DRAFTVERSIONFEBRUARY1,2008 PreprinttypesetusingLATEXstyleemulateapjv.03/07/07 ECCENTRICDOUBLEWHITEDWARFSASLISASOURCESINGLOBULARCLUSTERS B.WILLEMS1,V.KALOGERA1,A.VECCHIO1,2,N.IVANOVA3,F.A.RASIO1,J.M.FREGEAU1,K.BELCZYNSKI4,5 DraftversionFebruary1,2008 ABSTRACT Weconsidertheformationofdoublewhitedwarfs(DWDs)throughdynamicalinteractionsinglobularclus- ters. Such interactions can give rise to eccentric DWDs, in contrast to the exclusively circular population expectedtoformintheGalacticdisk. Weshowthatfora5-yearLaserInterferometerSpaceAntenna(LISA) missionanddistancesasfarastheLargeMagellanicCloud,multipleharmonicsfromeccentricDWDscanbe 8 detectedatasignal-to-noiseratiohigherthan8foratleastahandfulofeccentricDWDs,giventheirformation 0 0 rateandtypicallifetimesestimatedfromcurrentclustersimulations.Consequentlytheassociationofeccentric- 2 itywithstellar-massLISAsourcesdoesnotuniquelyinvolveneutronstars,asisusuallyassumed. Duetothe difficultyofdetecting(eccentric)DWDswithpresentandplannedelectromagneticobservatories,LISAcould n provideuniquedynamicalidentificationsofthesesystemsinglobularclusters. a J Subjectheadings:Stars: Binaries: Close,Stars: WhiteDwarfs,GravitationalWaves 7 1 1. INTRODUCTION trophysicswithLISA. InthisLetter,weexploretheformationofDWDswithnon- ] Double white dwarf binaries (DWDs) are the single most h zero eccentricities(e>0.01)and orbitalperiodsP<5000s abundantanda guaranteedclassofgravitationalwave(GW) p through dynamical interactions in GCs, and assess their de- sources for the Laser Interferometer Space Antenna (LISA; - tectabilityby LISA. Since mass-transferringbinariesare not o Benderetal.1998). Theirformationencompassestwomass- expectedto retain appreciableeccentricitiesforlongperiods r transfer phases, with at least one leading to a common- t of time, we focus our attention on detached DWDs. While s envelopephase shrinkingthe orbitto periodsof 0.1–1000hr the importanceofstellar-mass binariesin GCs forLISA has a (Han1998;Nelemansetal.2001). Onceformed,orbitalan- [ gularmomentumlossesdueto GWemissionandtidalinter- beendiscussedpreviously(Benacquista1999,Benacquistaet actionscontinuetoshrinktheorbituntiltheleastmassiveWD al. 2001,Benacquista2001),thepossiblepresenceofeccen- 3 tricDWDsintheLISAbandhasnotbeenconsideredbefore. v fillsitsRochelobeandtransfersmasstoitscompanion. De- 7 pendingonthebinarymassratio,themasstransferleadstoa 2. DOUBLEWHITEDWARFSINGLOBULARCLUSTERS 8 fastmergeroralong-livedphaseoforbitalexpansion(Marsh 2 etal. 2004;Gokhaleetal. 2007,andreferencestherein). Ivanova et al. (2006) recently explored the formation and 4 As the progenitors of DWDs are tight enough to undergo evolution of WD binaries in GCs using an updated version . multiplemass-transferphases,tidalforcesareusuallythought of the Monte Carlo code described in Ivanova et al. (2005). 5 to circularize their orbits long before they form a DWD. In this study, we adopt these sample simulations to address 0 7 DWDs are therefore expected to be born with circular or- therelevanceofeccentricDWDsasLISAsources. Thecode 0 bits. However, dynamical interactions in globular clusters combines a simple two-zone core-halo GC model with the : (GCs) form DWDs at enhanced rates compared to the field STARTRACK binarypopulationsynthesiscode(Belczynskiet v (Shara & Hurley 2002). Through these interactions, DWDs al.2007)andtheFEWBODYsmallN-bodyintegrator(Fregeau i X canformwithnon-zeroeccentricities;thesecansubsequently at al. 2005) to treat dynamical interactions between single bedampedbyGWemission,butcanalsobeexcitedbyfly-by starsandbinaries. r a encounterswithsingleandbinarystars. Contrarytocommon Inthe simulations, the two mostimportanttypesof stellar assumptions,theidentificationofeccentricstellar-massbina- interactions creating eccentric DWDs in the LISA band are riesintheLISAGWdatastreamthereforedoesnotuniquely exchangeinteractions and physical collisions. Exchange in- identifythemasneutronstar(NS)binaries. teractions are a subset of the broader class of binary-single Due to their inherent faintness, DWDs are difficult to de- andbinary-binaryinteractions. Eccentric LISA DWDsform tect at GC distances or beyond, even with the SDSS, HST, throughthis channelbyexchangesbetweensingle WDs and orJWST inthefuture. LISAcouldthereforeprovideunique binaries with a WD component. During the exchange, the dynamicalidentificationsofeccentricDWDsinGCs. Asthe two WDs become bound, while the companionto the initial potential of studying WD interiors through tidal effects in- binaryWDisejectedfromthesystem. Theresultingbinaries creases dramatically with increasing orbital eccentricity, ec- have average post-exchange eccentricities e 0.8. Alterna- ≃ centricDWDs offera uniqueopportunityto studystellar as- tively, physicalcollisionsbetweenWDs andred giantsform eccentricDWDswhenthegiant’senvelopegetsdisruptedand 1NorthwesternUniversity, DepartmentofPhysicsandAstronomy,2131 itsdegeneratecorebecomesboundtotheinitiallysingleWD TechDrive,Evanston,IL60208,USA (Lombardiet al. 2006). LISA DWDs formedthis way have 2SchoolofPhysicsandAstronomy,UniversityofBirmingham,Edgbas- average post-collosion eccentricities e 0.5. We stress that ton,BirminghamB152TT,UK ≃ the quoted average eccentricities are for LISA DWDs only 3CanadianInstituteforTheoreticalAstrophysics,UniversityofToronto, 60St.George,Toronto,ONM5S3H8,Canada (average post-exchange and post-collision eccentricities for 4NewMexicoStateUniversity,DepartmentofAstronomy,1320Frenger theentireDWDpopulationarelower). Mall,LasCruces,NM88003,USA In addition to direct formation, we find that non-zero ec- 5TombaughFellow centricities of DWDs can be significantly enhanced through 2 B.Willemsetal. TABLE1 TABLE2 GLOBULARCLUSTERMODELPARAMETERS. STATISTICALPROPERTIESOFECCENTRICLISADWDS. GCmodel nc trh Z σ1 vesc GCmodel hNi P1 P2 P3 Typical 105 1.0 0.005 10 40 Typical 0.39(0.01) 0.32(0.01) 0.06 0.01 47Tuc-like 2.5×105 3.0 0.0035 11.5 57 47Tuc-like 1.9(0.03) 0.84(0.03) 0.55 0.29 Terzan5-like 8×105 0.93 0.013 11.6 49 Terzan5-like 1.6(0.03) 0.80(0.03) 0.48 0.22 trhN(GOyTrE),.m—etaGllCicictyorZe,nounme-bdeimrdenensisoitnyalnvce(lopcci-t3y),dihsaplef-rmsioanssσr1el(akxmatsio- 1n),tiamnde ibnilNiptaiOerseTnPEtN.h—etsheaMstacetoanlrereanssuptmoNbnedsurtchohNDDiWoWfDDecsscwaernietthrpicraetLsleIeSnatAsattDtawWnoyDhgsaivarmenndotnPiimocises.sdoNentuepmcrtoabbbealres- escapevelocityvesc(kms- 1). atadistanceof10kpcwithSNR>8,fora5yearLISAmissionandasingle time-delayinterferometryobservable. oneormultiplebinary-singlestarscatteringevents.Thesein- teractionsareeitherdistantfly-byencountersorclose,strong, 47Tuc-like,and80%fortheTerzan5-likeGCmodel. andpossibly resonantinteractions. Evenfor distantencoun- The hatched probability distribution functions (PDFs) in ters(atmorethan 5–10timesthebinaryseparation)andini- Fig. 1 show the statistical properties of the eccentric LISA tialeccentricitiese<0.1,Heggie&Rasio(1996)findthatthe DWDsinthethreeGCsimulations. Forallthreemodels,the relative changein eccentricity can be as high as 30%. Con- number of systems decreases monotonicallywith increasing sequently, the absolute eccentricity change in a given inter- orbital frequency ν =1/P. The abrupt drop in the number actionisinsignificantforcircularbinaries,butincreaseswith ofsystemsatlogν.- 3.7correspondstotheadoptedcut-off thebinary’seccentricityandthecumulativeeffectofmultiple orbital period of 5000s for sources in the LISA band. The interactions. On the otherhand, close and possibly resonant orbital eccentricity distributions (affected by post-formation interactions are rare, but can change the binary eccentricity GWemission)typicallypeakate 0.1anddecreaseslowly drasticallyinjustasingleencounter. ≃ toward higher values. In all three cases, there is a non- The possibility of forming eccentric DWDs through the negligible fraction of systems with e up to 0.7. The ma- above channelsis in stark contrast to DWDs in the Galactic ≃ jority of systems consists of at least one WD more mas- diskwhichareexpectedtobeinvariablycircular. Despitethe manydynamicalinteractions,circularbinariesstilldominate sive than 0.6M⊙. The total mass distributions therefore DWDpopulationsinGCs,althoughwewillshowinthenext peak at re∼latively high masses M1+M2 1.0–1.2M⊙. The ≃ sectionthatthenumberofeccentricDWDsis non-ngligible. chirpmassdistributionspeakat 0.3–0.4M⊙. Thetyp- M≃ ical merger lifetime τ of an eccentric LISA DWD is 10– CircularDWDbinariescouldinprinciplealsobecomeeccen- e>0 100Myr.However,asubstantialnumberofsystemsalsohave tricviabinary-singlestarencounters,althoughtheefficiency considerablyshorterlifetimesdownto 1Myr. oftheprocessissmall(Heggie&Rasio1996). Nevertheless, ∼ all possible eccentricity inducing effects are included in the 4. IMPLICATIONSFORLISA simulationsdescribedbelow, regardlessof their relativeeffi- ciency. Since the detectabilityandpropertiesof GWsignals Gravitationalwavesfrombinarieswithnon-zeroeccentric- fromeccentricDWDs are independentofthe detailsof their ityproduceradiationatfrequenciesthatareintegermultiples formation,wedonotdistinguishbetweenthedifferentforma- ofthe orbitalfrequency. To computethe LISA sensitivityto tionchannelsintheremainderofthispaper. eccentric DWDs, we model GWs at the leading quadrupole order.The(angle-averaged)SNRatwhichagivenbinarycan 3. MONTECARLOSIMULATIONS bedetectedoveranobservationtimeT isthengivenby obs We use simulations by Ivanova et al. (2006) to estimate h (f ) 2 the expected number of eccentric DWDs in the LISA band S/N 2 = S/N 2 = c,n n dlnf , (10- 4–10- 1Hz). Theoutcomeofthesimulationsdependson D(cid:0) (cid:1) E Xn D(cid:0) (cid:1)nE Xn Z (cid:20)hrms(fn)(cid:21) n theadoptedGCmodelparameters(e.g.,clustercoredensity, (1) metallicity)andstellarandbinaryevolutionparameterizations wherenlabelstheharmonicsatfrequency f nν, n (e.g.,initialmassesandorbitalelements,efficiencyoforbital ≃ angularmomentumlossmechanisms).WeconsiderthreeGC h =(πd)- 1 2E˙ /f˙ 1/2 (2) c,n n n modelsrepresentativeofa typicalGalactic, 47Tuc-like, and (cid:0) (cid:1) Terzan 5-like GC (see Table 1), and GC ages in the range is the characteristic amplitude of the n-th harmonic, and of 10–13Gyr. All stellar andbinaryevolutionaryphasesare h2rms(fn)=(20/3)Sh(fn)fnistheroot-meansquarevalueofthe treatedasinthestandardmodelofIvanovaetal.(2006). noiseaveragedoversourcepositionandorientation;Sh(fn)is theone-sidednoisespectraldensity,whichwecomputedus- The maximum number of eccentric DWDs present in the LISA band at any given time in the 10–13Gyr time inter- ing Larson’s online sensitivity curve generator6. In Eq. (2), ˙ val is 2, 8, and 7, for the typical, 47 Tuc-like, and Terzan EnisthetimederivativeoftheenergyradiatedinGWsatfre- 5-like GC models, respectively. Using the total number of quency fn nν,whichtolowestorderis ≃ systemsformedin the10–13Gyrtime window,wecalculate E˙ =(32/5)(2πν )10/3g(n,e). (3) the formation rates of eccentric LISA DWDs in the typical, n M 47Tuc-like, andTerzan5-likeGCmodelsto be 10- 8yr- 1, Here,g(n,e)isafunctionoftheorbitaleccentricitygivenby 3 10- 8yr- 1,and 5 10- 8yr- 1. ForeachGC≃model,the ˙ p≃rob×ability that N ec≃cent×ric LISA DWDs are present at any Eq.(20)inPeters&Mathews(1963).Theterm fn≃ndν/dt, totheleadingquadrupoleorder,is giventimeinthe10–13Gyragerangeisfurthermorewellfit- tedbyaPoissondistributionwithmean N listedinTable2. f˙ =96(10π)- 1 5/3(2πν)11/3F(e), (4) Itfollowsthattheprobabilitytofindatlehasti1eccentricDWD n M intheLISAbandis32%forthetypicalGalactic,84%forthe 6http://www.srl.caltech.edu/∼shane/sensitivity/ Eccentricdoublewhitedwarfsinglobularclusters 3 FIG. 1.—Statistical properties ofeccentric DWDs(e>0.01) in the LISAband (P<5000s). Hatched histograms represent the entire eccentric DWD population; greyshadedhistogramsrepresentthesubpopulationofsystemsforwhichatleast2harmonicsareindividually observablebyLISAatSNR>8, assuminga5yearLISAmission,adistanceof10kpc,andasingletime-delayinterferometryobservable.AsthePDFsarenormalizedtounity,theverticalscale variesfrompaneltopanel. whereF(e)=[1+(73/24)e2+(37/96)e4]/(1- e2)7/2.Thelat- ter functiontendsto unity for e 0, but can becomeof the → orderofafewfore-valuestypicalofthesystemsproducedin −19 10 oursimulations,e.g.F(0.2)=1.29andF(0.5)=4.88. Using Eqs. (1)-(4), we compute the number of detectable eccentricLISAsources,requiringthatatleasttwoharmonics e beindividuallyobservableatSNRof8orhigher.Thisthresh- d oldisconservative,sincewestilldonotexactlyknowhowthe plitu10−20 dataanalysiswillbecarriedout(notethatintheLISAlitera- m a ture a SNR=5 thresholdis usuallyconsidered). Thechoice ofassumingthatatleasttwoharmonicsareobservablesepa- rately,isdictatedbyreasonsofrobustness: froman analysis 10−21 pointofview,aneccentricDWDlookslikeacollectionofn "circular" (one single harmonic) binaries, with suitable am- plitudesandfrequencies.Thereisgrowingevidencethatdata 10−3 10−2 analysisschemesthatarealreadyinhandforcircularbinaries frequency / Hz (cf. e.g. Arnaudetal. 2007,andreferencestherein)willper- FIG. 2.—LISAsensitivitytoeccentricDWDsinthetypicalGalacticGC formwellatsuchmoderate-to-highSNR.Wecanthereforebe model. Thesolidlinesshowtheangle-averagedlevelofthermsnoise(thin: instrumentalnoise;thick: totalinstrumentalandgalacticforegroundnoise). confidentthateven an analysis, comprisingof a first pass of SquaresindicatetheamplitudesofDWDsforwhichatleasttwoharmonics thedatatosearchfor"circular"binaries,followedbyasecond canbedetected individually atSNR>8. Somesquares correspond tothe stageinwhichonelooksforamoreconsistentfittothedata samesystematdifferenttimesinitsevolution.Forillustration,thesolidstars by allowing that some signals are actually harmonics of the andcircles representtheharmonicsofaDWDwithmassesM1=0.74M⊙ same system is available. In practice, it is quite conceivable andM2=0.86M⊙ attwodifferent stagesofitsevolution: atP=872.64s ande=0.33(stars)andP=2211.84sande=0.74(circles). Inbothcases4 thata full-blownanalysisallowingfornon-negligibleeccen- harmonicsareobservableindividuallyatSNR>8,butalln=1,...,20har- tricitycanbecarriedoutfromthebeginning. Inthiscasethe monicsareplottedforillustration.Theamplitudesarefora5-yearmission,a numberofdetectablesystems perGC presentedhere is con- GCdistanceof10kpc,andasingleMichelsonobservable. servativeandshouldberegardedasalowerlimit. In Fig. 2, we show the detectability results for the typical Galactic GC model, adoptinga distance of 10kpc. The am- TABLE3 plitudeontheverticalaxisisthecharacteristicamplitudeh OBSERVABLEBINARIES. c,n multipliedbythesquarerootofmin[1,df /dt (T /f )]to n × obs n GCmodel Ntot Nobsat3kpc Nobsat10kpc Nobsat50kpc consistently account for the frequency band swept by each source(allthesystemsinoursimulationssatisfythecondition Typical 29 24(28) 17(27) 1(7) 47Tuc-like 102 72(96) 36(90) 9(24) df /dt (T /f )<1forT =5yr).Theheightofeachdot n obs n obs Terzan5-like 140 112(133) 79(127) 4(36) × abovethenoiselevelthenrepresentstherelevantSNRvalue NOTE. — Total number of eccentric LISA DWDs Ntot and number of [cf. Eq. (1)]. We consider Tobs =5 yrs, the current require- systemsNobswithatleasttwoharmonicsindividuallydetectablebyLISAat mentfortheminimummissionlifetime.Wealsouseonlyone distancesof3,10,and50kpcwithacoherentSNR>8fora5yearLISA Michelsonobservableintheanalysis;ifallsixDopplerlinks mission.BothNtotandNobsareintegratedovera10–13Gyragerange.Note thatessentiallyallsystemsinGCsatGalacticdistancesaredetectablewith are available over T , the SNR would increase by a factor obs atleastoneharmonic;thisnumberisprovidedwithinparenthesisintheNobs √2. Notethatonlysystemsforwhichatleasttwoharmon- columns. ≃ icshave (S/N)2 1/2>8areplottedinthefigure. h ni range from 24–112 for d =3kpc and 17–79 for d =10kpc. In Table 3, we list the number of detectable eccentric It is quite remarkablethat LISA has a high enoughsensitiv- DWDs formed in the 10–13Gyr age range, for distances ity to observe these systems throughoutthe Galaxy and, for d =3,10,50kpc. Dependingon the GC model, the numbers a very limited regionof the parameterspace, even up to the 4 B.Willemsetal. Large Magellanic Cloud. The probability to find at least 1 from electromagnetic observations allows the derivation of detectableeccentricDWD is1%forthetypicalGalactic GC chirp masses of GC binaries from the amplitude of their modeland3%forboththe47Tuc-likeandTerzan5-likeGC GW signals. Unfortunately, the identification of sources models (see Table 2). If we require only one harmonic to with GCs is severely hampered by LISA’s limited angu- be detectable, the probabilities increase by a factor of 2. lar resolution. In particular, the error box in the sky is Comparedtothefullpopulation,thedetectablesampleso∼fec- 1.6(5mHz/ν)2(10/SNR)2deg2(e.g.Takahashi&Seto2002≃) centric DWDs furthermore favor higher orbital frequencies, compared to a typical GC’s size of <0.002deg2. Indepen- total system masses, and chirp masses, and shorter system dentchirpmassandsourcedistancedeterminationstherefore lifetimes; relevantorbitaleccentricitiesspanthe entirerange relycompletelyonmeasurementsofintrinsic changesofthe frome 0.1to0.8(greyPDFsinFig.1). GW frequency during the observation time (e.g. Takahashi ≃ & Seto 2002), assuming the drifts are entirely due to gravi- 5. DISCUSSION tational radiation with no significant contribution from tidal InthisLetter,weexaminetheformationofeccentricDWDs and/ormagneticspin-orbitcoupling(e.g.Stroeeretal.2005). intheLISAbandthroughdynamicalinteractionsinGCs. Al- Itisthereforeinterestingtonotethat,dependingontheirphys- though the majority of cluster DWDs are circular, the exis- icalparametersand the phaseof evolutionatwhich theyare tenceofthisnon-negligiblepopulationis instark contrastto observed,thesystemsinoursimulationscanproduceobserv- the Galactic disk population expected to consist exclusively ablefrequencydrifts. of circular binaries. Owing to the non-zero orbital eccen- We conclude that the identification of eccentric binaries tricities, the DWDs emit GWs at multiple harmonics of the with LISA does not necessary imply association with NSs. orbitalfrequency. Fora5-yearLISAmission,wefindanon- Since binary component masses are difficult to derive from negligible number of systems for which multiple harmonics GWdata, a cleardiscriminationbetweenNS-WD orNS-NS aredetectableindividuallyatSNR>8,allowingthedetermi- andDWDbinariesfromLISAdatamaynotbepossible.How- nationoftheorbitaleccentricityfromthe ratioof theampli- ever,thenumberofeccentricNS-WDbinariesinthethreeGC tudesofthedetectedharmonics. models is an order of magnitude lower than the number of As a proof of principle, we show that for standard stellar eccentricDWDs, and NS-NS binariesare evenscarcer. Sta- andbinaryevolutionassumptions,47TucandTerzan5type tistically, an eccentricity detection therefore favors a DWD clusters are the most efficient eccentric DWD factories. For interpretation over a NS-WD or NS-NS interpretation. The these two types of GCs, there is an 80% probability that at formation of eccentric DWDs in GCs thus provides an ex- least oneeccentric DWD is presentin the LISA bandatany citingopportunitytostudyWD physicsinaccessiblethrough giventime.ForatypicalGalacticGC,thisprobabilityis32%. DWDs in the Galactic disk. Tidal effects particularly be- Moreover,foraGCat10kpc,theprobabilitythatoneormore come increasingly important with increasing orbital eccen- systemsare presentwith at leasttwo harmonicsindividually tricity,eitherthroughtheexcitationofdynamicaloscillations detectable at SNR>8 is 1% for a typical Galactic GC, and orthroughtheireffectsontheorbitalevolution. LISAcould 3% for 47 Tuc and Terzan 5 type GCs. Considering there provideuniquedynamicalidentificationsofeccentricDWDs, are79knownGCswithin10kpcfromtheSun(Harris1996), openingup a unique astrophysicalwindow to stellar and bi- we thus expect at least a handful of eccentric DWDs to be naryevolution. detectable by LISA. There are furthermore3 GCs within 2– 3kpcand15GCswithin5kpcfromtheSun. Fortheseclus- ters,theSNRincreasesbyafactorof 2–3. Thesenumbers This work is partially supported by a Packard Foundation ∼ mayevidentlyvarysignificantlywiththeadoptedstellarand Fellowship, a NASA BEFS grant (NNG06GH87G), and a binaryevolutionmodelparameters. Sinceourgoalwastoas- NSF CAREER grant (AST-0449558) to VK. FAR and JMF sessthefeasibilityofforminganddetectingeccentricDWDs acknowledgesupportfromNASA grantsNNG06GI62Gand inGCs, wechoseparameterswhicharecommoninstandard NNG04G176G, and from NSF grant PHY-0601995. Clus- populationsynthesismodels. tersimulationswereperformedonCITA’sSunnyvalecluster, In principle, the independent knowledge of GC distances fundedbytheCFIandtheORF-RI. REFERENCES Arnaud,K.A.,etal.2007,arXiv:gr-qc/0701139 Ivanova, N., Heinke, C. O., Rasio, F. A., Taam, R. 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