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Mon.Not.R.Astron.Soc.000,000–000(0000) PrintedDecember11,2013 (MNLATEXstylefilev2.2) Population synthesis of triple systems in the context of mergers of carbon-oxygen white dwarfs 3 A.S. Hamers1⋆, O.R. Pols2, J.S.W. Claeys2 and G. Nelemans2,3 1 1LeidenObservatory,LeidenUniversity,POBox9513,NL-2300RALeiden,TheNetherlands 0 2DepartmentofAstrophysics/IMAPP,RadboudUniversityNijmegen,POBox9010,NL-6500GLNijmegen,TheNetherlands 2 3InstituteforAstronomy,KULeuven,Celestijnenlaan200D,3001Leuven,Belgium n a J AcceptedforpublicationinMNRAS2013January7.Received2013January7;inoriginalform2012November21 8 ] ABSTRACT R Hierarchical triple systems are common among field stars yet their long-term evolution is S poorlyunderstoodtheoretically.InsuchsystemsKozaicyclescanbeinducedintheinnerbi- h. narysystemduringwhichtheinnerorbiteccentricityandtheinclinationbetweenbothbinary p orbits vary periodically. These cycles, combined with tidal friction and gravitational wave - emission,cansignificantlyaffecttheinnerbinaryevolution.Toinvestigatetheseeffectsquan- o titativelyweperformapopulationsynthesisstudyoftriplesystemsandfocusonevolutionary r t pathsthatleadto mergersofcarbon-oxygen(CO)whitedwarfs(WDs), whichconstitutean s important candidate progenitor channel for type Ia supernovae (SNe Ia). We approach this a [ problem by Monte Carlo sampling from observation-based distributions of systems within the primary mass range 1.0−6.5M⊙ and inner orbit semi-major axes a1 and eccentricities 1 e satisfyinga 1−e2 >12AU,i.e.non-interactingintheabsenceofatertiarycomponent. v 1 1 1 9 Weevolvethese(cid:16)system(cid:17)sbymeansofanewlydevelopedalgorithmthatcouplesseculartriple 6 dynamicswithanexistingbinarypopulationsynthesiscode.Wefindthatthetertiarysignifi- 4 cantlyalterstheinnerbinaryevolutioninabout24%ofallsampledsystems.Inparticular,we 1 findseveralchannelsleadingtoCOWDmergers.Amongsttheseisanovelchannelinwhich . a collision occurs in wide inner binary systems as a result of extremely high eccentricities 1 0 inducedbyKozaicycles.WithassumptionsonwhichCOWDmergersleadtoaSNIaexplo- 3 sion we estimate the SNe Ia delay time distribution resultingfrom triples and compareto a 1 binarypopulationsynthesisstudyandtoobservations.Althoughwefindthatourtriplerateis : low,we havedetermineda lowerlimitofthetriple-inducedSNeIa rateandfurtherstudyis v neededthatincludestripleswithinitiallytighterinnerorbits. i X Keywords: binaries:general–celestialmechanics–methods:statistical–whitedwarfs r a 1 INTRODUCTION Kisseleva-Eggleton&Eggleton 2010; Fabrycky&Tremaine 2007). In particular, Fabrycky&Tremaine (2007) have shown Coeval stellar hierarchical triple systems, henceforth triple sys- thatKCTFisresponsible forproducing closeMSbinary systems tems, are known to be quite common among stellar systems in with periods P ∼ 3d, which is consistent with the observation the field (e.g. Tokovininetal. 2006; Raghavanetal. 2010). Such 1 that such systems are very likely (96%) orbited by a tertiary systems consist of an inner binary pair orbited by a distant star, (Tokovininetal. 2006). It remains to be seen, however, whether the tertiary. Due to secular gravitational effects high-amplitude KCTF is still effective in higher-mass triples (individual masses eccentricity oscillations, known as Kozai or Kozai-Lidov cycles m & 1.2M ). This is because their constituents have radiative (Kozai1962;Lidov1962),canbeinducedintheinnerbinarysys- ⊙ envelopes and are thus much less effective at dissipating tidal tem.Theseoscillationsoccurontimescalesthataretypicallymuch energythantheirlower-masscounterparts,whichhaveconvective longer than the orbital periods of both inner and outer binaries. envelopes (Zahn 1977). On the other hand, as such higher-mass One possible consequence of these eccentricity cycles is strong stars greatly increase in size and develop convective envelopes tidalfrictionandsubsequent orbitalshrinkage,aprocesswhichis duringtheir redgiant branch(RGB)and asymptoticgiant branch knownasKozaicycleswithtidalfriction(KCTF).KCTFhasbeen (AGB) phases tidal friction is expected to become much more studied in the context of (solar-mass) main sequence (MS) stars effective.ThusitispossiblethatKCTFiseffectiveatsignificantly (Mazeh&Shaham 1979; Eggleton&Kiseleva-Eggleton 2001; shrinking the inner binary system during the RGB/AGB phases, whereasthiswasnotthecaseduringtheMS. ⋆ E-mail:[email protected] Anotherpossibleconsequenceofhigh-amplitudeeccentricity 2 A.S. Hamers,O.R.Pols,J.S.W.Claeys and G. Nelemans cyclesinduced in(close)inner binarysystemsbyatertiaryisthe initial values of i , e and g , where g denotes the argument of tot 1 1 j enhancementoftheemissionofgravitationalwaves(GWs),apro- periastron.Otherquantitiesthatvaryperiodicallyareg (i.e.orbital j cess which is strongly dependent on the orbital eccentricity (e.g. precession,alsoknownasapsidalmotion)andthelongitudesofthe Peters 1964). Thompson (2011) has suggested that this mecha- ascendingnodesh . j nismcansignificantlyreducetheGWmergertimeinclosecarbon- In systems with large a /a the eccentric Kozai mech- 1 2 oxygen (CO) white dwarf (WD) binaries if they are orbited by a anism may apply in some cases (Lithwick&Naoz 2011, tertiarywiththeappropriateorbitalparameters.Thisscenarioim- Shappee&Thompson 2012). This mechanism is associated with pliesanincreaseintheexpectednumberofcloseCOWDsystems orbital“flips”(changeoforbitalinclinationfromretrogradetopro- thatmergewithinaHubbletime,whichisrelevantforbinarypop- gradeandviceversa)withpotentiallyveryhighinnerorbiteccen- ulation synthesis studiesinwhich theCO WD merger channel is tricities.A quantity that measures theimportance of thiseffect is considered asanimportant candidate progenitor channel for type the“octupoleparameter”ǫ definedby: oct Iasupernovae(SNeIa)(e.g.Ruiteretal.2009;Toonenetal.2012; m −m a e Claeysetal.inprep.). ǫoct = m1+m2a1 1−2e2, (2) In this work we investigate both scenarios of KCTF during 1 2 2 2 stellarevolutionandthemergingofCOWDsundertheinfluenceof where typically log10(|ǫoct|) & −2 indicates that the eccen- atertiarycompanionbymeansofapopulationsynthesisoftriples. tric Kozai mechanism could be important (Naozetal. 2011; Wetakeintoaccountlong-termdynamical,stellarandbinaryevo- Shappee&Thompson2012). lutionbymeansofanewlydeveloped algorithminwhichacode Inphysicallyrealisticsituationsthereexistsourcesofpreces- modelingtheseculardynamicalevolutionoftriplesiscoupledwith sion in the inner orbit other than those due to secular three-body thebinary population synthesis code Binary c(seeSect. 2.3for dynamicsalone(inourstudysucheffectsintheouterorbitareneg- details)tomodeltheinnerbinaryevolution.Werestrictourselves ligiblebecausea2isalwayssufficientlylarge).Wetakeintoaccount to intermediate mass triples with wide inner binary systems that thefollowingadditionalcontributionstoprecession:duetogeneral wouldnotinteractintheabsenceofatertiarycomponent.Forthese relativisticeffects(g˙1,GR),duetotidalbulgesinducedinphysically triplesweshowinthispaperthatthepresenceofthetertiarycom- extendedstarsinrelativecloseproximity(g˙1,tide)andduetointrin- ponentintroducestotheinnerbinarysystemvariousnovelevolu- sicspinrotation(g˙1,rotate).Thefactthatg˙1,GR,g˙1,tide andg˙1,rotate are tionaryscenarios. Inparticular, wefindanalternativepathway to all positive impliesthat the associated sources of precession pro- SNeIaasaresultofcollisionsofWDsinwideinnerbinarysys- moteprecessioninoneparticulardirectionandthereforesuppress temsinwhichtheeccentricityreachesextremelyhighvaluesasa the resonance between g1 and e1 during Kozai cycles. This phe- resultofKozaicycles. nomenonhasbeeninvestigatedindetailbyBlaesetal.(2002). Theorganizationofthispaperisasfollows.InSect.2wede- Furthermore,wheneverthedistancesbetweenthecomponents scribethetripleevolutionalgorithmdevelopedinthisworkandin intriplesystemsbecomecomparabletotheirradiitidaleffectsmay Sect.3weprovidedetailsoftwoexamplesystemsthatleadtoCO becomeimportant.Inourstudysucheffectsintheouterorbitcan WDmergersviatwo distincttriplechannels. Withthisalgorithm beneglected.However,intheinnerbinarysystemtidaleffectsmay weperformapopulationsynthesisstudyofwhichwediscussour become important even forlargesemi-major axeswithrespect to methodsinSect.4andpresenttheresultsinSect.5.Forthevarious theradii,a1/Ri(whereRidenotestheinnerbinarystellarradius),if COWDmergerchannelswefindweestimatetheexpectedtriple- theinnerorbitaleccentricityissufficientlyhigh.Inthiscaseatidal induced SNeIaratesand compare these tothebinary population torqueisinduced, eitherduetotheequilibriumtideordynamical synthesisstudyofClaeysetal.(inprep.)andtoratesinferredfrom tide(Zahn1977),thatallowsfortheexchangebetweenangularmo- observationsinSect.6.InSect.7werelateourresultstotheprevi- mentaofthespinandorbitintheinnerbinarysystem.Thisprocess ousworkofThompson(2011)anddiscusstheuncertaintiesofour continuesuntilcoplanarity,synchronizationandcircularizationare treatmentoftripleevolution.WeconcludeinSect.8. achieved(Hut1980).Itisdescribedquantitativelyintermsoforbit- averagedequationsfortheorbitalparametertimederivatives,a˙ , 1,TF e˙ andΩ˙ ,whereΩ isthespinfrequencyofinnerorbitstari 1,TF i,TF i andTFdenotestidalfriction. 2 HIERARCHICALTRIPLEEVOLUTIONALGORITHM Another effect we take into account isGW emission, which 2.1 Secularhierarchicaltripledynamics wemodelfortheinnerbinaryonly.Asaconsequenceoftheemis- sionofGWstheinnerbinaryiscircularizedandshrinks,whichis During Kozai cycles the mutual torque exerted by the inner and describedquantitativelybyorbit-averagedequationsfore˙ and outerbinaryorbitscausesanexchange ofangularmomentumbe- 1,GW a˙ . tweentheorbitswhiletheorbitalenergies,andthusthesemi-major 1,GW axes a , are conserved (e.g. Mardling&Aarseth 2001). Here we j useindex j = 1,2todenotequantitiespertainingtotheinnerand 2.2 Systemoffirst-orderdifferentialequations outer orbits respectively. As a result of the exchange of angular momentumtheorbitaleccentricitiese andthetotalmutual incli- We describe the secular Kozai cycles quantitatively in terms of j nationanglebetweenbothorbitsitotvaryperiodicallyontimescales coupled equations for g˙j,STD and e˙j,STD (where STD denotes sec- P ≫P ,whereP denotesorbitalperiod.Tolowestorderina /a ular three-body dynamics), that follow after orbital averaging of K 2 j 1 2 P isgivenby(Kinoshita&Nakai1999): thethree-body Hamiltonianexpressed inDelaunay’s action-angle K variables,applyingHamilton’sequationsofmotionandeliminating P =αP22m1+m2+m3 1−e2 3/2, (1) thelongitudesoftheascendingnodes(Harrington1968;Fordetal. K P m 2 1 3 (cid:16) (cid:17) 2000;Naozetal.2011).Weemployequationsforthesefourquan- where m and m denote the inner binary primary and secondary titiesaccuratetooctupoleorder,i.e.uptoandincludingthirdorder 1 2 mass respectively and m denotes the tertiary mass. The dimen- in a /a . In addition, an equation that expresses conservation of 3 1 2 sionless quantity α is of order unity and depends weakly on the totalangularmomentumdescribesthechangeofi . tot Triplepopulationsynthesisandwhitedwarfmergers 3 WethencombineallthephysicaleffectsdiscussedinSect.2.1 theinnerbinarysystemsuchthatitdoesnotfillitsRoche-lobein toformasystemofeightfirst-orderordinarydifferentialequations itsorbitaroundtheinnerbinarysystem,noritisinvolvedinaCE (ODEs)thatreads: phasewiththeinnerbinarysystem,ascenariowhichhasbeenex- ploredinearlierstudies(e.g.Iben&Tutukov1999).Thereforethe g˙ =g˙ +g˙ +g˙ +g˙ ; 1 1,STD 1,GR 1,tide 1,rotate latterprocessesarenotmodeledinthetriplealgorithm.  g˙2 =g˙2,STD;  aee˙˙˙θ˙121 ====eae˙˙˙121−,,,SST1TTFDD+;G+˙a˙e˙1(,1GG,TWF;++Ge˙1,GθW);+G˙ (G +G θ) ; (3) Eaolfqg.oE3rqIint.ihs3tmhs,oe.wlIvOnheDidocErhfdoseriosrtlhvtqoeeureidtrnueosruuasttrtiiienoffenpritonhofepneesaraytcscuhtoreentm,ivmeworefgesedtunieffscpeeeoroeafnnftthtiOhaeelDBesiEqonulausaottriilyoovnnecssr where θ≡ΩΩ˙˙21cos==(itoΩΩG˙˙t)121G,,aTT2FFnh,;d G1 j d1enote2s the o2rbit2al ang1ulair momen- ts1nip9ooe9tnc6cioo)fif.ucWtpahlleeelydhsaadwmveiesetihcghhntiheeecerdkaBrteocidhnitinahctreaelygrterrcasiutpaellltegsstsooiyffrfistttOhhememDOsEbDsaysE(cCsostomuohdlpveiuenedtri&nriongHuFttihionnerdedemwveaothrlaesulnh-. tum.Forexpressionsfortheprecessionquantitiesg˙1,GR,g˙1,tide and (2000),Blaesetal.(2002)andNaozetal.(2011)andfindverysim- g˙1,rotate werefertoBlaesetal.(2002),Smeyers&Willems(2001) ilarresultsfore1,itotanda1(whereapplicable)asfunctionoftime. and Fabrycky&Tremaine (2007) respectively; for the tidal evo- Wehavealsofoundnearexactagreementofthetimeevolutionof lution quantities e˙1,TF, a˙1,TF and Ω˙i,TF we refer to Hut (1981) and e1 and itot when comparing to analytical solutions which exist in fortheGWevolutionquantitiese˙1,GWanda˙1,GW werefertoPeters thequadrupoleorderapproximationandwiththefurtherrestriction (1964).ThemainassumptioninEq.3isthatallphysicalprocesses thatG ≫ G (Kinoshita&Nakai 1999).During each Binary c 2 1 that are described are independent such that the individual time timestepquantitiesnotincludedinthelefthandsideofEq.3such derivative termscanbe added linearly.Inaddition, intheexpres- as the inner binary masses and radii are assumed to be constant. sions for g˙1,tide, g˙1,rotate, e˙1,TF, a˙1,TF, Ω˙i,TF and θ˙ we assume copla- BecausetheBinary calgorithmusesadaptivetimestepstomatch narity of spin and orbit at all times, even though Kozai cycles in the rate of stellar and binary evolution it is always ensured that principleaffecttherelativeorientationsbetweenthespinandorbit thesequantitiesdonotchangesignificantlyduringeachiterationof angularmomentumvectorsandinturnamisalignmentofthesevec- thetripleevolutionalgorithm.Inaddition,ifneededthetriplealgo- torsaffectstheKozai cyclesthemselves(e.g.Correiaetal.2011). rithmdecreasestheBinary ctimestepsuchthattheinnerorbital Wejustifythisassumption bynotingthatforthemajorityofsys- angular momentumG does not change by more than 2%. Thus, 1 temsthatwestudytheorbitalangularmomentaofbothinnerand changestotheinnerorbitsemi-majoraxisandeccentricitydueto outerorbitsgreatlyexceedthespinangularmomentainmagnitude, secular three-body dynamics coupled with tidal friction and GW thereforethestellarspinscannotgreatlyaffecttheexchangeofan- emission are communicated tothe Binary calgorithm at appro- gularmomentumbetweenbothorbits. priatetimes. The expression for θ˙ in Eq. 3 is derived by differentiating TheversionofBinary cusedinthisworkenforcescircular- the relation that expresses conservation of total angular momen- izationattheonset of Roche-lobeoverflow (RLOF)andCEevo- tum, G2 = G2 +G2 +2G G θ, with respect to time and solv- tot 1 2 1 2 lution.Althoughthisassumptionisgenerallyjustifiedforisolated ing for θ˙ in terms of G and G˙ . Note that GW emission acts to j j binaries(Hurleyetal.2002),intriplesystemshigh-amplitude ec- change a and e and therefore affects the magnitude of G but 1 1 1 centricitycyclescouldbeinducedduringtheformerphase,inpar- not its direction. Although in principle tidal friction can also af- ticularifthetimescaleofthesecyclesiscomparabletoorsmaller fectthedirectionofG ifthestellarspinsandorbitarenotcopla- 1 thanthetimescaleof masstransferdrivenby RLOF.Anaccurate nar, with our assumption of coplanarity this is not the case. This treatmentof masstransfer ineccentricorbitsisbeyond thescope implies that, withGj = Lj 1−e2j, where Lj = Lj(aj,mi) isthe ofthiswork(seee.g.Sepinskyetal.2007)andwethereforechoose q orbital angular momentum in case of a circular orbit and which to disable the ODE solver routine whenever mass transfer driven is constant with regard to STD, the quantities G˙j are given by byRLOForCEevolutionensues.WhenevertheODEsolverrou- G˙ =− e / 1−e2 ·G ·e˙ . tineisactiveitisensuredthattidalevolutionandGWemissionin j j j j j,STD h (cid:16) (cid:17)i theinnerbinarysystemascalculatedbyBinary cdonotchange theinnerorbitalparameters.Thisisnecessarybecauseinthiscase 2.3 Couplingtobinaryalgorithm theseprocessesaretakenintoaccount bytheODEsolver routine throughtheequationsfore˙ anda˙ inEq.3.Inthismannerthecou- Thehierarchicaltripleevolutionalgorithmdevelopedinthiswork 1 1 plingofeccentricitycycleswithtidalfrictionandGWemissionin couples the system of differential equations Eq. 3 to the existing theinnerorbitiscalculatedconsistently. rapid binary population synthesis algorithm Binary c based on Hurleyetal. (2000, 2002) with updates described in Izzardetal. Iftheratioβ ≡ a /a islargetheeffectsofSTDarelikelyto 2 1 (2004, 2006, 2009) and Claeys et al. (in prep.). This binary star bequenched bydifferent processes intheinner binarysystem, in algorithmincludesstellarevolutionandavarietyofbinaryinterac- particularduetogeneral relativisticprecessioninverytightinner tionprocessessuchasmasslossduetostellarwinds,masstrans- binarysystems,asdiscussedinSect.2.1.Insuchsituationsitisnot ferandcommonenvelope(CE)evolution.WerefertoHurleyetal. necessarytosolveEq.3becausetheSTD-relatedtermsarenegligi- (2002)forfurtherdetailsofthisbinarypopulationsynthesisalgo- bleandtheinnerbinaryevolutioncanthereforebefullyhandledby rithm. The novel aspect of our approach is that the evolution of Binary c.Wethereforeintroduceacriterionthattemporarilydis- theinnerbinarysystemistakenintoaccountthroughthecoupling ablestheODEsolverroutineandleavestheinnerbinaryevolution to Binary c. We also take into account the evolution of the ter- toBinary cinsuchcases. Specifically, theroutineisdisabled if tiary,whichistreatedasbeingisolatedsuchthatitsmassmaybe β>10β ,whereβ isthevalueofβsuchthattheperiods crit,GR crit,GR computed from single stellar evolution. In thiswork weconsider associatedwithprecessionduetoSTD(Eq.1)andgeneralrelativity systemsinwhichatalltimesthetertiaryissufficientlydistantfrom (approximately givenbythereciprocal ofg˙ )areequal, which 1,GR 4 A.S. Hamers,O.R.Pols,J.S.W.Claeys and G. Nelemans wefindisgivenby(omittingnumericalfactorsoforderunity): quantities k /T and r2 are therefore calculated from precisely am,i i g,i thesameprescriptionasinBinary c,whichisrequiredforacon- 1/3 βcrit,GR≈ Gc2N!1/3a11/3 (m1m+3m2)2!1/3 (cid:16)11−−ee221(cid:17)1/2 sistenTthtreeactlmasesnict.al apsidal motion constant kam,i (i.e. not the ra- 2 tiok /T), which is required for evaluation of g˙ and g˙ (cid:16) (cid:17) am,i i 1,tide 1,rotate ≈5·102 a1 1/3 m3/M⊙ 1/3 (cid:16)1−e21(cid:17)1/3, (pSumteedyaesrsa&fuWnciltlieomnsof20m0a1s;sFaanbdrytcimkye&relTarteivmeationeth2e00Z7A),MiSs cforomm- (cid:18)AU(cid:19) (m1/M⊙+m2/M⊙)2! 1−e2 1/2 detailedstellarmodelsformetallicityZ=0.02calculatedbyClaret 2 (cid:16) (cid:17) (4) (2004).Therunofk withtimeisdeterminedbymeansoflinear am,i relations scaled to the time spent during each evolutionary stage wherecdenotesthespeedoflightandG denotesthegravitational N for all tabulated masses in Claret (2004). Subsequently, k is constant. am,i calculatedfor arbitrarymassby meansof linearinterpolation be- Outer orbital expansion due to mass loss in either inner or tweenadjacent massvalues. Furthermore, forlow-mass stars,i.e. outerbinarysystemsistakenintoaccountwiththeassumptionof m < 0.7M , theclassical apsidal motion constant istaken tobe fast,isotropicwinds(i.e.Jeansmode)suchthata (m +m +m ) i ⊙ 2 1 2 3 k = 0.14 corresponding ton = 3/2 polytropes as an estimate and e remain constant (Huang 1956, 1963). With this relation a am,i 2 forthese(nearly)fullyconvectivestars.Forhelium(He)MSstars new value of a iscomputed fromthechanges of m ,m andm 2 1 2 3 and WDs k is calculated as function of mass by means of in- duringeachtimestepofBinary c. am,i terpolationsoftabulateddatafromVila(1977).Lastly,forneutron Lastly,ateachBinary ctimestepthetriplesystemischecked starsk iscalculatedasfunctionofmassandradiusfromtheex- for dynamical stability by means of the stability criterion formu- am,i pressiongiveninHinderer(2008)andforblackholesk = 0as latedbyMardling&Aarseth(2001),includingtheadhocinclina- am,i appropriatefornon-spinningblackholes. tionfactor f =1−(0.3/π)i (withi expressedinradians).When- tot tot everβ≤β ,whereβ isgivenbythisstabilitycriterion,theSTD crit crit equations arenolonger strictlyvalidwedonot model thesubse- quentevolution.Thephaseofdynamicalinstabilitywouldneedto 3 TRIPLEEVOLUTIONEXAMPLES be computed with a three-body integrator code, which is beyond TodemonstratethetriplealgorithmpresentedinSect.2wedescribe thescopeofthiswork(seePerets&Kratter2012forarecentstudy indetailtheevolutionoftwotriplesystemsinwhichaninnerbi- oftriplesystemsthatbecomedynamicallyunstable). naryCOWDmergeroccursineitheratightcircularorbitorawide andhighlyeccentricorbit.Theinitialconditionsforthesetwosys- temsareselectedfromourfirsttriplepopulationsynthesissample 2.4 Fixedbinaryparametersandevaluationofvarious (TSM1,cf.Sect.4.2). physicalquantities Wedetailsomeofthefixedbinaryevolutionparametersusedinthis 3.1 Mergerinatightcircularorbit work.Wechoosequasi-solarmetallicity,i.e.Z = 0.02.Theinitial spin periods of the inner orbit components are computed from a Weconsideratriplesystemwithinitialparametersm =3.95M , 1 ⊙ formulagivenbyHurleyetal.(2000),whichisfittedfromdataof m = 3.03M , m = 2.73M , a = 19.7AU, a = 636.1AU, 2 ⊙ 3 ⊙ 1 2 therotationalspeedofMSstarsofLang(1992).TheCEparame- e = 0.23,e = 0.82,i = 116.0◦,g = 28.5◦ andg = 249.6◦. 1 2 tot 1 2 terαCE describestheefficiencyoftheconversionoforbitalenergy Fig.1showsa1,e1,β ≡ a2/a1,thetidalstrengthquantitykam,i/Ti intobinding energy withwhichtoshed theenvelope and itisset (Hut 1981; Hurleyetal. 2002) and i as a function of time. The tot tothecanonicalvalueofαCE = 1.InthisCEdescriptionanaddi- evolution is shown with both the triple algorithm enabled (triple tionalparameterλCEisrequired,whichisadimensionlessmeasure case; solid line) and disabled (binary case; dashed line), leaving of the relative density distribution within the envelope; here it is all inner binary evolution to Binary c. The latter binary case is determinedbymeansoffunctionalfitsasdetailedinClaeysetal. equivalenttothesituationinwhichnotertiaryispresent. (inprep.).AllotherparametersintrinsictotheBinary calgorithm Inthebinarycasethesemi-majoraxisincreasesatthetwomo- arealsosettoidenticalvaluesasinClaeysetal.(inprep.). mentswhenthebinarystarsevolvefromAGBstarstoCOWDs(at Inaddition,variousphysicalquantitiesarerequiredtoevalu- t ≈ 0.2Gyrandt ≈ 0.5Gyrrespectively)asaconsequenceofthe atetheright-handsidesinEq.3andintheremainderofthissec- associated wind mass loss. Here (i.e. in Binary c) it is assumed tionwedescribehowthesequantitiesareobtainedinourtripleal- thatthewindmasslossisfastandisotropicsuchthata (m +m ) 1 1 2 gorithm. The tidal evolution equations (i.e. expressions for a˙1,TF, remainsapproximately constant.Wheneitherstarintheinnerbi- e˙1,TF andΩ˙i,TF)containaparameterthatcapturestheefficiencyof narysystem enterstheRGBor AGBphase it develops aconvec- tidalfrictionanddepends onthestructureofthestar(Hut1981). tiveenvelopeanditsradiusincreasessignificantly.Thechangeof Thisparameteristheratiokam,i/Ti,wherekam,i istheclassicalap- theenvelope structureisreflectedbyasubstantial increaseofthe sidal motion constant of inner orbit star i and Ti a tidal friction tidal strength quantities kam,i/Ti as is shown in Fig. 1 (primary timescale.Wecomputekam,i/Tiaccordingtothesameprescription AGBphase). However, despitethelargekam,1/T1 andprimaryra- thatisused inBinary c(Hurleyetal.2002).Inthisprescription diusthere isno significant tidal frictionduring the primaryAGB adistinctionismadebetweenconvective,radiativeanddegenerate phasebecauseofthelargea andsmall(andconstant)e .Conse- 1 1 envelopesandkam,i/Ti forthesethreecasesiscomputedbasedon quently, in the binary case the CO WDs end their evolution in a resultsofRasioetal.(1996),Zahn(1977)andCampbell(1984)re- wideandnon-interactingbinarywitha ≈84AU. 1 spectively.Furthermore,thegyrationradiir2 arerequiredforthe Inthetriplecasehigh-amplitudeKozaicyclesareinduceddur- g,i evaluationofΩ˙ andarecomputedwithaprescriptioninwhich ingtheMSintheinnerorbitbythetertiarywithmaximaofe ≈ i,TF 1,max thestarsaresplitintotwoparts,consistingofthecoreandtheenve- 0.9.Octupoleordertermsareimportantbecause|ǫ |≈10−2.Nev- oct lope,detailedinHurleyetal.(2000)andHurleyetal.(2002).The ertheless,duringtheMStheeccentricityisnothighenoughtodrive Triplepopulationsynthesisandwhitedwarfmergers 5 20 100 binarycase 10 15 triplecase U 1 binarycase AU A (cid:144) 10 (cid:144)1 triplecase a1 a 0.1 5 0.01 0.001 0 0.2 0.5 1.0 2.0 5.0 10.0 220.0 220.5 221.0 221.5 222.0 222.5 223.0 t(cid:144)Gyr t(cid:144)Myr 0.0 106 -0.2 105 L -0.4 1 104 -e 1 Β H0 -0.6 1 1000 og l -0.8 binarycase 100 triplecase -1.0 10 0.2 0.5 1.0 2.0 5.0 10.0 220.0 220.5 221.0 221.5 222.0 222.5 223.0 t(cid:144)Gyr t(cid:144)Myr -8 -0.4 -10 primary H(cid:144)LLs -12 secondary -0.5 Ti L (cid:144) ot Hkam,i -14 Hcosit -0.6 10 -16 g o -0.7 l -18 -20 -0.8 220.0 220.5 221.0 221.5 222.0 222.5 223.0 220.0 220.5 221.0 221.5 222.0 222.5 223.0 t(cid:144)Myr t(cid:144)Myr Figure1.Severalquantitiesofinterestintheevolutionofthefirstexamplesystem(Sect.3.1).Shownasafunctionoftimearea1,e1,β ≡ a2/a1,kam,i/Ti anditot.Intheplotsinthefirstcolumnofthefirsttworowstheentireevolutionisshown;theotherplotscorrespondtotheendoftheprimarycorehelium burningphase,theprimaryAGBphaseandstartoftheprimaryCOWDphase.Solidline:triplecase;dashedline:binarycase(whereapplicable).Intheplot ofkam,i/Ti onlythetriplecaseisshown(thebinarycaseisverysimilar);inthisplotthesolidlineappliestotheprimary(i=1)andthedashedlinetothe secondary(i = 2).Notethattheevolutionisnotfullysampledinallplots,causingseveralkinks(inparticularintheplotsfore1 anditot)–moredetailed calculationsareperformedinternallyintheODEsolverroutinebutarenotshownhere(cf.Sect.2.3). significanttidalfrictionbecausethetidalstrengthquantityisvery increaseintheeffectivenessof orbitalshrinkage isduetotheex- small (k /T ∼ 10−18s−1) as aconsequence of theradiative en- pansionoftheprimaryfromaradiusofR ≈49R toR ≈497R am,i i 1 ⊙ 1 ⊙ velopes in the MS stars. During the primary RGB phase the pri- between t = 220Myr andt = 222.5Myr. Consequently a isre- 1 mary possesses a convective envelope (k /T ∼ 10−8s−1) but ducedtoa ≈ 7AUandtheorbitiscompletelycircularized.Note am,1 1 1 e isnothighenoughtotriggersignificanttidalfriction.However, thatforcompletecircularizationtooccurthedurationofthephase 1 tidalfrictiondoesbecomeeffectiveduringtheprimaryAGBphase inwhichk /T issubstantialmustbesufficientlylongcompared am,1 1 starting at t ≈ 220.6Myr and circularizes the inner orbit during totheKozaiperiodP (Eq.1),whichisthecaseforthisexample K thetimespanoffiveKozaicycles,wheresignificantorbitalshrink- system.Inothercasesoftriplesystems,however,P canbemuch K ageoccursateccentricitymaxima.Notethatduringthisphasethe 6 A.S. Hamers,O.R.Pols,J.S.W.Claeys and G. Nelemans 0 3000 binarycase -2 2000 triplecase L 1 e AU 1500 H-1 -4 (cid:144)a1 og10 binarycase 1000 l -6 triplecase -8 0.2 0.5 1.0 2.0 5.0 10.0 0 1 2 3 4 5 6 7 t(cid:144)Gyr t(cid:144)Gyr 1.0 30 0.5 L ot Β Hsit 0.0 o 20 c -0.5 15 -1.0 0.2 0.5 1.0 2.0 5.0 10.0 0 1 2 3 4 5 6 7 t(cid:144)Gyr t(cid:144)Gyr Figure2.QuantitiesofinterestforthesecondexamplesysteminwhichaninnerbinaryCOWDmergeroccursinawideandhighlyeccentricorbit. longer than the duration of the RGB/AGB phases, thus avoiding unlikely that the primary RGB and AGB phases (which occur at strongtidalfrictioneveniftheeccentricitymaximaarehigh. t≈0.3Gyrandt≈0.4Gyrrespectively)coincidewitheccentricity Shortly after KCTF during the primary AGB phase the pri- maxima. Suchacoincidence isindeed not thecase inthesecond mary has swelled up to R ≈ 611R , fills its Roche-lobe and examplesystem. 1 ⊙ invokes CE evolution, shrinking the orbit further to a1 ≈ 1AU AstheprimaryandsecondaryevolvetoCOWDswithoutin- (t ≈222.6Myr).Duringthisprocesstheratioβincreasessubstan- teracting the associated wind mass loss widens the orbit to a ≈ 1 tially to β ≈ 1321, which is large enough for general relativistic 2600AU.Although thisinner binarywindmasslossincreasesa 2 precession to fully quench any subsequent Kozai cycles. Conse- aswell,theratioβ = a /a decreaseswiththeassumptionoffast 2 1 quently, theinclinationangleitot isfrozentoitot ≈ 128◦.Thepri- isotropicwindssuchthata1(m1+m2)anda2(m1+m2+m3)arecon- mary emerges from the CE as a CO WD and the secondary as a stant(inthisexamplem alsoremainsconstant).Consequently,the 3 MSstar.Att ≈ 0.5Gyrthesecondary evolvestoanAGBstarit- ratioβdecreasestoβ≈18afterthesecondaryhasevolvedtoaCO self, invoking a second CE phase. A close CO WD binary with WD.AsthetertiaryevolvestoaCOWDandsubsequently loses a1 ≈ 0.01AU emerges from the CE. Due to GW emission the massatt ≈ 1.5Gyra2 increaseswhilea1 staysconstant, thusin- system subsequently merges at t ≈ 13.2Gyr in a circular orbit. creasingβagain.ThelatterprocessalsoincreasestheKozaiperiod. ThecombinedCOWDmass,1.45M⊙,exceedstheChandrasekhar Aftert ≈ 1.5Gyrthemaximumvaluesofcos(itot),whicharestill massMCh ≈1.4M⊙,thereforethemergercouldpotentiallyleadto negative(retrogradeorbit),slowlyincreaseandthecorresponding aSNIaexplosion(cf.Sect.6.2). eccentricitymaximaincreaseaswell.Ascos(i )approaches0the tot eccentricity maxima reach very high values until at t ≈ 7.0Gyr theorbitsbecomeprograde,whichisassociatedwithe reachinga 1 3.2 Mergerinawideeccentricorbit valueof1−e ≈ 10−7.4.Thisishighenoughtotriggeranorbital 1 Thesecondexamplesystemhasinitialparametersm1 = 3.26M⊙, collisionbetweenthetwowhitedwarfs,eventhougha1 ≈2600AU m =2.00M ,m =1.39M ,a =716.6AU,a =2.012·104AU, isverylarge.SuchacollisioncouldpotentiallyresultinaSNIaex- 2 ⊙ 3 ⊙ 1 2 e =0.78,e =0.64,i =93.4◦,g =150.0◦andg =151.2◦.The plosionasinthepreviousexample(cf.Sect.6.2),butnotethatthe 1 2 tot 1 2 evolutionofa ,e ,β≡a /a andi forthissystemasfunctionof natureofthemergerpriortoapossibleSNeIaeventisverydiffer- 1 1 2 1 tot timeisshowninFig.2.Themaindifferenceininitialparameters ent. inthisexamplesystemcomparedtothepreviousexampleisthata Onemightwonderwhetherinthissystemtidalfrictionissig- 1 anda arelarger.Theirratioβiscomparable,however.Becauseof nificant during close periastron passages before the collision oc- 2 thelargera theKozaiperiod P ≈ 100Myrismuchlongerthan curs.Forthisreasonwehaverecalculatedthissystemwiththetidal 2 K inthefirstexample(P ≈0.3Myr).Consequently,itismuchmore strength quantities k /T for radiative and degenerate damping K am,i i Triplepopulationsynthesisandwhitedwarfmergers 7 multipliedbyadhocfactorsof103 and106 toartificiallyincrease 12 theeffectiveness of tidal friction.Evenwiththesemultiplications Tokovininsample wefindthattheoutcomeisidentical,showingthattidalfrictionis 10 ns(4syoestetiPemSmOepscPotiUr.nt7aLt.nh2Ate)Ti.npIotOhpiNuslasSytYisotNnemTsy.HnWEtheSehIsSaisv:esMtpuEedryTfoaHrnmOdeDhdaSsviemfiolaurntdesntosfeoffreacltl H(cid:144)LlogPd210 2468 ×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× Before presenting the results of our population synthesis study × (Sect.5)wediscusshereourselectioncriteriaandsamplingmeth- 0 odswhichareessentialingredientsforsuchastudy. 0 2 4 6 8 10 log HP (cid:144)dL 10 1 4.1 Selectioncriteria 12 TSM1 Becausewearemainlyinterestedininnerbinarymergersinvolving 10 bsamteaC1trwOoefevWemonl1Duf,uotiwro=men.i6tna.T5gkheMaeaC⊙ln,oOwuwpWehpriecDlrhimlaainmiptdpiotraofonxfmiOtm1hNeaisteienWlsnyeetDcrtaobipnitnmutarh1re,eylsc=ptahrsie1me.b0aoorfMuysn⊙imdnagafroslyesr H(cid:144)LPd210 68 ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× anylowermasswillnotproduceasignificantnumberofCOWDs og withinaHubbletimeforametallicityofZ =0.02.Inadditionwe l 4 focus on triplesystems withwide inner binary systems such that 2 theinner binarystarsdonotinteract duringtheirevolutioninthe absenceofatertiary.Takingintoaccountpossibletidalfrictionthis 0 canbeachievedbyrequiringthattheinitialinnerorbitsemi-latus 0 2 4 6 8 10 rectuml1 ≡a1 1−e21 satisfiesl1 >l1,l ≡12AU. log10HP1(cid:144)dL sOpnts(o2yhreue0sordrts0woeies8rmOrsnee)btlcnsi,etitthneclawiyentltpihhoposotienaehcsbmrshe(sciiP(cid:16)oebrprii1rdsleltv,eesgelPeriimbddkmi2aeae)eet-tlso(cid:17)twadyahfbimeofabeiredngetponwnrclmtaeeaolemorueooogsnb.bfeb1iA0snts(o7aepaPcif2rernvl1y5cte/ehttasddrhetryor)eistsdpp=rtciiaielnffiopuem2iplctceiissiauactynsylwlsytdatytoisnretlifthdmoopesgfml1syve1d.sis00spoet.(euPtaf<Femar1ciaTl/gmstmodbi.nw1k)ie3/ngoti=Maetv(hsrrtiu⊙4isoniecpniiis<nhss)-. H(cid:144)LogPd210 110268 TSM2 ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× ×××× 6.5and l > 12AU decrease thenumber of systemsfrom725 to l 4 1 165, where a value of log (P /d) = 4 is taken to represent the 10 1 dividing value l = 12AU. The number of remaining systems 2 1,l (indicatedinthetopfigureofFig.3by dots) isnot largeenough 0 for direct sampling, therefore wechoose toutilize themethod of 0 2 4 6 8 10 MonteCarlosamplingfromobservation-baseddistributions.Inor- log HP (cid:144)dL der togaininformationontheuncertaintiesofour resultsweuse 10 1 twodistinctsamplingmethodsthataredescribedbelow. Figure 3. Inner (P1) and outer (P2) orbital periods of triple systems of theobservedtriplesampleofTokovinin(2008)(top)andthetwosynthetic triplesamplesdescribedinSect.4.2(middleandbottom).Intheobserva- 4.2 Samplingmethods tionalsamplesystemssatisfyingourselectioncriteriaaredenotedwithlarge (red)dots(•).Inthesyntheticsamplesthelargedots(•)denotethesystems 4.2.1 TSM1 that undergo an early MS inner binary mergerordestabilization and are Thefirsttriplesamplingmethod(TSM1)isbasedonthesupposi- thereforenotlikelytobeobserved.ThelinecorrespondstoP2/P1=5and tionthatahierarchicaltriplesystemiscomposedoftwouncorre- representsthetypicalcriticalvaluefordynamicalstability. latedbinarysystems.Themainadvantageofthisapproachisthat thestatisticsofbinaryparametersareknownwithgreatercertainty thanthoseoftriplesystems.Aprimarymass1.0 < m /M < 6.5 e ande fromathermaldistributiondN/de =2e asisappropriate 1 ⊙ 1 2 j j issampledfromaKroupaetal.(1993)initialmassfunction(IMF), forP >103dbinaries(Kroupa&Burkert2001).Twosemi-major j i.e.dN/dm ∝ m−2.70.Subsequently,twomassratios,q ≡ m /m axesarethensampledfromadistributionthatisflatinlog (a ),the 1 1 1 2 1 10 j and q ≡ m /(m +m ), are sampled independently from a uni- smallerofwhichisdesignateda andthelargerofwhichisdesig- 2 3 1 2 1 form distribution such that 0 < q ≤ 1, m > 0.01M and nateda .Thelowerandupperlimitsa anda inthedistributionof j 2 ⊙ 2 l u m > 0.01M ,consistentwithClaeysetal.(inprep.).Thelower thesemi-majoraxesarea =5R anda =5·106R forbothinner 3 ⊙ l ⊙ u ⊙ limitonthestellarmassesisinaccordancewiththelimitfoundby and outer orbits (Kouwenhovenetal. 2007). From these systems Kouwenhovenetal.(2007).Thesecondaryandtertiarymassesare werejectthosethatdonotsatisfyl = a 1−e2 > l = 12AU. 1 1 1 1,l subsequentlycomputedfromthesesampledmassratios.Wesample Lastly, we sample the initial orbital inclin(cid:16)ation(cid:17)angle i from a tot 8 A.S. Hamers,O.R.Pols,J.S.W.Claeys and G. Nelemans Tokovinin TSM1 TSM2 0.25 TokovininsampleHselectedL Mean SD Mean SD Mean SD m1/M⊙ 2.36 1.23 1.85 1.01 2.17 1.24 0.20 TSM1 m2/M⊙ 1.69 0.94 0.93 0.79 1.09 0.93 TSM2 m3/M⊙ 1.53 1.39 1.36 1.23 1.59 1.43 F 0.15 log10(P1/d) 5.13 0.73 5.47 0.81 5.77 1.29 PD log10(P2/d) 7.04 0.76 7.87 0.75 8.28 1.61 0.10 Table1.Meanandstandarddeviation(SD)ofmassesandorbitalperiods 0.05 oftheTokovinin(2008)samplesatisfyingtheselectioncriteria(Sect.4.1) andofbothsampledtriplepopulationsTSM1andTSM2. 0.00 1 2 3 4 5 6 log HP (cid:144)P L distribution that is uniform in cos(i ) with −1 < cos(i ) < 1 10 2 1 tot tot and the initial arguments of periastron of both orbits, g and g , 1 2 Figure4.Probabilitydensityfunctions(PDFs)ofP2/P1fortheTokovinin fromauniformdistributionwith0<gj <2π.Fromthetriplesys- sample with ourselection criteria applied (solid line), the TSM1sample tems obtained in this manner we subsequently reject the systems (dashedline)andtheTSM2sample(dottedline).FortheTokovininsample, that are not dynamically stable based on the stabilitycriterion of errorsbasedonPoissonstatisticsareindicated. Mardling&Aarseth(2001),consistentwithourtripleevolutional- gorithm(cf.Sect.2.3). theTokovininsamplewithourselectioncriteriaappliedarecom- paredtothetwosampledpopulations. Becauseofthesteepslope of −2.70 in the TSM1 IMF the mean masses in TSM1 are small 4.2.2 TSM2 comparedtothoseinTSM2,forwhichaSalpeter-likeslope(-2.35) In the second triple sampling method (TSM2) we use the mul- applies for m > 1.0M . Compared tothe Tokovinin sample the 1 ⊙ tiple system recipe developed by Eggleton (2009). This recipe massesofTSM1andTSM2appeartobeonthelowside,butitis is designed to reproduce the properties of a set of 4558 stellar important tonotethattheTokovinin sampleismagnitude-limited systems with Hipparcos magnitude brighter than 6 collected by ratherthanvolume-limited,thereforeitisstronglybiasedtowards Eggleton&Tokovinin (2008). It is followed until the second bi- moremassive systems. Thedistributions of the orbital periods of furcation,effectivelyrestrictingtomultiplesystemsthataretriple. TSM1andTSM2aresimilar,theorbitalperiodsbeingonaverage Theparametersgivenbytherecipearem1,m2,m3,P1andP2.The somewhathigherinTSM2comparedtothoseinTSM1.Theorbital primarymassdistributioninthisrecipeisdesignedtoresemblethe periodsoftheTokovininsample,notablytheouterorbitalperiods, Salpeter IMF (dN/dm1 ∝ m1−2.35) at large masses and turns over aresmallerthanthoseinthesampledpopulations.Thisisalsolikely for masses below 0.3M⊙. The mass ratio distribution is approxi- affectedbyobservationalbias,becauseverylongorbitalperiodson matelyflatandtheP2 distributionhasabroadpeakaround105d. theorderof107 daysorlongeraregenerallyverydifficulttomea- WerefertoEggleton(2009)forfurtherdetails.Thecorresponding sure. semi-major axes are computed from Kepler’s third law. The pa- Lastly, Fig. 4 compares the distributions of the ratio P /P 2 1 rametersnotprescribedbytherecipeareitot,ej andgj,whichare forthethreepopulations.ForTSM1andTSM2thesedistributions sampledfromthesamedistributionsasinTSM1above.Similarly aresimilar. Thedistribution of P /P of the Tokovinin sample is 2 1 tothemethodinTSM1systemsarerejectedifl1≤l1,l =12AUand peakedtowardssmallerratios,althoughhereobservationalbiasfor iftheydonotsatisfythestabilitycriterionofMardling&Aarseth largeperiodratiosmaybeimportant.Notealsothelargeerrorbars (2001). intheTokovininsampleasaconsequenceoftherathersmallnum- ber of systems in the Tokovinin sample that satisfy our selection criteria(165). 4.2.3 Propertiesandcomparisons Hereweelaborateonsomeofthepropertiesofthesampledandob- 4.2.4 Samplingprocedure servedpopulations.Fig.3(middleandbottom)showsthe(P ,P )- 1 2 diagramforasmallsample(360systems)oftheTSM1andTSM2 WesampleN =2·106triplesystemsforbothpopulationsTSM1 calc populations.TheupperboundariesofP andP inbothpopulations and TSM2. Using the triple algorithm described in Sect. 2 these 1 2 are quite different. In TSM1 P and P are limited by a and a , systemsareevolvedfromt=0,withallthreecomponentsstarting 1 2 l u whereasinTSM2suchsharpboundariesdonotexist.InTSM2the asZAMSstars,toaHubbletimet = 13.7Gyr.Inordertoretain H upperboundaryofP isdependentonP becauseinthissampling sufficientresolutioninhigher-masssystems(2.0 < m /M < 6.5) 2 1 1 ⊙ methodthelargestratioP /P isgivenbyP /P =2·106.These wesplitbothTSM1andTSM2intotwopartsof1·106systems,one 2 1 2 1 wide outer binary systems arevery rare, however. Fig.3 (middle with1.0< m /M < 2.0andonewith2.0 <m /M < 6.5.Inthe 1 ⊙ 1 ⊙ andbottom)alsoshowsthesystemsthataccordingtoourtripleal- resultspresentedinSect5systemsineachpartaregivenappropri- gorithmexperienceamergerintheinnerbinarysystemoradesta- ateweightsdeterminedbythemassfunctionofm toaccountfor 1 bilizationearlyontheMS(cf.Sects.5.1.1and5.3).Suchsystems thefact that theactual number of systemsoccurring inthe1.0 < arenotlikelypartofanobservedsampleoftriplesbecauseoftheir m /M < 2.0rangeislargerthanthatinthe2.0 < m /M < 6.5 1 ⊙ 1 ⊙ shortlifetimes.Becauseoftheirsmallnumber,removingthesesys- range(seeAppendixAfordetails).Severalkeyeventsintheevolu- tems does not significantly affect the distributions of masses and tionarekepttrackof,mostnotablytheonsetofsignificantKCTF, orbitalperiods(themeanvaluesareaffectedbytypically1%). CE evolution, a merger in the inner binary system and a desta- InTable1thedistributionsofthemassesandorbitalperiodsof bilization of the triple system. We thus obtain a catalogue of the Triplepopulationsynthesisandwhitedwarfmergers 9 k Description includesbothCEevolutiontriggeredinhighlyeccentricorbitsand CEevolutiontriggeredbyRLOF(incircularorbits).Thefraction 0 MS(m.0.7M⊙) f includes the possibility for multiple phases of CE evolution. 1 MS(m&0.7M⊙) CE Lastly,thequantity f isdefinedasthefractionofsystemsineach 2 Hertzsprunggap(HG) EC 3 redgiantbranch(RGB) channelinwhichamergeroccursduetoaneccentriccollision.We 4 coreheliumburning(CHeB) definesuchacollisiontooccur iftheinner binaryperiastrondis- 5 earlyasymptoticgiantbranch(EAGB) tanceissmallerthanorequaltothesumoftheinnerbinaryradii, 6 thermallypulsingAGB(TPAGB) i.e.ifa (1−e )≤R +R andwheninthecorrespondingcircular 1 1 1 2 7 nakedheliumstarMS(HeMS) casetherewouldnotbeacollision,i.e.a >R +R . 1 1 2 8 nakedheliumstarHertzsprunggap(HeHG) AnimportantresultofoursimulationsshowninTable3isthat 9 nakedheliumstargiantbranch(HeGB) in≈ 8%ofallsystemsaninnerbinarymergeroccurs,in≈ 6%of 10 heliumwhitedwarf(HeWD) all systemsno inner binary merger occurs but the orbit isshrunk 11 carbon-oxygenwhitedwarf(COWD) significantly and that ≈ 10% of all systems become dynamically 12 oxygen-neonwhitedwarf(ONeWD) unstableatsomepointintheevolution.Thismeansthatin≈24% 13 neutronstar(NS) 14 blackhole(BH) ofallsystemsthetertiarysignificantlyalterstheinnerbinaryevo- 15 masslessremnant lution whereas in the absence of the tertiary there would not be suchinteraction.Takingintoaccountthatsystemsinwhichthein- Table2.DescriptionofthedifferentstellartypesusedinFig.5.Identicalto nerbinarycomponentsmergeduringtheMS(≈4%)orinwhicha thetypesusedinHurleyetal.(2002). dynamicalinstabilityoccursduringtheMS(≈ 4%)arenotlikely to be observed (cf. Sect. 5.1) this estimate is reduced by ≈ 8%. This implies that the tertiary significantly alters the inner binary evolutionary outcomes where we distinguish between three main evolutionin≈ 16%/(1−0.08) ≈ 17%ofthe“observable”triples, channels:innerbinarymergers,noinnerbinarymergersandtriple i.e.triplesinwhichthehierarchicalstructureisnotdisruptedinan destabilizations.Inaverysmallnumberofsystems(55and44for earlystageintheevolution. TSM1andTSM2respectively)thecalculationoftheevolutioncan- A second important feature evident from Table 3 is that the notbecompletedduetoconvergenceerrorsintheODEsolverrou- resultsforTSM1andTSM2aregenerallysimilar,eventhoughthey tine(cf.Sect.2);thesesystemsareexcludedfromtheresultsbelow. have been sampled by different methods. To provide insight into InthesecasestheerrorsoccurjustpriortoaMSmergerorRGB+ the dependence of the probabilities of the main channels on the MSmerger.Becausetheaffectedsystemsconstituteatinyfraction initialparametersweshowinFig.6theseprobabilities(expressed ofallsystemsforwhichaMSmergerorRGB+MSmergerapplies asfractions)asfunctionoftheinitialparametersq ≡ m /m ,a , 1 2 1 1 thisdoesnotaffectourconclusions. βandi .WewillrefertothisfigureandTable3whendiscussing tot thethreemainchannelsinmoredetailbelow. 5 POPULATIONSYNTHESIS:RESULTS 5.1 Innerbinarymergers Wepresentthemainchannelsthatwefindinourpopulationsynthe- As a consequence of high-amplitude eccentricity cycles induced sisstudyinTable3,wherethefirsttwocolumnsshowtheprobabil- intheinnerorbitthecomponentsintheinnerbinarymayatsome itiesP(inpercent)ofthemainchannelsandseveralsubchannels pointmerge,eitherdirectlybecauseofanorbitalcollision(included ofbothsampledpopulationsTSM1andTSM2.Inadditiontothis inTable3inthefraction f )orindirectlyasaresultofstrongtidal table,Fig.5showsmoredetailedinformationonthelikelihoodof EC frictioninducedbyhigheccentricity,possiblyinvokingCEevolu- thedifferentcombinationsofinnerbinarymergerswherethestellar tion.Inthelattercasetheinnerbinarycaneithermergeduringthe typesusedaredefinedinTable2.Apost-MSmergerisdefinedasa CEorsurvivetheCEinatightorbitand(possiblyafteranotherCE mergerbetweenapost-MS(non-compactobject)primaryandany invokedbythesecondary)subsequentlymergeasaresultoforbital secondary,whereasacompactobjectmergerisdefinedasamerger energy loss due to GW emission. We separate further discussion betweenacompactobjectprimaryandanysecondary.Inaddition into mergers occurring on the MS, after the MS and mergers in- totheprobabilitiesofthechannelsinformationontheoccurrence volvingaprimarycompactobject. of KCTF, CE evolution and the nature of the merger (i.e. occur- ring in a circular or highly eccentric orbit) is given by means of thequantities f , f and f . Wedefine f asthefraction KCTF CE EC KCTF 5.1.1 MSmergers ofsystemsineachchannelinwhich|e˙ | > 10−18s−1 (suchthat 1,STD thereisasignificantchangeine duetoSTDinaHubbletime)and As Table 3 demonstrates most mergers occur on the MS and for 1 0.1|e˙ |<|e˙ |<|e˙ |(i.e.e˙ ande˙ areofcomparable almostallofthelatterthemergeroccursthroughaneccentriccol- 1,TF 1,STD 1,TF 1,STD 1,TF orderofmagnitude).Thiscanhappenatvariouspointsintheevolu- lision. In the latter case −4 . log (1−e ) . −5 just prior to 10 1 tionandisstronglyinfluencedbytheenvelopestructureoftheinner merger.IntheBinary calgorithmitisassumedthattheresultof binary stars because thisdetermines the effectiveness withwhich themerger isasingleMS starwithmassm +m ;depending on 1 2 tidalenergycanbedissipated.Inparticularifanyinnerbinarystar thelattervaluethefinalremnantiseitheraCOWD(mostcases), possessesaconvectiveenvelope(mostnotablylow-massMS,HG, an ONe WD or a NS (following a core-collapse supernova). We RGBandAGBstars)strongtidalfrictionislikelyife isalsohigh find that octupole terms in the STD are very important for MS 1 duetoKozaicycles.InTable3adistinctionismadebetweenthe mergers,whichisreflectedbythelargevaluesof|ǫ |(cf.Eq.2) oct typesofthestarinwhichtidalenergyisdissipatedfortheduration prior to merger for these systems; the latter quantity is narrowly ofKCTF.Furthermore, f isdefinedasthefractionofsystemsin distributedaround|ǫ | ≈ 10−1.5.Thiscanbeattributedtoacom- CE oct eachchannelinwhichCEevolutionisatsomepointinvoked.This binationofinitiallysmallsemi-majoraxisratiosβ ≡ a /a ,small 2 1 10 A.S.Hamers, O.R.Pols,J.S.W.Claeys and G.Nelemans 1 1 2 2 100 3 3 4 4 5 5 k1 6 k1 6 7 7 8 8 9 10-6 9 10 10 11 11 12 12 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12 k k 2 2 Figure5.Left:thenumberofsystemsinTSM1thatexperienceaninnerbinarymergerforeachoccurringcombinationofstellartypesjustpriortomerging, normalized tothetotalnumberofmergersystems(thatconstitute ≈ 8%ofallsampledtriplesystems).Right:thenumberofsystemsinTSM1inwhich themergeroccursduetoaneccentriccollision,againnormalizedtothetotalnumberofmergersystems.Thestellartypesareidenticaltotheonesusedin Hurleyetal.(2002)andarerepeatedforconvenienceinTable2. inner binary mass ratiosq ≡ m /m and high outer orbit eccen- 1.0 1 2 1 tricitiese . Fig. 6 indeed shows that the fraction of MS mergers, 2 fMSmerge, decreases significantly withincreasing q1 and is peaked 0.8 MSmergers towards small β around 0.5 . log10(β) . 2.0. The MS mergers MSdestabilizations are also characterized by high initial inclination angles (cf. Fig. 0.6 6).Notethatamaximumin f occursatinclinationsslightly F MSmerge D aboveitot =90◦,i.e.forretrogradeorbits,whichisconsistentwith C 0.4 Shappee&Thompson (2012) who find an asymmetry in cos(i ) tot towardsnegativevaluesfortheprobabilityoftheeccentricKozai mechanism to occur. With regards to the dependence of f 0.2 MSmerge ona itmightbeexpectedthatMSmergerslikelyoccurforsmall 1 initiala asatighterorbitmakesaneccentriccollisionmorelikely. 0.0 1 However,wefindthatMSmergersonaveragehavelargeinitiala -6 -5 -4 -3 -2 -1 0 1 withamaximum of fMSmerge around a1 ≈ 102.8AU ≈ 6·102AU log10Ht(cid:144)ΤMSL (cf.Fig.6).Thisimpliesthattheeffectofincreasingtheeccentric- itymaximawithdecreasingβ(andthusincreasinga )isstronger Figure7.CumulativedistributionsforTSM1ofthetimestofMSmerger 1 thanthe effect thateccentric mergersbecome more unlikely with (solid line) and MS destabilization (dashed line) normalized to the MS increasinga1ife1werefixed. timescale,estimatedasτMS=10(m1/M⊙)−2.8Gyr. As Table 3 shows, for a small fraction of MS mergers (i.e. about0.009ofallMSmergersforTSM1)themergerdoesnotoc- sionoccursduringthefirstfewKozaicycleswhichtypicallyhave curduetohigheccentricity,i.e.themergeroccursinacircularor- timescalesofafewMyr,asmallfractionoftheMSlifetimeofthe bit.InthesecasesKCTFisresponsibleforstrongorbitalshrinkage primarystarsconsidered inthetriplesample.Thetimeofmerger leadingtoatightinnerorbit,eventuallyresultinginamergerina is thus mainly determined by the Kozai period P ∝ (P /P )P circularorbitduetoGWemissionwithinaHubbletime.Thefrac- K 2 1 2 andhencetheorbitalperiods.Weindeedfindthatthedistribution tionofMSmergersystemsinwhichthisoccurs(0.009forTSM1) ofthetimesofMSmergersisverysimilartotheinitialdistribution isindeedcomparablewiththefractionofMSmergersystemsthat of the Kozai periods. The fact that most MS mergers occur early satisfy our KCTF criterion on the MS (0.010 for TSM1); this is on the MS makes it unlikely to observe these systems as triples. alsothecaseforTSM2.WefindthatsuchstrongKCTFontheMS AsimilarconclusionappliestotheMSdestabilizations(Sect.5.3) occursonly inlow-massinner binarysystems(m < 1.25M )in 1 ⊙ thathappenevenearlier. whichthecomponents possessconvective envelopes (i.e.thesce- ArelevantquestionfortheMSmergersiswhetherthemerged narioofFabrycky&Tremaine2007). MSremnant couldevolvetoabluestraggler star,i.e.astarmore The eccentricMS mergers, which constitute thebulkof MS luminousthanstarsattheMSturn-offpoint.Thistriplebluestrag- mergers, typically occur early on the MS, with the majority of gler scenario has been proposed by Perets&Fabrycky (2009). If mergers(about70%)occurringwithin10%oftheprimaryMSlife- theMSstarsmergeveryearlyintheevolutionthentheageofthe time (Fig. 7, solid line). This is because in many cases the colli- merged MS star would appear to be consistent withitsmass and

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Preview Population synthesis of triple systems in the context of mergers of carbon-oxygen white dwarfs

Mon.Not.R.Astron.Soc.000,000–000(0000) PrintedDecember11,2013 (MNLATEXstylefilev2.2) Population synthesis of triple systems in the context of mergers of carbon-oxygen white dwarfs 3 A.S. Hamers1⋆, O.R. Pols2, J.S.W. Claeys2 and G. Nelemans2,3 1 1LeidenObservatory,LeidenUniversity,POBox9513,NL-2300RALeiden,TheNetherlands 0 2DepartmentofAstrophysics/IMAPP,RadboudUniversityNijmegen,POBox9010,NL-6500GLNijmegen,TheNetherlands 2 3InstituteforAstronomy,KULeuven,Celestijnenlaan200D,3001Leuven,Belgium n a J AcceptedforpublicationinMNRAS2013January7.Received2013January7;inoriginalform2012November21 8 ] ABSTRACT R Hierarchical triple systems are common among field stars yet their long-term evolution is S poorlyunderstoodtheoretically.InsuchsystemsKozaicyclescanbeinducedintheinnerbi- h. narysystemduringwhichtheinnerorbiteccentricityandtheinclinationbetweenbothbinary p orbits vary periodically. These cycles, combined with tidal friction and gravitational wave - emission,cansignificantlyaffecttheinnerbinaryevolution.Toinvestigatetheseeffectsquan- o titativelyweperformapopulationsynthesisstudyoftriplesystemsandfocusonevolutionary r t pathsthatleadto mergersofcarbon-oxygen(CO)whitedwarfs(WDs), whichconstitutean s important candidate progenitor channel for type Ia supernovae (SNe Ia). We approach this a [ problem by Monte Carlo sampling from observation-based distributions of systems within the primary mass range 1.0−6.5M⊙ and inner orbit semi-major axes a1 and eccentricities 1 e satisfyinga 1−e2 >12AU,i.e.non-interactingintheabsenceofatertiarycomponent. v 1 1 1 9 Weevolvethese(cid:16)system(cid:17)sbymeansofanewlydevelopedalgorithmthatcouplesseculartriple 6 dynamicswithanexistingbinarypopulationsynthesiscode.Wefindthatthetertiarysignifi- 4 cantlyalterstheinnerbinaryevolutioninabout24%ofallsampledsystems.Inparticular,we 1 findseveralchannelsleadingtoCOWDmergers.Amongsttheseisanovelchannelinwhich . a collision occurs in wide inner binary systems as a result of extremely high eccentricities 1 0 inducedbyKozaicycles.WithassumptionsonwhichCOWDmergersleadtoaSNIaexplo- 3 sion we estimate the SNe Ia delay time distribution resultingfrom triples and compareto a 1 binarypopulationsynthesisstudyandtoobservations.Althoughwefindthatourtriplerateis : low,we havedetermineda lowerlimitofthetriple-inducedSNeIa rateandfurtherstudyis v neededthatincludestripleswithinitiallytighterinnerorbits. i X Keywords: binaries:general–celestialmechanics–methods:statistical–whitedwarfs r a 1 INTRODUCTION Kisseleva-Eggleton&Eggleton 2010; Fabrycky&Tremaine 2007). In particular, Fabrycky&Tremaine (2007) have shown Coeval stellar hierarchical triple systems, henceforth triple sys- thatKCTFisresponsible forproducing closeMSbinary systems tems, are known to be quite common among stellar systems in with periods P ∼ 3d, which is consistent with the observation the field (e.g. Tokovininetal. 2006; Raghavanetal. 2010). Such 1 that such systems are very likely (96%) orbited by a tertiary systems consist of an inner binary pair orbited by a distant star, (Tokovininetal. 2006). It remains to be seen, however, whether the tertiary. Due to secular gravitational effects high-amplitude KCTF is still effective in higher-mass triples (individual masses eccentricity oscillations, known as Kozai or Kozai-Lidov cycles m & 1.2M ). This is because their constituents have radiative (Kozai1962;Lidov1962),canbeinducedintheinnerbinarysys- ⊙ envelopes and are thus much less effective at dissipating tidal tem.Theseoscillationsoccurontimescalesthataretypicallymuch energythantheirlower-masscounterparts,whichhaveconvective longer than the orbital periods of both inner and outer binaries. envelopes (Zahn 1977). On the other hand, as such higher-mass One possible consequence of these eccentricity cycles is strong stars greatly increase in size and develop convective envelopes tidalfrictionandsubsequent orbitalshrinkage,aprocesswhichis duringtheir redgiant branch(RGB)and asymptoticgiant branch knownasKozaicycleswithtidalfriction(KCTF).KCTFhasbeen (AGB) phases tidal friction is expected to become much more studied in the context of (solar-mass) main sequence (MS) stars effective.ThusitispossiblethatKCTFiseffectiveatsignificantly (Mazeh&Shaham 1979; Eggleton&Kiseleva-Eggleton 2001; shrinking the inner binary system during the RGB/AGB phases, whereasthiswasnotthecaseduringtheMS. ⋆ E-mail:[email protected] Anotherpossibleconsequenceofhigh-amplitudeeccentricity 2 A.S. Hamers,O.R.Pols,J.S.W.Claeys and G. Nelemans cyclesinduced in(close)inner binarysystemsbyatertiaryisthe initial values of i , e and g , where g denotes the argument of tot 1 1 j enhancementoftheemissionofgravitationalwaves(GWs),apro- periastron.Otherquantitiesthatvaryperiodicallyareg (i.e.orbital j cess which is strongly dependent on the orbital eccentricity (e.g. precession,alsoknownasapsidalmotion)andthelongitudesofthe Peters 1964). Thompson (2011) has suggested that this mecha- ascendingnodesh . j nismcansignificantlyreducetheGWmergertimeinclosecarbon- In systems with large a /a the eccentric Kozai mech- 1 2 oxygen (CO) white dwarf (WD) binaries if they are orbited by a anism may apply in some cases (Lithwick&Naoz 2011, tertiarywiththeappropriateorbitalparameters.Thisscenarioim- Shappee&Thompson 2012). This mechanism is associated with pliesanincreaseintheexpectednumberofcloseCOWDsystems orbital“flips”(changeoforbitalinclinationfromretrogradetopro- thatmergewithinaHubbletime,whichisrelevantforbinarypop- gradeandviceversa)withpotentiallyveryhighinnerorbiteccen- ulation synthesis studiesinwhich theCO WD merger channel is tricities.A quantity that measures theimportance of thiseffect is considered asanimportant candidate progenitor channel for type the“octupoleparameter”ǫ definedby: oct Iasupernovae(SNeIa)(e.g.Ruiteretal.2009;Toonenetal.2012; m −m a e Claeysetal.inprep.). ǫoct = m1+m2a1 1−2e2, (2) In this work we investigate both scenarios of KCTF during 1 2 2 2 stellarevolutionandthemergingofCOWDsundertheinfluenceof where typically log10(|ǫoct|) & −2 indicates that the eccen- atertiarycompanionbymeansofapopulationsynthesisoftriples. tric Kozai mechanism could be important (Naozetal. 2011; Wetakeintoaccountlong-termdynamical,stellarandbinaryevo- Shappee&Thompson2012). lutionbymeansofanewlydeveloped algorithminwhichacode Inphysicallyrealisticsituationsthereexistsourcesofpreces- modelingtheseculardynamicalevolutionoftriplesiscoupledwith sion in the inner orbit other than those due to secular three-body thebinary population synthesis code Binary c(seeSect. 2.3for dynamicsalone(inourstudysucheffectsintheouterorbitareneg- details)tomodeltheinnerbinaryevolution.Werestrictourselves ligiblebecausea2isalwayssufficientlylarge).Wetakeintoaccount to intermediate mass triples with wide inner binary systems that thefollowingadditionalcontributionstoprecession:duetogeneral wouldnotinteractintheabsenceofatertiarycomponent.Forthese relativisticeffects(g˙1,GR),duetotidalbulgesinducedinphysically triplesweshowinthispaperthatthepresenceofthetertiarycom- extendedstarsinrelativecloseproximity(g˙1,tide)andduetointrin- ponentintroducestotheinnerbinarysystemvariousnovelevolu- sicspinrotation(g˙1,rotate).Thefactthatg˙1,GR,g˙1,tide andg˙1,rotate are tionaryscenarios. Inparticular, wefindanalternativepathway to all positive impliesthat the associated sources of precession pro- SNeIaasaresultofcollisionsofWDsinwideinnerbinarysys- moteprecessioninoneparticulardirectionandthereforesuppress temsinwhichtheeccentricityreachesextremelyhighvaluesasa the resonance between g1 and e1 during Kozai cycles. This phe- resultofKozaicycles. nomenonhasbeeninvestigatedindetailbyBlaesetal.(2002). Theorganizationofthispaperisasfollows.InSect.2wede- Furthermore,wheneverthedistancesbetweenthecomponents scribethetripleevolutionalgorithmdevelopedinthisworkandin intriplesystemsbecomecomparabletotheirradiitidaleffectsmay Sect.3weprovidedetailsoftwoexamplesystemsthatleadtoCO becomeimportant.Inourstudysucheffectsintheouterorbitcan WDmergersviatwo distincttriplechannels. Withthisalgorithm beneglected.However,intheinnerbinarysystemtidaleffectsmay weperformapopulationsynthesisstudyofwhichwediscussour become important even forlargesemi-major axeswithrespect to methodsinSect.4andpresenttheresultsinSect.5.Forthevarious theradii,a1/Ri(whereRidenotestheinnerbinarystellarradius),if COWDmergerchannelswefindweestimatetheexpectedtriple- theinnerorbitaleccentricityissufficientlyhigh.Inthiscaseatidal induced SNeIaratesand compare these tothebinary population torqueisinduced, eitherduetotheequilibriumtideordynamical synthesisstudyofClaeysetal.(inprep.)andtoratesinferredfrom tide(Zahn1977),thatallowsfortheexchangebetweenangularmo- observationsinSect.6.InSect.7werelateourresultstotheprevi- mentaofthespinandorbitintheinnerbinarysystem.Thisprocess ousworkofThompson(2011)anddiscusstheuncertaintiesofour continuesuntilcoplanarity,synchronizationandcircularizationare treatmentoftripleevolution.WeconcludeinSect.8. achieved(Hut1980).Itisdescribedquantitativelyintermsoforbit- averagedequationsfortheorbitalparametertimederivatives,a˙ , 1,TF e˙ andΩ˙ ,whereΩ isthespinfrequencyofinnerorbitstari 1,TF i,TF i andTFdenotestidalfriction. 2 HIERARCHICALTRIPLEEVOLUTIONALGORITHM Another effect we take into account isGW emission, which 2.1 Secularhierarchicaltripledynamics wemodelfortheinnerbinaryonly.Asaconsequenceoftheemis- sionofGWstheinnerbinaryiscircularizedandshrinks,whichis During Kozai cycles the mutual torque exerted by the inner and describedquantitativelybyorbit-averagedequationsfore˙ and outerbinaryorbitscausesanexchange ofangularmomentumbe- 1,GW a˙ . tweentheorbitswhiletheorbitalenergies,andthusthesemi-major 1,GW axes a , are conserved (e.g. Mardling&Aarseth 2001). Here we j useindex j = 1,2todenotequantitiespertainingtotheinnerand 2.2 Systemoffirst-orderdifferentialequations outer orbits respectively. As a result of the exchange of angular momentumtheorbitaleccentricitiese andthetotalmutual incli- We describe the secular Kozai cycles quantitatively in terms of j nationanglebetweenbothorbitsitotvaryperiodicallyontimescales coupled equations for g˙j,STD and e˙j,STD (where STD denotes sec- P ≫P ,whereP denotesorbitalperiod.Tolowestorderina /a ular three-body dynamics), that follow after orbital averaging of K 2 j 1 2 P isgivenby(Kinoshita&Nakai1999): thethree-body Hamiltonianexpressed inDelaunay’s action-angle K variables,applyingHamilton’sequationsofmotionandeliminating P =αP22m1+m2+m3 1−e2 3/2, (1) thelongitudesoftheascendingnodes(Harrington1968;Fordetal. K P m 2 1 3 (cid:16) (cid:17) 2000;Naozetal.2011).Weemployequationsforthesefourquan- where m and m denote the inner binary primary and secondary titiesaccuratetooctupoleorder,i.e.uptoandincludingthirdorder 1 2 mass respectively and m denotes the tertiary mass. The dimen- in a /a . In addition, an equation that expresses conservation of 3 1 2 sionless quantity α is of order unity and depends weakly on the totalangularmomentumdescribesthechangeofi . tot Triplepopulationsynthesisandwhitedwarfmergers 3 WethencombineallthephysicaleffectsdiscussedinSect.2.1 theinnerbinarysystemsuchthatitdoesnotfillitsRoche-lobein toformasystemofeightfirst-orderordinarydifferentialequations itsorbitaroundtheinnerbinarysystem,noritisinvolvedinaCE (ODEs)thatreads: phasewiththeinnerbinarysystem,ascenariowhichhasbeenex- ploredinearlierstudies(e.g.Iben&Tutukov1999).Thereforethe g˙ =g˙ +g˙ +g˙ +g˙ ; 1 1,STD 1,GR 1,tide 1,rotate latterprocessesarenotmodeledinthetriplealgorithm.  g˙2 =g˙2,STD;  aee˙˙˙θ˙121 ====eae˙˙˙121−,,,SST1TTFDD+;G+˙a˙e˙1(,1GG,TWF;++Ge˙1,GθW);+G˙ (G +G θ) ; (3) Eaolfqg.oE3rqIint.ihs3tmhs,oe.wlIvOnheDidocErhfdoseriosrtlhvtqoeeureidtrnueosruuasttrtiiienoffenpritonhofepneesaraytcscuhtoreentm,ivmeworefgesedtunieffscpeeeoroeafnnftthtiOhaeelDBesiEqonulausaottriilyoovnnecssr where θ≡ΩΩ˙˙21cos==(itoΩΩG˙˙t)121G,,aTT2FFnh,;d G1 j d1enote2s the o2rbit2al ang1ulair momen- ts1nip9ooe9tnc6cioo)fif.ucWtpahlleeelydhsaadwmveiesetihcghhntiheeecerdkaBrteocidhnitinahctreaelygrterrcasiutpaellltegsstsooiyffrfistttOhhememDOsEbDsaysE(cCsostomuohdlpveiuenedtri&nriongHuFttihionnerdedemwveaothrlaesulnh-. tum.Forexpressionsfortheprecessionquantitiesg˙1,GR,g˙1,tide and (2000),Blaesetal.(2002)andNaozetal.(2011)andfindverysim- g˙1,rotate werefertoBlaesetal.(2002),Smeyers&Willems(2001) ilarresultsfore1,itotanda1(whereapplicable)asfunctionoftime. and Fabrycky&Tremaine (2007) respectively; for the tidal evo- Wehavealsofoundnearexactagreementofthetimeevolutionof lution quantities e˙1,TF, a˙1,TF and Ω˙i,TF we refer to Hut (1981) and e1 and itot when comparing to analytical solutions which exist in fortheGWevolutionquantitiese˙1,GWanda˙1,GW werefertoPeters thequadrupoleorderapproximationandwiththefurtherrestriction (1964).ThemainassumptioninEq.3isthatallphysicalprocesses thatG ≫ G (Kinoshita&Nakai 1999).During each Binary c 2 1 that are described are independent such that the individual time timestepquantitiesnotincludedinthelefthandsideofEq.3such derivative termscanbe added linearly.Inaddition, intheexpres- as the inner binary masses and radii are assumed to be constant. sions for g˙1,tide, g˙1,rotate, e˙1,TF, a˙1,TF, Ω˙i,TF and θ˙ we assume copla- BecausetheBinary calgorithmusesadaptivetimestepstomatch narity of spin and orbit at all times, even though Kozai cycles in the rate of stellar and binary evolution it is always ensured that principleaffecttherelativeorientationsbetweenthespinandorbit thesequantitiesdonotchangesignificantlyduringeachiterationof angularmomentumvectorsandinturnamisalignmentofthesevec- thetripleevolutionalgorithm.Inaddition,ifneededthetriplealgo- torsaffectstheKozai cyclesthemselves(e.g.Correiaetal.2011). rithmdecreasestheBinary ctimestepsuchthattheinnerorbital Wejustifythisassumption bynotingthatforthemajorityofsys- angular momentumG does not change by more than 2%. Thus, 1 temsthatwestudytheorbitalangularmomentaofbothinnerand changestotheinnerorbitsemi-majoraxisandeccentricitydueto outerorbitsgreatlyexceedthespinangularmomentainmagnitude, secular three-body dynamics coupled with tidal friction and GW thereforethestellarspinscannotgreatlyaffecttheexchangeofan- emission are communicated tothe Binary calgorithm at appro- gularmomentumbetweenbothorbits. priatetimes. The expression for θ˙ in Eq. 3 is derived by differentiating TheversionofBinary cusedinthisworkenforcescircular- the relation that expresses conservation of total angular momen- izationattheonset of Roche-lobeoverflow (RLOF)andCEevo- tum, G2 = G2 +G2 +2G G θ, with respect to time and solv- tot 1 2 1 2 lution.Althoughthisassumptionisgenerallyjustifiedforisolated ing for θ˙ in terms of G and G˙ . Note that GW emission acts to j j binaries(Hurleyetal.2002),intriplesystemshigh-amplitude ec- change a and e and therefore affects the magnitude of G but 1 1 1 centricitycyclescouldbeinducedduringtheformerphase,inpar- not its direction. Although in principle tidal friction can also af- ticularifthetimescaleofthesecyclesiscomparabletoorsmaller fectthedirectionofG ifthestellarspinsandorbitarenotcopla- 1 thanthetimescaleof masstransferdrivenby RLOF.Anaccurate nar, with our assumption of coplanarity this is not the case. This treatmentof masstransfer ineccentricorbitsisbeyond thescope implies that, withGj = Lj 1−e2j, where Lj = Lj(aj,mi) isthe ofthiswork(seee.g.Sepinskyetal.2007)andwethereforechoose q orbital angular momentum in case of a circular orbit and which to disable the ODE solver routine whenever mass transfer driven is constant with regard to STD, the quantities G˙j are given by byRLOForCEevolutionensues.WhenevertheODEsolverrou- G˙ =− e / 1−e2 ·G ·e˙ . tineisactiveitisensuredthattidalevolutionandGWemissionin j j j j j,STD h (cid:16) (cid:17)i theinnerbinarysystemascalculatedbyBinary cdonotchange theinnerorbitalparameters.Thisisnecessarybecauseinthiscase 2.3 Couplingtobinaryalgorithm theseprocessesaretakenintoaccount bytheODEsolver routine throughtheequationsfore˙ anda˙ inEq.3.Inthismannerthecou- Thehierarchicaltripleevolutionalgorithmdevelopedinthiswork 1 1 plingofeccentricitycycleswithtidalfrictionandGWemissionin couples the system of differential equations Eq. 3 to the existing theinnerorbitiscalculatedconsistently. rapid binary population synthesis algorithm Binary c based on Hurleyetal. (2000, 2002) with updates described in Izzardetal. Iftheratioβ ≡ a /a islargetheeffectsofSTDarelikelyto 2 1 (2004, 2006, 2009) and Claeys et al. (in prep.). This binary star bequenched bydifferent processes intheinner binarysystem, in algorithmincludesstellarevolutionandavarietyofbinaryinterac- particularduetogeneral relativisticprecessioninverytightinner tionprocessessuchasmasslossduetostellarwinds,masstrans- binarysystems,asdiscussedinSect.2.1.Insuchsituationsitisnot ferandcommonenvelope(CE)evolution.WerefertoHurleyetal. necessarytosolveEq.3becausetheSTD-relatedtermsarenegligi- (2002)forfurtherdetailsofthisbinarypopulationsynthesisalgo- bleandtheinnerbinaryevolutioncanthereforebefullyhandledby rithm. The novel aspect of our approach is that the evolution of Binary c.Wethereforeintroduceacriterionthattemporarilydis- theinnerbinarysystemistakenintoaccountthroughthecoupling ablestheODEsolverroutineandleavestheinnerbinaryevolution to Binary c. We also take into account the evolution of the ter- toBinary cinsuchcases. Specifically, theroutineisdisabled if tiary,whichistreatedasbeingisolatedsuchthatitsmassmaybe β>10β ,whereβ isthevalueofβsuchthattheperiods crit,GR crit,GR computed from single stellar evolution. In thiswork weconsider associatedwithprecessionduetoSTD(Eq.1)andgeneralrelativity systemsinwhichatalltimesthetertiaryissufficientlydistantfrom (approximately givenbythereciprocal ofg˙ )areequal, which 1,GR 4 A.S. Hamers,O.R.Pols,J.S.W.Claeys and G. Nelemans wefindisgivenby(omittingnumericalfactorsoforderunity): quantities k /T and r2 are therefore calculated from precisely am,i i g,i thesameprescriptionasinBinary c,whichisrequiredforacon- 1/3 βcrit,GR≈ Gc2N!1/3a11/3 (m1m+3m2)2!1/3 (cid:16)11−−ee221(cid:17)1/2 sistenTthtreeactlmasesnict.al apsidal motion constant kam,i (i.e. not the ra- 2 tiok /T), which is required for evaluation of g˙ and g˙ (cid:16) (cid:17) am,i i 1,tide 1,rotate ≈5·102 a1 1/3 m3/M⊙ 1/3 (cid:16)1−e21(cid:17)1/3, (pSumteedyaesrsa&fuWnciltlieomnsof20m0a1s;sFaanbdrytcimkye&relTarteivmeationeth2e00Z7A),MiSs cforomm- (cid:18)AU(cid:19) (m1/M⊙+m2/M⊙)2! 1−e2 1/2 detailedstellarmodelsformetallicityZ=0.02calculatedbyClaret 2 (cid:16) (cid:17) (4) (2004).Therunofk withtimeisdeterminedbymeansoflinear am,i relations scaled to the time spent during each evolutionary stage wherecdenotesthespeedoflightandG denotesthegravitational N for all tabulated masses in Claret (2004). Subsequently, k is constant. am,i calculatedfor arbitrarymassby meansof linearinterpolation be- Outer orbital expansion due to mass loss in either inner or tweenadjacent massvalues. Furthermore, forlow-mass stars,i.e. outerbinarysystemsistakenintoaccountwiththeassumptionof m < 0.7M , theclassical apsidal motion constant istaken tobe fast,isotropicwinds(i.e.Jeansmode)suchthata (m +m +m ) i ⊙ 2 1 2 3 k = 0.14 corresponding ton = 3/2 polytropes as an estimate and e remain constant (Huang 1956, 1963). With this relation a am,i 2 forthese(nearly)fullyconvectivestars.Forhelium(He)MSstars new value of a iscomputed fromthechanges of m ,m andm 2 1 2 3 and WDs k is calculated as function of mass by means of in- duringeachtimestepofBinary c. am,i terpolationsoftabulateddatafromVila(1977).Lastly,forneutron Lastly,ateachBinary ctimestepthetriplesystemischecked starsk iscalculatedasfunctionofmassandradiusfromtheex- for dynamical stability by means of the stability criterion formu- am,i pressiongiveninHinderer(2008)andforblackholesk = 0as latedbyMardling&Aarseth(2001),includingtheadhocinclina- am,i appropriatefornon-spinningblackholes. tionfactor f =1−(0.3/π)i (withi expressedinradians).When- tot tot everβ≤β ,whereβ isgivenbythisstabilitycriterion,theSTD crit crit equations arenolonger strictlyvalidwedonot model thesubse- quentevolution.Thephaseofdynamicalinstabilitywouldneedto 3 TRIPLEEVOLUTIONEXAMPLES be computed with a three-body integrator code, which is beyond TodemonstratethetriplealgorithmpresentedinSect.2wedescribe thescopeofthiswork(seePerets&Kratter2012forarecentstudy indetailtheevolutionoftwotriplesystemsinwhichaninnerbi- oftriplesystemsthatbecomedynamicallyunstable). naryCOWDmergeroccursineitheratightcircularorbitorawide andhighlyeccentricorbit.Theinitialconditionsforthesetwosys- temsareselectedfromourfirsttriplepopulationsynthesissample 2.4 Fixedbinaryparametersandevaluationofvarious (TSM1,cf.Sect.4.2). physicalquantities Wedetailsomeofthefixedbinaryevolutionparametersusedinthis 3.1 Mergerinatightcircularorbit work.Wechoosequasi-solarmetallicity,i.e.Z = 0.02.Theinitial spin periods of the inner orbit components are computed from a Weconsideratriplesystemwithinitialparametersm =3.95M , 1 ⊙ formulagivenbyHurleyetal.(2000),whichisfittedfromdataof m = 3.03M , m = 2.73M , a = 19.7AU, a = 636.1AU, 2 ⊙ 3 ⊙ 1 2 therotationalspeedofMSstarsofLang(1992).TheCEparame- e = 0.23,e = 0.82,i = 116.0◦,g = 28.5◦ andg = 249.6◦. 1 2 tot 1 2 terαCE describestheefficiencyoftheconversionoforbitalenergy Fig.1showsa1,e1,β ≡ a2/a1,thetidalstrengthquantitykam,i/Ti intobinding energy withwhichtoshed theenvelope and itisset (Hut 1981; Hurleyetal. 2002) and i as a function of time. The tot tothecanonicalvalueofαCE = 1.InthisCEdescriptionanaddi- evolution is shown with both the triple algorithm enabled (triple tionalparameterλCEisrequired,whichisadimensionlessmeasure case; solid line) and disabled (binary case; dashed line), leaving of the relative density distribution within the envelope; here it is all inner binary evolution to Binary c. The latter binary case is determinedbymeansoffunctionalfitsasdetailedinClaeysetal. equivalenttothesituationinwhichnotertiaryispresent. (inprep.).AllotherparametersintrinsictotheBinary calgorithm Inthebinarycasethesemi-majoraxisincreasesatthetwomo- arealsosettoidenticalvaluesasinClaeysetal.(inprep.). mentswhenthebinarystarsevolvefromAGBstarstoCOWDs(at Inaddition,variousphysicalquantitiesarerequiredtoevalu- t ≈ 0.2Gyrandt ≈ 0.5Gyrrespectively)asaconsequenceofthe atetheright-handsidesinEq.3andintheremainderofthissec- associated wind mass loss. Here (i.e. in Binary c) it is assumed tionwedescribehowthesequantitiesareobtainedinourtripleal- thatthewindmasslossisfastandisotropicsuchthata (m +m ) 1 1 2 gorithm. The tidal evolution equations (i.e. expressions for a˙1,TF, remainsapproximately constant.Wheneitherstarintheinnerbi- e˙1,TF andΩ˙i,TF)containaparameterthatcapturestheefficiencyof narysystem enterstheRGBor AGBphase it develops aconvec- tidalfrictionanddepends onthestructureofthestar(Hut1981). tiveenvelopeanditsradiusincreasessignificantly.Thechangeof Thisparameteristheratiokam,i/Ti,wherekam,i istheclassicalap- theenvelope structureisreflectedbyasubstantial increaseofthe sidal motion constant of inner orbit star i and Ti a tidal friction tidal strength quantities kam,i/Ti as is shown in Fig. 1 (primary timescale.Wecomputekam,i/Tiaccordingtothesameprescription AGBphase). However, despitethelargekam,1/T1 andprimaryra- thatisused inBinary c(Hurleyetal.2002).Inthisprescription diusthere isno significant tidal frictionduring the primaryAGB adistinctionismadebetweenconvective,radiativeanddegenerate phasebecauseofthelargea andsmall(andconstant)e .Conse- 1 1 envelopesandkam,i/Ti forthesethreecasesiscomputedbasedon quently, in the binary case the CO WDs end their evolution in a resultsofRasioetal.(1996),Zahn(1977)andCampbell(1984)re- wideandnon-interactingbinarywitha ≈84AU. 1 spectively.Furthermore,thegyrationradiir2 arerequiredforthe Inthetriplecasehigh-amplitudeKozaicyclesareinduceddur- g,i evaluationofΩ˙ andarecomputedwithaprescriptioninwhich ingtheMSintheinnerorbitbythetertiarywithmaximaofe ≈ i,TF 1,max thestarsaresplitintotwoparts,consistingofthecoreandtheenve- 0.9.Octupoleordertermsareimportantbecause|ǫ |≈10−2.Nev- oct lope,detailedinHurleyetal.(2000)andHurleyetal.(2002).The ertheless,duringtheMStheeccentricityisnothighenoughtodrive Triplepopulationsynthesisandwhitedwarfmergers 5 20 100 binarycase 10 15 triplecase U 1 binarycase AU A (cid:144) 10 (cid:144)1 triplecase a1 a 0.1 5 0.01 0.001 0 0.2 0.5 1.0 2.0 5.0 10.0 220.0 220.5 221.0 221.5 222.0 222.5 223.0 t(cid:144)Gyr t(cid:144)Myr 0.0 106 -0.2 105 L -0.4 1 104 -e 1 Β H0 -0.6 1 1000 og l -0.8 binarycase 100 triplecase -1.0 10 0.2 0.5 1.0 2.0 5.0 10.0 220.0 220.5 221.0 221.5 222.0 222.5 223.0 t(cid:144)Gyr t(cid:144)Myr -8 -0.4 -10 primary H(cid:144)LLs -12 secondary -0.5 Ti L (cid:144) ot Hkam,i -14 Hcosit -0.6 10 -16 g o -0.7 l -18 -20 -0.8 220.0 220.5 221.0 221.5 222.0 222.5 223.0 220.0 220.5 221.0 221.5 222.0 222.5 223.0 t(cid:144)Myr t(cid:144)Myr Figure1.Severalquantitiesofinterestintheevolutionofthefirstexamplesystem(Sect.3.1).Shownasafunctionoftimearea1,e1,β ≡ a2/a1,kam,i/Ti anditot.Intheplotsinthefirstcolumnofthefirsttworowstheentireevolutionisshown;theotherplotscorrespondtotheendoftheprimarycorehelium burningphase,theprimaryAGBphaseandstartoftheprimaryCOWDphase.Solidline:triplecase;dashedline:binarycase(whereapplicable).Intheplot ofkam,i/Ti onlythetriplecaseisshown(thebinarycaseisverysimilar);inthisplotthesolidlineappliestotheprimary(i=1)andthedashedlinetothe secondary(i = 2).Notethattheevolutionisnotfullysampledinallplots,causingseveralkinks(inparticularintheplotsfore1 anditot)–moredetailed calculationsareperformedinternallyintheODEsolverroutinebutarenotshownhere(cf.Sect.2.3). significanttidalfrictionbecausethetidalstrengthquantityisvery increaseintheeffectivenessof orbitalshrinkage isduetotheex- small (k /T ∼ 10−18s−1) as aconsequence of theradiative en- pansionoftheprimaryfromaradiusofR ≈49R toR ≈497R am,i i 1 ⊙ 1 ⊙ velopes in the MS stars. During the primary RGB phase the pri- between t = 220Myr andt = 222.5Myr. Consequently a isre- 1 mary possesses a convective envelope (k /T ∼ 10−8s−1) but ducedtoa ≈ 7AUandtheorbitiscompletelycircularized.Note am,1 1 1 e isnothighenoughtotriggersignificanttidalfriction.However, thatforcompletecircularizationtooccurthedurationofthephase 1 tidalfrictiondoesbecomeeffectiveduringtheprimaryAGBphase inwhichk /T issubstantialmustbesufficientlylongcompared am,1 1 starting at t ≈ 220.6Myr and circularizes the inner orbit during totheKozaiperiodP (Eq.1),whichisthecaseforthisexample K thetimespanoffiveKozaicycles,wheresignificantorbitalshrink- system.Inothercasesoftriplesystems,however,P canbemuch K ageoccursateccentricitymaxima.Notethatduringthisphasethe 6 A.S. Hamers,O.R.Pols,J.S.W.Claeys and G. Nelemans 0 3000 binarycase -2 2000 triplecase L 1 e AU 1500 H-1 -4 (cid:144)a1 og10 binarycase 1000 l -6 triplecase -8 0.2 0.5 1.0 2.0 5.0 10.0 0 1 2 3 4 5 6 7 t(cid:144)Gyr t(cid:144)Gyr 1.0 30 0.5 L ot Β Hsit 0.0 o 20 c -0.5 15 -1.0 0.2 0.5 1.0 2.0 5.0 10.0 0 1 2 3 4 5 6 7 t(cid:144)Gyr t(cid:144)Gyr Figure2.QuantitiesofinterestforthesecondexamplesysteminwhichaninnerbinaryCOWDmergeroccursinawideandhighlyeccentricorbit. longer than the duration of the RGB/AGB phases, thus avoiding unlikely that the primary RGB and AGB phases (which occur at strongtidalfrictioneveniftheeccentricitymaximaarehigh. t≈0.3Gyrandt≈0.4Gyrrespectively)coincidewitheccentricity Shortly after KCTF during the primary AGB phase the pri- maxima. Suchacoincidence isindeed not thecase inthesecond mary has swelled up to R ≈ 611R , fills its Roche-lobe and examplesystem. 1 ⊙ invokes CE evolution, shrinking the orbit further to a1 ≈ 1AU AstheprimaryandsecondaryevolvetoCOWDswithoutin- (t ≈222.6Myr).Duringthisprocesstheratioβincreasessubstan- teracting the associated wind mass loss widens the orbit to a ≈ 1 tially to β ≈ 1321, which is large enough for general relativistic 2600AU.Although thisinner binarywindmasslossincreasesa 2 precession to fully quench any subsequent Kozai cycles. Conse- aswell,theratioβ = a /a decreaseswiththeassumptionoffast 2 1 quently, theinclinationangleitot isfrozentoitot ≈ 128◦.Thepri- isotropicwindssuchthata1(m1+m2)anda2(m1+m2+m3)arecon- mary emerges from the CE as a CO WD and the secondary as a stant(inthisexamplem alsoremainsconstant).Consequently,the 3 MSstar.Att ≈ 0.5Gyrthesecondary evolvestoanAGBstarit- ratioβdecreasestoβ≈18afterthesecondaryhasevolvedtoaCO self, invoking a second CE phase. A close CO WD binary with WD.AsthetertiaryevolvestoaCOWDandsubsequently loses a1 ≈ 0.01AU emerges from the CE. Due to GW emission the massatt ≈ 1.5Gyra2 increaseswhilea1 staysconstant, thusin- system subsequently merges at t ≈ 13.2Gyr in a circular orbit. creasingβagain.ThelatterprocessalsoincreasestheKozaiperiod. ThecombinedCOWDmass,1.45M⊙,exceedstheChandrasekhar Aftert ≈ 1.5Gyrthemaximumvaluesofcos(itot),whicharestill massMCh ≈1.4M⊙,thereforethemergercouldpotentiallyleadto negative(retrogradeorbit),slowlyincreaseandthecorresponding aSNIaexplosion(cf.Sect.6.2). eccentricitymaximaincreaseaswell.Ascos(i )approaches0the tot eccentricity maxima reach very high values until at t ≈ 7.0Gyr theorbitsbecomeprograde,whichisassociatedwithe reachinga 1 3.2 Mergerinawideeccentricorbit valueof1−e ≈ 10−7.4.Thisishighenoughtotriggeranorbital 1 Thesecondexamplesystemhasinitialparametersm1 = 3.26M⊙, collisionbetweenthetwowhitedwarfs,eventhougha1 ≈2600AU m =2.00M ,m =1.39M ,a =716.6AU,a =2.012·104AU, isverylarge.SuchacollisioncouldpotentiallyresultinaSNIaex- 2 ⊙ 3 ⊙ 1 2 e =0.78,e =0.64,i =93.4◦,g =150.0◦andg =151.2◦.The plosionasinthepreviousexample(cf.Sect.6.2),butnotethatthe 1 2 tot 1 2 evolutionofa ,e ,β≡a /a andi forthissystemasfunctionof natureofthemergerpriortoapossibleSNeIaeventisverydiffer- 1 1 2 1 tot timeisshowninFig.2.Themaindifferenceininitialparameters ent. inthisexamplesystemcomparedtothepreviousexampleisthata Onemightwonderwhetherinthissystemtidalfrictionissig- 1 anda arelarger.Theirratioβiscomparable,however.Becauseof nificant during close periastron passages before the collision oc- 2 thelargera theKozaiperiod P ≈ 100Myrismuchlongerthan curs.Forthisreasonwehaverecalculatedthissystemwiththetidal 2 K inthefirstexample(P ≈0.3Myr).Consequently,itismuchmore strength quantities k /T for radiative and degenerate damping K am,i i Triplepopulationsynthesisandwhitedwarfmergers 7 multipliedbyadhocfactorsof103 and106 toartificiallyincrease 12 theeffectiveness of tidal friction.Evenwiththesemultiplications Tokovininsample wefindthattheoutcomeisidentical,showingthattidalfrictionis 10 ns(4syoestetiPemSmOepscPotiUr.nt7aLt.nh2Ate)Ti.npIotOhpiNuslasSytYisotNnemTsy.HnWEtheSehIsSaisv:esMtpuEedryTfoaHrnmOdeDhdaSsviemfiolaurntdesntosfeoffreacltl H(cid:144)LlogPd210 2468 ×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× Before presenting the results of our population synthesis study × (Sect.5)wediscusshereourselectioncriteriaandsamplingmeth- 0 odswhichareessentialingredientsforsuchastudy. 0 2 4 6 8 10 log HP (cid:144)dL 10 1 4.1 Selectioncriteria 12 TSM1 Becausewearemainlyinterestedininnerbinarymergersinvolving 10 bsamteaC1trwOoefevWemonl1Duf,uotiwro=men.i6tna.T5gkheMaeaC⊙ln,oOwuwpWehpriecDlrhimlaainmiptdpiotraofonxfmiOtm1hNeaisteienWlsnyeetDcrtaobipnitnmutarh1re,eylsc=ptahrsie1me.b0aoorfMuysn⊙imdnagafroslyesr H(cid:144)LPd210 68 ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× anylowermasswillnotproduceasignificantnumberofCOWDs og withinaHubbletimeforametallicityofZ =0.02.Inadditionwe l 4 focus on triplesystems withwide inner binary systems such that 2 theinner binarystarsdonotinteract duringtheirevolutioninthe absenceofatertiary.Takingintoaccountpossibletidalfrictionthis 0 canbeachievedbyrequiringthattheinitialinnerorbitsemi-latus 0 2 4 6 8 10 rectuml1 ≡a1 1−e21 satisfiesl1 >l1,l ≡12AU. log10HP1(cid:144)dL sOpnts(o2yhreue0sordrts0woeies8rmOrsnee)btlcnsi,etitthneclawiyentltpihhoposotienaehcsbmrshe(sciiP(cid:16)oebrprii1rdsleltv,eesgelPeriimbddkmi2aeae)eet-tlso(cid:17)twadyahfbimeofabeiredngetponwnrclmtaeeaolemorueooogsnb.bfeb1iA0snts(o7aepaPcif2rernvl1y5cte/ehttasddrhetryor)eistsdpp=rtciiaielnffiopuem2iplctceiissiauactynsylwlsytdatytoisnretlifthdmoopesgfml1syve1d.sis00spoet.(euPtaf<Femar1ciaTl/gmstmodbi.nw1k)ie3/ngoti=Maetv(hsrrtiu⊙4isoniecpniiis<nhss)-. H(cid:144)LogPd210 110268 TSM2 ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× ×××× 6.5and l > 12AU decrease thenumber of systemsfrom725 to l 4 1 165, where a value of log (P /d) = 4 is taken to represent the 10 1 dividing value l = 12AU. The number of remaining systems 2 1,l (indicatedinthetopfigureofFig.3by dots) isnot largeenough 0 for direct sampling, therefore wechoose toutilize themethod of 0 2 4 6 8 10 MonteCarlosamplingfromobservation-baseddistributions.Inor- log HP (cid:144)dL der togaininformationontheuncertaintiesofour resultsweuse 10 1 twodistinctsamplingmethodsthataredescribedbelow. Figure 3. Inner (P1) and outer (P2) orbital periods of triple systems of theobservedtriplesampleofTokovinin(2008)(top)andthetwosynthetic triplesamplesdescribedinSect.4.2(middleandbottom).Intheobserva- 4.2 Samplingmethods tionalsamplesystemssatisfyingourselectioncriteriaaredenotedwithlarge (red)dots(•).Inthesyntheticsamplesthelargedots(•)denotethesystems 4.2.1 TSM1 that undergo an early MS inner binary mergerordestabilization and are Thefirsttriplesamplingmethod(TSM1)isbasedonthesupposi- thereforenotlikelytobeobserved.ThelinecorrespondstoP2/P1=5and tionthatahierarchicaltriplesystemiscomposedoftwouncorre- representsthetypicalcriticalvaluefordynamicalstability. latedbinarysystems.Themainadvantageofthisapproachisthat thestatisticsofbinaryparametersareknownwithgreatercertainty thanthoseoftriplesystems.Aprimarymass1.0 < m /M < 6.5 e ande fromathermaldistributiondN/de =2e asisappropriate 1 ⊙ 1 2 j j issampledfromaKroupaetal.(1993)initialmassfunction(IMF), forP >103dbinaries(Kroupa&Burkert2001).Twosemi-major j i.e.dN/dm ∝ m−2.70.Subsequently,twomassratios,q ≡ m /m axesarethensampledfromadistributionthatisflatinlog (a ),the 1 1 1 2 1 10 j and q ≡ m /(m +m ), are sampled independently from a uni- smallerofwhichisdesignateda andthelargerofwhichisdesig- 2 3 1 2 1 form distribution such that 0 < q ≤ 1, m > 0.01M and nateda .Thelowerandupperlimitsa anda inthedistributionof j 2 ⊙ 2 l u m > 0.01M ,consistentwithClaeysetal.(inprep.).Thelower thesemi-majoraxesarea =5R anda =5·106R forbothinner 3 ⊙ l ⊙ u ⊙ limitonthestellarmassesisinaccordancewiththelimitfoundby and outer orbits (Kouwenhovenetal. 2007). From these systems Kouwenhovenetal.(2007).Thesecondaryandtertiarymassesare werejectthosethatdonotsatisfyl = a 1−e2 > l = 12AU. 1 1 1 1,l subsequentlycomputedfromthesesampledmassratios.Wesample Lastly, we sample the initial orbital inclin(cid:16)ation(cid:17)angle i from a tot 8 A.S. Hamers,O.R.Pols,J.S.W.Claeys and G. Nelemans Tokovinin TSM1 TSM2 0.25 TokovininsampleHselectedL Mean SD Mean SD Mean SD m1/M⊙ 2.36 1.23 1.85 1.01 2.17 1.24 0.20 TSM1 m2/M⊙ 1.69 0.94 0.93 0.79 1.09 0.93 TSM2 m3/M⊙ 1.53 1.39 1.36 1.23 1.59 1.43 F 0.15 log10(P1/d) 5.13 0.73 5.47 0.81 5.77 1.29 PD log10(P2/d) 7.04 0.76 7.87 0.75 8.28 1.61 0.10 Table1.Meanandstandarddeviation(SD)ofmassesandorbitalperiods 0.05 oftheTokovinin(2008)samplesatisfyingtheselectioncriteria(Sect.4.1) andofbothsampledtriplepopulationsTSM1andTSM2. 0.00 1 2 3 4 5 6 log HP (cid:144)P L distribution that is uniform in cos(i ) with −1 < cos(i ) < 1 10 2 1 tot tot and the initial arguments of periastron of both orbits, g and g , 1 2 Figure4.Probabilitydensityfunctions(PDFs)ofP2/P1fortheTokovinin fromauniformdistributionwith0<gj <2π.Fromthetriplesys- sample with ourselection criteria applied (solid line), the TSM1sample tems obtained in this manner we subsequently reject the systems (dashedline)andtheTSM2sample(dottedline).FortheTokovininsample, that are not dynamically stable based on the stabilitycriterion of errorsbasedonPoissonstatisticsareindicated. Mardling&Aarseth(2001),consistentwithourtripleevolutional- gorithm(cf.Sect.2.3). theTokovininsamplewithourselectioncriteriaappliedarecom- paredtothetwosampledpopulations. Becauseofthesteepslope of −2.70 in the TSM1 IMF the mean masses in TSM1 are small 4.2.2 TSM2 comparedtothoseinTSM2,forwhichaSalpeter-likeslope(-2.35) In the second triple sampling method (TSM2) we use the mul- applies for m > 1.0M . Compared tothe Tokovinin sample the 1 ⊙ tiple system recipe developed by Eggleton (2009). This recipe massesofTSM1andTSM2appeartobeonthelowside,butitis is designed to reproduce the properties of a set of 4558 stellar important tonotethattheTokovinin sampleismagnitude-limited systems with Hipparcos magnitude brighter than 6 collected by ratherthanvolume-limited,thereforeitisstronglybiasedtowards Eggleton&Tokovinin (2008). It is followed until the second bi- moremassive systems. Thedistributions of the orbital periods of furcation,effectivelyrestrictingtomultiplesystemsthataretriple. TSM1andTSM2aresimilar,theorbitalperiodsbeingonaverage Theparametersgivenbytherecipearem1,m2,m3,P1andP2.The somewhathigherinTSM2comparedtothoseinTSM1.Theorbital primarymassdistributioninthisrecipeisdesignedtoresemblethe periodsoftheTokovininsample,notablytheouterorbitalperiods, Salpeter IMF (dN/dm1 ∝ m1−2.35) at large masses and turns over aresmallerthanthoseinthesampledpopulations.Thisisalsolikely for masses below 0.3M⊙. The mass ratio distribution is approxi- affectedbyobservationalbias,becauseverylongorbitalperiodson matelyflatandtheP2 distributionhasabroadpeakaround105d. theorderof107 daysorlongeraregenerallyverydifficulttomea- WerefertoEggleton(2009)forfurtherdetails.Thecorresponding sure. semi-major axes are computed from Kepler’s third law. The pa- Lastly, Fig. 4 compares the distributions of the ratio P /P 2 1 rametersnotprescribedbytherecipeareitot,ej andgj,whichare forthethreepopulations.ForTSM1andTSM2thesedistributions sampledfromthesamedistributionsasinTSM1above.Similarly aresimilar. Thedistribution of P /P of the Tokovinin sample is 2 1 tothemethodinTSM1systemsarerejectedifl1≤l1,l =12AUand peakedtowardssmallerratios,althoughhereobservationalbiasfor iftheydonotsatisfythestabilitycriterionofMardling&Aarseth largeperiodratiosmaybeimportant.Notealsothelargeerrorbars (2001). intheTokovininsampleasaconsequenceoftherathersmallnum- ber of systems in the Tokovinin sample that satisfy our selection criteria(165). 4.2.3 Propertiesandcomparisons Hereweelaborateonsomeofthepropertiesofthesampledandob- 4.2.4 Samplingprocedure servedpopulations.Fig.3(middleandbottom)showsthe(P ,P )- 1 2 diagramforasmallsample(360systems)oftheTSM1andTSM2 WesampleN =2·106triplesystemsforbothpopulationsTSM1 calc populations.TheupperboundariesofP andP inbothpopulations and TSM2. Using the triple algorithm described in Sect. 2 these 1 2 are quite different. In TSM1 P and P are limited by a and a , systemsareevolvedfromt=0,withallthreecomponentsstarting 1 2 l u whereasinTSM2suchsharpboundariesdonotexist.InTSM2the asZAMSstars,toaHubbletimet = 13.7Gyr.Inordertoretain H upperboundaryofP isdependentonP becauseinthissampling sufficientresolutioninhigher-masssystems(2.0 < m /M < 6.5) 2 1 1 ⊙ methodthelargestratioP /P isgivenbyP /P =2·106.These wesplitbothTSM1andTSM2intotwopartsof1·106systems,one 2 1 2 1 wide outer binary systems arevery rare, however. Fig.3 (middle with1.0< m /M < 2.0andonewith2.0 <m /M < 6.5.Inthe 1 ⊙ 1 ⊙ andbottom)alsoshowsthesystemsthataccordingtoourtripleal- resultspresentedinSect5systemsineachpartaregivenappropri- gorithmexperienceamergerintheinnerbinarysystemoradesta- ateweightsdeterminedbythemassfunctionofm toaccountfor 1 bilizationearlyontheMS(cf.Sects.5.1.1and5.3).Suchsystems thefact that theactual number of systemsoccurring inthe1.0 < arenotlikelypartofanobservedsampleoftriplesbecauseoftheir m /M < 2.0rangeislargerthanthatinthe2.0 < m /M < 6.5 1 ⊙ 1 ⊙ shortlifetimes.Becauseoftheirsmallnumber,removingthesesys- range(seeAppendixAfordetails).Severalkeyeventsintheevolu- tems does not significantly affect the distributions of masses and tionarekepttrackof,mostnotablytheonsetofsignificantKCTF, orbitalperiods(themeanvaluesareaffectedbytypically1%). CE evolution, a merger in the inner binary system and a desta- InTable1thedistributionsofthemassesandorbitalperiodsof bilization of the triple system. We thus obtain a catalogue of the Triplepopulationsynthesisandwhitedwarfmergers 9 k Description includesbothCEevolutiontriggeredinhighlyeccentricorbitsand CEevolutiontriggeredbyRLOF(incircularorbits).Thefraction 0 MS(m.0.7M⊙) f includes the possibility for multiple phases of CE evolution. 1 MS(m&0.7M⊙) CE Lastly,thequantity f isdefinedasthefractionofsystemsineach 2 Hertzsprunggap(HG) EC 3 redgiantbranch(RGB) channelinwhichamergeroccursduetoaneccentriccollision.We 4 coreheliumburning(CHeB) definesuchacollisiontooccur iftheinner binaryperiastrondis- 5 earlyasymptoticgiantbranch(EAGB) tanceissmallerthanorequaltothesumoftheinnerbinaryradii, 6 thermallypulsingAGB(TPAGB) i.e.ifa (1−e )≤R +R andwheninthecorrespondingcircular 1 1 1 2 7 nakedheliumstarMS(HeMS) casetherewouldnotbeacollision,i.e.a >R +R . 1 1 2 8 nakedheliumstarHertzsprunggap(HeHG) AnimportantresultofoursimulationsshowninTable3isthat 9 nakedheliumstargiantbranch(HeGB) in≈ 8%ofallsystemsaninnerbinarymergeroccurs,in≈ 6%of 10 heliumwhitedwarf(HeWD) all systemsno inner binary merger occurs but the orbit isshrunk 11 carbon-oxygenwhitedwarf(COWD) significantly and that ≈ 10% of all systems become dynamically 12 oxygen-neonwhitedwarf(ONeWD) unstableatsomepointintheevolution.Thismeansthatin≈24% 13 neutronstar(NS) 14 blackhole(BH) ofallsystemsthetertiarysignificantlyalterstheinnerbinaryevo- 15 masslessremnant lution whereas in the absence of the tertiary there would not be suchinteraction.Takingintoaccountthatsystemsinwhichthein- Table2.DescriptionofthedifferentstellartypesusedinFig.5.Identicalto nerbinarycomponentsmergeduringtheMS(≈4%)orinwhicha thetypesusedinHurleyetal.(2002). dynamicalinstabilityoccursduringtheMS(≈ 4%)arenotlikely to be observed (cf. Sect. 5.1) this estimate is reduced by ≈ 8%. This implies that the tertiary significantly alters the inner binary evolutionary outcomes where we distinguish between three main evolutionin≈ 16%/(1−0.08) ≈ 17%ofthe“observable”triples, channels:innerbinarymergers,noinnerbinarymergersandtriple i.e.triplesinwhichthehierarchicalstructureisnotdisruptedinan destabilizations.Inaverysmallnumberofsystems(55and44for earlystageintheevolution. TSM1andTSM2respectively)thecalculationoftheevolutioncan- A second important feature evident from Table 3 is that the notbecompletedduetoconvergenceerrorsintheODEsolverrou- resultsforTSM1andTSM2aregenerallysimilar,eventhoughthey tine(cf.Sect.2);thesesystemsareexcludedfromtheresultsbelow. have been sampled by different methods. To provide insight into InthesecasestheerrorsoccurjustpriortoaMSmergerorRGB+ the dependence of the probabilities of the main channels on the MSmerger.Becausetheaffectedsystemsconstituteatinyfraction initialparametersweshowinFig.6theseprobabilities(expressed ofallsystemsforwhichaMSmergerorRGB+MSmergerapplies asfractions)asfunctionoftheinitialparametersq ≡ m /m ,a , 1 2 1 1 thisdoesnotaffectourconclusions. βandi .WewillrefertothisfigureandTable3whendiscussing tot thethreemainchannelsinmoredetailbelow. 5 POPULATIONSYNTHESIS:RESULTS 5.1 Innerbinarymergers Wepresentthemainchannelsthatwefindinourpopulationsynthe- As a consequence of high-amplitude eccentricity cycles induced sisstudyinTable3,wherethefirsttwocolumnsshowtheprobabil- intheinnerorbitthecomponentsintheinnerbinarymayatsome itiesP(inpercent)ofthemainchannelsandseveralsubchannels pointmerge,eitherdirectlybecauseofanorbitalcollision(included ofbothsampledpopulationsTSM1andTSM2.Inadditiontothis inTable3inthefraction f )orindirectlyasaresultofstrongtidal table,Fig.5showsmoredetailedinformationonthelikelihoodof EC frictioninducedbyhigheccentricity,possiblyinvokingCEevolu- thedifferentcombinationsofinnerbinarymergerswherethestellar tion.Inthelattercasetheinnerbinarycaneithermergeduringthe typesusedaredefinedinTable2.Apost-MSmergerisdefinedasa CEorsurvivetheCEinatightorbitand(possiblyafteranotherCE mergerbetweenapost-MS(non-compactobject)primaryandany invokedbythesecondary)subsequentlymergeasaresultoforbital secondary,whereasacompactobjectmergerisdefinedasamerger energy loss due to GW emission. We separate further discussion betweenacompactobjectprimaryandanysecondary.Inaddition into mergers occurring on the MS, after the MS and mergers in- totheprobabilitiesofthechannelsinformationontheoccurrence volvingaprimarycompactobject. of KCTF, CE evolution and the nature of the merger (i.e. occur- ring in a circular or highly eccentric orbit) is given by means of thequantities f , f and f . Wedefine f asthefraction KCTF CE EC KCTF 5.1.1 MSmergers ofsystemsineachchannelinwhich|e˙ | > 10−18s−1 (suchthat 1,STD thereisasignificantchangeine duetoSTDinaHubbletime)and As Table 3 demonstrates most mergers occur on the MS and for 1 0.1|e˙ |<|e˙ |<|e˙ |(i.e.e˙ ande˙ areofcomparable almostallofthelatterthemergeroccursthroughaneccentriccol- 1,TF 1,STD 1,TF 1,STD 1,TF orderofmagnitude).Thiscanhappenatvariouspointsintheevolu- lision. In the latter case −4 . log (1−e ) . −5 just prior to 10 1 tionandisstronglyinfluencedbytheenvelopestructureoftheinner merger.IntheBinary calgorithmitisassumedthattheresultof binary stars because thisdetermines the effectiveness withwhich themerger isasingleMS starwithmassm +m ;depending on 1 2 tidalenergycanbedissipated.Inparticularifanyinnerbinarystar thelattervaluethefinalremnantiseitheraCOWD(mostcases), possessesaconvectiveenvelope(mostnotablylow-massMS,HG, an ONe WD or a NS (following a core-collapse supernova). We RGBandAGBstars)strongtidalfrictionislikelyife isalsohigh find that octupole terms in the STD are very important for MS 1 duetoKozaicycles.InTable3adistinctionismadebetweenthe mergers,whichisreflectedbythelargevaluesof|ǫ |(cf.Eq.2) oct typesofthestarinwhichtidalenergyisdissipatedfortheduration prior to merger for these systems; the latter quantity is narrowly ofKCTF.Furthermore, f isdefinedasthefractionofsystemsin distributedaround|ǫ | ≈ 10−1.5.Thiscanbeattributedtoacom- CE oct eachchannelinwhichCEevolutionisatsomepointinvoked.This binationofinitiallysmallsemi-majoraxisratiosβ ≡ a /a ,small 2 1 10 A.S.Hamers, O.R.Pols,J.S.W.Claeys and G.Nelemans 1 1 2 2 100 3 3 4 4 5 5 k1 6 k1 6 7 7 8 8 9 10-6 9 10 10 11 11 12 12 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12 k k 2 2 Figure5.Left:thenumberofsystemsinTSM1thatexperienceaninnerbinarymergerforeachoccurringcombinationofstellartypesjustpriortomerging, normalized tothetotalnumberofmergersystems(thatconstitute ≈ 8%ofallsampledtriplesystems).Right:thenumberofsystemsinTSM1inwhich themergeroccursduetoaneccentriccollision,againnormalizedtothetotalnumberofmergersystems.Thestellartypesareidenticaltotheonesusedin Hurleyetal.(2002)andarerepeatedforconvenienceinTable2. inner binary mass ratiosq ≡ m /m and high outer orbit eccen- 1.0 1 2 1 tricitiese . Fig. 6 indeed shows that the fraction of MS mergers, 2 fMSmerge, decreases significantly withincreasing q1 and is peaked 0.8 MSmergers towards small β around 0.5 . log10(β) . 2.0. The MS mergers MSdestabilizations are also characterized by high initial inclination angles (cf. Fig. 0.6 6).Notethatamaximumin f occursatinclinationsslightly F MSmerge D aboveitot =90◦,i.e.forretrogradeorbits,whichisconsistentwith C 0.4 Shappee&Thompson (2012) who find an asymmetry in cos(i ) tot towardsnegativevaluesfortheprobabilityoftheeccentricKozai mechanism to occur. With regards to the dependence of f 0.2 MSmerge ona itmightbeexpectedthatMSmergerslikelyoccurforsmall 1 initiala asatighterorbitmakesaneccentriccollisionmorelikely. 0.0 1 However,wefindthatMSmergersonaveragehavelargeinitiala -6 -5 -4 -3 -2 -1 0 1 withamaximum of fMSmerge around a1 ≈ 102.8AU ≈ 6·102AU log10Ht(cid:144)ΤMSL (cf.Fig.6).Thisimpliesthattheeffectofincreasingtheeccentric- itymaximawithdecreasingβ(andthusincreasinga )isstronger Figure7.CumulativedistributionsforTSM1ofthetimestofMSmerger 1 thanthe effect thateccentric mergersbecome more unlikely with (solid line) and MS destabilization (dashed line) normalized to the MS increasinga1ife1werefixed. timescale,estimatedasτMS=10(m1/M⊙)−2.8Gyr. As Table 3 shows, for a small fraction of MS mergers (i.e. about0.009ofallMSmergersforTSM1)themergerdoesnotoc- sionoccursduringthefirstfewKozaicycleswhichtypicallyhave curduetohigheccentricity,i.e.themergeroccursinacircularor- timescalesofafewMyr,asmallfractionoftheMSlifetimeofthe bit.InthesecasesKCTFisresponsibleforstrongorbitalshrinkage primarystarsconsidered inthetriplesample.Thetimeofmerger leadingtoatightinnerorbit,eventuallyresultinginamergerina is thus mainly determined by the Kozai period P ∝ (P /P )P circularorbitduetoGWemissionwithinaHubbletime.Thefrac- K 2 1 2 andhencetheorbitalperiods.Weindeedfindthatthedistribution tionofMSmergersystemsinwhichthisoccurs(0.009forTSM1) ofthetimesofMSmergersisverysimilartotheinitialdistribution isindeedcomparablewiththefractionofMSmergersystemsthat of the Kozai periods. The fact that most MS mergers occur early satisfy our KCTF criterion on the MS (0.010 for TSM1); this is on the MS makes it unlikely to observe these systems as triples. alsothecaseforTSM2.WefindthatsuchstrongKCTFontheMS AsimilarconclusionappliestotheMSdestabilizations(Sect.5.3) occursonly inlow-massinner binarysystems(m < 1.25M )in 1 ⊙ thathappenevenearlier. whichthecomponents possessconvective envelopes (i.e.thesce- ArelevantquestionfortheMSmergersiswhetherthemerged narioofFabrycky&Tremaine2007). MSremnant couldevolvetoabluestraggler star,i.e.astarmore The eccentricMS mergers, which constitute thebulkof MS luminousthanstarsattheMSturn-offpoint.Thistriplebluestrag- mergers, typically occur early on the MS, with the majority of gler scenario has been proposed by Perets&Fabrycky (2009). If mergers(about70%)occurringwithin10%oftheprimaryMSlife- theMSstarsmergeveryearlyintheevolutionthentheageofthe time (Fig. 7, solid line). This is because in many cases the colli- merged MS star would appear to be consistent withitsmass and

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