Tail effects in the 3PN gravitational wave angular momentum flux of compact binaries in quasi-elliptical orbits

Transcription

1 Chapr 5 Tail ffcs in h 3PN graviaional wav angular momnum flux of compac binaris in quasi-llipical orbis 5.1 Inroducion Asrophysical compac binaris in quasi-llipical orbis In h prvious hr chaprs w hav sudid h daa-analysis and compuaion of graviaional wav (GW) polarisaions from inspiralling compac binaris, bu wih orbis which wr quasi-circular. Indd, hs ar h mos plausibl sourcs for dcing GW wih h hlp of lasr inrfromrs lik LIGO, VIRGO and h plannd LISA. I has bn sablishd long ago, in a sminal work by Prs [47], ha compac binaris moving in llipical orbis wih larg rlaiv sparaion circularis undr radiaion racion du o loss of nrgy hrough mission of GW. Moivad by his, h 3.5PN phasing of inspiralling compac binaris moving in quasi-circular orbis is now compl and availabl for us in GW daa analysis [32, 163]. Rcn progrss in incrasing h snsiiviy of LIGO by an ordr of magniud o rach Advancd LIGO snsiiviy rndrs h complion of h highly accura phasing formula vry imly. This is bcaus prooyp binary sourcs of GW for ground-basd inrfromric GW dcors ar nuron sar (NS) or black hol (BH) binaris clos o hir mrgr phas and consqunly hav los all hir ccnriciy by h im h GW from hm nr h dcor bandwidh. Howvr, asrophysical invsigaions indicas h possibiliy of dcing binaris wih ccnriciy in h snsiiv bandwidh of boh rrsrial and spac-basd graviaional wav dcors. On such scnario is h Kozai oscillaions [48]. This occurs in h cors of dns globular clusrs whr graviaional inracions bwn pairs of binary BH sysms ar likly. Ths inracions can lad o an vnual jcion of on of h BH rsuling in h formaion of a sabl hirarchical ripl. This is a hr-body configuraion, whr wo closly bound BH orbi ach ohr whil h hird orbis h cnr-of-mass of h firs wo. Whn h wo orbial plans hav a larg inclinaion angl bwn hm, idal forcs from h our body incrass h ccnriciy of h innr sabl binary by causing an orbial 13

2 rsonanc. Th Kozai mchanism dscribd abov can lad o ccnriciis grar han.1 a h im GW from h innr binary nrs h bandwidh of Advancd LIGO [49]. Binaris comprising sllar mass BH ar simad o possss a hrmal disribuion of ccnriciis [164]. Eccnric binaris wih highr masss siuad in globular clusrs ar also ponial sourcs for LISA. Inrmdia mass ( 1 3 M ) BH binaris wih ccnriciis bwn.1 and.2 ar also xpcd o b gnrad by h Kozai mchanism [165]. Ths sysms will li in h bandwidh of LISA. Suprmassiv BH binaris, which ar h mos promising sourcs for LISA, can also mrg wihin Hubbl im wih high ccnriciis if h Kozai mchanism is in opraion [166]. Anohr asrophysical siuaion whr GW from ccnric binaris can b obsrvabl has bn dscribd by Davis, Lvan and King [167]. A NS-BH binary can bcom ccnric in h la sags of is inspiral. During h firs phas of mass ransfr, numrical sudis [167] show ha h NS is no disrupd and orbis h BH in a high ccnriciy orbi. Th rsuling sysm loss angular momnum via mission of GW which fall in h bandwidh of dcors lik Advancd LIGO. Evoluion undr radiaion racion drivs h wo bodis o conac which is followd by furhr phass of mass-ransfr. Rcnly, his mchanism was succssful in xplaining h ligh curv of h shor gamma-ray-burs GRB 5911 [168]. Rcnly, Grindlay. al [169] hav proposd ha shor GRBs ar producd by NS-NS binary mrgrs which ar formd in globular clusrs. Ths sysms hav h disinc faur ha hy possss high ccnriciis a shor orbial sparaions (s Figur 2 of [169]). Apar from globular clusrs, galaxis can also hos compac binaris which hav rsidual ccnriciy in hir la-inspiral phas. Asymmric kicks impard o NS a h im of hir birh will rsul in highly ccnric NS-NS binaris [17]. Th sam conclusion also applis o NS-BH and BH-BH binaris [17] Tmpla consrucion for ccnric binaris and daa analysis All h abov asrophysical paradigms clarly shows ha inspiralling compac binaris in quasi-llipic orbis ar also qui plausibl sourcs for boh ground and spac basd inrfromric GW dcors. Consrucion of mplas for ccnric binaris rquir an accura knowldg of h scular voluion of h GW phas and h voluion of h orbial lmns (lik smi-major axis, ccnriciy) undr radiaion racion. Th firs amp in his dircion was h classic works of Prs and Mahws [171, 47]. Afr compuing h imavragd (ovr an orbi) far-zon nrgy and angular momnum flux in h Nwonian limi, Prs and Mahws balancd hm wih h loss of binding nrgy and angular momnum of h Kplrian orbi. This allowd hm o obain h ra of dcay of h orbial lmns and showd ha ccnriciy dcays roughly by a facor of hr whn h smi-major axis of h orbi is halvd. Afr Prs and Mahws h voluion of h orbial lmns by his procdur was progrssivly xndd by Blanch and Schäfr o 1PN in [172] and 1.5PN in [172, 173] and finally o 2PN by Gopakumar and Iyr [174]. Whil [175, 173] rquir h 1PN accura orbial dscripion of Damour and Drull [176], [174] crucially mploys h gnralisd 2PN quasi-kplrian paramrizaion of h binary s orbial moion in ADM coordinas as givn in [177, 178, 179]. Mor rcnly, Damour, Gopakumar and Iyr [18] discussd an analyic mhod for consrucing high accuracy mplas for h GW signals from compac binaris in quasillipical orbis in hir inspiral phas. Thy go byond h compuaion of h slow scular 14

3 ffcs by h sandard avraging ovr h orbial priod and compu h addiional fas oscillaory conribuions byond h avrag scular conribuions. Using an improvd mhod of variaion of consans and working up o h lading radiaion racion ordr of 2.5PN, hy combin h hr im scals involvd in h llipical orbi cas -h orbial priod, priasron prcssion and radiaion racion im scals - wihou making h usual approximaion of raing h radiaiv im scal as an adiabaic procss. This was xndd o 3.5PN ordr in Rf [181]. Thr hav bn rlaivly fw xrciss in daa-analysis aspcs of GW from ccnric binaris. This is primarily bcaus inclusion of ccnriciy in h paramr spac lads o a larg incras in h numbr of mplas rquird o sarch for signals [182], which is consqunly accompanid by highr compuaional coss. To circumvn his issu, Marl and Poisson [183] compud h loss in vn ra if ccnric binaris ar sarchd wih circular mplas. Thy found ha vn hough h circular mplas ar no opimal filrs, hy will b fficin in dcing ccnric binaris. Furhr, h loss in signal-o-nois-raio incurrd incrass wih incrasing ccnriciy for a givn oal mass of h binary. Rcnly Tssmr and Gopakumar [184] rvisid h sam problm, bu usd a 2.5PN accura orbial voluion and adopd h phasing formalism dvlopd in [18] mniond abov. Th rsul of hir analysis (wih dcor Iniial LIGO) was ha mplas which modlld GW from binaris volving undr quadrupolar radiaion racion and whos orbis ar 2PN accura circular orbis ar vry fficin in sarching for ccnric binaris PN angular momnum flux: hrdiary rms Th gnraion problm of graviaional wavs for inspiralling compac binaris has bn compld a h hird pos-nwonian (3PN) ordr boh for h quaion of moion of h binary and for is far-zon radiaion fild. Rcnly, in a sris of wo rlad paprs [185, 186], h compuaion of h nrgy flux of graviaional wavs (GW) from inspiralling compac binaris moving in gnral non-circular orbis up o 3PN ordr was discussd. For non-circular orbis, in addiion o h consrvd nrgy and graviaional wav nrgy flux, h angular momnum flux nds o b known o drmin h phasing of quasi-ccnric binaris. As mniond bfor, a knowldg of h angular momnum flux of h sysm avragd ovr an orbi is mandaory o calcula h voluion of h orbial lmns of non-circular, in paricular, llipic orbis undr GW radiaion racion. In his chapr, w compu all h hrdiary rms in h angular momnum flux of inspiralling compac binaris moving in non-circular orbis up o 3PN ordr gnralising arlir work a 1.5PN ordr (ails and spin-orbi) by Scha fr and Rih [173]. Th hrdiary rms, unlik h insananous rms which ar funcions of h rardd im, dpnd on h dynamics of h sysm in is nir pas. Th 3PN hrdiary conribuion o angular momnum flux coms, apar from h ail rms, h ails of ails and ail-squard rms [135, 134]. Unlik h nrgy flux cas, h angular momnum flux also conains an inrsing mmory conribuion a 2.5PN. Using h angular momnum flux xprssion in conjuncion wih h 3PN accura hrdiary par of h nrgy flux obaind in Rf. [187, 185], w compu h hrdiary par of h voluion of h orbial lmns, smi-major axis a r, ccnriciy, man moion n and h priasron advanc paramr k. Evoluion of ohr rlad paramrs such as orbial priod P can b drivd from hs xprssions. W also provid h xprssions for h fluxs and voluion of orbial lmns in h limi of small ccnriciy up o 15

4 scond ordr in. All h rsuls of his chapr ar providd in rms of h PN paramr x = ( Gmω ) 2/3, rlad o h orbial frquncy ω, which hlps on o rcovr h circular orbi c 3 limi sraighforwardly. Th insananous rms in h 3PN angular momnum flux wr compud by Arun in [188]. Th rs of his chapr is organisd as follows. In Scion 5.2, w provid h gnral xprssion for h angular momnum flux in rms of radiaiv momns and rlaions bwn h radiaiv and sourc momns, kping only rms rlvan for compuaion of h hrdiary rms upo 3PN accuracy. Scion 5.3 rviws h soluion of h quaions of moion of compac binaris upo h accuracy w rquir for his chapr. In Scion 5.4, w provid h Fourir domain rprsnaions of h sourc mulipol momns and hir us in avraging h flux ovr h orbial im-scal. Scion 5.5 provids xplici xprssions of h hrdiary conribuions in rms of h Fourir ampliuds. In Scion 5.6 w giv dails of h numrical valuaion of hs conribuions. W provid h compl 3PN hrdiary rms in Scion 5.7 along wih rlvan chcks. In Scion 5.8, voluion of h orbial lmns du o hrdiary rms ar providd. Scion 5.9 compriss h conclusion and discussion of h rsuls of his chapr. Finally, in Scion 5.1, a lis of h Fourir cofficins of h Nwonian momns ar givn in rms of Bssl funcions. 5.2 Srucur of h hrdiary rms in h angular momnum flux Th compl 3PN accura angular momnum flux in h sourc s far-zon, wrin in rms of h symmric rac-fr (STF) mass and currn yp radiaiv mulipol momns (U L s and V L s) [13] can b found in Rf. [188]. Blow w provid h far zon angular momnum flux F J i upo 1PN ordr which will b sufficin o conrol h hrdiary par of F J i upo 3PN. F J i = G { 2 c ɛ 5 i jk 5 U jau (1) ka + 1 [ 1 c 2 63 U jabu (1) kab + 32 ]} 45 V jav (1) ka +. (5.1) Th dos indica rms which conribu o h 3PN insananous rms in F J i and w do no wri hm xplicily. U L and V L (wih L = i jk... a muli-indx composd of l indics, ach indx running from 1 o 3) ar h mass and currn yp radiaiv mulipol momns rspcivly and U (l) L and V(l) L dno hir lh im drivaivs. Th momns ar funcions of rardd im T R T R in radiaiv coordinas. ε c ipq is h usual Lvi-Civia symbol such ha ε 123 = +1. Th shorhand O(n) indicas ha h pos-nwonian rmaindr is of ordr of O(c n ). Using h mulipolar Pos-Minkowskian (MPM) formalism oulind in h prvious chapr, w r-xprss h radiaiv momns in Eq. (5.1) in rms of h sourc momns o an accuracy sufficin for h compuaion of h hrdiary par of h angular momnum flux up o 3PN. Th compl xprssions rquird o calcula h insananous rms wr alrady givn in h prvious chapr. Th rlaions conncing h diffrn radiaiv momns U L and V L o h corrsponding sourc momns I L and J L [124, 135, 134] rquird for his chapr ar givn blow. 16

5 For h mass yp momns w hav U i j (T R ) = I (2) i j (T R ) + 2GM c 3 ( GM ) 2 TR +2 c 3 +O(7), U i jk (T R ) = I (3) i jk (T R) + 2GM c 3 TR dvi (5) i j TR dv [ ( TR V ln [ (V) dv 2b ln 2 ( TR V 2b [ ( TR V ln 2b ) ) ) ] I (4) i j (V) 2 G 7 c ( 7 ln TR V 2b + 97 ] 6 ) I (5) i jk (V) + O(5), (5.2b) TR dvi (3) a<i (V)I(3) ] whr h brack <> dnos STF projcion. In h abov formulas, M is h oal ADM mass of h binary sysm. Th I L s and J L s ar h mass and currn-yp sourc momns, and I L, J L dno hir p-h im drivaivs. For h currn-yp momns, on h ohr hand, w find V i j (T R ) = J (2) i j (T R ) + 2GM TR [ ( TR V ) dv ln + 7 ] J (4) c 3 i j (V) + O(5). (5.3) 2b 6 Th radiaiv momns hav wo disinc conribuions. Th firs par which is a funcion only of h rardd im, T R = T R, ar h insananous rms. Th scond par ha c dpnds on h dynamics of h sysm in is nir pas [124] is rfrrd o as hrdiary conribuions and forms h subjc mar of his chapr. Th paramr b apparing in h logarihms of Eqs. (5.2) and (5.3) is a frly spcifiabl consan, having h dimnsion of im, nring h rlaion bwn h rardd im T R = T R/c in radiaiv coordinas and h corrsponding im ρ/c in harmonic coordinas (whr ρ is h disanc of h sourc in harmonic coordinas). Mor prcisly, w hav T R = ρ c 2 G M ( ) ρ ln. (5.4) c 3 c r W choos h consan b scaling h logarihm o b r o mach wih h choic mad in h c compuaion of ails-of-ails in [135]. From h xprssions for U L s and V L s, on can schmaically spli h oal conribuion o h angular momnum flux as h sum of h insananous and hrdiary rms. F J i = ( ) F J i + ( ) F J i. (5.5) ins hrd Sinc w do no discuss h insananous rms in h angular momnum flux hr, hy ar no givn byond h Nwonian ordr hr hough i is asy o wri hm down using h xprssions, Eqs (5.2) and (5.3) for h radiaiv momns. Th Nwonian insananous rm in F J i which will b calculad lar in his chapr for illusraing h Fourir dcomposiion mhod for calculaing h hrdiary rms is j>a (V) (5.2a) ( ) F J Nwonian i () = 2 G ins 5 c ε ipqi (2) 5 p j ()I(3) q j () (5.6) 17

6 Th hrdiary par can b dcomposd, as mniond in h arlir Scion, as (F J i) hrd = (F J i) ail + (F J i) ail(ail) + (F J i) (ail) 2 + (F J i) mmory, (5.7) Th quadraic-ordr ail ingrals ar xplicily givn by (using Eqs 5.2 and 5.3 in Eq 5.1) (F J i) ail () = G { 4GM + ε c 5 5c 3 i jk dτ ( ( ) I (2) ja () I(5) ka ( τ) I(3) ja () I(4) ka [ln ( τ)) τ + 11 ] 2r /c GM + 63c ε 5 i jk dτ ( ( ) I (3) jab () I(6) kab ( τ) I(4) jab () I(5) kab [ln ( τ)) τ + 97 ] 2r /c GM + ε 45c 5 i jk dτ ( ( ) J (2) ja () J(5) ka ( τ) J(3) ja () J(4) ka [ln ( τ)) τ + 7 ]}, 2r /c 6 (5.8) whil h cubic-ordr ails (proporional o M 2 ) ar + (F J i) ail(ail) () = G 4G 2 M 2 ε c 5 5c 6 i jk dτ ( I (2) ja ()I(6) ka ( τ) I(3) ja ()I(5) ka ( τ)) (5.9a) ( ) [ln 2 τ + 57 ( ) 2r /c 7 ln τ ], 2r /c 441 (F J i) (ail) 2() = G ( 8G 2 M 2 + ( ) ε c 5 5c 6 i jk dτ I (4) ja [ln ( τ) τ + 11 ]) 2r /c 12 ( + ( ) dτ I (5) ka [ln ( τ) τ + 11 ]), 2r /c 12 (5.9b) and finally, h mmory ingral is ( F J i )mmory () = G c 5 4G 35c 5 ε i jk I (3) ja () ( [ ] ) I (3) b<k I(3) a>b [ τ] dτ. (5.1) No ha in Eq. (5.1), a rm of h form G 4G ε c 5 35c 5 i jk I (2) ja () ( d [ ) d I (3) b<k a>b] I(3) [ τ] dτ has bn lf ou. This rm simply rducs o G 4G ε c 5 35c 5 i jk I (2) ja ()I(3) b<k ()I(3) a>b (), i.., an insananous rm. This rm, hrfor, has bn incorporad in h insananous par of h angular momnum flux and has bn compud in Rf. [188]. For h nrgy flux cas, h nir mmory conribuion bcoms insananous (s Rf. [185]). In h quaions Eq. (5.8), (5.9) rcall ha M is h consrvd mass monopol or oal ADM mass of h sourc. Th firs rm in (5.8) is h dominan ail a ordr 1.5PN whil h scond and hird rprsn h sub-dominan ails apparing boh a ordr 2.5PN. Th highr-ordr ails ar no givn sinc hy ar a las a 3.5PN ordr (s [135] for hir xprssions). Th wo cubic-ordr ails givn in Eqs. (5.9) ar boh a 3PN ordr. Th mmory rm appars a 2.5PN ordr. No ha h consan b scaling h logarihms in h ail ingrals in h radiaiv momns Eqs (5.2) & (5.3) has bn rplacd in h abov ail, ail-of-ail c. ingrals as (r /c). For simpliciy, w hav rplacd h symbol for h rardd im T R in radiaiv coordinas by. 18

7 5.3 Soluion of h quaions of moion of compac binaris Doubly-priodic srucur of h soluion To compu h ingrals apparing in Eqs. (5.8), (5.9) and (5.1), a knowldg of h voluion of h sourc is rquird. For his purpos, w nd o consruc h soluion of h quaion-of-moion of compac binaris. For his purpos, w rviw in his Scion, h gnral doubly-priodic srucur of h PN soluion, and h quasi-kplrian rprsnaion of h 1PN binary moion by mans of diffrn yps of ccnriciis. Th works [31, 189, 176] ar closly followd hr. If w nglc h radiaion racion rm a h 2.5PN ordr, h quaions of moion of a compac binary sysm up o h 3PN ordr admi n firs ingrals of h moion. Ths corrspond o h consrvd nrgy, angular and linar momna, and posiion of h cnr of mass [145, 146]. Whn rsricd o h fram of h cnr of mass, h quaions admi four firs ingrals associad wih h nrgy E and h angular momnum vcor J, givn a 3PN ordr by Eqs. (4.8) (4.9) of Rf. [149]. Th moion aks plac in a plan orhogonal o J. W dno by r = x h binary s orbial sparaion, and by v = v 1 v 2 h rlaiv vlociy (boh x and v li in h plan of moion). Th consrvd E and J ar funcions of r, ṙ 2, v 2 and x v (for dfininss w mploy h harmonic coordina sysm of [149] 1 ), and dpnd on h oal mass m = m 1 +m 2 and rducd mass µ = m 1 m 2 /m. Polar coordinas r, φ in h orbial plan ar usd o xprss E and h norm J = J as som xplici funcions of r, ṙ 2 and φ (v 2 = ṙ 2 + r 2 φ 2 ). Th lar funcions can b invrd (by mans of sraighforward PN iraion) o giv ṙ 2 and φ in rms of r and h consans of moion E and J. Thus, ṙ 2 = R[r; E, J], (5.11a) φ = G[r; E, J], (5.11b) whr R and G ar polynomials in 1/r, h dgr of which dpnds on h PN approximaion in qusion. A 3PN ordr, i is svnh dgr for boh R and G [19]. Th various cofficins of h powrs of 1/r ar hmslvs polynomials in E and J, and also, of cours, dpnd on m and h dimnsionlss rducd mass raio ν µ/m. For bound llipic-lik orbis, on can prov [189] ha h funcion R admis wo ral roos, r P and r A such ha r P < r A, which admi som non-zro fini Nwonian limis whn c, and rprsn rspcivly h radii of h orbi s priasron and apasron. Th ohr roos nd o zro in h limi c. Th binary s orbial priod, or im of rurn o h priasron, is obaind by ingraing h radial moion (w drop h dpndnc on E and J in h following, for simpliciy). P = 2 ra r P dr R[r]. (5.12) L us inroduc h man anomaly l and h man moion n by l = n( P ) (5.13) 1 All calculaions in his chapr will b don a h rlaiv 1PN ordr, and a ha ordr hr is no diffrnc bwn h harmonic and ADM coordinas. 19

8 n = 2π P, (5.14) whr P dnos h insan of passag o h priasron. For a givn valu of h man anomaly l, h orbial sparaion r is obaind by invrsion of h ingral quaion l = n r dr r P R [r ]. (5.15) This dfins h funcion r(l) which is a priodic funcion in l wih priod 2π. Using his in Eq. (5.11), h orbial phas φ can b obaind in rms of h man anomaly l by ingraing h angular moion as φ(l) = φ P + 1 n l dl G [ r(l ) ], (5.16) whr φ P dnos h valu of h phas a h insan P. Formally, h funcions r(l) and φ(l) compl h soluion bu dos no ak ino accoun h doubly-priodic naur of h problm. For his purpos, considr h advanc of priasron pr priod, i.., h incras in h orbial phas φ (modulo 2π) during a singl rurn o h priasron φ + 2π = 2 ra r P dr G[r] R[r], (5.17) L us dfin h fracional angl (i.. h angl dividd by 2π) of h oal advanc of h priasron pr orbial rvoluion, 2πK = 2π + φ. (5.18) Thus h prcssion of h priasron pr priod is givn by φ = 2π(K 1). As K nds o on in h limi c (as is chckd from h Nwonian limi), i is ofn convnin o dfin k K 1, which will hn nirly consiu h rlaivisic prcssion. If, lik h radial moion, w inroduc anohr man anomaly l φ and a man angular moion ω φ givn by l φ = ω φ ( P ) (5.19) ω φ = 2π P/K, (5.2) w find ha h wo man moions and man anomalis ar rlad by ω φ = K n (5.21) l φ = K l (5.22) In h cas of a circular orbi, whr h phas volvs linarly wih im, φ = G [r] = ω, whr ω is h orbial frquncy of h circular orbi givn by ω = K n = (1 + k) n. (5.23) In h gnral cas of a non-circular orbi w us h man angular moion ω = Kn (w 11

9 drop h subscrip φ) and o xplicily inroduc h linarly growing par of h orbial phas (5.16) by dcomposing i in h form φ = φ P + ω ( P ) + W(l) = φ P + K l + W(l). (5.24) Hr W(l) dnos a paricular funcion which is priodic in l (hnc, priodic in im wih priod P). From Eq (5.16), his funcion is givn in rms of h man anomaly l by W(l) = 1 n l dl [ G [ r(l ) ] ω ]. (5.25) Finally, h dcomposiion (5.24) xhibis clarly h doubly priodic naur of h binary moion, in rms of h man anomaly l wih priod 2π, and in rms of h priasron advanc K l wih priod 2π K. I is worh noing ha in Rfs. [191, 18] h noaion λ is usd; i corrsponds o λ = K l and will also occasionally b usd hr Quasi-Kplrian rprsnaion of h moion of compac binaris In our calculaions w shall also rquir h xplici soluion of h moion a 1PN ordr, in h form du o Damour & Drull [176]. Th soluion is givn in paramric form in rms of h ccnric anomaly u. Th radius r and h man anomaly l ar xprssd as r = a r (1 r cos u), l = u sin u. (5.26a) (5.26b) Th phas angl φ is givn by (h addiiv consan φ P has bn s qual o zro) whr h ru anomaly V is dfind by 2 φ = K V, (5.27) [( ) 1/2 1 + φ V = 2 arcan an u ]. (5.28) 1 φ 2 In h abov, K is h priasron advanc givn in gnral rms by Eq. (5.18), and a r is h smi-major axis of h orbi. No ha hr ar, in his paramrizaion a 1PN ordr, hr kinds of ccnriciis r, and φ (lablld afr h coordinas r, and φ). All hs ccnriciis diffr from on anohr by 1PN rms, whil h advanc of h priasron pr orbial rvoluion appars also saring a h 1PN ordr. Du o hs faurs, his rprsnaion is rfrrd o as h quasi-kplrian (QK) paramrizaion for h 1PN orbial moion of h binary. Th priodic funcion W of Eq. (5.25) now rads W = K (V l). (5.29) 2 W hav dnod h ru anomaly by V rahr han by h symbol v of arlir paprs o avoid confusion wih h rlaiv spd v. 111

10 Th abov soluion is closd by h xplici dpndnc of h orbial lmns in rms of h 1PN consrvd nrgy E and angular momnum J in h cnr-of-mass fram (akn as usual pr uni of h rducd mass µ). This is givn in Rf. [176]. No ha h smi-major axis a r and man moion n dpnd a 1PN ordr only on h consan of nrgy hrough a r = G m { ( E 2 ν ) } E, (5.3a) 2 c { ( 2 ( 2 E)3/2 15 n = 1 + G m 4 ν ) } E. (5.3b) 4 c 2 Posing h J/(Gm), h 1PN priasron prcssion simply rads whil h hr diffrn ccnriciis ar givn by r = = φ = K = c 2 h 2, (5.31) [ ( {1 + 2 E h ) E 2 ν c ν ]} 1/2, (5.32a) 2 c 2 h 2 [ ( 17 {1 + 2 E h ) E 2 ν c + 2 2ν ]} 1/2, (5.32b) 2 c 2 h 2 [ ( {1 + 2 E h ν ) E 2 c 6 ]} 1/2. (5.32c) 2 c 2 h 2 Noic h following simpl raios (valid a 1PN ordr) r = 1 + (8 3ν) E c, 2 (5.33a) φ = 1 + (8 2ν) E c, 2 (5.33b) r φ = 1 + ν E c. 2 (5.33c) Th binary orbi can b characrisd by ihr (E, J) or any wo of h quasi-kplrian orbial lmns. W choos (n, ) and lis h 1PN accura xprssions for h ohr orbial lmns in rms of n and, which w will rquir lar in h work. For his purpos w firs invr Eq. (5.3b) o obain h 1PN consrvd nrgy in rms of n. E = 1 ( G m n ) 2/3} {1 2 (G m n)2/3 + (15 ν), (5.34) c 3 Using his in Eq. (5.3a), w g a r = ( ) G m 1/3 { ( ) G m n 2/3} 1 + ( 3 + ν), (5.35) n 2 c 3 which is a 1PN xnsion of Kplr s law. From Eqs (5.33) w obain r and φ in rms of n 112

11 and. { r = 1 + (4 3 ( G m n ) 2/3} 2 ν), (5.36a) c 3 ( G m n ) 2/3} φ = {1 + (4 ν). (5.36b) c 3 Th priasron prcssion, givn by Eq. (5.31), also nds o b xprssd as a funcion of n and. To obain i, w us h following xprssion, asily obaind from Eq. (5.32b). 2E h 2 = (1 ) 2 {1 + Th abov rlaion, along wih Eq. (5.34) yilds ( G m n ) 2/3 1 ( ) } 9 + ν (17 7 ν) 2 c 3 4 (1 2 ) (5.37) K = (1 2 ) ( G m n c 3 ) 2/3 (5.38) 5.4 Fourir dcomposiion of h binary s mulipol momns Nwonian Angular Momnum flux Th mhod w shall us in his chapr is illusrad by h compuaion of h avragd angular momnum flux of compac binaris a Nwonian ordr using a Fourir dcomposiion of h Kplrian moion [171]. Th GW angular momnum flux rducs a Nwonian ordr o (from Eq. 5.6) 3 (F J i) = 2 5 ε (2) i jk I ja () I (3) ka (), (5.39) whr mans h Nwonian limi, h suprscrip (n) rfrs o im diffrniaions. I i j is, by consrucion, h symmric-rac-fr (STF) quadrupol momn a Nwonian ordr givn by I i j = µ x <i x j>. x i is h binary s orbial sparaion, and h angular bracks around indics indica h STF projcion: x <i x j> x i x j 1 3 δi j r 2. Howvr, h prsnc of h Lvi-Civia symbol nsurs ha h rac par of h symmrizd quadrupol momn cancls ou. Hnc, unlik h nrgy-flux calculaion in Rf. [185], in his chapr w will ignor h rac of h quadrupol momn. Thus I i j = µ x i x j. (5.4) Prs & Mahws [171, 47] obaind h xprssion of h (avragd) Nwonian flux for compac binaris on ccnric orbis by wo mhods. Th firs mhod was o dircly avrag in im Eq. (5.39) using h xprssion (5.4) compud for h Kplrian llips; h scond mhod was o dcompos h componns of h quadrupol momn ino discr Fourir sris using h known Fourir dcomposiion of h Kplr orbi (h wo mhods, 3 From now on w s c = 1 and G =

12 as xpcd, agrd on h rsul). In h scond mhod h quadrupol momn, which is a priodic funcion of im a Nwonian ordr, is hus dcomposd ino a Fourir sris I i j () = wih I i j = p= 2π Ii j i p l, (5.41a) dl 2π I i j i p l, (5.41b) whr l is h man anomaly of h binary moion, Eq. (5.13). Sinc I i j cofficins clarly saisfy I i j Eqs. (5.41) ino (5.39) w obain (F J i) = 2 5 = ( p) I i j p= q= is ral h Fourir ( dnos h complx conjuga). Insring (ip n) 2 (iq n) 3 ε i jk I ja I (q) ka i(p+q)l. (5.42) Afr his, w prform an avrag ovr on priod P. This mans h avrag ovr l = n ( P ) which is asily prformd wih h following ipl 2π dl 2π i p l = δ p,. (5.43) This immdialy yilds h avragd angular momnum flux in h form of h Fourir sris (F J i) = 4 5 i (p n) 5 ε i jk I ja I ka. (5.44) p=1 Using dimnsional analysis (and h known circular orbi limi) his flux is ncssarily of h form (F J i) = 32 m9/2 ν2 5 a f J() z 7/2 i, (5.45) whr ν = µ/m and a is h smi-major axis of h Kplr orbi, z i is an uni vcor paralll o h angular momnum of h binary (and prpndicular o h orbi lying in h x-y plan) and h funcion f J () is a dimnsionlss funcion dpnding only on h binary s ccnriciy. Th cofficin in fron of (5.45) is chosn in such a way ha f J () rducs o on in h circular orbi limi ( ). Thrfor, f J () = i 8 µ 2 a 4 p=1 p 5 ε i jk I ja I ka z i. (5.46) Th Fourir cofficins of h quadrupol momn ar xplicily givn by Eqs. (5.128) in Scion 5.1. f J () admis an algbraically closd-form xprssion which is crucial for h iming of h binary pulsar PSR [5], and givn by f J () = (1 2 ) 2. (5.47) 114

13 Th mhod of dcomposing h Nwonian momn of compac binaris as discr Fourir sris was usd in Rf. [173] o compu h ail a h dominan 1.5PN ordr. To xnd his rsul w nd o b mor sysmaic abou h Fourir dcomposiion of h (no ncssarily Nwonian) sourc mulipol momns Gnral srucur of h Fourir dcomposiion Th wo ss of sourc momns of h compac binary ar dnod by I L () and J L () following Rf. [126]. Th muli-indx noaion mans L i 1 i 2 i l, whr l is h numbr of indics or mulipolariy (which should no b confusd wih h man anomaly l). In his Scion w sudy h srucur of h mass and currn momns I L and say J L 1 (whr L 1 i 1 i 2 i l 1 is chosn in h currn momn for convninc rahr han L), a any PN ordr and for a compac binary sysm moving on a gnral non-circular orbi 4. Thir gnral srucur can b wrin as I L () = J L 1 () = l F k [r, ṙ, v 2 ] x <i 1 i k v i k+1 i l >, k= l 2 k= G k [r, ṙ, v 2 ] x <i 1 i k v i k+1 i l 2 ε i l 1>ab x a v b, (5.48a) (5.48b) whr x i = y i 1 yi 2 and vi = dx i /d = v i 1 vi 2 dno h rlaiv posiion and vlociy of h wo bodis (in a harmonic coordina sysm). In (5.48) w pos for insanc x i 1 i k x i1 x i k, and h angular bracks surrounding indics rfr o h usual symmric-rac-fr (STF) projcion wih rspc o hos indics. Using polar coordinas r, φ in h orbial plan (as in Sc ), h abov inroducd cofficins F k and G k dpnd on h masss and on r, ṙ and v 2 = ṙ 2 + r 2 φ 2. For quasi-llipic moion on can xplicily facoriz ou h dpndnc on h orbial phas φ by insring x = r cos φ, y = r sin φ, and v x = ṙ cos φ r φ sin φ, v y = ṙ sin φ + r φ cos φ. Furhrmor, using h xplici soluion of h moion (Sc ) r, ṙ and v 2, and hnc h F k s and G k s can b xprssd as priodic funcions of h man anomaly l = n ( P ), whr n = 2π/P. W hn find ha h abov gnral srucur of h mulipol momns can b xprssd in rms of h phas angl φ, as h following fini sum ovr som magnicyp indx m ranging from l o +l, I L () = J L 1 () = l m= l l m= l A (m) L (l) i m φ, B (m) L 1 (l) i m φ, (5.49a) (5.49b) involving som cofficins (m) A L and (m) B L 1 dpnding on h man anomaly l and which ar complx ( C). (Som of hs cofficins could b vanishing in paricular cass.) Th poin for our purpos is ha hs cofficins ar priodic funcions of l wih priod 2π. As plan. 4 Howvr h inrinsic spins of h compac objcs ar nglcd, so h moion aks plac in a fixd orbial 115

14 w can s, h srucur of h mass and currn momns I L and J L 1 is basically h sam, bu hir cofficins (m) A L and (m) B L 1 will hav a diffrn pariy, bcaus of h Lvi-Civia symbol nring h currn momn J L 1. To procd furhr, l us xploi h doubly priodic naur of h dynamics in h wo variabls λ K l and l (as rviwd in Sc ). Th phas is givn in full gnraliy by Eq. (5.24) whr W(l) is priodic in l. In h following i will b mor convnin o singl ou in h xprssion of h phas h purly rlaivisic prcssion of h priasron, namly λ l = k l whr k = K 1. W insr h xprssion of h phas variabl ino Eqs. (5.49) which yilds many facors modifying h cofficins of (5.49), bu in such a way ha hy rain h priodiciy in l. Hnc I L () = J L 1 () = l m= l l m= l I (m) L (l) i m k l, J (m) L 1 (l) i m k l, (5.5a) (5.5b) whr h cofficins (m) I L (l) and (m) J L 1 (l) ar 2π-priodic. This allows us o us a discr Fourir sris xpansion in h inrval l [, 2π] for ach of hs cofficins, i.., I (m) L (l) = J (m) L 1 (l) = p= p= I (p,m) L i p l, J (p,m) L 1 i p l, (5.51a) (5.51b) and h invrs rlaions ar I (p,m) L = J (p,m) L 1 = 2π 2π dl 2π I (m) L (l) i p l, dl 2π J (m) L 1 (l) i p l. (5.52a) (5.52b) W now hav h following final dcomposiions of h mulipol momns, I L () = J L 1 () = l p= m= l l p= m= l I (p,m) L i (p+m k) l, J (p,m) L 1 i (p+m k) l. (5.53a) (5.53b) Th momns I L and J L 1 bing ral, hir Fourir cofficins saisfy (p,m) I L = ( p, m) I L and (p,m)j L 1 = ( p, m) J L 1. Th prvious dcomposiions wr gnral, bu i is sill usful o inroduc a spcial noaion for h paricular cas of h Nwonian ordr, for which h rlaivisic prcssion k. In his limi, h usual priodic Fourir dcomposiion of h momns is rcovrd 116

15 [gnralizing Eqs. (5.41)], wih only on Fourir summaion ovr h indx p, so ha I L () = p= J L 1 () = IL i p l, (5.54a) JL 1 i p l. p= (5.54b) Th Nwonian Fourir cofficins ar qual o h sums ovr m of h doubly-priodic Fourir cofficins in Eqs. (5.53) whn akn in h Nwonian limi, namly IL = JL 1 = l (p,m) m= l l I L, (5.55a) J L 1. (p,m) m= l (5.55b) 5.5 Hrdiary conribuions in h angular momnum flux Th chniqu of h prvious Scion is applid o h compuaion of h ail ingrals in h angular momnum flux of compac binaris. Alhough h compuaions ar ffcivly don up o h 3PN lvl, h mhod w propos could in principl b implmnd a any PN ordr. W shall compu all h ail and ail-of-ail rms (5.8) (5.9) [i.. up o h 3PN ordr] avragd ovr h man anomaly l. Toghr wih h insananous rms rpord in Rf. [188] w shall obain h compl xprssion of h 3PN angular momnum flux. I is clar from Eqs. (5.8) (5.9) ha all h rms ncssia an valuaion a h rlaiv Nwonian ordr xcp h mass-yp quadrupolar ail rm firs rm in (5.8) which mus crucially includ h 1PN corrcions. W sar wih all h rms rquird a rlaiv Nwonian ordr and hn ackl h mor difficul 1PN quadrupolar ail rm Tails a rlaiv Nwonian ordr In his scion w considr h mass-yp quadrupolar ail rm in h angular momnum flux, h firs rm in Eq. (5.8). Howvr, w will no compu his rm a h PN ordr rquird for his chapr, bu a h rlaiv Nwonian ordr 5. This will srv as a simpl illusraion of h mhod w will us for compuing h highr-ordr ails. Th 1.5PN mass-quadrupol ail conribuion is, from Eq. (5.8) (F J i) mass quad ail = 4M 5 ε i jk [ ln + dτ ( ) τ 2r 5 W shall compu his rm a 1PN rlaiv ordr in Sc ( (2) I ja () (5) I ka ( τ) (3) I ja () (4) I ka ( τ) + 11 ], (5.56) 12 ) 117

16 whr h bracks rfr o h avrag ovr h man anomaly l as dfind by Eq. (5.43). Th rm (5.56) was alrady compud using a Fourir sris a Nwonian ordr in Rf. [173]; no ha h mhod of [173] is valid only for priodic moion and hus is applicabl only a h Nwonian lvl. In his Scion w rcovr h Nwonian rsul of [173]. Th Fourir dcomposiion of h Nwonian quadrupol momn was alrady givn in gnral form by Eqs. (5.41). Insring ha ino h flux (5.56), w valua h ail ingral by using h fac ha if l() = n ( P ) corrsponds o h currn im, hn l( τ) = l() n τ corrsponds o h rardd im τ. Nx w prform h avrag ovr h currn valu l() wih h hlp of h formula (5.43). W g (F J i) mass quad ail = 8M 5 i p= (p n) 7 ε i jk I ja I ka ( p) + ( ) τ dτ [ln ip n τ 2r + 11 ]. (5.57) 12 No h crucial rplacmn of h Fourir dcomposiion of h quadrupol momn I ka ( τ) a h rardd im τ in h ail ingral in Eq. (5.56) by h Fourir cofficins a h currn im, dfind by Eq. (5.41). This prmis us o ak h Fourir cofficins of I ka ousid h ail ingral. This rplacmn maks h rsul drivd blow xac only in a PN sns, as w hav nglcd h ffc of h binary s adiabaic voluion by radiaion racion in h pas. Consqunly, his rplacmn inroducs a rmaindr rm in Eq. (5.57) givn by h ordr of magniud of h adiabaic paramr ξ rad ω/ω 2. From Rfs. [124, 158], w know ha h abov rplacmn of h currn moion in h ail ingral is valid only modulo som rmaindr O(ξ rad ) or, mor prcisly, O(ξ rad ln ξ rad ). This rmaindr corrsponds o a corrcion rm of rlaiv 2.5PN ordr which is always ngligibl for our purposs (h 1.5PN ordr of h ails maks h corrcion rms du o h influnc of h binary s pas a 4PN ordr). To ackl h las facor in (5.57) which is h ail ingral in h Fourir domain, w us h closd-form formula + ( ) τ dτ i σ τ ln = 1 [ π 2r σ 2 sign(σ) + i( ln(2 σ r ) + C ) ], (5.58) whr σ p n, sign(σ) = ±1 and C =.577 dnos h Eulr consan. Insring Eq. (5.58) ino (5.57) w hav (F J i) mass quad ail = 8π M i (p n) 6 ε i jk I ja I ka 5. (5.59) p=1 No ha h rang of p s corrsponds o posiiv frquncis only. Th rmaining ail ingrals, givn by h scond and hird rms in Eq. (5.8), ar valuad in xacly h sam way. Wih h PN accuracy of h prsn calculaion hs ingrals ar ruly Nwonian so h mass ocupol momn I i jk and currn quadrupol momn J i j ar rquird a Nwonian ordr only. For simpliciy, w drop h suprcrip bcaus hr can b no confusion wih ohr rsuls. W hus nd o valua h im-avragd fluxs (F J i) mass oc ail = 2M 63 ε i jk + dτ ( I (3) jab () I(6) kab ( τ) I(4) jab () I(5) kab ( τ)) 118

17 (F J i) curr quad ail = 64M + 45 ε i jk [ ( τ ln 2r [ ln ) dτ ( J (2) ja ( ) τ 2r + 97 ], (5.6) 6 () J(5) ka ( τ) J(3) ja () J(4) ka ( τ)) ]. (5.61) No ha, as in h cas of h mass-quadrupol momn, h rac of J i j also dos no conribu o h angular momnum flux (s h argumn afr Eq. (5.39)). Howvr, for I i jk, w do hav o ak ino accoun is rac. Insring h Fourir dcomposiion of h momns, prforming h avrag using Eq. (5.43) and using h ingraion formula (5.58) givs us (F J i) mass oc ail = 4πM 63 i (p n) 8 ε i jk I (F J i) curr quad ail = 128πM 45 p=1 i p=1 (p n) 6 ε i jk J jab I kab ja J (5.62a) ka. (5.62b) In Sc. 5.6 w shall provid som numrical plos for h nhancmn ccnriciydpndn facors associad wih Eqs. (5.62), sinc hy do no hav a closd-form xprssion Tails-of-ails and ails squard A h 3PN ordr (i.. 1.5PN byond h dominan ail) h firs cubic non-linar inracion, bwn h quadrupol momn I i j and wo mass monopol facors M, appars. From Eqs. (5.9) w hav o compu h ail-of-ail conribuion and h so-calld ail squard on, + (F J i) ail(ail) = 4M2 5 ε i jk dτ ( I (2) ja ()I(6) ka ( τ) I(3) ja ()I(5) ka ( τ)) [ ( ) τ ln ( ) τ 2r 7 ln ], (5.63a) 2r 441 ( + ( ) (F J i) (ail) 2 = 8M2 τ 5 ε i jk dτ I (4) ja [ln ( τ) + 11 ]) 2r 12 ( + ( ) τ dτ I (5) ka [ln ( τ) + 11 ]). (5.63b) 2r 12 (5.63c) Boh rms ar valuad a rlaiv Nwonian ordr. W insr h Fourir dcomposiion of h Nwonian quadrupol momn (5.41) [again supprssing h suprscrip for simpliciy]. Th nw faur wih rspc o h ails is h apparanc of a logarihm squard in h ail-of-ail ingral (5.63). W hav again rplacd h moion in h infini pas of 119

18 h binary by h moion in h currn im (s h argumn following Eq. (5.57)). Th closd-form formula rquird o dal wih his rm is [compar wih Eq. (5.58)] + ( ) τ dτ i σ τ ln 2 = i { π 2 [ π 2r σ 6 2 sign(σ) + i( ln(2 σ r ) + C ) ] 2 }, (5.64) and wih his formula, oghr wih (5.58), w obain h rsul (F J i) ail(ail) = 8M2 5 i (p n) 7 ε i jk I ja I ka p=1 { π 2 6 2( ln(2p n r ) + C ) 2 57( + ln(2p n r ) + C ) } Th ail squard rm is radily compud wih (5.58) and is (F J i) (ail) 2 = 8M2 5 i (p n) 7 ε i jk I ja p=1 I ka { π ( Adding h wo rsuls (5.65a) and (5.66) w finally g ln(2p n r ) + C (5.65a) ) 2 }. (5.66) (F J i) ail(ail)+(ail) 2 = 8M2 5 i (p n) 7 ε i jk I ja p=1 I ka { 2π ln(2p n r ) C }. 294 (5.67a) No ha h conribuion from logarihms squard has canclld ou bwn h wo rms (5.65a) (5.66). Such cancllaion is known o occur for gnral sourcs [135]. No also ha h rsul (5.67a) sill dpnds on h arbirary lngh scal r. I is imporan o rac ou h fa of his consan and chck ha h compl angular momnum flux w obain a h nd (including all h insananous conribuions compud in [188]) is indpndn of r Th mass quadrupol ail a 1PN ordr In his subscion, w calcula h mass quadrupol ail a h rlaiv 1PN ordr, namly (F J i) mass quad ail = 4M 5 ε i jk + [ ln ( τ dτ ( I (2) ja ) 2r () I(5) ka ( τ) I(3) ja () I(4) ka ( τ)) ], (5.68) A h 1PN ordr (and similarly a any highr PN ordrs), w mus ak car of h doubly-priodic srucur of h soluion of h moion [Sc ], and dcompos h mulipol momns according o h gnral formulas (5.53). Hnc h 1PN mass quadrupol 12

19 momn I i j nring Eq. (5.68) is dcomposd as I i j () = 2 p= m= 2 I (p,m) i j i (p+m k) l, (5.69) wih doubly-indxd Fourir cofficins (p,m) I i j which ar valid hrough ordr 1PN. Th harmonics for which m = ±1 ar zro a h 1PN ordr, so Eq. (5.69) rducs o I i j () = p= { } I (p, 2) i j i (p 2k) l + I i j i p l + I i j i (p+2k) l, (5.7) (p,) (p,2) bu for our purposs, Eq. (5.69) is mor convnin, kping in mind ha h rms wih m = ±1 ar absn. As bfor w insr Eq. (5.69) ino Eq. (5.68) o obain [afr nglcing 2.5PN radiaion racion rms O (ξ rad ) mniond bfor] (F J i) mass quad ail = 4M 5 i ε i jk I (p,m) ja p,p ;m,m n 7 ( (p + mk) 2 (p + m k) 5 (p + mk) 3 (p + m k) 4) I (p,m ) ka i(p+p +(m+m )k)l + [ ( ) dτ i (p +m k) n τ τ ln + 11 ], 2r 12 (5.71) whr h summaions rang from o + for p and p, and from 2 o 2 for m and m. Th facors (p + mk) 2, (p + m k) 5 c. com from h im-drivaivs of h quadrupol momn. W lav h las wo facors in Eq. (5.71) as hy ar, namly h avrag ovr l of an lmnary doubly-priodic complx xponnial, and h Fourir ransform of h ail ingral. Th xprssion in Eq. (5.71) is o b calculad a h 1PN ordr. Sinc h rlaivisic advanc of h priasron k is a small 1PN quaniy, w firs valua Eq. (5.71) a linar ordr in k [i.., nglcing O(k 2 ) which is a las 2PN]. Afrwards w shall insr h xplici PN xprssions for h 1PN quadrupol momn and ADM mass. Th ncssary formulas for prforming h linar-ordr xpansion in k of h las wo facors in Eq. (5.71) ar providd blow. Th avrag w prform is ovr h orbial priod (im o rurn o h priasron) and so is dfind by 2π i (p+m k) l dl 2π i (p+m k) l. (5.72) Using h fac ha m k 1 sinc w ar in h limi whr k (hnc p + m k is nvr an ingr unlss k = ), w radily find m k if p i (p+m k) l p = + O(k 2 ). (5.73) 1 + i π m k if p = Th abov rsul dpnds only on whhr p is zro or no, and is ru for any ingr m, xcp ha whn m = h rsul (5.73) bcoms xac as hr is no rmaindr rm O(k 2 ) in his cas. 121

20 To compu h ail ingral givn by h las facor in Eq. (5.71), w xpand i a firs ordr in k, obaining + ( ) ( τ dτ i (p+m k) n τ ln = 1 m k ) + ( ) τ dτ ip n τ ln i m k 2r p 2r p 2 n + O(k2 ), (5.74) and w apply for h rmaining ingral in Eq (5.74) h formula Eq. (5.58). Using Eqs. (5.73) and (5.74) w can xplicily compu h ail xprssion Eq. (5.71) a firs ordr in k (h xnsion o highr ordr in k would in principl b sraighforward). Th rsul is lf in h form of h mulipl Fourir sris Eq. (5.71), ino which h rsuls (5.73) (5.74) hav bn insrd (w do no giv a mor xplici form for his rsul which is givn by a complicad Mahmaica xprssion). In h nx Scion w shall rxprss his sris in rms of som lmnary nhancmn funcions which w valua numrically Mmory Ingral a 2.5PN ordr Th mmory conribuion is, from Eq. (5.1), ( F J i )mm = 4 35 ε i jk I (3) ja () ( [ ] ) I (3) b<k I(3) a>b [ τ] dτ (5.75) in which h symmrisaion ovr k & a in h ingrand can b rmovd bcaus i is manifsly symmric and h raclssnss condiion can also b rmovd bcaus of h prsnc of ε i jk and h symmry of I i j. Fourir dcomposing I i j w g ( F J i )mm = 4 35 ε i jk (i p n) 3 I ja ipl p= ( ) i(q+r)n τ dτ q= (i q n) 3 I bk iql r= (i r n) 3 I ab irl (5.76) whr, lik in h ail ingrals w hav nglcd h adiabaic orbial voluion of h binary and rplacd i by h moion a h currn im. Th ingrand for h mmory dos no conain h log krnl, bu i bing highly oscillaory h crss and roughs cancl ou and h only conribuion coms from h boundaris. Howvr h infini pas conribuion (corrsponding o τ ) is zro if w assum saionariy in h pas. On prforming an avrag ovr an orbi w g using i(p+q+r)l = δ p+q+r,. (5.77) ( F J i )mm = 4 35 n8 ε i jk p= q= p 2 q 3 (p + q) 3 I ja I bk I (p+q) ab (5.78) 122

21 This, on simplificaion, rducs o ( F J i )mm = 8 35 n8 ε i jk p=1 q=1 p 2 q ((p 3 + q) 3 R[I ja I bk I ab ] (q) (p+q) ) (p q) 3 R[I ja I bk I ab ] (q) (p q) (5.79) whr R[x] sands for ral par of x. Th angular momnum flux conribuion from h mmory, in rms of h PN paramr x = (m ω) 2/3, bcoms ( F J i )mm = 32 ( ) 1 5 ν2 m x 7/2 28 x5/2 ρ J () z i (5.8) whr ρ J () is h nhancmn funcion corrsponding o h mmory and gos o zro as. I is givn by ρ J () = ε i jk p=1 q=1 ) p 2 q ((p 3 + q) 3 R[I ja I bk I ab ] (p q)3 R[I ja I bk I ab ] z i (5.81) (q) (p+q) (q) (p q) W find ha h funcions of h quadrupol momn Fourir cofficins apparing in ach of h wo rms in Eq. (5.81) ar pur imaginary and hrfor, w hav, lik h circular orbi cas, ρ J () =. (5.82) In h fuur, w would lik o look a his rsul in mor dail, spcially a proof of h vanishing of his rm in a im-domain calculaion. Also, h validiy of h assumpion of h rplacmn of h pas moion by h currn moion insid h mmory ingral nds o b rad mor rigorously, prhaps by h us of Fourir ransforms rahr han Fourir sris. 5.6 Numrical calculaion of h ail ingrals Dfiniion of h ccnriciy nhancmn facors W dfin hr som funcions of h ccnriciy by crain Fourir sris of h componns of h Nwonian mulipol momns I L and J L 1 for a Kplrian llips wih ccnriciy, smi-major axis a, frquncy n = 2π/P (such ha Kplr s law n 2 a 3 = m holds a Nwonian ordr). In h cnr of mass fram I L = µs l (ν)x <L> and J L 1 = µs l(ν)x <L 2 ε il 1>ab x a v b whr µ = m 1 m 2 /m = ν m. W pos s l (ν) X l ( ) l X l 1 1, whr, X 1 m 1 = ( ) 1 m ν, and, X 2 m 2 = ( ) 1 m ν. L us rscal h lar Nwonian momns in ordr o mak hm dimnsionlss I L µ a l s l (ν) Î L, (5.83a) J L 1 µ a l n s l (ν) Jˆ L 1. (5.83b) 123

22 φ J ( ) Figur 5.1: Variaion of ϕ J () wih h ccnriciy. In h circular orbi limi w hav ϕ J () = 1. Our firs nhancmn funcion is h Prs & Mahws [171, 47] funcion which w hav alrady xprssd in Eq. (5.46) as a Fourir sris [and which urns ou o admi h analyically closd form (5.47)]. This sris bcoms, in rms of h Fourir componns of h rscald quadrupol momn Î i j f J () = i 8 p=1 p 5 ε i jk Î jb Î kb zi, (5.84) and is such ha h avragd Nwonian angular momnum flux of compac binaris rads (F J i) = 32 5 ν2 m x 7/2 f J () z i. (5.85) In h abov w hav dfind, for fuur convninc, h frquncy-rlad PN paramr x = (m ω) 2/3 whr ω is h binary s orbial frquncy dfind for gnral orbis by Eq. (5.23). No ha in Eq. (5.85) (which is Nwonian) w can rplac ω by n (hnc x rducs o m/a). Nx, w dfin svral ohr nhancmn funcions of h ccnriciy which will prmi o usfully paramriz h ail rms a Nwonian ordr. Firs w pos ϕ J () = i 16 p=1 p 6 ε i jk Î jb Î kb zi. (5.86) Lik for f J () his funcion is dfind in such a way ha i nds o on in h circular orbi limi, whn. Howvr, unlik f J (), i dos no admi a closd-form xprssion, and w lav i in h form of a Fourir sris. Th funcion ϕ J () paramrizs h mass quadrupol ail a Nwonian ordr, in h sns ha w hav, from Eq. (5.59), (F J i) mass quad = 32 5 ν2 m x 7/2 [ 4π x 3/2 ϕ J () ] z i. (5.87) For circular orbis, ϕ J () = 1 and w rcogniz h cofficin 4π of h 1.5PN ail rm ( x 3/2 ) as compud analyically in Rfs. [173]. Th funcion ϕ J () has alrady bn compud 124

23 β J ( ) γ J ( ) Figur 5.2: Variaion of β J () (lf panl) and γ J () (righ panl) wih h ccnriciy. In h circular orbi limi w hav β J () = γ J () = 1. numrically from is Fourir sris (5.86) in Rf. [173]. Hr w show h plo of ϕ J () in Fig. 5.1 (s Sc for dails on h numrical compuaion) 6. W nx provid similar xprssions for h 2.5PN mass ocupol and currn quadrupol ails by posing β J () = 2 i 1643 γ J () = 8 i p=1 p=1 p 6 ε i jk p 8 ε i jk Î jab Î ˆ J ja J ˆ ka zi. kab zi, (5.88a) (5.88b) Ths funcions also nd o on whn (as will b chckd lar) and mos probably do no admi any closd-form xprssions. Th ail rms ( x 5/2 ) of Eqs. (5.6) rduc o (F J i) mass oc ail = 32 [ ] ν2 m x 7/2 216 π (1 4 ν) x5/2 β J () z i, (5.89) (F J i) curr quad ail = 32 [ π ] 5 ν2 m x 7/2 18 (1 4 ν) x5/2 γ J () z i. (5.9) Th numrical graphs of h funcions β J () and γ J () ar shown in Fig Two furhr nhancmn facors ar ndd for h ail-of-ail and ail squard ingrals (which ar Nwonian). Th firs of hs funcions looks vry much lik f J (), Eq. (5.84), in h sns ha is Fourir sris involvs odd powrs of h mods p. Namly w dfin F J () = i 32 p=1 p 7 ε i jk Î ja Î ka zi. (5.91) Thanks o his odd powr p 7 w find ha F J () admis lik for f J () an analyic closd 6 Our noaion is diffrn from h on in Rih & Schäfr [173]; h funcion ϕ RS () hr is rlad o our dfiniion by ϕ RS () = ϕ J ()/ f J (). In h prsn work i is br no o divid h various funcions by h Prs & Mahws funcion f J (). 125

24 χ J ( ) F J ( ) Figur 5.3: Variaion of χ J () (lf panl) and F J () (righ panl) wih h ccnriciy. In h righ panl, h xac xprssion of F J () givn by Eq. (5.92) is usd. In h circular orbi limi w hav χ J () = and F J () = 1. form which is F J () = (1 2 ) 5. (5.92) W nd anohr funcion whos Fourir ransform diffrs from h on of F J () by h prsnc of h logarihm of mods, namly χ J () = i 32 p=1 ( p ) p 7 ln ε i jk 2 Î ja Î ka zi. (5.93) Mos probably χ J () dos no admi any analyic form [hnc w nam i using h Grk alphab in conras o f J () and F J ()]. No ha χ J () has bn xcpionally dfind in such a way ha i vanishs whn. This is asily chckd sinc in h circular orbi posssss only on harmonic, which is h on for which p = 2, and consqunly h log-rm in χ J () bcoms zro. In Fig. 5.3 w show h numrical plo of h funcion χ J () [and also h on for F J ()]. In Fig. 5.3 w show h numrical plo of h funcion χ J () [and also h on for F J ()]. Wih h abov dfiniions h sum of ail-of-ail and ail squard conribuions obaind in Eq. (5.67a) bcoms limi (and a Nwonian ordr) h quadrupol momn I i j (F J i) ail(ail)+(ail) 2 = 32 {[ 5 ν2 m x 13/ π C 1712 ] 15 ln (4 ω r ) F J () 1712 } 15 χ J() z i. (5.94) (5.95) Th circular-orbi limi is rad off and sn o agr wih Eq. (5.9) in Rf. [135] or Eq. (12.7) in Rf. [116]. Finally w provid h mass quadrupol ail a 1PN ordr, whos compuaion is much mor involvd (s Sc ) as h Fourir sris Eq. (5.71) conains svral summaions, and dpnd on h inrmdia rsuls (5.73) and (5.74). Th compuaion mus also incor- 126

25 α J ( ) θ J ( ) Figur 5.4: Variaion of α J () (lf panl) and θ J () (righ panl) wih h ccnriciy. In h circular orbi limi w hav α J () = and θ J () = 1. pora h 1PN rlaivisic corrcion in h mass quadrupol momn and ADM mass; w provid hm in Eqs. (5.98) and (5.99) blow. Probably no simpl way xiss [i.. no simpl Fourir sris lik for insanc (5.93)] for xprssing h nw nhancmn funcions of ccnriciy which appar a h 1PN ordr. Howvr i can b asily chckd ha h 1PN rm is a linar funcion of h symmric mass raio ν, hnc w mus inroduc wo nhancmn funcions, dnod blow α J and θ J. As usual, w normaliz hm so ha α J () = 1 and θ J () = 1. W hus hav [xnding Eq. (5.87) a h 1PN ordr] (F J i) mass quad = 32 [ 5 ν2 m x {4π 5 ϕ J ( ) + π x α J( ) ]} 21 ν θ J( ) z i. (5.96) This quaion dfins h wo nhancmn funcions α J and θ J, and w us Mahmaica o compu hm as complicad Fourir dcomposiions, which will hn b dircly compud numrically using h mhod oulind in Sc Noic ha sinc w ar a h 1PN ordr w mus b spcific abou ccnriciy w us. W adopd hr h im ccnriciy nring h Kplr quaion (5.26b) in Sc Th ohr ccnriciis ar rlad o i by Eqs. (5.33) a h 1PN ordr. On h ohr hand, h frquncy-rlad PN paramr, givn by x = (m ω) 2/3, (5.97) crucially includs h 1PN rlaivisic corrcion coming from h priasron advanc K = 1 + k, hrough h dfiniion ω = n K of Sc All h 1PN corrcions arising from h formulas (5.73) and (5.74), h mulipol momns M and I i j, h us of h im ccnriciy and h spcific PN variabl x, ar incorporad in a Mahmaica program daling wih h dcomposiion (5.71) and usd o obain (5.96). Th plos of h nhancmn funcions α() and θ() ar givn in Fig Numrical valuaion of h Fourir cofficins L us now dscrib h numrical implmnaion of h compuaion of h Fourir cofficins of h mulipol momns ha lad o h numrical plos of h prvious Scion. W focus on h compuaion of h crucial cofficins (p,m) I i j a 1PN ordr which ar h mor difficul o obain. Th mass quadrupol momn wih 1PN accuracy is givn by [compar 127

26 wih h gnral srucur (5.48a)] { [ ( 29 I i j = µ 1 + v ) 14 ν + m ( 57 r + 87 )] ν x i x j ( ) 7 ν r 2 v i v j + ( ) } ν r ṙ x i v j, (5.98) whr x i and v i = dx i /d ar h rlaiv posiion and vlociy in harmonic coordinas, and r = x i (lik in Sc ). Equaion (5.98) is valid for non-spinning compac binaris on an arbirary quasi-kplrian orbi in h cnr-of-mass fram (s.g. [132]). Sinc w calcula ails wih 1PN rlaiv accuracy w nd o know how h ADM mass M rlas o h oal mass m = m 1 + m 2 a 1PN ordr, ( v 2 M = m [ 1 + ν 2 m r )]. (5.99) Wih h hlp of h quasi-kplrian rprsnaion [Sc ], h dpndnc of I i j on x i, v i, r, v 2 and ṙ is paramrizd by h ccnric anomaly u. Howvr, as xplaind prviously w rquir I i j (l) in h im domain o procd. Th sps of our numrical implmnaion can b schmaically xprssd as : 1. Firsly, ach componn of h 1PN mass quadrupol is xprssd in rms of h quasi-kplrian paramrs using Eqs. (5.26) (5.28). Th componns of h mass quadrupol bcom funcions of h ccnric anomaly u, and ar paramrizd by h man moion n and by on of h ccnriciis (w chos o b h im ccnriciy in Kplr s quaion (5.26b) 7 ) 2. Now w numrically invr h quaion for h man anomaly l = u sin u o obain h funcion u(l). This can b don ihr by using h sris rprsnaion in rms of Bssl funcions, 1 u = l + 2 s J s(s ) sin(s l), (5.1) (5.33). s=1 or numrically by finding h roo of l = u sin u. W find h lar mhod o b mor fficin and accura mhod and mploy i using h FindRoo rouin in Mahmaica. A abl of 2 poins of u and l bwn and 2π (for ach valu of ) was gnrad for his purpos. Th abov invrsion allows us o r-xprss all funcions of h ccnric anomaly u as funcions of h man anomaly l. If rquird, on can amp a mor accura implmnaion, in h fuur, for solving Kplr s quaion along h lins of [192]. 3. Thr is a disconinuiy in h u dpndnc of V in Eq. (5.28). To avoid i w us ( V(u) = u + 2 arcan β φ sin u 1 β φ cos u ), (5.11) 7 Th smi-major axis a r and h ohr ccnriciis r and φ ar dducd from n and using Eqs. (5.3) 128