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

0.57 MB·English

by  Adrian S. Hamers

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