Next to leading order gravitational wave emission and dynamical evolution of compact binary

D¨orteHansen

InstituteofTheoreticalPhysics,Friedrich-Schiller-UniversityJena,Max-Wien-Platz1,D-07743JenaE-mail:D.Hansen@uni-jena.de

Abstract.Compactbinarysystemswithspinningcomponentsareconsidered.Finitesizeeffectsduetorotationaldeformationaretakenintoaccount.Thedynamicalevolutionandnexttoleadingordergravitationalwaveformsarecalculated,takingintoaccounttheorbitalmotionuptothefirstpost-Newtonianapproximation.

PACSnumbers:04.25.Nx,04.30.-w,04.30.Tv,95.85.Sz

Submittedto:Class.QuantumGrav.

Nexttoleadingordergravitationalwaveemission1.Introduction

2

Inspirallingcompactbinarysystemsareamongthemostpromisingsourcesfortheemissionofgravitationalwavesdetectablewithpresentday’sgravitationalwaveinterferometers.EarthboundgravitationalwavedetectorssuchasGeo600,VIRGO,TAMAandLIGOaremostsensitiveatwavelengthsofabout10-1000Hz.Thiscorrespondsroughlytothelast10minutesoftheinspiralbeforefinalplounge.InthatregimeNewtonianmechanicsisnotvalidandpost-Newtonianapproximationmustbeapplied.AnearlierstageoftheinspiralprocesswillbecoveredbytheyettobebuiltLISAinterferometer.ItisexpectedthatLISAwillbesensibletogravitationalwaveswithfrequenciesfromabout10−1to10−4Hz.However,inordertoactuallydetectgravitationalwaveshighlyaccuratetemplatesareessentially.Post-NewtoniancorrectionsmustbeincludedintotheEoM.Moreover,computingthegravitationalwaveformsbeyondtheleadingorderapproximationhighermassandcurrentmultipolemomentsmustbetakenintoaccount.

Inthepastthestudyofclosecompactbinarieswasoftenbasedontheassumptionthatthestarscanbetreatedaspointlike,non-spinningobjects.Uponthisassumptionitispossibletoderiveananalytic,socalledquasi-KepleriansolutionfortheconservativepartoftheEoMuptothethirdpost-Newtonianapproximation[1].Dissipativeeffectsduetotheemissionofgravitationalwavesfirstappearattheorder(v/c)5,whichcorrespondstothe2.5post-Newtonianorder.

Inthepresenceofatleastonespinningcomponentapost-Newtonianspin-orbitcouplingwhichfirstappearsat1.5post-Newtonianorderleadstocomplifications,whichhampertheinvestigationofspinningcompactbinarysystemsenormeously.Ingeneral,neitherorbitalangularmomentumnorthestellarspinareconserved.UptonowananalyticsolutiontotheconservativepartoftheEoMincludingspin-orbitcouplinghasbeenfoundinafewspecialcasesonly(seee.g.[2]).

Thusfarnotmuchprogresshasbeenmadeintheinvestigationofclosebinarysystemsoffinitesizeobjects.Withintheframeworkofpost-Newtoniananalysisitisoftenarguedthatfinitesizeeffectsareneglegibleduringmostoftheinspiralprocessandwillbecomeimportantnotuntilthelastfeworbitsbeforethefinalplunge[3].Growinginterestintheroleoffinitesizeeffectscomesmainlyfromthesideofnumericalrelativity.Infact,thoughfinitesizeeffectsduetostellarrotationandoscillationincompactbinarysystemsareverysmalltheycanwellbeintheorderofthefirstpost-Newtoniancorrectionstotheorbitaldynamics.Theseseculareffects,beingofNewtonianorigin,accumulateoveralargenumberoforbitsandthus,seenatlongerterms,leadtosignificantphaseshiftsinthegravitationalwaves.TheinfluenceofstellaroscillationsonthedynamicalevolutionandleadingordergravitationalwaveemissionhasbeeninvestigatedbyKokkotasandSch¨afer[4]andLaiandHo[5]fornonrotating,polytropicneutronstarsandbyLaietal[6]andHansen[7]forRiemann-Sbinaries.InalltheseapproachestheanalysiswasbasedonNewtoniantheory,the2.5pNradiationreactiontermsbeingtheonlypost-Newtoniantermsincluded.

Nexttoleadingordergravitationalwaveemission3

Thispaperisdevotedtotheinvestigationoftheinfluenceoffinitesizeeffectsonthedynamicsandgravitationalwaveemissionbeyondtheleadingorderapproximation.Basically,perturbationsofthepointparticledynamicsariseduetostellaroscillations(mainlytidallydriven)androtationaldeformation.Hereweshallrestrictourselvestosystemsweretheapsidalmotionduetorotationaldeformationismuchlargerthantheonecausedbystellaroscillations.

Tobeginwith,letusnotethatthereis,strictlyspeaking,nospinningpointparticle.Aspinningobjectautomaticallygainsafinitesize(seee.g.[8]).Ithatbeenlongknownthatthecouplingofthenonvanishingstellarquadrupolemomenttotheorbitalmotion,whichisapurelyNewtonianeffect,leadstoanapsidalmotion.Foracoupleofclosemainsequencestarbinariesthisapsidalmotionhasbeenobservedtogreataccuracy(seee.g.ClaretandWillems[9]).Inthesesystemsapsidalmotionduetofinitesizeeffectsisconsiderablylargerthantherelativisticperiastronadvance.Anothergroupofbinarysystems,wherefinitesizeeffectsplayanimportantrole,arebinarypulsarssuchasPSRB1259-63,whichhasbeenfoundbyJohnstonetal.in1992[10].WhilethecompactcomponentofPSR1259-63isa47mspulsar,it’scompanionisaBestar,whosespin-inducedquadrupoledeformationleadstoanapsidalmotion.

However,itisclosecompactbinarysystemswhicharemostrelevantforgravitationalwavedetectors.Thisincludesnotonlyblackhole-blackhole,blackhole-neutronstarorneutronstar-neutronstarbinariesbut,ifLISAisreadytowork,alsoclosewhitedwarfbinaries.Ofcoursetherotationaldeformationofcompactstarsismuchsmallerthanitwouldbepossibleinnon-compactstars.However,atleastforneutronstar(NS)andwhitedwarf(WD)binariesoneshouldnotapriorineglectfinitesizeeffectsduetoquadrupoledeformation.Inparticularitiswellpossiblethattheperturbationsintroducedbythecouplingofthestellarquadrupolemomenttotheorbitalmotionisofthesameorderofmagnitudeasthefirstpost-Newtoniancorrections.Thisisassumedthroughoutthepaper.Theanalysisappliestocompactbinarysystems,whosespinningcomponentisaWDorafastrotatingNS.Inordertosimplifycalculationsitisassumedthatthespinisperpendiculartotheorbitalplane.Insection2theorbitalevolutionofaspinningcompactbinaryisstudieduptofirstpostNewtonianorderinthepointparticledynamics.TheEoMaswellasaparametric,quasi-Kepleriansolutionarederived.Insection3thenexttoleadingordergravitationalwaveformsarecalculatedexplicitly.Thelongtimeevolutionandtheinfluenceofthequadrupolecouplingtotheinspiralprocessisdiscussedinsection4.

2.The1pNorbitalmotionincludingspineffectsduetorotationaldeformation

In1985DamourandDeruelle[11]succeededinderivingananalyticsolutiontothe1pNEoMofapoint-particlebinary,whichexhibitsaremarkablesimilaritytothewellknownKeplerparametrizationinNewtoniantheoryandishencecalledquasi-Kepleriansolution.UsingthesamestrategyWexconsideredabinaryconsistingofapulsarand

Nexttoleadingordergravitationalwaveemission4

aspinningmainsequencecomponentatNewtonianorder[12].TreatingtheNewtoniancouplingbetweentherotationallydeformedstarandtheorbitaldynamicsasasmallperturbationhederivedaquasi-Kepleriansolutionuptofirstorderinthedeformationparameterq,whichwillbeintroducedinthefollowing.Inthisletterweshallextendhisinvestigations,takingintoaccounttheorbitaldynamicsuptofirstpost-Newtonianapproximation.Inordertoderiveananalyticsolutionweshallfurtherassumethatthemodificationsinducedbyfinitesizeeffectsareofthesameorderasthe1pNorbitalcorrections.Thatis,werestrictouranalysistocompactbinarysystemsconsistingofafastspinningneutronstarorwhitedwarfandanon-spinningcompactobject.Ofcourse,allresultscanbeappliedtoameansequencestar-compactstarbinaryintheNewtonianlimit.

Therotationaldeformationofaspinningstarofmassmcanbedescribedbysomeparameterq,whichisdefinedas[13]

mq:=

1

23

kRΩˆ2,Ω

ˆ=ΩGm/R3

.(2)

HereRdenotesthepolarradiusandΩtheangularvelocityoftherotatingstar.The

constantofapsidalmotionkstronglydependsonthedensitydistribution.Itvanishesifallmassisconcentratedinthecenterandtakesit’smaximalvaluekmax=0.75forahomogeneoussphere.NeutronstarsandlowmasswhitedwarfscanbeapproximatelymodelledbyanpolytropicEoSwithindexn=0.5...1andn≈1.5forneutronstarsandwhitedwarfs,respectively.Infact,forpolytropesthevalueofkisgivenby(seee.g.[15])

k=

1s·r)2

2r3

󰀒

3(ˆNexttoleadingordergravitationalwaveemission5

totheangularmomentumofthesystem,i.e.perpendiculartotheorbitalplane.InthesecasesEq.(3)takesarathersimpleform,

HMµq

q=−

G7−ν

2r+

E0

r22pN.

(5)

0c

≡EN+ComparingEpNwiththecouplingenergywefindthattheq-termoffersacontribution

comparabletothe1pNorbitalperturbationif

q

c2

≡rS.Ifq/r0ismuchlargerthantheSchwarzschildradiusrStheNewtonianquadrupolecontributionwillclearlydominate,whileforq/r0Keplerianorbitcomesfromthepost-Newtonian≪GcorrectionM/c2theleadingperturbationtotheterms.Thevalueofqcruciallydependsonthedensitydistribution(viak)andontheangularvelorcityoftherotatingstar.Thelateroneisboundedbythemass-sheddinglimit.ForaNewtonianstarwithapolytropicEoSonecanshowthatthemass-sheddinglimitisgivenby[17]

Ωmax=

󰀌

2

GmkR2Ωˆ2=2

3

max

󰀌Nexttoleadingordergravitationalwaveemission6

risetoperturbationswhichareconsiderablylargerthanthe1.5pNorder.Forspinningwhitedwarfsthingscanbedifferent.InthatcasethefinitesizecontributionduetorotationaldeformationcanbeequalorevenlargerthanthepNcontribution.

Thesepreliminaryconsiderationsshowedthattheremightexistclosecompactbinarysystemsforwhichthefinitesizeeffectsintroducedbyrotationaldeformationareinthesamerangeasthefirstpost-Newtoniancorrection.Employingthisassumptionweshallnowderivethequasi-Keplerianparametrizationatthefirstpost-Newtonianorderincludingleadingorderquadrupolecoupling(furtherondenotedasq-coupling).Introducingthereducedenergyandangularmomentum,E=:E/µandJ=:J/µ,respectively,the1pNconservedenergyincludingNewtonianq-couplingreads

󰀇󰀒󰀚qGM3v222

1+(3+ν)v+νr˙+E=

rc22r

r2

󰀒

1−3ν1−2B

r2

+

c2rD

(4−2ν),

󰀔

(8)

r˙2=A+

2

(3ν−1)

E󰀔,

c2

C=−J2+

1

(10)

c2

areconstants.InthestandardapproachofDamourandDeruelleitiscrucialthatDisoforderO(c−2)andthusasmallquantity.NowinourcaseDdependsnotonlyoncbutitisalsolinearinthedeformationparameterq.IfthespinningcomponentisgovernedbyasoftEoSthecorrectiontotheorbitalmotioninducedbytheq-couplingismuchlargerthanthe1pNcorrections.Thisisusualthecaseformain-sequencestarbinaries(seee.g.ClaretandWillems[9]).Forcompactstarsinclosebinarysystems,ontheotherhand,thecontributionoftheq-couplingcanbeofthesameorderasthe1pNorbitalcorrection.UnderthisassumptionwecanapplyDamourandDeruellesstrategystraightforwardly,derivingaquasi-Kepleriansolutionuptolinearorderofq.Thisyields

r=ar(1−ercosu),u−etsinu=n(t−t0),

󰀂󰀙

utanϕ=2(κ+1)arctan

1−eϕ

(11)

Nexttoleadingordergravitationalwaveemission

wheren,ar,er,etandeϕdependonthecoefficientsdefinedaboveas

󰀒

ADB

,e=e1−+ar=−ϕt2J2c2

󰀙󰀔󰀒

BD

,et=C+

2BJ2B2

−A3/B2,andκisgivenbyκ=

3G2M2

c2

+

q

7

n2=−

2E8E3

−

1

4c2E2c2

(ν−7),

󰀚󰀕ν

eϕ=er1−Eδ+

󰀉󰀔

,

Nexttoleadingordergravitationalwaveemission8

Note,thatH1pN=E=µE,sinceH1pNisconservedatthefirstpost-Newtonianorder.Ashasbeenalreadymentionedbefore,thequadrupoleinteractiontermisdefactoaNewtoniancorrectiontothepointparticleHamiltonian.Thisisimportanttokeepinmindwhen,forinstance,calculatingtheorbitalevolutionandgravitationalwaveemissionofbinarypulsarswithamainsequencestarcompanion,suchasPSRB1259-63[12].FortheHamiltonianequationsthatgovernthetimeevolutionofthebinarysystemonefinds

󰀇󰀏3ν−1GMµ2pr

−r˙=

µ2c2r2

󰀌󰀒󰀉󰀔2

p1ϕ2

p+1+(3+ν),r

µr22r

󰀌󰀒󰀌

p2p23q3ν−1ϕϕ2

1+p+p˙r=r

r2c2r3

󰀉󰀔

GM

,−

r3r2

p˙ϕ=0.Nowletusincludeleadingorderdissipativeeffectsintothatscheme.Ingeneral,the

leadingorderenergydissipationofamatterdistributionisgovernedbythetime-dependentradiationreactionHamiltonian[18]

Hreac(t)=

2G

ρ+

1

effect.Atthelevelofthesecondpost-Newtonianapproximationtherelativisticspin-spincouplingleadstoaprecessionoftheorbitalplane.Both,incorporatingspin-orbitaswellasinvestigatingthespin-spincoupling,isbeyondthisletter,whichisdevotedtothestudyoftheinfluenceofcertainfinitesizeeffectsonthedynamicalevolutionandgravitationalwaveemissionofthebinarysystem.

Inthecenterofmasssystemthe1pNHamiltonian,includingq-coupling,reads

󰀔󰀇󰀉2

pGMµ1142ϕ

−q+p+2pH1pN=rrr22r38µ3r4

󰀔

p2GMϕ

−.(17)+(3+ν)

µ2r2

5c

I(t)5ij

(3)

󰀒

pipj

r3

󰀔

.(20)

Nexttoleadingordergravitationalwaveemission

(3)

9

ItiscrucialtoconsiderIij(t)asafunctionoftime,andnotasafunctionofgeneralizedcoordinatesandmomenta,whencalculatingtheradiationreactionpartoftheEoMaccordingto

∂Hreac

.(p˙i)reac=−

∂piOnlyafterwardsIijcanbeexpressedasafunctionofpr,pϕ,randϕ.Explicitly,thecalculationyields[4],[7]

󰀏󰀌

8GM3ν

(p˙r)rad=,

r4c5νr

󰀌󰀏2GM3ν282

−p(p˙ϕ)rad=−r,νr3c5r2

󰀌

p28ϕ2

2p+6(r˙)rad=−r

νr2c5G2prpϕ3

(3)

q˜.

cc4

Applyingthis,theHamiltonianequationsgoverningtheevolutionofthebinarysystemincludingleadingorderradiationbackreactionread

󰀉󰀔󰀒

3+2νν3ν−1˙=p−,(22)r˜˜r1+22r˜15r˜

󰀇󰀔

p˜28νp˜ϕϕ2

p˜r+−,(23)ϕ˙=

2r˜r˜4󰀒󰀔2

p˜2p˜p˜23˜q3+2νϕϕϕ˙r=p˜1+−(3+ν)

r˜22r˜22

󰀔󰀒

8ν1

,(24)+

r˜3r˜4r˜󰀒󰀔8ν2˙ϕ=−p˜(25)−p˜2r.32r˜r˜

Neitherthetotalenergynortheorbitalangularmomentumisconserved,asindicatedbyEq.(25).

ThetimeevolutionofbinarysystemsdescribedbyEqs.(22)-(25)isfullydeterminedby3parameters:thesemi-majoraxisar,theorbitaleccentricityerandthedeformationparameterq,whichhavetobeknownfort=0.Startingthenumericalintegrationintheperiastron,i.e.atϕ(0)=0,theinitialvaluesforrandprfollowimmediatelyas

r(0)=r0=ar(0)(1−er(0)),

pr(0)=0.

Todeterminetheinitialvalueforpϕweusethat,atthebeginningoftheintegration,thetotalenergyofthesystemisgivenbytheconservativepartoftheHamiltonian,or,

p˜ϕ,r=

GM

Nexttoleadingordergravitationalwaveemission

usingthereducedenergy,E(0)=E=H1pN/µ.ItfollowsthenfromEq.(17)that¶

󰀒

󰀃q2orb22

(1−3ν)r(E)pϕ(0)2=2r0E1pN+2GMr01+0Nc2

󰀆orb22

+4(1−ν)GMr0EN+(6−ν)GM,

orb

whereENistheNewtonianenergyoftheorbit.

10

3.Higherordergravitationalwaveemission

IntheprevioussectionswehaveinvestigatedthedynamicalevolutionofaspinningcompactbinarysystematthefirstpostNewtonianapproximation.Nowweshallturnourattentiontothegravitationalwavesemittedbythesystem.Farawayfromthesourcethespacetimecanbeassumedtobeasymptoticallyflat,suchthatthemetricislocallyMinkowskian.Infact,inasymptoticallyflatspace-timesthegravitationalwavesemittedbyanisolatedbinarysystemsareexpectedtoobeyamultipoleexpansionoftheform(seee.g.[19])

󰀏l−2󰀌

4GT

hT=ij

c

󰀏l−1󰀌

8l

c

Dc4

∞󰀄

l=2m=−l

l󰀄

c

󰀏l−1

󰀂󰀌

1

󰀅

B2,lm

S(l)lm(t−D/c)Tij(Θ,Φ),

(27)

E2,lmB2,lm

whereTijandTijaretheso-calledpure-spintensor-sphericalharmonicsofelectricandmagnetictype.Theseharmonicsareorthonormalontheunitsphere.Infact,

ˆandΘ,ˆtheycanbedecomposedintoatermproportionaltointroducingunitvectorsΦ

ˆ⊗Θˆ−Φˆ⊗Φ)ˆand(Θˆ⊗Φˆ+Φˆ⊗Θ),ˆrespectively.Thatway,iftheTE/B2,lmare(Θ

¶Notethatpϕ→µpϕandpr→µpr.

Nexttoleadingordergravitationalwaveemission11

known,oneobtainsthepolarizationstatesh+andh×oftheradiationfieldfromEq.(27)withoutanyfurthercalculations.Thepure-spintensor-sphericalharmonicsneededherearegivenbyEqs.(A.7)-(A.16)intheappendix+.

Insection2thepointparticlecontributionwastakenintoaccountuptofirstpost-Newtonianapproximation,andthequadrupolecouplingterm,thoughpresentalreadyatNewtonianorder,wasassumedtobeofthesameorderasthe1pNcorrectionstothepointparticledynamics.Thatmeans,wehavetogobeyondtheleadingordergravitationalwaveformula.Consideringthedynamicsupto1pNrequirestheapplicationofthemultipoleexpansion(27)uptol=4forthemassmultipolemomentsanduptol=3forthecurrentmultipolemoments.Explicitly,neglectingallhigherorderterms,Eq.(27)isreducedto∗

󰀎3󰀑2󰀄󰀄GB2,2mE2,3m

S(2)2mT+I(3)3mT++h+,×=,×,×

cm=−3

m=−2

+1

c

h+,×+

(1)

1

Iijk

221c√

=−µ

󰀚󰀛

2s

−12(r·v)x󰀈ivj󰀉+11rv󰀈ivj󰀉+Iij,

42

(1−3ν)

v2

7c2

GM

(30)

1−4νεab󰀈ixj󰀉xavb,

Jijk=µ(1−3ν)εab󰀈kxixj󰀉xavb,

(33)(34)

wherebracketsdenotetheSTF-partofthecorrespondenttensor(seeappendixA).

Atthispointitisneccessarytoconsideramomentthecontributionofthestellarmass-quadrupolemoment.Sinceonecomponentofthebinaryisspinningandthus

s

automaticallygainsafinitesizetherecouldbe,inprinciple,acontributionofIijtothegravitationalwaveemissionofthesystem.Thiscontributionis,however,verysmall,unlesstheenergystoredintheinternalstellardegreesoffreedom,e.g.oscillationsof

¨sisthestar,iscomparabletotheorbitalenergy.FromnowonweshallassumethatIij

+

ˆ⊗Θˆ−Φˆ⊗Φ)-partˆˆ⊗Φ+ˆΦˆ⊗Θ)-part.ˆInthisrepresentation,h+isthe(ΘofEq.(27),whileh×isthe(Θ

∗FromnowonweomitthesupscriptTTfornotationalconvenience.

Nexttoleadingordergravitationalwaveemission12

eithertrivialorcanbeneglectedcomparedtoallothertermspresentinthecalculation.UsingtheNewtonianequationofmotionforapoint-particlebinary♯

v

˙=−GMr·v)2r3

r󰀌

1+

3q

r3c2

󰀒r

󰀇

GM(2

2

M

14

(1−3ν)

vG21

GM

7

3󰀒2󰀌

1+

3q

21

7c

2

(1−3ν)(r·v)2

rc2

,

I(3)

ijk=−µ

√

r5

(r·v)x󰀈ixjxk󰀉−21

GM

󰀔

rvGM

󰀈ivjxkxl󰀉+42

(r·v)2

r3

x󰀈ixjxkxl󰀉󰀇

r3

7

GM

r2

−15

1−4νGM

r3

(1−3ν)󰀚−4εxr·v

ab󰀈kxivj󰀉avb+3

23

󰀘

5󰀒−2󰀇

v−

GM

2r2

󰀏󰀉

+

1

r2(ν−10)

+

9

27r

((37−20ν)r2ϕ˙−(15+32ν)r˙2)󰀉󰀔

,I(2)21=0,

I(2)22=

󰀘5

µe−2iϕ󰀒2󰀇

r

˙2−r2ϕ˙2−GM

2r2

󰀏

−2irr˙ϕ˙

󰀉

♯Rememberthatqistreatedformallyasa1pNquantity.

v2

(35)

(40)(41)(42)

Nexttoleadingordergravitationalwaveemission

+1

r2

+GM

7

(1−3ν)(r˙4−r4ϕ˙4)−irr˙ϕ˙

󰀌10

r+18

π

󰀇

2I(3)33

=2ν(m1−m2)

󰀘

r

−v2

󰀉

+irϕ˙v−

7r

󰀉󰀔

,21

e−3iϕ󰀒2󰀇

r˙2−2GM−6r˙2+2r2ϕ˙2r

󰀉󰀔

,I(4)40

=2πGr2

−

M√

2

+

GM

63rr

󰀉󰀔,

I(4)44=

2

π

r2

+GM

r

−24r˙2+24r2ϕ˙2

S(2)20=S(2)22=0,󰀉󰀔

,S(2)21=

8

2ππ3󰀘

7

(1−3ν)GMµe−2iϕϕ˙(r˙−4irϕ˙).

Hereweusedthat

√

13

(44)

(45)

(48)

(49)(50)

()

Nexttoleadingordergravitationalwaveemission(28)onefinds,atleadingorder,Dc4

r󰀌

14

1+

3q󰀌

3q1+

G2r2

Notethat,inordertoemphasizethecharacteroftheq-coupling,theq-dependenttermshavebeenincludedintotheleadingordercomponentoftheradiationfield.Defining∆m≡m1−m2thefirstcorrectiontermsread

󰀇󰀌󰀒

GMDc

2µsinΘ

M2

GM6󰀇󰀌

GM

2r˙cos3(Φ−ϕ)r˙2−3r2ϕ˙2−2

4

󰀏󰀉󰀔

,(56)−6r˙2+2r2ϕ˙2

r

∆mGMDc5

22

󰀉󰀉

rϕ˙+−6r˙2+2r2ϕ˙2

rr

󰀇

GM

+r˙sin3(Φ−ϕ)r˙2−3r2ϕ˙2−2

1+cos2Θ

󰀌15+32ν

21(5+27ν)

7

9

h×

(0)

r

󰀒󰀇

GM

=µcosΘ4rr˙ϕ˙cos2(Φ−ϕ)−2sin2(Φ−ϕ)r˙2−r2ϕ˙2−

󰀏󰀉󰀔

.

G

(2)h+

=

+

rrr˙ϕ˙

GM

+

r2

󰀏󰀉r2ϕ˙2

2

3

󰀒

G2M2

µ(ν−10)

(1−3ν)v4+

GM

+

561−3ν

󰀒22

GM

µ(7cos4Θ−8cos2Θ+1)7

GM

2

2

r

(18r˙2+13r2ϕ˙2)+6v4

󰀔

r

(18r˙2−3r2ϕ˙2)−7

r

󰀉󰀔

Nexttoleadingordergravitationalwaveemission

+1−3ν

r2

+6r˙4−36r2r˙2ϕ˙2

r

−24r˙2

15

GM

+6r4ϕ˙4+

󰀖󰀛1−3ν22

−+24rϕ˙

(2)

G

h×=µcosΘ

󰀒

rr˙ϕ˙

3󰀉󰀔

(5+27ν)r2

+

GM

r˙2−

11+156ν

GM

7

(1−3ν)(r˙4−r4ϕ˙4)

7

󰀒1

+(1−3ν)µcosΘ−

GMϕ˙{r˙cos2(Φ−ϕ)}+4rϕ˙sin2(Φ−ϕ)

󰀏

−−24r˙2+24r2ϕ˙2

r

󰀏󰀉󰀌22

GM

(51r2ϕ˙2−18r˙2)+6r˙4+6r4ϕ˙4−36r2r˙2ϕ˙2+sin4(Φ−ϕ)7

r

󰀏2

7cosΘ−5−

r

󰀌󰀏󰀉󰀔

G2M2

+sin2(Φ−ϕ)−7.(59)(18r˙2−3r2ϕ˙2)+6r4ϕ˙4−6r˙4

r

Thepolarizationstatesofthegravitationalradiationfield,expressedintermsofgeneralizedcoordinatesandmomenta,canbefoundinappendixB.

6sin2Θ

4.Discussion

IthasbeenlongknownthatfinizesizeeffectsintroduceaperiastronshiftalreadyatthelevelofNewtoniantheory.Foracoupleofmainsequencestarbinariesthetotalapsidalmotionϕ˙tothasbeendeterminedfromobservationalevidence.Comparedwiththecontributionϕ˙relpredictedbyGRitbecameobviousthatinallsystemstheNewtonianperturbationsgivethedominantcontributiontoϕ˙tot(foranoverviewseee.g.[9]).Thisisduetothe”soft”equationsofstategoverningthestellarmatterofmainsequencestars.ForcompactstarbinariesNewtonianperturbationsareoftenneglected.Inparticular,itisoftenarguedthattheeffectofthespin-inducedquadrupoleistoosmallunlessthecompactstar(e.g.aneutronstar)isrotatingnearthemass-sheddinglimit[3].However,evenforNS-NSorNS-BHbinariesthisargumentdoesnotholdcompletely.IthasbeenshowninprevioussectionsthatforcloseNS-NSbinariesthepotentialenergyintroducedbythecouplingcanbeconsiderablylargerthanthecorresponding1.5pNorbitalcorrectionterms,thoughitissmallerbyafactor100ormorethanthe1pN

󰀉

Nexttoleadingordergravitationalwaveemission16

contribution.Thus,alreadyatthelevelofthefirstpost-NewtonianapproximationintheEoMtherotationaldeformationinducesanon-relativisticperiastronshift,whichaccumulatesoveralargenumberofperiods(figures3)and4).Inordertoobtainhighlyaccuratetemplatesitisthusdesirabletotakeintoaccountthesecorrectionsproperly,atleastforfastspinningneutronstarsinacompactbinarysystem.

Withthespace-boundlaserinterferometricdetectorLISAathandthefrequencybandaccessibletoobservationswillbeextendedtomuchlowerfrequencies(10−1to10−4Hz),whichenlargesthenumberofpossiblesourcesenormeously.Inparticular,withLISAnotonlyBH-BH,NS-NSandNS-BHbinariesshouldbedetectable,butalsowhitedwarfbinaries.Inparticulartothisclassofcompactbinariestheanalysisshowninthispaperapplies.IthasbeenarguedbyWillemsetal.inarecentpaper[21]that–contrarytoprevailingopinions–theremightexistaclassofeccentricgalacticdoublewhitedwarfs,whichareformedbyinteractionsintidalclusters.Willemsetal.showedthattidesandstellarrotationstronglydominatetheperiastronshiftatorbitalfrequencies≥1mHz.ThephaseshiftsinducedbytheseNewtonianperturbationsaremuchlargerthanthegeneralrelativisticcorrectionsthen.TheyconcludethatitisessentialtoincludephaseshiftsgeneratebyNewtonianperturbationsintothesignaltemplatesinordertonotbiasLISAsurveysagainsteccentricdoublewhitedwarfs.Generally,neglectingthecontributiontoϕ˙totinducedbyrotationaldeformationwillleadtoanoverestimationofthetotalmassderivedfromϕ˙tot.

Inthispaperthecompetinginfluencesoftherotationaldeformationandthe1pNcorrectiontermswereexaminedinmoredetail.Inparticular,wesucceededincalculatinga1pNquasi-Kepleriansolution,whichtakesintoaccountfinitesizeeffectsuptolinearorderinthequadrupoledeformationparameterq.Theresultsgiveninsection2arevalidaslongasq/J2isoftheorderO(c−2).ForwhitedwarfbinariesorbinarypulsarssuchasPSR1259-63theperiastronshiftinducedbyrotationaldeformationispossiblymuchlargerthanthegeneralrelativisticcontribution.Inthatcase,Eqs.(11-12)stillapplyinthelimitv/c→0.Insection3thepolarizationstatesofthegravitationalradiationfieldarecalculatedbeyondtheleadingorderapproximation.Fornon-spinningcompactbinariesthecorrespondingwaveformsareshowninfigures1and2.Inthesefigureswaveformscalculatedusingtheleadingorderexpressionsh+,×(0)arecomparedtothe

(1)(2)

1pNcorrectwaveformswiththenexttoleadingordercorrectionsh+,×andh+,×taken

(1)

intoaccount.Thefirstcorrection,h+,×,isnontrivialonlyfordifferentmassbinaries,i.e.forequal-massbinariesthefirstnon-vanishingcorrectiontotheleadingorderformulaappearsattheorderO(c−2).

Theinfluenceoftheq-couplingonthegravitationalwaveformsisshowninfigures3and4.Asexpected,thespin-inducedquadrupolemomentleadstoaphaseshiftcomparedtothepurepoint-particleGWemission.Moreover,thequadrupoledeformationofthespinningcompactobjectsspeedsuptheinspiralprocess,ashasbeenshowninfigures5and6foranequal-massbinaryinaslightlyellipticorbit.

Moreanalysisisneededinordertofullyunderstandtheimprintoffinitesizeeffectsontothegravitationalwavepatternofclosecompactbinarysystemsbeyondtheleadingorder.

Nexttoleadingordergravitationalwaveemission17

Inparticularitwouldbehighlydesirabletoincludethestellaroscillationmodesintothecalculations.Frompreviousworksitisexpectedthatinthesecasessocalledtidalresonanceswillhaveanimportantimpactontheinspiralprocessandthegravitationalwaveemissionofthebinary[4],[5],[7].Acknowledgements

IamgratefultoGerhardSch¨aferforhelpfuldiscussionsandcarefulreadingofthemanuscript.ThisworkissupportedbytheDeutscheForschungsgemeinschaft(DFG)throughSFB/TR7”Gravitationswellenastronomie”.AppendixA.Usefulrelations

ThemassandcurrentmultipolemomentsIlmandSlm(m=−l,...,l)thatareirreduciblydefinedwithrespecttotheorbitalangularmomentumaxisarerelatedtoIAlandJAlaccordingto

Ilm(t)=

16π

(l+1)(l+2)

󰀙

(l+1)(2l+1)!!

where,form≥0,

󰀙

2(l−1)l

lm∗

JAlYA,l

(A.2)

lmm

YA=(−1)(2l−1)!!l

4π(l−m)!(l+m)!

l|m|∗

332121

(A.3)(δ󰀈i1+iδ󰀈i1)···(δim+iδim)δim+1···δil󰀉,

and

lm

YA=(−1)mYAl

l

form<0.(A.4)

Thecomplexconjugatesaregivenby

Ilm∗=(−1)mIl−m,

Slm∗=(−1)mSl−m.

(A.5)

Thepure-spintensor-sphericalharmonicsareorthonormalontheunitsphere.Forthecomplexconjugatethefollowingrelationholds:

TE/B2,lm∗=(−1)mTE/B2,l−m.

Defining

ˆ⊗Θˆ−Φˆ⊗Φˆ,Υ+≡Θ

ˆ⊗Φˆ+Φˆ⊗ΘˆΥ−≡Θ

(A.7)(A.8)(A.6)

theexpressionsneededinthepaperread

󰀘󰀚󰀛

22iΦE2,22

(1+cosΘ)Υ++2icosΘΥ−,eT=

128π󰀘TE2,20=

π

sin2ΘΥ+,

Nexttoleadingordergravitationalwaveemission

󰀘󰀚󰀛

iΦB2,21

sinΘeiΥ+−cosΘΥ−,T=−

32π󰀘󰀚T

E2,33

18(A.9)(A.10)(A.11)(A.12)(A.13)(A.14)(A.15)(A.16)

TT

E2,31

=

B2,32

=−sinΘe256π󰀘󰀚=−e

128π󰀘󰀘π󰀘512π256π󰀘

sinΘe

2iΦ

iΦ

3iΦ

TB2,30=TT

E2,44

cosΘsin2ΘΥ−.sinΘee

2iΦ2

4iΦ

󰀚󰀛

22

2i(2cosΘ−1)Υ+−cosΘ(3cosΘ−1)Υ−,

󰀚󰀛

2

(1+cosΘ)Υ++2icosΘΥ−,

󰀛

(3cosΘ−1)Υ++2icosΘΥ−,

2

󰀛

(1+cosΘ)Υ++2icosΘΥ−,

2

==

E2,42

TE2,40=−

128π󰀘

256π

(7cos4Θ−8cos2Θ+1)Υ+.

󰀚󰀛

422

(7cosΘ−6cosΘ+1)Υ++icosΘ(7cosΘ−5)Υ−,

AppendixA.1.Symmetrictracefreetensors

Throughoutthispapersymmetric-tracefree3rdand4thranktensorsareused.Symmetrizingatensorofrankprequirestotaketheproperlyweightedsumoverallindexpermutations,

1≡T(i1...ip)=Tisymm

1...ip

(−1)k(2p−2k−1)!!

(2p−1)!!󰀃

󰀆

δabT(cii)+δbcT(aii)+δacT(bii),

5

T󰀈abcd󰀉=T(abcd)−

(A.20)

1

35

[δacδbd+δadδbc+δabδcd]T(iijj).(A.21)

Nexttoleadingordergravitationalwaveemission19

AppendixB.Expressionsforh+andh×intermsofgeneralizedcoordinatesandmomenta

Theexpressionsfortheleadingandnexttoleadingordercontributiontothepolarizationstatesofthegravitationalwavefield,h+andh×,read

󰀌󰀒󰀇2

p3qGµϕ(0)2

1+cos2(Φ−ϕ)p−h+=r

µ2r

󰀏󰀔󰀉󰀛sin2Θ2

GMµ

sin2(Φ−ϕ)−−,(B.1)rr22r2

󰀒pϕGMµ2G(1)

sinΘh+=

Mµ23

󰀌

p27ϕ2

−pr−

2r

󰀉󰀇

GMµ2

prcos(Φ−ϕ)22r2

GMµ21+cos2Θ

−2−

r2

󰀌

p27ϕ2

sin3(Φ−ϕ)−3pr+rr󰀎󰀏2

5(3ν−1)sin2Θ

(ν−10)−

Dc6r2r2

󰀉󰀔

GM+

r2

󰀌4󰀏2

1+cosΘpr+(10−ν)+5(3ν−1)

r2µ4r4

p2GMϕ

+

3

󰀌

p2prpϕGMϕ2

p+sin2(Φ−ϕ)r

7µ23

󰀌󰀒󰀇22

p2GMϕ222

18p−51sinΘ(1+cosΘ)cos4(Φ−ϕ)7r

24µ2r

󰀏󰀉2

p2pprpϕrϕ−6−µ4µ4r4r󰀌2󰀏󰀉󰀔pϕ1−3ν

+24+

µ2

󰀏󰀌4󰀏󰀉

pϕGM

+6+

r2r2µ4

󰀉󰀔

p2prpϕϕ

−+

µ2r

Nexttoleadingordergravitationalwaveemission

−1−3νGM

r2

1−3ν

󰀏󰀔µ2r2

r2

+6

󰀌p2r

µ2r2

󰀏2

20

−−

󰀚p

ϕ4+GM

2r2󰀉

r

−2

GMµ2󰀌3

(B.6)

r2

󰀏−2

󰀉

󰀏󰀉

Dc

󰀒󰀇2

pr

cosΘsin2(Φ−ϕ)−4

+2∆m

prpϕ

󰀒pϕ

µ2r2

GMµ2

r

2

Dc5

󰀇2

pϕ

+prsin(Φ−ϕ)p2+r−6p2r

+2

p2ϕ

2

󰀉

r2

+pϕ

r2

µ4

󰀌

p4ϕ4

pr−

󰀉

(10−ν)

Dc

󰀒󰀇

5

cosΘsin2(Φ−ϕ)6

p2ϕ

21+prpϕ

µ2r

r

󰀇

1−3νGMp2r2−sinΘcos4(Φ−ϕ)−

µ2rµ2r2

󰀌󰀇22

p2GMϕ2

51sinΘsin4(Φ−ϕ)7

12µ2r

󰀏󰀉2

p2p1−3νprpϕrϕ−6−µ4µ4r4r

󰀉󰀒2

ppprϕϕ

(7cos2Θ−5)+42µ2r

󰀌󰀇

p2p4G2M2ϕr2

18p−3−+sin2(Φ−ϕ)−7r

µ2rµ4r4

p2GMϕ2

(3cosΘ−1)cos2(Φ−ϕ)+46µ2r

r2

(3ν−1)7

Nexttoleadingordergravitationalwaveemission

themultipoleexpansion(27).DefiningF(u)≡1−ercosuoneobtains

S

(2)21

=

32π

1−e2r

π

1−e2r

2−2iϕ

3󰀘

7

(1−3ν)µEe󰀗

F(u)3

󰀚ersinu−4i

π

F(u)

󰀇

󰀗

1−

q

E

F(u)

δ+

F(u)

+

2(19ν−4)F(u󰀉󰀔

,

I(2)22=4

󰀘

5

µEe

−2iϕ

󰀚−1+3

F(u)2

+2i

er󰀗)3F(u)2+5q+

4e2rsin2

u

1−e2rsinu

F(u)1−e2r−

ercosu

42c

2

󰀕9(3ν−1)−3(51ν−115)F(u)2

−4(111ν−2)

1−e2r

󰀗

2π

F(u)

−i

󰀗

F(u)

󰀌

1−

5/62π

F(u)

1+

4(1−e2r)

1−e2r

+

4(1−e2r)

󰀇

21󰀘(1−3ν)µE

2

5

󰀒

F(u)6−

6

F(u)2+

5(1−e2r)

63

√

F(u)−

7−12e2r

F(u)3

−3i

er

󰀗

F(u)2

󰀇

4+

1

e2r)

9

󰀘

(1−3ν)µE2e−4iϕ7

󰀒

6−6

F(u)2−

27(1−+6i

er󰀗F(u)2

󰀇4+1

F(u)2

󰀉󰀅

F(u)4

.

21

(C.4)

(C.10)

Nexttoleadingordergravitationalwaveemission22

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Nexttoleadingordergravitationalwaveemission

0.03 0.02 0.01 0-0.01-0.02-0.03-0.04

23

0 0.5 1 1.5 2

Nexttoleadingordergravitationalwaveemission

0.02 0.015 0.01 0.005

0-0.005-0.01-0.015-0.02-0.025

0

1

2

3

4

5

24

Nexttoleadingordergravitationalwaveemission

0.04 0.03 0.02 0.01 0-0.01-0.02-0.03-0.04

25

0 10 20 30 40 50 60 70