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Post-1-Newtonian quadrupole tidal interactions in binary systems Justin E. Vines and Eánna E´. Flanagan Department of Physics, Cornell University, Ithaca, NY 14853, USA. (Dated: January 11, 2011) Weconsiderorbital-tidalcouplinginabinarystellarsystemtopost-1-Newtonianorder. Wederive theorbital equationsofmotion forbodieswithspinsandmassquadrupolemomentsandshowthat they conserve the total linear momentum of the binary. Momentum conservation also allows us to specialize ouranalysis to thesystem’s center-of-mass-energy frame; we find the binary’s relative equation of motion in this frame and also present a generalized Lagrangian from which it can be derived. Wethenspecializetothecaseinwhichthequadrupolemomentisadiabaticallyinducedby the tidal field. We show how the adiabatic dynamics for the quadrupole can be incorporated into ouractionprincipleandpresentthesimplifiedorbitalequationsofmotionandconservedenergyfor theadiabatic case. These results arerelevant to thegravitational wavesignal of inspiralling binary 1 neutron stars. 1 0 2 I. INTRODUCTION AND SUMMARY λ to 10%, with the largest source of error being post- ∼ 1-Newtonian (1PN) corrections to the tidal-orbital cou- n a A. Background and motivation pling [8, 9]. It is our goal in this paper to lay the foun- J dations for computing the tidal signal to 1PN accuracy. 9 Inspiralling and coalescing compact binaries present The modeling of tidal effects in GW signals from neu- tronstarbinariescanbedividedintothreeseparateprob- oneofthemostpromisingsourcesforground-basedgrav- ] lems: (i) to calculate the deformation response of each c itationalwave(GW)detectors[1]. Aprimarygoalinthe q measurement of GW signals from neutron star-neutron neutronstartothetidalfieldgeneratedbyitscompanion, - star (NSNS) and black hole-neutron star (BHNS) bina- (ii)tocalculatetheinfluenceofthetidaldeformationson r the system’s orbital dynamics, and (iii) to calculate the g riesis toprobethe neutronstarmatter equationofstate [ (EoS),whichiscurrentlyonlylooselyconstrainedbyelec- gravitationalwaveformemittedbythesystemandincor- tromagnetic observations in the relevant denisty range poratethecorrespondingradiationreactioneffectsinthe 2 v ρ 2–8 1014 g cm3 [2]. The EoS will leave its imprint orbitaldynamics. This paper is focused onsolvingprob- ∼ × lem (ii) to 1PN accuracy, while we leave problem (iii) to 9 onthe GW signalvia the effects of tidal coupling,as the 1 neutron star is distorted by the non-uniform field of its future work. A solution to problem (i) that is sufficient 9 companion. Many recent studies into the effects of the for our purposes has been given in Refs. [9–12]. 4 neutronstar EoSon binaryGW signals havebeen based Tidal Deformability Computations: While computing . 9 on numerical simulations of the fully relativistic hydro- tidal deformations of stars is a well studied problem in 0 dynamical evolution of NSNS and BHNS binaries (see Newtoniangravity[13],it hasrecentlybeenre-examined 0 e.g.thereviews[3,4]). Thesesimulationshavelargelyfo- in the context of fully relativistic stars by Hinderer et 1 cusedonthe binaries’lastfeworbitsandmerger,atGW al. [9, 10], Damour and Nagar [11], and Binnington and : v frequencies &500 Hz, and have investigated constraining Poisson [12]. These authors used fully relativistic mod- i neutron star structure via (for example) the GW energy els for the star’s interior, with various candidate equa- X spectrum [5], effective cutoff frequencies at merger [6], tionsofstate,tocalculatethe perturbationstothestar’s r a and tidal disruption signals [7]. equilibrium configuration produced by a static external AsrecentlyinvestigatedbyFlanaganandHinderer[8], tidal field. In Refs. [9, 10], the results were used to de- neutron star internal structure may also have a measur- terminethe(electric-type)quadrupolartidaldeformabil- able influence on the GW signalfrom the earlier inspiral ity λ, while Refs. [11, 12] extended the analysis and in- stage of a binary’s orbital evolution, at GW frequencies cludedallhigher-multipolarelectric-typeresponsecoeffi- .500Hz. While tidal coupling will produce only a small cients (for the mass multipole moments) as well as their perturbationtothe GWsignalinthislow-frequencyadi- magnetic-type analogues (for the current multipole mo- abatic regime,the tidal signalshould be relativelyclean, ments). While these results concern a star’s response to depending (at leading order) on a single parameter per- astatictidalfield,theyarestillapplicableforinspiralling taining to the neutron star structure. This tidal de- binaries atsufficiently low orbitalfrequencies; a star will formability parameter λ is the proportionality constant approximately maintain equilibrium with the instanta- between the applied tidal field and the star’s induced neous tidal field if the field changes adiabatically. quadrupole moment and is sensitive to the neutron star Orbital Dynamics including Quadrupole: In their EoS. The measurement scheme proposed in Ref. [8] is treatmentoftidaleffectsinbinaryGWsignalsinRef.[8], based on an analytical model for the tidal contribution Flangan and Hinderer used Newtonian gravity to treat to the GW signal, giving a linear perturbation to the (the conservativepartof)the system’s orbitaldynamics; GW phase proportional to λ. At GW frequencies .400 they estimated that 1PN corrections to the orbits would Hz,themodelshouldbesufficientlyaccuratetoconstrain modify the calculated tidal signal by 10%. A general ∼ 2 formalism for calculating 1PN corrections to the orbital canbecharacterizedbyderivativesofanexternalNewto- dynamics ofextended bodies hasbeen developedina se- nianpotentialφ (t,x); inabinarysystem,theleading- ext ries of papers [14, 15] by Damour, Soffel, and Xu (DSX) order potential is φ = GM/r, where G is Newton’s ext − and later extended by Racine and Flanagan[16]. In this constant, M is the mass of the companion, and r the paper,weapplythatformalismtodetermineandanalyze distance between the body and its companion. theexplicit1PNtranslationalequationsofmotionforbi- Thenon-uniformfieldofthecompanionproducestidal nary systems with quadrupolar tidal interactions. While forcesonthenon-sphericalbody(inadditiontotheusual suchequationsofmotionhavepreviouslybeenpresented 1/r2 force) according to in Refs. [17, 18], we find that those results differ from ours. We also extend previous results by analyzing 1PN 1 1 Mzï = M∂ + Qjk∂ + Qjkl∂ +... φ momentum conservation for the binary system (which − i 2 ijk 6 ijkl ext (cid:18) (cid:19) serves as a strong consistency check for our equations M Qij Qijk of motion, and one not satisfied by the results [17, 18]), GM + | | + | | +... ∼ r2 r4 r5 by specializing the equations of motion to the system’s (cid:20) (cid:21) center-of-mass-energy frame, and by formulating an ac- GM2 R2 R3 1+ +O , (1.2) tion principle for the orbital dynamics. ∼ r2 r2 r3 (cid:20) (cid:18) (cid:19)(cid:21) Gravitational Wave Emission: To calculate the emitted GW signal and radiation reaction effects, Flanagan where M is the mass of the body or its companion, as- and Hinderer [8] used the quadrupole radiation approx- sumed here to be roughly equal, and the derivatives of imation and associated 2.5PN radiation reaction forces. the external potential are evaluated at the body’s center To consistently generalize their calculation in Ref. [8] to of mass, xi = zi(t). The contributions to the net force (relative)1PNorder,it willbe necessaryto compute the fromthequadrupoleandhigher-ordermultipolesarethus 3.5PN corrections to the GW generation, in addition to seen to take the form of an expansion in R/r, the ratio the 1PN orbital corrections considered in this paper; we of the size of the body to the orbital separation. When leave these calculations to future work. this finite-size parameter is small, as in the early stages of binary inspiral, the tidal force is well approximated Theissueoftidaleffectsininspiral-stageNSNSbinary bythequadrupolartermalone. Amoredetailedaccount GW signals has also been recently addressed by means of Newtonian tidal forces and related results is given in of numerical relativity simulations in Refs. [19, 20], and Sec. II below. analytically,within the context of the effective one body In neutron star binaries, a quadrupole is induced by (EOB) formalism, by Damour and Nagar [21]. Ref. [21] differential forces resulting from the non-uniformfield of proposes a contribution to the EOB “radial potential” thecompanion. Intheadiabaticlimit,whentheresponse that should accountfor the same 1PNcorrections to the time scale of the body is much less than the time scale quadrupole-tidalinteractionconsideredin this paper. In onwhich the tidalfield changes,the induced quadrupole AppendixA,wedemonstratetheequivalenceoftheEOB will be given (to linear order in the tidal field) by Hamiltonian of Ref. [21] and our Lagrangian (1.12) for circular orbits; we also show how the EOB Hamiltonian Qij(t)= λ∂ ∂ φ (t,z), (1.3) can be extended to include 1PN tidal effects for non- − i j ext circular orbits. withtheconstantλbeingthetidaldeformability. Thisis related to the more often used dimensionless Love num- ber k by λ = 2k R5/3G, where R is the body’s radius 2 2 B. Newtonian tidal coupling [13]. Using Newtonian gravity to describe the orbital dy- Gravitational tidal coupling arises from the interac- namics and the adiabatic approximation to model the tion of the non-spherical components of a body’s matter stars’ induced quadrupoles, Flanagan and Hinderer [8] distribution with a non-uniform gravitational field. The calculated the effect of tidal interactions on the phase non-sphericity is characterized at leading order by the of the gravitational waveform emitted by an inspiralling body’s mass quadrupole moment, neutronstarbinaryandanalyzedthemeasurabilityofthe tidaleffects. TheyfoundthatAdvancedLIGOshouldbe 1 able to constrain the neutron stars’ tidal deformability Qij(t)= d3xρ(x¯ix¯j δij x¯ 2), (1.1) − 3 | | to λ (2.0 1037gcm2s2)(D/50Mpc) with 90% con- Z ≤ × fidence, for a binary of two 1.4 M neutron stars at a ⊙ withρ(t,x)beingthemassdensityandx¯i(t)=xi zi(t) distance D from the detector, using only the portion of − being the displacement from the body’s center of mass the signal with GW frequencies less than 400 Hz. The position xi = zi(t). The quadrupole is (at most) on the calculationsofλfora1.4M neutronstarinRefs.[9–12], ⊙ orderofQij MR2, with M being the body’s mass and using several different equations of state, give values in R its radius.∼ Higher-order deformations are described the range 0.03–1.0 1037gcm2s2, so that nearby events by the octupole, Qijk d3xρx¯ix¯jx¯k MR3, and mayallowAdvance×dLIGOtoplaceusefulconstraintson ∼ ∼ higher-order multipole moments. The non-uniform field candidate equations of state. R 3 Refs. [8, 9] estimate the fractional corrections to surpressedbyhigherpowersofR/randwhichweneglect the tidal signal due to several effects neglected by the in our analysis [16]. model of the GW phasing used in Ref. [8], namely, The first term in the bottom line represents the 1PN non-adiabaticity (.1%), higher-multipolar tidal cou- spin-orbit coupling [15]. Though it would be convenient pling (.0.7%),nonlinear hydrodynamic effects (.0.1%), to be able to specialize to the case of non-spinning bod- spin effects (.0.3%), nonlinear response to the tidal ies when studying tidal interactions, tidal fields will in field (.3%), viscous dissipation (negligible), and post- general lead to tidal torques on non-spherical bodies; a Newtonianeffects(.10%). Thelargestcorrections,from consistent 1PN treatment of tidal coupling thus requires post-Newtonian effects in the orbital dynamics and GW the inclusion of 1PN spin-orbit coupling terms. The fi- emission, will depend on the neutron star physics only nal term of the bottom line denotes 1PN contributions through the same tidal deformability parameter λ used to the EoMs from the bodies’ current quadrupoles and in the Newtonian treatment and thus can be easily in- higher current multipoles, which we will not include in corporated into the data anaysis methods outlined in our analysis. Refs. [8, 9]. TheDSXtreatmentof1PNcelestialmechanics[14,15] providesaframeworkforcalculatingtheorbitalEoMsfor bodieswitharbitrarilyhigh-ordermassandcurrentmul- C. Post-1-Newtonian corrections tipole moments. In Ref. [15], DSX applied their formalism to rederive the explicit 1PN EoMs for bodies with mass monopoles and current dipoles (spins). The calcu- Forinspriallingneutronstarbinarieswithatotalmass lationwasthenextendedtoincludetheeffectsofthebod- of 3M atorbitalfrequenciesof 200Hz(GWfrequen- ⊙ ∼ ∼ ies’ mass quadrupoles by Xu et al [17, 18]. Racine and cies of 400 Hz), the post-Newtonian expansion param- ∼ Flanagan (RF) [16] later reworked the DSX formalism eter v2/c2 GM/c2r is 0.1, so that the 1PN approxi- ∼ ∼ and presented explicit 1PN EoMs for bodies with arbi- mationiswellsuitedtodescribingrelativisticcorrections trarilyhigh-ordermultipoles,whichcanbespecializedto to the binary orbit. As discussed in depth in Sec. III be- thecaseofbodieswithonlyspinsandmassquadrupoles. low, the 1PN orbital dynamics of a binary system with Wehavefound,however,thatthefinalresultsofXuetal tidal coupling can be described by translational equa- andthoseofRFareindisagreementwitheachotherand tions of motion (EoMs) similar in form to the Newto- with our recent calculations. Some typos and omissions nian equations schematically represented in (1.2), giving leading to errorsin RF have been identified and are out- the center-of-mass acceleration of each constituent body lined in an upcoming erratum; the results given in this in terms of their positions and multipole moments. The paper agree with the corrected results of RF. In Sec. III 1PNequationsofmotionaddorder1/c2correctionterms below, we review the essential ideas of the DSX formal- to (1.2) which depend not only on the bodies’ positions ism, following the notations and conventions of RF, and and mass multipole moments, M, Qij, Qijk, etc., but weoutlinethefullprocedurebywhichourresultsforthe also on their velocities and current multipole moments, 1PN EoMs are derived. Si, Sij, etc. Expanding in both the post-Newtonian pa- rameterv2/c2andthefinitesizeparameterR/r,the1PN equations of motion can be written schematically as D. Summary of results M v2M v4M Mzï GM + +O ∼ r2 c2 r2 c4 r2 1. The M1-M2-S2-Q2 system (cid:20) (cid:18) (cid:19) Qij v2 Qij Qijk R3M +| | + | | +O | | Ourresultsconcernthe1PNgravitationalinteractions r4 c2 r4 r5 ∼ r3 r2 (cid:18) (cid:19) in a system of two bodies, which we label “1” and “2”. v Si v Sij v2R2M We model body 1 as an effective point particle, with + | | +O | | (1.4) c2r3 c2r4 ∼ c2 r2 r2 a mass monopole moment only, while we take body 2 (cid:18) (cid:19)(cid:21) to have additionally a spin and a mass quadrupole mo- In the top line, the first term gives the usual point- ment. We consistently work to linear order in the spin particle (monopole) force of Newtonian gravity, and the andquadrupole,andourresultscanthusbeeasilygener- second term represents its 1PN corrections. The last alized to the case of two spinning, deformable bodies by term of the top line denotes 2PN and higher post- interchanging particle labels. We initially assume noth- Newtonian order corrections, which we neglect in this ingaboutthebodies’internalstructureordynamics. Our paper. Point-particle EoMs are in fact currently known primaryassumptionisthevalidityofthe1PNapproxima- up through post-3-Newtonian order [22]. tiontogeneralrelativityinavacuumregionsurrounding In the second line of Eq. (1.4), we have first the New- the bodies. tonian quadrupole-tidal term, followed by its 1PN cor- The system’s orbital dynamics can be formulated rections(whicharethesubjectofthispaper),andfinally in terms of the bodies’ center-of-mass worldlines zi(t) 1 contributions from the octupole and higher mass multi- and zi(t) and their multipole moments: the mass 2 poles (and their post-Newtonian corrections) which are monopolesM (t)andM (t),thespinSi(t),andthemass 1 2 2 4 quadrupole Qij(t). (c.f. Sec. IIF). Eqs. (1.6) ensure that M satisfies the 2 2 The wordlines xi = zi(t) and xi = zi(t) parametrize 1PN evolution equation (3.26,3.27). The decomposition 1 2 the bodies’ positions to 1PN accruacy in a (confor- of the monopole M in (1.6a) is essential for properly 2 mally Cartesian and harmonic) ’global’ coordinate sys- formulating an action principle for the orbital dynam- tem (t,xi), which tends to an inertial coordinate system ics. The evolution of the spin Si is determined by the in Minkowski spacetime as x . The global coor- (Newtonian-order) tidal torque formula: | | → ∞ dinates and center-of-mass worldlines are defined more precisely in Sec. IIID. We use the following notation for S˙i = 3M1ǫijkQjanank+O(c−2), (1.7) the relative position and velocity: r3 zi =zi zi, r = z = δ zizj, ni =zi/r, 2− 1 | | ij as in (2.45). Finally, the 1PN translational equations of vi =z˙i, vi =z˙i, vi =vi vi, r˙ =vini, motion, which govern the evolution of the worldlines zi 1 1 2 2 p 2− 1 1 and zi, are of the form with dots denoting derivativeswith respectto the global 2 frame time coordinate t. The multipole moments—M1(t), M2(t), S2i(t), and z¨1i = F1i(zj,v1j,v2j,M1,M2,Sj,Qjk,Q˙jk,Q¨jk), Qi2j(t) for our truncated system—are defined in z¨2i = F2i(zj,v1j,v2j,M1,M2,Sj,Qjk,Q˙jk), Secs.IIIBandIIIEviaamultipoleexpansionofthe1PN metricinavacuumregionsurroundingeachbody[16]. In and are given explicitly by Eqs. (3.31). thecaseofweaklyself-gravitatingbodies,thesemoments In Sec. IV, we define and calculate the 1PN-accurate can be defined as intergrals of the stress-energy tensor mass dipole moment of the entire system Mi (t), given sys overthevolumeofthebodies,asinEqs.(3.12);thesedef- in Eq. (4.8). We find that the condition Mï =O(c−4), sys initionscoincidewiththoseoftheBlanchet-Damourmul- required by Einstein’s equations and reflecting the con- tipole moments introducedinRef. [23]andusedby DSX servation of the system’s total momentum, is satisfied [14,15]. Themassmultipolemoments(likeM1,M2,and as a consequence of the orbital EoMs (3.31); this serves Qij) are defined with 1PN accuracy, while the current as a non-trivial consistency check for our results. The 2 multipole moments (like Si) appear only in 1PN-order conservation of momentum also allows us to specialize 2 terms andthus needonly be definedwith Newtonianac- theEoMstothesystem’scenter-of-mass(-energy)(CoM) curacy. We will often denote the spin and quadrupole of frame, which can be defined by the condition Mi = 0 sys body 2 by S and Qij, dropping the “2” labels. as in Sec. VA. The two EoMs (3.31) for the worldlines i zi and zi can then be traded for the single EoM for 1 2 the CoM-frame relative position zi = zi zi, given in 2. General equations of motion and orbital Lagrangian Eq. (5.9). 2 − 1 Ourresultscanbemostcompactlysummarizedbygiv- The equations of motion for the monopoles M (t) and ing a Lagrangian formulation of the CoM-frame orbital 1 M (t), the spin Si(t), and the worldlines zi(t) and zi(t) dynamics, as discussed in Sec. (VB). We find that the 2 1 2 are determined by Einstein’s equations alone, while that CoM-frame orbital EoM (5.9) can be derived from the for the quadrupole Qij(t) will depend on the details of Euler-Lagrangeequation body 2’s internal dynamics and can initially be left unspecified. The mass monopole of body 1, the effective ∂ d ∂ d2 ∂ + =0, (1.8) point particle, is found to be conserved to 1PN order: ∂zi − dt∂vi dt2∂ai Lorb (cid:18) (cid:19) M˙ =O(c−4), (1.5) 1 with a generalized Lagrangian given by orb L while that of body 2 is not. As discussed in Sec. IIIF, the1PN-accuratemassmonopoleM canbedecomposed = + + , (1.9a) 2 orb M S Q L L L L according to with the monopole part, M = nM +c−2 Eint+3U +O(c−4). (1.6a) 2 2 2 Q Here, nM2 is the New(cid:0)tonian-order(cid:1)(rest mass) contribu- = µv2 + µM + µ 1−3ηv4 (1.9b) tion, which is conserved: LM 2 r c2 8 (cid:20) nM˙ =0. (1.6b) M M 2 + ηr˙2+(3+η)v2 +O(c−4), The 1PN contributions to (1.6a) involve the Newtonian 2r − r (cid:18) (cid:19)(cid:21) potential energy of the quadrupole-tidal interaction, the spin part, 3M U = 1Qijninj, (1.6c) Q − 2r3 χ 2M χ andtheNewtonianinternalenergyofbody2,E2int,whose LS = c21ǫabcSavb r2 nc+ 21ac +O(c−4), (1.9c) evolution is governedby the rate at which the tidal field (cid:20) (cid:21) does work on the body [24]: and the quadrupole part, 3M E˙int = AQ˙ijninj, (1.6d) 2 2r3 5 3M 1 M M = 1Qabnanb+ Qab nanb A v2+A r˙2+A +A vavb+A r˙navb LQ 2r3 c2(r3 (cid:20) (cid:18) 1 2 3 r (cid:19) 4 5 (cid:21) M M + Q˙ab A navb+A r˙nanb +Eint A v2+A +O(c−4). (1.9d) r2 6 7 2 8 9 r (cid:20) (cid:21)) (cid:2) (cid:3) We use here the notation ization of the derivatives of the Newtonian potential in Eq. (1.3), and λ is the 1PN-accurate tidal deformability. M =M1+ nM2, χ1 =M1/M, χ2 = nM2/M, With the quadrupole given by (1.10), the tidal torque µ=M nM /M, η =χ χ =µ/M, (1.7) vanishes (c.f. Eq. (2.46)), so that the spin Si is 1 2 1 2 constant; thus, in the adiabatic limit, we can specialize withM beingthetotal(conserved)Newtonianrestmass, tothe caseSi =0withoutgeneratinginconsistencies. In µthereducedmass,andηthesymmetricmassratio. The Sec. VIA, we show that the adiabatic evolution for the dimensionless coefficients A –A appearing in (1.9d) are quadrupole(1.10)canbederivedfromaLagrangianthat 1 9 functions only of the mass ratios χ and χ given by adds to the orbital Lagrangianan internal elastic poten- 1 2 (5.10e). tial energy term which is quadratic in the 1PN-accurate quadrupole: 3. Adiabatic approximation for the induced quadrupole 1 The above results concerning the orbital EoM and its = orb[zi,Qij] QabQab+O(c−4), (1.11) L L − 4λ Lagrangianformulation are valid regardless of the internal structure of body 2, i.e. for arbitrary evolution of its quadrupole Qij(t). In Sec. VI, we discuss a simple adiabatic model for the evolution of Qij. In the adiabatic with orb given by Eq. (1.9) with Si = 0, and with limit,thequadrupolerespondstotheinstantaneoustidal Eint =L (1/4λ)QabQab + O(c−2). (Note that any addi- 2 field according to tional constant contribution to the internal energy Eint 2 can be included as a 1PN contribution to the constant Qij(t)=λGij(t), (1.10) nM in Eq. 1.6a.) Substituting the solution (1.10) for 2 2 Qij(t) into this Lagrangian, we obtain a simplified La- where Gij(t), given by Eq. (6.6b), is the quadrupolar grangianinvolving only the CoM-frame relative position 2 gravito-electric tidal moment of body 2, a 1PN general- zi(t): µv2 µM Λ µ M Λ Λ M Λ [zi]= + 1+ + θ v4+ v2 θ +ξ +r˙2 θ +ξ + θ +ξ (1.12) L 2 r r5 c2 0 r 1 1r5 2 2r5 r 3 3r5 (cid:18) (cid:19) (cid:26) (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)(cid:27) with Λ=(3χ /2χ )λ, and with the dimensionless coeffi- The orbital EoM resulting from this Lagranigan, which 1 2 cients can also be found by substituting the adiabatic solution (1.10) for Qij(t) directly into the general orbital θ0 = (1 3η)/8, EoM (5.9), is given by Eq. (6.10). The conserved en- − θ = (3+η)/2, ergy E(z,v) derivedfrom the Lagrangian(1.12) is given 1 by Eq. (6.13). In the case of circular orbits, we find the θ = η/2, 2 gauge-invariantenergy-frequency relationship θ = 1/2, 3 − ξ = (χ /2)(5+χ ), 1 1 2 ξ = 3(1 6χ +χ2), 2 − − 2 2 ξ = 7+5χ . (1.13) 3 2 − 6 1 9χ λω10/3 (9+η)(Mω)2/3 11χ λω4 E(ω)=µ(Mω)2/3 + 1 + + 1(3+2χ +3χ2) . (1.14) −2 2χ M5/3 24 c2 4χ 2 2 Mc2 (cid:20) 2 2 (cid:21) This result, along with others from this paper, will be bodies. We consider in particular the case of a binary useful in calculating the phasing of GW signals from in- system containing a point particle (body 1) and an ex- spiralling neutron star binaries. tendeddeformablestar(body2),workingtoquadrupolar order in the star’s multipole series. We also discuss an action principle formulation of the orbitaldynamics, the E. Notation and Conventions processofenergytransferbetweenthe gravitationalfield andthedeformablebody,andtheevolutionofthebody’s We use units where Newton’s constant is G = 1, but spin due to tidal torques. Finally, we discuss the evolu- retain factors of the speed of light c, with 1/c2 serv- tionofthe body’stidally inducedquadrupolemomentin ing as the formal expansion parameter for the post- the adiabatic limit. Newtonian expansion. We use lowercase latin letters a,b,i,j,... for indices of(three-dimensional)spatialten- sors. Spatial indices are contracted with the Eucliden A. Field equations metric, viwi = δ viwj, with up or down placement of ij the indices having no meaning. We use uppercase latin In Newtonian physics, the scalar potential φ(t,x) letters to denote multi-indices: L denotes the l indices obeys the Poissonequation, a a ...a , K denotes the k indices b b ...b , etc. For a 1 2 l 1 2 k given vector vi or for the partial derivative operator ∂ , 2φ=4πρ, (2.1) i ∇ we use multi-indices or explicit sequences of indices to denote their tensorial powers: with ρ(t,x) being the rest mass density of matter. The influence ofthe gravitationalfieldonmatter isdescribed vL = va1a2...al =va1va2...val, by the test particleaccelerationxï = ∂ φ, or moregen- i − ∂ = ∂ =∂ ∂ ...∂ , (1.15) erally, by Euler’s equation supplimented by the continu- K b1b2...bk b1 b2 bk ity equation (the conservation of mass), and, for example, vij = vivj. We also use v2 = vii danisdtin∇ct2t=ens∂oiris. ofMvualrtyi-iningdricaensk,areMa,lMsoau,Msedab,fo.r..se,twsitohf ∂t(ρvi)+∂j(ρvivj +tij)=−ρ∂iφ, (2.2) ML = Ma1a2···al denoting the te{nsor of rank l. W}e use the Einstein summation convention for both individual ρ˙+∂ (ρvi)=0, (2.3) i indices and multi-indices. Derivatives with respect to a time coordinate t are denoted by ∂ or by overdots. with vi(t,x) being the matter’s velocity field, and t We use angular brackets to denote the symmetric, tij(t,x) the material stress tensor. Together, Eqs. (2.1- trace-free (STF) projection of tensors [14]: 2.3) provide a complete description of Newtonian gravitational interactions. However, they do not in general 1 T<ab> = T(ab) δabTcc, form a closed set of evolution equations for the fields φ, − 3 ρ,vi,andtij;oneneedsalsotospecifythematter’sinter- 3 T<abc> = T(abc) δ(abTc)dd, (1.16) nal dynamics, in particular concerning the stress tensor − 5 t . (Inthe simplestcases,onecanfix tij byanalgebraic ij andsoon,withparenthesesdenotingthesymmetricpro- equation, like tij = 0 for ‘dust’ or tij = pδij with p(ρ) jection. ForaSTFtensorSL =S<L> andgeneraltensor being the pressure for an isentropic perfect fluid.) Still, TL, note that SLTL =SLT<L>. one can derive many useful results, like the form of the translationalequationsofmotionforasystemofgravitat- ing bodies, while leaving the matter’s internal dynamics II. NEWTONIAN TIDAL INTERACTIONS unspecified. In this section, we review the standard treatment of tidal coupling in Newtonian theory [13]; the post-1- B. Multipole moments Newtonian treatment given subsequent sections makes extensiveuseoftheseNewtonian-orderresults. Wedefine We consider N isolated celestial bodies, i.e. regions of themultipolemomentsandtidalmomentsofanextended space containing matter (ρ = 0) surrounded by regions 6 object and use them to derive the orbital (or transla- of vacuum (ρ = 0), and label the bodies by an index A, tional) equations of motion for systems of gravitating with 1 A N. The potential that is locally generated ≤ ≤ 7 by body A, whichwe will callthe internalpotential φint, about which the multipole expansion is centered to be A is given by the standard solution to (2.1) as an integral fixed to the body’s center of mass position: over the volume of the body: 1 zi (t)= d3xρ(t,x)xi, (2.10) ρ(t,x′) A M φint(t,x)= d3x′ . (2.4) A Z A − x x′ ZA | − | which is equivalent to (2.9b). Next comes the l = 2 Wecanexpressthispotentialasamultipoleseriesaround terminvolvingthebody’smassquadrupoletensorQij(t) A a (moving) point xi =zi (t) by using the Taylor series (2.9c), which we have renamed Mij Qij to accord A A → A with commonconvention. The higher-orderterms in the 1 ∞ ( 1)l 1 multipole series are supressed by increasing powers of = − (x′ z )L∂ , (2.5) x x′ l! − A L x z thefinitesizeparameterR/x¯ ,whereRisthesizeofthe | − | Xl=0 | − A| body. | | We note that the continuity equation(2.3) implies the where L = a a ...a is a spatial multi-index, denoting here tensorial1p2owerslof the vector (x′ z )i and of the constancy of the body’s total mass, M˙A =0. The Euler A − equation (2.2) similarly constrains the evolution of the operator ∂ (c.f. Eq. (1.15)). Using the Taylor series i mass dipole Mi, thus determining the body’s transla- (2.5)in(2.4)allowsustowritetheinternalpotential,for A points xi exterior to the body, in the form tionalequationofmotion,asdiscussedinSec.IIDbelow. Eqs. (2.2)and(2.3)donot,however,fully determinethe ∞ ( 1)l 1 evolutionofthequadrupoleandhighermultipoles,which φiAnt(t,x)=− −l! MAL(t)∂L x z (t) , (2.6) will depend on the body’s internal dynamics. A l=0 | − | X with C. Tidal moments ML(t)= d3xρ(t,x)[x z (t)]<L>. (2.7) A − A Having described the potential generated by an iso- ZA lated body with its multipole moments, we can similarly The quantities ML are the mass multipole moments of describe the potential felt by the body with tidal mo- A the body about the worldline zi (t). They are symmet- ments. Given a collection of several bodies indexed by A ric, trace-free spatial tensors of varying order l, ML = B,eachgivingriseits ownintrinsicpotentialofthe form A Ma1a2...al. The STF property follows from the fact that (2.6), we define the external potential felt by a given A ∂ x−1 is an STF tensor, because partial derivatives body A to be the sum of the potentials due to all the L com| m|ute, and because ∂ x−1 = 2 x−1 = 0. As other bodies: jj themultipole momentsML|ar|econtra∇cte|d|withtheSTF tensors ∂ x zA −1 in (2A.6), only their STF parts will φeAxt = φiBnt (2.11) L contribute|to−the p|otentialφint;thisiswhytheSTFpro- BX6=A A jection (denoted by angular brackets) has been included The body’s tidal moments1 GL (t) are then defined as in the definition of the multipole moments in (2.7). g,A coefficients in the Taylor expansion of the external po- The leading-order terms in the multipole series (2.6) tential about the center-of-mass position zi : can be written more explicitly as A ∞ 1 φiAnt =−Mx¯A − 21QiAj∂ij x1¯ +... (2.8) φeAxt(t,x)=− l!GLg,A(t)[x−zA(t)]L, (2.12) | | | | Xl=0 with GL (t)= ∂ φext(t,x) . (2.13) g,A − L A x=zA(t) M = d3xρ, (2.9a) A (cid:12) ZA (cid:12) 0=Mi = d3xρx¯i, (2.9b) A 1 The subscript g, standing for ‘global,’ has been included here ZA to avoid confusion with tidal moments introduced in our post- 1 Qij =Mij = d3xρ x¯ix¯j x¯ 2δij , (2.9c) Newtonian treatment below. We introduce there a set of body- A A ZA (cid:18) − 3| | (cid:19) frame tidal moments GLA, defined in an accelerated reference frameattached tothebody, andasetofglobal-frametidal mo- agnenderx¯aite=dxbiy−thzeAit.otFailrmstaissstohfethmeobnoodpyoleMter(m2.9(al)=, a0n)d, tmideanltsmGomLg,Aen,tdsedfienfiendedininan(2(.a1s3s)ymcopintoctidicealwlyit)hintehretilaalttferarmate.NTewhe- A tonian order. While we have chosen to work exclusively in an givingrisetoaCoulomb-typepotentialin(2.8). Wehave inertial frame in our Newtonian treatment here, an analogous omitted the dipole term (l=1) in (2.8) because Mi can A Newtoniantreatmentthatusesaccelerated framescanbefound always be made to vanish by choosing the point zi (t) inSec.IIIofRef.[15]. A 8 Like the multipole moments, the tidalmoments areSTF find the following EoMs: tensors,GL =G<L>,ascanbeseenfromthedefinition g,A g,A 1 1 1 (2.13) and the fact that 2φext = 0 everywhere outside M zï = M M ∂(1) + M Qjk∂(1) ∇ A 1 1 1 2 i z z 2 1 ijk z z the bodies B = A. We see that G is simply (minus) 1 2 1 2 g,A | − | | − | thepotentiala6tthebody’scenter,andGig,A isthewould- M zï = M M ∂(2) 1 + 1QjkM ∂(2) 1 be testparticleacceleration−∂iφeAxt. Forl≥2,theGLg,A 2 2 2 1 i |z2−z1| 2 1 ijk|z2−z1| arehigher-orderderivativesofthepotentialthatwillgive Defining the radius and unit vector associated with the rise to tidal forces on a nonspherical body. relative position, The tidal moments of a body A can be expressed in terms of the multipole moments ofthe other bodies B = 6 zi =zi zi, r = z , ni =zi/r, (2.17) A by combining (2.6), (2.11), and (2.13), with the result 2− 1 | | and using the general identity, GL = ∂ φint(t,x) g,A − L B x=zA(t) B6=A 1 n<L> X (cid:12) ∂ =( 1)l(2l 1)!! , (2.18) ∞ ( 1)k (cid:12) 1 Lr − − rl+1 = − MK∂(A) , (2.14) k! B KL zA zB these EoMs can be written as B6=Ak=0 | − | X X where ∂i(A) =∂/∂zAi , and ∂KL =∂b1...∂bk∂a1...∂al. M1z¨1i = −MMM2z¨2i 15M = 1 2n + 1Qjkn<ijk> (2.19) r2 i 2r4 D. Translational equations of motion In this form, it is evident that the total momentum pi = M z˙i+M z˙i isconserved,andthatthesetwoEoMsforzi 1 1 2 2 1 Aprimaryadvantageofthe languageofmultipole and andzi canbetradedfortheEoMoftherelativeposition 2 tidalmomentsisthatitallowsonetotakethePDEs(2.1- zi =zi zi: 2.3) governing the evolution of the fields ρ, vi, tij, and 2− 1 φ and extract from them ODEs for the center-of-mass M 15M zï = ni Qjkn<ijk> (2.20) worldlines zi (t) of a collection of gravitating bodies. To −r2 − 2M r4 A 2 this end, we consider a body A with multipole moments M 3M MLdefinedby(2.7),inthepresenceofanexternalpoten- = ni Qjk(5nijk 2δijnk) A −r2 − 2M r4 − tial φext generated by other bodies B = A according to 2 A 6 (2.12) and (2.14). The body’s translational EoM can be where M = M +M is the total mass. In the second 1 2 found by applying two time derivatives to the definition line, we have used Eq. (1.16) to expand n<ijk> and the of zi in (2.10), using the Euler and continuity equations fact that Qjk is STF. A (2.3) and (2.2), and intergrating by parts. The result is With Eq. (2.20), we have reduced the description of an expressionfor the body’s center-of-massacceleration, the binary system’s translational dynamics to a single ODE. Still, we can only solve this ODE if we know the M zï = d3xρ∂ φext, (2.15) time evolution of the quadrupole moment Qij(t), which A A − i A requires a detailed model of body 2’s interior. We will Z describe a simple adiabatic model for Qij in Sec. IIG which can be rewritten in terms of the body’s multipole below. and tidal moments by using (2.12) and (2.7): ∞ M zï = 1MLGiL (2.16) E. Action principle A A l! A g,A l=0 X 1 As for any Newtonian system, a Lagrangian for a col- = MAGig,A+ 2QjAkGigj,kA+... lection of gravitating bodies can be constructed from =T U,withT beingthekineticenergyandU thepo- L − Thefirstterminthesecondlinerepresentstheforcethat tential energy. The total kinetic energy receves separate would act on a freely falling test mass M , while the contributions from each body, T = T , and each T A A A A second term gives the leading-order tidal force. can be split into a contribution from the center-of-mass P TorendertheEoM(2.16)fullyexplicit,onecanusethe motion of the body and an internal contribution: expressions for the tidal moments (2.14) in terms of the 1 1 multipole moments and worldlines of the other bodies. T = d3xρv2 = M z˙2 +Tint, (2.21) Consideringthecaseofatwo-bodysystemA=1,2,with A 2ZA 2 A A A body1havingonlyamonopolemomentM ,andbody 2 1 having a monopole M2 and a quadrupole Q1ij ≡ Qi2j, we TAint = 2ZAd3xρ(v−z˙A)2. (2.22) 9 Here,wehaveusedthefactthat d3xρvi =M z˙i ,im- (We have omitted the internal Lagrangian for body 1 A A A plied by the continuity equation (2.3) and the definition as it is completely decoupled from the rest of the sys- of the center-of-mass position zRi in (2.10). The gravi- tem.) Varying this action with respect to zi and zi A 1 2 tational potential energy U = U can be similarly leadstotheir EoMsfoundin(2.19). Alternately,varying A A split into external and internal parts: with respect to their separation zi = zi zi gives the P 2 − 1 relative EoM (2.20), while the system’s center of mass 1 UA = 2ZAd3xρφ=UAext+UAint, (2.23) t(Mhe1Lz1aig+raMng2iaz2ni)(/2M.27i)stfoouthnedcetontebre-ocf-ymclaics.sfrSapmecei,awlihzienrge 1 M zi +M zi =0, gives Uint = d3xρφint, (2.24) 1 1 2 2 A 2 A ZA µz˙2 µM Uext = 1 d3xρφext = 1 ∞ 1MLGL . (2.25) L= 2 + r −UQ+Li2nt, (2.29) A 2 A −2 l! A g,A ZA l=0 X with µ=M M /M being the reduced mass. 1 2 In the last line, we have used (2.12) to express φext in While leaving the functions int(qα,q˙α) and Qij(qα) A L2 2 2 2 termsofthetidalmomentsand(2.7)forthedefinitionof unspecified, we can still use the Lagrangian (2.27) to the multipole moments. writedownEoMsforbody2’sinternalconfigurationvari- While the system’s total potential energy will also re- ables q2α: cieve non-gravitational contributions from the internal d ∂ int ∂ int 1 ∂Qij structureofeachbody, wecanlumpthese contributions, L2 = L2 + Gij . (2.30) along with Tint and Uint as defined above, into an inter- dt ∂q˙α ∂qα 2 g,2 ∂qα A A 2 2 2 nal Lagrangian int for each body. We can then write LA which will be useful in the next subsection. the total Lagrangianfor an N-body system as ∞ 1 1 1 = M z˙2 + MLGL + int . (2.26) F. Energy L 2 A A 2 l! A g,A LA ! A l=0 X X Continuing to specialize to the two-body M -M -Q Thebodies’center-of-massworldlineszi (t)enterthisLa- 1 2 2 A case,wecanconstructaconservedenergyforthesystem grangian through the translational kinetic energy terms from and through the external gravitational potential energy terms (via the tidal moments); the internal Lagrangians ∂ ∂ ∂ LcoiAnnts,trhuocwtieovne.r,Tahreeindinetpwenildlebnet foufntchteiownsoroldflsinomesezsAie,tboyf E =T +U = ∂zL˙1iz˙1i + ∂zL˙2iz˙2i + ∂qL˙2αq˙2α−L, (2.31) internalconfiguratioLnAvariablesqα (andtheirtimederiva- with summationoverα implied. Using the (CoM-frame) A tives) for each body, which will include e.g. Euler angles Lagrangian(2.29), we find for the orientation of the body, vibrational mode ampli- tudes,etc.,dependingonthemodelofthebody’sinternal µz˙2 µM E = +U +Eint, (2.32) structure. (Infullgenerality,theproperinternalconfigu- 2 − r Q 2 rationvariables qα are the fields ρ, vi and tij, subject to A where U is given by (2.28), and the internal energy of (2.10) as a constraint.) The bodies’ multipole moments Q ML, for l 2, appearing in the gravitational potential body 2 is given by A ≥ energy terms in (2.26), will be functions of these same ∂ int internal variables qα. Together, the zi and qα, for all Eint(qα,q˙α)= L2 q˙α int (2.33) bodies A, form a coAmplete set of dynamAical varAiables for 2 2 2 ∂q˙2α 2 −L2 the N-body system. Varying the action S = dt with respectto the worldlineszi reproducesthe tranLslational Thisinternalenergywillgenerallyhaveseveralcontribu- A R tions: internal gravitational potential energy, rotational EoMs (2.16). Determining the evolution of the variables kinetic energy, vibrational kinetic and potential energy, qα, and hence the moments ML for l 2, will require a mAodel for int(qα,q˙α) and MLA(qα). ≥ thermal energy, etc. Nonetheless, the rate at which en- LA A A A A ergy is exchanged between the interior of body 2 and its Specializingtothetwo-bodyM -M -Q caseandusing 1 2 2 surroundings, via the gravitational tidal interaction, is (2.14),(2.17),and(2.18),theLagrangian(2.26)becomes a function only of the orbital separation, M , and Qij. 1 1 1 M M Differentiating (2.33) with respect to time and using the L= 2M1z˙12+ 2M2z˙22+ 1r 2 −UQ+Li2nt. (2.27) EoM (2.30) for the internal variables q2α, we find where UQ is the potentialenergy ofthe quadrupole-tidal E˙int = q˙α d ∂Li2nt ∂Li2nt = 1Gij ∂Qijq˙α interaction: 2 2 dt ∂q˙α − ∂qα 2 g,2 ∂qα 2 (cid:18) 2 2 (cid:19) 2 1 3M 1 3M U = QijGij , Gij = 1n<ij>. (2.28) = Gij Q˙ij = 1nijQ˙ij (2.34) Q −2 g,2 g,2 r3 2 g,2 2r3 10 The energy transfer described by (2.34) is often referred to linear order in the tidal deformability λ. to as tidalheating (see e.g.[25]). This expressionfor the The internal energy Eint (2.33), in the adiabatic ap- 2 power delivered to the body is valid (in the quadrupolar proximation, is given by approximation)regardlessof the body’s internaldynam- 1 ics. Using (2.34) and the orbital EoM (2.20), one can Eint = QabQab, (2.40) 2 4λ confirm that the binary system’s total energy (2.32) is conserved. up to a constant,and satisfies the tidalheating equation (2.34)byvirtueofEq.(2.35). (Weshouldnotethatthere is actually no ‘heating’ taking place here, as this model G. Adiabatic approximation neglects dissapative effects and is completely conserva- tive.) Using Eq. (2.35), the binary system’s total energy While we have thus far left unspecified the internal E (2.32) can be written as dynamics for the deformable body 2, which determines the evolution of the quadrupole, we now specialize our µz˙2 µM 3λM1 E = 1+ , (2.41) analysistothecasewhereQij(t)isadiabaticallyinduced 2 − r 2r5M (cid:18) 2(cid:19) by the tidal field. This will lead to a closed system of in the adiabatic model, and is conserved by the orbital evolutionequationsforthebinary. Intheadiabaticlimit, EoM (2.38). Then, using z˙2 = r2ω2 and Eq. (2.39), we whenthebody’sinternaldynamicaltimescalesaremuch find less than the orbital period, the qudrupole will respond to the instaneous tidal field according to µ 9λM ω10/3 E(ω)= (Mω)2/3 1 1 , (2.42) Qij(t)=λGigj,2(t), (2.35) −2 (cid:18) − M2M5/3 (cid:19) for the circular-orbitenergy-frequency relationship. whereλisthetidaldeformability,andGab (t)isthetidal g,2 tensorgiveninEq.(2.28). AsdiscussedinSec.IBandin more detail in Refs. [8, 9], the relation (2.35) should be H. Spin validto 1%forneutronstarbinariesatGWfrequencies ∼ .400 Hz. In anticipation of the 1PN treatment of the binary’s Theadiabaticevolutionofthequadrupolecanbeinco- proratedintoouractionprinciple(2.29)bytakingQij(t) orbital dynamics, in which a body’s angular momentum tobeourloneinternalconfigurationvariable(qα),andby (or spin) has a direct influence on the orbit, it will be taking the internal Lagrangian int to contain2 a simple usefultodiscusstheevolutionofanextendedbody’sspin L2 atNewtonianorder. ThespinofabodyAaboutitsCoM quadratic potential energy cost for the quadrupole: worldline zi is defined by A µz˙2 µM [zi,Qij] = + U + int (2.36) L 2 r − Q L2 Sa(t)=ǫabc d3xρ(t,x) xb zb(t) vc(t,x), (2.43) = µz˙2 + µM + 1Gab Qab 1 QabQab. A Z (cid:2) − A (cid:3) 2 r 2 g,2 − 4λ with ρ being the mass density and vc the velocity field. Takingatimederivativeofthisequation,usingtheEuler Varyingthe actionwithrespectto the orbitalseparation zi still gives the orbital EoM (2.20), and varying with and continuity equations (2.2) and (2.3) and the defini- respect to the quadrupole Qij reproduces the adiabatic tion of zAi in (2.10), and integrating by parts, we find evolutionequation(2.35). UsingEq.(2.35)toreplaceQij and Eq. (2.28) for Gag,b2, the Lagrangian can can written S˙Aa = −ǫabc d3xρ(xb−zAb)∂cφeAxt solely in terms of zi as Z ∞ 1 [zi]= µz˙2 + µM 1+ 3λM1 . (2.37) = ǫabc l!MAbLGcgL,A. (2.44) L 2 r 2r5M Xl=0 (cid:18) 2(cid:19) Inthesecondline,wehaveusedthedefinitionsofMLand ThisLagrangianleadstotheorbitalEoM(2.20)withQij A GL in (2.7) and (2.13). This formula gives the torque replaced by its adiabatic value (2.35), g,A on the body due to tidal forces. As it is not directly Mni 9λM relevanttoourpurposes,wewillnotdiscussaLagrangian ai = 1+ 1 , (2.38) − r2 r5M formulation of the Newtonian rotational dynamics. (cid:18) 2(cid:19) Applying Eq.(2.44) to the M -M -Q system, we find 1 2 2 which shows that the tidal coupling results in an attrac- that the tidal torque on body 2 is given by tive force. For circularorbits,withai = rω2ni, we find the radius-frequency relationship − S˙2a =ǫabcQbdGcgd,2 (2.45) r(ω)= M1/3 1 3λM1ω10/3 , (2.39) with the tidal tensor Gcgd,2 given by (2.28). Eq. (2.45) is ω2/3 − M M5/3 valid (in the quadrupolar approximation) regardless of (cid:18) 2 (cid:19)