Next to leading order gravitational wave emission and dynamical evolution of compact binary systems with spin
aa r X i v : . [ g r- q c ] N ov Next to leading order gravitational wave emissionand dynamical evolution of compact binary systemswith spin
D¨orte Hansen
Institute of Theoretical Physics, Friedrich-Schiller-University Jena,Max-Wien-Platz 1, D-07743 JenaE-mail:
Abstract.
Compact binary systems with spinning components are considered. Finitesize effects due to rotational deformation are taken into account. The dynamicalevolution and next to leading order gravitational wave forms are calculated, takinginto account the orbital motion up to the first post-Newtonian approximation.PACS numbers: 04.25.Nx, 04.30.-w, 04.30.Tv, 95.85.Sz
Submitted to:
Class. Quantum Grav. ext to leading order gravitational wave emission
1. Introduction
Inspiralling compact binary systems are among the most promising sources for theemission of gravitational waves detectable with present day’s gravitational waveinterferometers. Earth bound gravitational wave detectors such as Geo600, VIRGO,TAMA and LIGO are most sensitive at wavelengths of about 10-1000 Hz. Thiscorresponds roughly to the last 10 minutes of the inspiral before final plounge. Inthat regime Newtonian mechanics is not valid and post-Newtonian approximation mustbe applied. An earlier stage of the inspiral process will be covered by the yet to bebuilt LISA interferometer. It is expected that LISA will be sensible to gravitationalwaves with frequencies from about 10 − to 10 − Hz. However, in order to actuallydetect gravitational waves highly accurate templates are essentially. Post-Newtoniancorrections must be included into the EoM. Moreover, computing the gravitationalwaveforms beyond the leading order approximation higher mass and current multipolemoments must be taken into account.In the past the study of close compact binaries was often based on the assumption thatthe stars can be treated as pointlike, non-spinning objects. Upon this assumption it ispossible to derive an analytic, so called quasi-Keplerian solution for the conservative partof the EoM up to the third post-Newtonian approximation [1]. Dissipative effects due tothe emission of gravitational waves first appear at the order ( v/c ) , which correspondsto the 2.5 post-Newtonian order.In the presence of at least one spinning component a post-Newtonian spin-orbit couplingwhich first appears at 1.5 post-Newtonian order leads to complifications, which hamperthe investigation of spinning compact binary systems enormeously. In general, neitherorbital angular momentum nor the stellar spin are conserved. Up to now an analyticsolution to the conservative part of the EoM including spin-orbit coupling has beenfound in a few special cases only (see e.g. [2]).Thus far not much progress has been made in the investigation of close binary systems offinite size objects. Within the framework of post-Newtonian analysis it is often arguedthat finite size effects are neglegible during most of the inspiral process and will becomeimportant not until the last few orbits before the final plunge [3]. Growing interest inthe role of finite size effects comes mainly from the side of numerical relativity. In fact,though finite size effects due to stellar rotation and oscillation in compact binary systemsare very small they can well be in the order of the first post-Newtonian corrections tothe orbital dynamics. These secular effects, being of Newtonian origin, accumulate overa large number of orbits and thus, seen at longer terms, lead to significant phase shifts inthe gravitational waves. The influence of stellar oscillations on the dynamical evolutionand leading order gravitational wave emission has been investigated by Kokkotas andSch¨afer [4] and Lai and Ho [5] for nonrotating, polytropic neutron stars and by Lai etal [6] and Hansen [7] for Riemann-S binaries. In all these approaches the analysis wasbased on Newtonian theory, the 2.5pN radiation reaction terms being the only post-Newtonian terms included. ext to leading order gravitational wave emission et al. in 1992 [10]. Whilethe compact component of PSR 1259-63 is a 47 ms pulsar, it’s companion is a Be star,whose spin-induced quadrupole deformation leads to an apsidal motion.However, it is close compact binary systems which are most relevant for gravitationalwave detectors. This includes not only black hole-black hole, black hole-neutron staror neutron star-neutron star binaries but, if LISA is ready to work, also close whitedwarf binaries. Of course the rotational deformation of compact stars is much smallerthan it would be possible in non-compact stars. However, at least for neutron star(NS) and white dwarf (WD) binaries one should not a priori neglect finite size effectsdue to quadrupole deformation. In particular it is well possible that the perturbationsintroduced by the coupling of the stellar quadrupole moment to the orbital motion is ofthe same order of magnitude as the first post-Newtonian corrections. This is assumedthroughout the paper. The analysis applies to compact binary systems, whose spinningcomponent is a WD or a fast rotating NS. In order to simplify calculations it is assumedthat the spin is perpendicular to the orbital plane. In section 2 the orbital evolution of aspinning compact binary is studied up to first post Newtonian order in the point particledynamics. The EoM as well as a parametric, quasi-Keplerian solution are derived. Insection 3 the next to leading order gravitational waveforms are calculated explicitly.The long time evolution and the influence of the quadrupole coupling to the inspiralprocess is discussed in section 4.
2. The 1pN orbital motion including spin effects due to rotationaldeformation
In 1985 Damour and Deruelle [11] succeeded in deriving an analytic solution to the1pN EoM of a point-particle binary, which exhibits a remarkable similarity to the wellknown Kepler parametrization in Newtonian theory and is hence called quasi-Kepleriansolution. Using the same strategy Wex considered a binary consisting of a pulsar and ext to leading order gravitational wave emission
4a spinning main sequence component at Newtonian order [12]. Treating the Newtoniancoupling between the rotationally deformed star and the orbital dynamics as a smallperturbation he derived a quasi-Keplerian solution up to first order in the deformationparameter q , which will be introduced in the following. In this letter we shall extendhis investigations, taking into account the orbital dynamics up to first post-Newtonianapproximation. In order to derive an analytic solution we shall further assume thatthe modifications induced by finite size effects are of the same order as the 1pN orbitalcorrections. That is, we restrict our analysis to compact binary systems consisting ofa fast spinning neutron star or white dwarf and a non-spinning compact object. Ofcourse, all results can be applied to a mean sequence star-compact star binary in theNewtonian limit.The rotational deformation of a spinning star of mass m can be described by someparameter q , which is defined as [13] mq := 12 Z dV ′ ρ ( r ′ ) (cid:2) r ′ − s · r ′ ) (cid:3) = ∆ I, (1)where m = R ρ ( r ′ ) dV ′ is the stellar mass and ∆ I is the difference of the moments ofinertia parallel and perpendicular to the spin axis ˆ s , respectively. In particular, forrotating fluids, q is given by (see e.g. [14]) q = 23 kR ˆΩ , ˆΩ = Ω p Gm/R . (2)Here R denotes the polar radius and Ω the angular velocity of the rotating star. Theconstant of apsidal motion k strongly depends on the density distribution. It vanishesif all mass is concentrated in the center and takes it’s maximal value k max = 0 .
75 for ahomogeneous sphere. Neutron stars and low mass white dwarfs can be approximatelymodelled by an polytropic EoS with index n=0 . . . . n ≈ . k is given by (see e.g.[15]) k = 12 (∆ ( n ) − , where the function ∆ ( n ) has been introduced by Chandrasekhar [16].The quadrupole deformation gives rise to a quadrupole coupling, which modifies theorbital dynamics as well as the gravitational wave emission of the binary. Neglectingcontributions arising from tidally induced stellar oscillations the quadrupole couplingcan be described by a Hamiltonian H q = G M µq r (cid:20) s · r ) r − (cid:21) , (3)where r is the orbital separation, while M and µ denote total and reduced mass,respectively. As we shall see in the following this contribution will lead to a periastronadvance already at Newtonian order.In general, the direction of the star’s spin won’t be conserved, which complicates theanalysis enormeously. However, one might assume scenarios where the spin is parallel ext to leading order gravitational wave emission H q = − G M µq r , (4)and we shall restrict to this special situation further on.Before we proceed it is important to compare the perturbation introduced by quadrupolecoupling with the first post-Newtonian correction terms to the EoM. Let us assume aspinning binary system in a circular orbit. Not taking into account the q -coupling theorbital energy reads ‡E orb = − G M µ r + 7 − ν G M µr c ≡ E N + E pN . (5)Comparing E pN with the coupling energy we find that the q -term offers a contributioncomparable to the 1pN orbital perturbation if qr ≈ G M c ≡ r S . If q/r is much larger than the Schwarzschild radius r S the Newtonian quadrupolecontribution will clearly dominate, while for q/r ≪ G M /c the leading perturbationto the Keplerian orbit comes from the post-Newtonian correction terms. The value of q crucially depends on the density distribution (via k ) and on the angular velorcity of therotating star. The later one is bounded by the mass-shedding limit. For a Newtonianstar with a polytropic EoS one can show that the mass-shedding limit is given by [17]Ω max = (cid:18) (cid:19) / r GmR , (6)where R is the polar radius of the star. Thus, for polytropic stars q is bounded by q max = 23 kR ˆΩ max = (cid:18) (cid:19) kR . For a typical neutron star with polytropic index n = 1 and polar radius R = 10 kmthe maximal value of q is q max = 5 . , while for a white dwarf with n = 1 . § q max = 28563 km . Now the post-Newtonian approximation is valid only outside the innermost stable circular orbit k .For finite size binaries the orbital separation should be considerably larger than thisvalue. To be more precisely, since we do not consider mass overflow the spinning starshould remain well inside it’s Roche volume throughout this calculation. Taking thisinto account it becomes clear that for close NS-BH and NS-NS binaries the contributionof the quadrupole coupling is by a factor 100 or more smaller than the first post-Newtonian correction. However, at least for sufficiently fast rotation the q -term gives ‡ Note, that E = H . § For n = 1 the constant of apsidal motion is k = 0 . n = 1 . k = 0 . k In isotrope coordinates the radius of the last stable orbit of a test particle orbiting a Schwarzschild BHis given by r ISCO = 5 G M /c . For compact binaries a calculation of r ISCO in full General Relativityhas not been successfull until now, but post-Newtonian analysis suggests a value around 5 G M /c . ext to leading order gravitational wave emission q -coupling).Introducing the reduced energy and angular momentum, E =: E /µ and J =: J /µ ,respectively, the 1pN conserved energy including Newtonian q -coupling reads E = v − G M r h q r i + 1 c (cid:20)
38 (1 − ν ) v + G M r (cid:26) (3 + ν ) v + ν ˙ r + G M r (cid:27)(cid:21) . (7)Since the spin is parallel to the orbital angular momentum J the orbital plane is invariantin space. We are thus encouraged to introduce polar coordinates r and ϕ in the usualway. Inserting v = ˙ r + r ˙ ϕ into Eq. (7) one finds, after a little algebra, the 1pNexact EoM to be˙ ϕ = Jr (cid:20) − − νc E − G M c r (4 − ν ) (cid:21) , (8)˙ r = A + 2 Br + Cr + Dr , (9)where A = 2 E (cid:20) ν − Ec (cid:21) ,B = G M (cid:20) ν − Ec (cid:21) ,C = − J + 1 c (cid:2) − ν ) EJ + (5 ν − G M (cid:3) ,D = G M q + (8 − ν ) G M J c (10)are constants. In the standard approach of Damour and Deruelle it is crucial that D is of order O ( c − ) and thus a small quantity. Now in our case D depends not only on c but it is also linear in the deformation parameter q . If the spinning component isgoverned by a soft EoS the correction to the orbital motion induced by the q -couplingis much larger than the 1pN corrections. This is usual the case for main-sequence starbinaries (see e.g. Claret and Willems [9]). For compact stars in close binary systems,on the other hand, the contribution of the q -coupling can be of the same order as the1pN orbital correction. Under this assumption we can apply Damour and Deruellesstrategy straightforwardly, deriving a quasi-Keplerian solution up to linear order of q .This yields r = a r (1 − e r cos u ) , u − e t sin u = n ( t − t ) , (11) ϕ = 2( κ + 1)arctan "s e ϕ − e ϕ tan u . (12) ext to leading order gravitational wave emission n, a r , e r , e t and e ϕ depend on the coefficients defined above as a r = − BA − C J , e ϕ = e t (cid:20) − AB (cid:18) ν − G M c + DJ (cid:19)(cid:21) , (13) e r = e t (cid:20) − DA BJ (cid:21) , e t = s − AB (cid:20) C + BDJ (cid:21) (14)and n = p − A /B , and κ is given by κ = 3 G M J (cid:20) c + q J (cid:21) . (15)As it turns out, the parameter which has to be small for this solution to hold is δ ≡ q/J ∝ /c . Expressing the parameters of the quasi-Keplerian solution in terms ofthe 1pN conserved energy and δ , we find a r = − G M E − G M δ + G M c ( ν − ,n = − E G M (cid:20) − ν − Ec (cid:21) e t = e r (cid:20) E (cid:26) δ + 8 − νc (cid:27)(cid:21) ,e ϕ = e r h − E n δ + νc oi . The quasi-Keplerian solution given above describes the dynamics of a spinning compactbinary with Newtonian quadrupole coupling at the first post-Newtonian order. However,according to GR the system looses energy due to the emission of gravitational waves,beginning at the order ( v/c ) in post-Newtonian approximation schemes. There arebasically two ways to study the dynamical evolution of the binary system. In oneapproach the gravitational wave emission is considered as a secular effect and thedissipative terms do not enter the EoM. This allows the derivation of the quasi-Keplerian parametrization given above. Now we shall follow the other approach, wherethe radiation reaction terms are included into the EoM. In a first step we derive theHamiltonian formulation to the conservative system. Energy loss due to emission ofgravitational waves is incorporated by a time dependent radiation reaction Hamiltonianin a second step.The total Hamiltonian of a spinning compact binary system up to first post-Newtonianapproximation reads H pN = H N + H q + H pN , (16)where H N and H pN denote the point-particle Hamiltonian at Newtonian and first post-Newtonian order, respectively. If we would extend our analysis up to 2pN, other spin-depended terms would be present: at 1.5pN order the relativistic spin-orbit couplingenters into the Hamiltonian. It is, among others, responsible for the Lense-Thirring ext to leading order gravitational wave emission q -coupling, reads H pN = 12 µ (cid:20) p r + p ϕ r (cid:21) − G M µr − G M µ r q + 1 c (cid:20) ν − µ (cid:26) p r + 2 p r p ϕ r + p ϕ r (cid:27) − G M r (cid:26) (3 + 2 ν ) p r µ + (3 + ν ) p ϕ µr (cid:27) + G M µ r (cid:21) . (17)Note, that H pN = E = µE , since H pN is conserved at the first post-Newtonian order.As has been already mentioned before, the quadrupole interaction term is de facto a Newtonian correction to the point particle Hamiltonian. This is important to keepin mind when, for instance, calculating the orbital evolution and gravitational waveemission of binary pulsars with a main sequence star companion, such as PSR B1259-63[12]. For the Hamiltonian equations that govern the time evolution of the binary systemone finds ˙ r = p r µ (cid:20) µ c (cid:26) ν − (cid:18) p r + p ϕ r (cid:19) − G M µ r (3 + 2 ν ) (cid:27)(cid:21) , (18)˙ ϕ = p ϕ µr (cid:20) µ c (cid:26) ν − (cid:18) p r + p ϕ r (cid:19) − G M µ r (3 + ν ) (cid:27)(cid:21) , ˙ p r = p ϕ µr − G M µr (cid:18) q r (cid:19) + 1 c (cid:20) ν − µ p ϕ r (cid:18) p r + p ϕ r (cid:19) + G M µr − G M µr (cid:26) (3 + 2 ν ) p r + 3(3 + ν ) p ϕ r (cid:27)(cid:21) , ˙ p ϕ = 0 . Now let us include leading order dissipative effects into that scheme. In general, theleading order energy dissipation of a matter distribution is governed by the time-dependent radiation reaction Hamiltonian [18] H reac ( t ) = 2 G c I (3) ij ( t ) Z dV (cid:20) π i π j ρ + 14 πG ∂ i U ∂ j U (cid:21) , (19)where I ij is the symmetric tracefree mass quadrupole tensor of the matter distribution, ρ is the coordinate rest-mass density, π i the momentum density and U is the gravitationalpotential that satisfies the Poisson equation with source term ρ . Since we are treating the q -dependent terms as being formally of first post-Newtonian order, the components ofthe binary can be considered as pointlike objects troughout the calculation of H reac ( t ).Thus in the center of mass frame the radiation reaction Hamiltonian describing theleading order energy dissipation due to gravitational wave emission is given by H reac ( t ) = 2 G c I (3) ij ( t ) (cid:20) p i p j µ − G M µ x i x j r (cid:21) . (20) ext to leading order gravitational wave emission I (3) ij ( t ) as a function of time, and not as a function of generalizedcoordinates and momenta, when calculating the radiation reaction part of the EoMaccording to ( ˙ p i ) reac = − ∂H reac ∂q i , ( ˙ q i ) reac = ∂H reac ∂p i . Only afterwards I (3) ij can be expressed as a function of p r , p ϕ , r and ϕ . Explicitly, thecalculation yields [4],[7]( ˙ p r ) rad = 83 G p r r c (cid:18) G M ν − p ϕ νr (cid:19) , ( ˙ p ϕ ) rad = − G p ϕ νr c (cid:18) G M ν r + 2 p ϕ r − p r (cid:19) , ( ˙ r ) rad = − G νr c (cid:18) p r + 6 p ϕ r (cid:19) , ( ˙ ϕ ) rad = − G p r p ϕ νr c . (21)For numerical calculations it is useful to introduce scaled variables such that G = c = 1.The corresponding scaling is given by p r = µc ˜ p r , p ϕ = G M µc ˜ p ϕ , r = G M c ˜ r, H = µc ˜ H, q = G M c ˜ q. Applying this, the Hamiltonian equations governing the evolution of the binary systemincluding leading order radiation back reaction read˙˜ r = ˜ p r (cid:20) ν − (cid:26) ˜ p r + ˜ p ϕ ˜ r (cid:27) − ν ˜ r (cid:21) − ν ˜ r (cid:20) p r + 6 ˜ p ϕ ˜ r (cid:21) , (22)˙ ϕ = ˜ p ϕ ˜ r (cid:20) ν − (cid:26) ˜ p r + ˜ p ϕ ˜ r (cid:27) − ν ˜ r (cid:21) − ν p r ˜ p ϕ ˜ r , (23)˙˜ p r = ˜ p ϕ ˜ r − r (cid:20) q r (cid:21) + 3 ν −
12 ˜ p ϕ ˜ r (cid:20) ˜ p r + ˜ p ϕ ˜ r (cid:21) − ν r ˜ p r −
32 (3 + ν ) ˜ p ϕ ˜ r + 1˜ r + 8 ν p r ˜ r (cid:20) − ˜ p ϕ ˜ r (cid:21) , (24)˙˜ p ϕ = − ν p ϕ ˜ r (cid:20) r + 2 ˜ p ϕ ˜ r − ˜ p r (cid:21) . (25)Neither the total energy nor the orbital angular momentum is conserved, as indicatedby Eq. (25).The time evolution of binary systems described by Eqs. (22)-(25) is fully determined by3 parameters: the semi-major axis a r , the orbital eccentricity e r and the deformationparameter q , which have to be known for t = 0. Starting the numerical integration inthe periastron, i.e. at ϕ (0) = 0, the initial values for r and p r follow immediately as r (0) = r = a r (0)(1 − e r (0)) , p r (0) = 0 . To determine the initial value for p ϕ we use that, at the beginning of the integration,the total energy of the system is given by the conservative part of the Hamiltonian, or, ext to leading order gravitational wave emission E (0) = E = H pN /µ . It follows then from Eq. (17) that ¶ p ϕ (0) = 2 r E pN + 2 G M r (cid:20) q r (cid:21) + 1 c (cid:2) (1 − ν ) r ( E orbN ) +4(1 − ν ) G M r E orbN + (6 − ν ) G M (cid:3) , where E orbN is the Newtonian energy of the orbit.
3. Higher order gravitational wave emission
In the previous sections we have investigated the dynamical evolution of a spinningcompact binary system at the first post Newtonian approximation. Now we shall turnour attention to the gravitational waves emitted by the system. Far away from thesource the space time can be assumed to be asymptotically flat, such that the metric islocally Minkowskian. In fact, in asymptotically flat space-times the gravitational wavesemitted by an isolated binary systems are expected to obey a multipole expansion ofthe form (see e.g. [19]) h T Tij = GDc P ijkm ( N ) ∞ X l =2 "(cid:18) c (cid:19) l − (cid:18) l ! (cid:19) I ( l ) kmA l − ( t − D/c ) N A l − + (cid:18) c (cid:19) l − (cid:18) l ( l + 1)! (cid:19) ε pq ( k J ( l ) m ) pA l − ( t − D/c ) N q N A l − , (26)where h T Tij is the symmetric-tracefree (STF) part of the metric perturbation h ij , thebrackets denote symmetrization and D is the source-observer distance. The unit vector N points from the binary to the observer and A l = a a . . . a l ( a i = 1 , ,
3) is a multi-index. I A l and J A l are the STF mass and current multipole moments that parameterizethe radiation field in a Cartesian coordinate frame. However, if the direction of theangular momentum is conserved, it is more suitable to use STF-multipole moments I lm and S lm , m = − l, . . . , l , that are irreducibly defined with respect to the axis of angularmomentum. The relation of these to the Cartesian multipole components are givenin Eqs. (A.1) and (A.2) in the appendix. From Eq. (26) it is then derived that theradiation field h T Tij , expressed in terms of time derivatives of I lm and S lm , is given by h T Tij = GDc ∞ X l =2 l X m = − l "(cid:18) c (cid:19) l − I ( l ) lm ( t − D/c ) T E ,lmij (Θ , Φ)+ (cid:18) c (cid:19) l − S ( l ) lm ( t − D/c ) T B ,lmij (Θ , Φ) , (27)where T E ,lmij and T B ,lmij are the so-called pure-spin tensor-spherical harmonics of electricand magnetic type. These harmonics are orthonormal on the unit sphere. In fact,introducing unit vectors ˆΦ and ˆΘ, they can be decomposed into a term proportional to( ˆΘ ⊗ ˆΘ − ˆΦ ⊗ ˆΦ) and ( ˆΘ ⊗ ˆΦ + ˆΦ ⊗ ˆΘ), respectively. That way, if the T E/B ,lm are ¶ Note that p ϕ → µp ϕ and p r → µp r . ext to leading order gravitational wave emission h + and h × of the radiation field from Eq.(27) without any further calculations. The pure-spin tensor-spherical harmonics neededhere are given by Eqs. (A.7)-(A.16) in the appendix + .In section 2 the point particle contribution was taken into account up to first post-Newtonian approximation, and the quadrupole coupling term, though present alreadyat Newtonian order, was assumed to be of the same order as the 1pN correctionsto the point particle dynamics. That means, we have to go beyond the leadingorder gravitational wave formula. Considering the dynamics up to 1pN requires theapplication of the multipole expansion (27) up to l = 4 for the mass multipole momentsand up to l = 3 for the current multipole moments. Explicitly, neglecting all higherorder terms, Eq. (27) is reduced to ∗ h + , × = GDc " X m = − I (2)2 m T E , m + , × + 1 c ( X m = − I (3)3 m T E , m + , × + X m = − S (2)2 m T B , m + , × ) + 1 c ( X m = − I (4)4 m T E , m + , × + X m = − S (3)3 m T B , m + , × ) (28)or, in a more convenient form, h + , × = h (0)+ , × + 1 c h (1)+ , × + 1 c h (2)+ , × . (29)Note that according to Eq. (28) except for I (2)2 m all other time derivatives and multipolemoments are required only at leading order.Let us start by noting that, in the center-of-mass system, the mass and current multipolemoments of a two-body system are given by I ij = µx h i x j i (cid:20) − ν ) v c − − ν c G M r (cid:21) + µ (1 − ν )21 c h − r · v ) x h i v j i + 11 r v h i v j i i + I sij , (30) I ijk = − µ √ − ν x h i x j x k i , (31) I ijkl = µ (1 − ν ) x h i x j x k x l i , (32) J ij = − µ √ − νε ab h i x j i x a v b , (33) J ijk = µ (1 − ν ) ε ab h k x i x j i x a v b , (34)where brackets denote the STF-part of the correspondent tensor (see appendix A).At this point it is neccessary to consider a moment the contribution of the stellarmass-quadrupole moment. Since one component of the binary is spinning and thusautomatically gains a finite size there could be, in principle, a contribution of I sij to thegravitational wave emission of the system. This contribution is, however, very small,unless the energy stored in the internal stellar degrees of freedom, e.g. oscillations ofthe star, is comparable to the orbital energy. From now on we shall assume that ¨ I sij is + In this representation, h + is the ( ˆΘ ⊗ ˆΘ − ˆΦ ⊗ ˆΦ)-part of Eq. (27), while h × is the ( ˆΘ ⊗ ˆΦ+ ˆΦ ⊗ ˆΘ)-part. ∗ From now on we omit the supscript
T T for notational convenience. ext to leading order gravitational wave emission ♯ ˙ v = − G M r r to calculate I (3)3 m , I (4)4 m , J (2)2 m and J (3)3 m and the 1pN equation of motion in theform˙ v = − G M r r (cid:18) q r (cid:19) + G M r c (cid:20) r (cid:26) G M r (4 + 2 ν ) − v (1 + 3 ν ) + 3 ν r · v ) r (cid:27) +(4 − ν )( r · v ) v i to calculate I (2) ij one finds, after some lengthy calculations (see also [20]) I (2) ij = 2 µv h i v j i (cid:20) − ν ) v c + 54 ν − G M rc (cid:21) +2 µx h i v j i
25 + 9 ν G M r c ( r · v ) − µx h i x j i G M r (cid:20) (cid:18) q r (cid:19) + 61 + 48 ν v c − c (1 − ν ) ( r · v ) r − (10 − ν ) G M rc (cid:21) , (35) I (3) ijk = − µ √ − ν (cid:20) G M r ( r · v ) x h i x j x k i − G M r v h i x j x k i + 6 v h i v j v k i (cid:21) , (36) I (4) ijkl = 4 µ (1 − ν ) (cid:20) v h i v j v k v l i − G M r v h i v j x k x l i + 42 G M r ( r · v ) v h i x j x k x l i + G M r x h i x j x k x l i (cid:26) G M r + 3 v r −
15 ( r · v ) r (cid:27)(cid:21) , (37)while the time derivatives of the current multipole moments read J (2) ij = µ √ − ν G M r ε ab h i x j i x a v b , (38) J (3) ijk = 2 µ G M r (1 − ν ) h − ε ab h k x i v j i x a v b + 3 r · v r ε ab h k x i x j i x a v b i . (39)In particular, within this approximation the q coupling is only relevant for the secondtime derivative of the mass quadrupole tensor. Using the relation between the two classesof multipole moments given by Eqs. (A.1) and (A.2) we get the following expressionsfor the time derivatives of I lm and S lm , now in polar coordinates: I (2)20 = 4 µ r π (cid:20) − (cid:26) v − G M r (cid:18) q r (cid:19)(cid:27) + 1 c (cid:26) G M r ( ν − ν − v + G M r ((37 − ν ) r ˙ ϕ − (15 + 32 ν ) ˙ r ) (cid:27)(cid:21) , (40) I (2)21 = 0 , (41) I (2)22 = r π µe − iϕ (cid:20) (cid:26) ˙ r − r ˙ ϕ − G M r (cid:18) q r (cid:19) − ir ˙ r ˙ ϕ (cid:27) (42) ♯ Remember that q is treated formally as a 1pN quantity. ext to leading order gravitational wave emission
13+ 1 c (cid:26) (10 − ν ) G M r + G M r (3(15 + 32 ν ) ˙ r − (11 + 156 ν ) r ˙ ϕ )+ 97 (1 − ν )( ˙ r − r ˙ ϕ ) − ir ˙ r ˙ ϕ (cid:18) ν ) G M r + 187 (1 − ν ) v (cid:19)(cid:27)(cid:21) ,I (3)30 = 0 , (43) I (3)31 = 4 ν ( m − m ) r π e − iϕ (cid:20) ˙ r (cid:26) G M r − v (cid:27) + ir ˙ ϕ (cid:26) v − G M r (cid:27)(cid:21) , (44) I (3)33 = 2 ν ( m − m ) r π e − iϕ (cid:20) (cid:26) ˙ r − G M r − r ˙ ϕ (cid:27) ˙ r + ir ˙ ϕ (cid:26) G M r − r + 2 r ˙ ϕ (cid:27)(cid:21) , (45) I (4)40 = 221 (1 − ν ) µ r π (cid:20) G M r − G M r (cid:8)
18 ˙ r + 13 r ˙ ϕ (cid:9) + 6 v (cid:21) , (46) I (4)41 = I (4)43 = 0 , (47) I (4)42 = 263 √ π (1 − ν ) µe − iϕ (cid:20) − G M r + G M r (18 ˙ r − r ˙ ϕ )+6 r ˙ ϕ − r + 3 ir ˙ r ˙ ϕ (cid:26) v − G M r (cid:27)(cid:21) , (48) I (4)44 = 29 r π
14 (1 − ν ) µe − iϕ (cid:20) G M r + G M r ( −
18 ˙ r + 51 r ˙ ϕ ) + 6 ˙ r − r ˙ r ˙ ϕ + 6 r ˙ ϕ + ir ˙ r ˙ ϕ (cid:26) G M r −
24 ˙ r + 24 r ˙ ϕ (cid:27)(cid:21) , (49) S (2)20 = S (2)22 = 0 , (50) S (2)21 = 83 r π m − m ) G M ν ˙ ϕe − iϕ , (51) S (3)30 = − r π
105 (1 − ν ) G M µ ˙ r ˙ ϕ, (52) S (3)31 = S (3)33 = 0 , (53) S (3)32 = 23 r π − ν ) G M µe − iϕ ˙ ϕ ( ˙ r − ir ˙ ϕ ) . (54)Here we used that √ − ν = ( m − m ) / M . Since only S (2)21 , I (3)31 and I (3)33 contribute to the first correction to the leading order quadrupole formula h (1)+ , × , itbecomes clear that h (1)+ , × vanishes for equal mass binary systems. In that case the firstnontrivial correction to h (0)+ , × is of order 1 /c .It is possible to derive, after some straightforward but rather lengthy calculations,analytic expressions for the time derivatives of the multipole moments in terms of e r , E and δ ; the corresponding relations are given in appendix C. For our purposes it is,however, more useful to express the polarization states of the gravitational radiationfield in terms of generalized coordinates and velocities or – in Hamiltonian formulation– in terms of generalized coordinates and momenta. Inserting above relations in Eq. ext to leading order gravitational wave emission Dc G h (0)+ = (1 + cos Θ) µ (cid:20) cos 2(Φ − ϕ ) (cid:26) ˙ r − r ˙ ϕ − G M r (cid:18) q r (cid:19)(cid:27) +2 r ˙ r ˙ ϕ sin 2(Φ − ϕ ) i − µ sin Θ (cid:20) ˙ r + r ˙ ϕ − G M r (cid:18) q r (cid:19)(cid:21) , (55) Dc G h (0) × = µ cos Θ (cid:20) r ˙ r ˙ ϕ cos 2(Φ − ϕ ) − − ϕ ) (cid:26) ˙ r − r ˙ ϕ − G M r (cid:18) q r (cid:19)(cid:27)(cid:21) . Note that, in order to emphasize the character of the q -coupling, the q -dependent termshave been included into the leading order component of the radiation field. Defining∆ m ≡ m − m the first correction terms read Dc G h (1)+ = ∆ m M µ sin Θ (cid:20) G M ˙ ϕ sin(Φ − ϕ ) + 3 cos Θ − (cid:26)(cid:18) G M r − v (cid:19) ˙ r cos(Φ − ϕ ) − (cid:18) v − G M r (cid:19) r ˙ ϕ sin(Φ − ϕ ) (cid:27) − Θ4 (cid:26) r cos 3(Φ − ϕ ) (cid:18) ˙ r − r ˙ ϕ − G M r (cid:19) − r ˙ ϕ sin 3(Φ − ϕ ) (cid:18) G M r − r + 2 r ˙ ϕ (cid:19)(cid:27)(cid:21) , (56) Dc G h (1) × = sin 2Θ2 ∆ m M µ (cid:20) r ˙ ϕ cos(Φ − ϕ ) (cid:26) G M r − v (cid:27) + ˙ r sin(Φ − ϕ ) (cid:26) v − G M r (cid:27) + r ˙ ϕ − ϕ ) (cid:26) G M r − r + 2 r ˙ ϕ (cid:27) + ˙ r sin 3(Φ − ϕ ) (cid:26) ˙ r − r ˙ ϕ − G M r (cid:27)(cid:21) . (57)Note that these expressions depend on the mass difference and vanish for equal massbinaries. The next corrections to the gravitational waveforms read Dc G h (2)+ = 1 + cos Θ2 µ (cid:20) cos 2(Φ − ϕ ) (cid:26) (10 − ν ) G M r + 97 (1 − ν )( ˙ r − r ˙ ϕ )+ G M r (cid:18)
15 + 32 ν r −
11 + 156 ν r ˙ ϕ (cid:19)(cid:27) + r ˙ r ˙ ϕ − ϕ ) (cid:26)
103 (5 + 27 ν ) G M r + 18(1 − ν ) v (cid:27)(cid:21) + sin Θ2 µ (cid:20) ( ν − G M r −
97 (1 − ν ) v + G M r (cid:16) (37 − ν ) r ˙ ϕ − (15 + 32 ν ) ˙ r (cid:17)i − − ν µ (7 cos Θ − Θ + 1) (cid:20) G M r − G M r (18 ˙ r + 13 r ˙ ϕ ) + 6 v (cid:21) + 1 − ν µ (7 cos Θ − Θ + 1) h cos 2(Φ − ϕ ) n r ˙ ϕ − ˙ r )+ G M r (18 ˙ r − r ˙ ϕ ) − G M r (cid:27) − r ˙ r ˙ ϕ sin 2(Φ − ϕ ) (cid:26) v − G M r (cid:27)(cid:21) ext to leading order gravitational wave emission
15+ 1 − ν µ sin Θ(1 + cos Θ) (cid:20) cos 4(Φ − ϕ ) (cid:26) G M r + 6 ˙ r − r ˙ r ˙ ϕ +6 r ˙ ϕ + G M r (51 r ˙ ϕ −
18 ˙ r ) (cid:27) − r ˙ r ˙ ϕ sin 4(Φ − ϕ ) (cid:26) G M r −
24 ˙ r +24 r ˙ ϕ oi − − ν G M µ ˙ ϕ (2 cos Θ − h r ˙ ϕ cos 2(Φ − ϕ ) − ˙ r sin 2(Φ − ϕ ) i , (58) Dc G h (2) × = µ cos Θ (cid:20) r ˙ r ˙ ϕ − ϕ ) (cid:26)
103 (5 + 27 ν ) G M r + 18 v (1 − ν ) (cid:27) − sin 2(Φ − ϕ ) (cid:26) (10 − ν ) G M r + G M r (cid:18)
15 + 32 ν r −
11 + 156 ν r ˙ ϕ (cid:19) + 97 (1 − ν )( ˙ r − r ˙ ϕ ) (cid:27)(cid:21) + (1 − ν ) µ cos Θ (cid:20) − G M ˙ r ˙ ϕ sin Θ+ 3 cos Θ − G M ˙ ϕ { ˙ r cos 2(Φ − ϕ ) } + 4 r ˙ ϕ sin 2(Φ − ϕ ) (cid:27) − sin Θ12 (cid:26) r ˙ r ˙ ϕ cos 4(Φ − ϕ ) (cid:18) G M r −
24 ˙ r + 24 r ˙ ϕ (cid:19) + sin 4(Φ − ϕ ) (cid:18) G M r + G M r (51 r ˙ ϕ −
18 ˙ r ) + 6 ˙ r + 6 r ˙ ϕ − r ˙ r ˙ ϕ (cid:19)(cid:27) − Θ − (cid:26) r ˙ r ˙ ϕ cos 2(Φ − ϕ ) (cid:18) v − G M r (cid:19) + sin 2(Φ − ϕ ) (cid:18) − G M r + G M r (18 ˙ r − r ˙ ϕ ) + 6 r ˙ ϕ − r (cid:19)(cid:27)(cid:21) . (59)The polarization states of the gravitational radiation field, expressed in terms ofgeneralized coordinates and momenta, can be found in appendix B.
4. Discussion
It has been long known that finize size effects introduce a periastron shift already at thelevel of Newtonian theory. For a couple of main sequence star binaries the total apsidalmotion ˙ ϕ tot has been determined from observational evidence. Compared with thecontribution ˙ ϕ rel predicted by GR it became obvious that in all systems the Newtonianperturbations give the dominant contribution to ˙ ϕ tot (for an overview see e.g. [9]). Thisis due to the ”soft” equations of state governing the stellar matter of main sequence stars.For compact star binaries Newtonian perturbations are often neglected. In particular,it is often argued that the effect of the spin-induced quadrupole is too small unlessthe compact star (e.g. a neutron star) is rotating near the mass-shedding limit [3].However, even for NS-NS or NS-BH binaries this argument does not hold completely. Ithas been shown in previous sections that for close NS-NS binaries the potential energyintroduced by the coupling can be considerably larger than the corresponding 1.5pNorbital correction terms, though it is smaller by a factor 100 or more than the 1pN ext to leading order gravitational wave emission − to 10 − Hz),which enlarges the number of possible sources enormeously. In particular, with LISAnot only BH-BH, NS-NS and NS-BH binaries should be detectable, but also white dwarfbinaries. In particular to this class of compact binaries the analysis shown in this paperapplies. It has been argued by Willems et al. in a recent paper [21] that – contrary toprevailing opinions – there might exist a class of eccentric galactic double white dwarfs,which are formed by interactions in tidal clusters. Willems et al. showed that tidesand stellar rotation strongly dominate the periastron shift at orbital frequencies ≥ ϕ tot induced by rotational deformation will lead to an overestimation ofthe total mass derived from ˙ ϕ tot .In this paper the competing influences of the rotational deformation and the 1pNcorrection terms were examined in more detail. In particular, we succeeded in calculatinga 1pN quasi-Keplerian solution, which takes into account finite size effects up to linearorder in the quadrupole deformation parameter q . The results given in section 2 are validas long as q/J is of the order O ( c − ). For white dwarf binaries or binary pulsars suchas PSR 1259-63 the periastron shift induced by rotational deformation is possibly muchlarger than the general relativistic contribution. In that case, Eqs. (11-12) still applyin the limit v/c →
0. In section 3 the polarization states of the gravitational radiationfield are calculated beyond the leading order approximation. For non-spinning compactbinaries the corresponding waveforms are shown in figures 1 and 2. In these figureswaveforms calculated using the leading order expressions h + , × (0) are compared to the1pN correct waveforms with the next to leading order corrections h (1)+ , × and h (2)+ , × takeninto account. The first correction, h (1)+ , × , is nontrivial only for different mass binaries, i.e.for equal-mass binaries the first non-vanishing correction to the leading order formulaappears at the order O ( c − ).The influence of the q -coupling on the gravitational waveforms is shown in figures 3 and4. As expected, the spin-induced quadrupole moment leads to a phase shift comparedto the pure point-particle GW emission. Moreover, the quadrupole deformation of thespinning compact objects speeds up the inspiral process, as has been shown in figures 5and 6 for an equal-mass binary in a slightly elliptic orbit.More analysis is needed in order to fully understand the imprint of finite size effects ontothe gravitational wave pattern of close compact binary systems beyond the leading order. ext to leading order gravitational wave emission Acknowledgements
I am grateful to Gerhard Sch¨afer for helpful discussions and careful reading of themanuscript. This work is supported by the Deutsche Forschungsgemeinschaft (DFG)through SFB/TR7 ”Gravitationswellenastronomie”.
Appendix A. Useful relations
The mass and current multipole moments I lm and S lm ( m = − l, . . . , l ) that areirreducibly defined with respect to the orbital angular momentum axis are related to I A l and J A l according to I lm ( t ) = 16 π (2 l + 1)!! s ( l + 1)( l + 2)2( l − l I A l ( t ) Y lm ∗ A l , (A.1) S lm ( t ) = − πl ( l + 1)(2 l + 1)!! s ( l + 1)( l + 2)2( l − l J A l Y lm ∗ A l , (A.2)where, for m ≥ Y lmA l = ( − m (2 l − s l + 14 π ( l − m )!( l + m )! ( δ h i + iδ h i ) · · · ( δ i m + iδ i m ) δ i m +1 · · · δ i l i , (A.3)and Y lmA l = ( − m Y l | m |∗ A l for m < . (A.4)The complex conjugates are given by I lm ∗ = ( − m I l − m , S lm ∗ = ( − m S l − m . (A.5)The pure-spin tensor-spherical harmonics are orthonormal on the unit sphere. For thecomplex conjugate the following relation holds: T E/B ,lm ∗ = ( − m T E/B ,l − m . (A.6)Defining Υ + ≡ ˆΘ ⊗ ˆΘ − ˆΦ ⊗ ˆΦ , Υ − ≡ ˆΘ ⊗ ˆΦ + ˆΦ ⊗ ˆΘthe expressions needed in the paper read T E , = r π e i Φ h (1 + cos Θ) Υ + + 2 i cos Θ Υ − i , (A.7) T E , = r π sin Θ Υ + , (A.8) ext to leading order gravitational wave emission T B , = − r π sin Θ e i Φ h i Υ + − cos Θ Υ − i , (A.9) T E , = − r π sin Θ e i Φ h (1 + cos Θ) Υ + + 2 i cos Θ Υ − i , (A.10) T E , = r π sin Θ e i Φ h (3 cos Θ − Υ + + 2 i cos Θ Υ − i , (A.11) T B , = − r π e i Φ h i (2 cos Θ − Υ + − cos Θ(3 cos Θ − Υ − i , (A.12) T B , = r π cos Θ sin Θ Υ − . (A.13) T E , = r π sin Θ e i Φ h (1 + cos Θ) Υ + + 2 i cos Θ Υ − i , (A.14) T E , = r π e i Φ h (7 cos Θ − Θ + 1) Υ + + i cos Θ(7 cos Θ − Υ − i , (A.15) T E , = − r π (7 cos Θ − Θ + 1) Υ + . (A.16) Appendix A.1. Symmetric tracefree tensors
Throughout this paper symmetric-tracefree 3 rd and 4 th rank tensors are used.Symmetrizing a tensor of rank p requires to take the properly weighted sum over allindex permutations, T ( i ...i p ) = T symmi ...i p ≡ p ! X permut. T i ...i p . (A.17)The tracefree part of the tensor T i ...i p is calculated according to [22] T h i ...i p i = [ p/ X k =0 a pk δ ( i i · · · δ i k − i k T symmi k +1 ...i p ) α α ...α k α k , (A.18)with a pk = p !(2 p − − k (2 p − k − p − k )!(2 k )!! . (A.19)In particular, 3 rd and 4 th rank STF tensors are given by T h abc i = T ( abc ) − (cid:2) δ ab T ( cii ) + δ bc T ( aii ) + δ ac T ( bii ) (cid:3) , (A.20) T h abcd i = T ( abcd ) −
17 [ δ ab T ( cdii ) + δ ac T ( bdii ) + δ ad T ( bcii ) + δ bc T ( adii ) + δ bd T ( acii ) + δ cd T ( abii ) ]+ 135 [ δ ac δ bd + δ ad δ bc + δ ab δ cd ] T ( iijj ) . (A.21) ext to leading order gravitational wave emission Appendix B. Expressions for h + and h × in terms of generalized coordinatesand momenta The expressions for the leading and next to leading order contribution to the polarizationstates of the gravitational wave field, h + and h × , read h (0)+ = GµDc (cid:26) Θ µ (cid:20) cos 2(Φ − ϕ ) (cid:26) p r − p ϕ r − G M µ r (cid:18) q r (cid:19)(cid:27) +2 p r p ϕ r sin 2(Φ − ϕ ) i − sin Θ µ (cid:20) p r + p ϕ r − G M µ r (cid:18) q r (cid:19)(cid:21)(cid:27) , (B.1) h (1)+ = GDc ∆ m M µ sin Θ (cid:20) p ϕ r sin(Φ − ϕ ) (cid:26) G M µ r + 3 cos Θ − (cid:18) G M µ r − p r − p ϕ r (cid:19)(cid:27) + 3 cos Θ − p r cos(Φ − ϕ ) (cid:26) G M µ r − p r − p ϕ r (cid:27) − Θ2 (cid:26) p r cos 3(Φ − ϕ ) (cid:18) p r − p ϕ r − G M µ r (cid:19) − p ϕ r sin 3(Φ − ϕ ) (cid:18) G M µ r − p r + p ϕ r (cid:19)(cid:27)(cid:21) , (B.2) h (2)+ = GµDc ( sin Θ14 " G M r ( ν − − ν − µ (cid:18) p r + p ϕ r (cid:19) + G M µ r (cid:26) ν ) p r + (121 + 8 ν ) p ϕ r (cid:27)(cid:21) + 1 + cos Θ14 cos 2(Φ − ϕ ) (cid:20) G M r (10 − ν ) + 5(3 ν − (cid:18) p r µ − p ϕ µ r (cid:19) + G M µ r (cid:26) − ν p ϕ r − ν ) p r (cid:27)(cid:21) + 1 + cos Θ7 sin 2(Φ − ϕ ) p r p ϕ µ r (cid:20) ν − µ (cid:18) p r + p ϕ r (cid:19) − − ν G M r (cid:21) + 1 − ν
24 sin Θ(1 + cos Θ) (cid:20) cos 4(Φ − ϕ ) (cid:26) G M r − G M µ r (cid:18) p r − p ϕ r (cid:19) +6 (cid:18) p r µ − p r p ϕ µ r + p ϕ µ r (cid:19)(cid:27) − p r p ϕ µ r sin 4(Φ − ϕ ) (cid:26) G M r +24 (cid:18) p ϕ µ r − p r µ (cid:19)(cid:27)(cid:21) + 1 − ν
42 (7 cos Θ − Θ + 1) h cos 2(Φ − ϕ ) (cid:26) − G M r + G M µ r (cid:18) p r − p ϕ r (cid:19) + 6 (cid:18) p ϕ µ r − p r µ (cid:19)(cid:27) − p r p ϕ µ r sin 2(Φ − ϕ ) (cid:26) (cid:18) p r µ + p ϕ µ r (cid:19) − G M r (cid:27)(cid:21) ext to leading order gravitational wave emission − − ν
56 (7 cos Θ − Θ + 1) " G M r + 6 (cid:18) p r µ + p ϕ µ r (cid:19) − G M µ r (cid:18) p r + 13 p ϕ r (cid:19)(cid:21) − − ν Θ − G M p ϕ µ r h p ϕ r cos 2(Φ − ϕ ) − p r sin 2(Φ − ϕ ) i(cid:27) . (B.3)For h × one finds h (0) × = 2 GµDc cos Θ (cid:20) sin 2(Φ − ϕ ) (cid:26) − p r µ + p ϕ µ r + G M r (cid:18) q r (cid:19)(cid:27) +2 p r p ϕ µ r cos 2(Φ − ϕ ) (cid:21) , (B.4) h (1) × = GDc ∆ m M µ sin 2Θ2 (cid:20) p ϕ r cos(Φ − ϕ ) (cid:26) G M µ r − p r − p ϕ r (cid:27) + p r sin(Φ − ϕ ) (cid:26) p r + p ϕ r − G M µ r (cid:27) + p ϕ r cos 3(Φ − ϕ ) (cid:26) G M µ r − p r + 2 p ϕ r (cid:27) + p r sin 3(Φ − ϕ ) (cid:26) p r − p ϕ r − G M µ r (cid:27)(cid:21) , (B.5) h (2) × = GµDc cos Θ (cid:20) sin 2(Φ − ϕ ) (cid:26)
57 (1 − ν ) µ (cid:18) p r − p ϕ r (cid:19) + G M µ r (cid:18)
37 (23 + 8 ν ) p r − − ν p ϕ r (cid:19) − G M r (10 − ν ) (cid:27) (B.6)+ p r p ϕ µ r cos 2(Φ − ϕ ) (cid:26)
107 (3 ν − µ (cid:18) p r + p ϕ r (cid:19) −
221 (227 − ν ) G M r (cid:27) − − ν p r p ϕ µ r sin Θ cos 4(Φ − ϕ ) (cid:26) G M r + 24 (cid:18) p ϕ µ r − p r µ (cid:19)(cid:27) − − ν
12 sin Θ sin 4(Φ − ϕ ) (cid:26) G M r + G M µ r (cid:18) p ϕ r − p r (cid:19) +6 (cid:18) p r µ − p r p ϕ µ r + p ϕ µ r (cid:19)(cid:27) − − ν Θ G M r p r p ϕ µ r − − ν
42 (7 cos Θ − (cid:20) p r p ϕ µ r cos 2(Φ − ϕ ) (cid:26) (cid:18) p r µ + p ϕ µ r (cid:19) − G M r (cid:27) + sin 2(Φ − ϕ ) (cid:26) − G M r + G M µ r (cid:18) p r − p ϕ r (cid:19) + 6 (cid:18) p ϕ µ r − p r µ (cid:19)(cid:27)(cid:21) + 1 − ν Θ − G M r (cid:26) p r p ϕ µ r cos 2(Φ − ϕ ) + 4 p ϕ µ r sin 2(Φ − ϕ ) (cid:27)(cid:21) . Appendix C. Higher order gravitational wave forms: Analytic expressions
Using the quasi-Keplerian parametrization derived in section 2 it is possible to calculateanalytic expressions for the times derivatives of the STF multipole moments entering in ext to leading order gravitational wave emission F ( u ) ≡ − e r cos u one obtains S (2)21 = 323 r π m − m ) ν ( − E ) / p − e r F ( u ) , (C.1) S (3)30 = − µE r π
105 (1 − ν ) e r sin u p − e r F ( u ) , (C.2) S (3)32 = 83 r π − ν ) µE e − iϕ p − e r F ( u ) h e r sin u − i p − e r i , (C.3)while the time derivatives of the mass multipole moments read I (2)20 = − µE r π (cid:20) − F ( u ) (cid:26) − q a r F ( u ) (cid:27) + EF ( u ) δ + E c n ν − − ν − F ( u ) + 2(19 ν − F ( u ) + 4( ν −
26) 1 − e r F ( u ) (cid:27)(cid:21) , (C.4) I (2)22 = 4 r π µEe − iϕ h − F ( u ) − e r sin uF ( u ) + 2 i e r p − e r sin uF ( u ) + 5 q a r F ( u ) − Eδ ( F ( u ) + 4 e r sin uF ( u ) + 2 i e r p − e r sin uF ( u ) (cid:18) e r − e r − e r cos uF ( u ) (cid:19)) + E c n ν − − ν − F ( u ) + 42(8 ν − − e r (3 ν − F ( u ) − ν − − e r F ( u ) + 2 ie r sin u p − e r F ( u ) (253 − ν − ν − e r cos u +(213 ν − e r + 9(3 ν − e r cos u (cid:1)oi , (C.5) I (3)31 = 8 r π
35 ( m − m ) ν ( − E ) / e − iϕ " e r sin uF ( u ) − i p − e r F ( u ) (cid:18) − / F ( u ) (cid:19) , (C.6) I (3)33 = 8 r π
21 ( m − m ) ν ( − E ) / e − iϕ (cid:20) − e r sin uF ( u ) (cid:26) − e r ) F ( u ) (cid:27) + i p − e r F ( u ) (cid:26) − / F ( u ) + 4(1 − e r ) F ( u ) (cid:27) , (C.7) I (4)40 = 821 r π − ν ) µE (cid:20) − F ( u ) − F ( u ) + 5(1 − e r ) F ( u ) (cid:21) , (C.8) I (4)42 = 863 √ π (1 − ν ) µE e − iϕ (cid:20) − F ( u ) − − e r F ( u ) + 3(1 − e r ) F ( u ) − i e r p − e r sin uF ( u ) (cid:26) F ( u ) (cid:27) , (C.9) I (4)44 = 49 r π − ν ) µE e − iϕ (cid:20) − F ( u ) + 43 − e r F ( u ) − − e r ) F ( u ) + 48(1 − e r ) F ( u ) +6 i e r p − e r sin uF ( u ) (cid:26) F ( u ) + 8(1 − e r ) F ( u ) (cid:27) . (C.10) ext to leading order gravitational wave emission [1] A. Gopakumar R.-M. Memmesheimer and G. Sch¨afer. Third post-Newtonian accurate generalizedquasi-Keplerian parametrization for compact binaries in eccentric orbits. Phys. Rev. D ,70:104011, 2004.[2] C. K¨onigsd¨orffer and A. Gopakumar. Post-Newtonian accurate parametric solution to thedynamics of spinning compact binaries in eccentric orbits: The leading order spin-orbitinteraction.
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Lectures on General Relativity, Brendels Summer Institute in TheoreticalPhysics 1964, Vol. 1 , chapter 2, page 287. New Jersey, 1965. ext to leading order gravitational wave emission -0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0 0.5 1 1.5 2 PSfrag replacements t/P h × [ G M /Dc ] h + [ G M / D c ] h (0)+ h (1 pN )+ Figure 1. h + -component of a non-spinning, equal mass binary system with ˜ a r =40 , e r = 0 .
3. Plotted are the waveform according to the leading order quadrupoleformula and and the 1pN-corrected waveform. Observer-dependent parameters: Φ = π/ , Θ = π/ -0.03-0.02-0.01 0 0.01 0.02 0.03 0 0.5 1 1.5 2 PSfrag replacements t/P h × [ G M / D c ] h + [ G M /Dc ] h (0) × h (1 pN ) × Figure 2. h × -component of a non-spinning, equal mass binary system. All parametersare the same as in Fig. 1. ext to leading order gravitational wave emission -0.025-0.02-0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0 1 2 3 4 5 PSfrag replacements t/P h + [ G M / D c ] ˜ q = 0 ˜ q = 4 Figure 3.
Influence of the quadrupole coupling on the gravitational wave emission.The 1pN correct h + -component emitted by a non-spinning binary with semi-major axisand eccentricity ˜ a r = 40 and e r = 0 .
3, respectively, is compared with the correspondingwaveform emitted by a NS-NS binary with ˜ q = 4. The masses are m = 3 m . Observer-dependent angles are Φ = π/ , Θ = π/ -0.07-0.06-0.05-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0 1 2 3 4 5 PSfrag replacements t/P h + [ G M / D c ] ˜ q = 0 ˜ q = 4 Figure 4.
Influence of the quadrupole coupling on the gravitational wave emission.The 1pN correct h + -component emitted by a non-spinning, equal mass binary withsemi-major axis and eccentricity ˜ a r = 40 and e r = 0 .
6, respectively, is comparedwith the corresponding waveform emitted by a NS-NS binary with ˜ q = 4. Observer-dependent angles are Φ = π/ , Θ = π/ ext to leading order gravitational wave emission -0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 0 10 20 30 40 50 60 70 PSfrag replacements h + [ G M / D c ] t/P Figure 5. h + -component of the gravitational wave field emitted by a non-spinning,equal mass binary during the inspiral process (initial values ˜ a r (0) = 50 , e r (0) = 0 . π/ , Θ = π/ -0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 0 10 20 30 40 50 60 70 PSfrag replacements h + [ G M / D c ] t/P Figure 6. h + -component of the gravitational wave field emitted by an equal massbinary with ˜ q = 4 during the inspiral process (initial values ˜ a r (0) = 50 , e r (0) = 0 . π/ , Θ = π/π/