Gravitational Waves from Long-Duration Simulations

arXiv:astro-ph/9911525v26Jun2000GravitationalWavesfromLong-DurationSimulationsoftheDynamicalBarInstabilityKimberlyC.B.New∗DepartmentofPhysics,DrexelUniversity,Philadelphia,PA19104JoanM.CentrellaDepartmentofPhysics,DrexelUniversity,Philadelphia,PA19104JoelE.TohlineDepartmentofPhysics&Astronomy,LouisianaStateUniversity,BatonRouge,LA70803(toappearinPhysicalReviewD)Compactastrophysicalobjectsthatrotaterapidlymayencounterthedynamical“barinstability.”Thebar-likede-formationinducedbythisrotationalinstabilitycausestheobjecttobecomeapotentiallystrongsourceofgravitationalradiation.Wehavecarriedoutasetoflong-durationsimula-tionsofthebarinstabilitywithtwoEulerianhydrodynamicscodes.Ourresultsindicatethattheremnantofthisinsta-bilityisapersistentbar-likestructurethatemitsalong-livedgravitationalradiationsignal.PACSnumbers:04.30.Db,04.40.Dg,95.30.Lz,97.60.-sI.INTRODUCTIONThedirectdetectionofgravitationalradiationpresentsoneofthegreatestscientificchallengesofourday.WithinterferometerssuchasLIGO,VIRGO,GEO,andTAMA[1]expectedtobeoperatinginthenextfewyears,andanewgenerationofsphericalresonantmassdetectorsunderstudy[2,3],thecalculationofthesignalsexpectedfromvariousastrophysicalsourceshasahighpriority.Accuratecalculationsofthewaveformsareneededtoen-ableboththedetectionandidentificationofsources[4].Inparticular,shortdurationburstsareexpectedtobemoredifficulttodetectthanlonger-livedsignals.Oneinterestingclassofsourcesincludesrapidlyro-tatingcompactobjectsthatdeveloptherotationally-induced“barinstability”.Thisinstabilityderivesitsnamefromthebar-likedeformationitinduces.Theresul-tantobjectispotentiallyastrongsourceofgravitationalradiationbecauseofitshighlynonaxisymmetricstruc-ture.Examplesofcompactastrophysicalobjectsthatmayrotaterapidlyenoughtoencounterthisinstabilityincludestellarcoresthathaveexpendedtheirnuclearfuelandarepreventedfromundergoingfurthercollapsebycentrifugalforces[5–10];aneutronstarspunupbyaccre-tionfromabinarycompanion[11,12];andtheremnantofacompactbinarymerger[13,14].Suchglobalrotationalinstabilitiesinfluidsarisefromnonaxisymmetricmodese±imϕ,wherem=2isknownasthe“barmode”.Itisconvenienttoparametrizethembyβ=Trot/|W|,(1)whereTrotistherotationalkineticenergyandWisthegravitationalpotentialenergy[15–17].Inthispaper,wefocusonthedynamicalbarinstability,whichisdrivenbyNewtonianhydrodynamicsandgravity,andisexpectedtobethefastestgrowingmode.Itoperatesforfairlylargevaluesofthestabilityparameterββdandde-velopsonatimescaleontheorderoftherotationperiodoftheobject.Fortheuniformdensity,incompressible,uniformlyrotatingMaclaurinspheroids,βd≈0.27.Inthecaseofdifferentiallyrotatingfluidswithapolytropicequationofstate,them=2dynamicalstabilitylimitβd≈0.27hasbeennumericallydeterminedtobevalidforinitialangularmomentumdistributionsthataresimilartothoseofMaclaurinspheroids[16–19];seealso[20,21].(Wenotethatwhenβisgreaterthansomecriticalvalueβsβd,asecularinstabilitycanarisefromdissipativeprocessessuchasgravitationalradiationreactionandvis-cosity.Whenthisinstabilityarises,itdevelopsonthetimescaleoftherelevantdissipativemechanism,whichcanbeseveralrotationperiodsorlonger[12].Inrecentyears,muchworkhasalsobeencarriedoutonvariousothermodesinrotatingrelativisticstarsasdetectablesourcesofgravitationalradiation;see[22]forareviewandreferences.)ThefirstnumericalsimulationsofthedynamicalbarinstabilitywerecarriedoutbyTohline,Durisen,andMc-Collough(TDM;[23])inthecontextofstarformation.Usingapolytropicequationofstate,P=KρΓ=Kρ1+1/N,(2)withpolytropicindexN=1.5,theyevolveddifferentiallyrotatingaxisymmetricmodelswitha3-DEulerianhydro-dynamicscode,orhydrocode,inNewtoniangravity.Inallmodelswithinitialβ≥0.30,them=2modegrewtononlinearamplitudesandatwo-armedspiralpatternwasproducedasmassandangularmomentumwereshedfromtheendsofthebar.Numerousothersimulationshaveconfirmedthisbasicscenariofortheevolutionofthebarmodeintothenonlinearregime;seeSec.IIforreferencesandfurtherdiscussion.Morerecently,thesetechniqueshavebeenextendedtothecontextofrapidlyrotating,compactobjectsin1theNewtonianregime,withthegravitationalwavescal-culatedinthequadrupolelimit.Thisisareasonablefirstapproximationforanobject,suchasacentrifugally-hungstellarcorewithadensityintermediatebetweenthatofwhitedwarfsandneutronstars,withinitialmassM=1.4M⊙andradiusR∼100km,andhenceGM/Rc2∼0.02.Centrellaandcollaboratorsusedbothsmoothparticlehydrodynamics(SPH)andEulerianfi-nitedifferencehydrodynamicstoevolvethebarinsta-bilityinamodelwithN=1.5[24,25].Inalloftheirrunsthegravitationalwavesignalwasarelativelyshortdurationburstlastingforseveralbarrotationperiods,andthesystemevolvedtoanearlyaxisymmetriccentralcoresurroundedbyaflattened,disk-likehalo.New[26]carriedoutasimilarstudy,withanimprovedversionofTohline’sEuleriancode[27].Hersimulationemployedasymmetryconditionthatonlypermittedthegrowthofevenmodesm;seeSec.II.Thissimulationproducedafinalstatewithapersistentbar-likecore,whichyieldedagravitationalwavesignalofmuchlongerdurationthanthatfoundbyCentrellaandcollaborators.Giventherequirementsofreliablewaveformsforthedetectionandidentificatio