Formation of Millisecond Pulsars with Heavy White Dwarf Companions - Extreme Mass Transfer on Sub-Thermal Timescales

9 Pages·0.13 MB·English

by  T. M. Tauris

ACCEPTEDBYTheAstrophysicalJournalLetters,DECEMBER20,1999
PreprinttypesetusingLATEXstyleemulateapjv.04/03/99
FORMATIONOFMILLISECONDPULSARSWITHHEAVYWHITEDWARFCOMPANIONS.
EXTREMEMASSTRANSFERONSUB-THERMALTIMESCALES
THOMASM.TAURIS1,EDWARDP.J.VANDENHEUVEL1,GERRITJ.SAVONIJE1
AcceptedbyTheAstrophysicalJournalLetters,December20,1999
ABSTRACT
We have performed detailed numerical calculations of the non-conservative evolution of close X-ray binary
systems with intermediate-mass(2.0 – 6.0M⊙) donorstars and a 1.3M⊙ accreting neutronstar. We calculated
the thermal response of the donor star to mass loss, in order to determine its stability and follow the evolution
0 of the mass transfer. Under the assumption of the “isotropic re-emission model” we demonstrate that in many
0 casesitispossibleforthebinarytopreventaspiral-inandsurviveahighlysuper-Eddingtonmass-transferphase
0 (1≪M˙/M˙ <105) on a sub-thermal timescale, if the convective envelope of the donor star is not too deep.
Edd
2 ThesesystemsthusprovideanewformationchannelforbinarymillisecondpulsarswithheavyCOwhitedwarfs

n and relatively short orbital periods (3–50 days). However, we conclude that to produce a binary pulsar with a
a O-Ne-Mg white dwarf or P ∼1 day (e.g. PSR B0655+64)the abovescenario doesnotwork, and a spiral-in
orb
J phaseisstillconsideredthemostplausiblescenariofortheformationofsuchasystem.

3 Subjectheadings:stars: evolution,mass-loss,neutron—whitedwarfs—binaries:close

1
v 1. INTRODUCTION tributedtotheirinabilitytotransfermassinsuchastablemode
3 that the system becomes a persistent long-lived X-ray source
Recentlyalargenumberofbinarymillisecondpulsars(BM-
1 (vandenHeuvel1975;Kalogera&Webbink1996). Belowwe
SPs)withrelativelyheavywhitedwarf(WD)companionshave
0 investigateforthese systemshowthe stability ofthe RLO de-
been reported. These pulsars form a distinct class of BM-
1 pends on the evolutionary status of the donor (and hence the
SPs (cf. Table 1) which are characterized by relatively slow
00 spinperiods(Pspin≃10- 200ms)andhighperiodderivatives: orbitalperiod)attheonsetofmasstransfer.
0 10- 20<P˙ <10- 18. IthasbeensuggestedthatsuchBMSPs
spin 2.1. Numericalcomputations
/ evolved througha common envelope and spiral-in phase (e.g.
h
van den Heuvel 1994). This gives a natural explanation for WehavecalculatedtheevolutionofalargenumberofX-ray
p
- theircloseorbitsandthepresenceofarelativelyheavyCO/O- binarieswithadonorstarofmass2≤M2/M⊙<6anda1.3M⊙
o Ne-MgWD,ifthemasstransferwasinitiatedwhilethedonor accretingneutronstar. Boththeradiusofthedonorstaraswell
r star (the progenitorof the WD) ascended the AGB. However, as its Roche-lobe are functionsof time and mass (as a conse-
t
s it has been argued that a neutron star engulfed in a common quence of nuclear burning, magnetic braking and other tidal
a
envelopemightexperiencehypercriticalaccretionandthereby spin-orbit couplings). We used an updated version of Eggle-
:
v collapseintoablackhole(e.g. Chevalier1993;Brown1995). ton’snumericalcomputercode(Polsetal. 1998)tokeeptrack
Xi IfthispictureiscorrectthentheseBMSPscannothaveformed ofthe stellar evolutionandincludeda numberofbinaryinter-
inacommonenvelopeandspiral-inphase. actionstocarefullyfollowthedetailsofthemass-transferpro-
ar Hereweinvestigateanalternativescenarioforproducingthe cess. Foralldonorstarsconsideredherewe assumeda chem-
mildlyrecycledBMSPswithHeorCOWDincloseorbits,in ical composition of X=0.70 and Z=0.02, and a mixing-length
which a 2–6M⊙ donorstar, with a non-(or partly)convective parameter of α=l/Hp =2.0. We refer to Tauris & Savonije
envelope,transfersmassonasub-thermaltimescaleandyetin (1999)foradetaileddescriptionofourcomputercode.
adynamicallystablemode.
2.2. Highlysuper-Eddingtonmasstransfer
2. STABILITYCRITERIAANDMODEOFMASSTRANSFER Themaximumaccretionrateontoaneutronstarisgivenap-
The stability and nature of the mass transfer is very impor- proximatelybytheEddingtonlimitforsphericalaccretionofa
tantinbinarystellarevolution.Itdependsontheresponseofthe hydrogengas,M˙Edd=1.5×10- 8M⊙ yr- 1. Ifthemass-transfer
˙
mass-losing donorstar and of the Roche-lobe(e.g. Paczynski rate fromthe donorstar, M is largerthan thislimit, radiation
2
1976;Soberman,Phinney&vandenHeuvel1997). Themass pressurefromtheaccretedmaterialwillcausetheinfallingmat-
transferisstableaslongasthedonorstar’sRoche-lobecontin- tertobeejectedfromthesystematarate:
uestoenclosethestar. Otherwiseitisunstableandproceedson |M˙|=|M˙ |- M˙ ≃|M˙ | ifM˙ ≫M˙
2 Edd 2 2 Edd
theshortestunstabletimescale. In systems with very large mass-transfer rates, matter piles
As long as the mass of the donor, M2 is less than 1.8M⊙ up around the neutron star and presumably forms a growing,
the mass transfer will be dynamically stable for all initial or- bloatedcloudengulfingalargefractionoftheaccretiondisk.A
bital periods (e.g. Tauris & Savonije 1999). These LMXBs system will only avoid a spiral-in if it manages to evaporate
aretheprogenitorsoftheBMSPswithaheliumWDcompan- the bulk of the transferred matter via the liberated accretion
ion. TheobservationalabsenceofX-raybinarieswithRoche- energy. This would require the radius of the accretion cloud,
˙ ˙
lobefillingcompanionsmoremassivethan∼2M⊙hasbeenat- rcl > RNS|M2|/MEdd in order for the liberated accretion en-
1CenterforHigh-EnergyAstrophysics,andAstronomicalInstitute“AntonPannekoek”,UniversityofAmsterdam,Kruislaan403,NL-1098SJAmsterdam,The
Netherlands;[email protected],[email protected],[email protected]
1
2 FORMATIONOFMILLISECONDPULSARSWITHHEAVYWHITEDWARFCOMPANIONS
ergytoejectthetransferedmaterial(∼0.1M˙Eddc2∼> 12M˙2v2esc, isexhausted(g). The0.602M⊙corethenhasachemicalcom-
wherev2 =2GM /r ;R istheradiusoftheneutronstar). position of 19% C, 79% O and 2% Ne. It is surroundedby a
esc NS cl NS
If thematerialwhichisto beejected comescloserto theneu- 0.016M⊙ envelope (16% H, 82% He, 1% N14). The central
tronstaritwillhavetoomuchnegativebindingenergyinorder densityisρc=4.13×104gcm- 3andR2=0.21R⊙. Fromhere
fortheliberatedaccretionenergytoexpelit. Atthesametime onthestarcontractsandsettlesasahotCOwhitedwarf.
r must be smaller than the Roche-lobe radius of the neutron We have now demonstrateda scenariofor producinga BMSP
cl
star duringthe entireevolution,if formationofa commonen- withthesamecharacteristicsasthoselistedinTable1.
velope(CE)istobeavoided2.
A simple isotropic re-emission model will approximately re-
3.2. TheP –M diagram
main valid for our scenario. In this model it is assumed that orb WD
matter flows over conservatively from the donor star to the InFig.2wehaveplottedthecalculatedfinalorbitalperiods
vicinity of the neutron star before it is ejected with the spe- as a function of the mass of the white dwarf companion (the
cific orbitalangularmomentumoftheneutronstar. Assuming remnantofthedonor)foragiveninitialmassofthedonorstar.
thistobethecasewefindthat,evenforextremelyhighmass- ThevaluesfortheBMSPswhichoriginatedfromabinarywith
˙ ˙
transferrates(|M2|>104MEdd),thesystemcanavoidaCEand alow-masscompanion(theformerLMXBswithM2∼<1.8M⊙
spiral-inevolution. locatedontheupperbranch)aretakenfromTauris&Savonije
(1999). These white dwarfs are expected to be helium WD –
3. RESULTS unlesstheinitialorbitalperiodwasverylarge(Porb>150days)
soa relativelyheavyheliumcoredevelopedpriortotheRLO,
3.1. Acasestudy:M2=4.0M⊙ andPorb=4.0days inwhichcasetheheliumcorelaterignitedformingaCOWD.
InFig.1weshowtheevolutionofabinaryinitiallyconsist- ThefinalproductofX-raybinarieswithM2>2M⊙areseento
ingofaneutronstarandazeroagemain-sequencecompanion deviatesignificantlyfromthe low-massbranch. Thereasonis
starwithmassesMNS=1.3M⊙ andM2=4.0M⊙,respectively, theformersystemshadalargemassratio,q≡M2/MNS which
andinitialorbitalperiodPorb=4.0days. causedthebinaryseparationtoshrinkinitiallyuponmasstrans-
At the age of t =176.6 Myr the companion has evolved to fer–cf. Fig.1. Suchsystemsonly“survive”themass-transfer
fillitsRoche-lobe(R2=8.95R⊙;Teff=9550K)andrapidmass phaseif the envelopeof the donoris radiativeor slightly con-
transferisinitiated(A).Thedonorstarhasjustevolvedpastthe vective.Thissetsanupperlimitontheinitialorbitalperiodfor
MShookintheHR-diagramandisburninghydrogeninashell a given system. If the donoris in a wide binary it developsa
arounda0.56M⊙heliumcore.Priortothemass-transferphase, deep convectiveenvelopepriorto filling its Roche-lobeand it
aradiation-drivenwind(|M˙2|∼4×10- 10M⊙ yr- 1)hascaused willthereforeexpandrapidlyinresponsetomasslosswhich,in
thedonortodecreaseitsmassslightly(M2=3.99M⊙)andcon- combinationwiththeorbitalshrinking,willresultintheforma-
sequentlyresultedinaslightwideningoftheorbit(P =4.02 tionofaCEanda(tidallyunstable)spiral-inevolution.
orb
days).OncethedonorfillsitsRoche-lobeitisseentolosemass InFig.3weshowhowthefinalorbitalperiodandthemassof
ataveryhighrateof|M˙2|≃4×10- 5M⊙yr- 1=2.7×103M˙Edd. theWD dependson theinitial orbitalperiodfora binarywith
Atthisstagethedonorhasonlydevelopedaverythinconvec- M2=4.0M⊙. WenoticethatthequestionofinitiatingRLObe-
tive envelope of size Zconv =0.015R⊙, so its envelope is still foreoraftertheterminationofhydrogencoreburning(caseA
radiativeandwillthereforeshrinkinresponsetomassloss. At or early case B, respectively) is important for these relations.
t =176.7Myrits radiushasdecreasedtoa minimumvalueof For initial P <2.4 days (case A RLO), Pf decreases with
orb orb
3.38R⊙, but now Zconv =0.97R⊙. At this point (s) the donor increasingPorb. Thereasonissimplythatinthesesystemsthe
hasamassof1.76M⊙,Teff=5640K,andPorb=1.59days.The donor star is still on the the main-sequence and the mass of
donorexpandsagain,butshortlythereafter,itsrateofexpansion its helium core, at the onset of RLO, increases strongly with
˙ ˙
slowsdowncausing|M |todecreaseto∼10M . P andthereforetheamountofmaterialtobetransferred(the
2 Edd orb
The mass transferceases (B) when the donorhas an age of donor’senvelope)decreases with P . Since the orbitwidens
orb
178.3Myr. AtthisstagePorb=8.11days,R=6.68R⊙,Zconv= efficientlyneartheendofthemasstransfer,whenthemassra-
0.07R⊙andTeff=12700K.Themassofthedonoris0.618M⊙. tio between donorand accretorhas been inverted(cf. Fig. 1),
Itstillhasa0.56M⊙ heliumcore,butnowonlya0.06M⊙ en- PofrbwillalsodecreaseasafunctionofinitialPorb. However,for
velope consisting of 16% H and 82% He. The mass-transfer P >2.4days(earlycase BRLO)the finalorbitalperiodin-
orb
phase(A–B)lastsrelativelyshort: t =1.7Myr,andhencethe creaseswithinitialorbitalperiodasexpected–thecoremassof
neutronstarwillonlyaccrete:∆MNSX=tXM˙Edd=0.03M⊙.This the donoronly increasesslightly (due to hydrogenshell burn-
˙
leadstorelativelargevaluesofPspinandPspinforthe(mildly)re- ing)asafunctionofinitialPorb.
cycledpulsar. Asthemass-transferrateisalwayshighlysuper-
EddingtonduringtheRLO,andtheaccretingneutronstarwill
3.3. Theinitial(M ,P )parameterspace
beenshroudedbyathick(bloated)disk,itisdoubtfulwhetherit 2 orb
willbeobservableasanX-raybinaryduringthisphase–except In Fig. 4 we outline the results of our work in a diagram
verybrieflyjustattheonsetandneartheendoftheRLO. showing the fate of a binary as a function of its initial P
orb
We followedtheevolutionofthedonorstarfurtheron. The and the value of M . We conclude that X-ray binaries with
2
donorcontinuesto burnhydrogeninits lightenvelope. Att = 2∼<M2/M⊙<6canavoidaspiral-inandCEevolutionifPorb
186.4Myr(f)theheliumburningis finallyignited(L /L > isbetween1–20days,dependingonM . IftheinitialP istoo
He H 2 orb
10) in the core which now has a mass of 0.596M⊙. After 70 short,thesystemswillobviouslyenteraCEphase,sincethese
Myr(t=253.8MyrsincetheZAMS)thecore-heliumburning systemalwaysdecreasetheirorbitalseparationwhenthemass
2Note,thatifnoefficientcoolingprocessesarepresentintheaccretiondiskthentheincomingmatterretainsitsnet(positive)energyandiseasilyejectedinthe
formofawindfromthedisk(Narayan&Yi1995;Blandford&Begelman1999).Evenifthearrivinggasisabletocool,interactionsbetweenthereleasedradiation
fromthisprocessandtheinfallinggasmayalsohelptoejectthematter.Inbothcasesrclcanbesmallerthanestimatedabove.
TAURIS,VANDENHEUVEL&SAVONIJE 3
transferisinitiated3. Ontheotherhand,iftheinitialP istoo progenitorcandidateforaBMSPwithaheavyWD.
orb
large the donor develops a deep convective envelope prior to ItisseenfromFig.2thatwecannotreproducethesystems
RLO and a runaway mass-transfer event is unavoidable, also with very massive O-Ne-Mg WD or the short orbital periods
leading to a CE formation. For systems with M2 ∼< 1.8M⊙ (∼<3days)observedinsomesystemswithaCOWD.Wethere-
and initial P <1 day the outcome is a BMSP with an ultra foreconcludethatthesebinariesmostlikelyevolvedthrougha
orb
low-massdegeneratehydrogenstar(e.g. PSRJ2051–0827,cf. CE phase where frictional torques were responsible for their
Ergma,Sarna&Antipova1998). present short Pf (cf. thin lines in Fig. 2 and gray area in
orb
Fig. 4). These systems thereforeseem to originatefrombina-
4. DISCUSSION rieswhichinitiallyhadarelativelylargeP andcaseCRLO–
orb
We have now demonstrated how to form a BMSP with a otherwiseifPorb wassmallthestellarcomponentswouldhave
relativelyheavy(Heor CO)WD companionwithoutevolving coalescedeitherinthespiral-inprocess,orasaresultofgravita-
through a CE phase. If a substantial fraction of BMSPs have tionalwaveradiationshortlythereafter(typicallywithin1Gyr
evolved through a phase with super-Eddington mass transfer forsystemssurvivingcaseBRLOandspiral-in).
onasub-thermaltimescale(afewMyr),thiswilleliminatethe
needforalongX-rayphase. Thiswouldthereforehelpsolving We thanktheParkesMultibeamSurveyteamandtheSwin-
the birthrate problembetween BMSPs and LMXBs (Kulkarni burne Pulsar Group for releasing binary parameters prior to
&Narayan1988)forsystemswithPf <50days. publication. We appreciatecommentsfromEneErgmaonthe
orb
It has recently been suggested (Podsiadlowski & Rappaport issue of formingBHWD systems. T.M.T.wouldlike to thank
1999; King & Ritter 1999) that Cygnus X-2 descended from Gerry Brown for discussions and hospitality at Stony Brook.
anintermediate-massX-raybinaryviaascenariowhichresem- T.M.T. acknowledges the receipt of a Marie Curie Research
blestheonedescribedhere. We confirmthatCygnusX-2isa GrantfromtheEuropeanCommission.
REFERENCES
BlandfordR.D.&BegelmanM.C.,1999,MNRAS303,L1 NarayanR.&YiI.,1995,ApJ.444,231
BrownG.E.,1995,ApJ.440,270 Paczynski B., 1976, in: Structure and Evolution in Close Binary Systems,
ChevalierR.A.,1993,ApJ.411,L33 eds: P.P.Eggleton,S.Mitton,J.Whealan,Proc.IAUSymp.73,Dordreicht,
EdwardsR.,etal.,1999,inpreparation Reidel,p.75
ErgmaE.,SarnaM.J.&Antipova,J.,1998,MNRAS300,352 PodsiadlowskiP.&RappaportS.,1999,ApJ.submitted,astro-ph/9906045
KalogeraV.&WebbinkR.F.,1996,ApJ.458,301 Pols O.R., Schröder K.P., Hurley J.R., Tout C.A. & Eggleton P.P., 1998,
Kippenhahn R. & Weigert A., 1990, Stellar Structure and Evolution, A&A MNRAS298,525
Library,Springer-Verlag SobermanG.E.,PhinneyE.S.&vandenHeuvelE.P.J.,1997,A&A327,620
KingA.R.&RitterH.,1999,MNRAS,309,253 TaurisT.M.&SavonijeG.J.,1999,A&A350,928
KulkarniS.R.&NarayanR.,1988,ApJ.335,755 vandenHeuvelE.P.J.,1975,ApJ.198,L109
ManchesterR.N.,etal.,1999,toappearin:IAUColloq.177,PulsarAstronomy vandenHeuvelE.P.J.,1994,A&A291,L39
–2000andBeyond,eds:M.Krameretal.(ASPConf.Series).
3Inthesenarrowbinariestheamountofavailableorbitalenergy(apossibleenergysourceforprovidingtheoutwardejectionoftheenvelope)issmallandhence
theneutronstarismostlikelytospiralintowardthe(unevolved)coreofthedonor,formingaThorne-Zy˙kow-likeobject. Inthatcasetheneutronstarwillprobably
undergohypercriticalaccretionandcollapseintoablackhole.
4 FORMATIONOFMILLISECONDPULSARSWITHHEAVYWHITEDWARFCOMPANIONS
TABLE 1
OBSERVED PULSARS WITH A“HEAVY” WD COMPANION
˙
PSR P f Mobs P P
orb WD spin spin
(days) (M ) (M ) (ms)
⊙ ⊙
J1904+04∗ 15.75 0.0046 0.27 71.1 ···
J1810–2005∗ 15.01 0.0085 0.34 32.8 1.3×10- 19
J1453–58∗ 12.42 0.13 1.07 45.3 ···
J0621+1002 8.319 0.0271 0.540 28.9 <8×10- 20
J1022+1001 7.805 0.0833 0.872 16.5 4.2×10- 20
J2145–0750 6.839 0.0242 0.515 16.1 2.9×10- 20
J1603–7202 6.309 0.00881 0.346 14.8 1.4×10- 20
J1157–5112∗∗ 3.507 0.2546 >1.20 43.6 <9×10- 19
J1232–6501∗ 1.863 0.0014 0.175 88.3 1.0×10- 18
J1435–60∗ 1.355 0.14 1.10 9.35 ···
B0655+64 1.029 0.0714 0.814 196 6.9×10- 19
J1756–5322∗∗ 0.453 0.0475 0.683 8.87 ···
∗Newpulsar,ParkesMultibeamSurvey(Manchesteretal. 1999).
∗∗Newpulsar,Edwardsetal. (1999).
NOTE.—MWobDs iscalculatedassumingMNS=1.4M⊙andi=60◦.
TAURIS,VANDENHEUVEL&SAVONIJE 5
FIG.1.—TheevolutionofanX-raybinarywithM2=4.0M⊙andPorb=4.0days.TheleftpanelshowstheevolutionofPorbasafunctionofM2(timeisincreasing
totheright).Thecentralpanelgivesthemass-lossrateofthedonorasafunctionofitsagesincetheZAMS.Therightpanelshowstheevolutionofthemass-losing
donor(solidline)inanHR-diagram. Thedottedlinerepresentstheevolutionarytrackofasingle4.0M⊙star.Thelettersinthedifferentpanelscorrespondtoone
anotheratagivenevolutionaryepoch–seetextforfurtherexplanation.
FIG.2.—ThefinalPorbasafunctionofWDmassfordifferentBMSPs.Nexttoeachcurveisgiventheinitialmassofthedonorstar(theprogenitoroftheWD)
usedinourevolutionarycalculations. ThefreeparameterineachcurveistheinitialPorb(attheonsetoftheRLO).Thecurvesingraycolorrepresenttheformaion
ofBMSPswithheliumWD,whiletheblackcurvesareBMSPswithCOWD.TheopencirclesonsomeofthecurvesindicatethetransitionfromcaseAtoearly
caseBRLOmasstransfer(i.e.whetherornotthedonorburnedhydrogeninthecoreattheonsetoftheRLO,Kippenhahn&Weigert1990).Thethinlinesshowthe
calculatedparametersforsystemswhichevolvedthroughaCEandspiral-inphasescenarioassuminganefficiencyparameterofηCEλ=1.0(e.g. vandenHeuvel
1994).The12observedBMSPswitha“heavy”WDcompanionaremarkedwithastar,seeTable1.
FIG.3.—Thedependenceoffinalorbitalperiod(top)andmassoftheWD(bottom),ontheinitialorbitalperiod,Porb.
FIG. 4.—Thisplotillustratestheallowedparameterspace(whitearea)forproducingBMSPswithoutevolvingthroughaCEphase. IfM2>1.8M⊙ andthe
donorhasadeepconvectiveenvelopeattheonsetofmasstransfer(i.e.Porbislarge)thesystemwillevolveintoaCEandspiral-inphase.Thisisalsothecaseifthe
initialperiodisveryshortandM2>1.8M⊙.Inthelattercasetheneutronstarmaycollapseintoablackhole.
6 FORMATIONOFMILLISECONDPULSARSWITHHEAVYWHITEDWARFCOMPANIONS
FIG.1.—
TAURIS,VANDENHEUVEL&SAVONIJE 7
FIG.2.—
8 FORMATIONOFMILLISECONDPULSARSWITHHEAVYWHITEDWARFCOMPANIONS
FIG.3.—
TAURIS,VANDENHEUVEL&SAVONIJE 9
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)O(cid:0)(cid:1)-N(cid:0)(cid:1)e(cid:0)(cid:1)-M(cid:0)(cid:1)g(cid:0)(cid:1)(cid:0)(cid:1)
0 (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
WD
0
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
0
1
Common Envelope
CO WD & spiral-in
)
s CO WD
y
0
a
0
case C RLO
d
1
(
case B RLO
b
r
o
P
He WD
0

1
l
a
ti CO WD
i
n
i
converging sys.
1
He WD /
coalescense BH
H-deg.
1 2 3 4 5 6 7
M (M )
2
FIG.4.—