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Astronomy&Astrophysicsmanuscriptno.15777 c ESO2011 (cid:13) January5,2011 The eccentricity distribution of compact binaries I.Kowalska1,T.Bulik1,2,K.Belczynski1,3,M.Dominik1,andD.Gondek-Rosinska4,2 1 AstronomicalObservatory,UniversityofWarsaw,AlUjazdowskie4,00-478Warsaw,Poland 2 NicolausCopernicusAstronomicalCenter,Bartycka18,00716,Warsaw,Poland 3 Dept.ofPhysicsandAstronomy,UniversityofTexas,Brownsville,TX78520,USA 4 InstituteofAstronomy,UniversityofZielonaGo´ra,ul.Lubuska2,65-265ZielonaGo´ra,Poland Received;accepted 1 ABSTRACT 1 0 Context.The current gravitational wave detectors have reached their operational sensitivity and are nearing detection of compact 2 objectbinaries.Inthecomingyears,weexpectthattheAdvancedLIGO/VIRGOwillstarttakingdata.Atthesametime,thereare plansforthirdgenerationground-baseddetectorssuchastheEinsteinTelescope,andspacedetectorssuchasDECIGO. n Aims.Wediscusstheeccentricitydistributionofinspiralcompactobjectbinariesduringtheyinspiralphase.Weanalyzetheexpected a distributionsofeccentricitiesatthreefrequenciesthatarecharacteristicofthreefuturedetectors:AdvancedLIGO/VIRGO(30Hz), J EinsteinTelescope(3Hz),andDECIGO(0.3Hz). 4 Methods.We use the StarTrack binary population code to investigate the properties of the population of compact binaries in formation. We evolve their orbits until the point that they enter a given detector sensitivity window and analyze the eccentricity ] distributionatthattime. O Results.WefindthattheeccentricitiesofBH-BHandBH-NSbinariesarequitesmallwhenenteringtheAdvancedLIGO/VIRGO C detectorwindowforallconsideredmodelsofbinaryevolution.EveninthecaseoftheDECIGOdetector,thetypicaleccentricities . ofBH-BHbinariesarebelow10−4,andtheBH-NSeccentricitiesaresmallerthan10−3.SomefractionofNS-NSbinariesmayhave h significanteccentricities.Withintherangeofconsideredmodels,wefoundthatafractionofbetween0.2%and2%NS-NSbinaries p willhaveaneccentricityabove0.01fortheAdvancedLIGO/VIRGOdetectors.FortheETdetector,thisfractionisbetween0.4%and - 4%,andfortheDECIGOdetectoritliesbetween2%and27%. o r Keywords.binaries–gravitationalwaves t s a [ 1. Introduction The observed NS-NS binaries have merger times 3 T % 100 Myr. However, a significant fraction of the merg v As the interferometric gravitational wave detectors LIGO populationofthemergingNS-NSmayoriginateintheso-called 1 and VIRGO (Harry&theLIGOScientificCollaboration 2010; ultra-compactNS-NSbinaries,whichhavemuchshortermerger 1 Acerneseetal. 2006) reach their design sensitivities, the first timescales T - 100 Myr (Belczynskietal. 2002a). In 5 merg detection of gravitational waves has become more imminent. addition, the small number of known pulsars are indicative of 0 . Both detectors will undergo serious improvements to increase therebeingasignificantfractionofveryeccentricbinaries. 0 their sensitivity (Smith&LIGOScientificCollaboration 2009; Since no BH-NS nor BH-BH binaries are known, we can 1 Spalliccietal.2005). only rely on evolutionaryconsiderationswhen estimating their 0 It is therefore important to investigate the propertiesof the numberandproperties.Ithasbeenfoundthattheirnumberde- 1 primary candidate sources for detection, namely compact ob- pends very strongly on the outcome of the common envelope : v ject binaries. There have been a number of papers dealing phasewhenthesecondaryisontheHertzsprunggap.Thisphase Xi with several properties of the population of compact binaries will very likely end up as a merger and the formation of a (e.g., Nelemans&vandenHeuvel 2001; Voss&Tauris 2003; Thorne-Zytkowobject.However,ithasbeendemonstratedthat r a DeDonder&Vanbeveren 2004; Sipior&Sigurdsson 2002; inalow-metallicityenvironmentthecommonenvelopemergers Pfahletal. 2005; Dewietal. 2002, 2005; Bogomazovetal. may be (to some extent) avoided and the BH-BH formation is 2007; Kieletal. 2010). In particular, Abadieetal. (2010) pre- veryeffective(Belczynskietal.2010b). sentedtheirestimateddetectionrates.Inaddition,themassspec- Theeccentricityofacompactobjectbinarymaypotentially trum (Gondek-Rosin´skaetal. 2007), and even spin properties be derived by analyzing the inspiral signal, provided that the (Schnittman 2004; Mandel&O’Shaughnessy 2010) have been eccentricity is significant. In this paper, we investigate the ec- studied.Inthispaper,wepresentyetanotheraspectofthemerg- centricity distributions in the frequency band of the currently ingcompactobjectbinarypopulation:thedistributionoftheec- workingandfuturedetectorsofgravitationalwaves.Forthecur- centricity. rently working detectors (LIGO and VIRGO), we assume that From radio observations, we currently know of only six the sensitivity of the detectors will allow us to measure the compact object binaries with merger timescales shorter than a signal for the frequencies starting at 30 Hz. This may not be Hubble time, all of them NS-NS (neutron star - neutron star) accurate for the current state of these instruments, but it does systems,andweknowofnoBH-NS(blackhole-neutronstar) accurately represent the predicted sensitivity of the Advanced norBH-BH(blackhole-blackhole)system.TheknownNS-NS LIGO/VIRGO detectors. We consider two future detectors: systemsarelistedalongwiththeirorbitalparametersinTable1. the Einstein Telescope (VanDenBroeck 2010) and DECIGO 2 I.Kowalskaetal.:Theeccentricitydistributionofcompactbinaries (Kawamura2006;Setoetal.2001).FortheEinsteinTelescope, set of modelsin whichthe magnitudeofthe NSkicksis lower we assume that binariesshall be detectablefrom 3 Hz, and for by a factor of 2, to σ = 132.5km/s, as some observationsand DECIGO we assume that the lowest frequency detectable is empiricallybasedargumentsseemtoindicatethatnatalkicksin 0.3Hz.Inallcases,thesehavetobetreatedasindicativenum- closebinariesarelowerthanforsinglestars(Dessartetal.2006; bersthatroughlydescribetheseinstruments. Kitauraetal. 2006). The BH kicks are decreasedin the similar In section 2, we describe the model used to investigate the fashionasinmodelswiththefullNSkicks.Thestandardvalue populationofcompactobjectbinaries.Section3presentsthere- ofσparameterisdenotedbyKandthesmallervaluebyk.The sults for the currentand future gravitationalwave detectors. In detailed list of models considered in this paper is presented in section4,wesummarizeanddiscusstheresults. Table2.ModelAZKisastandardsetofparametersdescribedin detailbyBelczynskietal.(2002b). 2. Themodel Table2:Thelistofmodelsofstellarevolutionusedinthepaper. 2.1.Compactobjectbinarypopulationmodel Model Metallicity σ[kms 1] HG Tomodelthepopulationofcompactobjectbinaries,weusedthe − AZK Z 265.0 + StarTrackpopulationsynthesiscode(Belczynskietal.2002b). BZK Z⊙ 265.0 - It perform a suite of Monte Carlo simulations of the stellar AZk Z⊙ 132.5 + evolution of stars in environments of two typical metallicities: BZk Z⊙ 132.5 - Z = Z = 0.02andZ = 10%Z = 0.002(e.g.,Belczynskietal. AzK 10%⊙Z 265.0 + 2010b⊙).Inthesecalculations,w⊙eemployedtherecentestimates BzK 10%Z⊙ 265.0 - ofmasslossrates(Belczynskietal.2010a).Wecalculateapop- Azk 10%Z⊙ 132.5 + Bzk 10%Z⊙ 132.5 - ulation of 2 million massive binary stars, tracking the ensuing ⊙ formation of relativistic binary compact objects: double neutron stars (NS-NS), double black hole binaries (BH-BH), and mixed systems (BH-NS). Our modeling utilizes updated stellar and binary physics, including results from supernova sim- 2.2.Evolutionoforbits ulations (Fryer&Kalogera 2001) and compact object forma- The evolution of the orbit of compact object binary under tion (Timmesetal. 1996), incorporatingelaborate mechanisms the influence of gravitational radiation had been calculated by for treating stellar interactions such as mass transfer episodes Peters&Mathews(1963);Peters(1964).Inthequadrupoleap- (Belczynskietal. 2008) or tidal synchronization and circular- proximation,theorbitdecaysas ization (Hut1981). We place special emphasisonthe common envelopeevolutionphase (Webbink 1984), which is crucialfor close double compact object formation because the attendant da β 1+73/24e2+37/96e4 = Ψ(e), Ψ(e)= , (1) mass transferpermitsan efficienthardeningof the binary.This dt −a3 (1 e2)7/2 orbitalcontractioncanbesufficientlyefficienttocausetheindi- − vidual stars in the binary to coalesce and form a single highly where a is the great semi-axis, e is the eccentricity of binary, rotating object, thereby preventing additional binary evolution M1 isthemassofthefirstcomponent,M2 isthemassofsecond andtheformationofadoublecompactobject.Becauseofsignif- component,and icantradialexpansion,starscrossingtheHertzsprunggap(HG) veryfrequentlyinitiateacommonenvelopephase.HGstarsdo 64G3µM2 M M not have a clear entropy jump at the core-envelope transition β= , µ= 1 2 . (2) 5 c5 M +M (Ivanova&Taam2004);ifsuchastaroverflowsitsRochelobe 1 2 andinitiatesacommonenvelopephase,theinspiralisexpected Whiletheeccentricitydecaysas to lead to a coalescence (Taam&Sandquist 2000). In particular,ithasbeenestimatedthatforasolarmetallicityenvironment de 19 β (1+121/304e2)e (e.g.,ourGalaxy),properlyaccountingfortheHGgapmaylead = Φ(e), Φ(e)= . (3) toareductioninthemergerratesofBH-BHbinariesby 2 3 dt −12a4 (1 e2)5/2 ∼ − − orders of magnitude (Belczynskietal. 2007). In contrast, in a Using the above formulae we can express the fundamental low metallicity environment this suppression is much less se- gravitationalwavefrequencyasafunctionoftheeccentricity vere ( 1 order of magnitude; Belczynskietal. (2010b)). The details∼of the common envelope phase are not yet fully under- 2 (1 e2)3/2 121 stood,thusinwhatfollowsweconsidertwosetofmodels,one fGW(e)= P e−18/19 [1+ 304e2]−1305/2299c03/2, (4) 0 thatdoesnottakeintoaccountthesuppression(optimisticmod- els: marked with A), and another that assumes the maximum wherec0 =(e012/19[1+ 132014e20]1305/2299)(1−e20)−1,P0istheinitial suppression(pessimistic models: markedwith B). Solar metal- orbital period, and fGW(e) is the first non-zero harmonic. The licityand10%ofsolarmetallicityarelabeledwithZandz,re- gravitationalwavefrequencyistwicetheorbitalfrequency,i.e., spectively.InthecaseofNSs,weadoptnatalkickdistributions fGW =2forb = P2orb. fromobservationsofsingleGalacticpulsars(Hobbsetal.2005) We present the evolution of eccentricity as a function of withσ = 265km/s.However,forBHswedrawkicksfromthe gravitational wave frequency in Figure 1 for a binary neutron same distribution (but at a lower magnitude), which is inverse starwithcomponentsofequalmassesof1.4M .Theinitialfre- proportionalto the amountof fall back expected at BH forma- quency corresponds to a semi-major axis such⊙that the merger tion(e.g.,Fryer&Kalogera2001).Inparticular,formostmas- timeissettobeT = 104Myr.Figure1containsseveraldif- merg siveBHsthatformwiththefullfallback(directBHformation), ferent cases of evolution in the plane stretched by eccentricity the amountof natalkick is zero. In addition,we test one more andgravitationalwavefrequency. I.Kowalskaetal.:Theeccentricitydistributionofcompactbinaries 3 Table1:Knownmergingcompactobjectbinaries Name P [h] Presente T [Gyr] eat0.3Hz eat3Hz eat30Hz Ref. orb merge J0737-3039A/B 2.454 0.088 0.085 4.5 10 5 4 10 6 3.5 10 7 Burgayetal.(2003) − − − B2127+11C 8.05 0.681 0.2 2.9×10 4 2.6× 10 5 2.3×10 6 Andersonetal.(1990) − − − J1906+0746 3.98 0.085 0.3 2.6×10 5 2.3×10 6 2 ×10 7 Lorimeretal.(2006) − − − B1913+16 7.752 0.617 0.3 2.2×10 4 1.9×10 5 1.7× 10 6 Weisberg&Taylor(2005) − − − J1756-2251 7.67 0.181 1.7 2.6×10 5 2.5×10 6 2.2×10 7 Faulkneretal.(2005) − − − B1534+12(=J1537+1155) 10.098 0.274 2.7 3.6×10 5 3.2×10 6 2.8×10 7 Wolszczan(1991) − − − × × × 00 1100 --11 1100 ee --22 1100 AAZZKK AAZZkk AAzzKK AAzzkk --33 1100 00 1100 --11 1100 ee --22 1100 BBZZKK BBZZkk BBzzKK BBzzkk --33 1100 --66 --44 --22 00 22 --66 --44 --22 00 22 --66 --44 --22 00 22 --66 --44 --22 00 22 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 ff ff ff ff GGWW GGWW GGWW GGWW Fig.2:ThepropertiesofthepopulationofdoubleneutronstarsobtainedusingtheStarTrackcode.Theplotshowsonlythebinaries thatwillmergewithintheHubbletime.Solidlinescorrespondtoevolutionarytracksforinitialgravitationalwavesfrequenciesfrom f =10 8Hz(firstlinefromtheleft-handside)to f =102Hz(firstlinefromtheright-handside). 0 − 0 objectbinariesinFigures2-4inthespacespannedbytheini- 1 tial eccentricityand initialgravitationalwave frequency,which is twice theorbitalfrequency.Eachpanelin thesesfigurescor- 0.8 respondstoadifferentmodellabeledaslistedinTable2. The case of the NS-NS systems is shown in Figure 2. The 0.6 boundaryoftheregionpopulatedbythesystemsontheleft-hand e sidecorrespondstotherequirementthatweonlyconsiderbina- 0.4 ries that merge within a Hubble time. The bulk of the binaries shown in each panel correspond to those that have undergone oneCEphaseintheirevolution.Thetoprowcorrespondstothe 0.2 modelsAZK,AZk,AzK,andAzk,inwhichweallowthebina- riestocrossthroughthecommonenvelopewiththedonoronthe 0 10-7 10-6 10-5 10-4 10-3 10-2 Hertzsprunggap,denotedby”+”inTable2.Thesebinariesmay undergo a second common envelope phase with a helium star f GW companion.AtthesecondCEstage,theorbitistightenedeven Fig.1: We present ten cases of eccentricity evolution, starting moreleadingtoformationofthestripeinthediagramstretching with different values of e from e = 0.1 (first line from the from fGW 10−2 Hz at e 10−2. In these models, the initial ≈ ≈ bottom) to e = 0.99 (first line from the top). Initial semi- distributioninthespaceofgravitationalwavefrequencyversus major axis is chosen such that a binarywill mergewithin time eccentricityisbimodal.Theinfluenceofthevalueofthekickve- T =10Gyrineachcase. locityhasasmallimpactontheshapeofdistributionspresented merg inFigure2ascanbeseenbycomparingthedatainplotslabeled aseitherK-largekicksork-smallkicks. ForBH-NSsystems,presentedinFigure3,andBH-BH bi- 3. Results naries, in Figure 4, we present the results of six out of eight models, since in models BZK and BZk, almost no binariesare 3.1.Propertiesofthebinariesatformationtime formedinoursimulationsthatinvolve2 106initialbinaries.For WestartwithaninitialpopulationcreatedusingtheStarTrack BH-NSandBH-BHbinaries,theforma×tionofultra-compactbi- code. We present the properties of the population of compact naries is not expected. The formation of NS-NS ultra-compact 4 I.Kowalskaetal.:Theeccentricitydistributionofcompactbinaries 00 1100 --11 1100 ee --22 1100 AAZZKK AAZZkk AAzzKK --33 1100 00 1100 --11 1100 ee --22 1100 BBzzKK AAzzkk BBzzkk --33 1100 --66 --44 --22 00 22 --66 --44 --22 00 22 --66 --44 --22 00 22 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 ff ff ff GGWW GGWW GGWW Fig.3:Thepropertiesofthepopulationofneutronstar-blackholesystemsobtainedusingtheStarTrackcode.Theplotshows onlythebinariesthatwillmergewithintheHubbletime.Solidlinescorrespondtoevolutionarytracksforinitialgravitationalwave frequenciesfrom f =10 8Hz(firstlinefromtheleft-handside)to f =102Hz(firstlinefromtheright-handside). 0 − 0 systems in veryclose orbitsis the consequenceof the finalCE and 3 Hz (top horizontalaxis) correspondingapproximatelyto episode,which isinitiatedby a low-masshelium(2-4 M )star the ET and DECIGO detectors. The results for the Advanced anditsNScompanion(1.4M ).Sincethedonorisabout⊙twice LIGO/VIRGOcanbeeasilyobtainedbyrescalingthehorizontal as massive as its companion,⊙the CE phase is initiated by the axis. non-stablemasstransferandtheorbitsignificantlydecreasedin Theshapeoftheeccentricitydistributionsatthemomentthat size. For more massive BH-BH/BH-NS binaries, helium stars the binaryentersthe givendetectorbandfollowsfromthe cor- areonaveragemoremassive(M >3 4M )anddonotexpand respondinginitialdistribution.However,onemustnotethatfor (so no CE phase), and even if a low−mas⊙s helium star forms, eachtypeofbinarythereisadifferentnaturaltimescaleandfre- then its companion is a BH (M > 3M ), so most likely in- quency,becauseofthedifferentmassscalesofeachbinary. steadofCEtheRLOFisstableanddoes⊙notleadtoorbitalde- WepresenttheresultsfortheDECIGOdetectorandaddap- cay (mass ratio 1). Very few systems (e.g., models AzK or ∼ propriatenumbersfortheETinparentheses.FortheNS-NSbi- Azk) produce ultra-compact BH-NS/BH-BH binaries for very nariesshowninthetoppanelofFigure5,thedistributionisei- special cases of binary evolution. In the case of BH-BH bina- thercenteredone 10 4(ET:10 5)forthemodelsBZK,BZk, ries,showninFigure4wepresentonlysixmodels,sincemod- ≈ − − BzK, and Bzk, where we do not allow the formation of ultra- els BZK and BZk do not lead to the formation of BH-BH bi- compact binaries in a second CE phase. The remaining mod- naries (Belczynskietal. 2007). In all models, there is an en- elsAZK, AZk,AzK,andAzk containanothercomponentcen- hanceddensityofsystemsformedwithe 0.1atapproximately 10−5Hz < fGW < 10−4Hz. In these syst≈ems, the second black nteernetdrreopuregshelnytathteeu≈ltra1-0c−o4m(pEaTc:t1b0in−a3r)i.eTshthisatadhdaviteioenxaplecroiemncpeod- holehasformedviadirectcollapse.Whentreatingthedirectcol- two episodesof mass transfer in their evolutionaryhistory and lapse, we assume that 10% of the mass escapes in the form of werealreadyverytightatthesecondsupernovaexplosion.The neutrinosand possibly gravitationalwaves. Hence, the gravita- mixed BH-NS binaries, shown in the middle panel of Figure 5 tionalmassoftheBHis10%lowerthanthebaryonmassofthe exhibitadistributionofeccentricitycenteredate 3 10 5(ET: cthoellaspyssitnemgsstawr.erTehcisiricnutlraordizuecdesinasthmeamllaescscetrnatnriscfietyr p≈ri0o.r1tosinthcee b3o×tto1m0−6p)a,nwelhiolfeFthigeuercec5enltireicbietytwBeHen-BeHbin1a0rie6s≈(,EsTh×:o1w0n−7i)natnhde collapseandtheformationofthesecondBH. e 10 4(ET:10 5). ≈ × − − − − ≈× FortheAdvancedLIGO/VIRGOdetectorswhereweassume 3.2.Eccentricitywhenbinaryentersdetectorband thatthe lowfrequencyboundaryliesat 30Hz, theeccentric- ≈ ities are even smaller. It follows from equation 4 that the dis- Forthedetectionofgravitationalwaves,itisimportanttoknow tributions are shifted by a factor of 10 19/18 for each factor of − the eccentricity of a binary at the time it enters the sensitivity ten in frequency. Thus, the values of eccentricity in the case windowofthedetector.Weconsiderthreecasesthatcorrespond of Advanced LIGO/VIRGO type detectors are consistent with approximatelytothreetypesofdetectors.InFigure5,weshow e = 0 and we can safely assume that all BH-NS and BH-BH theeccentricitydistributionsat0.3Hz(bottomhorizontalaxis) binariesarecircularwithoutanylossofsensitivity. I.Kowalskaetal.:Theeccentricitydistributionofcompactbinaries 5 00 1100 --11 1100 ee --22 1100 AAZZKK AAZZkk AAzzKK --33 1100 00 1100 --11 1100 ee --22 1100 BBzzKK AAzzkk BBzzkk --33 1100 --66 --44 --22 00 22 --66 --44 --22 00 22 --66 --44 --22 00 22 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 ff ff ff GGWW GGWW GGWW Fig.4:ThepropertiesofthepopulationofdoubleblackholesobtainedusingtheStarTrackcode.Theplotshowsonlythebinaries thatwillmergewithintheHubbletime.Solidlinescorrespondtoevolutionarytracksforinitialgravitationalwavesfrequenciesfrom f =10 8Hz(firstlinefromtheleft-handside)to f =102Hz(firstlinefromtheright-handside). 0 − 0 Intable3,wepresentthefractionofbinarieswitheccentric- Table 3: Fraction of the compact binaries with eccentricity itiesabove0.01atthetimeofenteringthedetectorband,tohelp greaterthan10−2.Toptablecorrespondstodoubleneutronstars quantifytheextentofthelargeeccentricitytailsofthedistribu- (NS-NS), middle to the mixed systems (BH-NS), and the bot- tionspresentedinFigure5.Thisfractiondoesnotreflectthede- tomtothebinaryblackholes(BH-BH).Wepresentthefraction tectability of eccentricity (ShapiroKey&Cornish 2010),which ofbinariesatthemomentofenteringdifferentfrequencybands forrealisticdistributionsofbinarieswillbediscussedinaforth- (30Hz, 3 Hz, and0.3 Hz).In brackets,we includethe number comingpaper. ofthese systemsin thesimulationN.We onlylisted resultsfor modelsthatarenon-zero.Thenumberofdigitsshownisforfor- mattingonly,andtherelativesamplingerrorisN 1/2. − 4. Summary Wehavepresentedtheeccentricitydistributionsofcompactob- NS-NS ject binaries at three frequencies immediately before merger. 30Hz 3Hz 0.3Hz The propertiesof the compactobjectbinarieshavebeen calcu- AZK 0.60% (51) 1.32% (112) 11.13% (945) latedusingtheStarTrackpopulationsynthesiscode.We have BZK 1.27% (36) 2.33% (66) 6.52% (185) found that the eccentricity distributions of the compact object AZk 0.16% (27) 0.38% (64) 10.37% (1732) BZk 0.30% (15) 0.75% (37) 2.22% (110) binariesdonotdependstronglyontheassumedmodelofbinary AzK 0.29% (25) 0.96% (83) 21.74% (1880) evolution. Any dependence has been found to be the strongest BzK 1.87% (13) 4.02% (28) 9.33% (65) forbinaryneutronstars,whosedistributionsmaybeeithersingle Azk 0.26% (37) 0.57% (81) 26.91% (3799) ordoublepeaked.Theextrapeakcorrespondstoultra-compact Bzk 1.74% (21) 3.31% (40) 7.79% (94) NS-NSbinariesthathaveundergoneanadditionalCEphaseim- BH-NS mediatelybeforeformingthesecondNS. 30Hz 3Hz 0.3Hz Tomaketheresultseasiertouseinthesimulations,wehave AZK 0.29% (2) 0.73% (5) 3.05% (21) fittedtheresultingdistributionsofeccentricitywithasinglelog- AZk 0.15% (2) 0.54% (7) 0.61% (8) normaldistributioninthecaseofBH-BHandBH-NSbinaries AzK 0.56% (14) 0.96% (24) 3.96% (99) 1 (x µ)2 BzK 0.68% (10) 1.23% (18) 3.63% (53) f(x)= exp − , (5) Azk 0.35% (15) 0.78% (34) 2.81% (122) σ√2π − 2σ2 ! Bzk 0.33% (8) 0.91% (22) 1.53% (37) wherex=loge,µisthemean,andσisthevariance. BH-BH For the NS-NS eccentricities, we used a sum of two log- 30Hz 3Hz 0.3Hz normaldistributions with two weights, since the distribution is AZK 0.31% (1) 0.62% (2) 1.87% (6) doublepeaked,givenby AzK 0.02% (3) 0.02% (4) 0.13% (23) w (x µ )2 (1 w) (x µ )2 BzK 0.15% (2) 0.15% (2) 0.46% (6) f(x)= exp − 1 + − exp − 2 ,(6) Azk 0.00% (1) 0.01% (2) 0.03% (6) σ √2π − 2σ2  σ √2π − 2σ2  1  1  2  2  6 I.Kowalskaetal.:Theeccentricitydistributionofcompactbinaries log(e) ET Table4:Parametersoflognormaldistributionfittedtoresultsof -7 -6 -5 -4 -3 -2 -1 modelAZKwithasymptoticstandarderrors. 1.2 AZK NS-NS 1 AZk AzK σ : 0.47 0.04 ge 0.8 BAzzKk σw21:: 00..4369±±00..0033 dN/dlo 00..46 BBBZZzKkk µ1 -3.803.3±H0z.04 -4.839±H±z0.04 -5.9340±H0z.04 µ -2.32 0.03 -3.38 0.03 -4.43 0.03 2 ± ± ± 0.2 BH-NS σ: 0.55 0.03 0 ± 0.3Hz 3Hz 30Hz -7 -6 -5 -4 -3 -2 -1 0 µ -4.18 0.04 -5.23 0.03 -6.27 0.03 log(e) DECIGO ± ± ± BH-BH log(e) ET σ: 0.70 0.03 ± 0.3Hz 3Hz 30Hz -7 -6 -5 -4 -3 -2 -1 1.2 µ -4.91 0.08 -5.95 0.03 -7.06 0.03 ± ± ± AZK 1 AZk AzK 0.8 BzK the neutron star. Second, the initial kicks at formation of BHs e Azk are lower than in the case of NS, so the initialeccentricitiesof g dlo 0.6 Bzk BH-BH systems are typically lower than in the case of NS-NS N/ ones. d 0.4 The eccentricities of the mixed BH-NS systems are larger than in the case of BH-BH ones. However,the numberof sys- 0.2 temsissmallenoughtoensurethatbyneglectingtheeccentricity 0 we do not decrease the sensitivity of Advanced LIGO/VIRGO -7 -6 -5 -4 -3 -2 -1 0 detectors.FortheET-likedetector,someeccentricsystemsmay log(e) DECIGO bedetected.InthecaseoftheDECIGO-likedetector,thenum- ber of systems with eccentricities above 0.01 lies between 3% log(e) ET and4%. -7 -6 -5 -4 -3 -2 -1 Theeccentricityof NS-NSsystemsare largerthanthose of 1.2 binariescontainingBHs.Givenamuchlargerexpecteddetection AZK 1 AZk rateforET,thismeansthatthereshouldbeasignificantnumber AzK ofNS-NSbinarieswithdetectableeccentricities.Finally,inthe 0.8 BzK caseofDECIGOafractionofbetween2%and27%oftheNS- e Azk og Bzk NSbinarieshaveeccentricitiesabove0.01.Moreover,theshape N/dl 0.6 of the eccentricity distribution of NS-NS binaries will depend d 0.4 on the existence ofan evolutionaryscenario leadingto the for- mationofultra-compactbinaries.Thus,themeasurementofthe 0.2 eccentricitydistributionisaninterestingtoolforprobingthede- tailsofNS-NSformationscenarios. 0 -7 -6 -5 -4 -3 -2 -1 0 log(e) DECIGO Acknowledgments Fig.5:Distributionofeccentricityfor(NS-NS-toppanel,BH- This work was supported by the EGO-DIR-102-2007; the NS middle panel and BH-BH bottom panel) seen at 0.3 Hz FOCUS 4/2007 Program of Foundation for Polish Science, (DECIGO-likedetectors)andat3Hz(ET-likedetectors).Solid thePolishgrantsN N203511238,DPN/N176/VIRGO/2009,N thick line corresponds to standard model (AZK). Dashed and N203 302835, N N203 404939 and by CompStar a Research dottedlinesindicateothermodels. NetworkingProgrammeoftheEuropeanScienceFoundation. where x = loge,µ isthemeanofthefirstpeak,µ isthemean 1 2 References ofthesecondpeak,σ isthevarianceofthefirstdistribution,σ 1 2 isthevarianceoftheseconddistribution,andwistheweight. Abadie,J.,Abbott,B.P.,Abbott,R.,etal.2010,ClassicalandQuantumGravity, WeusedtheMarquardt-Levenbergalgorithmtofindthepa- 27,173001 Acernese,F.,Amico,P.,Al-Shourbagy,M.,etal.2006,ClassicalandQuantum rameters,andestimatetheasymptoticstandarderrorofeachof Gravity,23,63 them. The results of the fits are shown in Table 4. The widths Anderson,S.B.,Gorham,P.W.,Kulkarni,S.R.,Prince,T.A.,&Wolszczan,A. ofthedistributionsσarethesameforeachfrequencybandand 1990,Nature,346,42 onlythecentroidsmove. Belczynski,K.,Bulik,T.,Fryer,C.L.,etal.2010a,ApJ,714,1217 Belczynski,K.,Bulik,T.,&Kalogera,V.2002a,ApJ,571,L147 TheeccentricityoftheBH-BH binariesinallthreecasesis Belczynski,K.,Dominik,M.,Bulik,T.,etal.2010b,ApJ,715,L138 negligible.Thisisduetotwofactors.First,atthefrequenciesof Belczynski,K.,Kalogera,V.,&Bulik,T.2002b,ApJ,572,407 interesttheBH-BHsystemsaremuchclosertocoalescencethan Belczynski,K.,Kalogera,V.,Rasio,F.A.,etal.2008,ApJS,174,223 I.Kowalskaetal.:Theeccentricitydistributionofcompactbinaries 7 Belczynski,K.,Taam,R.E.,Kalogera,V.,Rasio,F.A.,&Bulik,T.2007,ApJ, 662,504 Bogomazov,A.I.,Lipunov,V.M.,&Tutukov,A.V.2007,AstronomyReports, 51,308 Burgay,M.,D’Amico,N.,Possenti,A.,etal.2003,Nature,426,531 DeDonder,E.&Vanbeveren,D.2004,NewA,9,1 Dessart,L.,Burrows,A.,Ott,C.D.,etal.2006,ApJ,644,1063 Dewi,J.D.M.,Podsiadlowski,P.,&Pols,O.R.2005,MNRAS,363,L71 Dewi,J.D.M.,Pols,O.R.,Savonije,G.J.,&vandenHeuvel,E.P.J.2002, MNRAS,331,1027 Faulkner,A.J.,Kramer,M.,Lyne,A.G.,etal.2005,ApJ,618,L119 Fryer,C.L.&Kalogera,V.2001,ApJ,554,548 Gondek-Rosin´ska, D., Bulik, T., & Belczyn´ski, K. 2007, Advances in Space Research,39,285 Harry,G.M.&theLIGOScientificCollaboration.2010,ClassicalandQuantum Gravity,27,084006 Hobbs,G.,Lorimer,D.R.,Lyne,A.G.,&Kramer,M.2005,MNRAS,360,974 Hut,P.1981,A&A,99,126 Ivanova,N.&Taam,R.E.2004,ApJ,601,1058 Kawamura,S.2006,AstronomicalHerald,99,490 Kiel,P.D.,Hurley,J.R.,&Bailes,M.2010,MNRAS,406,656 Kitaura,F.S.,Janka,H.,&Hillebrandt,W.2006,A&A,450,345 Lorimer,D.R.,Stairs,I.H.,Freire,P.C.,etal.2006,ApJ,640,428 Mandel, I. & O’Shaughnessy, R. 2010, Classical and Quantum Gravity, 27, 114007 Nelemans,G.&vandenHeuvel,E.P.J.2001,A&A,376,950 Peters,P.C.1964,PhysicalReview,136,1224 Peters,P.C.&Mathews,J.1963,PhysicalReview,131,435 Pfahl,E.,Podsiadlowski,P.,&Rappaport,S.2005,ApJ,628,343 Schnittman,J.D.2004,Phys.Rev.D,70,124020 Seto, N.,Kawamura, S., &Nakamura, T.2001, Physical Review Letters, 87, 221103 ShapiroKey,J.&Cornish,N.J.2010,ArXive-prints Sipior,M.S.&Sigurdsson,S.2002,ApJ,572,962 Smith, J. R. & LIGO Scientific Collaboration. 2009, Classical and Quantum Gravity,26,114013 Spallicci, A. D. A. M., Aoudia, S., de Freitas Pacheco, J., Regimbau, T., & Frossati,G.2005,ClassicalandQuantumGravity,22,461 Taam,R.E.&Sandquist,E.L.2000,ARA&A,38,113 Timmes,F.X.,Woosley,S.E.,&Weaver,T.A.1996,ApJ,457,834 VanDenBroeck,C.2010,ArXive-prints Voss,R.&Tauris,T.M.2003,MNRAS,342,1169 Webbink,R.F.1984,ApJ,277,355 Weisberg, J.M.&Taylor, J.H.2005,inAstronomical Society ofthe Pacific Conference Series, Vol. 328, Binary Radio Pulsars, ed. F. A. Rasio & I.H.Stairs,25–+ Wolszczan,A.1991,Nature,350,688