitt, Deser and Misner (ADM)
mass of the source and hence undergoes relativistic corrections given by
$M=m(1-\eta x/2)$. The constant $\tau_{0}$ is related to an arbitrary length
scale $r_{0}$ by $\tau_{0}=r_{0}/c$ and was originally introduced in the MPM
formalism.

The required accuracy of mass octupole moment is 1.5PN which is,

$\displaystyle U_{ijk}$ $\displaystyle=U_{ijk}^{\rm inst}+U_{ijk}^{\rm
hered},$ (4)

and both the parts separately read

$\displaystyle U_{ijk}^{\rm inst}(U)$ $\displaystyle=I^{(3)}_{ijk}(U)+{G\over
c^{5}}\left\\{-{4\over 3}I^{(3)}_{a\langle i}I^{(3)}_{jk\rangle a}-{9\over
4}I^{(4)}_{a\langle i}I^{(2)}_{jk\rangle a}+{1\over 4}I^{(2)}_{a\langle
i}I^{(4)}_{jk\rangle a}-{3\over 4}I^{(5)}_{a\langle i}I^{(1)}_{jk\rangle
a}+{1\over 4}I^{(1)}_{a\langle i}I^{(5)}_{jk\rangle a}+{1\over
12}I^{(6)}_{a\langle i}I_{jk\rangle a}\right.$ $\displaystyle\left.+{1\over
4}I_{a\langle i}I^{(6)}_{jk\rangle a}+{1\over 5}\varepsilon_{ab\langle
i}\left[-12J^{(2)}_{ja}I^{(3)}_{k\rangle b}-8I^{(2)}_{ja}J^{(3)}_{k\rangle
b}-3J^{(1)}_{ja}I^{(4)}_{k\rangle b}-27I^{(1)}_{ja}J^{(4)}_{k\rangle
b}-J_{ja}I^{(5)}_{k\rangle b}-9I_{ja}J^{(5)}_{k\rangle b}\right.\right.$
$\displaystyle\left.\left.-{9\over 4}J_{a}I^{(5)}_{jk\rangle
b}\right]+{12\over 5}J_{\langle
i}J^{(4)}_{jk\rangle}+4\left[W^{(2)}I_{ijk}-W^{(1)}I_{ijk}^{(1)}+3\,I_{\langle
ij}Y_{k\rangle}^{(1)}\right]^{(3)}\right\\}+\,\mathcal{O}\left(\frac{1}{c^{6}}\right),$
(5a) $\displaystyle U_{ijk}^{\rm hered}(U)$ $\displaystyle={2GM\over
c^{3}}\int_{-\infty}^{U}d\tau\left[\ln\left({U-\tau\over
2\tau_{0}}\right)+{97\over 60}\right]I^{(5)}_{ijk}(\tau)+{G\over
c^{5}}\left\\{\int_{-\infty}^{U}d\tau\left[-{1\over 3}I^{(3)}_{a\langle
i}(\tau)I^{(4)}_{jk\rangle a}(\tau)\right.\right.$
$\displaystyle\left.\left.-{4\over 5}\varepsilon_{ab\langle
i}I^{(3)}_{ja}(\tau)J^{(3)}_{k\rangle
b}(\tau)\right]\right\\}+\,\mathcal{O}\left(\frac{1}{c^{6}}\right).$ (5b)

As the other two mass type mulitpole moments $U_{ijkl}$ and $U_{ijklm}$ appear
in the LMF at 1PN and 2PN respectively, the desired accuracy for these two are
1.5 PN and 0.5PN respectively, which read,

$\displaystyle U_{ijkl}$ $\displaystyle=U_{ijkl}^{\rm inst}+U_{ijkl}^{\rm
hered},$ (6)

$\displaystyle U_{ijkl}^{\rm inst}(U)$
$\displaystyle=I^{(4)}_{ijkl}(U)+{G\over c^{3}}\left\\{-{21\over
5}I^{(5)}_{\langle ij}I_{kl\rangle}-{63\over 5}I^{(4)}_{\langle
ij}I^{(1)}_{kl\rangle}-{102\over 5}I^{(3)}_{\langle
ij}I^{(2)}_{kl\rangle}\right\\}+\,\mathcal{O}\left(\frac{1}{c^{5}}\right),$
(7a) $\displaystyle U_{ijkl}^{\rm hered}(U)$ $\displaystyle={G\over
c^{3}}\left\\{2M\int_{-\infty}^{U}d\tau\left[\ln\left({U-\tau\over
2\tau_{0}}\right)+{59\over 30}\right]I^{(6)}_{ijkl}(\tau)+{2\over
5}\int_{-\infty}^{U}d\tau I^{(3)}_{\langle
ij}(\tau)I^{(3)}_{kl\rangle}(\tau)\right\\}+\,\mathcal{O}\left(\frac{1}{c^{5}}\right).$
(7b) $\displaystyle U_{ijklm}^{\rm inst}(T_{R})$
$\displaystyle=I^{(4)}_{ijklm}(T_{R})+\,\mathcal{O}\left(\frac{1}{c^{4}}\right).$
(8)

Among the current type moments, current quadrupole moment is needed to be
evaluated at 2.5PN order.

$\displaystyle V_{ij}$ $\displaystyle=V_{ij}^{\rm inst}+V_{ij}^{\rm hered},$
(9)

The instantaneous and the hereditary parts of the current quadrupole moments
read

$\displaystyle V_{ij}^{\rm inst}(U)$ $\displaystyle=J^{(2)}_{ij}(U)+{G\over
7\,c^{5}}\left\\{4J^{(2)}_{a\langle i}I^{(3)}_{j\rangle a}+8I^{(2)}_{a\langle
i}J^{(3)}_{j\rangle a}+17J^{(1)}_{a\langle i}I^{(4)}_{j\rangle
a}-3I^{(1)}_{a\langle i}J^{(4)}_{j\rangle a}+9I_{a\langle i}I^{(5)}_{j\rangle
a}-3I_{a\langle i}J^{(5)}_{j\rangle a}\right.$ $\displaystyle\left.-{1\over
4}J_{a}I^{(5)}_{ija}-7\varepsilon_{ab\langle i}J_{a}J^{(4)}_{j\rangle
b}+{1\over 2}\varepsilon_{ac\langle i}\left[3I^{(3)}_{ab}I^{(3)}_{j\rangle
bc}+{353\over 24}I^{(2)}_{j\rangle bc}I^{(4)}_{ab}-{5\over
12}I^{(2)}_{ab}I^{(4)}_{j\rangle bc}+{113\over 8}I^{(1)}_{j\rangle
bc}I^{(5)}_{ab}\right.\right.$ $\displaystyle\left.\left.-{3\over
8}I^{(1)}_{ab}I^{(5)}_{j\rangle bc}+{15\over 4}I_{j\rangle
bc}I^{(6)}_{ab}+{3\over 8}I_{ab}I^{(6)}_{j\rangle
bc}\right]+14\,\left[\varepsilon_{ab\langle i}\left(-I_{j\rangle
b}^{(3)}W_{a}-2I_{j\rangle b}Y_{a}^{(2)}+I_{j\rangle
b}^{(1)}Y_{a}^{(1)}\right)\right.\right.$
$\displaystyle\left.\left.+3J_{\langle
i}Y_{j\rangle}^{(1)}-2J_{ij}^{(1)}W^{(1)}\right]^{(2)}\right\\}+\,\mathcal{O}\left(\frac{1}{c^{6}}\right)\,,$
(10a) $\displaystyle V_{ij}^{\rm hered}(U)$ $\displaystyle={2GM\over
c^{3}}\int_{-\infty}^{U}d\tau\left[\ln\left({U-\tau\over
2\tau_{0}}\right)+{7\over
6}\right]J^{(4)}_{ij}(\tau)+\,\mathcal{O}\left(\frac{1}{c^{6}}\right)\,.$
(10b)

For current octupole moment the two contributions are the following

$\displaystyle V_{ijk}$ $\displaystyle=V_{ijk}^{\rm inst}+V_{ijk}^{\rm
hered},$ (11)

where $V_{ijk}^{\rm inst}$ and $V_{ijk}^{\rm hered}$ are given in terms of the
source multipole moments as

$\displaystyle V_{ijk}^{\rm inst}(U)$ $\displaystyle=J^{(3)}_{ijk}(U)+{G\over
c^{3}}\left\\{{1\over 10}\varepsilon_{ab\langle i}I^{(5)}_{ja}I_{k\rangle
b}-{1\over 2}\varepsilon_{ab\langle i}I^{(4)}_{ja}I^{(1)}_{k\rangle
b}-2J_{\langle
i}I^{(4)}_{jk\rangle}\right\\}+\,\mathcal{O}\left(\frac{1}{c^{5}}\right),$
(12a) $\displaystyle V_{ijk}^{\rm hered}(U)$ $\displaystyle={2GM\over
c^{3}}\int_{-\infty}^{U}d\tau\left[\ln\left({U-\tau\over
2\tau_{0}}\right)+{5\over
3}\right]J^{(5)}_{ijk}(\tau)+\,\mathcal{O}\left(\frac{1}{c^{5}}\right).$ (12b)

For one other current type moments $V_{ijkl}$, we only need the instantaneous
part to obtain LMF at 2.5PN, which reads,

$\displaystyle V_{ijkl}^{\rm inst}(T_{R})$
$\displaystyle=J^{(4)}_{ijkl}(T_{R})+\mathcal{O}\left(\frac{1}{c^{3}}\right).$
(13)

Using Eqs. (2)-(13) we obtain the closed form expression for LMF at 2.5PN in
terms of the source multipole moments. Similar to the radiative moments, the
LMF also admits a decomposition into instantaneous and hereditary parts. The
instantaneous and the hereditary parts indicate two distinct physical
processes and their evaluations need separate treatments. Thus, for our
convenience, we explicitly write the two types of contributions (
instantaneous and hereditary) to linear momentum flux separately as follows,

$\displaystyle{\mathcal{F}_{i}}=\left({\mathcal{F}_{i}}\right)_{\rm
inst}+\left({\mathcal{F}_{i}}\right)_{\rm hered},$ (14)

where the instantaneous part up to 2.5PN in terms of the source multipole
moments is (see Eq.(2.2) of ref. Mishra et al. (2012))

$\displaystyle\left({\mathcal{F}_{i}}\right)_{\rm inst}$ $\displaystyle=$
$\displaystyle\frac{G}{c^{7}}\,\left\\{\frac{2}{63}\,I^{(4)}_{ijk}\,I^{(3)}_{jk}+\frac{16}{45}\,\varepsilon_{ijk}I^{(3)}_{ja}\,J^{(3)}_{ka}+{1\over
c^{2}}\left[\frac{1}{1134}\,I^{(5)}_{ijkl}\,I^{(4)}_{jkl}+\frac{4}{63}\,J^{(4)}_{ijk}\,J^{(3)}_{jk}+\frac{1}{126}\,\varepsilon_{ijk}I^{(4)}_{jab}\,J^{(4)}_{kab}\right]\right.$
$\displaystyle\left.+{1\over
c^{4}}\left[\frac{1}{59400}\,I^{(6)}_{ijklm}\,I^{(5)}_{jklm}+\frac{2}{945}\,J^{(5)}_{ijkl}\,J^{(4)}_{jkl}+\frac{2}{14175}\,\varepsilon_{ijk}I^{(5)}_{jabc}\,J^{(5)}_{kabc}\right]\right.$
$\displaystyle\left.+{G\over c^{5}}\left[{2\over
63}\left(I_{ijk}^{(4)}\left[{1\over 7}\,I_{a\langle j}^{(6)}I_{k\rangle
a}-{4\over 7}\,I_{a\langle j}^{(5)}I_{k\rangle a}^{(1)}-I_{a\langle
j}^{(4)}I_{k\rangle a}^{(2)}-{4\over 7}\,I_{a\langle j}^{(3)}I_{k\rangle
a}^{(3)}+{1\over 3}\,\varepsilon_{ab\langle j}I_{k\rangle
a}^{(5)}J_{b}\right.\right.\right.\right.$
$\displaystyle\left.\left.\left.\left.+4\left[W^{(2)}I_{jk}-W^{(1)}I_{jk}^{(1)}\right]^{(3)}\right]+I_{jk}^{(3)}\left[-{43\over
12}\,I_{a\langle i}^{(4)}I_{jk\rangle a}^{(3)}-{17\over 12}\,I_{a\langle
i}^{(3)}I_{jk\rangle a}^{(4)}-3\,I_{a\langle i}^{(5)}I_{jk\rangle
a}^{(2)}\right.\right.\right.\right.$
$\displaystyle\left.\left.\left.\left.+{1\over 2}\,I_{a\langle
i}^{(2)}I_{jk\rangle a}^{(5)}-{2\over 3}\,I_{a\langle i}^{(6)}I_{jk\rangle
a}^{(1)}+{1\over 2}\,I_{a\langle i}^{(1)}I_{jk\rangle a}^{(6)}+{1\over
12}\,I_{a\langle i}^{(7)}I_{jk\rangle a}+{1\over 4}\,I_{a\langle
i}I_{jk\rangle a}^{(7)}+{1\over 5}\,\varepsilon_{ab\langle
i}\left(-12\,J_{j\mathrm{d}{a}}^{(3)}I_{k\rangle
b}^{(3)}\right.\right.\right.\right.\right.$
$\displaystyle\left.\left.\left.\left.\left.-12\,I_{j\mathrm{d}{a}}^{(3)}J_{k\rangle
b}^{(3)}-15\,J_{j\mathrm{d}{a}}^{(2)}I_{k\rangle
b}^{(4)}-35\,I_{j\mathrm{d}{a}}^{(2)}J_{k\rangle
b}^{(4)}-4\,J_{j\mathrm{d}{a}}^{(1)}I_{k\rangle
b}^{(5)}-36\,I_{j\mathrm{d}{a}}^{(1)}J_{k\rangle
b}^{(5)}-J_{j\mathrm{d}{a}}I_{k\rangle
b}^{(6)}-9\,I_{j\mathrm{d}{a}}J_{k\rangle
b}^{(6)}\right.\right.\right.\right.\right.$
$\displaystyle\left.\left.\left.\left.\left.-{9\over
4}\,J_{\mathrm{d}{a}}I_{jk\rangle b}^{(6)}\right)+{12\over 5}\,J_{\langle
i}J_{jk\rangle}^{(5)}+4\,\left[W^{(2)}I_{ijk}-W^{(1)}I_{ijk}^{(1)}+3\,I_{\langle
ij}Y_{k\rangle}^{(1)}\right]^{(4)}\right]\right)+{16\over
45}\,\varepsilon_{ijk}\left(I_{jp}^{(3)}\right.\right.\right.$
$\displaystyle\left.\left.\left.\left[{4\over 7}\,J_{a\langle
k}^{(3)}I_{p\rangle a}^{(3)}+{8\over 7}\,I_{a\langle k}^{(3)}J_{p\rangle
a}^{(3)}+3\,J_{a\langle k}^{(2)}I_{p\rangle a}^{(4)}+{5\over 7}\,I_{a\langle
k}^{(2)}J_{p\rangle a}^{(4)}+{26\over 7}\,J_{a\langle k}^{(1)}I_{p\rangle
a}^{(5)}-{6\over 7}\,I_{a\langle k}^{(1)}J_{p\rangle a}^{(5)}+{9\over
7}\,J_{a\langle k}I_{p\rangle a}^{(6)}\right.\right.\right.\right.$
$\displaystyle\left.\left.\left.\left.-{3\over 7}\,I_{a\langle k}J_{p\rangle
a}^{(6)}-{1\over 28}\,J_{a}I_{kpa}^{(6)}-\varepsilon_{ab\langle
k}J_{\mathrm{d}{a}}J_{p\rangle b}^{(5)}+{1\over 14}\,\varepsilon_{ac\langle
k}\left({425\over 24}\,I_{p\rangle bc}^{(3)}I_{ab}^{(4)}+{31\over
12}\,I_{p\rangle bc}^{(4)}I_{ab}^{(3)}+{173\over 6}\,I_{p\rangle
bc}^{(2)}I_{ab}^{(5)}\right.\right.\right.\right.\right.$
$\displaystyle\left.\left.\left.\left.\left.-{19\over 24}\,I_{p\rangle
bc}^{(5)}I_{ab}^{(2)}+{143\over 8}\,I_{p\rangle bc}^{(1)}I_{ab}^{(6)}+{15\over
4}\,I_{p\rangle bc}I_{ab}^{(7)}+{3\over 8}\,I_{p\rangle
bc}^{(7)}I_{ab}\right)+2\left[\varepsilon_{ab\langle k}\left(-I_{p\rangle
b}^{(3)}W_{a}-2\,I_{p\rangle
b}Y_{a}^{(2)}\right.\right.\right.\right.\right.\right.$
$\displaystyle\left.\left.\left.\left.\left.\left.+I_{p\rangle
b}^{(1)}Y_{a}^{(1)}\right)+3\,J_{\langle
k}Y_{p\rangle}^{(1)}-2\,J_{kp}^{(1)}W^{(1)}\right]^{(3)}\right]+J_{kp}^{(3)}\left[{1\over
7}\,I_{a\langle j}^{(6)}I_{p\rangle a}-{4\over 7}\,I_{a\langle
j}^{(5)}I_{p\rangle a}^{(1)}-I_{a\langle j}^{(4)}I_{p\rangle a}^{(2)}-{4\over
7}\,I_{a\langle j}^{(3)}I_{p\rangle a}^{(3)}\right.\right.\right.\right.$
$\displaystyle\left.\left.\left.\left.+{1\over 3}\,\varepsilon_{ab\langle
j}I_{p\rangle
a}^{(5)}J_{b}+4\left[W^{(2)}I_{jp}-W^{(1)}I_{jp}^{(1)}\right]^{(3)}\right]\right)+{1\over
1134}\,I_{jkl}^{(4)}\left(-20\,I_{\langle
ij}^{(3)}I_{kl\rangle}^{(3)}-{84\over 5}\,I_{\langle
ij}^{(5)}I_{kl\rangle}^{(1)}-33\,I_{\langle
ij}^{(4)}I_{kl\rangle}^{(2)}\right.\right.\right.$
$\displaystyle\left.\left.\left.-{21\over 5}\,I_{\langle
ij}^{(6)}I_{kl\rangle}\right)+{1\over
126}\,\varepsilon_{ijk}\,I_{jpq}^{(4)}\left(-{2\over
5}\,\varepsilon_{ab\langle k}I_{p\mathrm{d}{a}}^{(5)}I_{q\rangle
b}^{(1)}+{1\over 10}\,\varepsilon_{ab\langle
k}I_{p\mathrm{d}{a}}^{(6)}I_{q\rangle b}-{1\over 2}\,\varepsilon_{ab\langle
k}I_{p\mathrm{d}{a}}^{(4)}I_{q\rangle b}^{(2)}-2\,J_{\langle
k}I_{pq\rangle}^{(5)}\right)\right.\right.$ $\displaystyle\left.\left.+{4\over
63}\,J_{jk}^{(3)}\left(-{2\over 5}\,\varepsilon_{ab\langle
i}I_{j\mathrm{d}{a}}^{(5)}I_{k\rangle b}^{(1)}+{1\over
10}\,\varepsilon_{ab\langle i}I_{j\mathrm{d}{a}}^{(6)}I_{k\rangle b}-{1\over
2}\,\varepsilon_{ab\langle i}I_{j\mathrm{d}{a}}^{(4)}I_{k\rangle
b}^{(2)}-2\,J_{\langle i}I_{jk\rangle}^{(5)}\right)\right]+{\cal
O}\left({1\over c^{6}}\right)\right\\},$

where,

$\displaystyle\left[W^{(2)}I_{ij}-W^{(1)}I_{ij}^{(1)}\right]^{(3)}$
$\displaystyle=\left[2\,W^{(4)}I_{ij}^{(1)}+W^{(5)}I_{ij}-W^{(1)}I_{ij}^{(4)}-2\,W^{(2)}I_{ij}^{(3)}\right]\,,$
(16) $\displaystyle\left[W^{(2)}I_{ijk}-W^{(1)}I_{ijk}^{(1)}+3\,I_{\langle
ij}Y_{k\rangle}^{(1)}\right]^{(4)}$
$\displaystyle=\left[W^{(6)}I_{ijk}+3\,W^{(5)}\,I_{ijk}^{(1)}+2\,W^{(4)}\,I_{ijk}^{(2)}-3\,W^{(2)}\,I_{ijk}^{(4)}\right.$
$\displaystyle\left.-2\,W^{(3)}\,I_{ijk}^{(3)}-W^{(1)}I_{ijk}^{(5)}+3\,I_{\langle
ij}Y_{k\rangle}^{(5)}+12\,I_{\langle ij}^{(1)}Y_{k\rangle}^{(4)}\right.$
$\displaystyle\left.+18\,I_{\langle ij}^{(2)}Y_{k\rangle}^{(3)}+12\,I_{\langle
ij}^{(3)}Y_{k\rangle}^{(2)}+3\,I_{\langle
ij}^{(4)}Y_{k\rangle}^{(1)}\right]\,,$ (17)
$\displaystyle\left[\varepsilon_{ab\langle i}\left(-I_{j\rangle
b}^{(3)}W_{a}-2\,I_{j\rangle b}Y_{a}^{(2)}+I_{j\rangle
b}^{(1)}Y_{a}^{(1)}\right)\right]^{(3)}$ $\displaystyle=\varepsilon_{ab\langle
i}\left(-I_{j\rangle b}^{(6)}W_{a}-3\,I_{j\rangle
b}^{(5)}W_{a}^{(1)}-3\,I_{j\rangle b}^{(4)}W_{a}^{(2)}-I_{j\rangle
b}^{(3)}W_{a}^{(3)}\right.$ $\displaystyle\left.-2\,I_{j\rangle
b}Y_{a}^{(5)}-5\,I_{j\rangle b}^{(1)}Y_{a}^{(4)}-3\,I_{j\rangle
b}^{(2)}Y_{a}^{(3)}+I_{j\rangle b}^{(3)}Y_{a}^{(2)}+I_{j\rangle
b}^{(4)}Y_{a}^{(1)}\right)\,,$ (18) $\displaystyle\left[3\,J_{\langle
i}Y_{j\rangle}^{(1)}-2\,J_{ij}^{(1)}W^{(1)}\right]^{(3)}$
$\displaystyle=\left[3\,J_{\langle
i}Y_{j\rangle}^{(4)}-2\,J_{ij}^{(1)}W^{(4)}-6\,J_{ij}^{(2)}W^{(3)}-6\,J_{ij}^{(3)}W^{(2)}\right.$
$\displaystyle\left.-2\,J_{ij}^{(4)}W^{(1)}\right]\,.$ (19)

In the above equations, symmetric trace-free projections of the mulitpole
moments are denoted by the angular brackets ($\langle\rangle$) around their
indices and the underlined indices are excluded while taking the projection.
And the hereditary contribution to LMF at 2.5PN is

$\displaystyle{\left({\mathcal{F}_{i}}\right)_{\rm hered}}$ $\displaystyle=$
$\displaystyle\frac{4\,G^{2}\,M}{63\,c^{10}}\,I_{ijk}^{(4)}(U)\int_{0}^{\infty}d\tau\left[\ln\left({\tau\over
2\tau_{0}}\right)+{11\over
12}\right]I^{(5)}_{jk}(U-\tau)+\frac{4\,G^{2}\,M}{63\,c^{10}}\,I_{jk}^{(3)}(U)\int_{0}^{\infty}d\tau\left[\ln\left({\tau\over
2\tau_{0}}\right)+{97\over 60}\right]I^{(6)}_{ijk}(U-\tau)$
$\displaystyle+{32\,G^{2}\,M\over
45\,c^{10}}\,\varepsilon_{ijk}\,I_{ja}^{(3)}(U)\int_{0}^{\infty}d\tau\left[\ln\left({\tau\over
2\tau_{0}}\right)+{7\over 6}\right]J^{(5)}_{ka}(U-\tau)+{32\,G^{2}\,M\over
45\,c^{10}}\,\varepsilon_{ijk}\,J_{ka}^{(3)}(U)\int_{0}^{\infty}d\tau\left[\ln\left({\tau\over
2\tau_{0}}\right)+{11\over 12}\right]I^{(5)}_{ja}(U-\tau)$
$\displaystyle+{G^{2}\,M\over
567\,c^{12}}\,I_{ijkl}^{(5)}(U)\int_{0}^{\infty}d\tau\left[\ln\left({\tau\over
2\tau_{0}}\right)+{97\over 60}\right]I^{(6)}_{jkl}(U-\tau)+{G^{2}\,M\over
567\,c^{12}}\,I_{jkl}^{(4)}(U)\int_{0}^{\infty}d\tau\left[\ln\left({\tau\over
2\tau_{0}}\right)+{59\over 30}\right]I^{(7)}_{ijkl}(U-\tau)$
$\displaystyle+{G^{2}\,M\over
63\,c^{12}}\,\varepsilon_{ijk}\,I_{jab}^{(4)}(U)\int_{0}^{\infty}d\tau\left[\ln\left({\tau\over
2\tau_{0}}\right)+{5\over 3}\right]J^{(6)}_{kab}(U-\tau)+{G^{2}\,M\over
63\,c^{12}}\,\varepsilon_{ijk}\,J_{kab}^{(4)}(U)\int_{0}^{\infty}d\tau\left[\ln\left({\tau\over
2\tau_{0}}\right)+{97\over 60}\right]I^{(6)}_{jab}(U-\tau)$
$\displaystyle+{8\,G^{2}\,M\over
63\,c^{12}}\,J_{ijk}^{(4)}(U)\int_{0}^{\infty}d\tau\left[\ln\left({\tau\over
2\tau_{0}}\right)+{7\over 6}\right]J^{(5)}_{jk}(U-\tau)+{8\,G^{2}\,M\over
63\,c^{12}}\,J_{jk}^{(3)}(U)\int_{0}^{\infty}d\tau\left[\ln\left({\tau\over
2\tau_{0}}\right)+{5\over 3}\right]J^{(6)}_{ijk}(U-\tau)+{\cal O}\left({1\over
c^{6}}\right).$

##  III Orbital dynamics of the compact binary source

In the previous section, we have provided an explicit closed form expression
of the far-zone linear momentum flux from a compact binary system in terms of
various source multipole moments. Here we specialize to the case of a non-
spinning compact binary system in quasi-elliptical orbits, with the component
masses $m_{1}$ and $m_{2}$ with $m_{1}\geq m_{2}$, the total mass
$m=m_{1}+m_{2}$, and the symmetric mass ratio, $\eta=m_{1}m_{2}/m^{2}$. Since
the binary constituents are nonspinning, its motion is completely confined in
a plane with a relative separation,

$\displaystyle\mathbf{x}=\mathbf{x_{1}}-\mathbf{x_{2}}=r\,\mathbf{\hat{n}},$
(21)

with $r=|{\bf x}|$, $\mathbf{x_{1}}$ and $\mathbf{x_{2}}$ are the position
vectors of the component masses, and $\mathbf{\hat{n}}$ is the unit vector
along the relative separation vector. In polar coordinates,

${\bf\hat{n}}=\frac{\bf
x}{r}={\cos{\phi}}\,{\bf\hat{e}_{x}}+{\sin{\phi}}\,{\bf\hat{e}_{y}}\,,$ (22)

where $\phi$ is the orbital phase of the binary, and $\bf\hat{e}_{x}$ and
$\bf\hat{e}_{y}$ are the unit vectors along $x$ and $y$ axes. The relative
velocity and acceleration for the system are the following,

$\displaystyle\mathbf{v}$ $\displaystyle={{\rm d}\mathbf{x}\over{\rm
d}t}=(\dot{r}\cos{\phi}-r\dot{\phi}\sin\phi)\,{\bf\hat{e}_{x}}+(\dot{r}\sin{\phi}+r\dot{\phi}\cos\phi)\,{\bf\hat{e}_{y}}\,,$
(23) $\displaystyle\mathbf{a}$ $\displaystyle={{\rm d}\mathbf{v}\over{\rm
d}t}={{\rm d^{2}}\mathbf{x}\over{\rm d^{2}}t}.$ (24)

To calculate the 2.5PN accurate LMF, we need the time derivative of the source
multipole moments. Hence we need 2.5PN accurate equations of motion for the
compact binary system. We use the same from ref. Blanchet and Iyer (2003);
Arun et al. (2008a) in the center of mass frame. The equation of motion can be
used to write down the following expressions in order to obtain the
derivatives of the multipole moments,

$\displaystyle\dot{v}$ $\displaystyle={{\bf a}\cdot{\bf v}\over v},$ (25)
$\displaystyle\ddot{r}$ $\displaystyle={1\over
r}\left[\left(v^{2}-\dot{r}^{2}\right)+{\bf a}\cdot{\bf x}\right],$ (26)

where the $\dot{r}$ and $\ddot{r}$ denote the first and the second time
derivatives of the orbital separation $r$ respectively and we denote the
magnitude of the orbital velocity by $v=|{\bf v}|$.

To evaluate the instantaneous and the hereditary contributions to the LMF, we
would also need the explicit expressions for the various multipole moments for
compact binaries moving in quasi-elliptical orbits. These are obtained from
the long algebraic computations (see ref. Blanchet et al. (2002) for details)
using MPM-PN formalism Blanchet (2006). The expressions are too long to be
explicitly quoted here. Hence we point to the Refs. Blanchet and Iyer (2005);
Arun et al. (2008a) for those.

##  IV Instantaneous contribution to the Linear momentum flux

With all the ingredients provided in the previous sections we now compute the
instantaneous contribution to the LMF using the source multipole moments.
First we calculate all the time derivatives of the source multipole moments
using the equation of motion as quoted in Ref. (Arun et al. (2008a)) at 2.5PN.
Next we perform all the contractions in Eq. (II) and the resulting
instantaneous linear momentum flux in terms of dynamical variables
$(r,\dot{r},v,{\mathbf{v}},{\mathbf{x}})$ is given by,

$\displaystyle\left({\mathcal{F}_{i}}\right)_{\rm inst}$
$\displaystyle=\frac{64}{105}\frac{G^{3}m^{4}}{r^{4}c^{7}}\sqrt{1-4\eta}\eta^{2}\Bigg{[}\Bigg{(}\dot{r}\Big{[}\frac{3}{2}\frac{Gm}{r}-\frac{45}{8}\dot{r}^{2}+\frac{55}{8}v^{2}\Big{]}+\frac{1}{c^{2}}\Bigg{(}\frac{G^{2}m^{2}}{r^{2}}\dot{r}\Big{[}-\frac{295}{24}+\frac{1}{12}\eta\Big{]}+\frac{Gm}{r}\dot{r}\Big{[}\Big{(}-\frac{4358}{48}+\frac{239}{12}\eta\Big{)}v^{2}$
$\displaystyle+\Big{(}\frac{12301}{144}-\frac{73}{9}\eta\Big{)}\dot{r}^{2}\Big{]}+v^{4}\dot{r}\Big{[}\frac{851}{24}-\frac{779}{24}\eta\Big{]}-v^{2}\dot{r}^{3}\Big{[}\frac{1417}{12}-\frac{1877}{24}\eta\Big{]}+\dot{r}^{5}\Big{[}\frac{1843}{24}-\frac{259}{6}\eta\Big{]}\Bigg{)}+\frac{1}{c^{4}}\Bigg{(}\dot{r}^{7}\Big{[}\frac{35}{22}+\frac{123565}{264}\eta$
$\displaystyle-\frac{5495}{33}\eta^{2}\Big{]}+\frac{G^{3}m^{3}}{r^{3}}\dot{r}\Big{[}\frac{52781}{1584}+\frac{15773}{264}\eta-\frac{607}{264}\eta^{2}\Big{]}+\frac{G^{2}m^{2}}{r^{2}}\dot{r}\Big{[}\Big{(}-\frac{803423}{1584}+\frac{5685}{32}\eta-\frac{14437}{528}\eta^{2}\Big{)}\dot{r}^{2}+\Big{(}\frac{189413}{396}$
$\displaystyle-\frac{754361}{3168}\eta+\frac{26527}{396}\eta^{2}\Big{)}v^{2}\Big{]}-\frac{Gm}{r}\dot{r}\Big{[}\Big{(}\frac{329405}{528}-\frac{102581}{96}\eta+\frac{21603}{88}\eta^{2}\Big{)}\dot{r}^{4}-\Big{(}\frac{292697}{352}-\frac{113233}{72}\eta+\frac{463801}{1056}\eta^{2}\Big{)}v^{2}\dot{r}^{2}$
$\displaystyle+\Big{(}\frac{76409}{352}-\frac{458683}{1056}\eta+\frac{178873}{1056}\eta^{2}\Big{)}v^{4}\Big{]}-v^{2}\dot{r}^{4}\Big{[}\frac{5363}{132}+\frac{150719}{132}\eta-497\eta^{2}\Big{]}+v^{4}\dot{r}^{2}\Big{[}\frac{1327}{33}+\frac{900359}{1056}\eta-\frac{5240}{11}\eta^{2}\Big{]}$
$\displaystyle+v^{6}\Big{[}\frac{85}{44}-\frac{187945}{1056}\eta+\frac{9371}{66}\eta^{2}\Big{]}\Bigg{)}+\frac{Gm}{rc^{5}}\eta\Bigg{(}\frac{701}{90}v^{6}-\frac{51137}{96}v^{4}\dot{r}^{2}+\frac{41611}{40}v^{2}\dot{r}^{4}-\frac{49219}{96}\dot{r}^{6}-\frac{4}{15}\frac{G^{3}m^{3}}{r^{3}}$
$\displaystyle+\frac{G^{2}m^{2}}{r^{2}}\Big{[}\frac{1237}{90}v^{2}-\frac{6607}{180}\dot{r}^{2}\Big{]}-\frac{Gm}{r}\Big{[}\frac{4261}{120}v^{4}+\frac{8397}{40}v^{2}\dot{r}^{2}-\frac{3778}{15}\dot{r}^{4}\Big{]}\Bigg{)}\Bigg{)}\frac{\mathbf{x}_{i}}{r}$
$\displaystyle-\Bigg{(}\frac{Gm}{r}+\frac{25}{4}v^{2}-\frac{19}{4}\dot{r}^{2}+\frac{1}{c^{2}}\Bigg{(}\frac{Gm}{r}\Big{[}\Big{(}\frac{2699}{48}+\frac{5}{24}\eta\Big{)}\dot{r}^{2}-\Big{(}\frac{907}{16}-\frac{81}{8}\eta\Big{)}v^{2}\Big{]}-\frac{G^{2}m^{2}}{r^{2}}\Big{(}\frac{21}{2}+\frac{17}{18}\eta\Big{)}+v^{4}\Big{[}\frac{925}{48}-\frac{259}{12}\eta\Big{]}$
$\displaystyle-v^{2}\dot{r}^{2}\Big{[}\frac{245}{3}-\frac{1285}{24}\eta\Big{]}+\dot{r}^{4}\Big{[}\frac{2663}{48}-\frac{697}{24}\eta\Big{]}\Bigg{)}+\frac{1}{c^{4}}\Bigg{(}\frac{Gm}{r}\Big{[}v^{4}\Big{(}-\frac{47107}{352}+\frac{214813}{1056}\eta-\frac{80173}{1056}\eta^{2}\Big{)}+v^{2}\dot{r}^{2}\Big{(}\frac{199993}{352}$
$\displaystyle-\frac{163247}{176}\eta+\frac{221851}{1056}\eta^{2}\Big{)}-\dot{r}^{4}\Big{(}\frac{120737}{264}-\frac{694453}{1056}\eta+\frac{42029}{396}\eta^{2}\Big{)}\Big{]}+\frac{G^{2}m^{2}}{r^{2}}\Big{[}v^{2}\Big{(}\frac{955835}{4752}-\frac{8047}{144}\eta+\frac{3327}{176}\eta^{2}\Big{)}$
$\displaystyle-\dot{r}^{2}\Big{(}\frac{381131}{1584}-\frac{62105}{1584}\eta-\frac{10855}{1584}\eta^{2}\Big{)}\Big{]}+\frac{G^{3}m^{3}}{r^{3}}\Big{[}\frac{236347}{4752}+\frac{1913}{44}\eta-\frac{1103}{396}\eta^{2}\Big{]}+v^{6}\Big{[}\frac{7741}{264}-\frac{54215}{528}\eta+\frac{5741}{66}\eta^{2}\Big{]}$
$\displaystyle-v^{4}\dot{r}^{2}\Big{[}\frac{73439}{528}-\frac{38405}{66}\eta+\frac{12765}{44}\eta^{2}\Big{]}+v^{2}\dot{r}^{4}\Big{[}\frac{30271}{132}-\frac{39085}{48}\eta+\frac{25607}{88}\eta^{2}\Big{]}-\dot{r}^{6}\Big{[}\frac{61339}{528}-\frac{88925}{264}\eta+\frac{24391}{264}\eta^{2}\Big{]}\Bigg{)}$
$\displaystyle-\frac{Gm\eta}{rc^{5}}\dot{r}\Bigg{(}\frac{157787}{480}v^{4}-\frac{39869}{60}v^{2}\dot{r}^{2}+\frac{31913}{96}\dot{r}^{4}+\frac{Gm}{r}\Big{[}\frac{10773}{40}v^{2}-\frac{99277}{360}\dot{r}^{2}\Big{]}+\frac{737}{36}\frac{G^{2}m^{2}}{r^{2}}\Bigg{)}\Bigg{)}\mathbf{v}_{i}\Bigg{]}.$
(27)

As a consitency check, we confirm that Eq. (IV) agrees with the non-spinning
contribution to the LMF quoted in ref. Racine et al. (2009). One may notice
here that the component of the linear momentum flux along the radial direction
(i.e. the term associated to the radial direction, $\mathbf{x}$) depends on
$\dot{r}$ and hence in case of quasi-circular orbit contributions from these
terms are 0 and the emission of linear momentum is along the direction of the
relative velocity vector, $\mathbf{v}$. Although this is true only upto
relative 2PN order.

The above expression for linear momentum flux is given in terms of generic
dynamical variables $r,\dot{r}$ and $v$. While specializing to the case of
quasi-elliptical orbits, it is convenient to express these dynamical variables
in terms of the parameters associated with quasi-elliptical orbits, namely the
generalized quasi-Keplerian representation (QKR) of the orbital dynamics. One
needs 2.5PN QKR to compute the 2.5PN LMF in terms of the orbital parameters.
In the next section we briefly start with the description of the
parametrization of Keplerian orbits followed by its PN generalization, the
quasi-Keplerian (QK) representation.

##  V Keplerian and Quasi-Keplerian parametrization

The Keplerian parametrization for the Newtonian motion of a compact binary
system is widely used in describing celestial mechanics. In polar coordinates
and in the center of mass frame, the parametrization is given by,

$\displaystyle r_{\rm N}=a_{N}(1-e_{N}\cos u)\,,$ (28a)
$\displaystyle\phi_{\rm N}=V_{N}\,,$ (28b) $\displaystyle l_{\rm
N}=n(t-t_{0})=u-e\sin u\,,$ (28c) $\displaystyle v=V_{\rm N}(u)\equiv
2\arctan\left[\left(\frac{1+e}{1-e}\right)^{2}\tan\Big{(}\frac{u}{2}\Big{)}\right]\;,$
(28d)

where subscript N denotes the Newtonian quantities. $r_{N}$ and $\phi_{N}$
together define the relative separation vector,
$\mathbf{r_{N}}=r_{N}(\cos\phi_{N},\sin\phi_{N},0)$. The semi-major axis of
the orbit is $a_{N}$ with an eccentricity $e_{N}$. Both of these can be
written in terms of the conserved orbital energy and angular momentum which
completely define the orbits. Here $u,v,l$ are the eccentric, true, and mean
anomalies and $n$ is the mean motion, $n=2\pi/P$, where P is the radial
orbital (periastron to periastron) period.

Having discussed the Keplerian representation (KR), we now describe the PN
extension of the KR, the quasi-Keplerian representation at 2.5PN. In 1985,
Damour and Deruelle generalized this parametrization up to 1PN Damour and
Deruelle (1985) and proposed a “Keplerian like parametrization”. Later in
refs. Damour and Schäfer (1988); Schäfer and Wex (1993); Wex (1995) the 2PN
extension of the parametric solution has been quoted. 3PN extension of the
same is discussed in ref. Memmesheimer et al. (2004). 3PN QK parametrization
is obtained considering 3PN conservative contributions to the binary motion
and it admits very similar expressions as the Keplerian one but with more
complex structure. In order to obtain 2.5PN accurate LMF, it is sufficient to
use 2PN accurate QKR of the orbital motion. Hence, in this section we only
describe the 2PN QKR of the conservative dynamics as an extension of Eqs.
(28),

$\displaystyle r=a_{r}(1-e_{r}\cos u)\,,$ (29a)
$\displaystyle\phi=\lambda+W(l;n,e_{t})\,,$ (29b)
$\displaystyle\lambda=(1+k)n(t-t_{0})+c_{\lambda}\,,$ (29c) $\displaystyle
W(l;n,e_{t})=(1+k)(v-l)+\frac{f_{4\phi}}{c^{4}}\sin
2v+\frac{g_{4\phi}}{c^{4}}\sin 3v\,,$ (29d) $\displaystyle
l=n(t-t_{0})+c_{l}=u-e_{t}\sin
u+\frac{g_{4t}}{c^{4}}(v-u)+\frac{f_{4t}}{c^{4}}\sin v\,,$ (29e)
$\displaystyle v=V(u)\equiv
2\arctan\left[\left(\frac{1+e_{\phi}}{1-e_{\phi}}\right)^{2}\tan\Big{(}\frac{u}{2}\Big{)}\right]\,.$
(29f)

The expressions of the functions $f_{4t}$, $g_{4t}$, $f_{4\phi}$, $f_{6\phi}$
and $g_{4\phi}$ are given in ref. Memmesheimer et al. (2004). $a_{r}$ is some
2PN equivalent “semi-major axis”. Unlike the KR, in QKR, there appear three
eccentricities $e_{r},e_{t}$ and $e_{\phi}$, instead of one to completely
parameterize the motion. These eccentricities can also be written in terms of
the 2PN conserved energy and the angular momentum. In the literature, it has
been found to be convenient to use only $e_{t}$ and the mean motion $n$ as the
constants of motion and express all the dynamical variables in terms of these
two Damour and Gopakumar (2006). Additionally one uses a combination of total
mass and $n$ given by $\zeta=Gmn/c^{3}$, as a PN expansion parameter. However,
it is equivalent to use $x$ ( $\sim\Big{(}\frac{Gm\omega}{c^{3}}\Big{)}^{2/3}$
with $\omega$ being the orbital frequency) and $e_{t}$ instead of $\zeta$ and
$e_{t}$. Since the convenient choice for a binary moving in quasi-circular
orbit would be to use $x$ as the PN expansion parameter, we stick to the
variable $x$ to express all our quantities here. One may notice that the PN
expansion of LMF, Eq. (IV) are expressed as a series in $1/c$. One can easily
use the relation between $n,c,\zeta$ and $x$ in order to get the correct
expression of the concerned quantity at every PN order. For the convenience of
the readers, we provide the explicit expression, used in our calculation, for
$\zeta$ in terms of $x,e_{t}$ below,

$\displaystyle\zeta$
$\displaystyle={x^{3/2}\over(1-e_{t}^{2})^{3}}\Bigg{\\{}1-3e_{t}^{2}+3e_{t}^{4}-e_{t}^{6}+x\Big{(}-3+6e_{t}^{2}-3e_{t}^{4}\Big{)}+x^{2}\Big{[}-\frac{9}{2}+7\eta+\Big{(}-\frac{33}{4}-\frac{\eta}{2}\Big{)}e_{t}^{2}+\Big{(}\frac{51}{4}-\frac{13\eta}{2}\Big{)}e_{t}^{4}\Big{]}\Bigg{\\}}\,.$
(30)

##  VI Instantaneous LMF for compact binaries in terms of quasi-Keplerian
parameters in the small eccentricity limit

We have provided all the necessary ingredients to compute LMF from a compact
binary moving in quasi-elliptical orbit in terms of its orbital elements. As a
next step, we re-express the instantaneous contribution to the LMF in terms of
QK parameters. To be precise, we use Eqs. (29) to re-express Eq. (IV) in terms
of $\\{x,e_{t},u,~{}\&~{}\phi\\}$. For the instantaneous contributions we do
not assume the orbital eccentricity to be small and quote the complete closed
form expression valid for arbitrary values of eccentricity. We find it
convenient to express the quantities in terms of $\phi$ and $u$ both which
helps to obtain the circular orbit limits quite straightforwardly. Finally, we
quote the instantaneous contribution to the LMF emitted by a non-spinning
compact binary system in a quasi-elliptical orbit.

$\displaystyle\mathcal{F}_{\rm
inst}=\frac{464}{105}\frac{c^{4}}{G}\sqrt{1-4\eta}\eta^{2}x^{11/2}{1\over(1-e_{t}^{2})^{2}(1-e_{t}\cos
u)^{11}}\begin{bmatrix}\mathcal{F}_{\cos\phi}^{\rm
inst}&\mathcal{F}_{\sin\phi}^{\rm
inst}\end{bmatrix}\begin{bmatrix}\cos\phi&\sin\phi\\\
\sin\phi&-\cos\phi\end{bmatrix}\begin{bmatrix}{\bf\hat{e}_{x}}\\\
{\bf\hat{e}_{y}}\end{bmatrix}$ (31)

$\displaystyle\mathcal{F}_{\cos\phi}^{\rm inst}=\mathcal{F}_{\cos\phi;N}^{\rm
inst}+x~{}\mathcal{F}_{\cos\phi;1PN}^{\rm
inst}+x^{2}~{}\mathcal{F}_{\cos\phi;2PN}^{\rm
inst}+x^{5/2}~{}\mathcal{F}_{\cos\phi;2.5PN}^{\rm inst}$ (32)

Where, the explicit contributions at different PN orders are the following,

$\displaystyle\mathcal{F}_{\cos\phi;N}^{\rm inst}=$ $\displaystyle{e_{t}\over
58}(1-e_{t}^{2})^{2}(1-e_{t}\cos u)^{4}(9-6e_{t}^{2}-4e_{t}\cos
u+e_{t}^{2}\cos 2u)\sin u$ (33) $\displaystyle\mathcal{F}_{\cos\phi;1PN}^{\rm
inst}=$ $\displaystyle-{e_{t}\over 4176}(1-e_{t}^{2})(1-e_{t}\cos u)^{2}\sin
u\Big{[}52\Big{(}183+29\eta\Big{)}+e_{t}^{2}\Big{(}7396-13190\eta-7e_{t}^{2}[2986-2727\eta]+e_{t}^{4}[10920-7407\eta]\Big{)}$
$\displaystyle-4e_{t}\cos u\Big{(}6948-1754\eta-
e_{t}^{2}[7369-2586\eta]+e_{t}^{4}[3193-832\eta]\Big{)}+2e_{t}^{2}\cos
2u\Big{(}1804-95\eta+4e_{t}^{2}[404-67\eta]$ $\displaystyle-
e_{t}^{4}[648-363\eta]\Big{)}-e_{t}^{3}\Big{(}4\cos[3u][145+2\eta+e_{t}^{2}(251-2\eta)]-3e_{t}\cos[4u][46+\eta+e_{t}^{2}(20-\eta)]\Big{)}\Big{]}$
(34) $\displaystyle\mathcal{F}_{\cos\phi;2PN}^{\rm inst}=$
$\displaystyle-\Bigg{(}e_{t}\Bigg{[}-{2473\over 5742}-{483137\over
45936}\eta-{1645\over 12528}\eta^{2}\Bigg{]}+e_{t}^{2}\Bigg{[}{1537571\over
34452}+{462007\over 45936}\eta-\frac{481315}{137808}\eta^{2}\Bigg{]}\cos
u+e_{t}^{3}\Bigg{[}-{2106595\over 22968}$ $\displaystyle+{15373855\over
275616}\eta+{26021\over 9504}-\Bigg{(}{233005\over 5742}-{4004795\over
275616}\eta+{1027127\over
275616}\eta^{2}\Bigg{)}\cos[2u]\Bigg{]}+e_{t}^{4}\Bigg{[}\Bigg{(}{1327349\over
34452}-{2556193\over 22968}\eta$ $\displaystyle+{929135\over
34452}\eta^{2}\Bigg{)}\cos u+\Bigg{(}{120299\over 11484}-{207\over
638}\eta+{8803\over
68904}\eta^{2}\Bigg{)}\cos[3u]\Bigg{]}+e_{t}^{5}\Bigg{[}\frac{334223}{2088}-\frac{51247793}{1102464}\eta-\frac{25252307}{1102464}\eta^{2}$
$\displaystyle+\Bigg{(}\frac{1826369}{34452}-\frac{19955}{594}\eta+\frac{1280395}{137808}\eta^{2}\Bigg{)}\cos[2u]+\Bigg{(}\frac{38389}{17226}-\frac{603781}{367488}\eta+\frac{61339}{1102464}\eta^{2}\Bigg{)}\cos[4u]\Bigg{]}+e_{t}^{6}\Bigg{[}\Bigg{(}-\frac{14490317}{68904}$
$\displaystyle+\frac{65655203}{275616}\eta-\frac{13503565}{275616}\eta^{2}\Bigg{)}\cos
u-\Bigg{(}\frac{1571803}{45936}-\frac{6006811}{551232}\eta-\frac{37231}{551232}\eta^{2}\Bigg{)}\cos[3u]-\Bigg{(}\frac{599}{1392}-\frac{166543}{551232}\eta$
$\displaystyle+\frac{335}{50112}\eta^{2}\Bigg{)}\cos[5u]\Bigg{]}+e_{t}^{7}\Bigg{[}-\frac{6995893}{183744}-\frac{45896101}{551232}\eta+\frac{25446325}{551232}\eta^{2}+\Bigg{(}{25213079\over
551232}-{22646075\over 2204928}\eta$ $\displaystyle-{21076241\over
2204928}\eta^{2}\Bigg{)}\cos[2u]+\Bigg{(}{164939\over 551232}-{26933\over
20416}\eta-{46213\over 551232}\eta^{2}\Bigg{)}\cos[4u]+\Bigg{(}{685\over
61248}-{20551\over 734976}\eta+{91\over
734976}\eta^{2}\Bigg{)}\cos[6u]\Bigg{]}$
$\displaystyle+e_{t}^{8}\Bigg{[}\Bigg{(}{137465\over 1584}-{13144529\over
137808}\eta+{4279205\over 137808}\eta^{2}\Bigg{)}\cos u+\Bigg{(}{1923539\over
91872}+{393389\over 275616}\eta-{142867\over
275616}\eta^{2}\Bigg{)}\cos[3u]+\Bigg{(}{281\over 8352}$
$\displaystyle+{89009\over 275616}\eta+{335\over
25056}\eta^{2}\Bigg{)}\cos[5u]\Bigg{]}+e_{t}^{9}\Bigg{[}-{9437\over
1056}+{26338153\over 367488}\eta-{4153147\over
122496}\eta^{2}-\Bigg{(}{58459811\over 1102464}+{1107319\over 367488}\eta$
$\displaystyle-{6780793\over
1102464}\eta^{2}\Bigg{)}\cos[2u]-\Bigg{(}{373913\over 275616}+{87121\over
367488}\eta-{835\over 1102464}\eta^{2}\Bigg{)}\cos[4u]+\Bigg{(}{1891\over
122496}-{10073\over 367488}\eta$ $\displaystyle-{91\over
367488}\eta^{2}\Bigg{)}\cos[6u]\Bigg{]}+e_{t}^{10}\Bigg{[}\Bigg{(}{274789\over
11484}+{179717\over 275616}\eta-{1525295\over 275616}\eta^{2}\Bigg{)}\cos
u-\Bigg{(}{7345\over 4176}+{146081\over 551232}-{16189\over
50112}\eta^{2}\Bigg{)}\cos[3u]$ $\displaystyle+\Bigg{(}{617\over
4176}+{8275\over 551232}\eta-{335\over
50112}\eta^{2}\Bigg{)}\cos[5u]\Bigg{]}+e_{t}^{11}\Bigg{[}-{173407\over
15312}-{543457\over 45936}\eta+{184673\over 22968}\eta^{2}+\Bigg{(}{3565\over
696}+{4368883\over 734976}\eta$ $\displaystyle-{1584883\over
734976}\eta^{2}\Bigg{)}\cos[2u]+\Bigg{(}{3031\over 15312}-{29509\over
91872}\eta+{2521\over 91872}\eta^{2}\Bigg{)}\cos[4u]-\Bigg{(}{15\over
2552}-{1493\over 734976}\eta-{91\over
734976}\eta^{2}\Bigg{)}\cos[6u]\Bigg{]}\Bigg{)}\sin u$ $\displaystyle+{1\over
232}\sqrt{1-e_{t}^{2}}e_{t}(1-e_{t}^{2})(5-2\eta)(1-e_{t}\cos u)^{3}\sin
u(252-10e_{t}\cos u-e_{t}^{2}(256-24\cos[2u])-e_{t}^{3}(11\cos u-\cos[3u]))$
(35) $\displaystyle\mathcal{F}_{\cos\phi;2.5PN}^{\rm inst}=$
$\displaystyle{1\over
10440}(1-e_{t}^{2})^{2}\eta\Bigg{[}-20508+12700e_{t}\cos[u]-e_{t}^{2}(82035-149499\cos[2u])-e_{t}^{3}(56418\cos
u+5878\cos[3u])$
$\displaystyle+e_{t}^{4}(235851-271998\cos[2u]-1129\cos[4u])+e_{t}^{5}(46952\cos
u+4179\cos[3u]+\cos[5u])-e_{t}^{6}(136804$
$\displaystyle-124158\cos[2u]-1401\cos[4u]-29\cos[6u])\Bigg{]}$ (36)

$\displaystyle\mathcal{F}_{\sin\phi}^{\rm inst}=\mathcal{F}_{\sin\phi;N}^{\rm
inst}+x~{}\mathcal{F}_{\sin\phi;1PN}^{\rm
inst}+x^{2}~{}\mathcal{F}_{\sin\phi;2PN}^{\rm
inst}+x^{5/2}~{}\mathcal{F}_{\sin\phi;2.5PN}^{\rm inst}$ (37)

Where, the explicit contributions at different PN orders are the following,

$\displaystyle\mathcal{F}_{\sin\phi;N}^{\rm inst}=$ $\displaystyle{1\over
29}(1-e_{t}^{2})^{5/2}(1-e_{t}\cos u)^{4}\Big{(}\Big{[}29-4e_{t}\cos
u-(22+3\cos 2u)e_{t}^{2}\Big{]}\cos\phi\Big{)}$ (38)
$\displaystyle\mathcal{F}_{\sin\phi;1PN}^{\rm inst}=$ $\displaystyle-{1\over
2088}(1-e_{t}^{2})^{3/2}(1-e_{t}\cos
u)^{2}\Big{(}10848+4556\eta+e_{t}^{2}\Big{(}39054-27676\eta-
e_{t}^{2}[49881-30512\eta-3e_{t}^{2}(3953-2794\eta)]\Big{)}$ $\displaystyle-
e_{t}\cos u\Big{(}49878-10136\eta-
e_{t}^{2}[26067-6446\eta]-e_{t}^{4}[8259-2394\eta]\Big{)}+2e_{t}^{2}\cos[2u]\Big{(}6708-1396\eta-
e_{t}^{2}(3255-1276\eta)$
$\displaystyle-3e_{t}^{4}(719-4\eta)\Big{)}-3e_{t}^{3}\cos[3u](13+70\eta-
e_{t}^{2}[589+22\eta])-3e_{t}^{4}\cos[4u](151-4\eta+e_{t}^{2}[65-14\eta])\Big{)}$
(39) $\displaystyle\mathcal{F}_{\sin\phi;2PN}^{\rm inst}=$
$\displaystyle{1\over 116}(1-e_{t}^{2})(5-2\eta)(1-e_{t}\cos
u)^{3}\Bigg{(}638-1588e_{t}^{2}+896e_{t}^{4}+(468-458e_{t}^{2}+47e_{t}^{4})e_{t}\cos
u-(130-136e_{t}^{2})e_{t}^{2}\cos[2u]$
$\displaystyle-3(4-e_{t}^{2})e_{t}^{3}\cos[3u]\Bigg{)}+{1\over
1102464}\sqrt{1-e_{t}^{2}}\Bigg{(}\Bigg{[}-33742320+31536912\eta+2353616\eta^{2}$
$\displaystyle+e_{t}^{2}(226059048-75675888\eta-33433600\eta^{2})-e_{t}^{4}(27954300+173455350\eta-122385830\eta^{2})-e_{t}^{6}(334128948$
$\displaystyle-474234540\eta+159091188\eta^{2})+e_{t}^{8}(173846220-240441438\eta+79270710\eta^{2})-e_{t}^{10}(8196120-29456064\eta$
$\displaystyle+12233808\eta^{2})+\Big{(}e_{t}(10048432-85118736\eta+18798496\eta^{2})-e_{t}^{3}(526309724-447425412\eta+78251360\eta^{2})$
$\displaystyle+e_{t}^{5}(685925776-543187848\eta+79109344\eta^{2})-e_{t}^{7}(106001292-55794972\eta+9227568\eta^{2})-e_{t}^{9}(57233736$
$\displaystyle-53777688\eta+9259920\eta^{2})\Big{)}\cos
u+\Big{(}e_{t}^{2}(109669752-18730992\eta+602064\eta^{2})-e_{t}^{4}(3309896+38646144\eta$
$\displaystyle-13112728\eta^{2})-e_{t}^{6}(143996252-103863315\eta+20703521\eta^{2})+e_{t}^{8}(26676538+6644574\eta+4561366\eta^{2})$
$\displaystyle+e_{t}^{10}(8117568-2160713\eta+1910583\eta^{2})\Big{)}\cos[2u]+\Big{(}-e_{t}^{3}(64308804-12941340\eta-2757537\eta^{2})+e_{t}^{5}(66212382$
$\displaystyle-10732134\eta+3410608\eta^{2})-e_{t}^{7}(11366016+12462156\eta-46648\eta^{2})+e_{t}^{9}(9854478+5904870\eta-628440\eta^{2})\Big{)}\cos[3u]$
$\displaystyle+\Big{(}e_{t}^{4}(12816100-2181642\eta+407458\eta^{2})-e_{t}^{6}(8141924-486948\eta+579380\eta^{2})-e_{t}^{8}(3024980+1254162\eta-122770\eta^{2})$
$\displaystyle-
e_{t}^{10}(1400904-195072\eta-94296\eta^{2})\Big{)}\cos[4u]+\Big{(}-e_{t}^{5}(1457262-654318\eta+33360\eta^{2})+e_{t}^{7}(429228+760464\eta$
$\displaystyle+61176\eta^{2})+e_{t}^{9}(897354+34578\eta-51576\eta^{2})\Big{)}\cos[5u]+\Big{(}e_{t}^{6}(1332-154947\eta-87\eta^{2})+e_{t}^{8}(67230-99966\eta+4530\eta^{2})$
$\displaystyle-
e_{t}^{10}(48960-37509\eta+879\eta^{2})\Big{)}\cos[6u]\Bigg{]}\Bigg{)}$ (40)
$\displaystyle\mathcal{F}_{\sin\phi;2.5PN}^{\rm inst}=$ $\displaystyle-{1\over
10440}(1-e_{t}^{2})^{5/2}\eta\sin
u\Big{[}890669e_{t}-1329517e_{t}^{3}+476628e_{t}^{5}-e_{t}^{2}(446788-390148e_{t}^{2})\cos
u$
$\displaystyle+e_{t}^{3}(24447-2667e_{t}^{2})\cos[2u]-2320e_{t}^{4}\cos[3u]-600e_{t}^{5}\cos[4u]\Big{]}$
(41)

We find that in the circular orbit limit where $e_{t}\rightarrow 0$, Eq. (31)
agrees with the 2PN LMF expression provided in Eq. (1) of Blanchet et al.
(2005) except the 1.5PN term, which is a hereditary contribution discussed
later in sec. VII. We also confirm that in the same limit, our instantaneous
contribution at 2.5 PN in Eq. (31), agrees with Eq. (3.13) of ref. Mishra et
al. (2012) provided an additional post-adiabatic contribution (which is given
in Eq. (65)) is added. Next, we cross check various limiting cases of these
results. In the Newtonian limit, Eq. (31) does not represent Eq. (2.23) of
ref. Fitchett (1983) in its present form. By accurately replacing $u$ with
$\phi$, we recover the correct limit. To be noted here, the replacement of $u$
in terms of $\phi$ at each successive order, leads to aperiodic terms, having
linear or quadratic dependence on $\phi$ along with all the periodic terms in
$\phi$. Here, one would readily agree that only the periodic terms in $\phi$
contributes to the expression of the recoil for quais-circular orbits, since
the other contributions are functions of eccentricity and hence 0. The
aperiodic terms in $\phi$ arise due to the fact that in this coordinate system
defined by $(a_{r},e_{t}~{}\&~{}\phi)$, along with the mass asymmetry that
gives rise to the emission of LMF in the first place, there is another
asymmetry in the orbital motion. Since the origin of the reference frame is at
one of the foci of the elliptical orbits, the velocity at the pericenter is
not the same as the velocity at the apocenter. As a result, there is a flux of
linear momenta emitted along the preferred (towards y-axis in ref. Fitchett
(1983))direction. This effect is discussed at the Newtonian order in Eq.
(2.23) of ref. Fitchett (1983). In order to avoid these aperiodic terms which
give rise to the diverging terms w.r.t time, we choose to keep all the
expressions in terms of $x,e_{t},u$ and $\phi$. Having discussed the
instanteneous contributions, we now focus on the hereditary contribution at
1.5PN and 2.5PN in the next section.

##  VII Hereditary contribution to the linear momentum flux

Due to non-linearity of the Einstein’s field equations, the time varying
source moments couple to themselves and to the others. This give rise to the
hereditary contributions which depend on the entire history of the system
Blanchet and Damour (1992). The leading order hereditary interaction between
the mass quadrupole moment ($I_{ij}$) and mass monopole (M or the ADM mass)
appears at relative 1.5PN order. In order to estimate the 2.5PN accurate LMF,
we need to calculate the 1.5PN and 2.5PN hereditary contributions to it. The
explicit contributions to the hereditary part from various source type
multipole moments are quoted in Eq. (II).

There are different methods proposed in the literature to compute the
hereditary contributions. The first one is a semi-analytical method in the
frequency domain, proposed in ref. Arun et al. (2008b). It is based on the
Fourier decomposition of Keplerian motion Peters and Mathews (1963b). The
general prescription of this decomposition at arbitrary PN order is discussed
in Arun et al. (2008b). The Fourier decompositions of the multipole moments at
Newtonian order simply read,

$\displaystyle I_{L}(U)$
$\displaystyle=\sum_{p=-\infty}^{\infty}\mathcal{I}_{L}e^{ip\ell},$ (42)
$\displaystyle J_{L-1}(U)$
$\displaystyle=\sum_{p=-\infty}^{\infty}\mathcal{J}_{L-1}e^{ip\ell},$ (43)

with the inverse relation to be

$\displaystyle\mathcal{I}_{L}=\frac{1}{2\pi}\int_{0}^{2\pi}d\ell
I_{L}(U)e^{-ip\ell}$ (44)
$\displaystyle\mathcal{J}_{L-1}=\frac{1}{2\pi}\int_{0}^{2\pi}d\ell
J_{L-1}(U)e^{-ip\ell}.$ (45)

All the Fourier coefficients in Eqs. (44) & (45) can easily be obtained as
combinations of Bessel functions. With the correct normalization factors
depicted in Eqs (5.1a) & (5.1b) of ref. Arun et al. (2008b), these
coefficients are quoted in Appendix A of the same. This procedure can very
well be adopted in order to compute hereditary contributions. However another
method is proposed in ref. Boetzel et al. (2019) where the hereditary
integrals are executed in time domain. We employ this method here in order to
obtain the 2.5PN hereditary contribution to LMF,
$\left({\mathcal{F}_{i}}\right)_{\rm hered}$. The closed form expressions of
the 2.5 PN hereditary contributions [see Eq. (II)], consist of several
integrations on the various combinations of the source moments over time
starting from the remote past to the current retarded time. All the terms are
evaluated following the similar procedure. In this method, we first obtain
every integrals in terms of QK variables ${x,e_{t},l,\lambda}$ and then
perform the time integrations. For example we choose the 1st term in Eq. (II)
and for the reader’s convenience we rewrite it here, in Eq. (46).

$\displaystyle\Big{(}(\mathcal{F}_{i})_{\rm hered}\Big{)}_{1}$
$\displaystyle=\frac{4\,G^{2}\,m}{63\,c^{10}}\,I_{\langle
ijk\rangle}^{(4)}(t)\int_{0}^{\infty}d\tau\left[\ln\left({\tau\over
2\tau_{0}}\right)+{11\over 12}\right]I^{(5)}_{\langle jk\rangle}(t-\tau)$ (46)

To evaluate this integration, we first perform the contraction, $I_{\langle
ijk\rangle}^{(4)}(t)I^{(5)}_{\langle jk\rangle}(t-\tau)$ to rewrite them as a
sum over all their non-zero components explicitly in terms of the QK
variables. As a result, this integral is expressed as a sum over a few
integrals of the standard form

$\displaystyle\int_{0}^{\infty}d\tau e^{i[\alpha
l(t-\tau)+\beta\lambda(t-\tau)]}\left[\ln\left({\tau\over
2\tau_{0}}\right)+{11\over 12}\right],$ (47)

which can be reduced to

$\displaystyle e^{i[\alpha l(t)+\beta\lambda(t)]}\int_{0}^{\infty}d\tau
e^{-i[\alpha l(\tau)+\beta\lambda(\tau)]}\left[\ln\left({\tau\over
2\tau_{0}}\right)+{11\over 12}\right],$ (48)

with the fact that if $\ell(t)=n(t-t_{0})$ at the current time $t$, then at a
retarded time ($t-\tau$), $\ell(t-\tau)$ and $\lambda(t-\tau)$ are simply
$(\ell(t)-n\tau)$ and $\lambda(t)-\lambda(\tau)$ respectively where $n$ is the
mean motion. The above integrals can be solved using the standard formula
given by

$\displaystyle\int_{0}^{\infty}d\tau
e^{i\sigma\tau}\ln\Big{(}\frac{\tau}{2r_{0}}\Big{)}=-\frac{1}{\sigma}\Big{[}\frac{\pi}{2}Sign(\sigma)+i\Big{(}\ln(2|\sigma|r_{0})+\gamma_{E}\Big{)}\Big{]},$
(49)

with $\sigma$ being a combination of $\alpha,\beta$, $\gamma_{E}$ (the Euler
constant) and the function $Sign(\sigma)=\pm 1$. In this whole computation, we
restrict ourselves in the small eccentricity limit and provide the results
accurate up-to $\mathcal{O}(e_{t})$. Furthermore, since the effects of
radiation reaction on the various variables ${x,e_{t},l,\lambda}$ starts
appearing at relative 2.5PN order, hence neglected for the computation of
these integrals. All the other integrals are similarly computed to find the
complete hereditary contributions at 2.5PN. The two non-zero components of the
hereditary contributions to the linear momentum flux in the small eccentricity
limit read as,

$\displaystyle\mathcal{F}_{\rm
hered}=\frac{464}{105}\frac{c^{4}}{G}\sqrt{1-4\eta}\eta^{2}x^{7}\begin{bmatrix}\mathcal{F}_{\cos\phi}^{\rm
hered}&\mathcal{F}_{\sin\phi}^{\rm
hered}\end{bmatrix}\begin{bmatrix}\cos\phi&\sin\phi\\\
\sin\phi&-\cos\phi\end{bmatrix}\begin{bmatrix}{\bf\hat{e}_{x}}\\\
{\bf\hat{e}_{y}}\end{bmatrix},$ (50)

$\displaystyle\mathcal{F}_{\cos\phi}^{\rm hered}=$ $\displaystyle
x^{3/2}\mathcal{F}_{\cos\phi;1.5PN}^{\rm
hered}+x^{5/2}\mathcal{F}_{\cos\phi;2.5PN}^{\rm hered},$ (51)

$\displaystyle\mathcal{F}_{\cos\phi;1.5PN}^{\rm hered}=$
$\displaystyle-2\log{n\over\omega_{0}}-e_{t}\Bigg{(}\cos[u]\Bigg{[}{11277\over
8410}+{95951\over 841}\log 2-{463563\over 6728}\log 3+{523\over
29}\log{n\over\omega_{0}}\Bigg{]}-{39\over
29}\pi\sin[u]\Bigg{)}-e_{t}^{2}\Bigg{(}{48657\over 16820}$
$\displaystyle+{2343\over 58}\log{n\over\omega_{0}}+{945141\over 1682}\log
2-{4582737\over 13456}\log 3-{2767\over
464}\pi\sin[2u]+\cos[2u]\Bigg{[}{126007\over 16820}-{463589\over 1682}\log 2$
$\displaystyle-{388557\over 6728}\log 3+{78125\over 464}\log 5+{2653\over
58}\log{n\over\omega_{0}}\Bigg{]}\Bigg{)},$ (52)
$\displaystyle\mathcal{F}_{\cos\phi;2.5PN}^{\rm hered}=$
$\displaystyle-{196573\over 50460}+{38917\over 25230}\eta-\Big{(}{32835\over
841}-{109740\over 841}\eta\Big{)}\log 2+\Big{(}{77625\over 3364}-{66645\over
841}\eta\Big{)}\log 3+\Big{(}{904\over 87}+{1400\over
261}\eta\Big{)}\log\Big{(}{n\over\omega_{0}}\Big{)}$
$\displaystyle+e_{t}\Bigg{(}\cos u\Bigg{[}-{218401\over 12615}+{783692\over
37845}\eta+\Bigg{(}{1307629\over 2523}-{664381\over 7569}\eta\Bigg{)}\log
2-\Bigg{(}{8470827\over 53824}+{24589467\over 53824}\eta\Bigg{)}\log 3$
$\displaystyle-\Bigg{(}{234375\over 1856}-{703125\over 1856}\eta\Bigg{)}\log
5+\Bigg{(}{5569\over 87}+{18127\over
261}\eta\Bigg{)}\log\Big{(}{n\over\omega_{0}}\Big{)}\Bigg{]}-\Bigg{(}{323099\over
16704}+{25325\over 5568}\eta\Bigg{)}\pi\sin
u\Bigg{)}+e_{t}^{2}\Bigg{(}-{100693\over 1740}$ $\displaystyle+{17524951\over
302760}\eta+\Bigg{[}{39737\over 18}-{4714343\over 841}\eta\Bigg{]}\log
2-\Bigg{[}{1099125\over 3712}-{23785335\over 107648}\eta\Bigg{]}\log
3-\Bigg{[}{26328125\over 33408}-{26328125\over 11136}\eta\Bigg{]}\log 5$
$\displaystyle+\Bigg{[}{533\over 6}+{17341\over
87}\eta\Bigg{]}\log\Big{(}{n\over\omega_{0}}\Big{)}+\cos[2u]\Bigg{[}-{196331\over
25230}+{25418317\over 302760}\eta-\Bigg{(}{20853451\over 15138}-{44982391\over
7569}\eta\Bigg{)}\log 2-\Bigg{(}{75584331\over 107648}$
$\displaystyle+{25176087\over 107648}\eta\Bigg{)}\log 3+\Bigg{(}{1203125\over
1152}-{76484375\over 33408}\eta\Bigg{)}\log 5+\Bigg{(}{5225\over
58}+{6893\over
29}\eta\Bigg{)}\log\Big{(}{n\over\omega_{0}}\Big{)}\Bigg{]}-\pi\sin[2u]\Bigg{[}{921673\over
11136}+{1016891\over 33408}\eta\Bigg{]}\Bigg{)}.$ (53)

$\displaystyle\mathcal{F}_{\sin\phi}^{\rm hered}=$ $\displaystyle
x^{3/2}\mathcal{F}_{\sin\phi;1.5PN}^{\rm
hered}+x^{5/2}\mathcal{F}_{\sin\phi;2.5PN}^{\rm hered},$ (54)

Where the contributions are

$\displaystyle\mathcal{F}_{\sin\phi;1.5PN}^{\rm hered}=$
$\displaystyle{309\over 58}\pi+e_{t}\Bigg{(}{5361\over 116}\pi\cos
u+\Bigg{(}{124698\over 4205}+{95951\over 841}\log 2-{463563\over 6728}\log
3+{407\over 29}\log\Big{(}{n\over\omega_{0}}\Big{)}\Bigg{)}\sin
u\Bigg{)}+e_{t}^{2}\Bigg{(}{23967\over 232}\pi$ $\displaystyle+{51961\over
464}\pi\cos[2u]+\Bigg{[}{219667\over 1682}+{193375\over 1682}\log
2-{1983609\over 6728}\log 3+{78125\over 464}\log 5+{3587\over
58}\log\Big{(}{n\over\omega_{0}}\Big{)}\Bigg{]}\sin[2u]\Bigg{)},$ (55)
$\displaystyle\mathcal{F}_{\sin\phi;2.5PN}^{\rm hered}=$
$\displaystyle-{2663\over 116}\pi-{2185\over
87}\pi\eta+e_{t}\Bigg{(}-\Bigg{[}{628025\over 5568}+{4862903\over
16704}\eta\Bigg{]}\pi\cos u+\Bigg{[}-{5025971\over 25230}-{21195599\over
302760}\eta-\Bigg{(}{1643716\over 2523}$ $\displaystyle-{2639701\over
7569}\eta\Bigg{)}\log 2+\Bigg{(}{13725027\over 53824}+{16058907\over
53824}\eta\Bigg{)}\log 3+\Bigg{(}{234375\over 1856}-{703125\over
1856}\eta\Bigg{)}\log 5-\Bigg{(}{7958\over 87}$ $\displaystyle+{1645\over
29}\eta\Bigg{)}\log\Big{(}{n\over\omega_{0}}\Big{)}\Bigg{]}\sin
u\Bigg{)}+e_{t}^{2}\Bigg{(}-{4328065\over 33408}\pi-{26893253\over
33408}\pi\eta-\Bigg{[}{1738843\over 33408}\pi+{30155227\over
33408}\pi\eta\Bigg{]}\cos[2u]$ $\displaystyle+\Bigg{[}-{16060223\over
25230}-{30465373\over 67280}\eta-\Bigg{(}{9123587\over 7569}+{5419900\over
7569}\eta\Bigg{)}\log 2+\Bigg{(}{162475587\over 107648}-{56210697\over
107648}\eta\Bigg{)}\log 3$ $\displaystyle-\Bigg{(}{14890625\over
33408}-{16484375\over 33408}\eta\Bigg{)}\log 5-\Bigg{(}{8280\over
29}+{29548\over
87}\eta\Bigg{)}\log\Big{(}{n\over\omega_{0}}\Big{)}\Bigg{]}\sin[2u]\Bigg{)},$
(56)

where

$\displaystyle\omega_{0}$
$\displaystyle=\frac{1}{\tau_{0}}\exp\Bigg{[}\frac{5921}{1740}+\frac{48}{29}\log
2-\frac{405}{116}\log 3-\gamma_{E}\Bigg{]}.$

##  VIII Post-adiabatic corrections at 2.5PN order

In the section V we have discussed the conservative quasi-Keplerian
description of the binary motion and in the following two sections, obtain the
instantaneous and the hereditary contributions to the LMF corresponding to the
conservative dynamics where the basic assumption is that the PN parameter x
and the eccentricity parameter $e_{t}$ are the two intrinsic constants of
motion. Apart from these two, there are two more extrinsic constants
$c_{\ell}$ and $c_{\lambda}$ associated to the initial values of the two
parameters $\ell$ and $\lambda$ respectively (see Eq. 29). Now in order to
obtain LMF at 2.5PN we need to incorporate the effect of radiation reaction on
these constants which makes x and $e_{t}$ to be a time varying function and
the equations for the angles $\ell$ and $\lambda$, i.e. Eq. (29e) & (29c)
modifies to

$\displaystyle l(t)=$
$\displaystyle\int_{t_{0}}^{t}n(t^{\prime})dt^{\prime}+c_{l}(t)$ (57)
$\displaystyle\lambda(t)=$
$\displaystyle\int_{t_{0}}^{t}(1+k(t^{\prime}))n(t^{\prime})dt^{\prime}+c_{\lambda}(t)$
(58)

In order to obtain these complete solutions, one uses PN accurate equation of
motion, Eq. (23) with the correct dissipative contributions(see Ref. Boetzel
et al. (2019) for a detailed prescription) which were neglected in case of
obtaining QK representation given in Eq. (29). Hence, every quantity
describing the binary motion have a two scale decomposition: a slow(radiation
reaction time scale) secular drift (will be represented by a bar) and a fast
periodic oscillations (orbital timescale)(will be represented by a tilde)

$\displaystyle x(t)$ $\displaystyle=\bar{x}(t)+\tilde{x}(t),$ (59)
$\displaystyle e_{t}(t)$ $\displaystyle=\bar{e}_{t}(t)+\tilde{e}_{t}(t),$ (60)
$\displaystyle c_{l}(t)$ $\displaystyle=\bar{c}_{l}(t)+\tilde{c}_{l}(t),$ (61)
$\displaystyle c_{\lambda}(t)$
$\displaystyle=\bar{c}_{\lambda}(t)+\tilde{c}_{\lambda}(t).$ (62)

We do not show the detailed computation of these quantities here. We simply
quote the expressions for the periodic contributions,
$\tilde{x},\tilde{e}_{t}\tilde{c}_{l}~{}\&~{}\tilde{c}_{\lambda}$ in terms of
their secular counter parts
$\bar{x},\bar{e}_{t}\bar{c}_{l}~{}\&~{}\bar{c}_{\lambda}$ following ref.
Boetzel et al. (2019).

$\displaystyle\tilde{x}(t)=\;$
$\displaystyle\eta\bar{x}^{7/2}\bar{e_{t}}\bigg{[}80\sin(\bar{l})+\frac{1436}{15}\bar{e_{t}}\sin(2\bar{l})$
$\displaystyle+\bar{e_{t}}^{2}\left(\frac{4538}{15}\sin(\bar{l})+\frac{6022}{45}\sin(3\bar{l})\right)\bigg{]}$
$\displaystyle+\mathcal{O}(\bar{x}^{9/2})\,,$ (63a)
$\displaystyle\tilde{e_{t}}(t)=\;$
$\displaystyle-\eta\bar{x}^{5/2}\bigg{[}\frac{64}{5}\sin(\bar{l})+\frac{352}{15}\bar{e_{t}}\sin(2\bar{l})$
$\displaystyle+\bar{e_{t}}^{2}\left(\frac{1138}{15}\sin(\bar{l})+\frac{358}{9}\sin(3\bar{l})\right)\bigg{]}$
$\displaystyle+\mathcal{O}(\bar{x}^{7/2})\,,$ (63b)
$\displaystyle\tilde{l}(t)=\;$
$\displaystyle-\eta\bar{x}^{5/2}\bigg{[}\frac{64}{5\bar{e_{t}}}\cos(\bar{l})+\frac{352}{15}\cos(2\bar{l})$
$\displaystyle+\bar{e_{t}}\left(\frac{1654}{15}\cos(\bar{l})+\frac{358}{9}\cos(3\bar{l})\right)$
$\displaystyle+\bar{e_{t}}^{2}\left(\frac{694}{15}\cos(2\bar{l})+\frac{1289}{20}\cos(4\bar{l})\right)\bigg{]}$
$\displaystyle+\mathcal{O}(\bar{x}^{7/2})\,,$ (63c)
$\displaystyle\tilde{\lambda}(t)=\;$
$\displaystyle-\eta\bar{x}^{5/2}\bigg{[}\frac{296}{3}\bar{e_{t}}\cos(\bar{l})+\frac{199}{5}\bar{e_{t}}^{2}\cos(2\bar{l})\bigg{]}$
$\displaystyle+\mathcal{O}(\bar{x}^{7/2})\,.$ (63d)

As it is evident from the above relations, that the periodic contributions
$\tilde{x},\tilde{e}_{t}\tilde{c}_{l}~{}\&~{}\tilde{c}_{\lambda}$ starts at
2.5PN and fully oscillatory in nature and thus correctly describe the post
adiabatic corrections to the dynamical variables. We adopt these relations to
obtain the 2.5PN post adiabatic corrections to the components of LMF. We use
the instanteneous contributions to LMF at Newtonian order (in Eqs. (31)),
substitute,

$\displaystyle x$ $\displaystyle\rightarrow\bar{x}+\tilde{x}\,,$ (64a)
$\displaystyle e_{t}$ $\displaystyle\rightarrow\bar{e_{t}}+\tilde{e_{t}}\,,$
(64b) $\displaystyle l$ $\displaystyle\rightarrow\bar{l}+\tilde{l}\,,$ (64c)
$\displaystyle\lambda$
$\displaystyle\rightarrow\bar{\lambda}+\tilde{\lambda}\,,$ (64d)

and replace $\tilde{x},\tilde{e}_{t}\tilde{c}_{l}~{}\&~{}\tilde{c}_{\lambda}$
by their slowly evolving counter parts to obtain the post adiabatic
corrections. We find the post adiabatic corrections at 2.5PN to have the
following closed-form expressions,

$\displaystyle\mathcal{F}^{\rm
PA}=\frac{464}{105}\frac{c^{4}}{G}\sqrt{1-4\eta}\eta^{3}x^{8}\begin{bmatrix}\mathcal{F}_{\cos\phi}^{\rm
PA}&\mathcal{F}_{\sin\phi}^{\rm
PA}\end{bmatrix}\begin{bmatrix}\cos\phi&\sin\phi\\\
\sin\phi&-\cos\phi\end{bmatrix}\begin{bmatrix}{\bf\hat{e}_{x}}\\\
{\bf\hat{e}_{y}}\end{bmatrix},$ (65)

$\displaystyle\mathcal{F}_{\cos\phi}^{\rm PA}=$ $\displaystyle{16\over
435}(-54-507\bar{e_{t}}\cos u+\bar{e_{t}}^{2}[2225+20232\cos[2u]]),$ (66a)
$\displaystyle\mathcal{F}_{\cos\phi}^{\rm PA}=$ $\displaystyle{2\over
1305}\bar{e_{t}}\sin u(315948+620003\bar{e_{t}}\cos u).$ (66b)

To be noted here that this post adiabatic correction applies only to the
Newtonian terms and in rest of the contributions we can safely replace all the
variables simply by their secular counter parts.

##  IX Total LMF and Log absorption

In the previous sections, we have presented a closed form expressions for the
various contributions to the LMF components in terms of
${\bar{x},\bar{e_{t}},\bar{u}~{}\&~{}\bar{\phi}}$. For simplicity, we first
re-express the total LMF components in terms of
$\bar{x},\bar{e_{t}},\bar{l},\bar{\lambda}$ in stead of
${\bar{x},\bar{e_{t}},\bar{u}~{}\&~{}\bar{\phi}}$. For this purpose we use
Eqs. (25a) & (25b) of Ref. Boetzel et al. (2019).

Now, as the readers would agree that the hereditary components have some
dependence on the arbitrary constant $\tau_{0}$. We find that this arbitrary
constant can be reabsorbed by a redefinition of the mean anomaly by,

$\displaystyle\xi$
$\displaystyle=\bar{l}-\frac{3GM}{c^{3}}\bar{n}\ln\Big{(}\frac{\bar{x}}{x_{0}^{\prime}}\Big{)}\,,$
(67)

with $M$ to be the ADM mass and
$x_{0}^{\prime}=\Big{(}\frac{Gm\omega_{0}}{c^{3}}\Big{)}^{2/3}$. This serves
simply as a constant shift to the time coordinate and hence $\xi$ and $l$
follow the same evolution equation, $d\xi/dt=d\bar{l}/dt=\bar{n}$. Similarly,
the phase $\bar{\lambda}$ evaluated at the shifted time has the following
form,

$\displaystyle\bar{\lambda}_{\xi}=$
$\displaystyle\;\bar{\lambda}-\frac{3GM}{c^{3}}(1+\bar{k})\bar{n}\ln\Big{(}\frac{\bar{x}}{x_{0}^{\prime}}\Big{)}\,.$
(68)

With this shift in the coordinate time, one can also redefine the phase
variable as $\psi$. The relation between these variables are given by,

$\displaystyle\bar{l}=$
$\displaystyle\;\xi+3\left(\bar{x}^{3/2}-\bar{x}^{5/2}\left(3+\frac{\eta}{2}\right)\right)\ln\Big{(}\frac{\bar{x}}{x_{0}^{\prime}}\Big{)}\,,$
(69a) $\displaystyle\phi=$
$\displaystyle\;\psi+\bigg{(}\bar{x}^{3/2}\left(3+6\bar{e_{t}}\cos(\xi)\right)$
$\displaystyle+\bar{x}^{5/2}\left(-\frac{3\eta}{2}+6\bar{e_{t}}(2-\eta)\cos(\xi)\right)\bigg{)}\ln\Big{(}\frac{\bar{x}}{x_{0}^{\prime}}\Big{)}$
$\displaystyle-9\bar{x}^{3}\bar{e_{t}}\sin(\xi)\ln^{2}\Big{(}\frac{\bar{x}}{x_{0}^{\prime}}\Big{)}\,.$
(69b)

As a result of this redefinition, Eq. (50) together with the instantaneous
parts in Eqs. (31) and the post adiabatic corrections in Eq. (65) give rise to
the total LMF from the system, given in Eq. ((70)) below,

$\displaystyle\mathcal{F}=\frac{464}{105}\frac{c^{4}}{G}\sqrt{1-4\eta}\eta^{2}x^{11/2}\begin{bmatrix}\mathcal{F}_{\cos[\lambda_{\xi}]}^{\rm
Tot}&\mathcal{F}_{\sin[\lambda_{\xi}]}^{\rm
Tot}\end{bmatrix}\begin{bmatrix}\cos[\lambda_{\xi}]&\sin[\lambda_{\xi}]\\\
\sin[\lambda_{\xi}]&-\cos[\lambda_{\xi}]\end{bmatrix}\begin{bmatrix}{\bf\hat{e}_{x}}\\\
{\bf\hat{e}_{y}}\end{bmatrix}$ (70)

$\displaystyle\mathcal{F}_{\cos[\lambda_{\xi}]}^{\rm Tot}=$
$\displaystyle{125\over 58}e_{t}\sin[\xi]-e_{t}x\Bigg{(}{929\over
348}+{5977\over 1044}\eta\Bigg{)}\sin[\xi]-e_{t}x^{2}\Bigg{(}{389\over
1276}-{337585\over 45936}\eta-{918679\over
137808}\eta^{2}\Bigg{)}\sin[\xi]+e_{t}x^{3/2}\Bigg{(}12\pi\sin[\xi]$
$\displaystyle-\Bigg{[}{11277\over 8410}+{95951\over 841}\log 2-{463563\over
6728}\log 3\Bigg{]}\cos[\xi]\Bigg{)}+x^{5/2}\Bigg{(}{106187\over
50460}-{117774\over 4205}\eta-\Bigg{[}{32835\over 841}-{109740\over
841}\eta\Bigg{]}\log 2$ $\displaystyle+\Bigg{[}{77625\over 3364}-{66645\over
841}\eta\Bigg{]}\log 3+e_{t}\Bigg{[}\Bigg{(}{464114\over 12615}-{937549\over
2523}\eta+\Bigg{[}{1307629\over 2523}-{664381\over 7569}\eta\Bigg{]}\log
2-\Bigg{[}{8470827\over 53824}+{24589467\over 53824}\eta\Bigg{]}\log 3$
$\displaystyle-\Bigg{[}{234375\over 1856}-{703125\over 1856}\eta\Bigg{]}\log
5\Bigg{)}\cos[\xi]-\Bigg{(}{200123\over 16704}+{334669\over
5568}\eta\Bigg{)}\pi\sin[\xi]\Bigg{]}\Bigg{)}$ (71)
$\displaystyle\mathcal{F}_{\sin[\lambda_{\xi}]}^{\rm Tot}=$ $\displaystyle
1+{199\over 29}e_{t}\cos[\xi]-x\Bigg{(}{452\over 87}+{1139\over
522}\eta+e_{t}\Bigg{[}{2653\over 116}+{12785\over
522}\eta\Bigg{]}\cos[\xi]\Bigg{)}+x^{3/2}\Bigg{(}{309\over
58}\pi+e_{t}\Bigg{[}{5361\over 116}\pi\cos[\xi]+\Bigg{[}{124698\over 4205}$
$\displaystyle+{95951\over 841}\log 2-{463563\over 6728}\log
3\Bigg{]}\sin[\xi]\Bigg{]}\Bigg{)}+x^{2}\Bigg{(}-{71345\over
22968}+{36761\over 2088}\eta+{147101\over
68904}\eta^{2}-e_{t}\Bigg{[}{3010489\over 34452}-{1623695\over 11484}\eta$
$\displaystyle-{2793017\over
68904}\eta^{2}\Bigg{]}\cos[\xi]\Bigg{)}+x^{5/2}\Bigg{(}-{2663\over
116}\pi-{2185\over 87}\pi\eta+e_{t}\Bigg{[}-\Bigg{(}{628025\over
5568}+{4682903\over 16704}\eta\Bigg{)}\pi\cos[\xi]+\Bigg{(}-{3097214\over
12615}+{4905709\over 12615}\eta$ $\displaystyle-\Bigg{(}{1446706\over
2523}-{664381\over 7569}\eta\Bigg{)}\log 2+\Bigg{(}{11241027\over
53824}+{24589467\over 53824}\eta\Bigg{)}\log 3+\Bigg{(}{234275\over
1856}-{703125\over 1856}\eta\Bigg{)}\log 5\Bigg{)}\sin[\xi]\Bigg{]}\Bigg{)}$
(72)

To be noted here, although we have computed the total LMF in terms of the
secular counter part of all the variables but we have removed the bar while
quoting the final expression of the same in Eq. (70). From now onward we will
quote all the quantities in terms of the secular variables but represent them
without the bar.

One may wish to re-express the above equation in terms of the new phase
variable $\psi$ and $\xi$ using Eq. (IX) below. As a consistency check, in the
limit $e_{t}\rightarrow 0$, above expression reduces to the estimate of LMF
from a binary moving in a quasi-circular orbit (see Eq. (1) of ref. Blanchet
et al. (2005)) once $\lambda_{\xi}$ is replaced by $\psi$.

$\displaystyle\lambda_{\xi}=$
$\displaystyle\;\psi-\Bigg{[}2e_{t}\sin(\xi)+\frac{5}{4}e_{t}^{2}\sin(2\xi)+\bar{x}\left((10-\eta)e_{t}\sin(\xi)+\left(\frac{31}{4}-\eta\right)e_{t}^{2}\sin(2\xi)\right)+x^{2}\Big{(}\frac{1}{12}\left(624-235\eta+\eta^{2}\right)e_{t}\sin(\xi)$
$\displaystyle+\frac{1}{24}\left(969-326\eta+2\eta^{2}\right)e_{t}^{2}\sin(2\xi)\Big{)}-x^{5/2}\eta\left(\frac{128}{5}+\frac{888}{5}e_{t}\cos(\xi)+\frac{1}{45}e_{t}^{2}(10728+8935\cos(2\xi))\right)\Bigg{]}\,,$
(73)

##  X Accumulated LMF

Having discussed the flux of linear momentum, we now compute the total Linear
momentum loss in the binary evolution. Total loss of linear momentum through
the binary evolution over the inspiralling quasi-elliptic orbits can be
obtained by integrating the LMF,

$\displaystyle\frac{d\mathcal{P}_{i}}{dt}=\mathcal{F}_{i},$ (74)
$\displaystyle\mathcal{P}_{i}=\int_{-\infty}^{t}\mathcal{F}_{i}dt^{\prime}.$
(75)

In order to perform the integration in Eq. (75) we follow the prescription for
the computation of the memory terms provided in Ref. Ebersold et al. (2019)
and extend it to the accuracy needed here. As one would immediately identify
by replacing Eq. (70) in Eq. (75), that every term is of the form

$\displaystyle\int_{-\infty}^{t}dt^{\prime}x^{p}e_{t}^{q}e^{i(s\lambda_{\xi}+r\xi)},$
(76)

where, $p,q,r,s$ are different integers depending on the particular term. From
the definition of $\lambda_{\xi}$, we remind the readers that
$\lambda_{\xi}=(1+k)\xi$ and $\xi=nt$. We quote the final result for
integration in Eq. (76) below,

$\displaystyle-\frac{\mathrm{i}}{n(r+s(1+k))}\Bigg{(}1-\frac{3\mathrm{i}\chi(T_{R})}{8(r+s(1+k))}-\frac{\mathrm{i}\chi(T_{R})}{(r+s(1+k))}\Bigg{[}-\frac{p}{4}+\frac{19q}{48}\Bigg{]}\Bigg{)}x^{p}\,e^{q}\,\mathrm{e}^{\mathrm{i}(s\lambda_{\xi}+r\xi)}.$
(77)

We also give the detailed computation in Appendix A. Furthermore, we observe
that there are two types of terms. In the first case, when $r\neq-s$, we
identify as the fast oscillatory terms, preserving the usual PN notion in
their expansion. On the other hand, when $r=-s$, the terms oscillate on the
periastron precession timescale hence slowly oscillating. As a result they are
enhance by 1PN. We quote our final accumulated LMF expressions in Eq. (78). As
evident from Eq. (78), we observe a relative -1PN and a 0.5PN term appearing
in the expression due to the argument presented above. This effect is also
seen in case of slow oscillatory memory Ebersold et al. (2019). These terms
are not present in case of quasi-circular orbits.

$\displaystyle\mathcal{P}=-\frac{464}{105}\sqrt{1-4\eta}\eta^{2}cmx^{4}\begin{bmatrix}\mathcal{P}_{\cos[\lambda_{\xi}]}&\mathcal{P}_{\sin[\lambda_{\xi}]}\end{bmatrix}\begin{bmatrix}\cos[\lambda_{\xi}]&\sin[\lambda_{\xi}]\\\
\sin[\lambda_{\xi}]&-\cos[\lambda_{\xi}]\end{bmatrix}\begin{bmatrix}{\bf\hat{e}_{x}}\\\
{\bf\hat{e}_{y}}\end{bmatrix}$ (78)

$\displaystyle\mathcal{P}_{\cos[\lambda_{\xi}]}$
$\displaystyle=\frac{91}{116x}e_{t}\cos[\xi]+1-e_{t}\Bigg{(}\frac{1195}{522}+\frac{301}{232}\eta\Bigg{)}\cos[\xi]+\sqrt{x}e_{t}\Bigg{(}\frac{1323}{232}\pi\cos[\xi]+\Bigg{[}\frac{2737}{580}+\frac{364}{45}\eta\Bigg{]}\sin[\xi]\Bigg{)}+x\Bigg{(}-\frac{452}{87}-\frac{1139}{522}\eta$
$\displaystyle+e_{t}\Bigg{[}\frac{506521}{18729}-\frac{396779}{25056}\eta+\frac{2429}{928}\eta^{2}\Bigg{]}\cos[\xi]\Bigg{)}+x^{3/2}\Bigg{(}\frac{309}{58}\pi+e_{t}\Bigg{[}\Bigg{(}-\frac{67663}{6264}-\frac{19729}{783}\eta\Bigg{)}\pi\cos[\xi]+\Bigg{(}-\frac{1721569}{50460}+\frac{40249}{841}\log
2$ $\displaystyle-\frac{174069}{6728}\log
3-\frac{786671}{9396}\eta-\frac{6146}{783}\eta^{2}\Bigg{)}\sin[\xi]\Bigg{]}\Bigg{)}+x^{2}\Bigg{(}-\frac{71345}{22968}+\frac{36761}{2088}\eta+\frac{147101}{68904}\eta+e_{t}\Bigg{[}\frac{13818}{145}\pi\eta\sin[\xi]$
$\displaystyle+\Bigg{(}-\frac{10384631}{30624}+\frac{5356178951}{62013600}\eta+\frac{15582709}{413424}\eta^{2}+\frac{17003}{2784}\eta^{3}\Bigg{)}\cos[\xi]\Bigg{]}\Bigg{)}+x^{5/2}\Bigg{(}-\frac{2663\pi}{116}-\frac{2185\pi}{87}\eta+e_{t}\Bigg{[}\Bigg{(}\frac{10102429\pi}{33408}$
$\displaystyle-\frac{80835007\pi}{300672}\eta-\frac{138103\pi}{2349}\eta^{2}\Bigg{)}\cos[\xi]+\Bigg{(}\frac{5811883}{25230}+\frac{9725212199}{12261780}\eta-\frac{2842859}{15660}\eta^{2}+\frac{603757}{23490}\eta^{3}-\Bigg{(}\frac{1751699}{10092}$
$\displaystyle-\frac{169934}{7569}\eta\Bigg{)}\log
2+\Bigg{(}\frac{2908071}{107648}+\frac{26744067}{107648}\eta\Bigg{)}\log
3+\Bigg{(}\frac{234375}{3712}-\frac{703125}{3712}\eta\Bigg{)}\log
5\Bigg{)}\sin[\xi]\Bigg{]}\Bigg{)},$
$\displaystyle\mathcal{P}_{\sin[\lambda_{\xi}]}$
$\displaystyle=\frac{91e_{t}}{116x}\sin[\xi]-e_{t}\Bigg{(}\frac{7097}{1044}+\frac{301}{232}\eta\Bigg{)}\sin[\xi]-\sqrt{x}e_{t}\Bigg{(}\Bigg{[}\frac{2737}{580}+\frac{364}{45}\eta\Bigg{]}\cos[\xi]-\frac{1323}{232}\pi\sin[\xi]\Bigg{)}$
$\displaystyle+xe_{t}\Bigg{(}\frac{77426}{2349}-\frac{18215}{25056}\eta+\frac{2429}{928}\eta^{2}\Bigg{)}+x^{3/2}e_{t}\Bigg{(}\Bigg{[}\frac{625897}{12615}+\frac{786671}{9396}\eta+\frac{6146}{783}\eta^{2}+\frac{55702}{841}\log
2-\frac{144747}{3364}\log 3\Bigg{]}\cos[\xi]$
$\displaystyle-\Bigg{[}\frac{124997}{3132}+\frac{19729}{783}\eta\Bigg{]}\pi\sin[\xi]\Bigg{)}-x^{2}e_{t}\Bigg{(}\frac{13818}{145}\pi\eta\cos[\xi]+\Bigg{[}\frac{81690155}{275616}-\frac{1564168363}{31006800}\eta-\frac{11651279}{826848}\eta^{2}$
$\displaystyle-\frac{17003}{2784}\eta^{3}\Bigg{]}\sin[\xi]\Bigg{)}-x^{5/2}\Bigg{(}\frac{106187}{50460}+\frac{97522}{4205}\eta-\Bigg{[}\frac{32835}{841}-\frac{109740}{841}\eta\Bigg{]}\log
2+\Bigg{[}\frac{77625}{3364}-\frac{66645}{841}\eta\Bigg{]}\log 3$
$\displaystyle+e_{t}\Bigg{[}\Bigg{(}\frac{35146787}{100920}+\frac{5917848347}{12261780}\eta-\frac{2842859}{15660}\eta^{2}+\frac{603757}{23490}\eta^{3}+\Bigg{[}\frac{2029853}{10092}-\frac{494447}{7569}\eta\Bigg{]}\log
2-\Bigg{[}\frac{5678271}{107648}$
$\displaystyle+\frac{22434867}{107648}\eta\Bigg{]}\log
3-\Bigg{[}\frac{234375}{3712}-\frac{703125}{3712}\eta\Bigg{]}\log
5\Bigg{)}\cos[\xi]-\Bigg{(}\frac{3575993}{11136}-\frac{28032817}{300672}\eta-\frac{138103}{2349}\eta^{2}\Bigg{)}\pi\sin[\xi]\Bigg{]}\Bigg{)}.$

As an algebraic test, in the limit $e_{t}\rightarrow 0$, we recover the
correct circular orbit result presented in Ref. Mishra et al. (2012) by
replacing $\lambda_{\xi}$with $\psi$ where $\psi$ is the orbital phase in case
of compact binaries in quasi-circular orbits.

##  XI Conclusion

In this paper we compute rate of loss of linear momentum in the far-zone of a
nonspinning inspiralling compact binary system in quasi-elliptical orbits. We
use QK representation of the orbital variable at 2.5PN. We quote the linear
momentum flux accurate upto $\mathcal{O}(e_{t})$ at 2.5PN. We also provide a
closed form expression for the accumulated linear momentum over the binary
evolution. Unlike the linear momentum flux, we observe an additional -1PN and
a 0.5PN term appearing in expression for the total linear momentum (see Eq.
(78)). This is a very similar effect as seen in computation of the slow
oscillatory memory presented in ref Ebersold et al. (2019).

In most of the previous literature Mishra et al. (2012); Blanchet et al.
(2005), Eq. (75) is used to compute the recoil of the center of mass of the
binary where the recoil velocity, $V_{i}=-\mathcal{P}_{i}/m$. Although,
recently in ref. Blanchet and Faye (2019), the authors have argued that the
momentum balance equation for the compact binary system will have an
additional contribution from the flux associated to the center of mass
position. Hence identification of $-\mathcal{P}_{i}/m$ to the recoil velocity
may lead to underestimation of the same. We postpone the calculation of the
center of mass flux and the recoil for the future work.

##  XII Acknowledgement

We would like to thank K. G. Arun for priming the author for the topic and for
subsequent fruitful discussions. We also acknowledge Chandra Kant Mishra for
his valuable suggestions, inputs and cross checking some of the initial
computations. Moreover, we thank B. Iyer, G. Faye, M. Favata, C. M. Will for
stimulating discussions. Furthermore, we appreciate all the help from the
computational facilities at the Institute of Mathematical Sciences, India
during the initial phase of this work and the Atlas Computational Cluster team
at the Albert Einstein Institute in Hanover for their beyond excellent
assistance.

##  Appendix A Integrals used to compute the accumulated LMF

We extensively use the method developed for the oscillatory memory integrals
in ref. Ebersold et al. (2019) to compute the accumulated linear momentum over
the binary evolution. However, to achieve the desired accuracy for our
calculation, we extend the procedure to one higher order. We provide the
complete prescription here. For convenience we set $G=c=1$ in this appendix.
We define the standard integral form (Eq. (76)) that has to be computed as

$\displaystyle\mathcal{S}_{\text{osc.}}$
$\displaystyle=\int_{-\infty}^{T_{R}}\mathrm{d}t\,x^{p}(t)\,e_{t}^{q}(t)\,\mathrm{e}^{\mathrm{i}(s\lambda_{\xi}+r\xi)}\,.$
(79)

The eccentric orbit is assumed to evolve only with the secular radiation-
reaction equations Peters and Mathews (1963a); Peters (1964) given by

$\displaystyle\frac{\mathrm{d}x}{\mathrm{d}t}=$
$\displaystyle\frac{c^{3}\eta}{Gm}\frac{x^{5}}{(1-e_{t}^{2})^{7/2}}\left(\frac{64}{5}+\frac{584}{15}e_{t}^{2}+\frac{74}{15}e_{t}^{4}\right)\,,$
(80a) $\displaystyle\frac{\mathrm{d}e_{t}}{\mathrm{d}t}=$
$\displaystyle-\frac{c^{3}\eta}{Gm}\frac{e_{t}\,x^{4}}{(1-e_{t}^{2})^{5/2}}\left(\frac{304}{15}+\frac{121}{15}e_{t}^{2}\right)\,,$
(80b)

with $e_{t}=1$ in the remote past when $x=0$. We ignore all the astrophysical
process such as mass loss during the binary evolution.

We rewrite the evolution Eq. (80a) accurate upto the leading order in $x$ and
keep terms at $\mathcal{O}(e_{t}^{2})$,

$\displaystyle\frac{\mathrm{d}x(t)}{\mathrm{d}t}=\frac{64\eta
x^{5}(t)}{5m}\left[1+\frac{157}{24}e_{t}^{2}(t)\right]\,,$ (81)

and integrate it over a time interval up to some coalescence time $T_{C}$,
where the orbital frequency and therefore $x$ tends to infinity:

$\displaystyle\int_{t}^{T_{C}}\mathrm{d}t=\int_{x(t)}^{\infty}\frac{\mathrm{d}x(t)}{(\mathrm{d}x/\mathrm{d}t)}\,.$
(82)

Thereby, we find an explicit relation between the orbital frequency (related
to $x$) and time $t$:

$\displaystyle
T_{C}-t=\frac{5m}{256\eta}\frac{1}{x^{4}(t)}\left[1-\frac{157}{43}e_{t}^{2}(t)\right]\,.$
(83)

Considering only the leading order, we find

$\displaystyle
e_{t}(t)=e_{t}(T_{R})\left(\frac{x(T_{R})}{x(t)}\right)^{19/12}\,.$ (84)
$\displaystyle
x(t)=\frac{1}{4}\left(\frac{5m}{\eta(T_{C}-t)}\right)^{1/4}\left[1-\frac{157}{172}\,e_{t}^{2}(T_{R})\left(\frac{T_{C}-t}{T_{C}-T_{R}}\right)^{19/24}\right]\,.$
(85)

For our convenience we introduce a new integration variable
$y=\frac{T_{R}-t}{T_{C}-T_{R}}$ and the different time-dependent quantities in
terms of $y$ and their values at the current time $T_{R}$. For $x$ we find

$\displaystyle
x(y)=x(T_{R})(1+y)^{-1/4}\left[1-\frac{157}{172}\,e_{t}^{2}(T_{R})\left((1+y)^{19/24}-1\right)\right]\,,$
(86)

and for the eccentricity we find

$\displaystyle e_{t}(y)=e_{t}(T_{R})\left(1+y\right)^{19/48}\,.$ (87)

Note that in the remote past as $y\rightarrow\infty$, the eccentricity evolves
until it reaches the maximum value of 1. Furthermore, the redefined mean
anomaly $\xi(t)$ in terms of $y$ and its value at the current time is the
following,

$\displaystyle\xi(t)=\xi(T_{C})-\int^{t}_{T_{C}}\mathrm{d}t^{\prime}\,n(t^{\prime})=\xi(T_{C})-\frac{1}{m}\int^{t}_{T_{C}}\mathrm{d}t^{\prime}\,x^{3/2}(t^{\prime})=\xi(T_{C})-\frac{(T_{C}-T_{R})}{m}\int_{-1}^{y}\mathrm{d}y^{\prime}\,x^{3/2}(y^{\prime})\,.$
(88)

Now inserting $x(y)$ given in Eq. (86) we find,

$\displaystyle\xi(t)=\xi(T_{C})-\frac{8(T_{C}-T_{R})x^{3/2}(T_{R})}{5m}(1+y)^{5/8}\left[1-\frac{471}{11696}e_{t}^{2}(T_{R})\left(15(1+y)^{19/24}-34\right)\right]\,,$
(89)

where $\xi(T_{C})$ is the value of $\xi$ at the coalescence. Hence, the mean
anomaly at the current time $T_{R}$ is given by

$\displaystyle\xi(T_{R})=$
$\displaystyle\;\xi(T_{C})-\frac{8(T_{C}-T_{R})x^{3/2}(T_{R})}{5m}\left[1+\frac{8949}{11696}e_{t}^{2}(T_{R})\right]\,.$
(90)

Now using Eqs. (89) & (90) we find $\xi(t)$ in terms of $\xi(T_{R})$ and $y$,

$\displaystyle\xi(t)=\xi(T_{R})-\frac{8(T_{C}-T_{R})x^{3/2}(T_{R})}{5m}\left[(1+y)^{5/8}-1\right]\left[1-\frac{471}{11696}e_{t}^{2}(T_{R})\frac{19-34(1+y)^{5/8}+15(1+y)^{17/12}}{(1+y)^{5/8}-1}\right]\,,$
(91)

where $x(T_{R})$ and $e_{t}(T_{R})$ denote their respective current values of
$x$ and $e_{t}$.

Now we introduce a dimensionless “adiabatic parameter” $\chi(T_{R})$, related
to the the inspiral rate at the current retarded time $T_{R}$ which is defined
as the ratio between the current period and the time difference between the
current time and the coalescence,

$\displaystyle\chi(T_{R})=\frac{1}{n(T_{R})(T_{C}-T_{R})}\,,$ (92)

where $n(T_{R})=x^{3/2}(T_{R})/m$ at leading order. We re-write the the
adiabatic parameter $\chi(T_{R})$ in terms of $x(T_{R})$ and $e(T_{R})$,

$\displaystyle\chi(T_{R})=\frac{256\eta}{5}x^{5/2}(T_{R})\left[1+\frac{157}{43}e_{t}^{2}(T_{R})\right]\,.$
(93)

We express $\xi(t)$ in Eq. (91) in terms of $\chi(T_{R})$,

$\displaystyle\xi(t)=\xi(T_{R})-\frac{8}{5\chi(T_{R})}\left[(1+y)^{5/8}-1\right]\left[1-\frac{471}{11696}e_{t}^{2}(T_{R})\frac{19-34(1+y)^{5/8}+15(1+y)^{17/12}}{(1+y)^{5/8}-1}\right]\,.$
(94)

Now using Eqs. (86), (87) & (94) , we rewrite Eq. 79 as an integral over $y$:

$\displaystyle J_{\textnormal{mem}}=$
$\displaystyle\;(T_{C}-T_{R})\int_{0}^{\infty}\mathrm{d}y\,x^{p}(y)\,e_{t}^{q}(y)\,\mathrm{e}^{\mathrm{i}(s\lambda_{\xi}(y)+r\xi(y))}$
$\displaystyle=$
$\displaystyle\;(T_{C}-T_{R})\,\mathrm{e}^{\mathrm{i}(r+s(1+k))\xi(T_{R})}\int_{0}^{\infty}\mathrm{d}y\,x^{p}(y)\,e_{t}^{q}(y)\exp\biggl{\\{}-\frac{8\mathrm{i}(r+s(1+k))}{5\chi(T_{R})}\left[(1+y)^{5/8}-1\right]$
$\displaystyle\times\left[1-\frac{471}{11696}e_{t}^{2}(T_{R})\frac{19-34(1+y)^{5/8}+15(1+y)^{17/12}}{(1+y)^{5/8}-1}\right]\biggr{\\}}\,.$
(95)

Schematically the above integral can be written as,

$\displaystyle
J_{\textnormal{mem}}\sim\int_{0}^{\infty}\mathrm{d}y\,f(y)\exp\left[\frac{\mathrm{i}}{\chi(T_{R})}g(y)\right]\,.$
(96)

We use the technique, integrate by parts, use result from the following type
of integral:

$\displaystyle\int\mathrm{d}y\,\mathrm{e}^{\mathrm{i}\sigma
g(y)}=-\frac{\mathrm{i}}{\sigma g^{\prime}(y)}\mathrm{e}^{\mathrm{i}\sigma
g(y)}-\frac{g^{\prime\prime}(y)}{[\sigma^{2}g^{\prime}(y)^{3}]}\mathrm{e}^{\mathrm{i}\sigma
g(y)}+\mathcal{O}(g^{\prime}(y)^{-3})\,.$ (97)

This formula is valid as long as $g^{\prime}(y)$ is sufficiently large.
Integrating Eq. (96) by parts we get

$\displaystyle J_{\textnormal{mem}}$ $\displaystyle\sim
f(y)\left[\Bigg{(}-\frac{\mathrm{i}\chi(T_{R})}{g^{\prime}(y)}-\frac{\chi(T_{R})^{2}g^{\prime\prime}(y)}{[g^{\prime}(y)]^{3}}\Bigg{)}\exp\left[\frac{\mathrm{i}}{\chi(T_{R})}g(y)\right]\right]_{0}^{\infty}+\mathrm{i}\chi(T_{R})\int_{0}^{\infty}\mathrm{d}y\,\frac{f^{\prime}(y)}{g^{\prime}(y)}\exp\left[\frac{\mathrm{i}}{\chi(T_{R})}g(y)\right]+\mathcal{O}(\chi(T_{R})^{3})\,.$
(98) $\displaystyle\sim
f(y)\left[\Bigg{(}-\frac{\mathrm{i}\chi(T_{R})}{g^{\prime}(y)}-\frac{\chi(T_{R})^{2}g^{\prime\prime}(y)}{[g^{\prime}(y)]^{3}}\Bigg{)}\exp\left[\frac{\mathrm{i}}{\chi(T_{R})}g(y)\right]\right]_{0}^{\infty}+\mathrm{i}\chi(T_{R})\times\frac{f^{\prime}(y)}{g^{\prime}(y)}\left[-\frac{\mathrm{i}\chi(T_{R})}{g^{\prime}(y)}\exp\left[\frac{\mathrm{i}}{\chi(T_{R})}g(y)\right]\right]_{0}^{\infty}+\mathcal{O}(\chi(T_{R})^{3})$
(99) $\displaystyle\sim
f(y)\left[\Bigg{(}\frac{\mathrm{i}\chi(T_{R})}{g^{\prime}(y)}+\frac{\chi(T_{R})^{2}g^{\prime\prime}(y)}{[g^{\prime}(y)]^{3}}\Bigg{)}\exp\left[\frac{\mathrm{i}}{\chi(T_{R})}g(y)\right]\right]_{y=0}+\mathrm{i}\chi(T_{R})\times\frac{f^{\prime}(y)}{g^{\prime}(y)}\left[\frac{\mathrm{i}\chi(T_{R})}{g^{\prime}(y)}\exp\left[\frac{\mathrm{i}}{\chi(T_{R})}g(y)\right]\right]_{y=0}+\mathcal{O}(\chi(T_{R})^{3})$
(100) $\displaystyle\sim
f(y)\left[\frac{\mathrm{i}\chi(T_{R})}{g^{\prime}(y)}\exp\left[\frac{\mathrm{i}}{\chi(T_{R})}g(y)\right]\Bigg{(}1-\mathrm{i}\chi(T_{R})\frac{g^{\prime\prime}(y)}{g^{\prime}(y)^{2}}+\mathrm{i}\chi(T_{R})\frac{f^{\prime}(y)}{f(y)g^{\prime}(y)}\Bigg{)}\right]_{y=0}$
(101)

Now in our case the different functions are the following,

$\displaystyle f^{\prime}(y)\Bigg{|}_{y=0}$
$\displaystyle=\Bigg{[}px^{p-1}e^{q}\frac{\mathrm{d}x}{dy}+qx^{p}e^{q-1}\frac{\mathrm{d}e}{dy}\Bigg{]}_{y=0}=-\frac{p}{4}x^{p}e_{t}^{q}+\frac{19}{48}qx^{p}e_{t}^{q}$
(102) $\displaystyle g^{\prime}(y)\Bigg{|}_{y=0}$ $\displaystyle=-(r+s(1+k))$
(103) $\displaystyle g^{\prime\prime}(y)\Bigg{|}_{y=0}$
$\displaystyle=\frac{3}{8}(r+s(1+k))$ (104)

As $y$ approaches infinity in the remote past, the quantity
$f(y)=x^{p}(y)e^{q}(y)$ and $f^{\prime}(y)$ both approach zero. since at early
times $x\rightarrow 0$ (as the frequency reaches zero) and maximum achievable
value for the eccentricity can be 1 in our model. While evaluating the terms
at $y=0$, we recover $x$ and $e_{t}$ at the current time and $g(0)=0$. The
derivative $g^{\prime}(y)$ in the denominator evaluated at $y=0$ is
effectively 1 multiplied by some constants. We keep terms at order
$\mathcal{O}(\chi(T_{R})^{2})$ and the higher-order contributions can be
safely ignored. We find the final result for the integral in Eq. (79) as

$\displaystyle\mathcal{S}_{\text{osc.}}$
$\displaystyle=-(T_{c}-T_{R})\,x^{p}\,e^{q}\,\mathrm{e}^{\mathrm{i}(s\lambda_{\xi}+r\xi)}\frac{\mathrm{i}\chi(T_{R})}{(r+s(1+k))}\Bigg{(}1-\frac{3\mathrm{i}\chi(T_{R})}{8(r+s(1+k))}-\frac{\mathrm{i}\chi(T_{R})}{(r+s(1+k))}\Bigg{[}-\frac{p}{4}+\frac{19q}{48}\Bigg{]}\Bigg{)}$
$\displaystyle=-\frac{\mathrm{i}}{n(r+s(1+k))}\Bigg{(}1-\frac{3\mathrm{i}\chi(T_{R})}{8(r+s(1+k))}-\frac{\mathrm{i}\chi(T_{R})}{(r+s(1+k))}\Bigg{[}-\frac{p}{4}+\frac{19q}{48}\Bigg{]}\Bigg{)}x^{p}\,e^{q}\,\mathrm{e}^{\mathrm{i}(s\lambda_{\xi}+r\xi)}\,,$
(105)

This allows us to compute the oscillatory hereditary integrals in Eq. (76).

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