Eective two-body approach to the hierarchical three-body problem quadrupole to 1PN Adrien Kuntzab Francesco Serraab Enrico Trincheriniab

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Eﬀective two-body approach to the hierarchical three-body problem: quadrupole

to 1PN

Adrien Kuntz a,b, Francesco Serra a,b, Enrico Trincherini a,b∗

aScuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, Italy

bINFN Sezione di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy

Abstract

Many binary systems of interest for gravitational-wave astronomy are orbited by a third distant

body, which can considerably alter their relativistic dynamics. Precision computations are needed

to understand the interplay between relativistic corrections and three-body interactions. We use an

eﬀective ﬁeld theory approach to derive the eﬀective action describing the long time-scale dynamics

of hierarchical three-body systems up to 1PN quadrupole order. At this level of approximation,

computations are complicated by the backreaction of small oscillations on orbital time-scales as

well as deviations from the adiabatic approximation. We address these diﬃculties by eliminating

the fast modes through the method of near-identity transformations. This allows us to compute for

the ﬁrst time the complete expression of the 1PN quadrupole cross-terms in generic conﬁgurations

of three-body systems. We numerically integrate the resulting equations of motion and show that

1PN quadrupole terms can aﬀect the long term dynamics of relativistic three-body systems.

∗E-mail: adrien.kuntz@sns.it,francesco.serra@sns.it,enrico.trincherini@sns.it

arXiv:2210.13493v1 [gr-qc] 24 Oct 2022

Contents

1 Introduction 2

1.1 Dictionaryofsymbols .................................... 4

2 The EFT approach to the relativistic hierarchical three body problem 5

2.1 Contactelements....................................... 7

3 The point-particle EFT to quadrupolar order 8

3.1 Integratingoutfastmodes.................................. 8

3.2 Center-of-mass and relative coordinates in boosted frame . . . . . . . . . . . . . . . . 10

3.3 AveragingtheLagrangian .................................. 10

3.4 Matching ........................................... 12

4 Double-averaged Lagrangian up to order v2ε5/214

5 Numerical solution to the LPE 15

6 Conclusions and Outlook 18

A Averaging through near-identity transformations 20

B Conservation of contact semi-major axis 23

C Backreaction and deviations form adiabaticity 24

C.1 Long-timescale and short-timescale Lagrangians . . . . . . . . . . . . . . . . . . . . . . 24

C.2 1PN quadrupolar cross-terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

D Lagrange Planetary Equations 28

E From the three-body center-of-mass frame to the inner binary rest frame 29

F Quadrupole-squared terms 31

G From contact elements to orbital elements 34

1 Introduction

The evolution of binary systems can be signiﬁcantly aﬀected by relativistic eﬀects as well as by the

presence of a third celestial body. The interplay between these eﬀects is especially relevant in view

of recent observations of triple systems with pulsars [1] as well as future detections of relativistic

three-body systems expected with the gravitational-wave (GW) spatial interferometer LISA [2,3].

Three-body eﬀects have been studied to understand the merger rate of compact objects [4–10], the

transits of exoplanets [11–13] or the evolution of triple star systems [14–17]. Triple systems often

come in a hierarchical setting where one of the constituents is far away from the other two. These

conﬁgurations are conveniently described in terms of an ”inner binary” elliptic motion composed of

the two closest objects, and an ”outer binary” elliptic motion made of the inner binary itself and the

outer object, as depicted schematically in Fig 1. The long-timescale evolution of these orbits presents

many interesting features already at the Newtonian level [18], the most well-known example being the

Kozai-Lidov (KL) oscillations of the eccentricity of the inner binary [19,20]. These eﬀects are usually

studied through the method of double averaging [18], in which quick oscillations on time-scales of order

of the orbital periods are eliminated from the dynamics, leaving only information about evolution on

timescales larger than the orbital periods. This method is known to work whenever the system is away

from resonances; the latter make the whole analysis much less straightforward, see e.g. [21].

When relativistic corrections to the Newtonian dynamics are considered, in the so-called Post-

Newtonian (PN) expansion, the long time-scale dynamics undergoes radical changes, even when the

system remains into a hierarchical conﬁguration. For instance, the periastron precession can suppress

high eccentricity oscillations when the precession timescale is much shorter than the timescale char-

acterizing KL oscillations [4,22–25]. Even if the system can be described by the inner and the outer

orbit, studying the eﬀects of relativistic corrections on its evolution over long time-scales remains in

general quite diﬃcult especially due to eﬀects at higher-orders in the PN expansion [9], which will

usually leave an imprint on the dynamics after a parametrically large time. It is thus important

to compute three-body relativistic interactions to a high accuracy in order to evolve these systems

on long timescales. One option is to approach the problem with a numerical relativistic three-body

solver [26–30], however this method is heavily time-consuming. On the other hand, analytic methods

are sparse at best [9,24,25,31–34].

Motivated by these developments and by the ensuing diﬃculties, we recently introduced a new

Eﬀective Field Theory (EFT) approach to the relativistic, hierarchical three-body problem [35]. The

central idea of this method is to take advantage of the two small parameters characterizing hierarchical

three-body systems: the typical velocity of the inner bodies divided by light speed, v(c= 1), and

the ratio ε=a/a3between the semimajor axis of the inner binary aand the one of the outer binary

a3. Thanks to the double perturbative expansion with respect to these parameters, it is possible to

match the dynamics of a hierarchical three-body system to a simpler two-body interaction, in which

the inner binary is described as a single point-particle endowed with multipole moments. This is

achieved by employing the EFT techniques developed for the relativistic two-body problem [36,37]

and by performing the averaging procedure at the level of the Lagrangian. Most noticeably, the EFT

approach exploits symmetries that are manifest in the eﬀective Lagrangian, restricting the form of

the allowed interaction terms. Moreover, working with a single functional rather than with several

equations of motion makes it simpler to setup a systematic study of the three body system.

In [35] we presented the EFT setup and derived the eﬀective Lagrangian describing the system

on long time-scales up to 1PN dipole order, i.e. up to order v2ε3/2beyond the leading Newtonian

interaction. Instead, in the present work we extend this computation up to 1PN quadrupolar order,

i.e. v2ε5/2beyond leading order. At this order, computations are substantially more complex with

respect to our previous study [35]. A ﬁrst source of complexity is due to the averaging procedure. At

lower orders the averaging can be performed in the so-called adiabatic approximation, i.e. neglecting

variations of slowly evolving variables during the average over the period of both orbits. Instead, when

accounting for terms of mixed quadrupolar and 1PN order, deviations from adiabaticity must be taken

into account. In addition to this, backreaction from quickly oscillating terms that are suppressed in

amplitude will also aﬀect the averaging, contrarily to what happened at lower orders. We address

these complications by following the method of near-identity transformations [38], which allows to

consistently implement the averaging procedure to any order of accuracy. While several authors

already studied quadrupolar couplings at 1PN order [9,31,32,34,39–42], we are aware of only three

which took into account these deviations from the adiabatic approximation [31–34]. However, we

believe that we give in this work the ﬁrst complete expressions of quadrupolar 1PN terms. Indeed,

only the particular case of a circular outer orbit is considered in [31–33] neglecting some PN interactions

that we describe in this work. On the other hand, the derivation in [34] reports a puzzling result that

we will mention in Section 4.

Another source of complexity lies in ﬁnding the suitable dynamical variables to eﬃciently package

relativistic corrections in our results. While the idea at the core of our approach of identifying the

inner binary to a spinning point-particle with multipole moments is very intuitive, providing a deﬁnite

relation between the variables describing the inner binary and the parameters of the eﬀective point-

particle is subtle in practice. For example, in our previous work [35] we showed how the choice of a Spin

Supplementary Condition (a gauge condition on the spin tensor of the eﬀective point-particle [43,44]) is

related to the center-of-mass choice of the inner binary. In the present computations, two new similar

subtleties arise. The ﬁrst one concerns the deﬁnition of osculating elements describing both inner

and outer orbits. In our previous work, we followed the usual convention and used the osculating

orbital elements deﬁned as the parameters of the ellipse instantaneously tangent to the trajectory

(described with positions and velocities). However, at 1PN quadrupolar order we ﬁnd that it becomes

more convenient to use osculating contact elements, which are deﬁned through momenta rather than

velocities [45]. Since PN corrections induce a non-trivial relation between momentum and velocity,

these two sets of osculating elements will diﬀer in general. We will give in Section 2.1 the precise

deﬁnition of contact elements, and we will elaborate more on their diﬀerence with respect to orbital

elements in Appendix G. One remarkable conclusion of the present analysis is that, while the slowly

evolving part of the contact semimajor axes are conserved throughout the evolution of the system (as

is common in long-timescale evolution of triple systems [18]), their orbital counterpart features small

variations over long time-scales, which oﬀers a new point of view on earlier ﬁndings of [31–34].

The second subtle point in the matching between inner binary and point-particle is that the

quantities describing the inner binary system are inherently deﬁned in the rest frame of its center

of mass, which is accelerating because of the presence of the third body. This entails non-trivial

relations between the absolute positions of the inner binary components, deﬁned in the rest frame

of the three-body center-of-mass, and the relative quantities deﬁned in the binary rest frame, as we

show in Section 3.2. As far as we know, this point went so far unnoticed in the relativistic three-body

literature. While this step just amounts to a redeﬁnition of the osculating elements of the inner binary,

it proves to be crucial in order to perform correctly the matching procedure described in Section 3.4.

Let us now describe in more detail the organization of this article. We will begin by summarising

our ”eﬀective two-body” approach in Section 2, which is deﬁned by four main steps. Steps 1 and 2

will then be performed in Section 3, where the core of our matching procedure is explained. In order

to keep the discussion as simple as possible, we have deferred the computation of beyond-adiabatic

corrections to Appendix Cand use only the ﬁnal result of this appendix in the main text. We then

perform the two ﬁnal steps of our approach in Section 4, where we integrate out the gravitational ﬁeld

due to the outer object. Finally, in Section 5we provide a numerical solution implementing the new

relativistic interaction derived in the present work and we show how it inﬂuences the long time-scale

dynamics in the case of a particular three-body system. The rest of the Appendices are devoted to: a

presentation of the averaging procedure that we employ (Appendix A); a discussion of the conservation

of the semimajor axis (Appendix B); a review of the Lagrange Planetary Equations (Appendix D);

the derivation of the expressions connecting the absolute coordinates of the two inner bodies in the

Figure 1: Illustration of the ”eﬀective two-body” description and of osculating elements. The inner binary is replaced

with a point-particle whose spin and multipole moments are related to the osculating elements of the inner orbit.

three body rest frame to their relative coordinates in the inner binary rest frame (Appendix E); an

independent computation of the so-called quadrupole-squared terms of [46] (Appendix F); and the

relation between contact and orbital elements (Appendix G).

1.1 Dictionary of symbols

As a preliminary, we deﬁne the quantities that characterize the three body problem and list the

symbols that we are going to use through the work. We will work out the results in terms of the

contact elements, the variables in which the PN Hamiltonian has the simplest expression. With

respect to the usual Newtonian orbital elements, contact elements include PN corrections. We review

their precise deﬁnition in Section 2and examine their diﬀerence from orbital elements in Appendix G.

•y1,v1,y2,v2: positions and velocities of the two constituents of the inner orbit, of masses m1

and m2;

•y3,v3: position and velocity of the external perturber, of mass m3;

•YCM,VCM : position and velocity of the center-of-mass of the inner binary, deﬁned in Eq (96);

•r,v: relative variables in the inner binary instantaneous rest frame, deﬁned in Section 3.2. Note

that r=y1−y2and v=v1−v2only at lowest PN order, as explained in Section 3.2;

•pA(A= 1,2,3), PCM,p: conjugate momenta to the positions of the three bodies, position of

the center of mass, and relative separation between two inner bodies;

•r=|r|,n=r/r,R=YCM −y3,R=|R|,N=R/R,V=VCM −v3;

•m=m1+m2is the mass of the inner binary, µ=m1m2/m is the reduced mass of the inner

binary, X1=m1/m,X2=m2/m and ν=µ/m are its characteristic mass ratios;

•a[a3]: contact semimajor axis of the inner [outer] orbit;

•e[e3]: contact eccentricity of the inner [outer] orbit;

•E=m−GNmµ/(2a): total (relativistic) energy of the inner binary;

•M=E+m3is the total mass of the eﬀective two-body system. Similarly, µ3=m3E/M is its

reduced mass, X3=m3/M,XCM =E/M and ν3=µ3/M are the mass ratios characterizing the

outer orbit;

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Eectivetwo-bodyapproachtothehierarchicalthree-bodyproblem:quadrupoleto1PNAdrienKuntza;b,FrancescoSerraa;b,EnricoTrincherinia;b*aScuolaNormaleSuperiore,PiazzadeiCavalieri7,56126,Pisa,ItalybINFNSezionediPisa,LargoPontecorvo3,56127Pisa,ItalyAbstractManybinarysystemsofinterestforgravitational-waveastro...

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Eective two-body approach to the hierarchical three-body problem quadrupole to 1PN Adrien Kuntzab Francesco Serraab Enrico Trincheriniab

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