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Binary neutron stars Equilibrium models beyond spatial conformal flatness

Equilibria of binary neutron stars in close circular orbits are computed numerically in a waveless formulation: The full Einstein-relativistic-Euler system is solved on an initial hypersurface to obtain an asymptotically flat form of the 4-metric and an ex

Binaryneutronstars:Equilibriummodelsbeyondspatialconformal atness

K¯ojiUry¯u,1Fran¸coisLimousin,2JohnL.Friedman,1EricGourgoulhon,2andMasaruShibata3

1

DepartmentofPhysics,UniversityofWisconsin-Milwaukee,P.O.Box413,Milwaukee,WI53201

2

Laboratoiredel’UniversetdesesTh´eories,UMR8102duC.N.R.S.,

ObservatoiredeParis,F-92195MeudonCedex,France

3

DepartmentofEarthScienceandAstronomy,GraduateSchoolofArtsandSciences,

UniversityofTokyo,Komaba,Meguro,Tokyo153-8902,Japan

(Dated:February7,2008)Equilibriaofbinaryneutronstarsinclosecircularorbitsarecomputednumericallyinawavelessformulation:ThefullEinstein-relativistic-Eulersystemissolvedonaninitialhypersurfacetoobtainanasymptotically atformofthe4-metricandanextrinsiccurvaturewhosetimederivativevanishesinacomovingframe.Twoindependentnumericalcodesaredeveloped,andsolutionsequencesthatmodelinspiralingbinaryneutronstarsduringthe nalseveralorbitsaresuccessfullycomputed.Thebindingenergyofthesystemnearits nalorbitdeviatesfromearlierresultsofthirdpost-Newtonianandofspatiallyconformally atcalculations.Thenewsolutionsmayserveasinitialdataformergersimulationsandasmembersofquasiequilibriumsequencestogenerategravitationalwavetemplates,andmayimproveestimatesofthegravitational-wavecuto frequencysetbythelastinspiralorbit.

PACSnumbers:04.25.Dm,04.25.Nx,04.30.Db,04.40.Dg

arXiv:gr-qc/0511136v2 21 Oct 2006

Introduction:Equilibriaofclosebinaryneutronstarsincircularorbits,constructednumerically,havebeenstudiedasamodelofthe nalseveralorbitsofbinaryinspiralpriortomerger(seeforareview).Thesenu-mericalsolutionshavebeenusedasinitialdatasetsformergersimulationsinquasi-equilibriumsequences,toestimategravitationalwaveforms4];andtodeterminethecuto frequencyoftheinspiralwavesTomaintainequilibriumcircularorbitsingeneralrel-ativityonemustintroduceanapproximationthatelim-inatesthebackreactionofgravitationalradiation.Anansatzofthiskindisthewavelessapproximationpro-posedbyIsenberg[6].Oneofhisproposalswastochooseaconformally atspatialgeometrymaximallyembeddedinaspacetime.Asaresultthegravitational eldisnolongerdynamical; eldequationsforthemetriccompo-nentsbecomeellipticequations.WilsonandMathewslaterrediscoveredthiswavelessapproximationandap-pliedittonumericalcomputationsofbinaryinspiralsAlthoughtheIsenberg-Wilson-Mathews(IWM)for-mulationhasbeenwidelyusedformodelingbinaryneu-tronstarandbinaryblackholeinspiralinthepastdecade[5,7,8,9,11,theerrorassociatedwithitscon-formally at3-geometrywasstudiedonlyforstationaryaxisymmetricsystemsInmodelsofbinaryneutronstars,theestimatederrorintheorbitalangularveloc-ity, ,isseveralpercent[3,implyingacomparabledeviationfromcircularorbitsNewwavelessformu-lations,incorporatingagenericformofthemetric,aresuitableforaccuratecomputationofbinarycompactob-jects17].Inthisletter,wepresentthe rstresultsofnumericalcomputationsforbinaryneutronstarsmod-eledinoneoftheseformulationsFormulationofthewavelessspacetime:Thenewfor-mulationexactlysolvestheEinstein-Eulersystem

writtenin3+1formonaspacelikehypersurface.Wefollownotation[23]usedin[17].ThespacetimeM=R×Σisfoliatedbythefamilyofspacelikehypersur-faces,Σt={t}×Σ.Thefuture-pointingnormalnαtoΣtisrelatedtothetimelikevectortα(thetangent ttocurvest→(t,x),x∈Σ)bytα=αnα+βα,whereαisthelapse,andwheretheshiftβαsatis esβαnα=0.Aspatialmetricγab(t)de nedonΣtisequaltotheprojec-tiontensorγαβ=gαβ+nαnβrestrictedtoΣt.Intermsofaconformalfactorψandaconformallyrescaledspa-tialmetricγ ab=ψ 4γab,themetricgαβtakestheform,ds2= α2dt2+ψ4γ ij(dxi+βidt)(dxj+βjdt),inachart{t,xi}.Aconditiontospecifytheconformaldecomposi-tionisdetγ ab=detfab,wherefabisa atmetric.Inourwavelessformulationweimpose,ascoordi-nateconditions,maximalslicing(K=0)andthespa- tiallytransverseconditionDbγ ab=0(theDiracgauge

18]),whereDbisthecovariantderivativewithre-specttothe atmetric.Wethenrestricttime-derivativetermsinthisgaugetoguaranteethatallcomponentsofthe eldequationareellipticequations,andhencethatallmetriccomponents,includingthespatialmet-ric,haveCoulomb-typefallo Whileitisfoundtobesu cienttoimposethecondition tγ ab=O(r 3)tohaveCoulomb-typefallo intheasymptotics,weim-poseastrongercondition: tγ ab=0.Fortheotherquantities,weimposehelicalsymmetry:spacetimeand uidvariablesaredraggedalongbythehelicalvectorkα=tα+ φα.Forexample,thetimederivativeofex-trinsiccurvatureKabisexpressedas tKab= £ φKab.Theresulting eldequationsaresolvedonasliceΣ0.TheHamiltonianconstraint,momentumconstraint,spa-tialtraceandspatialtracefreepartoftheEinsteinequa-tionare,respectively,regardedasellipticequationsforψ,βa,αandhab:=γ ab fab,whiletheextrinsiccur-

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