Theoretical uncertainties of the type Ia supernova rate

0.66 MB·

by  J. S. W. Claeys

Astronomy&Astrophysicsmanuscriptno.SNIa-popsyn (cid:13)c ESO2014
January14,2014
Theoretical uncertainties of the type Ia supernova rate
J.S.W.Claeys1,O.R.Pols1,R.G.Izzard2,J.Vink3,andF.W.M.Verbunt1
1 DepartmentofAstrophysics/IMAPP,RadboudUniversityNijmegen,P.O.Box9010,6500GLNijmegen,TheNetherlands;
2 ArgelanderInstituteforAstronomy,UniversityofBonn,AufdemHu¨gel71,D-53121Bonn,Germany;
3 AntonPannekoekInstitute/GRAPPA,UniversityofAmsterdam,POBox94249,1090GEAmsterdam,TheNetherlands;
e-mail:[email protected], [email protected]
Received20September2013;accepted12January2014
4 ABSTRACT
1
0 It is thought that type Ia supernovae (SNe Ia) are explosions of carbon-oxygen white dwarfs (CO WDs). Two main evolutionary
2 channelsareproposedfortheWDtoreachthecriticaldensityrequiredforathermonuclearexplosion:thesingledegeneratescenario
(SD),inwhichaCOWDaccretesfromanon-degenerate companion, andthedoubledegeneratescenario(DD),inwhichtwoCO
n
WDsmerge.However,itremainsdifficulttoreproducetheobservedSNIaratewiththesetwoscenarios.
a
J WithabinarypopulationsynthesiscodewestudythemainevolutionarychannelsthatleadtoSNeIaandwecalculatetheSNIarates
andtheassociateddelaytimedistributions.WefindthattheDDchannelisthedominantformationchannelforthelongestdelaytimes.
3
TheSDchannelwithhelium-richdonorsisthedominantchannelattheshortestdelaytimes.Ourstandardmodelrateisafactorfive
1
lowerthantheobservedrateingalaxyclusters.

Weinvestigatetheinfluenceofill-constrainedaspectsofsingle-andbinary-starevolutionanduncertaininitialbinarydistributionson
]
R therateoftypeIaSNe.Thesedistributions,aswellasuncertaintiesinbothheliumstarevolutionandcommonenvelopeevolution,have
thegreatestinfluenceonourcalculatedrates.InefficientcommonenvelopeevolutionincreasestherelativenumberofSDexplosions
S
suchthatforα = 0.2theydominatetheSNIarate.Ourhighest rateisafactor threelessthanthegalaxy-clusterSNIarate,but
ce
h. compatiblewiththeratedeterminedinafield-galaxydominatedsample.IfweassumeunlimitedaccretionontoWDs,tomaximize
p thenumberofSDexplosions,ourrateiscompatiblewiththeobservedgalaxy-clusterrate.
-
o Keywords.Stars:evolution-Binaries:general-Supernovae:general
r
t
s
a 1. Introduction servers to detect them as SSXS. Moreover, during the super-
[
novaexplosionsome of the materialfrom the donorstar is ex-
1 Type Ia supernovae (SNe Ia) are important astrophysical phe- pected to be mixed with the ejecta, which has never been con-
v nomena. On the one hand they drive galactic chemical evolu- clusively observed(Leonard2007; Garc´ıa-Senzetal. 2012). In
5 tion as the primary source of iron, on the other hand they are some cases NaD absorption lines have been observed, which
9 widelyusedascosmologicaldistanceindicators(Phillips1993; can be interpreted as circumstellar material from the donor
8 Riessetal. 1996, 1998;Perlmutteretal. 1999)becauseoftheir star (Patatetal. 2007; Sternbergetal. 2011). Furthermore, the
2 homogeneouslightcurves.Evenso,theexactprogenitorevolu- donor star is expected to survive the supernova explosion and
1. tion remains uncertain.It is generallyaccepted that SNe Ia are to have a high space velocity. In the case of the 400 year old
0 thermonuclear explosions of carbon-oxygenwhite dwarfs (CO Tychosupernovaremnant,wewouldexpecttoobservethissur-
4 WDs; Nomoto1982;Bloometal. 2012).The explosioncanbe viving star. Nevertheless, no such object has been unambigu-
1 triggered when the CO WD reaches a critical density, which ously identified (Ruiz-Lapuenteetal. 2004; Kerzendorfetal.
v: is reached when the mass approachesthe Chandrasekhar mass 2009; Schaefer&Pagnotta 2012). Some SNe and supernova
i (MCh =1.4M⊙,fornon-rotatingWDs).SinglestarsformaCO remnants show evidence of interaction with circumstellar ma-
X WDwithamassuptoabout1.2M⊙ (Weidemann2000).Toex- terial which links them to the SD channel(Hamuyetal. 2003;
r plainhowtheWDthenreachesMChtwomainchannelsarepro- Chiotellisetal. 2012; Dildayetal. 2012). However,a variation
a posed:thesingledegeneratechannel(SD; Whelan&Iben1973; on the DD channel, in which the merger occurs during or
Nomoto1982),inwhichtheWD accretesmaterialfromanon- shortlyaftertheCEphase(Hamuyetal.2003;Chevalier2012;
degeneratecompanion;andthedoubledegeneratechannel(DD; Sokeretal. 2013), cannot be excluded as the progenitor chan-
Webbink 1984; Iben&Tutukov 1984), in which two CO WDs nelfor these SNe. Additionally,SN 2011fe,a SN Ia which ex-
merge. ploded in a nearby galaxy, showed no evidence of interaction
Both channels cannot fully explain the observed properties with circumstellar material, in radio, X-ray and optical (e.g.
of SNe Ia (Howell 2011). WDs in the SD channel burn ac- Marguttietal.2012;Chomiuketal.2012;Patatetal.2013)and
cretedmaterialintocarbonandoxygen,andthereforethesesys- pre-explosionimagesexcludemosttypesofdonorsstarsofthe
tems should be observed as supersoft X-ray sources (SSXS, SDchannel(Lietal.2011a).
vandenHeuveletal. 1992), notenoughof which are observed
to explain the number of SNe Ia (Gilfanov&Bogda´n 2010; These difficulties in reconciling the observational evidence
DiStefano2010).However,Nielsenetal.(2013)arguethatonly with the predictions of the SD channel are avoided in the DD
a small amount of circumstellar mass loss is able to obscure channel, but this channel has its own difficulties. In particular,
these sources, making it difficult or even impossible for ob- Nomoto&Kondo(1991)showthatthemergerproductevolves
1
J.S.W.Claeysetal.:TheoreticaluncertaintiesofthetypeIasupernovarate
towards an accretion-induced collapse (AIC) rather than to-
wardsanormalSNIa.Onlyrecently,somestudiesindicatethat
1.2
the merger can lead to a SN Ia explosion under certain cir- Hurley et al. 2000
cumstances (Pakmoretal. 2010, 2011, 2013; Shenetal. 2013; Karakas et al. 2002
1.1
Danetal. 2013). Finally, binary population synthesis studies Our model
show that neither channel reproduces the observed SN Ia rate
1.0
(e.g. Ruiteretal. 2009; Mennekensetal. 2010; Toonenetal.
2012;Boursetal.2013;Chenetal.2012).
)⊙0.9
M
We investigate with a binary population synthesis (BPS)
(
codetheprogenitorevolutiontowardsSNeIathroughthecanon- Mf0.8
icalChandrasekharmasschannels:theSDandtheDDchannels.
Ourcodemakesitpossibletostudylargestellarpopulations,to 0.7
calculateSNIaratesandtocomparethesewiththeobservedrate
ofSNeIa.Weanalysenotonlythedifferentevolutionarychan- 0.6
nels,butalso–inthecaseoftheSDchannel–determinegeneral
propertiesofthedonorstaratthemomentoftheexplosion,and– 0.5
0 1 2 3 4 5 6 7
inthecaseoftheDDchannel–ofthemergerproduct.TheSNIa Mi(M )
⊙
rate hasbeenstudiedby severalgroupsthroughbinarypopula-
Fig.1.Initial(M)versusfinal(M )massofsinglestarsthatbe-
tionsynthesis,howevertheyusedonlyastandardmodel,with- i f
come CO WDs for different TPAGB models (Sect.2.1.1). The
outinvestigatingingreatdetailtheuncertaintiesintheirmodel,
dottedlineshowstheresultsoftheHurleyetal.(2000)-models,
ortheyinvestigatedoneuncertaintyinbinaryevolution,suchas
the full line shows the results for the fits to the models of
the commonenvelopeevolution(e.g.Yungelson&Livio2000;
Karakasetal.(2002),andthedashedlineshowsourmodel.
Ruiteretal. 2009; Wangetal. 2009a, 2010; Mennekensetal.
2010; Toonenetal. 2012). We performa studyof the mainpa-
rameters that play a role in the evolution towards type Ia SNe
whichareill-constrained.We studytheeffectsonthepredicted
2.1.1. TP-AGBmodels
rateandontheprogenitorevolutiontowardsSNeIa.
In Sect.2, the binary population synthesis code and our Theevolutionofthecore,luminosityandradiusoftheTP-AGB
modelaredescribed.InSect.3,thegeneralprogenitorevolution starversustimearebasedonthemodelsofKarakasetal.(2002),
as it follows from our standard model is outlined. The result- with the prescriptions described in Izzardetal. (2004, 2006).
ing SN Ia rate is discussed in Sect.4, while Sect.5 and 6 treat Because the core masses are determined without taking over-
theeffectsofvaryingtheparametersontheprogenitorevolution shootingduringpreviousevolutionphases into account,a fit is
and the total rate. Finally, Sect.7 discusses the validity of the madetotakeovershootingintoconsideration,basedonthecore
Chandrasekharmodel. massescalculatedbyHurleyetal.(2002)duringtheearly-AGB
(E-AGB). Up to and including the first thermalpulse, the core
mass is calculated by the formulae described by Hurleyetal.
(2002).Asmoothfunctionisbuiltintoguaranteeacontinuous
transition of the radius between the E-AGB and the TP-AGB.
2. Binarypopulationsynthesis COWDsformfrominitialstarswithmassesupto6.2M⊙,which
correspondstoamaximumCOWDofabout1.15M (Fig.1).
⊙
We employtherapidbinaryevolutioncodebinary c/nucsyn
basedontheworkofHurleyetal.(2000,2002)withupdatesde-
2.1.2. Heliumstarevolution
scribedinIzzardetal.(2004,2006,2009)andbelow.Wediscuss
thekeyassumptionsofourmodelofsingleandbinarystarevo- Inourmodel,ifahydrogen-richstarisstrippedofitsouterlayers
lution.InTable1welisttheassumptionsinourstandardmodel after the main-sequence(MS) but beforethe TP-AGB, and the
andgiveanoverviewofpossibleotherassumptions,andthede- helium core is not degenerate, a helium star is formed. If the
pendence of the theoretical SN Ia rate on them is discussed in exposedcoreisdegenerateaHeWDismade.Theprescriptions
Sects.5and6.Thiscodeservesasthebasisofourbinarypopu- todescribetheevolutionofnakedheliumstarsarediscussedin
lationsynthesis(BPS)calculations. Hurleyetal.(2000).However,becauseofitsimportanceforthe
formationoftypeIaSNeweemphasizesomedetails.
A distinction is made between three phases of helium star
2.1.Singlestarevolution evolution:He-MS,theequivalentoftheMS,whenheliumburns
in the centre of the helium star; He-HG, the equivalent of
Oursinglestarmodelsareanalyticfitstodetailedstellarmodels, the Hertzsprung gap (HG), when He-burning moves to a shell
describedinHurleyetal.(2000),withupdatesofthethermally aroundtheHe-depletedcore;andHe-GB,the equivalentofthe
pulsating asymptotic giant branch (TP-AGB) which are fits to giantbranch(GB), whenthe heliumstar hasa deepconvective
themodelsofKarakasetal.(2002),asdescribedinIzzardetal. envelope.Ifanon-degenerateheliumcoreisexposedduringthe
(2006).InthepresentstudyweadoptametallicityZof0.02.We E-AGB, a star on the He-HG forms, otherwise a He-MS star
assumethatonlyaCOWDcanexplodeasaSNIabasedonthe forms.
workofNomoto&Kondo(1991) whofoundthatan ONe WD A He-MS star becomes a He-WD when its mass is less
that reaches M undergoesAIC. CO WDs are formedonly in than0.32M ,theminimummassnecessarytoburnhelium.The
Ch ⊙
lowandintermediatemassbinarysystems,withaprimarymass boundary between a helium star forming a CO WD or a ONe
upto10M ,thereforewelimitourdiscussiontothismassrange. WDisdeterminedbythemassofthestarattheonsetoftheHe-
⊙
2
J.S.W.Claeysetal.:TheoreticaluncertaintiesofthetypeIasupernovarate
Table1.Physicalparametersofsingleandbinaryevolutioninourmodelanddistributionfunctionsofinitialbinaryparametersfor
thestandardmodelandtheirvariations(usedinSects.5and6).
Nameparameter Reference Standardmodel Variation
η(R75) Eq.1 0.5 5
TP-AGBwind Eq.2-3 HPT00 KLP02,R75(η=1),B95,VL05
α Eq.6 1.5 5
BH
B Eq.7 0.0 103
wind
Q a Table2 0.25 0.5
crit,HG
Q Table2 Q =Q =0.25 Q =Q =1.28
crit,He−stars crit,He−HG crit,HG crit,He−HG crit,He−GB
α Eq.8 1.0 0.2-10
ce
CEaccretionb Eq.8 0.0 0.05M
⊙
λ Eq.9&AppendixA variable 1
ce
λ Eq.9&AppendixA 0.0 0.5
ion
σ Eq.14 10 1,20,∞c
γ Eq.15 M /M 2,M /M
RLOF d a a d
γ Eq.15 M /M 2
wind a d
η ,η Eq.B.3&B.5 HKN96 1
H He
ψ(M ) Sect.2.3 KTG93 S98,K01,C03,B03
1,i
φ(q)∝qx Sect.2.3 x=0 −1≤ x≤1
i i
Notes.(a) Onlyinthecaseofmasstransferwithanon-degenerateaccretor.(b)OnlywhenthecompanionisaMSstar.(c) Thesymbol∞implies
conservativemasstransfertoalltypesofstars.
References.B03=Belletal.(2003),B95=Bloecker (1995),C03=Chabrier(2003),HKN96=Hachisuetal.(1996),HPT00=Hurleyetal.
(2000),HPT02=Hurleyetal. (2002),K01=Kroupa (2001),KLP02=Karakasetal.(2002),KTG93=Kroupaetal.(1993),R75=Reimers
(1975),S98=(Scalo1998),VL05=vanLoonetal.(2005).
HG. The mass of the star on the He-HG shouldexceed1.6M dependingonwhichofthetwoisstronger,
⊙
in order to form a dense enoughcore to burn carbon and form
anONe-core,whichisbasedondetailedmodels(carbonignites M˙ =max(M˙ ,M˙ ). (5)
off-centrewhen M & 1.08M ,Polsetal.1998).Theevo- He−star WR R
CO,core ⊙
lutionofaHe-HGstarwith M < 1.6M leadstotheformation
⊙
ofaCOWDwithamassgreaterthan1.2M⊙unlessitsenvelope 2.2.Binarystarevolution
islostbywindorbinarymasstransfer.
Binaryevolutioncansignificantlyimpacttheevolutionofthein-
dividualstars in a binarysystem. Binary evolutionis primarily
2.1.3. Windmassloss
determinedbytheinitialsemi-majoraxis(a)andthemassesof
Weadoptforbothlow-andintermediate-massstarstheprescrip- thetwostars.Inthewidestsystems(a&105R⊙)thestarsdonot
tionbasedonReimers(1975)duringtheHGandbeyond,multi- interact,butevolveasthoughtheyweresingle.Inclosersystems
pliedbyafactorη, (a . 105R⊙) interaction with the wind of the companion star
can alter the evolution.Inevencloser systems(a . 3·103R )
⊙
R L M
M˙ =η·4.0·10−13 ⊙ M yr−1, (1) Roche-lobeoverflow(RLOF)orcommonenvelope(CE)evolu-
R ⊙
R⊙ L⊙ M tion often has dramatic consequences for the further evolution
ofthetwostars(Hurleyetal.2002).Thesedifferentinteractions
where M is the mass, R the radius and L the luminosity of the
arediscussedbelow.
star, and η = 0.5 in our standard model (Table 1). During the
TP-AGB we use the prescription based on Vassiliadis&Wood We assume the initial eccentricity(ei) is zero,based on the
workofHurleyetal.(2002)whoshowthattheevolutionofclose
(1993),
binarypopulationsisalmostindependentoftheinitialeccentric-
log M˙VW =−11.4+0.0125 P0 −100max M −2.5,0.0 ,ity.Weusethesubscriptsdandaforthedonorandaccretingstar,
M⊙yr−1! d " M⊙ #! respectively,andifortheinitialcharacteristicsofthestaronthe
(2) zero-agemain-sequence(ZAMS).Weusethesubscripts1and2
whereP istheMirapulsationperiodindays(d),givenby fortheinitiallymoremassivestar,theprimary,andinitiallyless
0
massivestar,thesecondary,respectively.ThemassratioM /M
a d
P M R
log 0 =min 3.3,−2.7−0.9log +1.94log . isdenotedbyQ,whiletheinitialmassratioqiis M2,i/M1,i.
(cid:18) d (cid:19) " M⊙# "R⊙#!
(3)
The wind is limited by the steady superwind rate, M˙ = 2.2.1. Interactionwithastellarwind
SW
1.36(L/L )M yr−1.
⊙ ⊙
Masslostintheformofastellarwindisaccretedbythecompan-
In our model the wind of helium stars is either Reimers-
ion at a rate that dependson the mass loss rate, the velocityof
like (Eq.1) or Wolf-Rayet-(WR) like (Hamannetal. 1995;
thewindandthedistancetothecompanionstar.Todescribethis
Hamann&Koesterke1998),
processandtocalculatetherateM˙ atwhichmaterialisaccreted
a
M˙ =10−13(L/L )1.5M yr−1, (4) by the companion star, we use the Bondi-Hoyle prescription
WR ⊙ ⊙
3
J.S.W.Claeysetal.:TheoreticaluncertaintiesofthetypeIasupernovarate
Table2.ThecriticalmassratioQ forstableRLOFfordiffer- stable.InTable2welistthevaluesofthecriticalmassratiosfor
crit
ent types of donor stars, in the case of a non-degenerateand a the different phases of the evolution of a star during accretion
degenerateaccretor. ontoeitheranon-degenerateoradegeneratestar.
RLOFfromaMSdonorisgenerallystable,becausearadia-
tive MS star shrinksin reactionto mass loss. When materialis
Typeofdonor Non-degenerate Degenerate transferredtoaMScompanion,RLOFisstableonlyforcertain
accretora,b accretora mass ratios because a radiative accreting star expands and can
MS,M>0.7M c 0.625d(MSaccretor) – fill its own Roche lobe. If the mass ratio is less than 0.625 the
⊙
MS,M<0.7M c 1.44 1.0 systemevolvesintoacontactsystemandweassumethetwoMS
⊙
HG 0.25 0.21 stars merge (deMinketal. 2007). The stability criteria for the
GB,AGBe 2.13/ 1.67−x+2 Mc,d 5 0.87 othertypesofdonorstarstransferringmasstoanon-degenerate
He-HG (cid:20) 0.25 (cid:16) Md (cid:17) (cid:21) 0.21 companionarecalculatedaccordingtoHurleyetal.(2002).
To calculate Q of WDs, we assume that during RLOF
He-GB 1.28 0.87 crit
barelyanythingisaccretedontotheWD,β= 0.01,andthespe-
WD – 1.6
cificangularmomentumoftheejectedmaterialisthatoftheor-
Notes.(a)Thesymbol–indicatesthatnovalueisdefined.(b)Basedon bitoftheaccretingstar(Hachisuetal.1996).Becauseofthelow
Hurleyetal.(2002).(c)WedistinguishbetweenMSstars(M>0.7M ) accretionefficienciesofWDs(AppendixB),thecriticalmassra-
⊙
and low-mass MS stars (M < 0.7M ), because the latter are almost tioforstablemasstransferdecreases.Ouradoptedcriticalmass
⊙
completely convective, and react differently to mass loss and gain. ratios for mass transfer from non-degenerate donor stars onto
(d) Based on deMinketal. (2007). (e) At solar metallicity x ≈ 0.3 a WD are calculated with the formulae from Sobermanetal.
(Hurleyetal.2000,2002).
(1997,Chiotellispriv.comm.).
2.2.3. Commonenvelopeevolution
(Bondi&Hoyle1944;Hurleyetal.2002),morespecifically,
If the donor star is an evolved star and the mass ratio of
2 the Roche-lobe overflowing system is less than its critical
GM α 1
M˙ =min 0.8M˙ ,−1 a BH M˙ , (6) value (Table2), the system evolves into a CE. We use the α-
a  d  v2  2a2 (1+v2 )3/2 d
with M˙ themasslossratefrodmthedonorstarre,l etheeccentric- pwrheiscchritphteiobnintodidnegsecnriebregyth(iEsbcinodm)polfetxhephCaEseis(Wcoembbpianrked1w98it4h),thine
d orbital energy (E ) of the system to calculate the amount the
orb
ity,athesemi-majoraxis,v thewindvelocityofthedonorstar
d orbit should shrink in order to lose the envelope of the donor
whichwe assumeequals0.25timestheescapevelocityaspro-
star.Morespecifically,α isdefinedby,
ce
posedby Hurleyetal. (2002), and v = |v /v |, with v the
rel orb d orb
orbital velocity. The Bondi-Hoyle accretion efficiency parame- E =α (E −E ), (8)
bind,i ce orb,f orb,i
ter,α ,issetto3/2inourstandardmodel(Table1).Whenboth
BH
stars lose mass through a wind they are treated independently, wherethesubscriptsiandfcorrespondtothestatesbeforeand
nointeractionofthetwowindsisassumed. aftertheCEphase,respectively,andwhere
SomeobservedRSCVnsystemsshowthatthelessevolved
star is the more massive of the binary before RLOF occurs, M M
E =−G d env,d , (9)
therefore it has been suggested that wind mass loss is tidally bind,i λ R !
ce d
enhanced by a close companion (Tout&Eggleton 1988). It is
uncertainifthisphenomenonoccursforothertypesofbinaries. and M , M andR arethemass,theenvelopemass,andthe
d env,d d
Toapproximatethiseffectthefollowingformulaisimplemented radiusofthedonor,respectively,andGisthegravitationalcon-
(Tout&Eggleton1988), stant.
The CE efficiency parameter,α , describeshow efficiently
ce
1 R 6 theenergyistransferredfromtheorbittotheenvelope.Itsvalue
M˙ = M˙ 1+B ·max , , (7)
R wind "2 R #  is expected to be between 0 and 1, although it could be larger
whereRListheRocheradiusofthestar(EggLleton1983).Inour tehrgany.1Inifoaunrosthtaenrdeanredrgmyosdoeulrwceeitsakaveaαilcaeb=le,1su(sceheaTsanbulecl1e)a.rTehne-
standardmodelBwind =0(Table1). parameter λce depends on the relative mass distribution of the
envelope, and is not straightforward to define (Ivanova 2011).
According to Ivanova (2011) its value is close to 1 for low-
2.2.2. StabilityofRLOF
mass red giants and therefore in many studies taken as such.
Whetherornotmasstransferisstabledependson:1)thereaction In our model λ is variable and is dependent on the type of
ce
ofthedonor,becauseastarwithaconvectiveenveloperesponds star, its mass and luminosity. We use a prescription based on
differentlytomasslossthanastarwitharadiativeenvelope;2) Dewi&Tauris(2000, ourAppendixA),whichgivesa valueof
theevolutionoftheorbit,whichitselfdependsontheaccretion λ between0.25and0.75forHGstars,between1.0and2.0for
ce
efficiency(β = M˙ /M˙ )andthemassratioQ = M /M ;and3) GB and AGB stars and λ = 0.5 for helium stars. Because of
a d a d ce
the reaction of the accreting star. In our model the criterion to theshorttimescaleofthe CEphase(e.g.Passyetal.2012), we
determine whether mass transfer is stable is given by a critical assume in our standard model that no mass is accreted by the
massratioQ ,whichdependsonthetypesofdonorandaccret- companionstar(Table1).
crit
ingstars(Table2).DuringRLOFthemassratioQofthebinary Additional energy sources can boost the envelope loss,
systemiscomparedwith Q :if Q < Q masstransferisdy- e.g.the ionization energy of the envelope. The extended enve-
crit crit
namicallyunstableandaCEphasefollows,otherwiseRLOFis lope can become cool enough that recombination of hydrogen
4
J.S.W.Claeysetal.:TheoreticaluncertaintiesofthetypeIasupernovarate
occursin the outerlayers. In ourmodelthe fractionof thisen- theBPScodeisuptoaboutafactorthree(forMSdonors)and
ergywhichisusedisexpressedbyλ ,whichisbetween0and1 a factor five (for HG donors) larger than the maximum from
ion
(AppendixA).Inourstandardmodelthiseffectisnotconsidered thedetailedstellar evolutioncode,withthedurationofthermal
andλ =0(Table1). timescale mass transfer correspondingly shorter. Additionally,
ion
wefindsimilardurationsoftheentiremasstransferphasecalcu-
latedwithbothcodes.
2.2.4. StableRLOF
In addition, the accretor adjusts its structure to the ac-
When mass transfer is stable (Q > Q , Table2), the mass cretedmass.Ifthemasstransferrateishigherthanthethermal
crit
transfer rate is calculated as a function of the ratio of the stel- timescalerateoftheaccretor,theaccretingstarisbroughtoutof
lar radius of the donor R and the Roche radius R (based on thermalequilibrium,resultinginexpansionandadditionalmass
d L
Whyte&Eggleton1980), lossfromtheaccretor.Consequently,duringRLOFthefraction
oftransferredmaterialthatisaccretedisnottakentobeconstant,
R 3 M 2 butdependsonthethermaltimescaleoftheaccretor.Formoder-
M˙ = f·3.0·10−6 log d min d,5.0 M yr−1. (10)
⊙ atelyunevolvedstars(starsontheMS,HGorheliumstars)the
"RL#! " M⊙ #!
accretionefficiencyβiscalculatedinourmodelasfollows
The last, mass-dependent factor in the equation is defined by
Hurleyetal.(2002)forstabilityreasons.InHurleyetal.(2002) β≡ M˙a =min σM˙KH,a,1 , (14)
thedimensionlessfactor f is1,howeverthisunderestimatesthe M˙ M˙ !
d d
rate when mass transfer proceeds on the thermal timescale. In
whereσisaparameterforwhichweassumeavalueof10inour
our model f is not a constant, but depends on the stability of
standardmodel(asinHurleyetal.2002).Moreover,inthecase
masstransfer.If f islarge,theradiusstaysclosetotheRochera-
ofaccretionontoaMSorHe-MSstar,rejuvenationisassumed
diusandmasstransferisself-regulating.However,inourmodel
numericalinstabilitiesarisewhenlogR /R . 10−3andtheen- (Hurleyetal.2002).Theinternalstructureofthestarischanged
d L
andnewfuelismixedintotheburningregion,whichresultsin
velope of the donor star is small. Therefore, a function is de-
astarthatappearsyounger.Iftheaccretorisanevolvedstaron
fined which forces the radius to follow the Roche radius more
the GB or AGB, mass transfer is assumed to be conservative
closely during thermal-timescale mass transfer (f = 1000)and
(β = 1)becausea convectivestar shrinksasa reactionto mass
more loosely during nuclear timescale mass transfer (f=1). A
gain. Accretion onto a WD is a special situation because of its
smoothtransitionisimplementedbetweenthetwoextremeval-
degeneracyandthisisbediscussedseparatelybelow.
ues of f. To test the stability of the function to calculate the
masstransferrate,a binarygridwas runof 503 binarysystems When material is lost from the system it removes angular
momentum.Angularmomentumlossisdescribedwithaparam-
withbinary c/nucsyn,withinitialprimarymassesbetween2
eterγ thatexpressesthespecificangularmomentumofthelost
and 10M , secondary masses between 0.5 and 10M and ini-
⊙ ⊙
tialseparationsbetween3and104R .Themasstransferratesof materialintermsoftheaveragespecificorbitalangularmomen-
⊙
tum,asfollows:
systemsevolvingthroughtheSD channelwithahydrogen-rich
donorwereusedtooptimizethisfunctionforcomputationalsta-
J˙ M˙
bility.Theresultingfunctionisgivenby orb =γ(1−β) d . (15)
J M +M
orb d a
1000 Q<1and InourstandardmodelweassumethatduringstableRLOFmate-
f = 1000 1 lnQ 2 (11) rialislostbyisotropicre-emission,removingthespecificorbital
max1, Q ·exp−2(0.15)  Q>1. angularmomentumoftheaccretor(γ= Md/Ma,Table1).
Becausemasstransferisdynamicallystable,themasstrans-
2.2.5. StableRLOFontoaWD
fer rate is capped at the thermal timescale mass transfer rate
M˙ ,givenby Because a WD is degenerate, it burns accreted material stably
KH
M˙ = Md M yr−1, (12) onlyoverasmallrangeofmasstransferrateswhichcorresponds
KH,d τKH ⊙ toapproximately10−7M⊙/yrwhenhydrogen-richmaterialisac-
creted and 10−6M /yr when helium-rich material is accreted
withτ thethermaltimescaleofthedonorstar, ⊙
KH (Nomoto1982).Ifthemasstransferrateistoolow,thematerial
MM′ R L isnotburntimmediatelyandalayerofmaterialisdepositedon
τ =1.0·107 ⊙ ⊙yr, (13)
KH M 2 R L thesurface.Thislayerburnsunstably,resultinginnovae,and,if
⊙
themasstransferrateistoohigh,theWDcannotburnalltheac-
and M′ either the mass of the star, if the star is on the MS or cretedmaterial.AccordingtoNomoto(1982)theaccretedmate-
He-MS,ortheenvelopemassofthestarotherwise. rialformsanenvelopearoundtheWDandbecomesaredgiant-
In order to test the mass transfer rate calculated with our like stellar object with a degenerate core and, generally, a CE
BPScode,asanextstepasetofbinarysystemsandtheirmass subsequentlyforms.Hachisuetal.(1996)proposethattheburn-
transferrateswerecalculatedwiththeourBPScodeandwitha ingmaterialontopoftheWDdrivesawindwhichblowsaway
detailed binary stellar evolution code (STARS, Eggleton 1971, therestoftheaccretedmaterial.Theaccretedmaterialburnsat
2006;Polsetal.1995;Glebbeeketal.2008)andcompared.We therateofstableburning,butcontactisavoidedandmasstrans-
simulatedagridwithprimarymassesbetween2and6M ,sec- ferremainsstable.Wetakethelatterpossibilityintoaccount(for
⊙
ondarymassesvariedbetween0.7and3.5M andorbitalperi- adescriptionseeAppendixB).Thematerialejectedthroughthe
⊙
odsvariedbetween2and4days(tocheckbinarysystemswith wind from the WD removes specific angular momentum from
both MS and HG donors) and a more massive binary system theWD(γ = M /M ).TheabovealsoholdsforWDsaccreting
d a
of 12 and 7M at orbital periods of 3.5 and 4 days. We find throughawind,butthematerialtransferredtotheWDisbased
⊙
that the resulting maximum mass transfer rate computed with ontheBondi-Hoyleaccretionefficiency(Eq.6).
5
J.S.W.Claeysetal.:TheoreticaluncertaintiesofthetypeIasupernovarate
2.3.Binarypopulationsynthesis ofRLOFandthestellartypesattheonsetofmasstransfer.The
latterillustratestheinfluenceofdifferentaspectsofbinaryevo-
We simulate N × N × N binary systems in logM -
M1,i M2,i ai 1,i lutionandinitialbinarydistributions(Sects.5and6).Inthefol-
logM - loga space, with M and M the initial masses of
2,i i 1,i 2,i lowingsectionsthemassratioqisM /M werethesuffixes1and
the primary and secondary stars and a the initial semi-major 2 1
i 2denotetheinitiallymoreandlessmassivestar,respectively.
axis of the binary systems. The volume of each cell in the pa-
rameter space is δM δM δa. To compute the SN rate each
1,i 2,i i
system is assigned a weightΨ accordingto the initial distribu- 3.1.Doubledegeneratechannel
tionsofbinaryparameters.WenormalizetheSNratetothetotal
massofthestarsinourgrid, Thedoubledegeneratechannel(DD)needstwoCOWDs,with
a combined mass greater than M in a short enough orbit
Ch
M1,i,maxM2,i,maxai,max (a . 4R ), to merge within a Hubble time. Consequently, the
⊙
M = (M +M )ΨδM δM δa , (16)
total 1,i 2,i 1,i 2,i i rateofthischanneldependsonthenumberofsystemsthat,after
MX1,i,min MX2,i,min aXi,min phasesofstableRLOFand/orCEevolution,areinsuchashort
orbit.TheresultsofPakmoretal.(2010,2011)indicatethatthe
where
violentmergeroftwoCOWDsonlyresultsinanexplosionfora
– M =0.1M andM =80M , limitedrangeofmassratios.Incombinationwithpossiblemass
1,i,min ⊙ 1,i,max ⊙
– M = 0.01M (Kouwenhovenetal. 2007) and lossduringthemerger,thisrestrictioncanreducetheSNIarate
2,i,min ⊙
M = M , fromtheDDchannelbyaboutafactoroffive(Chenetal.2012).
2,i,max 1,i
– a = 5R and a = 5 · 106R (Kouwenhovenetal. However,inthisworkwedonotimposearestrictiononthemass
i,min ⊙ i,max ⊙
2007)and ratioofthetwoCOWDs.
– ΨistheinitialdistributionofM , M anda. Multiple formation channels can form close double
1,i 2,i i
WD systems (Mennekensetal. 2010; Toonenetal. 2012).
WeassumethatΨisseparable,namely
Mennekensetal. (2010) distinguish between the common en-
velope channel, which needs two consecutive CE phases, and
Ψ(M ,M ,a)=ψ(M )φ(M )χ(a), (17)
1,i 2,i i 1,i 2,i i the Roche-lobe overflow channel, in which a phase of stable
Roche-lobe overflow is followed by a CE phase. Additionally,
where
Toonenetal. (2012) discuss the formation reversal channel
– ψ(M ) is the initial distribution of primary masses from where the secondaryforms a WD first. Fig.2 shows the distri-
1,i
Kroupaetal.(1993,Table1), bution of systems that evolve towards SNe Ia according to the
– φ(M )istheinitialsecondarymassesdistribution,whichwe DDchannelintermsoftheirinitialseparationa,initialprimary
2,i i
assumeisflatinM /M and mass M andinitialmassratio q. We distinguishsix mainre-
2,i 1,i 1,i i
– χ(a) is the initial distribution of semi-major axes, which gions:two lowerregionsA andB, with a/R .300,inwhich
i i ⊙
we assumeisflatinloga (O¨pik1924;Kouwenhovenetal. the primary fills its Roche lobe duringthe HG; two intermedi-
i
2007). ate regions C and D, with 300 . a/R . 1000, in which the
i ⊙
primaryfills its Roche lobeduringthe GB or E-AGB;and two
Wecalculatethedelaytimedistribution(DTD),whichisthe upperregionsEandF,witha/R &1000,inwhichoneormore
i ⊙
SNrateasafunctionoftimeperunitmassofstarsformedina CEphasesareneededandtheprimaryfillsitsRochelobeduring
starburstatt=0,asfollows: theTP-AGB.
M1,i,max M2,i,max ai,maxδ(SNIa)ΨδM δM δa
DTD(t)= M1,i,min M2,i,min ai,min 1,i 2,i i ,
P P P M δt Closesystems:regionsAandB. If the initial separation is
total
(18) shorter than about 300R⊙ the primary first fills its Roche lobe
whereδ(SNIa) = 1ifthebinarysystemleadstoa SNIaevent ontheHGwhenthestarhasaradiativeenvelope.Thefirstphase
during a time interval t to t +δt, otherwise δ(SNIa) = 0. We ofmasstransferisthusstableandaCEisavoided.
assumeallstarsareformedinbinaries,whichisanoverestimate The separation and the mass of the secondary determine
of the binaryfraction,becausefor low-massstars the observed whetherthesecondphaseofmasstransfer,fromthesecondary,
fraction of system in binaries is less than 50% (Lada 2006). is stable. The separation determines both the moment the sec-
However,inintermediatemassstars,Kouwenhovenetal.(2007) ondaryfillsitsRochelobeandthestellartypeofthesecondary
findabestfitwith100%binariesandDucheˆne&Kraus(2013) at RLOF. The initially closest systems with the least massive
concludethatdifferentsurveysareconsistentwithamultiplicity secondaries avoid a CE and two CO WDs form in a short or-
higherthan50%.Additionally,Kobulnicky&Fryer(2007)and bit without any CE phase (region A), while the wider systems
Sanaetal. (2012) find thatmorethan70%ofmassive starsare experiencea CE phase when the secondaryfills its Rochelobe
inbinarysystems. (regionB).
TheresultsofSect.3arecalculatedbysimulatingagridwith
RegionA consistsofsystems thatavoida CE phase during
M between 2.5 and 9M , M between 1M and M and a
1,i ⊙ 2,i ⊙ 1 i theirentireevolution.Therangeofinitialmassratiosthatmake
between5and5·103R ,withN = N = N = N =150.
⊙ M1,i M2,i ai SNe Ia is stronglyrestricted (Fig.2), becausesystemswith ini-
tialmassratioslargerthan0.46formdoubleWDsystemsinan
orbit too wide to merge in a Hubble time. Because of this re-
3. Binaryprogenitorevolution
striction,only0.7%ofthesystemsthatbecomeaSNIaviathe
In this section we discuss the general progenitor evolution of DD channel follow this evolutionary channel. However, when
thedifferentSNIachannels,SDandDD,andtheircontribution RLOF is followed with a detailed stellar evolution code, some
according to our standard model (Sect.2). We describe binary of these systems with a mass ratio smaller than 0.46 that con-
evolution in terms of the number of stable or unstable phases tainanaccretingMS-starareexpectedtoformacontactsystem
6
J.S.W.Claeysetal.:TheoreticaluncertaintiesofthetypeIasupernovarate
104 10-1 104 10-1
F TP-AGB 10-2 F 10-2
E y E y
103 E-AGB sit 103 sit
n n
)⊙ D C GB de )⊙ C D de
a(R i 10-3 ber a(R i 10-3 ber
m m
102 B1 Nu 102 Nu
10-4 B1 10-4
B2 B2
A HG A
101 10-5 101 10-5
2 3 4 5 6 7 8 9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
M1,i (M ) M2,i/M1,i
⊙
Fig.2. Initial separation (a) versus initial mass of the primary star (M, left) and versus initial mass ratio (M /M , right) of
i i 2,i 1,i
systems that form SNe Ia through the DD channel in our standard model (Sect.2). Number density (greyscale) represents the
numberofsystemsnormalizedtothetotalnumberofsystemsformingaSNIathroughtheDDchannel.Linesshowtheminimum
separation (assuming q = 1) for which the primary fills its Roche lobe in a certain evolution stage, as indicated. Symbols A-F
indicatedifferencesintheevolutionstagewhentheprimaryfillsitsRochelobeandinthenumberofCEphasesnecessarytoevolve
toaSNIa:A=HG,noCE;B=HG,1CE(thedifferencebetweenB andB isbasedonwhichofthetwostarsfirstformsaWD);
1 2
C=GB,1CE;D=E-AGB,1CE;E=TP-AGB,1CE;F=TP-AGB,2CEs.
(deMinketal.2007)whicheventuallymergesbeforetheforma- that make SNe Ia is restricted because the lifetime of the cor-
tionoftwoCOWDs. respondingheliumstarhastobelongenoughforthesecondary
Region B represents the most common evolutionary chan- to evolve. This restriction originatesfrom the necessity for the
nel and corresponds to the Roche-lobe overflow channel secondary to fill its Roche lobe before the primary becomes a
(Mennekensetal. 2010; Toonenetal. 2012). In this region the WD.
primarystartsmasstransferduringtheHGandaCE phaseoc- Systems with initial separations in the gap between region
curs when the secondary fills its Roche lobe. This channel ac- B and regionsC and D do notform a doubleWD system. The
countsfor84%ofSNeIaformedthroughtheDDchannelinour primaryfillsitsRochelobeattheendoftheHG orontheGB,
model,asimilarfractiontoMennekensetal.(2010).Binarysys- whichleadstoaCEphaseinwhichthetwostarsmerge.
temswithinitialmasseslowerthan2.9M donotformamerger
⊙
productwith a combined mass greater than 1.4M , while sys-
⊙
temswithprimarymasseshigherthan8.2M formanONeWD. WidesystemsthatundergooneCEphase:regionsC,Dand
⊙
Binarysystemswithmassratioslowerthan0.25haveanunsta- E. Insystemswithseparationslongerthanabout300R⊙thepri-
blefirstmasstransferphaseandmerge. maryfillsitsRochelobeaftertheHGwhilehavingaconvective
InregionBwedistinguishbetweenregionsB andB ,based envelopewhich results in a CE phase. The primary becomes a
1 2
on which of the two stars becomes a WD first. In general, the WD immediately (region E) or after subsequent evolution as a
initiallymoremassivestarisexpectedtoformthefirstWDofthe helium star (regions C and D). Afterwards, the secondary fills
binarysystem(regionB ).However,sometimestheevolutionof its Roche lobe. In regions C, D and E mass transfer from the
1
the secondarycatchesup with the primarybecause ofprevious secondary is stable, permitting accretion onto the initial WD.
binaryinteractionandbecomesaWDfirst(regionB2),whichis However,theWDcannotreach MCh beforethesecondaryloses
the formationreversalchannel(Toonenetal. 2012). RegionB its entire envelopeand the binary becomes a short-perioddou-
2
contains49%ofthe systemsformingregionB(41%ofallDD bleWDsystem.Thisimposesarestrictionontherangeofinitial
systems). mass ratiosthatmake SNe Ia, which determinesthe separation
ThesystemsofregionB haveprimarymassesbetween2.9 afterthephaseofstableRLOF,becausetheorbitshouldbeshort
2
and5M andmassratiosbetween0.67and1.Whenahydrogen- enoughwhenthe secondarybecomesa WD. Thischannelpro-
⊙
richstarlosesitsenvelopebeforetheTP-AGBitbecomesahe- duces12.8%ofDD progenitors(1.9%inregionC,6.3%inre-
liumstarbeforeevolvingintoaWD(Sect.2.1).Inthiscasethe gionDand4.6%inregionE).Thedifferencebetweenthethree
resultingheliumstarshaveamassbetween0.5and0.85M .The regionsis the stellar type of the primariesat the onset of mass
⊙
timeittakesfortheselow-massheliumstarstobecomeaWDis transfer:inregionCtheprimaryisontheGB,inregionDonthe
long(betweenabout40and160Myr)whichgivesthesecondary, E-AGBandinregionEontheTP-AGB(Fig.2).
which has increased in mass, time to evolve and fill its Roche The ranges of the initial masses of the three evolutionary
lobe.Itbecomesthemostmassiveheliumstarofthebinarysys- channelsaredefinedbythenecessityforbothstarstobecomea
temafterthesubsequentCEphase.TheshortorbitaftertheCE COWD(upperlimitonthemasses)andformamassiveenough
phaseallowsthesecondarytofillitsRochelobeagainandevolve merger(lowerlimitonthemasses).Insomesystemswithamass
intothefirstWDofthebinarysystem.Afterwards,theprimary ratioclosetoone(q &0.93)thesecondaryisalreadyevolvedat
i
fills its Roche lobe for the second time and becomes the sec- the momentthe primaryfills its Roche lobe and duringthe CE
ond WD of the binary system. The range of primary masses phasebothenvelopesarelost.ThesecondarystillfillsitsRoche
7
J.S.W.Claeysetal.:TheoreticaluncertaintiesofthetypeIasupernovarate
lobeaftertheprimarybecomesaWD,butasaheliumstar,viz.
ontheHe-HG.
6 100
The systems with initial separation in the gap between re-
MS
gionsDandEdonotformdoubleWDsystems.Thesesystems
survive the first CE phase and form a helium star with a non- 5 10-1
evolvedcompanion.However,becauseofthelongerinitialsep-
aration,theheliumstarfillsitsRochelobeagainasaHe-giant. ty
si
This leads to a second CE phase, during which the two stars )⊙4 10-2 en
merge. M HG d
M (d3 10-3 mber
WidesystemsthatundergotwoCEphases:regionF. When u
N
theinitialseparationislongerthanabout1000R thefirstmass
⊙
transferphase starts whenthe primaryis on the TP-AGB. This 2 10-4
leadstounstablemasstransferandthereforeaCEphase,during GB
whichtheseparationdecreases.TheprimarybecomesaCOWD
immediately. Subsequently, the secondary fills its Roche lobe 1 10-5
0.5 1.0 1.5 2.0 2.5 3.0
which also results in unstable mass transfer. This evolutionary log a (R )
⊙
channel is similar to the common envelope channel discussed
inMennekensetal.(2010)andToonenetal.(2012).Thischan- Fig.3.Donormass(Md)vs.separation(a)atthemomentoffor-
nelproduces2.4%of the DD systems. These systems have the mationoftheCOWDofthesystemsproducingSNeIathrough
firstphaseofmasstransferduringtheTP-AGB(Fig.2),however the SDH channel,for differentCO WD masses: 1.15M⊙ (solid
thisevolutionarychannelalso occurswhenmass transferstarts line), 1.0M⊙ (dash-dotted line) and 0.8M⊙ (dotted line). The
duringtheE-AGB.Nevertheless,thesystemswhichstartRLOF greyscaleshowshowtheseregionsarepopulatedinourstandard
duringthe E-AGBand have two consecutiveCE phasesgener- model.Labelsindicatethedifferentstellartypesofdonorsstars.
allymergebeforetheformationofadoubleWDsystemandonly
accountfor0.1%ofallDDsystems.
In systems with longer initial separations (a & 2500R ),
⊙
bothstarsdonotfilltheirRochelobe,andthereforedonotform siblerangesindonormassandseparationforwhichWDswith
doubleWDsystemsinashortorbit(Fig.2). amassof0.8,1.0or1.15M⊙ cangrowtoMCh.
In Fig.3 three main regionsare distinguished with M <
WD
1.15M .Thedonorstarsintheregionwithloga/R . 1.2and
⊙ ⊙
3.2.Singledegeneratechannel 1.5 . M /M . 5.2 start transferringmass to the WD during
d ⊙
theMS,thedonorstarsintheregionwith1.0.loga/R . 1.5
The single degenerate channel needs a CO WD and a non- ⊙
and2.4. M /M .3.0duringtheHG,andthoseintheregion
degeneratecompanionwhichprovidesenoughmasstotheWD d ⊙
with2.0.loga/R .2.5and1.15. M /M .1.4duringthe
at a high enough rate. We distinguish between hydrogen- and ⊙ d ⊙
GB.ForfuturereferencewenamethesechannelstheWD+MS,
helium-richcompanions.
WD+HG,WD+RGchannel,respectively,asinMennekensetal.
(2010).
3.2.1. SDwithhydrogen-richdonor(SD ) Fig.3 in this article can be compared with Fig.3 of
H
Han&Podsiadlowski (2004), which was calculated with a de-
A hydrogen-rich companion can be in any evolutionary stage
tailed binary stellar evolution code and which discussed the
between the MS and the AGB. The stability criterion for mass
WD+MS channel. Like Han&Podsiadlowski (2004) we find
transfer (Table2) and the rate at which mass is transferred de-
thatSNeIacanoriginatefromaccretingWDswithdonormasses
termine which donor stars transfer enough material to the WD
between 1.5 and 3.5M , but in addition we find systems with
⊙
tomakeSNe Ia.Evolvedstars(GBorAGB) havea convective
donormassesgreaterthan3.5M ,whichtheyfindtobecomedy-
⊙
envelope,whichresultsinasmallercriticalmassratioforstable
namicallyunstableandhencedonotbecomeSNeIa.Thisarises
mass transfer comparedto non-evolvedstars (MS or HG) with
becauseofadifferenttreatmentofthestabilityofRLOFbetween
aradiativeenvelope(Table2).Consequently,donorsontheMS
thetwocodes.FortheWD+RGchannelwecompareourresults
can be more massive than evolved donors without the system
withFig.2ofWang&Han(2010)whofinddonormassesdown
evolvingintoaCE.Inaddition,themasstransferratedetermines
to0.6M .However,thesesystemsdonotbecomeaSNIawithin
⊙
the amountof materialthatis accretedby the WD. Ideally,the
aHubbletimeandthereforearenotshowninourFig.3.
masstransferrateisabout10−7M yr−1(Nomoto1982,Sect.2).
⊙ The greyscale in Fig.3 depicts the number density at WD
MasstransferisgenerallyfasterfromGBdonorsthanfromMS
formationofthesystemsevolvingthroughtheSD channelwith
donorswiththesamemass.Consequently,initiallylessmassive H
ourstandardmodel.Tounderstandwhyourstandardmodeldoes
starsontheGBthanstarsontheMScandonateenoughmassto
notformWD+MSsystemsovertheentiremassandseparation
aCOWDtogrowtoM (Fig.3).
Ch range shown in Fig3, a closer look at the distribution of the
ThecontoursinFig.3showtheresultsofasimulationwith
initial characteristics of the systems which evolve into a SN Ia
our BPS code in logM - logM - loga space as discussed
1,i 2,i i throughtheSD channelisnecessary(Fig.4).Inthisfigurethree
in Sect.2.3 with N = 125, in which one of two stars is a WD H
regionsaredistinguished:systemswithshort(A ),intermediate
at t = 0 and the other is on the ZAMS. The initial masses H
(B )andlonginitialseparations(C ).
range between 0.7 and 1.15M for the WD and between 0.7 H H
⊙
and 6M for the companion MS star, the separation is varied
⊙
betweentheminimumseparationforaMSstartofillitsRoche Closesystems:regionA . Systemswithinitialseparationbe-
H
lobeand103R andweassumee =0.Thefigureshowsthepos- tweenbetween20and70R havethefirstphaseofRLOFdur-
⊙ i ⊙
8
J.S.W.Claeysetal.:TheoreticaluncertaintiesofthetypeIasupernovarate
104 100 104 100
TP-AGB
10-1 10-1
CH y CH y
103 E-AGB sit 103 sit
n n
) e ) e
⊙ GB d ⊙ d
a(R i BH 10-2 ber a(R i BH 10-2 ber
m m
102 Nu 102 Nu
10-3 10-3
AH AH
HG
101 10-4 101 10-4
2 3 4 5 6 7 8 9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
M1,i (M ) M2,i/M1,i
⊙
Fig.4. As Fig.2 for systems that form SNe Ia through the SD channel. Symbols A , B and C indicate groups which are
H H H H
distinguished based on the evolutionary stage when the primary fills its Roche lobe and the number of CE phases necessary to
evolvetoaSNIa:A =HG,noCE;B =endofHG/GB,1CE;C =TP-AGB,1CE.
H H H
ing the HG, which is stable. Subsequently a WD forms in a
short orbit because of the initially low mass ratio. Afterwards
5.0 100
thesystemsevolvethroughtheWD+MSchannel(Fig.3).Only He-MS
1% of the SDH systems evolve through this channel, because 4.5
of the small initial mass ratios involved and the possibility of 10-1
the formation of a contact system because of the reaction of 4.0
thesecondaryduringaccretion(cf.regionAoftheDDchannel, ty
Sect.3.1). )⊙3.5 10-2 ensi
Widesystems:regionsBHandCH. Themostcommonevolu- M(M d32..05 10-3 mber d
tionarychannel(99%)oftheSD channelisthroughaCEafter u
H N
whichtheWDisinashortorbitwithanunevolvedcompanion. 2.0
The systems of region BH follow the WD+MS channel, while 1.5 He-HG 10-4
the systems of region C follow the WD+HG channel or the
H
WD+RGchannel. 1.0
In region B , systems have initial separations longer than −0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 10-5
H log a (R )
300R andtheprimaryfillsitsRochelobeattheendoftheHG
⊙ ⊙
ortheonsetoftheGB.AftertheCEphaseaheliumstarforms,
Fig.5.AsFig.3forsystemsatformationoftheWDandHe-MS
whichsubsequentlyoverflowsitsRochelobestablyandevolves
binarywhichproduceSNeIathroughtheSD channel.
He
intoaWD.ThesecondaryfillsitsRochelobeduringtheMSand
followstheWD+MSchannel.Thisevolutionarychannelisfol-
lowedby99%oftheSD systems,with21%startingRLOFat
H
the end of the HG and 78%duringthe GB. A star with an ini- largerthanabout1.1M cangrowtoM withanevolveddonor
⊙ Ch
tialmassbetween5.7and8.2M⊙ evolvesintoaCOWDwitha star.Additionally,theWD+RGchannelisalmostnon-existentas
massbetween0.8and0.95M⊙whenstrippedofitshydrogenen- binarysystemswithamassiveWD (& 1.0M⊙)andalow-mass
velopeattheendoftheHGorduringtheGB.Thisexplainswhy MS star (. 1.5M ) rarely form with separations between 100
⊙
thegreyscaleinFig.3islimitedinthemassrangeofthedonors. and300R withtheassumedCEefficiencyandthereforecannot
⊙
Binarysystemswithinitialseparationslongerthan400R⊙,after contributetotheWD+RGchannel.
a CE phase, as well as systems with initial separations shorter Systems with initial separations longer than about
than200R⊙,afteraphaseofstablemasstransfer,formaWDbi- 3000R⊙ do not experience RLOF and therefore do not be-
narysystemthatistoowidetogothroughtheWD+MSchannel comeaSNIa.However,thesesystemsstillinteractintheform
(Fig.3). ofastellarwind.Nevertheless,inourstandardmodelwindmass
In region CH, systems have initial separations longer than transferisinsufficientforaCOWDtogrowtoMCh(Sect.2.2.1).
1000R andtheprimaryfillsitsRochelobeduringtheTP-AGB
⊙
andformsa CO WD immediately.After the CE phase the sec-
3.2.2. SDwithhelium-richdonors(SD )
ondaryfillsitsRochelobeasanevolvedstarontheHGorGB. He
0.3%oftheSD systemsfollowthisevolutionarychannel.The Helium rich donors must be massive enough (M & 1M )
H ⊙
number of systems going through this evolutionary channel is to transfer enough material to the WD to make a SN Ia.
limitedbecauseonlysystemswithCOWDswithinitialmasses Consequently, their H-rich progenitor stars must be massive
9
J.S.W.Claeysetal.:TheoreticaluncertaintiesofthetypeIasupernovarate
104 100 104 100
TP-AGB
CHe 10-1 CHe 10-1
y y
103 E-AGB sit 103 sit
n n
) e ) e
⊙ GB d ⊙ d
a(R i 10-2 ber a(R i 10-2 ber
102 BHe Num 102 BHe Num
10-3 10-3
AHe AHe
HG
101 10-4 101 10-4
2 3 4 5 6 7 8 9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
M1,i (M ) M2,i/M1,i
⊙
Fig.6. As Fig.2 of systems that form SNe Ia through the SD channel. Symbols A , B and C indicate groups which are
He He He He
distinguishedbasedontheevolutionarystagewhentheprimaryfillsitsRochelobe,thenumberofCEphasesnecessarytoevolve
toaSNIaandthetypeofdonorstartransferringmasstotheCOWD:A =HG,noCE,withHe-MSdonors;B =HG,1CE,
He He
withHe-HGdonors;C =TP-AGB,1CE.
He
enough initially (at least 4M ). The initial helium star can be theyfindslightlyhigherdonormassesintheWD+He-HGchan-
⊙
non-evolved (He-MS) or evolved (He-HG or He-GB). Fig.5 nel, which in our modeldo not transfer enoughmaterialto the
shows the results of a simulation similar to that of Sect.3.2.1, WDcompanion.
with one of the stars at t = 0 a WD and the other star on the ThegreyscaleofFig.5representsthenumberdensityofthe
zero-ageHe-MS.Therangesofbothmasses,theseparationand systems at formation of a WD and helium star binary for the
theresolutionareequaltothosediscussedinSect.3.2.1. SD channel.Belowwedescribethedifferentprogenitorevolu-
He
Fig.5 depicts the helium donor masses versus the separa- tionary channels, distinguishingbetween systems with initially
tion of the systems with an initial WD mass ≤ 1.15M⊙ which short separations which have He-MS donors (AHe) or He-HG
becomeSNe Ia. Two regionsare distinguished.The leftregion donors(B ) andsystemswith initially longseparations(C ).
He He
with loga/R⊙ . 0.2 and 0.9 . Md/M⊙ . 4.5 shows systems ThesedifferentgroupscanbedistinguishedinFig.6.
that start RLOF to the WD during the He-MS. The middle re-
gion with 0.0 . loga/R . 0.85 and 1.05 . M /M . 1.5
⊙ d ⊙
showsdonorstarsthatstartRLOFtotheWDduringtheHe-HG. Closesystems:regionsAHeandBHe. Systemswithinitialsep-
He-GBdonorsdonotformSNeIawhentheinitialWDmassis arations between 25 and 300R⊙ have primaries that fill their
less than 1.15M , because the evolutionary timescale of these Roche lobe during the HG when they have radiative envelopes
⊙
starsistooshorttotransfermuchmaterialtoaWD. which results in stable mass transfer. After mass transfer a he-
lium star is formed which becomes a WD without interaction
He-MSstarsmoremassivethan1.6M explodeascorecol-
⊙
or after a phase of stable RLOF. Afterwards a CO WD and a
lapse SNe (CC SNe) if they evolve as single stars (Polsetal.
MS star, which has increased in mass, remain. Because of the
1998). Thispotentially allowssome binarysystems to produce
greatdifference in mass between the two stars a CE phase fol-
both a SN Ia, from the CO WD, and subsequently a core col-
lows whenthe secondaryfills its Roche lobe duringthe HG or
lapse SN fromthe remainingHe star. However,He-MSdonors
beyond, but before the TP-AGB. Subsequently a CO WD and
with these masses transfer much material to the WD and have
heliumstarinashortorbitremain.Whenthesecondaryfillsits
masseslowerthan1.6M whentheCOWDreachesM .Inour
⊙ Ch
Roche lobe for the second time the WD increases in mass and
modelthestructureofHe-MSadaptsduringRLOFandthestar
reachesM .
evolves further as though it started its evolution with this new Ch
mass. These initially massive helium stars then become a WD ThemaindifferencebetweenregionsAHeandBHeisthemo-
instead of exploding as a CC SN. However, if the He-MS star mentthecompanionstarfillsitsRochelobeasaheliumstar,i.e.
does not adapt to its new mass during RLOF, we find that one during the He-MS at the shortest separations (region AHe) and
binarysysteminourgridproducesbothaSNIaandaCCSN. duringtheHe-HGatthelongestseparations(regionBHe).
Our Fig.5 should be compared with Fig.8 in Wangetal. The initial primary mass and mass ratio of both groupsare
(2009b), which depicts the systems evolving through the determinedbytheneedtoformaCOWD(uppermassandmass
WD+He-MS and WD+He-HG channel calculated with a de- ratio limit) and a massive enough merger product (lower mass
tailedbinarystellarevolutioncode.Bothfiguresindicatethathe- and mass ratio limit). The lower primarymass boundaryof re-
liumdonorswithmassesdowntoabout0.9M⊙ transferatleast gionAHeislowerthanofregionBHe,becauselessmassiveWDs
0.25M⊙ to a 1.15M⊙ WD. However,we find that helium stars cangrowto MChwithHe-MSdonors(Fig.5).
moremassivethan3M canalsotransferthisamountstablyto Of the SD systems, 48% follow the evolutionary chan-
⊙ He
aWD,whileWangetal.(2009b)findthatRLOFisdynamically nelofregionA .Therangeofinitialseparationsislimitedby
He
unstable in these binary systems, because of differences in the the small range of radii of He-MS stars (Fig.5). Systems with
stability criteria of RLOF between the two codes. In addition, shorterseparationsthan25R mergeduringtheCEphasebefore
⊙
10