#  Third order post-Newtonian gravitational radiation from two-body
scattering. Instantaneous Energy and Angular momentum radiations

Gihyuk Cho<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  Subhajit Dandapat
<EMAIL_ADDRESS>Department of Astronomy and Astrophysics, Tata
Institute of Fundamental Research, Mumbai 400005, India  Achamveedu Gopakumar
Department of Astronomy and Astrophysics, Tata Institute of Fundamental
Research, Mumbai 400005, India

(August 28, 2024)

###### Abstract

We compute the third post-Newtonian (3PN) accurate instantaneous contributions
to the radiated gravitational wave (GW) energy and angular momentum arising
from the hyperbolic passages of non-spinning compact objects. The present
computations employ 3PN-accurate instantaneous contributions to the far-zone
energy and angular momentum fluxes and the 3PN-accurate Keplerian type
parametric solution for compact binaries in hyperbolic orbits.

##  I introduction

The routine detection of transient GW events that arise from merging black
hole (BH) binaries in bound orbits has inaugurated the era of GW astronomy
Abbott  _et al._ (2019, 2021); Venumadhav  _et al._ (2020). Further,
observations of a neutron star binary coalescence in GWs, and many
electromagnetic frequency windows have provided a peak into the benefits of
the multi-messenger GW astronomy Abbott  _et al._ (2017); Poggiani (2019);
Monitor  _et al._ (2017). In contrast, compact binaries in unbound orbits can
provide transient GW burst events in the LIGO,LISA and IPTA GW frequency
windows García-Bellido and Nesseris (2018); Mukherjee  _et al._ (2020); Kocsis
_et al._ (2006); Burke-Spolaor  _et al._ (2019). Interestingly, GW burst
events due to hyperbolic encounters of neutron stars may even be accompanied
by electromagnetic flares Tsang (2013). Therefore, there are on-going PN
efforts to characterize both the dynamics and associated GW emission aspects
of compact binaries in hyperbolic orbits in general relativity Cho  _et al._
(2018); Bae  _et al._ (2020); Bini  _et al._ (2021a, b); Bini and Geralico
(2021a, b).

The present effort extends the classic computations of the radiated
energy($\Delta{\cal E}$) and angular momentum ($\Delta{\cal J}$) during
hyperbolic encounters of non-spinning compact objects to 1PN order Hansen
(1972); Blanchet and Schäfer (1989); Junker and Schäfer (1992). Recall that PN
approximation allows us to write, for example, the orbital dynamics of non-
spinning compact binaries as corrections to Newtonian equations of motion in
powers of $(v/c)^{2}\sim G\,M/(c^{2}\,r)$, where $v,M,$ and $r$ are the
velocity, total mass and relative separation of the binary Blanchet (2014);
Porto (2016) with the gravitation constant $G$ and speed of light $c$. Note
that the expressions, available in Refs.Blanchet and Schäfer (1989); Junker
and Schäfer (1992), provided the next-to-leading order (1PN) contributions to
$\Delta{\cal E}$ and $\Delta{\cal J}$, influenced by Refs. Ruffini and Wheeler
(1971); Wagoner and Will (1976). The present computation provides 3PN-accurate
‘instantaneous’ contributions to $\Delta{\cal E}$ and $\Delta{\cal J}$ with
the help of Refs. Arun  _et al._ (2008, 2009); Cho  _et al._ (2018) in the
modified Harmonic gauge. It turned out that at PN orders beyond the 1PN, the
radiative moments and the resulting far-zone fluxes have two distinct
contributions Blanchet (2014). One part of the radiative moments and their
fluxes depends only at the usual retarded time and it is customary to refer
these terms as the “instantaneous contributions”. In contrast, the second part
depends on the dynamics of compact binary in its entire past and therefore
these contributions are usually termed as the “hereditary
contributions”Blanchet and Damour (1988, 1992). In this paper, we focus our
efforts on the instantaneous contributions.
The present computations are not a straightforward extensions to 3PN order of
what are done in Refs. Blanchet and Schäfer (1989); Junker and Schäfer (1992).
This is mainly because of logarithmic terms that appear at the 3PN corrections
to far-zone energy and angular momentum fluxes associated with compact
binaries in non-circular orbits Arun  _et al._ (2008, 2009). We provide a
prescription to compute these logarithmic integrals in terms of Clausen
function of order two Lewin (1991).

The manuscript is structured in the following way. The way of our PN-accurate
computations and underlying formalism to obtain $\Delta{\cal E}$ and
$\Delta{\cal J}$ and the results are presented in Sec. II. And also we briefly
present parabolic limit and the implications of the bremsstrahlung limit. The
detail of our computations are presented in the Appendix. A, and our PN-
accurate expressions in terms of energy and angular momentum is given in
Appendix. B while Sec. III provides a brief summary and on-going
investigations.

##  II 3PN accurate Instantaneous Contributions to $\Delta{\cal E}$ and
$\Delta{\cal J}$

By the matching between multipolar post-Minkowskian (MPM) expansion and PN
expansion Blanchet (1998); Poujade and Blanchet (2002), gravitational fluxes
$\cal{F}$ (either energy or angular momentum) can be expressed in terms of
mechanical variables describing binaries such as mass, radial distance and
velocities, of which explicit expression can be found in Eq.(5.2) in Arun  _et
al._ (2008) in standard/modified harmonic gauge. Additionally, once the 3PN
accurate quasi-Keplerian solution i.e. the mechanical variables in time
(implicitly), obtained as in Cho  _et al._ (2018), the fluxes can be fully
written as a function of time $t$. Here, we briefly show the PN structure of
the fluxes upto 3PN($1/c^{6}$) order,

$\displaystyle\mathcal{F}(t)=\mathcal{F}_{\rm{inst}}(t)+\mathcal{F}_{\rm{hered}}(t)\,,$
(1a) where
$\displaystyle\mathcal{F}_{\rm{inst}}=\mathcal{F}_{\text{N}}+\frac{1}{c^{2}}\mathcal{F}_{\text{1PN}}+\frac{1}{c^{4}}\mathcal{F}_{\text{1PN}}+\frac{1}{c^{6}}\mathcal{F}_{\text{1PN}}+\mathcal{O}(1/c^{7})\,,$
(1b)
$\displaystyle\mathcal{F}_{\rm{hered}}=\frac{1}{c^{3}}\,\mathcal{F}_{\text{tail}}+\frac{1}{c^{6}}\,\mathcal{F}_{\text{tail(tail)}}+\frac{1}{c^{6}}\mathcal{F}_{\text{tail}^{2}}+\mathcal{O}(1/c^{7})\,.$
(1c)

By $\mathcal{F}_{\rm{hered}}$ (hereditary contribution), we mean all
contribution that is dependent on the past history of binaries Blanchet and
Damour (1988, 1992). Otherwise, it is called instantaneous contribution
denoted as $\mathcal{F}_{\rm{inst}}$. Thus, total radiations
$\int^{+\infty}_{-\infty}dt\,\mathcal{F}$ also have both instantaneous and
hereditary contributions and the same PN structure. Note that total radiations
are observables, hence gauge invariant but each instantaneous/hereditary
contribution is not gauge invariant because of an ambiguous separation of the
long and short scales leaving coordinate dependence via $r_{0}$ (which will be
seen shortly). The full result should not be dependent on the scale $r_{0}$,
and hence recover gauge invariance. The leading (Newtonian) order of energy
and angular momentum radiations $\Delta\cal{E}$, $\Delta\cal{J}$ (hence
leading order of instantaneous part because hereditary part starts at 1.5PN
order), were computed in Ref. Hansen (1972) for the first time. Its extension
upto 1PN, (hence still instantaneous) was made in Blanchet and Schäfer (1989);
Junker and Schäfer (1992). The higher PN orders including hereditary
contribution have never been treated so far. As one of serial works in the
line of completing 3PN accurate radiations, we compute and complete the
instantaneous contribution first as what follows.
In Sec.II.1, we will explain how the computation goes by an example at leading
order influenced by Ref. Hansen (1972); Blanchet and Schäfer (1989). All
computation is similar to the leading order one except the logarithmic terms
that appear at the 3PN order. The detailed way of tackling these logarithmic
terms is provided in Appendix A. These computations are repeated to obtain
angular momentum radiation in Sec. II.3. Thereafter, we explain why our
instantaneous contributions to the radiated energy and angular momentum during
hyperbolic encounters are exact up to 3PN order and explore their limiting
cases.

###  II.1 Newtonian order $\Delta{\cal E}$ and $\Delta{\cal J}$ Computations

We begin by explaining the procedure of Ref. Hansen (1972); Blanchet and
Schäfer (1989) for computing the radiated energy and angular momentum in GWs
during hyperbolic encounters at the Newtonian order. The natural starting
point is the familiar Newtonian(leading) order far-zone GW energy flux
$\mathcal{F}_{\text{N}}$ for compact binaries in generic orbits Blanchet and
Schäfer (1989)

$\displaystyle\mathcal{F}_{\text{N}}$
$\displaystyle=\frac{8}{15}\frac{G^{3}M^{2}\mu^{2}}{c^{5}r^{4}}\left(12\,v^{2}-11\dot{r}^{2}\right)\,.$
(2)

The approach of Ref. Blanchet and Schäfer (1989) requires us to employ the
standard Keplerian parametric solution for compact binaries in Newtonian
hyperbolic orbits, available in Refs. Damour and Deruelle (1985); Klioner
(2016).This is for expressing the total orbital velocity $v$, the radial
velocity $\dot{r}$ and the radial separation in terms of various elements of
the Keplerian parametric solution for hyperbolic orbits. The underlying
parametric solution for hyperbolic orbits reads

$\displaystyle r$ $\displaystyle=a\,(e\,\cosh u-1)$ (3a)
$\displaystyle\phi-\phi_{0}$
$\displaystyle=2\,\tan^{-1}\left(\sqrt{\frac{e+1}{e-1}}\,\tanh\frac{u}{2}\right)$
(3b) $\displaystyle n\,(t-t_{0})$ $\displaystyle=e\,\sinh u-u\,,$ (3c)

where $r$, $\phi$ and $t$ are radial orbital separation, the angular variable
of the reduced mass $\mu$ around the total mass $M$ and coordinate time,
respectively. Further, the eccentric anomaly parameter $u$ has the range
$-\infty<u<\infty$ while $\phi_{0}$ and $t_{0}$ denote some initial value of
$\phi$ and $t$. The familiar Newtonian semi-major axis $a$, orbital
eccentricity $e$ and the mean motion $n$ are given in terms of the conserved
reduced energy $E=\frac{\mathcal{E}}{\mu}$ and reduced angular momentum
$j=\frac{\cal J}{\mu\,G\,M}$,

$\displaystyle a$ $\displaystyle=\frac{GM}{2E}$ $\displaystyle e^{2}$
$\displaystyle=1+2\,E\,j^{2}$ $\displaystyle n$
$\displaystyle=\frac{(2E)^{3/2}}{GM}\,.$

It is fairly straightforward to express $\dot{r}^{2}$, $v^{2}$ in terms of
$E,j^{2},(e\cosh u-1)$ with the help of

$\displaystyle\dot{r}^{2}$
$\displaystyle=\left(\frac{dr}{du}\bigg{/}\frac{dt}{du}\right)^{2}$ (4a)
$\displaystyle\dot{\phi}^{2}$
$\displaystyle=\left(\frac{d\phi}{du}\bigg{/}\frac{dt}{du}\right)^{2}$ (4b)
$\displaystyle v^{2}$ $\displaystyle=\dot{r}^{2}+r^{2}\,\dot{\phi}^{2}.$ (4c)

This leads to

$\displaystyle\dot{r}^{2}$ $\displaystyle=\bigg{(}1+\frac{2}{(e\cosh
u-1)}-\frac{(2Eh^{2})}{(e\cosh u-1)^{2}}\bigg{)}(2E)\,,$ (5a) $\displaystyle
v^{2}$ $\displaystyle=\left(1+\frac{2}{(e\cosh u-1)}\right)(2E).$ (5b)

Using Eqs. (5), we can express the instantaneous energy flux as a polynomial
in $(1-e\,\cosh u)^{-1}$ and the final expression reads

$\displaystyle\mathcal{F}_{\text{N}}$
$\displaystyle=\frac{1}{n}\,\frac{du}{dt}\sum_{N=3}^{5}\frac{\bar{\alpha}_{N}(E,j)}{(e\cosh
u-1)^{N}}\,,$ (6) $\displaystyle\bar{\alpha}_{N}(e_{t})$
$\displaystyle=\frac{\nu^{2}}{G}\,\left(\frac{E}{c}\right)^{5}\alpha_{N}(E,j)\,.$
(7)

In the above equation $\nu$ is defined as dimensionless mass parameter of the
binary, namely $\nu={m_{1}\,m_{2}}/{(m_{1}+m_{2})^{2}}$, where $m_{1}$ and
$m_{2}$ are the masses of the binary configuration. The coefficients
$\alpha_{N}(E,j)$ at the Newtonian order are given by

$\displaystyle\alpha_{3}$ $\displaystyle=\frac{256}{15}$
$\displaystyle\alpha_{4}$ $\displaystyle=\frac{512}{15}$
$\displaystyle\alpha_{5}$ $\displaystyle=\frac{5632}{15}\,E\,j^{2}.$ (8)

The fact that we have parametrized the far-zone Newtonian energy flux in terms
of Newtonian hyperbolic orbital description allows us to write the total
radiated energy in GWs at the Newtonian order as

$\displaystyle\Delta\mathcal{E}_{\text{N}}$
$\displaystyle=\int_{-\infty}^{\infty}\mathcal{F}_{\text{N}}\,dt\,,$
$\displaystyle=\frac{1}{n}\sum_{N=3}^{5}\bar{\alpha}_{N}(E,j)\,\int_{-\infty}^{\infty}\frac{du}{(e_{t}\cosh
u-1)^{N}}\,.$ (9)

Clearly, we can easily obtain the desired expression for the quadrupolar order
$\Delta\mathcal{E}$ during hyperbolic encounters if we can compute the three
integrals that appear on the right-hand side of Eq. (II.1). With the help of
Refs. Hansen (1972); Blanchet and Schäfer (1989), we find

$\displaystyle\int_{-\infty}^{\infty}\frac{du}{(e\cosh u-x)^{N}}$ (10)
$\displaystyle=\frac{2}{(N-1)!}\bigg{[}\bigg{(}\frac{d}{dx}\bigg{)}^{N-1}\bigg{(}\frac{1}{\sqrt{e^{2}-x^{2}}}\,\arccos\bigg{(}-\frac{x}{e}\bigg{)}\bigg{)}\bigg{]}\,.$

This leads to

$\displaystyle\Delta\mathcal{E}_{\text{N}}$
$\displaystyle=\frac{2M\,\nu^{2}}{15c^{5}\,j^{7}}\,\bigg{[}\sqrt{e^{2}-1}\bigg{(}\frac{602}{3}+\frac{673\,e^{2}}{3}\bigg{)}$
$\displaystyle+\bigg{(}96+292\,e^{2}+37\,e^{4}\bigg{)}\arccos\bigg{(}-\frac{1}{e}\bigg{)}\bigg{]}\,,$
(11a)

where we have used the following Newtonian accurate relation that connects $E$
to $j$ and $e$, namely $E=\frac{e^{2}-1}{2\,j^{2}}$, to obtain the above
result. Indeed, our expression is fully consistent with Ref. Hansen (1972);
Blanchet and Schäfer (1989); Junker and Schäfer (1992).

We now move onto explain briefly how Ref. Junker and Schäfer (1992) computed
the quadrupolar order contributions to the radiated angular momentum during
hyperbolic encounters of non-spinning compact objects. We begin by the
Newtonian order angular momentum flux $\mathcal{G}_{\text{N}}$ for compact
binaries in generic orbits Blanchet and Schäfer (1989):

$\displaystyle\mathbfcal{G}_{\text{N}}$
$\displaystyle=\frac{8}{5}\,\mathbf{\tilde{L}_{N}}\,\frac{G^{2}M\mu^{2}}{c^{5}\,r^{3}}\left\\{2v^{2}-3\dot{r}^{2}+2\,\frac{GM}{r}\right\\}$
(12)

where $\mathbf{\tilde{L}_{N}}=\mathbf{r\times v}$ stands for the scaled
Newtonian angular momentum vector. The fact that the the orbital angular
momentum vector remains a constant as we consider only non-spinning compact
binaries allows to us to compute an expression for the angular momentum flux
$d{\cal J}/dt$ from the above equation. Thereafter, we pursue the steps
involved in the $\Delta{\cal E}$ computations and this leads to

$\displaystyle\mathcal{G}_{\text{N}}$
$\displaystyle=\frac{1}{n}\frac{du}{dt}\sum_{N=2}^{4}\frac{\bar{A}_{N}(E,j)}{(e\cosh
u-1)^{N}}\,,$ (13) $\displaystyle\bar{A}_{N}(e_{t})$
$\displaystyle=\frac{M\,j\,E^{4}\,\nu^{2}}{c^{5}}\,A_{N}(E,j)\,.$ (14)

The three constant coefficients are given by

$\displaystyle A_{2}$ $\displaystyle=-\frac{128}{5}\,,$ $\displaystyle A_{3}$
$\displaystyle=0\,,$ $\displaystyle A_{4}$
$\displaystyle=\frac{768}{5}\,E\,j^{2}\,.$ (15)

The radiated angular momentum $\Delta\mathcal{J}_{\text{N}}$ at the Newtonian
order becomes

$\displaystyle\Delta\mathcal{J}_{\text{N}}$
$\displaystyle=\int_{-\infty}^{\infty}\mathcal{G}_{\text{N}}\,dt\,,$
$\displaystyle=\frac{1}{n}\sum_{N=2}^{4}\bar{A}_{N}(E,j)\,\int_{-\infty}^{\infty}\frac{du}{(e_{t}\cosh
u-1)^{N}}.$ (16)

Clearly, these integrals are similar to those we tackled earlier and this
eventually leads to

$\displaystyle\Delta\mathcal{J}_{\text{N}}$
$\displaystyle=\frac{8\,GM^{2}\nu^{2}}{5\,c^{5}\,j^{4}}\,\,\bigg{\\{}(13+2\,e^{2})\,\sqrt{e^{2}-1}$
$\displaystyle+(8+7\,e^{2})\arccos\left(-\frac{1}{e}\right)\bigg{\\}}.$ (17)

We have verified that the above expression is fully consistent with Ref.
Hansen (1972); Blanchet and Schäfer (1989); Junker and Schäfer (1992). We now
move on to extend these calculations to 3PN order while focusing on the
instantaneous contributions.

###  II.2  3PN-accurate instantaneous contributions to the radiated energy

It should be obvious that we require two crucial ingredients for our 3PN-
accurate $\Delta{\cal E}$ computation. The first ingredient is the 3PN
accurate ‘instantaneous’ contributions to the far-zone GW energy flux from
non-spinning compact binaries in non-circular orbits Blanchet (2014). These
instantaneous contributions depend only on the state of the binary at the
usual retarded time and they appear usually at the Newtonian, 1PN, 2PN, 2.5PN
and 3PN orders. In contrast, the hereditary contributions, as the name
suggests, are sensitive to the binary dynamics at all epochs prior to the
usual retarded time and appear at 1.5PN (relative) order for the first time
Blanchet (2014). In this paper, we focus our efforts on the 3PN-accurate
‘instantaneous’ contributions to the far-zone fluxes, as given by Eqs. (5.2)
in Ref. Arun  _et al._ (2008) in the Modified harmonic (MH) coordinates. A
close inspection of these contributions reveal that $\Delta{\cal E}$
computation at 3PN order will be demanding due to the presence of certain
‘logarithmic’ terms, as evident from Eq.(5.2e) of Ref. Arun  _et al._ (2008).
The second ingredient for the present computation is the 3PN-accurate
generalized quasi-Keplerian parametric solution for compact binaries in
hyperbolic orbits, derived in Ref. Cho  _et al._ (2018).

Note that Ref. Cho  _et al._ (2018) provided a parametric way to track 3PN-
accurate conservative trajectory of compact binaries in hyperbolic orbits.
This effort extended the 1PN-accurate derivation of Keplerian type parametric
solution for hyperbolic motion that employed the arguments of analytic
continuation Damour and Deruelle (1985). At the 3PN order, the radial motion
$r(t)$ is conveniently parametrized as

$\displaystyle r=a_{r}(e_{r}\cosh u-1)$ (18a) $\displaystyle
n(t-t_{0})=e_{t}\sinh
u-u+\left(\frac{f_{4t}}{c^{4}}+\frac{f_{6t}}{c^{6}}\right)V$
$\displaystyle+\left(\frac{g_{4t}}{c^{4}}+\frac{g_{6t}}{c^{6}}\right)\sin
V+\frac{h_{6t}}{c^{6}}\sin 2V+\frac{i_{6t}}{c^{6}}\sin 3V,$ (18b)

where $u$ is the eccentric anomaly while $a_{r},\ e_{r},\ e_{t},\ n$ and
$t_{0}$ are certain PN-accurate semi-major axis, radial eccentricity, time
eccentricity, mean motion, and initial epoch, respectively. In addition, we
have several orbital functions like $g_{4t},g_{6t},f_{4t},f_{6t},i_{6t}$ and
$h_{6t}$ that appear at 2PN and 3PN orders. Further, the angular motion is
described by

$\displaystyle\phi-\phi_{0}=(1+k)\,\bigg{\\{}V+\left(\frac{f_{4\phi}}{c^{4}}+\frac{f_{6\phi}}{c^{6}}\right)\sin
2V$
$\displaystyle+\left(\frac{g_{4\phi}}{c^{4}}+\frac{g_{6\phi}}{c^{6}}\right)\sin
3V+\frac{h_{6\phi}}{c^{6}}\sin 4V+\frac{i_{6t}}{c^{6}}\sin 5V\bigg{\\}}$ (19a)
where $\displaystyle
V=2\,\tan^{-1}\left(\sqrt{\frac{e_{\phi}+1}{e_{\phi}-1}}\,\tanh\frac{u}{2}\right)\,.$
(19b)

In the above expressions, $k$ and $e_{\phi}$ denote PN-accurate rate of
periastron advance, and certain angular eccentricity, respectively.
Additionally, we have several orbital functions like
$f_{4\phi},\leavevmode\nobreak\ f_{6\phi},\leavevmode\nobreak\
g_{4\phi},\leavevmode\nobreak\ g_{6\phi},\leavevmode\nobreak\ i_{6\phi}$ and
$h_{6\phi}$ that appear at 2PN and 3PN orders. We note that Ref. Cho  _et al._
(2018) provided 3PN-accurate expressions for various orbital elements and
functions in terms of $E$ and $j$.

It is straightforward but tedious to compute 3PN-accurate expressions for
$\dot{r}^{2},\ v^{2}$ in terms of $E,\ j^{2},\ (e_{t}\cosh u-1)$. These
dynamical variables that appear in the far-zone energy flux is computed by
employing the following relations

$\displaystyle\frac{dt}{du}$ $\displaystyle=\frac{\partial t}{\partial
u}+\frac{\partial t}{\partial v}\,\frac{dv}{du}\,,$ (20a)
$\displaystyle\dot{r}^{2}$
$\displaystyle=\left(\frac{dr}{du}\bigg{/}\frac{dt}{du}\right)^{2}\,,$ (20b)
$\displaystyle\dot{\phi}^{2}$
$\displaystyle=\left(\frac{d\phi}{dv}\,\frac{dv}{du}\bigg{/}\frac{dt}{du}\right)^{2}\,,$
(20c) $\displaystyle v^{2}$ $\displaystyle=\dot{r}^{2}+r^{2}\,\dot{\phi}^{2}.$
(20d)

In what follows, we display 1PN-accurate parametric expressions for
$\dot{r}^{2}$, $v^{2}$ and $G\,M/r$ for introducing the reader to the
structure of these expressions

$\displaystyle\dot{r^{2}}_{\text{MH}}$
$\displaystyle=\bigg{\\{}1+\frac{2}{(e_{t}\cosh
u-1)}-\frac{(2Ej^{2})}{(e_{t}\cosh
u-1)^{2}}\bigg{\\}}(2E)+\frac{(2E)^{2}}{c^{2}}\bigg{\\{}-\frac{3}{4}+\frac{9\nu}{4}+\frac{1}{(e_{t}\cosh
u-1)^{2}}\bigg{[}-2+(2Ej^{2})$
$\displaystyle\bigg{(}-\frac{7}{2}-\frac{\nu}{2}\bigg{)}+2\nu\bigg{]}+\frac{1}{(e_{t}\cosh
u-1)}\bigg{[}-\frac{3}{2}+\frac{9\nu}{2}\bigg{]}\bigg{\\}}\,,$ (21a)
$\displaystyle v^{2}_{\text{MH}}$
$\displaystyle=\left\\{1+\frac{2}{(e_{t}\cosh
u-1)}\right\\}(2E)+\frac{(2E)^{2}}{c^{2}}\bigg{\\{}-\frac{3}{4}+\frac{9\nu}{4}+\frac{(2Ej^{2})\nu}{(e_{t}\cosh
u-1)^{3}}+\frac{2(\nu-1)}{(e_{t}\cosh u-1)^{2}}\,,$
$\displaystyle+\frac{1}{(e_{t}\cosh
u-1)}\bigg{[}-\frac{3}{2}+\frac{9\nu}{2}\bigg{]}\bigg{\\}}\,,$ (21b)

$\displaystyle\frac{G\,M}{r}$ $\displaystyle=\frac{2\,E}{(e_{t}\,\cosh
u-1)}+\frac{(2E)^{2}}{c^{2}}\,\bigg{\\{}\frac{1}{(1-e_{t}\,\cosh
u)}\left[\frac{9}{4}-\frac{5\,\nu}{4}\right]+\frac{1}{(1-e_{t}\,\cosh
u)^{2}}\,\bigg{[}4-\frac{3}{2}\,\nu\bigg{]}\bigg{\\}}.$ (21c)

The explicit 3PN-accurate expressions for these dynamical variables are
provided at https://github.com/subhajittifr/hyperbolic_flux.

We are now in a position to replace dynamical variables $r,\dot{r}$ and
$\dot{\phi}$ that appear in the 3PN-accurate instantaneous far-zone energy
flux expression, given by Eqs. (5.2) of Ref. Arun  _et al._ (2008), with the
3PN-accurate version of the above equations. The associated 3PN-accurate
expression for the radiated energy during hyperbolic encounters reads

$\displaystyle\Delta\mathcal{E}=\int^{+\infty}_{-\infty}dt\,\mathcal{F}=\int^{+\infty}_{-\infty}du\,\left(\frac{dt}{du}\right)\mathcal{F}.$
(22)

The use of 3PN-accurate expressions for $r,\dot{r}$ and $\dot{\phi}$ in terms
of $E,h,$ and $(1-e_{t}\,\cosh u)$, as evident from our Eqs. (21), in the Eqs.
(5.2) of Ref. Arun  _et al._ (2008) leads to

$\displaystyle\mathcal{F}$
$\displaystyle=\frac{1}{n}\,\frac{du}{dt}\,\bigg{\\{}\sum_{N=3}^{11}\frac{\bar{\alpha}_{N}(E,j)}{(e_{t}\cosh
u-1)^{N}}$ $\displaystyle+\sum_{N=5}^{9}\bar{\beta}_{N}(E,j)\frac{\sinh
u}{(e_{t}\cosh u-1)^{N}}$
$\displaystyle+\sum_{N=5}^{9}\bar{\gamma}_{N}(E,j)\,\frac{\ln(e_{t}\cosh
u-1)}{(e_{t}\cosh u-1)^{N}}\bigg{\\}},$ (23)

where we may write in a compact manner the above constant coefficients as

$\displaystyle\bar{W}_{N}(E,j)=\frac{\nu^{2}}{G}\left(\frac{E}{c}\right)^{5}\,W_{N}(E,j)\,,$

where $W_{N}$ stands for $\bar{\alpha},\bar{\beta},\bar{\gamma}$. We do not
list all these lengthy coefficients in the manuscript and they are provided in
an ancillary Mathematica file at
(https://github.com/subhajittifr/hyperbolic_flux). However, we list below one
of them to show the typical structure of these coefficients

$\displaystyle\alpha_{3}(E,j)$
$\displaystyle=\frac{256}{15}+\frac{E}{c^{2}}\bigg{(}\frac{13184}{105}-\frac{128\,\nu}{21}\bigg{)}+\frac{E^{2}}{c^{4}}\bigg{(}\frac{51328}{63}$
$\displaystyle+\frac{5504\,\nu}{315}+\frac{640\,\nu^{2}}{63}\bigg{)}+\frac{E^{3}}{c^{6}}\bigg{(}\frac{5451968}{1155}+\frac{1803328\nu}{3465}$
$\displaystyle+\frac{181792\,\nu^{2}}{3465}-\frac{8000\,\nu^{3}}{693}\bigg{)}$
(24)

We note that all $\bar{\beta}_{N}$ terms appear at the 2.5PN order while
$\bar{\gamma}_{N}$ terms are accompanied by the logarithmic terms at the 3PN
order. Additionally, we have incorporated the dependence on the constant $\log
r_{0}$ into the coefficients $\bar{\alpha}_{N}$. Recall that $r_{0}$ is the
gauge dependent length scale appearing in the definition of source multiple
moments Blanchet and Iyer (2005) as discussed in Ref. Arun  _et al._ (2008).
We have indeed verified that the 1PN-accurate version of these coefficients
match with Ref. Blanchet and Schäfer (1989).

We now move to tackle how these coefficients contribute to the instantaneous
3PN-accurate expression for the radiated energy during hyperbolic encounters.
It is fairly straightforward to infer that all the 2.5PN terms, namely
$\bar{\beta}_{N}$ terms, do not contribute to $\Delta{\cal E}$ expression and
this is because

$\displaystyle\int_{-\infty}^{\infty}\frac{\sinh u}{(e_{t}\cosh
u-1)^{N}}\,du=0\,.$ (25)

This is obviously due to the fact that these integrands are an odd function in
$u$. The integration of $\bar{\alpha}_{N}$ terms should be straightforward as
they are essentially similar to the terms that we confronted at the Newtonian
order. Therefore, we employ Eq. (10) to compute $\Delta{\cal E}$ contributions
that arise from nine $\bar{\alpha}_{N}$ terms in Eq. (II.2).

Clearly, it is very tricky to tackle the logarithmic terms and we pursued an
entirely new line of investigations compared to what was done in Ref. Arun
_et al._ (2008) for the eccentric orbits. After some detailed efforts, we were
able to obtain analytic expressions for these integrals though the final
expressions were too lengthy to list here. However, we were eventually able to
obtain the most simplified form of these integrals with the help of Clausen
identity as detailed in the Appendix A. This allowed us to tackle the
$\bar{\gamma}_{N}$ contributions to the 3PN-accurate $\Delta{\cal E}$
expression as

$\displaystyle\int_{-\infty}^{\infty}\frac{\ln(e\cosh u-x)}{e\cosh
u-x}\,du=\frac{2}{\sqrt{e^{2}-x^{2}}}$
$\displaystyle\times\bigg{[}\text{Cl}_{2}\bigg{(}2\arccos\bigg{(}-\frac{x}{e}\bigg{)}\bigg{)}+\arccos\bigg{(}-\frac{x}{e}\bigg{)}$
$\displaystyle\times\ln\bigg{(}\frac{2(e^{2}-x^{2})}{e}\bigg{)}\bigg{]},$ (26)

where $\text{Cl}_{2}(x)$ is the Clausen function of order $2$ given by the
integral

$\displaystyle\text{Cl}_{2}(x)=-\int_{0}^{x}dy\,\ln\left|2\sin\frac{y}{2}\right|\,.$
(27)

Interestingly, the Clausen function admits the following Fourier series
representation:

$\displaystyle\text{Cl}_{2}(x)=\sum_{k=1}^{\infty}\frac{\sin kx}{k^{2}}\,.$
(28)

The crucial integrals that are associated with the ${\bar{\gamma}_{N}}$
coefficients in Eq. (II.2) can now tackled by noting that the integrals

$\int_{-\infty}^{\infty}\frac{\ln(e\cosh u-1)}{(e\cosh u-1)^{N}}\,du\,,$

can be computed after taking successive derivatives of Eq. (II.2) with
respective to $x$ and then finally taking $x\to 1$.

With these inputs and the following 3PN accurate expression of $E$ in terms of
$e_{t}$ and $j$ Cho  _et al._ (2018),

$\displaystyle
E=\frac{e_{t}^{2}-1}{2\,j^{2}}+\frac{1}{c^{2}\,j^{4}}\bigg{[}-\frac{9}{8}+e_{t}^{2}\bigg{(}\frac{13}{4}-\frac{3\,\nu}{4}\bigg{)}+e_{t}^{4}\bigg{(}-\frac{17}{8}+\frac{7\,\nu}{8}\bigg{)}-\frac{\,\nu}{8}\bigg{]}+\frac{1}{c^{4}\,j^{6}}\bigg{[}-\frac{81}{16}+\frac{7\,\nu}{16}-\frac{\,\nu^{2}}{16}+e_{t}^{4}\bigg{(}-\frac{339}{16}$
$\displaystyle+\frac{213\,\nu}{16}-\frac{35\,\nu^{2}}{16}\bigg{)}+e_{t}^{2}\bigg{(}\frac{243}{16}-\frac{29\,\nu}{16}+\frac{3\,\nu^{2}}{16}\bigg{)}+e_{t}^{6}\bigg{(}\frac{177}{16}-\frac{191\,\nu}{16}+\frac{33\,\nu^{2}}{16}\bigg{)}\bigg{]}+\frac{1}{c^{6}\,j^{8}}\bigg{[}-\frac{3861}{128}+\bigg{(}\frac{8833}{384}$
$\displaystyle-\frac{41\pi^{2}}{64}\bigg{)}\,\nu+\frac{5\,\nu^{2}}{64}-\frac{5\,\nu^{3}}{128}+e_{t}^{6}\bigg{(}\frac{4641}{32}-\frac{19037\,\nu}{96}+\frac{1119\,\nu^{2}}{16}-\frac{231\,\nu^{3}}{32}\bigg{)}+e_{t}^{2}\bigg{(}\frac{2105}{32}+\bigg{(}\frac{157553}{3360}-\frac{41\pi^{2}}{32}\bigg{)}\,\nu$
$\displaystyle-\frac{105\,\nu^{2}}{16}+\frac{\,\nu^{3}}{32}\bigg{)}+e_{t}^{4}\bigg{(}-\frac{7479}{64}+\bigg{(}\frac{16123}{2240}+\frac{123\pi^{2}}{64}\bigg{)}\,\nu-\frac{313\,\nu^{2}}{32}+\frac{105\,\nu^{3}}{64}\bigg{)}+e_{t}^{8}\bigg{(}-\frac{8165}{128}+\frac{15515\,\nu}{128}$
$\displaystyle-\frac{3435\,\nu^{2}}{64}+\frac{715\,\nu^{3}}{128}\bigg{)}\bigg{]}\,,$
(29)

we finally obtain the 3PN-accurate instantaneous contributions to $\Delta{\cal
E}$ in terms of $e_{t}$ and $j$,

$\displaystyle\Delta\mathcal{E}$
$\displaystyle=\frac{2}{15}\,\frac{M\,\nu^{2}}{j^{7}\,c^{5}}\Big{(}\mathcal{I}_{N}^{\text{MH}}+\frac{1}{c^{2}\,j^{2}}\,\mathcal{I}_{\text{1PN}}^{\text{MH}}+\frac{1}{c^{4}\,j^{4}}\,\mathcal{I}_{\text{2PN}}^{\text{MH}}+\frac{1}{c^{6}\,j^{6}}\,\mathcal{I}_{\text{3PN}}^{\text{MH}}\Big{)}\,.$
(30)

where contributions to $\Delta{\cal E}$ that appear at the Newtonian, 1PN, 2PN
and 3PN orders read

$\displaystyle\mathcal{I}_{N}^{\text{MH}}$
$\displaystyle=\sqrt{e_{t}^{2}-1}\,\bigg{[}\,\frac{602}{3}+\frac{673\,e_{t}^{2}}{3}\bigg{]}+\arccos\left(-\frac{1}{e_{t}}\right)\,\bigg{[}96+292\,e_{t}^{2}+37\,e_{t}^{4}\bigg{]}\,,$
(31a) $\displaystyle\mathcal{I}_{\text{1PN}}^{\text{MH}}$
$\displaystyle=\sqrt{e_{t}^{2}-1}\,\Bigg{[}\frac{153263}{70}-\frac{1547\,\nu}{3}+e_{t}^{2}\,\bigg{(}\frac{271849}{70}-\frac{13799\,\nu}{6}\bigg{)}+e_{t}^{4}\,\bigg{(}-\frac{288513}{280}-2\,\nu\Bigg{)}\Bigg{]}$
(31b)
$\displaystyle+\arccos\left(-\frac{1}{e_{t}}\right)\,\Bigg{[}\frac{6578}{7}-168\,\nu+e_{t}^{2}\,\bigg{(}\frac{31013}{7}-1982\,\nu\bigg{)}+e_{t}^{4}\,\bigg{(}-\frac{223}{4}-\frac{1483\,\nu}{2}\bigg{)}+e_{t}^{6}\,\bigg{(}-\frac{15219}{56}+74\,\nu\bigg{)}\Bigg{]}\,,$
$\displaystyle\mathcal{I}_{\text{2PN}}^{\text{MH}}$
$\displaystyle=\sqrt{e_{t}^{2}-1}\,\Bigg{[}\frac{405300022}{19845}-\frac{2947852\,\nu}{315}+\frac{1173\,\nu^{2}}{4}+e_{t}^{2}\,\bigg{(}\frac{6673495637}{158760}-\frac{114248429\,\nu}{2520}+\frac{66217\,\nu^{2}}{8}\bigg{)}$
$\displaystyle+e_{t}^{4}\,\bigg{(}-\frac{3823800817}{158760}+\frac{41499527\,\nu}{5040}+\frac{3619\,\nu^{2}}{2}\bigg{)}+e_{t}^{6}\,\bigg{(}\frac{39802111}{7840}-\frac{67328\,\nu}{35}-103\,\nu^{2}\bigg{)}\Bigg{]}$
$\displaystyle+\arccos\left(-\frac{1}{e_{t}}\right)\Bigg{[}\frac{1636769}{189}-\frac{74435\,\nu}{21}+48\,\nu^{2}+e_{t}^{2}\bigg{(}\frac{8638156}{189}-\frac{762901\,\nu}{21}+\frac{9851\,\nu^{2}}{2}\bigg{)}$
$\displaystyle+e_{t}^{4}\,\bigg{(}-\frac{554104}{63}-\frac{350943\,\nu}{28}+\frac{48063\,\nu^{2}}{8}\bigg{)}+e_{t}^{6}\,\bigg{(}-\frac{1324649}{336}+\frac{611613\,\nu}{112}-\frac{1779\,\nu^{2}}{2}\bigg{)}$
$\displaystyle+e_{t}^{8}\,\bigg{(}\frac{1224929}{672}-\frac{10070\,\nu}{7}+185\,\nu^{2}\bigg{)}\Bigg{]}\,,$
(31c) $\displaystyle\mathcal{I}_{\text{3PN}}^{\text{MH}}$
$\displaystyle=\sqrt{e_{t}^{2}-1}\,\Bigg{[}\frac{6713608}{1575}+\frac{17868572\,e_{t}^{2}}{525}+\frac{19300553\,e_{t}^{4}}{525}+\frac{17525209\,e_{t}^{6}}{3150}\Bigg{]}\,\log\left[\frac{2\,r_{0}\,(e_{t}^{2}-1)}{G\,M\,j^{2}\,e_{t}}\right]-\Bigg{[}\frac{54784}{35}+\frac{465664\,e_{t}^{2}}{21}$
$\displaystyle+\frac{4426376\,e_{t}^{4}}{105}+\frac{1498856\,e_{t}^{6}}{105}+\frac{31779\,e_{t}^{8}}{70}\Bigg{]}\,\Bigg{\\{}\log\left[\frac{2\,G\,M\,\,j^{2}}{e_{t}\,r_{0}}\right]\,\arccos\left(-\frac{1}{e_{t}}\right)+\text{Cl}_{2}\left[2\,\arccos\left(-\frac{1}{e_{t}}\right)\right]\Bigg{\\}}$
$\displaystyle+\sqrt{e_{t}^{2}-1}\,\Bigg{[}\frac{1959816183329}{8731800}+\Bigg{(}-\frac{69960810223}{317520}+\frac{11632643\,\pi^{2}}{3360}\Bigg{)}\,\nu+\frac{9109459\,\nu^{2}}{560}+\frac{689\,\nu^{3}}{24}$
$\displaystyle+e_{t}^{2}\Bigg{(}\frac{11238026145523}{17463600}+\Bigg{(}-\frac{264708911281}{317520}+\frac{998227\,\pi^{2}}{140}\Bigg{)}\,\nu+\frac{518878433\,\nu^{2}}{2520}-\frac{641483\,\nu^{3}}{48}\Bigg{)}$
$\displaystyle+e_{t}^{4}\,\Bigg{(}-\frac{10459843311391}{139708800}+\Bigg{(}\frac{28976695225}{254016}+\frac{99671\,\pi^{2}}{896}\Bigg{)}\,\nu+\frac{143029027\,\nu^{2}}{2880}-16908\,\nu^{3}\Bigg{)}+e_{t}^{6}\,\Bigg{(}\frac{135148514527}{739200}$
$\displaystyle+\Bigg{(}-\frac{15436846447}{211680}-\frac{12303239\,\pi^{2}}{13440}\Bigg{)}\,\nu+\frac{12729151\,\nu^{2}}{5040}+\frac{8501\,\nu^{3}}{3}\Bigg{)}+e_{t}^{8}\Bigg{(}-\frac{112472361473}{4139520}$
$\displaystyle+\frac{244676087\,\nu}{8820}-\frac{432849\,\nu^{2}}{140}-\frac{1147\,\nu^{3}}{3}\Bigg{)}\Bigg{]}+\arccos\left(-\frac{1}{e_{t}}\right)\Bigg{[}\,\frac{20510192533}{207900}+\Bigg{(}-\frac{69631105}{756}+1599\pi^{2}\Bigg{)}\,\nu$
$\displaystyle+\frac{145195\,\nu^{2}}{28}+9\,\nu^{3}+e_{t}^{2}\,\Bigg{(}\frac{269134761733}{415800}+\Bigg{(}-\frac{736708039}{1080}+\frac{55637\,\pi^{2}}{8}\Bigg{)}\,\nu+\frac{21177007\,\nu^{2}}{168}-\frac{22283\,\nu^{3}}{4}\Bigg{)}$
$\displaystyle+e_{t}^{4}\,\Bigg{(}\frac{125593677691}{554400}+\Bigg{(}-\frac{2952969469}{10080}+\frac{172405\,\pi^{2}}{64}\Bigg{)}\,\nu+\frac{112045205\,\nu^{2}}{672}-\frac{381255\,\nu^{3}}{16}\Bigg{)}+e_{t}^{6}\,\Bigg{(}\frac{115347955537}{1108800}$
$\displaystyle+\Bigg{(}\frac{7251474763}{60480}-\frac{174619\,\pi^{2}}{128}\Bigg{)}\,\nu-\frac{2843351\,\nu^{2}}{64}+3155\,\nu^{3}\Bigg{)}+e_{t}^{8}\,\Bigg{(}\frac{22565112667}{492800}+\Bigg{(}-\frac{37470739}{672}-\frac{12177\,\pi^{2}}{128}\Bigg{)}\,\nu$
$\displaystyle+\frac{2686573\,\nu^{2}}{112}-2075\,\nu^{3}\Bigg{)}+e_{t}^{10}\,\Bigg{(}-\frac{484326439}{39424}+\frac{502175\,\nu}{28}-\frac{25061\,\nu^{2}}{4}+518\,\nu^{3}\Bigg{)}\Bigg{]}\,.$
(31d)

We have found the full agreement with Eq.(1.4) of Ref. Herrmann  _et al._
(2021) upto 3PN($v^{7}$) order in bremsstrahlung limit. Note that since the
choice of the relation between $e_{t}$, $E$ and $j$ (Eq. II.2) was made in
favor of convenience of computation under MH gauge condition Cho  _et al._
(2018), hence the apparent expression of Eq. (31) (and also Eq. (35)) is not
gauge invariant. We have verified that the above expression is fully
consistent with Eqs. (C9) of Ref. Bini  _et al._ (2021a) up-to 2PN order which
required us to express $e_{t}$ and $j$ in terms of $e_{r}$ and $a_{r}$ to 2PN
order in the MH gauge.

We note that it is customary to characterize hyperbolic encounters with the
help of an impact parameter $b$ and an eccentricity parameter as noted in Ref.
Junker and Schäfer (1992). Therefore, we provide 3PN accurate relation that
connects $j$ to $b$ and $e_{t}$ with the help of Ref. Cho  _et al._ (2018).

$\displaystyle j^{2}$
$\displaystyle=\frac{b}{G\,M}\,\sqrt{e_{t}^{2}-1}\,\bigg{\\{}1+\bigg{[}1+\frac{\,\nu}{6}+e_{t}^{2}\bigg{(}-4+\frac{17\,\nu}{6}\bigg{)}\bigg{]}\rho+\bigg{[}-\frac{17}{4}-\frac{61\,\nu}{24}-\frac{\,\nu^{2}}{8}+e_{t}^{4}\bigg{(}-\frac{49}{4}+\frac{307\,\nu}{24}-\frac{95\,\nu^{2}}{24}\bigg{)}$
$\displaystyle+e_{t}^{2}\bigg{(}8+\frac{23\,\nu}{4}-\frac{5\,\nu^{2}}{12}\bigg{)}\bigg{]}\rho^{2}+\bigg{[}-\frac{23}{4}+\bigg{(}\frac{1375}{48}-\frac{41\pi^{2}}{64}\bigg{)}\,\nu-\frac{2213\,\nu^{2}}{72}+\frac{\,\nu^{3}}{12}+e_{t}^{2}\bigg{(}-\frac{125}{2}+\bigg{(}\frac{141199}{1680}$
$\displaystyle-\frac{123\pi^{2}}{64}\bigg{)}\,\nu+\frac{607\,\nu^{2}}{24}+\frac{5\,\nu^{3}}{4}\bigg{)}+e_{t}^{4}\bigg{(}\frac{333}{4}-\frac{557\,\nu}{16}-\frac{595\,\nu^{2}}{24}+\frac{95\,\nu^{3}}{12}\bigg{)}+e_{t}^{6}\bigg{(}-\frac{271}{3}+\frac{7339\,\nu}{48}-\frac{6391\,\nu^{2}}{72}$
$\displaystyle+\frac{217\,\nu^{3}}{12}\bigg{)}\bigg{]}\rho^{3}\bigg{\\}}\,,$
(32)

where $\rho=\frac{1}{\sqrt{e_{t}^{2}-1}}\,\frac{G\,M}{b\,c^{2}}\,.$ We now
move on the briefly list our approach to compute PN-accurate the radiated
angular momentum during hyperbolic encounters that extends the 1PN-accurate
effort of Ref. Junker and Schäfer (1992); Bini  _et al._ (2021a).

###  II.3 3PN-accurate instantaneous contributions to the radiated angular
momentum

The crucial input that is required for our $\Delta{\cal J}$ computation is the
3PN-accurate instantaneous contributions to the far-zone GW angular momentum
flux from compact binaries in non-circular orbits, given by Eqs. (3.4) in Ref.
Arun  _et al._ (2009) and therefore in the MH gauge. The dynamical variables
that appear in these PN-contributions that includes $|\mathbf{\tilde{L}_{N}}|$
are expressed in terms of $E,j$ and $(e_{t}\,\cosh u-1)$ to 3PN order with the
help of Ref. Cho  _et al._ (2018). The resulting 3PN extension of Eq. (13)
that provides 3PN-accurate instantaneous contributions to the scalar far-zone
GW angular momentum flux may be written as

$\displaystyle\Delta\mathcal{J}$
$\displaystyle=\frac{1}{n}\,\frac{du}{dt}\,\bigg{\\{}\sum_{N=2}^{10}\frac{\bar{A}_{N}(E,j)}{(e_{t}\cosh
u-1)^{N}}$ $\displaystyle+\sum_{N=4}^{8}\bar{B}_{N}(E,j)\,\frac{\sinh
u}{(e_{t}\cosh u-1)^{N}}$
$\displaystyle+\sum_{N=4}^{8}\bar{\Gamma}_{N}(E,j)\frac{\ln(e_{t}\cosh
u-1)}{(e_{t}\cosh u-1)^{N}}\bigg{\\}}\,,$ (33)

where

$\displaystyle\bar{Y}_{N}(E,j)=\frac{\nu^{2}\,E^{4}\,M\,j}{c^{5}}\,Y_{N}(E,j)\,,$

where $Y$ stands for $A,B,\Gamma$. It is obvious that we can pursue the
similar arguments, detailed in Sec. II.2, for computing
$\int^{+\infty}_{-\infty}\,dt\,\mathcal{G}(j,e_{t},u)$. This leads to the
following 3PN-accurate instantaneous contributions to the radiated angular
momentum during hyperbolic encounters of non-spinning compact objects

$\displaystyle\Delta\mathcal{J}$
$\displaystyle=\frac{8}{5\,c^{5}}\,\frac{GM^{2}\,\nu^{2}}{j^{4}}\,\Big{(}\mathcal{H}_{N}^{\text{MH}}+\frac{1}{c^{2}\,j^{2}}\,\mathcal{H}_{\text{1PN}}^{\text{MH}}+\frac{1}{c^{4}\,j^{4}}\,\mathcal{H}_{\text{2PN}}^{\text{MH}}+\frac{1}{c^{6}\,j^{6}}\,\mathcal{H}_{\text{3PN}}^{\text{MH}}\Big{)},$
(34)

where the individual contributions that appear at Newtonian, 1PN, 2PN and 3PN
orders are given by

$\displaystyle\mathcal{H}^{\text{MH}}_{\text{N}}$
$\displaystyle=\sqrt{e_{t}^{2}-1}\,\Bigg{(}13+2e_{t}^{2}\Bigg{)}+\arccos\left(-\frac{1}{e_{t}}\right)\,\Bigg{(}8+7e_{t}^{2}\Bigg{)}\,,$
(35a) $\displaystyle\mathcal{H}^{\text{MH}}_{\text{1PN}}$
$\displaystyle=\sqrt{e_{t}^{2}-1}\Bigg{[}\frac{14759}{168}+e_{t}^{2}\,\Bigg{(}\frac{11153}{336}-\frac{1975\,\nu}{36}\Bigg{)}-\frac{847\,\nu}{18}+e_{t}^{4}\,\Bigg{(}-\frac{62}{7}+4\,\nu\Bigg{)}\Bigg{]}+\arccos\left(-\frac{1}{e_{t}}\right)\Bigg{[}\frac{1777}{42}$
(35b)
$\displaystyle+e_{t}^{2}\,\Bigg{(}\frac{3649}{42}-\frac{241\,\nu}{3}\Bigg{)}+e_{t}^{4}\,\Bigg{(}-\frac{5713}{336}+\frac{5\,\nu}{12}\Bigg{)}-18\,\nu\Bigg{]}\,,$
$\displaystyle\mathcal{H}^{\text{MH}}_{\text{2PN}}$
$\displaystyle=\sqrt{e_{t}^{2}-1}\Bigg{[}\frac{38409857}{68040}-\frac{3668237\,\nu}{7560}+\frac{785\,\nu^{2}}{18}+e_{t}^{2}\,\Bigg{(}\frac{2630029}{17010}-\frac{13151567\,\nu}{15120}+\frac{4423\,\nu^{2}}{12}\Bigg{)}+e_{t}^{4}\,\Bigg{(}-\frac{853417}{3360}$
$\displaystyle+\frac{1691057\,\nu}{5040}-\frac{617\,\nu^{2}}{18}\Bigg{)}+e_{t}^{6}\,\Bigg{(}\frac{2480}{63}-\frac{755\,\nu}{14}+10\,\nu^{2}\Bigg{)}+\Bigg{]}+\arccos\left(-\frac{1}{e_{t}}\right)\,\Bigg{[}\frac{326917}{1134}-\frac{27031\,\nu}{126}+10\,\nu^{2}$
$\displaystyle+e_{t}^{2}\,\Bigg{(}\frac{414821}{756}-\frac{29055\,\nu}{28}+298\,\nu^{2}\Bigg{)}+e_{t}^{4}\,\Bigg{(}-\frac{1154087}{3024}+\frac{72917\,\nu}{336}+\frac{941\,\nu^{2}}{12}\Bigg{)}+e_{t}^{6}\,\Bigg{(}\frac{99103}{2016}-\frac{12847\,\nu}{336}+\frac{3\,\nu^{2}}{2}\Bigg{)}\Bigg{]}\,,$
$\displaystyle\mathcal{H}^{\text{MH}}_{\text{3PN}}$
$\displaystyle=\sqrt{e_{t}^{2}-1}\,\Bigg{[}\frac{99724}{315}+\frac{351067\,e_{t}^{2}}{315}+\frac{210683\,e_{t}^{4}}{630}\Bigg{]}\,\log\left[\frac{2\,r_{0}\,(e_{t}^{2}-1)}{G\,M\,j^{2}\,\,e_{t}}\right]-\Bigg{[}\frac{13696}{105}+\frac{98012\,e_{t}^{2}}{105}+\frac{23326\,e_{t}^{4}}{35}+\frac{2461\,e_{t}^{6}}{70}\Bigg{]}$
$\displaystyle\times\,\Bigg{\\{}\log\left[\frac{2\,G\,M\,j^{2}}{r_{0}\,e_{t}}\right]\,\arccos\left(-\frac{1}{\,e_{t}}\right)+\text{Cl}_{2}\left[2\,\arccos\left(-\frac{1}{\,e_{t}}\right)\right]\Bigg{\\}}+\sqrt{\,e_{t}^{2}-1}\,\Bigg{[}\frac{55475721271}{8382528}+\Bigg{(}-\frac{17854035221}{1905120}$
$\displaystyle+\frac{313363\,\pi^{2}}{1920}\Bigg{)}\,\nu+\frac{636197\,\nu^{2}}{540}-\frac{103\,\nu^{3}}{18}+e_{t}^{2}\,\Bigg{(}\frac{550589812147}{83825280}+\Bigg{(}-\frac{49917375859}{3810240}+\frac{541733\pi^{2}}{3840}\Bigg{)}\,\nu$
$\displaystyle+\frac{12681271\,\nu^{2}}{1890}-\frac{33883\,\nu^{3}}{36}\Bigg{)}+e_{t}^{4}\,\Bigg{(}-\frac{276385167053}{335301120}+\Bigg{(}\frac{11543781001}{1905120}-\frac{22427\pi^{2}}{1920}\Bigg{)}\,\nu-\frac{116779321\,\nu^{2}}{60480}$
$\displaystyle-\frac{21775\,\nu^{3}}{144}\Bigg{)}+e_{t}^{6}\,\Bigg{(}\frac{12794620753}{8279040}-\frac{197812189\,\nu}{70560}+\frac{9014755\,\nu^{2}}{8064}-\frac{23497\,\nu^{3}}{288}\Bigg{)}+\,e_{t}^{8}\Bigg{(}-\frac{13784}{77}+\frac{2999\,\nu}{6}$
$\displaystyle-254\,\nu^{2}+28\,\nu^{3}\Bigg{)}\Bigg{]}+\arccos\left(-\frac{1}{\,e_{t}}\right)\Bigg{[}\frac{4577461991}{1247400}+\Bigg{(}-\frac{21428779}{4536}+\frac{369\pi^{2}}{4}\Bigg{)}\,\nu+\frac{7853\,\nu^{2}}{18}$
$\displaystyle+e_{t}^{2}\,\Bigg{(}\frac{13811878057}{1247400}+\Bigg{(}-\frac{86352541}{5670}+\frac{5781\pi^{2}}{32}\Bigg{)}\,\nu+\frac{646651\,\nu^{2}}{126}-488\,\nu^{3}\Bigg{)}+\,e_{t}^{4}\,\Bigg{(}\frac{602403517}{831600}+\Bigg{(}\frac{150579449}{60480}$
$\displaystyle+\frac{6273\pi^{2}}{256}\Bigg{)}\,\nu+\frac{21055\,\nu^{2}}{16}-\frac{4289\,\nu^{3}}{6}\Bigg{)}+e_{t}^{6}\,\Bigg{(}\frac{2135052803}{1108800}+\Bigg{(}-\frac{1094353}{672}-\frac{615\pi^{2}}{128}\Bigg{)}\,\nu+\frac{8513\,\nu^{2}}{168}+\frac{565\,\nu^{3}}{12}\Bigg{)}$
$\displaystyle+e_{t}^{8}\,\Bigg{(}-\frac{94124017}{709632}+\frac{757831\,\nu}{2016}-\frac{306977\,\nu^{2}}{2688}+\frac{129\,\nu^{3}}{32}\Bigg{)}\Bigg{]}.$
(35c)

Note that the first line contributions in
$\mathcal{H}^{\text{MH}}_{\text{3PN}}$ are due to the log terms in the far-
zone angular momentum flux. We have verified that our expressions are
consistent with Eqs. (E6) of Ref. Bini  _et al._ (2021a) at the 2PN order. We
now explain why our instantaneous results can be treated to be exact up to 3PN
order.

###  II.4 On the exact nature of our 3PN results

We note that two crucial inputs are required to compute 3PN-accurate
expressions for the instantaneous contributions to $\Delta{\cal E}$ and
$\Delta{\cal J}$. The first input is the 3PN-accurate generalized quasi-
Keplerian parametric solution for compact binaries in hyperbolic orbits Cho
_et al._ (2018). This solution, presented in the modified harmonic(MH) gauge,
provided analytic expressions for the angular and radial dynamical variables
of the 3PN accurate conservative dynamics of compact binaries in hyperbolic
orbits. With the help of Ref. Cho  _et al._ (2018), we write schematically
analytic expressions for these dynamical variables as

$\displaystyle\begin{split}r&=\mathcal{R}(j,\,e_{t},\,u)\,,\\\\[2.15277pt]
\phi&=\mathcal{P}(j,\,e_{t},\,u)\,,\\\\[2.15277pt]
\dot{r}&=\mathcal{S}(j,\,e_{t},\,u)\,,\\\\[2.15277pt]
\dot{\phi}&=\mathcal{Q}(j,\,e_{t},\,u)\,,\end{split}$ (36)

where $r$ and $\dot{r}$ stand for the radial orbital separation and its time
derivative. Further, $\phi$ denotes the angular variable of the reduced mass
$\mu$ around the total mass $M$ while $\dot{\phi}$ is the time derivative of
the above orbital phase. For the present discussion, we employed certain time
eccentricity $e_{t}$ and the reduced angular momentum as orbital parameters to
characterize the hyperbolic orbit. The temporal evolution arises via the
eccentric anomaly $u$ and it is related to coordinate time $t$ via the PN-
accurate Kepler equation that we symbolically write as

$\displaystyle t$ $\displaystyle=\mathcal{T}(j,\,e_{t},\,u)\,.$ (37)

Let us emphasize that above parametric solution incorporates only the
conservative temporal evolution of orbital variables to the 3PN order.

The second crucial ingredient for our present computation is the 3PN-accurate
instantaneous contributions to the energy and angular momentum fluxes, given
by Eqs. (5.2) of Ref. Arun  _et al._ (2008) and Eqs. (3.4) of Ref. Arun  _et
al._ (2009), in the MH gauge. For the present discussion, we write these
fluxes Schematically as

$\displaystyle\begin{split}\mathcal{F}&=\mathcal{F}(r\,,\dot{r}\,,\dot{\phi})\,,\\\\[4.30554pt]
\mathcal{G}&=\mathcal{G}(r\,,\dot{r}\,,\dot{\phi})\,.\end{split}$ (38)

Following Refs. Blanchet and Schäfer (1989); Junker and Schäfer (1992), we
estimate the radiated energy and angular momentum during hyperbolic encounters
by integrating the above fluxes from $t=-\infty$ to $t=\infty$. The analytic
treatment of these integrals require us to express the dynamical variables
that appear in Eqs. (36) and Eqs. (37) with the help of PN-accurate Keplerian
parametric solution of Ref. Cho  _et al._ (2018). This leads to

$\displaystyle\Delta\mathcal{E}:=\int^{+\infty}_{-\infty}\,dt\,\mathcal{F}(j,e_{t},u)\,,$
(39a)
$\displaystyle\Delta\mathcal{J}:=\int^{+\infty}_{-\infty}\,dt\,\mathcal{G}(j,e_{t},u)\,.$
(39b)

The above approach is appropriate as it is customary to write these fluxes as
$\mathcal{F}=-\frac{d\mathcal{E}}{dt}$ and
$\mathcal{G}=-\frac{d\mathcal{J}}{dt}$. However, the far-zone energy and
angular momentum fluxes and the time derivatives of orbital energy and angular
momentum are related to each other modulo certain total time derivatives that
appear at the 2.5PN order Iyer and Will (1993, 1995); Gopakumar  _et al._
(1997).This is why the above equalities hold in an orbital averaged sense in
the case of bound elliptical orbits Blanchet and Schäfer (1989).

When we pursue the computations of $\Delta{\cal E}$ and $\Delta{\cal J}$ to
3PN order, there are certain subtleties that we need to address. This is
related to the fact that both $h$ and $e_{t}$ vary with time due to the
gravitational radiation reaction effects that appear at the 2.5PN order. This
implies that the temporal evolution in the above integrands occur not only
through $u$ but also through $j$ and $e_{t}$. However, the perturbative nature
of GW emission allows us to write

$\displaystyle\begin{split}e_{t}(t)&=e_{t0}+\frac{\delta
e_{t}(t)}{c^{5}}+\mathcal{O}(c^{-7})\,,\\\\[4.30554pt]
j(t)&=j_{0}+\frac{\delta j(t)}{c^{5}}+\mathcal{O}(c^{-7})\,,\end{split}$ (40)

where $e_{t0}$ and $j_{0}$ are the values of time eccentricity and the scaled
angular momentum at periastron (defined by $u=0$) and hence constants.
Therefore, we could ignore temporal evolution in $e_{t}$ and $j$ while
computing $\Delta{\cal E}$ and $\Delta{\cal J}$ expressions upto 2PN order.
Further, the resulting expressions involve only the constant scaled orbital
angular momentum and time-eccentricity along with the two mass parameters
$m_{1}$ and $m_{2}$. However, a close look of Eqs. (40) and its implications
for the integrands of Eqs. (39) reveal that the dissipative corrections
$\delta e_{t}$, $\delta j$ are required if we plan to obtain 3PN extension of
Refs. Blanchet and Schäfer (1989); Junker and Schäfer (1992).

This mainly arises due to the structure of the relative acceleration at 2.5PN
order which may be written as $\ddot{\bf x}={\bf
a}_{2.5PN}(r,\dot{r},\dot{\phi},{\bf r},{\bf v})$ and this ensures both
$d{\cal E}/dt\neq 0;d{\cal J}/dt\neq 0$ at the 2.5PN order and hence contain
terms of ${\cal O}(1/c^{5})$. Therefore, we may try to parametrise the orbital
dynamics at 2.5PN in the following manner

$\displaystyle\begin{split}r&=\mathcal{R}_{0}+\frac{1}{c^{5}}\partial\mathcal{R}_{0}\cdot\delta\,,\\\\[2.15277pt]
t&=\mathcal{T}_{0}+\frac{1}{c^{5}}\partial\mathcal{T}_{0}\cdot\delta+\frac{1}{c^{5}}\,C_{t}(t)\,,\\\\[2.15277pt]
\phi&=\mathcal{P}_{0}+\frac{1}{c^{5}}\partial\mathcal{P}_{0}\cdot\delta+\frac{1}{c^{5}}\,C_{\phi}(t)\,,\\\\[2.15277pt]
\dot{r}&=\mathcal{S}_{0}+\frac{1}{c^{5}}\partial\mathcal{S}_{0}\cdot\delta\,,\\\\[2.15277pt]
\dot{\phi}&=\mathcal{Q}_{0}+\frac{1}{c^{5}}\partial\mathcal{Q}_{0}\cdot\delta\,,\end{split}$
(41)

where we used a few short hand notation such that

$\displaystyle\mathcal{R}_{0}$ $\displaystyle=\mathcal{R}(j_{0},e_{0t},u)\,,$
$\displaystyle\partial\mathcal{R}_{0}$
$\displaystyle=\\{\frac{\partial}{\partial
j}\mathcal{R}(j_{0},e_{0t},u),\frac{\partial}{\partial
e_{t}}\mathcal{R}(j_{0},e_{0t},u)\\}\,,$ $\displaystyle\delta$
$\displaystyle=\\{\delta j,\delta e_{t}\\}\,,$

and similar notational conventions apply for the other dynamical variables.
The above expressions are influenced by the improved ‘method of variation of
constants’, detailed in Ref. Damour  _et al._ (2004) that provided a way to
include the effects of quadrupolar GW emission on the 2PN-accurate Keplerian
type parametric solution for eccentric compact binaries. Further, the two new
variables $C_{t}$ and $C_{\phi}$ that appear at the 2.5PN order are influenced
by the $c_{l}$ and $c_{m}$ variables of Ref. Damour  _et al._ (2004).It is not
difficult to argue that the temporal evolution of these new variables should
follow

$\displaystyle\frac{dC_{t}}{dt}$
$\displaystyle=\delta\cdot\Big{(}\frac{\partial_{u}\partial\mathcal{R}_{0}}{\partial_{u}\mathcal{R}_{0}}-\frac{\partial_{u}\partial\mathcal{T}_{0}}{\partial_{u}\mathcal{T}_{0}}-\frac{\partial\mathcal{S}_{0}\,\partial_{u}\mathcal{T}_{0}}{\partial_{u}\mathcal{R}_{0}}\Big{)}$
$\displaystyle+\frac{d\delta}{dt}\cdot\Big{(}-\partial\mathcal{T}_{0}+\frac{\partial\mathcal{R}_{0}\,\partial_{u}\mathcal{T}_{0}}{\partial_{u}\mathcal{R}_{0}}\Big{)}\,,$
$\displaystyle\frac{dC_{\phi}}{dt}$
$\displaystyle=\frac{d\delta}{dt}\cdot(-\partial\mathcal{P}_{0}+\frac{\partial\mathcal{R}_{0}\,\partial_{u}\mathcal{P}_{0}}{\partial_{u}\mathcal{R}_{0}})$
$\displaystyle+\delta\cdot$
$\displaystyle\Big{(}\partial\mathcal{Q}_{0}-\frac{\partial\mathcal{S}_{0}\,\partial_{u}\mathcal{P}_{0}}{\partial_{u}\mathcal{R}_{0}}-\frac{\partial_{u}\partial\mathcal{P}_{0}}{\partial_{u}\mathcal{T}_{0}}+\frac{\partial_{u}\mathcal{P}_{0}\partial_{u}\partial\mathcal{R}_{0}}{\partial_{u}\mathcal{R}_{0}\partial_{u}\mathcal{T}_{0}}\Big{)}\,.$

The fact that ${\bf a}_{\rm 2.5PN}$ does not explicitly depend on $\phi$ and
$t$ ensure that $C_{t}$ and $C_{\phi}$ can not contribute to the variations in
${\cal E}$ and ${\cal J}$ at 2.5PN order. In other words, we may write the
total time derivative of the conserved energy at 2.5PN to be

$\displaystyle\frac{d\mathcal{E}}{dt}=\dot{\mathcal{E}}_{0}+\delta\cdot\partial\dot{\mathcal{E}}_{0}\,,$
(42)

where $\dot{\mathcal{E}}_{0}$ stands for 2.5PN accurate energy flux, evaluated
at the periastron $e_{t}=e_{t0}$ and $j=j_{0}$. It is easily seen that
$\delta\cdot\partial\dot{\mathcal{E}}_{0}$ represents radiation reaction
correction to leading order radiation, which is induced by the deflection.
Note that $\dot{\mathcal{E}}_{0}$ is even function in time, because the choice
of $z$ axis (perpendicular to the orbital plane) in the opposite way should be
formally equivalent to time reverse operation $t\leftrightarrow-t$, which
cannot make any difference in the result and in other words, there is no
radiation reaction in $\dot{\mathcal{E}}_{0}$). Further,
$\partial\dot{\mathcal{E}}_{0}$ is also even in that they are just partial
derivatives of $\dot{\mathcal{E}}_{0}$ with respect to the $e_{t},\,j$, so
they can not affect to its time dependency structure. On the other hand,
$\delta(t)$ comes from time integration of $\dot{\mathcal{E}}_{0}$ and
therefore it should be odd in time. We may now conclude that the reaction
correction $\delta\cdot\partial\dot{\mathcal{E}}_{0}$ is also odd function.
Hence, when it comes to total radiation, which involves the integration from
$t=-\infty$ to $t=+\infty$, $\delta\cdot\partial\dot{\mathcal{E}}_{0}$ does
not contribute to $\Delta\mathcal{E}$ (likewise to $\Delta\mathcal{J}$) at
all.

These arguments ensure that we can express the various PN contributions to the
$\Delta\mathcal{E}$ and $\Delta\mathcal{J}$ integrands in terms of variables
that are associated with the Keplerian type solution to the PN-accurate
conservative dynamics of hyperbolic encounters. In other words, we are fully
justified to ignore Newtonian order GW emission induced variations in
$\Delta\mathcal{E}$ and $\Delta\mathcal{J}$ while characterizing the orbital
dynamics during the 3PN order $\Delta\mathcal{E}$ and $\Delta\mathcal{J}$
computations. Therefore, we employ $e_{t}$ and $j$ to characterize PN-accurate
hyperbolic orbits and can treat them as constant parameters during computing
3PN-accurate expressions for the radiated energy and angular momentum. This is
why we have not considered the radiation reaction while we computed the 3PN
accurate instantaneous radiation, and hence this argument fully completes our
computation.

###  II.5 Parabolic limit

Here, we list the exact values of the parabolic limit $e_{t}=e=1$, or $E=0$ of
the total radiations.

$\displaystyle\Delta\mathcal{E}(E=0)=\frac{M\,\nu^{2}}{15}\Bigg{[}\frac{850\pi}{j^{7}}+\frac{7\pi(5763-3220\,\nu)}{4j^{9}}$
$\displaystyle+\frac{\pi(29198255+378\,\nu(-86017+18270\,\nu))}{336\,j^{11}}$
(43)
$\displaystyle+\frac{\pi}{80640\,j^{13}}\Big{(}179020439969+440\nu\big{(}-361091813$
$\displaystyle+3587787\pi^{2}+945(105219-10780\,\nu)\,\nu\big{)}$
$\displaystyle-13003119360\log\left(2\,\frac{G\,M}{r_{0}}\,j^{2}\right)\Big{)}\Bigg{]}\,,$
$\displaystyle\Delta\mathcal{J}(E=0)=\frac{8\,GM^{2}\,\nu^{2}}{5}\Bigg{[}\frac{15\pi}{j^{4}}-\frac{5\,\pi\,(-1077+940\,\nu)}{48j^{6}}$
$\displaystyle+\frac{\pi\,\left(1307683-2782332\,\nu+1005480\,\nu^{2}\right)}{2592\,j^{8}}$
(44)
$\displaystyle+\frac{1}{322560\,j^{10}}\Big{(}\pi\,\big{(}5567205457+20(-301863524$
$\displaystyle+4719141\,\pi^{2})\,\nu+2200128840\,\nu^{2}-371498400\,\nu^{3}\big{)}$
$\displaystyle-569479680\,\pi\log\left(2\,\frac{G\,M}{r_{0}}\,j^{2}\right)\Big{)}\Bigg{]}\,.$

We have verified that these expressions are in full agreement with the
parabolic limit of the radiated energy and angular momentum during one radial
period of an eccentric binary available in Refs. Arun  _et al._ (2008, 2009).

###  II.6 Implications of Post-Bremsstrahlung Expansion

We now probe the implications of the post-bremsstrahlung limit of our
$\Delta{\cal E}$ and $\Delta{\cal J}$ expressions. Recall that the
bremsstrahlung limit arises by allowing the eccentricity parameter
$\rightarrow\infty$ as noted in Ref. Blanchet and Schäfer (1989), and we are
exploring the implications of eccentricity corrections to such bremsstrahlung
limit of our 3PN order hyperbolic expressions. This effort is also influenced
by the fact that it is rather difficult to obtain closed form expressions for
$\Delta{\cal E}$ and $\Delta{\cal J}$ expressions even when the leading order
hereditary contributions are included Bini and Geralico (2021a). Therefore, it
is reasonable that our ongoing effort to obtain fully 3PN-accurate expressions
for the radiated energy and angular momentum in GWs during hyperbolic
encounters will not be exact in orbital eccentricity. In what follows, we
probe the implications of post-bremsstrahlung expansion of our 3PN order
hyperbolic $\Delta{\cal E}$ and $\Delta{\cal J}$ expressions with respect to
their bound orbit counterparts. These counterparts are essentially 3PN-
accurate instantaneous $\delta{\cal E}$ and $\delta{\cal J}$ expressions that
provide the radiated energy and angular momentum during one radial period of
an eccentric binary. These expressions can easily be obtained from Refs. Arun
_et al._ (2008, 2009), and are exact in orbital eccentricity. Further, we find
it convenient to employ the Newtonian eccentricity parameter
$e=\sqrt{1+2\,E\,j^{2}}$ to characterize the both the bound and unbound far-
zone quantities to ensure that the same eccentricity parameter is used in our
comparisons. We display the relevant $\Delta{\cal E}$ and $\Delta{\cal J}$
expressions in Appendix. B. Further, the explicit 3PN-accurate instantaneous
post-bremsstrahlung $\Delta\cal E$ and $\Delta\cal J$ expressions for
hyperbolic encounters are available at
https://github.com/subhajittifr/hyperbolic_flux ,where $\Delta\cal E$ is
expanded up-to $\mathcal{O}(1/{e^{5}})$ while $\Delta\cal J$ has been expanded
up-to $\mathcal{O}(1/{e^{7}})$.

In Fig. 1, we plot the fractional differences between Refs. Arun  _et al._
(2008, 2009) based $\delta{\cal E}$ and $\delta{\cal J}$ 3PN order expression
that are exact in $e$ and the post-bremsstrahlung expansion of our
$\Delta\mathcal{E}$, $\Delta\mathcal{J}$ expressions while allowing $e\leq 1$.
These plots reveal that the post-bremsstrahlung versions of our
$\Delta\mathcal{E}$, and $\Delta\mathcal{J}$ expressions provide excellent
proxies to compact binaries both in parabolic and high eccentric (bound)
orbits. These post-bremsstrahlung approximants are substantially different
from their eccentric counterparts near their circular limits. This is most
likely due to the presence of $\frac{1}{e}$ terms and its multiples and
similar conclusions are drawn from plots where we change values of $\nu$ and
the dimensionless $h\,c$ as evident from Fig. 2. We infer that our post-
bremsstrahlung $\Delta\mathcal{E}$ and $\Delta\mathcal{J}$ expressions, which
should be accurate to describe large eccentric hyperbolic orbit, are not only
convergent at the parabolic limit ($e=1$) but also highly eccentric cases
($e\lesssim 1$). It will be interesting to explore if numerical relativity
simulations display a similar behaviour. This natural convergence at parabolic
limit is contrary to the attempt to cover parabolic limit adopted in Sec. IX
in Ref. Bini and Damour (2017), where the physical quantities are divergent at
parabolic limit, so the parabolic limit is incorporated by numerical
methodologies such as fitting and Pade approximation.

Figure 1: The fractional difference between the 3PN accurate instantaneous
fluxes (i.e
$\frac{\Delta\mathcal{Q}_{\text{hyp}}-\Delta\mathcal{Q}_{\text{ecc}}}{\Delta\mathcal{Q}_{\text{hyp}}}$,
where $\mathcal{Q}=\mathcal{E},\mathcal{J}$) vs Newtonian eccentricity.
Hyperbolic fluxes are expanded in the post-bremsstrahlung limit ($e\to\infty$)
while the elliptic fluxes have been kept in their original forms, namely in
terms of $e,\,j,\nu$. The blue curve represents the fractional difference of
Energy fluxes while the red curve represents the fractional difference of
Angular Momentum fluxes. The zoomed in version focuses on the $0.7$ to $1$
eccentricity range. The hyperbolic energy flux, as noted in the text, is
expanded up-to $\mathcal{O}(1/{e^{5}})$ while angular momentum flux has been
expanded up-to $\mathcal{O}(1/{e^{7}})$. Additionally, we have chosen
$c^{2}\,j^{2}=10$, $\frac{G\,M}{2\,r_{0}}\,j^{2}=1/5$ and $\nu=1/4$ for these
plots. Figure 2:  Plots that are similar to what are displayed in Fig. 1. For
these plots, we let $c^{2}\,j^{2}=10$, $\frac{G\,M}{2\,r_{0}}\,j^{2}=1/5$ and
$\nu=1/20$. Clearly, our conclusions are rather independent of $\nu$ values.

##  III Summary and On-going Efforts

We have provided explicit expressions for the 3PN-accurate instantaneous
contributions to the radiated energy and angular momentum during hyperbolic
encounters. These computations, pursued in the time-domain, are not
straightforward extensions of the classic 1PN-accurate efforts by Schäfer and
his collaborators Blanchet and Schäfer (1989); Junker and Schäfer (1992). This
is essentially due to the presence of certain logarithmic terms in the 3PN
order contributions to the far-zone flux expressions for the general orbits
Arun  _et al._ (2008). Additionally, we explored the implications of post-
Bremsstrahlung expansion of our results from the perspective of eccentric
orbits.

There are on-going efforts to compute hereditary contributions to the
$\Delta{\cal E}$ and $\Delta{\cal J}$ expressions that are accurate to 3PN
order, influenced by Ref. Bini  _et al._ (2021b). We are also pursuing efforts
to compare our PN-accurate results with those arising from Numerical
RelativityDamour  _et al._ (2014); Bae  _et al._ (2020). This will be helpful
to explore the validity of PN approximation while exploring GWs from
hyperbolic encounters. Further, these efforts may allow us to develop a
prescription for describing GW emission aspects of highly eccentric compact
binaries with constructs that arise from our present and on-going PN-accurate
hyperbolic computations.

## Acknowledgments

We thank Gerhard Schäfer and Luciano Rezzolla for helpful discussions. G.C is
supported by the ERC Consolidator Grant “Precision Gravity: From the LHC to
LISA,” provided by the European Research Council (ERC) under the European
Union’s H2020 research and innovation programme (grant No. 817791). S.D and
A.G acknowledge the support of the Department of Atomic Energy, Government of
India, under project identification # RTI 4002. A.G is grateful for the
financial support and hospitality of the Pauli Center for Theoretical Studies
and the University of Zurich.

##  Appendix A Hyperbolic Log Integrals

We provide the details of evaluating certain 3PN order log integrals that are
crucial for our results. We begin with the following expression

$\displaystyle I(x,y):=\int^{+\infty}_{-\infty}\,du\,\frac{\ln(e\cosh
u-x)}{e\cosh u-y}.$ (45)

The goal is to first take a differentiation followd by an integration of
$I(x,y)$ with respect to $x$. Differentiating the above expression with
respect to $x$ yields

$\displaystyle-\partial_{x}\,I(x,y)$
$\displaystyle=\int^{+\infty}_{-\infty}du\,\frac{1}{e\,\cosh
u-x}\,\frac{1}{e\cosh u-y}\,,$ $\displaystyle=J_{1}(x,y)+J_{2}(x,y),$ (46a)
where $\displaystyle J_{1}(x,y)$
$\displaystyle:=\frac{2}{\sqrt{e^{2}-x^{2}}\,(x-y)}\,\arccos\left(-\frac{x}{e}\right)\,,$
(46b) $\displaystyle J_{2}(x,y)$
$\displaystyle:=\frac{2}{\sqrt{e^{2}-y^{2}}\,(y-x)}\,\arccos\left(-\frac{y}{e}\right)\,.$
(46c)

Antiderivative of $-J_{2}$ with respect to $x$ is

$\displaystyle
I_{2}:=-\int\,dx\,J_{2}=\frac{2\,\ln(y-x)}{\sqrt{e^{2}-y^{2}}}\,\arccos\left(-\frac{y}{e}\right)\,.$
(47)

To integrate $J_{1}$, we make a change of variable
$f:=\arccos\left(-\frac{x}{e}\right)$, then

$\displaystyle I_{1}$ $\displaystyle:=-\int dx\,J_{1}=\int
df\,\frac{2\,f}{e\cos
f+y}=\frac{2\,i}{\sqrt{e^{2}-y^{2}}}\,\bigg{[}\text{Li}_{2}(g)-\text{Li}_{2}(g^{*})\bigg{]}$
$\displaystyle-\frac{2}{\sqrt{e^{2}-y^{2}}}\left[\arccos\left(-\frac{y}{e}\right)\,\log\left(-\,\frac{e^{2}-x\,y+\sqrt{e^{2}-x^{2}}\,\sqrt{e^{2}-y^{2}}}{4\,e\,(x-y)}\right)\right]+\alpha(y;e),$
(48)

where

$\displaystyle g$
$\displaystyle=\frac{(y-i\,\sqrt{e^{2}-y^{2}})}{e}\,\frac{(e+y)\sqrt{e^{2}-x^{2}}-(e+x)\sqrt{e^{2}-y^{2}}}{(e+y)\,\sqrt{e^{2}-x^{2}}+(e+x)\,\sqrt{e^{2}-y^{2}}}\,,$
(49a) $\displaystyle g^{*}$
$\displaystyle=\frac{(y+i\,\sqrt{e^{2}-y^{2}})}{e}\,\frac{(e+y)\sqrt{e^{2}-x^{2}}-(e+x)\sqrt{e^{2}-y^{2}}}{(e+y)\,\sqrt{e^{2}-x^{2}}+(e+x)\,\sqrt{e^{2}-y^{2}}}\,.$
(49b)

Therein, $\text{Li}_{2}$ is the polylogarithm function of order 2. These
results are valid up to modulo a constant $\alpha(y;e)$ that is independent of
$x$. To determine the expression of $\alpha$, let us consider $x\to-\infty$
behavior such that

$\displaystyle\lim_{x\to-\infty}I(x,y)=\lim_{x\to-\infty}\frac{2\,\log(-x)}{\sqrt{e^{2}-y^{2}}}\,\arccos\left(-\frac{y}{e}\right).$
(50)

An, it can be easily checked that

$\displaystyle\lim_{x\to-\infty}I(x,y)=\lim_{x\to-\infty}I_{2}(x,y)\,,$ (51)

which implies

$\displaystyle\lim_{x\to-\infty}I_{1}(x,y)=0\,,$ (52)

to keep $I=I_{1}+I_{2}$ hold. Making use of this condition at $x\to-\infty$ on
Eq. (A), we reach

$\displaystyle\alpha(y;e)=\frac{2\,i}{\sqrt{e^{2}-y^{2}}}\left\\{\frac{\pi^{2}}{6}-\text{Li}_{2}\left[-\frac{(i\,y+\sqrt{e^{2}-y^{2}}\,)^{2}}{e^{2}}\right]-i\,\arccos\left(-\frac{y}{e}\right)\,\log\left[\frac{4\,(y-i\,\sqrt{e^{2}-y^{2}})}{e}\right]\right\\}.$
(53)

In the above we used
$\lim_{x\to-\infty}\text{Li}_{2}(g^{*})=\frac{\pi^{2}}{6}$. Thus, we finally
obtain

$\displaystyle\allowdisplaybreaks I$
$\displaystyle=\frac{2\,i}{\sqrt{e^{2}-y^{2}}}\bigg{\\{}\frac{\pi^{2}}{6}+\text{Li}_{2}(g)-\text{Li}_{2}(g^{*})-\text{Li}_{2}\left(-\frac{(iy+\sqrt{e^{2}-y^{2}})^{2}}{e^{2}}\right)-i\,\arccos\left(-\frac{y}{e}\right)$
$\displaystyle\times\log\Bigg{[}\frac{(y-i\,\sqrt{e^{2}-y^{2}})\,(e^{2}-x\,y+\sqrt{e^{2}-x^{2}}\,\sqrt{e^{2}-y^{2}})}{e^{2}}\Bigg{]}\Bigg{\\}}.$
(54)

Now, we focus on a special condition on Eq. (A) where $x\to y$, the integral
we are interested to compute. Note that when $y\to x$, there are some
divergences because of $\sim\,\log(y-x)$ term in $I_{1}$, but with proper
calculus, these divergences are cancelled while taking $y\to x$ limit. we get

$\displaystyle\allowdisplaybreaks\int_{-\infty}^{\infty}\,du\,\frac{\ln(e\cosh
u-x)}{e\cosh u-x}$
$\displaystyle=\frac{2\,i}{\sqrt{e^{2}-x^{2}}}\,\Bigg{\\{}\frac{\pi^{2}}{6}-\text{Li}_{2}\left(-\frac{(i\,x+\sqrt{e^{2}-x^{2}})^{2}}{e^{2}}\right)-i\,\arccos\left(-\frac{x}{e}\right)$
$\displaystyle\times\,\log\left[\frac{2\,(e^{2}-x^{2})\,(x-i\,\sqrt{e^{2}-x^{2}})}{e^{2}}\right]\Bigg{\\}}.$
(55)

Eq. (A) can be further simplified using the usual definition of Clausen
function $\text{Cl}_{2}(x)$ Lewin (1991) of order 2 - given by the integral

$\displaystyle\text{Cl}_{2}(x)=-\int_{0}^{x}dx\,\log\left|2\sin\frac{x}{2}\right|$
(56)

There exist a nice expression of Dilogaritm function :$\text{Li}_{2}(x)$ in
terms of Clausen function, given by

$\displaystyle\text{Li}_{2}(\text{e}^{i\,\theta})=\frac{\pi^{2}}{6}-\frac{1}{4}\,|\theta|(2\pi-|\theta|)+i\,\text{Cl}_{2}(\theta),\enspace|\theta|\leq
2\pi.$ (57)

By virtue of Eq. (57) we obtain the most simplified form of the integral
$I(x)$. Our final expression is

$\displaystyle\int_{-\infty}^{\infty}\,du\,\frac{\ln(e\cosh u-x)}{e\cosh
u-x}=\frac{2}{\sqrt{e^{2}-x^{2}}}\left\\{\text{Cl}_{2}\left(-\frac{x}{e}\right)+\arccos\left(-\frac{x}{e}\right)\ln\left(2\,e-\frac{2\,x^{2}}{e}\right)\right\\}.$
(58)

##  Appendix B  Energy and angular momentum radiations in terms of energy and
angular momentum

Since the time eccentricity $e_{t}$ is not gauge invariant, we provide
maximally gauge invariant expressions of $\Delta\mathcal{E}$ and
$\Delta\mathcal{J}$ in terms of energy $E$ and angular momentum $j$. Note that
it could be never possible to provide fully gauge invariant expression,
because instantaneous part of $\Delta\mathcal{E}$, $\Delta\mathcal{J}$ is not
gauge invariant, since they depend on the separation of the scale $r_{0}$
differentiating short/long scales. Except the issue of $r_{0}$ which will be
removed by hereditary contribution, the following expressions are gauge
invariant. Here, we use the Newtonian eccentricity $e$ as a shorthand symbol
of $\sqrt{1+2\,E\,j^{2}}$ without implying any geometric meaning.

$\displaystyle\Delta\mathcal{E}$
$\displaystyle=\frac{2}{15}\,\frac{M\,\nu^{2}}{j^{7}\,c^{5}}\Big{(}\mathcal{I}_{N}^{\text{MH}}+\frac{1}{j^{2}\,c^{2}}\,\mathcal{I}_{\text{1PN}}^{\text{MH}}+\frac{1}{j^{4}\,c^{4}}\,\mathcal{I}_{\text{2PN}}^{\text{MH}}+\frac{1}{j^{6}\,c^{6}}\,\mathcal{I}_{\text{3PN}}^{\text{MH}}\Big{)}\,,$
(59)

where

$\displaystyle\mathcal{I}_{N}^{\text{MH}}$
$\displaystyle=\sqrt{e^{2}-1}\,\Bigg{[}\frac{602}{3}+\frac{673e^{2}}{3}\Bigg{]}+\arccos\left(-\frac{1}{e}\right)\,\Bigg{[}96+292\,e^{2}+37\,e^{4}\Bigg{]}\,,$
(60a) $\displaystyle\mathcal{I}_{\text{1PN}}^{\text{MH}}$
$\displaystyle=\sqrt{e^{2}-1}\Bigg{[}\frac{108+12\,\nu}{e^{2}}+\Bigg{(}\frac{90754}{35}-\frac{2185\,\nu}{6}\Bigg{)}+e^{2}\,\Bigg{(}\frac{141439}{70}-\frac{11441\,\nu}{6}\Bigg{)}+e^{4}\,\Bigg{(}\frac{89907}{280}-\frac{1117\,\nu}{2}\Bigg{)}\Bigg{]}$
$\displaystyle+\arccos\left(-\frac{1}{e}\right)\Bigg{[}\Bigg{(}\frac{11177}{7}-95\,\nu\Bigg{)}+e^{2}\,\Bigg{(}\frac{37785}{14}-\frac{3051\,\nu}{2}\Bigg{)}+e^{4}\,\Bigg{(}\frac{2817}{4}-\frac{2283\,\nu}{2}\Bigg{)}+e^{6}\,\Bigg{(}\frac{2393}{56}-\frac{111\,\nu}{2}\Bigg{)}\Bigg{]}\,,$
(60b) $\displaystyle\mathcal{I}_{\text{2PN}}^{\text{MH}}$
$\displaystyle=\sqrt{e^{2}-1}\Bigg{[}\frac{1}{e^{4}}\,\Bigg{(}-\frac{243}{2}-27\,\nu-\frac{3\,\nu^{2}}{2}\Bigg{)}+\frac{1}{e^{2}}\,\Bigg{(}\frac{207423}{112}-\frac{2515\,\nu}{56}-\frac{179\,\nu^{2}}{16}\Bigg{)}+\Bigg{(}\frac{8409586747}{317520}$
$\displaystyle-\frac{9858601\,\nu}{720}-\frac{1569\,\nu^{2}}{4}\Bigg{)}+e^{2}\,\Bigg{(}\frac{10125905183}{635040}-\frac{38505773\,\nu}{1440}+4098\,\nu^{2}\Bigg{)}+e^{4}\,\Bigg{(}-\frac{154205167}{158760}-\frac{36653233\,\nu}{5040}$
$\displaystyle+\frac{11581\,\nu^{2}}{2}\Bigg{)}+e^{6}\,\Bigg{(}\frac{509759}{1960}-\frac{680471\,\nu}{1120}+\frac{12693\,\nu^{2}}{16}\Bigg{)}\Bigg{]}+\arccos\left(-\frac{1}{e}\right)\Bigg{[}\frac{52925837}{3024}-\frac{158383\,\nu}{24}-\frac{4787\,\nu^{2}}{16}$
$\displaystyle+e^{2}\,\Bigg{(}\frac{36730439}{1512}-\frac{4415855\,\nu}{168}+\frac{3877\,\nu^{2}}{2}\Bigg{)}+e^{4}\,\Bigg{(}\frac{479005}{288}-\frac{3069361\,\nu}{224}+\frac{25641\,\nu^{2}}{4}\Bigg{)}+e^{6}\,\Bigg{(}-\frac{5885}{84}$
$\displaystyle-\frac{3489\,\nu}{2}+\frac{4339\,\nu^{2}}{2}\Bigg{)}+e^{8}\,\Bigg{(}\frac{745}{12}-\frac{11965\,\nu}{224}+\frac{925\,\nu^{2}}{16}\Bigg{)}\Bigg{]}\,,$
(60c) $\displaystyle\mathcal{I}_{\text{3PN}}^{\text{MH}}$
$\displaystyle=\sqrt{e^{2}-1}\Bigg{[}\frac{6713608}{1575}+\frac{17868572\,e^{2}}{525}+\frac{19300553\,e^{4}}{525}+\frac{17525209e^{6}}{3150}\Bigg{]}\,\log\left[\frac{2\,r_{0}\,(e^{2}-1)}{G\,M\,j^{2}\,e}\right]-\Bigg{[}\frac{54784}{35}$
$\displaystyle+\frac{465664\,e^{2}}{21}+\frac{4426376\,e^{4}}{105}+\frac{1498856\,e^{6}}{105}+\frac{31779\,e^{8}}{70}\Bigg{]}\,\Bigg{\\{}\log\left[\frac{2\,G\,M\,j^{2}}{r_{0}\,e}\right]\,\arccos\left(-\frac{1}{e}\right)$
$\displaystyle+\text{Cl}_{2}\left[2\,\arccos\left(-\frac{1}{e}\right)\right]\Bigg{\\}}+\sqrt{e^{2}-1}\,\Bigg{[}\frac{1}{4\,e^{6}}\,\Bigg{(}729+243\,\nu+27\,\nu^{2}+\,\nu^{3}\Bigg{)}+\frac{1}{e^{4}}\,\Bigg{(}-\frac{1106055}{448}$
$\displaystyle-\frac{63027\,\nu}{448}+\frac{10159\,\nu^{2}}{448}+\frac{167\,\nu^{3}}{192}\Bigg{)}+\frac{1}{e^{2}}\,\Bigg{(}\frac{15311843}{672}+\Bigg{(}-\frac{2862985}{378}+\frac{123\pi^{2}}{2}\Bigg{)}\,\nu-\frac{426791\,\nu^{2}}{672}-\frac{79\,\nu^{3}}{4}\Bigg{)}$
$\displaystyle+\frac{40066564486859}{139708800}+\Bigg{(}-\frac{205531241131}{635040}+\frac{13414913\pi^{2}}{3360}\Bigg{)}\,\nu+\frac{833407139\,\nu^{2}}{40320}+\frac{206305\,\nu^{3}}{192}$
$\displaystyle+e^{2}\,\Bigg{(}\frac{8653194752669}{23284800}+\Bigg{(}-\frac{243329285581}{423360}+\frac{2172959\pi^{2}}{280}\Bigg{)}\,\nu+\frac{310440959\,\nu^{2}}{2240}-\frac{4703\,\nu^{3}}{48}\Bigg{)}$
$\displaystyle+e^{4}\,\Bigg{(}\frac{10874967898483}{46569600}+\Bigg{(}-\frac{13311394987}{141120}-\frac{995521\pi^{2}}{896}\Bigg{)}\,\nu+\frac{131367287\,\nu^{2}}{1344}-\frac{1086249\,\nu^{3}}{64}\Bigg{)}$
$\displaystyle+e^{6}\,\Bigg{(}\frac{723666251779}{19958400}+\Bigg{(}\frac{19905131933}{1270080}-\frac{12303239\pi^{2}}{13440}\Bigg{)}\,\nu+\frac{284599649\,\nu^{2}}{20160}-\frac{261641\,\nu^{3}}{24}\Bigg{)}+e^{8}\Bigg{(}\frac{248312131}{4139520}$
$\displaystyle-\frac{11427029\,\nu}{28224}+\frac{3374859\,\nu^{2}}{4480}-\frac{56047\,\nu^{3}}{64}\Bigg{)}\Bigg{]}+\arccos\left(-\frac{1}{e}\right)\Bigg{[}\frac{642008180503}{3326400}+\Bigg{(}-\frac{21604997}{126}+\frac{15785\pi^{2}}{8}\Bigg{)}\,\nu$
$\displaystyle+\frac{7846337\,\nu^{2}}{1344}+\frac{12633\,\nu^{3}}{32}+e^{2}\,\Bigg{(}\frac{2827833857903}{6652800}+\Bigg{(}-\frac{6912058987}{12096}+\frac{124763\pi^{2}}{16}\Bigg{)}\,\nu+\frac{266844577\,\nu^{2}}{2688}$
$\displaystyle+\frac{26815\,\nu^{3}}{16}\Bigg{)}+e^{4}\,\Bigg{(}\frac{1246592422921}{3326400}+\Bigg{(}-\frac{3120760105}{12096}+\frac{112709\pi^{2}}{64}\Bigg{)}\,\nu+\frac{174794687\,\nu^{2}}{1344}-\frac{91945\,\nu^{3}}{8}\Bigg{)}$
$\displaystyle+e^{6}\,\Bigg{(}\frac{31441963853}{277200}+\Bigg{(}\frac{86715535}{6048}-\frac{211027\,\pi^{2}}{128}\Bigg{)}\,\nu+\frac{7483281\,\nu^{2}}{224}-\frac{245475\,\nu^{3}}{16}\Bigg{)}+e^{8}\,\Bigg{(}\frac{5518709201}{1478400}+\Bigg{(}\frac{2020985}{1344}$
$\displaystyle-\frac{12177\pi^{2}}{128}\Bigg{)}\,\nu+\frac{296053\,\nu^{2}}{112}-\frac{95255\,\nu^{3}}{32}\Bigg{)}+e^{10}\,\Bigg{(}\frac{2075735}{118272}-\frac{745\,\nu}{12}+\frac{45467\,\nu^{2}}{896}-\frac{407\,\nu^{3}}{8}\Bigg{)}\Bigg{]}.$
(60d)

Similarly, the 3PN-accurate instantaneous contributions to the radiated
angular momentum terms of $e$ and $j$ become

$\displaystyle\Delta\mathcal{J}$
$\displaystyle=\frac{8}{5\,c^{5}}\,\frac{GM^{2}\,\nu^{2}}{j^{4}}\Big{(}\mathcal{H}_{N}^{\text{MH}}+\frac{1}{j^{2}\,c^{2}}\,\mathcal{H}_{\text{1PN}}^{\text{MH}}+\frac{1}{j^{4}\,c^{4}}\,\mathcal{H}_{\text{2PN}}^{\text{MH}}+\frac{1}{j^{6}\,c^{6}}\,\mathcal{H}_{\text{3PN}}^{\text{MH}}\Big{)}\,,$
(61)

where

$\displaystyle\mathcal{H}^{\text{MH}}_{\text{N}}$
$\displaystyle=\sqrt{e^{2}-1}\,\Bigg{[}13+2e^{2}\Bigg{]}+\arccos\left(-\frac{1}{e}\right)\,\Bigg{[}8+7e^{2}\Bigg{]}\,,$
(62a) $\displaystyle\mathcal{H}^{\text{MH}}_{\text{1PN}}$
$\displaystyle=\sqrt{e^{2}-1}\,\Bigg{[}\frac{9+\,\nu}{e^{2}}+\Big{(}\frac{11525}{168}-\frac{1451\,\nu}{36}\Big{)}+e^{2}\,\Bigg{(}\frac{10313}{336}-\frac{2065\,\nu}{36}\Bigg{)}+e^{4}\,\Bigg{(}\frac{109}{28}-\frac{5\,\nu}{4}\Bigg{)}\Bigg{]}$
$\displaystyle+\arccos\left(-\frac{1}{e}\right)\Bigg{[}\frac{4877}{84}-\frac{65\,\nu}{4}+e^{2}\,\Bigg{(}\frac{869}{21}-\frac{419\,\nu}{6}\Bigg{)}+e^{4}\,\Bigg{(}\frac{4283}{336}-\frac{71\,\nu}{6}\Bigg{)}\Bigg{]}\,,$
(62b) $\displaystyle\mathcal{H}^{\text{MH}}_{\text{2PN}}$
$\displaystyle=\sqrt{e^{2}-1}\Bigg{[}-\frac{1}{8\,e^{4}}\Bigg{(}81+18\,\nu+\,\nu^{2}\Bigg{)}+\frac{1}{e^{2}}\,\Bigg{(}\frac{41169}{448}-\frac{11839\,\nu}{672}-\frac{109\,\nu^{2}}{64}\Bigg{)}+\frac{139894943}{272160}-\frac{17944223\,\nu}{30240}$
$\displaystyle+\frac{1915\,\nu^{2}}{144}+e^{2}\,\Bigg{(}-\frac{18473111}{272160}-\frac{201089\,\nu}{540}+\frac{25595\,\nu^{2}}{96}\Bigg{)}+e^{4}\,\Bigg{(}-\frac{13557}{560}-\frac{870131\,\nu}{10080}+\frac{15701\,\nu^{2}}{144}\Bigg{)}$
$\displaystyle+e^{6}\,\Bigg{(}\frac{3245}{4032}-\frac{327\,\nu}{224}+\frac{51\,\nu^{2}}{64}\Bigg{)}\Bigg{]}+\arccos\left(-\frac{1}{e}\right)\Bigg{[}\frac{2051491}{4536}-\frac{46297\,\nu}{126}-\frac{79\,\nu^{2}}{12}+e^{2}\Bigg{(}\frac{752767}{6048}$
$\displaystyle-\frac{121675\,\nu}{224}+\frac{4433\,\nu^{2}}{24}\Bigg{)}+e^{4}\,\Bigg{(}-\frac{28297}{378}-\frac{24611\,\nu}{168}+\frac{783\,\nu^{2}}{4}\Bigg{)}+e^{6}\,\Bigg{(}\frac{1327}{504}-\frac{3653\,\nu}{224}+\frac{337\,\nu^{2}}{24}\Bigg{)}\Bigg{]}\,,$
(62c) $\displaystyle\mathcal{H}^{\text{MH}}_{\text{3PN}}$
$\displaystyle=\sqrt{e^{2}-1}\,\Bigg{[}\frac{99724}{315}+\frac{351067e^{2}}{315}+\frac{210683e^{4}}{630}\Bigg{]}\,\log\left[\frac{r_{0}\,(e^{2}-1)}{G\,M\,j^{2}\,e}\right]-\Bigg{[}\frac{13696}{105}+\frac{98012e^{2}}{105}+\frac{23326e^{4}}{35}+\frac{2461e^{6}}{70}\Bigg{]}$
$\displaystyle\times\Bigg{\\{}\log\left[\frac{2\,G\,M\,j^{2}}{r_{0}\,e}\right]\,\arccos\left(-\frac{1}{e}\right)+\text{Cl}_{2}\left[2\,\arccos\left(-\frac{1}{e}\right)\right]\Bigg{\\}}+\sqrt{e^{2}-1}\,\Bigg{[}\frac{1}{16\,e^{6}}\Bigg{(}243+81\,\nu+9\,\nu^{2}+\,\nu^{3}\Bigg{)}$
$\displaystyle+\frac{1}{e^{4}}\Bigg{(}-\frac{257553}{1792}+\frac{16035\,\nu}{1792}+\frac{22507\,\nu^{2}}{5376}+\frac{121\,\nu^{3}}{768}\Bigg{)}+\frac{1}{e^{2}}\,\Bigg{(}\frac{27158603}{32256}+\Bigg{(}-\frac{165077725}{290304}+\frac{41\pi^{2}}{8}\Bigg{)}\,\nu$
$\displaystyle-\frac{718787\,\nu^{2}}{32256}-\frac{237\,\nu^{3}}{512}\Bigg{)}+\frac{387361378703}{67060224}+\Bigg{(}-\frac{337799063429}{30481920}+\frac{340423\pi^{2}}{1920}\Bigg{)}\,\nu+\frac{878650477\,\nu^{2}}{483840}+\frac{344695\,\nu^{3}}{4608}$
$\displaystyle+e^{2}\Bigg{(}\frac{1005116464409}{167650560}+\Bigg{(}-\frac{23645568707}{3048192}+\frac{512213\pi^{2}}{3840}\Bigg{)}\,\nu+\frac{134389529\,\nu^{2}}{34560}-\frac{708305\,\nu^{3}}{2304}\Bigg{)}+e^{4}\Bigg{(}\frac{432742948027}{335301120}$
$\displaystyle+\Bigg{(}\frac{5045091193}{7620480}-\frac{44567\pi^{2}}{1920}\Bigg{)}\,\nu+\frac{60548279\,\nu^{2}}{60480}-\frac{886135\,\nu^{3}}{1152}\Bigg{)}+e^{6}\Bigg{(}-\frac{1172569271}{24837120}+\frac{17091961\,\nu}{1128960}+\frac{4248733\,\nu^{2}}{32256}$
$\displaystyle-\frac{687113\,\nu^{3}}{4608}\Bigg{)}+e^{8}\Bigg{(}-\frac{6973}{354816}-\frac{3245\,\nu}{32256}+\frac{3379\,\nu^{2}}{3584}-\frac{249\,\nu^{3}}{512}\Bigg{)}\Bigg{]}+\arccos\left(-\frac{1}{e}\right)\Bigg{[}\frac{26951805241}{5702400}$
$\displaystyle+\Bigg{(}-\frac{503311993}{72576}+\frac{3239\pi^{2}}{32}\Bigg{)}\,\nu+\frac{12801961\,\nu^{2}}{16128}+\frac{2479\,\nu^{3}}{64}+e^{2}\Bigg{(}\frac{83623027931}{9979200}+\Bigg{(}-\frac{408126835}{36288}+\frac{6355\pi^{2}}{32}\Bigg{)}\,\nu$
$\displaystyle+\frac{15335105\,\nu^{2}}{4032}-\frac{3907\,\nu^{3}}{48}\Bigg{)}+e^{4}\Bigg{(}\frac{6781095059}{1663200}+\Bigg{(}-\frac{18560153}{24192}-\frac{615\pi^{2}}{256}\Bigg{)}\,\nu+\frac{5148791\,\nu^{2}}{2688}-\frac{71389\,\nu^{3}}{96}\Bigg{)}$
$\displaystyle+e^{6}\Bigg{(}\frac{254451209}{3326400}+\Bigg{(}\frac{1448255}{6048}-\frac{615\pi^{2}}{128}\Bigg{)}\,\nu+\frac{391981\,\nu^{2}}{1344}-\frac{16867\,\nu^{3}}{48}\Bigg{)}+e^{8}\Bigg{(}-\frac{227005}{709632}-\frac{14411\,\nu}{2016}+\frac{29727\,\nu^{2}}{1792}$
$\displaystyle-\frac{2693\,\nu^{3}}{192}\Bigg{)}\Bigg{]}.$ (62d)

## References

  * Abbott  _et al._ (2019) B. Abbott, R. Abbott, T. Abbott, S. Abraham, F. Acernese, K. Ackley, C. Adams, R. Adhikari, V. Adya, C. Affeldt, _et al._ , Physical Review X 9, 031040 (2019).
  * Abbott  _et al._ (2021) R. Abbott, T. Abbott, S. Abraham, F. Acernese, K. Ackley, A. Adams, C. Adams, R. Adhikari, V. Adya, C. Affeldt, _et al._ , Physical Review X 11, 021053 (2021).
  * Venumadhav  _et al._ (2020) T. Venumadhav, B. Zackay, J. Roulet, L. Dai, and M. Zaldarriaga, Physical Review D 101, 083030 (2020).
  * Abbott  _et al._ (2017) B. P. Abbott, R. Abbott, T. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. Adhikari, V. Adya, _et al._ , Physical review letters 119, 161101 (2017).
  * Poggiani (2019) R. Poggiani (Ligo Scientific, Virgo), PoS FRAPWS2018, 013 (2019).
  * Monitor  _et al._ (2017) F. G.-R. B. Monitor, L. S. Collaboration, V. Collaboration, _et al._ , arXiv preprint arXiv:1710.05834 (2017).
  * García-Bellido and Nesseris (2018) J. García-Bellido and S. Nesseris, Physics of the dark universe 21, 61 (2018).
  * Mukherjee  _et al._ (2020) S. Mukherjee, S. Mitra, and S. Chatterjee, arXiv preprint arXiv:2010.00916 (2020).
  * Kocsis  _et al._ (2006) B. Kocsis, M. E. Gáspár, and S. Marka, The Astrophysical Journal 648, 411 (2006).
  * Burke-Spolaor  _et al._ (2019) S. Burke-Spolaor, S. R. Taylor, M. Charisi, T. Dolch, J. S. Hazboun, A. M. Holgado, L. Z. Kelley, T. J. W. Lazio, D. R. Madison, N. McMann, _et al._ , The Astronomy and Astrophysics Review 27, 1 (2019).
  * Tsang (2013) D. Tsang, The Astrophysical Journal 777, 103 (2013).
  * Cho  _et al._ (2018) G. Cho, A. Gopakumar, M. Haney, and H. M. Lee, Physical Review D 98, 024039 (2018).
  * Bae  _et al._ (2020) Y.-B. Bae, H. M. Lee, and G. Kang, The Astrophysical Journal 900, 175 (2020).
  * Bini  _et al._ (2021a) D. Bini, T. Damour, and A. Geralico, arXiv preprint arXiv:2107.08896 (2021a).
  * Bini  _et al._ (2021b) D. Bini, T. Damour, A. Geralico, S. Laporta, and P. Mastrolia, Physical Review D 103, 044038 (2021b).
  * Bini and Geralico (2021a) D. Bini and A. Geralico, arXiv preprint arXiv:2108.05445 (2021a).
  * Bini and Geralico (2021b) D. Bini and A. Geralico, arXiv preprint arXiv:2108.02472 (2021b).
  * Hansen (1972) R. O. Hansen, Phys. Rev. D 5, 1021 (1972).
  * Blanchet and Schäfer (1989) L. Blanchet and G. Schäfer, Monthly Notices of the Royal Astronomical Society 239, 845 (1989).
  * Junker and Schäfer (1992) W. Junker and G. Schäfer, Monthly Notices of the Royal Astronomical Society 254, 146 (1992).
  * Blanchet (2014) L. Blanchet, Living reviews in relativity 17, 1 (2014).
  * Porto (2016) R. A. Porto, Phys. Rept. 633, 1 (2016), arXiv:1601.04914 [hep-th] .
  * Ruffini and Wheeler (1971) R. Ruffini and J. A. Wheeler, ESRO 52, 45 (1971).
  * Wagoner and Will (1976) R. V. Wagoner and C. Will, The Astrophysical Journal 210, 764 (1976).
  * Arun  _et al._ (2008) K. Arun, L. Blanchet, B. R. Iyer, and Q. Moh’d SS, Physical Review D 77, 064035 (2008).
  * Arun  _et al._ (2009) K. Arun, L. Blanchet, B. R. Iyer, and S. Sinha, Physical Review D 80, 124018 (2009).
  * Blanchet and Damour (1988) L. Blanchet and T. Damour, Phys. Rev. D 37, 1410 (1988).
  * Blanchet and Damour (1992) L. Blanchet and T. Damour, Phys. Rev. D 46, 4304 (1992).
  * Lewin (1991) L. Lewin,  _Structural properties of polylogarithms_ , 37 (American Mathematical Soc., 1991).
  * Blanchet (1998) L. Blanchet, Class. Quant. Grav. 15, 1971 (1998), arXiv:gr-qc/9801101 .
  * Poujade and Blanchet (2002) O. Poujade and L. Blanchet, Phys. Rev. D 65, 124020 (2002), arXiv:gr-qc/0112057 .
  * Damour and Deruelle (1985) T. Damour and N. Deruelle, Annales de l’I.H.P. Physique théorique 43, 107 (1985).
  * Klioner (2016) S. A. Klioner, arXiv preprint arXiv:1609.00915 (2016).
  * Blanchet and Iyer (2005) L. Blanchet and B. R. Iyer, Physical Review D 71, 024004 (2005).
  * Herrmann  _et al._ (2021) E. Herrmann, J. Parra-Martinez, M. S. Ruf, and M. Zeng, Phys. Rev. Lett. 126, 201602 (2021), arXiv:2101.07255 [hep-th] .
  * Iyer and Will (1993) B. R. Iyer and C. M. Will, Physical review letters 70, 113 (1993).
  * Iyer and Will (1995) B. R. Iyer and C. M. Will, Physical Review D 52, 6882 (1995).
  * Gopakumar  _et al._ (1997) A. Gopakumar, B. R. Iyer, and S. Iyer, Physical Review D 55, 6030 (1997).
  * Damour  _et al._ (2004) T. Damour, A. Gopakumar, and B. R. Iyer, Physical Review D 70, 064028 (2004).
  * Bini and Damour (2017) D. Bini and T. Damour, Phys. Rev. D 96, 064021 (2017), arXiv:1706.06877 [gr-qc] .
  * Damour  _et al._ (2014) T. Damour, F. Guercilena, I. Hinder, S. Hopper, A. Nagar, and L. Rezzolla, Phys. Rev. D 89, 081503 (2014), arXiv:1402.7307 [gr-qc] .