Understanding Compact Object Formation and Natal Kicks. III. The case of Cygnus X-1

1.3 MB·English

by  Tsing-Wai Wong

DRAFTVERSIONJANUARY5,2012
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
UNDERSTANDINGCOMPACTOBJECTFORMATIONANDNATALKICKS.III.THECASEOFCYGNUSX-1
TSING-WAIWONG1,FRANCESCAVALSECCHI1,TASSOSFRAGOS2,VASSILIKIKALOGERA1
1CenterforInterdisciplinaryExplorationandResearchinAstrophysics(CIERA)&DepartmentofPhysicsandAstronomy,NorthwesternUniversity,
2145SheridanRoad,Evanston,IL60208,USA;[email protected],[email protected],[email protected]
2Harvard-SmithsonianCenterforAstrophysics,60GardenSt.,Cambridge,MA02138,USA;[email protected]
DraftversionJanuary5,2012
ABSTRACT
2
In recent years, accurate observationalconstraints become available for an increasing number of Galactic
1
X-raybinaries. Togetherwithpropermotionmeasurements,wecouldreconstructthefullevolutionaryhistory
0
of X-ray binaries back to the time of compact object formation. In this paper, we present the first study of
2
thepersistentX-raysourceCygnusX-1thattakesintoaccountofallavailableobservationalconstraints. Our

n analysis accounts for three evolutionaryphases: orbital evolution and motion through the Galactic potential
a aftertheformationofblackhole(BH),andbinaryorbitaldynamicsatthetimeofcorecollapse. Wefindthat
J the mass of the BH immediate progenitor is 15.0- 20.0 M⊙, and at the time of core collapse, the BH has
4 potentially received a small kick velocity of ≤77 km s- 1 at 95% confidence. If the BH progenitor mass is
less than∼17M⊙, a nonzero natalkick velocityis requiredto explainthecurrentlyobservedpropertiesof
] CygnusX-1. Since the BH hasonlyaccretedmass fromitscompanion’sstellar wind, the negligibleamount
E
ofaccretedmassisimpossibletoexplaintheobservationallyinferredBHspinofa∗>0.95,andtheoriginof
H
this extreme BH spin must be connected to the BH formation itself. Right after the BH formation, we find
h. thattheBHcompanionisa19.8- 22.6M⊙ mainsequencestar,orbitingtheBHataperiodof4.7- 5.2days.
Furthermore, recent observationsshow that the BH companionis currently super-synchronized. This super-
p
synchronismindicatesthatthestrengthoftidesexertedontheBHcompanionshouldbeweakerbyafactorof
-
o atleasttwocomparedtotheusuallyadoptedstrength.
r Subjectheadings:binaries:close—X-rays:binaries—X-rays:individual(CygnusX-1)
t
s
a
[
1. INTRODUCTION cases. Both donors in GRO J1655-40 and XTE J1118+480
are transferring mass to the BH under Roche lobe overflow,
2 In recent years, the number of observed black hole (BH)
v X-raybinaries(XRBs)hasgrownsignificantly. Forthesebi- whereas the BH in Cygnus X-1 is accreting mass from the
5 naries,thereexistsawealthofobservationinformationabout stellarwindofitscompanion.
8 theircurrentphysicalstate: BHanddonormasses,orbitalpe- Theplanofthepaperisasfollows. InSection2,wereview
5 riod,donor’spositionontheH-Rdiagramandsurfacechem- CygnusX-1’scurrentlyavailableobservationalconstraints.A
5 generaloutlineoftheanalysisusedtoreconstructthesystem’s
icalcomposition, transientor persistentX-rayemission, and
. evolutionaryhistoryispresentedinSection3,whileindivid-
7 Roche lobe overflow (RLO) or wind-driven character of the
ualstepsoftheanalysisarediscussedinmoredetailinSection
0 mass transfer (MT) process. Furthermore, proper motions
4- 7. InSection8,wederiveconstraintsontheformationof
1 have been measured for a handful of these binaries (e.g.
theBH.Thefinalsectionisdevotedtoasummaryofourre-
1 Mirabeletal. 2001, 2002; Mirabel&Rodrigues 2003). To-
: gether with the earlier measurements of center-of-mass ra- sultsanddiscussionofsomeoftheassumptionsintroducedin
v ouranalysis.
dialvelocitiesanddistances,wecanobtaininformationabout
i
X the three-dimensionalkinematicpropertiesofthese binaries.
2. OBSERVATIONALCONSTRAINTSFORCYGNUSX-1
r Given this plethora of observation results, the current ob-
a servedsampleofBHXRBsprovidesuswithauniqueoppor- CygnusX-1wasfirstdetectedinAerobeesurveysin1964
tunitytounderstandtheformationandevolutionofBHsinbi- by Bowyeretal. (1965). Soon after the discovery, it was
naries.Thispaperisthethirdinaserieswhereweinvestigate identified as an XRB, which consisted of a compact object
in detail the BH formation in XRBs, especially focusing on and a visible star HDE 226868 (Murdin&Webster 1971;
themassrelationshipbetweenBHsandtheirimmediatepro- Webster&Murdin 1972; Bolton 1972a). Spectroscopic ob-
genitorsand the possible BH natal kick magnitudeimparted servationsledWalborn(1973)toclassifyHDE226868asan
duringthecorecollapseevent. O9.7Iabsupergiant.Bregmanetal.(1973)estimatedthedis-
In the firstpaperof this series, Willemsetal. (2005, here- tancetobe2.5kpcandsetalowerlimitof1kpc,basedonthe
afterPaperI)showedhowusingthecurrentlyavailablecon- colorsoffieldstarsinthevicinityofthesupergiant. Usinga
straints one could uncover the evolution history of an XRB combinationofdatafromDavidDunlapObservatory(DDO)
from the present state back to the time just prior to the core andtheRoyalgreenwichObservatory,Bolton(1972b)derived
collapse event. They applied their analysis to the BH XRB the orbital period, eccentricity and systemic radial velocity
GROJ1655-40.Inthesecondpaper,Fragosetal.(2009,here- (V0)tobe5.5995±0.0009days,0.09±0.02and- 6.0±0.1
afterPaperII)performedthesameanalysisforthecaseofthe kms- 1, respectively. BasedontheabsenceofX-rayandop-
BHXRBXTEJ1118+480.Inthiswork,wefocusonthecase ticaleclipses,theauthorgavealowerlimitof7.4M⊙ onthe
of the BH XRB CygnusX-1. The mass transfer mechanism massofthecompactobject. Thisimpliedthecompactobject
inCygnusX-1isdifferentfromtheXRBsstudiedinprevious wastoomassivetobeawhitedwarforaneutronstar. Thus,
theauthorconfirmedWebster&Murdin(1972)’sfindingthat
2 T.-W.Wongetal.
thecompactobjectisaverystrongBHcandidate. 1 is a persistent X-ray source. Since the supergiant is cur-
Usingtheorbitalperiodobtainedfromspectrometryanda rently not overfilling its Roche lobe (Gies&Bolton 1986a),
range in the assumed degree of Roche filling of the super- the observedX-raysare mainly poweredby the accretion of
giant,Gies&Bolton(1982,1986a)foundalowermasslimit stellarwind. TheX-rayluminosityofCygnusX-1variesbe-
of7M forthecompactobject.Thisconfirmedthatthecom- tween two discrete levels, namelythe "hard(low) state" and
⊙
pact object observed in Cygnus X-1 was a BH. The same the"soft(high)state". Asthesystemspendsmostofitstime
authors also refined the orbital period and eccentricity to be (∼90%,seeCadolleBeletal.2006)inthehardstate, wefo-
5.59974±0.00008and0.021±0.013,respectively,andmea- cus on the hard state X-ray luminosity (L ). Fronteraetal.
X
suredV to be - 2.0±0.7km s- 1. Ninkovetal. (1987) used (2001) observed Cygnus X-1 with the Narrow Field Instru-
0
therelationshipbetweentheequivalentwidthoftheHγ spec- mentsoftheBeppoSAXsatelliteatdifferentepochsin1996.
trallineandtheabsolutemagnitudeofearly-typesupergiants TheauthorsobtainedtheLX (0.5- 200keV)andtheextrapo-
toestimatethedistanceas2.5±0.3kpc. latedbolometricluminosity(L )as2.0×1037and2.4×1037
bol
Herreroetal. (1995) performed a detailed spectroscopic ergs- 1,respectively,assumingadistanceof2kpc. Usingob-
analysis on the supergiant, and derived the masses to be servational data obtained by the Compton Gamma Ray Ob-
10.1M⊙ and17.8M⊙ fortheBH (MBH) andthe supergiant servatory(CGRO)between1991and2000,McConnelletal.
(M2), respectively, if an orbitalinclinationangle of 35◦ was (2002)derivedLboltobe(1.62–1.70)×1037ergs- 1,withthe
assumed. Using the Isaac Newton telescope, LaSalaetal. distancetothesourcefixedat2kpc.CadolleBeletal.(2006)
(1998) measured the orbital period as 5.5997±0.0001 days observedCygnusX-1withtheInternationalGamma-RayAs-
andV0 as- 5.4±0.1kms- 1. Withalltheaccumulatedradial trophysicsLaboratory(INTEGRAL)between2002and2004
velocitymeasurementsandtheirownspectroscopyofthesu- and measured L (20- 100 keV) as 6.5×1036 erg s- 1, as-
X
pergiant,Brocksoppetal.(1999)refinedtheorbitalperiodto suming a distance of 2.4 kpc. The authorsalso gave L as
bol
5.599829±0.000024. ThepropermotionofCygnusX-1was 2.2×1037ergs- 1.
observedwiththeVeryLargeBaselineInterferometry(VLBI) For the systemic parameters relevant to our analysis, we
between 1988 and 2001 (Lestradeetal. 1999; Stirlingetal. adoptthe mostrecentobservationalconstraints, with the ex-
2001; Mirabel&Rodrigues 2003). During this period, the ception of L . We consider all the L values mentioned
system’s position shifted at a rate of - 4.2±0.2 mas yr- 1 in above, assumboilng they represent the typboilcal X-ray variabil-
rightascension(R.A.)and- 7.6±0.2masyr- 1indeclination ity range for this system. After rescaling their values to the
(dec.). Meanwhile, a trigonometric parallax of 0.73±0.30 parallaxdistancemeasurementbyReidetal.(2011)andcon-
maswasalsomeasuredwithVLBI,whichgaveadistanceof sideringthe uncertaintyin thatdistance, we adoptL to be
bol
1.4+- 00..94kpc(Lestradeetal.1999). (1.17- 2.35)×1037ergs- 1.Foreaseofreference,ouradopted
By studying the spectra obtained with the 0.9 m coudé observationalconstraintsaresummarizedinTable1.
feed telescope of Kitt Peak National Observatory, the 2.1
m telescope of University of Texas McDonald Observatory, 3. OUTLINEOFANALYSISMETHODOLOGY
andthe1.9mtelescopeofUniversityofTorontoDavidDun-
Inouranalysis,weassumethatCygnusX-1formedinthe
lap Observatory between 1998 and 2002, Giesetal. (2003)
Galacticdisk,fromtheevolutionofanisolatedprimordialbi-
derived V as - 7.0±0.5 km s- 1 and estimated M /M
0 BH 2 naryatsolarmetallicity. Infact,Mirabel&Rodrigues(2003)
≈ 0.36±0.05. Caballero-Nievesetal. (2009) examined
suggestthatCygnusX-1belongstoCygnusOB3(CygOB3),
the supergiant’s ultraviolet spectra from the Hubble space
whichisanOBassociationlocatedclosetotheGalacticplane.
telescope. Their results gave masses of 23+8 and 11+5
- 6 - 3 WealsoassumethatnomasstransferviaRochelobeoverflow
M for the supergiant and the BH, respectively. On the
⊙ occurredintheevolutionaryhistoryofthisbinary.
otherhand,Shaposhnikov&Titarchuk(2007)usedtheX-ray
According to our current understanding, in order to form
quasi-periodicoscillationandspectralindexrelationshipand
a∼15M⊙ stellar BH atsolar metalicity,the BH progenitor
deducedMBH tobe8.7±0.8M⊙,whichoverlappedwiththe in the primordialbinary needs to be more massive than 120
lowerendoftheM rangederivedbyCaballero-Nievesetal.
BH M⊙ (Belczynskietal. 2010). Such a massive star loses its
(2009).
hydrogenrichenvelopeviastellarwind,andexposesitsnaked
Recently, Reidetal. (2011) measured the trigonometric
heliumcore. Attheendofnuclearevolution,itcollapsesinto
parallax of Cygnus X-1 with the National Radio Astron-
aBH.Duringthecorecollapseevent,theorbitisalteredbythe
omy Observatory’s Very Long Baseline Array (VLBA) and
asymmetric mass loss from the system and a possible recoil
founda distance of 1.86+- 00..1121 kpc. The authorsalso reported kick imparted to the BH. If the binary survives through the
proper motion measurements of Cygnus X-1, which were corecollapseevent,angularmomentumlossviagravitational
- 3.78±0.06mas yr- 1 in R.A. and - 6.40±0.12mas yr- 1 in radiationandtidaleffectscausestheorbittoshrink,although
dec. Meanwhile, Xiangetal. (2011) studied the X-ray dust wind massloss leads to orbitalexpansion. In the meantime,
scatteringhaloofCygnusX-1anddeterminedthedistanceto the more evolved BH companionis losing mass via its own
be 1.81±0.09 kpc, after considering the compatibility with stellarwindatahigherrate. ThesystembecomesaBHXRB
theparallaxresult.Buildingonthetrigonometricparallaxdis- whentheBHcapturesanon-negligibleamountofmassfrom
tance measurement of Reidetal. (2011), Oroszetal. (2011) itscompanion’sstellarwind.
performed optical data modeling of CygnusX-1, and found Inthispaper,we restrictourselvesto the formationof BH
themassofthesupergianttobe19.2±1.9M⊙ andtheblack XRBsthroughtheaboveevolutionarychannel. Likethefirst
hole to be 14.8±1.0 M⊙. Using the results of Reidetal. two papers, our goal is to track the evolutionary history of
(2011)and Oroszetal. (2011), Gouetal. (2011) determined Cygnus X-1 back to the time just prior to the core collapse
that CygnusX-1 hosts a near-extreme Kerr BH, with a spin event. Our analysis incorporates a number of calculations
parametera∗>0.95. whichcanbesummarizedinfoursteps.
UnlikemostoftheXRBsknowntohostaBH,CygnusX- First, we identifythecurrentevolutionarystageofthe BH
UnderstandingCompactObjectformation.III. 3
Table1
PropertiesofCygnusX-1
Parameter Notation Value References
Distance(kpc) d 1.86+0.12 (9)
- 0.11
Galacticlongitude(deg) l 71.3 (2)
Galacticlatitude(deg) b +3.1 (2)
PropermotioninR.A.(masyr- 1) µR.A. - 3.78±0.06 (9)
Propermotionindecl.(masyr- 1) µdecl. - 6.40±0.12 (9)
Systemicvelocity(kms- 1) V0 - 7.0±0.5 (5)
Orbitalperiod(days) Porb 5.599829±0.000016 (1)
Orbitaleccentricity eorb 0.018±0.003 (8)
Inclinationangle i 27.06±0.76 (8)
Blackholemass(M⊙) MBH 14.81±0.98 (8)
Blackholespin a∗ >0.95 (10)
Companionmass(M⊙) M2 19.16±1.90 (8)
CompanionRadius(R⊙) R2 16.50±0.84 (8)
CompanionLuminosity(L⊙) L2 (1.91–2.75)×105 (8)
CompanionEffectivetemperature(K) Teff 30000–32000 (8)
Companionsurfacerotationspeed(kms- 1) Vrotsini 95±6 (7)
BolometricluminosityoftheX-raysource(ergs- 1) Lbol (1.3- 2.1)(cid:16)1.86dkpc(cid:17)2×1037 (3),(4),(6)
References.—(1)Brocksoppetal.1999,(2)Lestradeetal.1999,(3)Fronteraetal.2001,(4)McConnelletal.2002,(5)Giesetal.2003,(6)CadolleBelet
al.2006,(7)Caballero-Nievesetal.2009,(8)Oroszetal.2011,(9)Reidetal.2011,(10)Gouetal.2011
companion,sothatalltheobservationalconstraintsaresatis- eachsuccessfulsequence.
fied. UndertheassumptionthattheBHcompanionmasshas Finally, we follow the orbitalevolutionof these simulated
notbeenalteredby masstransferin the past, we modelit as binaries to the current epoch. Our calculation accounts for
an isolated star. Using a stellar evolutioncode, we calculate tides, wind mass loss, wind accretion onto the BH, and or-
a grid of evolutionary sequences of isolated stars at differ- bital angular momentum loss via gravitational radiation. At
ent zero age main sequence (ZAMS) masses. We examine theendofthecalculations,werequireagreementbetweenthe
eachsequencetofindwhetherthereexistsapointintimethat observedandcalculatedorbitalperiodandeccentricity.
thecalculatedstellarproperties,i.e. mass,radius,luminosity
andeffectivetemperature,areallsimultaneouslyinagreement 4. MODELINGTHEBHCOMPANION
withthecurrentlyobservedpropertiesoftheBHcompanion.
Under the assumption that the companion mass has not
If such a period of time exists, we classify that sequence as
beenalteredby masstransferin itspast, we modelthecom-
"successful".ThecurrentageoftheBHcompanioncanbees-
panionasanisolatedstarusingamodifiedversionofthestel-
timatedfromthesesuccessfulsequences,andthetimeexpired
larevolutioncodeEZ(originallydevelopedbyPaxton2004).
sincetheBHformationcanthenbederivedbysubtractingthe
We calculate the evolution of our stellar models at solar
approximatelifetimeoftheBHprogenitor.
metallicity, which is the same metallicity that Oroszetal.
Next,weconsiderthekinematicevolutionaryhistoryofthe
(2011)usedinderivingthepropertiesoftheBHcompanion.
XRBintheGalacticpotential.Startingfromthecurrentloca-
Whenweplacethecompanion’sobservationalconstraintson
tion, we follow the methodologyof Gualandrisetal. (2005)
anH-Rdiagram,wefindthatthecurrentlocationofthecom-
and use the observedthree-dimensionalvelocity to trace the
paniondoesnotseemto beconsistentwithanyevolutionary
Galactic motionof CygnusX-1 backwardin time. Together
trackscalculated by the stellar evolution code. As shown in
withtheconstraintsonthecurrentageofthesystemderived
Figure1,thecompanionisoverluminousforastarofitsmass.
in the first step, this allows us to determinethe location and
ThiscannotbeexplainedbyearliermasstransferfromtheBH
velocityofthebinaryatthetimeofBHformation(wedenote
progenitor to the companion. Braun&Langer (1995) stud-
theseas"birth"locationandvelocity). Bysubtractingthelo-
iedtheeffectsofmassaccretionontomassivemainsequence
cal Galactic rotational velocity at the "birth" location from
stars,andfoundthattheaccretingstarswouldnotappearover-
the systems’s center-of-mass velocity, we derive constraints
luminous for their new masses during the rest of their main
onthepeculiarvelocityofthebinaryrightaftertheformation
sequencelifetime. Ifmassaccretionleadstoasocalled"reju-
oftheBH.
venation"oftheaccretingstar,whichmeansitscentralhydro-
In the third step, we analyze the orbital dynamics of the
genabundancesubstantiallyincreases,itwouldhavethesame
corecollapseeventduetomasslossandpossiblenatalkicks
luminosityasastarofitsnewmass. Ifrejuvenationdoesnot
imparted to the BH. In this paper, we refer to the instants
occur,the accretingstar would appearunderluminousforits
right before and after the formation of the BH by the terms
newmassduringtherestofitsmainsequencelifetime.
"pre-SN"and"post-SN",respectively. Westartwiththecon-
One possible solution for matching the observedcompan-
strainedparameterspaceof(M ,M )derivedinthefirststep
BH 2 ion’sluminosityis increasingthe coreovershootingparame-
andperformaMonteCarlosimulationscanningoverthepa-
ter α to ∼0.45. Although this value is relatively high, it
rameterspaceofthepre-SNbinaryproperties.Thisparameter ov
is not unphysical. Claret (2007) comparedthe data from 13
spaceislimitedbyrequirementsoforbitalangularmomentum
double-lineeclipsingbinarysystemswiththeoreticalpredic-
andenergyconservations,andbythepost-SNbinarypeculiar
tions of stellar modeling and found α could be as high as
velocity constraint derived in the second step. This calcu- ov
0.6formassivestars. We varyα from0.35to0.5,insteps
lation yields a population of simulated post-SN binaries for ov
of0.01. We notethattheneedforsuchhighervaluesofα
ov
4 T.-W.Wongetal.
thewindmasslossrateofthecompanioninourmodels.F
wind
isaparametersuchthatM˙ neverexceeds- 0.8M˙ ,andα
acc 2 wind
is the accretionefficiency,which variesbetween1.5and 2.0
(Boffin&Jorissen 1988). A and e are the orbitalsemi-
orb orb
majoraxisandeccentricity,respectively.A isderivedfrom
orb
themeanmeasuredorbitalperiodP ,whichis
orb
G(M +M )P2 31
A = BH 2 orb , (3)
orb 4π2
(cid:20) (cid:21)
where M is the companion mass in our models. e is set
2 orb
equal to the mean measured orbital eccentricity. V de-
wind
notesthewindvelocity. V2 equalstoV2 /V2 ,whereV2
BH wind BH
is the orbitalvelocitysquare of the BH and is approximated
as G(M + M )/A . We adopt the spherically symmetric
BH 2 orb
windvelocitylawgiveninLamers&Cassinelli(1999),
β
R
V (r) = V + (V - V ) 1 - 2 (4)
wind esc ∞ esc
r
(cid:18) (cid:19)
where r is the distance from the companion to the BH and
Figure1. TheevolutionarytracksforisolatedstarsontheH-Rdiagram.On is set equal to A . β is a free parameter varying from 0.6
orb
eachtrack, themassofthestarinM⊙ isindicated atvarious points. The to 1.6 (Gies&Bolton 1986b; Lamers&Leitherer 1993), in
grayshapedarearepresentstheobservationalconstraintsoftheBHcompan-
ion. AtTeff≈31000K,themodelwithaninitialmassof20M⊙ (dashed steps of 0.1. V∞ is the wind velocity at infinity, while Vesc
line)wouldhaveamass≈19.64M⊙,whichisingoodagreementwiththe istheeffectiveescapevelocityatthesurfaceofthecompan-
measuredmassofthecompanion. However,itdoesnotmatchthemeasured ion. Within the typical range of O star surface temperature,
luminosity.Ontheotherhand,themodelwithaninitialmassof30M⊙(dot- V isscaledas2.65V (Kudritzki&Puls2000). Following
tthedeslitnaer)dmoeastcnhotthmeamtcehastuhreecdolmumpainnoiosnit’ys.aTthTeeffm≈od3e1l0w0i0thKa,nbiuntittihaelmmaassssooff La∞mers&Cassinelli(1e9sc99),
20M⊙andαov=0.45(solidline)couldmatchboththemeasuredmassand
luminosityofthecompanionatTeff≈31000K. Vesc= 2(1- Γe)GM2/R2, (5)
inthemodelingofmassivestarsmayverywellbeconnected
where p
tothesignificantpresenceofinternalrotationandassociated σ L
rotationalmixing.Effectivelyincreasingαovleadstostronger Γe = 4πceGM2 (6)
internalmixing and in a way allows the stellar modelto be- 2
havemorelikearotatingmodel. is the mass correcting factor for the radiative force due to
Besides the observational constraints on the companion’s electron scattering, and c is the speed of light in vacuum.
properties,therearethreeadditionalconstraints.Thefirstone Lamers&Leitherer(1993)scaled theelectronscatteringco-
comesfromthefactthatthecompanioniscurrentlynotover- efficientperunitmassσe as
filling its Roche lobe (Gies&Bolton 1986a). Thus, we re-
1 + qǫ
quirethestellarradiusR2inourmodelstobe σe = 0.401 1 + 3ǫ , (7)
R ≤ A r +∆R, (1) (cid:18) (cid:19)
2 orb Egg
whereq is the fractionof He++ and (1 – q) is the fractionof
where rEgg is the effective Roche lobe radius given by He+,withq=1ifT ≥35,000K,q=0.5if30,000K≤T
Eggleton (1983). Here, we make an approximation that the eff eff
<35,000K,andq=0ifT <30,000K.Theabundanceratio
orbit is circular and synchronized. The parameter ∆R is a eff
ǫ=He/(H+He)isfixedat0.15,whichisappropriateforan
constant accountingfor the differencein the calculated stel- OstarwithaspectraltypeofClassI.UsingM˙ fromequa-
larradiiamongstellarevolutioncodes(Valsecchietal.2010). acc
tion(2),wefollowBelczynskietal.(2008)andcalculatethe
Weset∆Rto2.5R .
⊙ bolometricluminosityresultingfromthecompanion’sstellar
Another constraint is that the calculated bolometric lumi-
windbeingaccretedontotheBHas
nosity(L )resultingfromthestellarwindaccretionprocess
bol
needstofallwithintheobservationalrange,whichis(1.17- 1GM M˙
2.35)×1037ergs- 1. ByadoptingtheBondi&Hoyle(1944) Lbol= BH acc, (8)
2 R
accretion model and following Belczynskietal. (2008), the acc
orbital-averagedaccretionrateisgivenby where R denotes the radius of the accretor. For the case
acc
ofBH,R istheradiusoftheinnermoststablecircularor-
M˙ =- Fwind GMBH 2 αwind M˙2 (2) bit,whichacwc ecalculatewithEquation(2.21)inBardeenetal.
acc 1- e2orb(cid:18) Vw2ind (cid:19) 2A2orb(1+V2)3/2 ((1G9o7u2e).taGl.i2v0e1n1)th,eweobasdeorpvtatthioenmalelydiainnfear∗re=d0s.9p7inanad∗fi>nd0.95
q
tHheer1e,σMraBnHgiesothfethBeHobmsearsvsaitnioonuarlmcoondsetlrsa,inwt,hiinchstveaprsieosfw0i.t0h9in8 Racc=2.57 MBH km. (9)
M
M⊙. Since the total mass that the BH could have accreted (cid:18) ⊙ (cid:19)
from its companion stellar wind is negligible, MBH in each This calculated Lbol needs to fall within the observational
evolutionarysequenceisfixedthroughoutouranalysis. M˙ is range.
2
UnderstandingCompactObjectformation.III. 5
Figure2. Systemicbehavioroftwoselectedevolutionarysequences,whichhavethesameαov=0.44,MBH=14.81M⊙,αwind=1.5,andβ=1.0. Sequence
1(solid)and2(dashed)haveM2,zamsof21and22M⊙,respectively. TheleftpanelshowstheevolutionarytracksontheH-Rdiagram,whilethemiddlepanel
illustratesthebehaviorsofthemassandtheradiusofthestar.Therightpanelshowsthevariationsofthecalculatedstellarluminosityand fL,where fLisdefined
inEquation(10). Thegrayshadedareasrepresenttheobservationalconstraintsontherelevantquantities,andthethickpartoftheevolutionarytracksindicates
thepartofthesequencethattheobservationalconstraintsontheH-Rdiagramaresatisfied.
Thelastadditionalconstraintisthattheobservationalcon-
straintsonL andL havetobeevaluatedatthesamedistant
2 bol
estimation.Toexaminethis,wecalculatetheratio
L L - 1
f = 2 bol , (10)
L 105L 1037ergs- 1
⊙
(cid:18) (cid:19)(cid:18) (cid:19)
which is independent of distance. From Figure1 in
Oroszetal.(2011),L2is2.09×105L⊙atTeff=30000K,and
is2.51×105L⊙atTeff=32000K,assumingadistanceof1.86
kpc. TogetherwiththemeasuredrangeofL rescaledatthe
bol
samedistanceestimation,theupperandlowerlimitsof f are
L
1.01and1.90,respectively.Wecanassurethatbothluminos-
ityconstraintsareevaluatedatthesamedistanceestimationif
f fallswithinthatrange.
L
In order to find the current evolutionary stage of the BH
companion,we apply these constraintsto a set of evolution-
arysequences, whichcoverthe parameterspace of thecom-
panion’sZAMSmass(M ),α ,M ,α ,andβ. For
2,zams ov BH wind
each sequence, we find whether there exists a point in time
that the calculated properties simultaneously satisfy all ob- Fsuigccuersesf3u.lsTehqeuepnacreasm.eter space ofM2,zams, αov, and MBH covered byall
servationalconstraints: the BH companion’smass, luminos-
ity, temperature, and radius, L , f , and not overfilling the
bol L
RochelobeoftheBHcompanion. SimilartotheRochelobe
sincetheBHformation(t )by
constraint,wealsoconsideranuncertaintyof±2.5R⊙ inthe sys
calculatedstellarradiiwhenweapplytheobservationalcon- t = t - t , (11)
sys 2 BH
straintoftheBHcompanion’sradius.Ifsuchaperiodoftime
exists,weclassifythatevolutionarysequenceas"successful". where tBH is the approximate lifetime of the BH progenitor.
ThebehaviorofsomerelevantparametersisillustratedinFig- WefollowBelczynskietal.(2010)tocalculateMBH andtBH
ure2 for two selected successful sequences that are chosen fordifferentprogenitorsusingthestellarevolutioncodeSSE
mainly to provide a clear and instructive picture. The dis- (Hurleyetal.2000),andadoptingthemasslossprescriptions
playedsequencesthereforedonotrepresentourbestpossible whichwereclassifiedas"Vinketal. Winds". Thecalculated
matchesto theobservedpropertiesof CygnusX-1. Figure3 tBHarefitasafunctionofMBH,
showstheparameterspaceofM ,α ,andM covered
byallsuccessfulsequences. For2,zαamwisnd aonvdβ, theBsHuccessful tBH = MMB⊙H +3.341,
sequencescoveredtheentireallowedparameterspace,which 106yrs 2
are1.5≤αwind≤2.0and0.6≤β≤1.6. 19.26- 4.902 MMB⊙H +0.3841 MMB⊙H
The current age of the BH companion could be derived (cid:16) (cid:17) (cid:16) (cid:17) (12)
from the time interval at which all observationalconstraints for MBH ≥9.5 M⊙. Figure4 shows the variationsof t2 and
are satisfied. Assuming that the BH progenitorand its com- t againstM . We find that t is between 4.8 and 7.6
BH 2,zams sys
panionformedatthe same time, we couldcomputethetime Myr.
6 T.-W.Wongetal.
Figure4. Thevariations oft2 (circles) against M2,zams. Thegrayshaded Figure5. Upperpanels: Thegreydots illustrate thepossible locations of
regionindicatestherangeoftBHforthecorrespondingsuccessfulsequences, CygnusX-1atthebirthtimeoftheBH,obtainedfrom3,000integrationsof
which is calculated byEquation (12). The difference between t2 and tBH itstrajectorybackwardsintime.Theinitialconditionsoftheintegrationsare
givestsys. generatedrandomlyusingthemethodologydescribedinSection5.Theplus
signs indicate the current location ofCygnus X-1, derived from the mean
distanceof1.86kpc. ThecrossesrepresentthecurrentlocationofCygOB3
5. KINEMATICHISTORYINTHEGALAXY center,withanadopteddistanceof2kpc. Lowerpanel: Thedistributionof
Here, we assume that CygnusX-1 formed in the Galactic post-SNpeculiarvelocitiesVpec,postSNagainstthetimeexpiredsincetheBH
formation.
disk.TheconsiderationofCygOB3beingtheparentassocia-
tionofCygnusX-1isdiscussedinSection9.2.Giventheob- timeofBHformation,obtainedfromintegrating3,000trajec-
servedpositionandmeasuredpropermotionofCygnusX-1, toriesbackwardsintime.Asthereisnotrajectorycrossingthe
wederivethepost-SNpeculiarvelocityofthebinary’scenter- Galacticplaneandtheendpointsofalltrajectoriesfallwithin
of-massbytracingitsorbitintheGalaxybacktothetimeof 110pcfromtheGalacticplane,weconsidereachendpointas
BHformation. Wedescribethemotionofthebinarywithre- apossiblebirthsiteoftheBH.Thepost-SNpeculiarvelocity
specttoaright-handCartesianreferenceframe,whoseorigin Vpec,postSN of the binary is obtained by subtracting the local
coincides with the Galactic center. The Z axis points to the Galactic rotationalvelocityfromthe center-of-massvelocity
northernGalactic pole, while the X axis points in the direc- ofthebinaryatthebirthsites. WefindVpec,postSNrangesfrom
tionfromtheprojectedpositionoftheSunontotheGalactic 22to32kms- 1 andistimeindependent. Thedistributionof
planetotheGalacticcenter. Inthisreferenceframe,theSun V againstthetimeexpiredsincetheformationofthe
pec,postSN
islocatedat(X⊙,Y⊙,Z⊙)=(- 8.5,0,0.03)kpc(Joshi2007; BHaredisplayedinFigure5.
Ghezetal.2008;Gillessenetal.2009;Reidetal.2009),and
has a peculiar motion (U ,V ,W ) = (11.1, 12.24, 7.25) 6. ORBITALDYNAMICSATCORECOLLAPSE
⊙ ⊙ ⊙
kms- 1(Schönrichetal. 2010). CygnusX-1 is currently lo- For each of the successfulsequence, we performa Monte
cated at a distance of 1.86+0.12 kpc from the Sun, with a Carlosimulationwhichconsistsoftwentymillionpre-SNbi-
- 0.11
Galactic longitude l = 71.3◦, and a Galactic latitude b = naries. ThepropertiesoftheBHprogenitor’scompanionare
3.1◦ (Lestradeetal. 1999; Reidetal. 2011). This means takenfromthestellarmodelofthatsequence,atthetimewhen
CygnusX-1iscurrently∼130kpcabovetheGalacticplane. theageofthestarisequaltot .Duringasupernova(SN)ex-
BH
To model the Galaxy, we adopt the Galactic potential of plosion,themasslossfromthesystemandpossiblythekick
Carlberg&Innanen(1987)withupdatedmodelparametersof imparted to the BH change the binary’s orbital parameters.
Kuijken&Gilmore(1989). Theequationsgoverningthesys- Thepre-andpost-SNcomponentmasses,orbitalsemi-major
tem’smotionin the Galaxyareintegratedbackwardin time, axis, and orbital eccentricity are related by the conservation
up to the time correspondingto the currentsystem’s age t lawsoftheorbitalenergyandangularmomentum.Inthefol-
sys
given by the successful sequences. We follow the method- lowings,we addthe subscripts"preSN"and"postSN" tothe
ology of Gualandrisetal. (2005) to initialize the parameters notationsoftheorbitalelementstodistinguishbetweentheir
fortheintegration,whichaccountsfortheuncertaintiesinthe valuesjustpriorandrightaftertheSNexplosionthatformed
estimated distance and measured velocity components. We theBH.
generatethe initialsystem’s positionby the Galactic coordi- We start with seven free parameters: the BH immediate
nates (l,b) and a random distance drawing from a Gaussian (He-rich)progenitormass (M ), pre-SN orbitalsemi-major
He
distribution.Wegenerateinitialsystem’svelocitybydrawing axis(A )andeccentricity(e ),themeananomaly(m),
preSN preSN
randomly the proper motions (µ ,µ ) and heliocentric the magnitude (V ) and direction (θ, φ) of the kick veloc-
R.A. decl. k
radialvelocity(V )fromGaussiandistributions. Thecurrent ity impartedto the BH. θ is the polar angle of the kick with
0
system’sageisuniformlydistributedbetween4.8and7.6Myr respect to the relative orbital velocity of the BH progenitor
(seeFigure4). justpriortotheSNexplosion,andφisthecorrespondingaz-
Figure5showsthepossiblepositionsofCygnusX-1atthe imuthalangle(seeFigure1inKalogera2000,foragraphical
UnderstandingCompactObjectformation.III. 7
representation).Thefirstfiveparametersaredrawnfromuni- BHimmediateprogenitorcanbeapproximatedbyEquations
formdistributions,whilethelasttwoaredrawnfromisotropic (3) in Fryer&Kalogera (1997), since we assume that it is
distributions. Itisobviousthattheprogenitormustofcourse a Helium star. Second, the pre-SN spin of the BH imme-
bemoremassivethantheBH, butthereisnoabsoluteupper diate progenitor and its companion need to be less than the
limit for the progenitormass. We adopt MHe ≤20 M⊙, and breakupangularvelocityΩc≈(GM/R3)1/2.Asthecalculated
provideadiscussiononthisupperlimitinSection9.1. stellar radiusR2 associateswith an uncertainty∆R=2.5R⊙
The relations between pre- and post-SN parameters have (Valsecchietal.2010),
beenderivedbyHills(1983):
Vk2 + VH2e,preSN + 2VkVHe,preSNcosθ Ω = GM2 1+3∆R (20)
= G(M +M ) 2- 1 , (13) c s R32 (cid:18) 2 R2 (cid:19)
BH 2
r A
(cid:18) postSN(cid:19)
G(M +M )A (1- e2 ) fortheBHcompanion.
BH 2 postSN postSN
=r2 V2sin2θcos2φ+ sinψ(V +V cosθ)
k He,preSN k
7. ORBITALEVOLUTIONAFTERTHESNEXPLOSION
(cid:16)- V cosψsinθsin(cid:2)φ 2 , (14)
k The orbital evolution of the simulated binaries, which are
Here, r is the orbital separation bet(cid:3)we(cid:17)en the BH progenitor generatedfromtheMonteCarlosimulationsdescribedinSec-
tion6,iscalculateduptothecurrentepoch. Aftertheforma-
anditscompanionatthetimeofSNexplosion,
tionoftheBH,theorbitalparametersofthebinaryaresubject
r = A (1- e cosE ), (15) to secular changesdueto the tidal torqueexertedbythe BH
preSN preSN preSN
onits companion,and dueto the loss oforbitalangularmo-
whereE istheeccentricanomaly,andisrelatedtomas mentum via gravitational radiation and stellar wind. Since
m = E - esinE. (16) thetidalinteractionsdependonboththeorbitalandrotational
propertiesoftheMScompanion,thestar’srotationalangular
VHe,preSNistherelativepre-SNorbitalvelocityoftheBHpro- velocity(Ω)rightafterSNexplosionthatformedtheBHen-
genitor, terstheproblemasanadditionalunknownquantity. Herewe
assumetherotationalangularvelocityoftheBHcompanionis
V = G(M +M ) 2- 1 1/2. (17) unaffectedbythe SNexplosion,andispseudo-synchronized
He,preSN He 2 r A tothepre-SNorbitalfrequency.Thesystemofequationsgov-
(cid:20) (cid:16) preSN(cid:17)(cid:21) erning the tidal evolution of the orbital semi-major axis A,
TheangleψisthepolarangleofthepositionvectoroftheBH
eccentricitye,andtheBHcompanion’srotationalangularve-
withrespecttoitspre-SNorbitalvelocityinthecompanion’s locityΩhasbeenderivedbyHut(1981):
frame.Itisrelatedtothepre-SNparametersas
r2VH2e,preSNsin2ψ = G(MHe+M2)ApreSN(1- e2preSN). (18) dA = - 6k2MBHMBH+M2 R2 8
Sincethecorecollapseisinstantaneous,rremainsunchanged. (cid:18)dt (cid:19)tides T M2 M2 (cid:18) A (cid:19)
Thisgivesaconstraint × A f e2 - 1- e2 3/2f e2 Ω ,
r = A (1- e cosE ) (1- e2)15/2 1 2 n
preSN preSN preSN (cid:20) (cid:21)
= A (1- e cosE ), (19) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (21)
postSN postSN postSN
whichneedstobesatisfiedwith|cosEpostSN|61. de = - 27k2MBHMBH+M2 R2 8
The mass loss from the system and a natal kick imparted dt T M M A
(cid:18) (cid:19)tides 2 2 (cid:18) (cid:19)
atottthheeBbiHnacrayn’sincdeuncteeraopfomsta-SssN. pItescumliaagrnvietulodceitiys(dVepteecr,mpositnSeNd) × e f e2 - 11 1- e2 3/2 f e2 Ω ,
(1- e2)13/2 3 18 4 n
byfollowingEquations(28)–(32)in PaperI, and isrequired
(cid:20) (cid:21)
tofallwithintherangederivedinSection5,whichis22- 32 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) ((cid:1)22)
km/s. dΩ k M 2M R2 R 6
In addition, there are two more restrictions on the proper- =3 2 BH 2 2 2
dt T M I A
tiesofpre-andpost-SNbinarycomponents.First,werequire (cid:18) (cid:19)tides (cid:18) 2 (cid:19) 2 (cid:18) (cid:19)
thatbothcomponentshavetofitwithintheirpre-andpost-SN n Ω
Roche lobe at periapsis. We impose this condition to avoid ×(1- e2)6 f2 e2 - 1- e2 3/2f5 e2 n .
complicationsarisingfrommasstransferinducedchangesin (cid:20) (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (23)
thestellarstructureoftheMScompanion,thatlaterbecomes
theBH companionofthe XRB. To calculatethe Rochelobe
radius of each component in eccentric pre- and post-SN or- Here, k2 and I2 are the apsidal-motionconstantand moment
bits, we adopt the fitting formulaeof Sepinskyetal. (2007). of inertiaof the MScompanion,respectively. T is a charac-
When calculating the pre-SN Roche lobe radii, we assume teristicstimescalefortheorbitalevolutionduetotides,andn
thatthe pre-SNorbitis pseudo-synchronized. Again, dueto =2π/Porbisthemeanorbitalangularvelocity.Thecoefficient
the differencein calculatedstellar radiiamongstellar evolu- functions f e2 fori=i,2,...,5aregiveninEquations(11)
i
tion codes, we consider an uncertainty of ±2.5 R⊙ on the inHut(1981).AstheBHcompanioninCygnusX-1isamas-
companion radius (Valsecchietal. 2010). The radius of the siveMSstar(cid:0)wi(cid:1)tharadiativeenvelope,thefactork /T canbe
2
8 T.-W.Wongetal.
approximatedas
k R R 5/2
2 =1.9782×104 2 ⊙
T R A
(cid:18) (cid:19)rad (cid:18) ⊙(cid:19)(cid:18) (cid:19)
M 1/2 M +M 5/6
× 2 BH 2 E yr- 1. (24)
2
M M
(cid:18) ⊙(cid:19) (cid:18) 2 (cid:19)
TheconstantE comesfromafittothetablesinClaret(2004),
2
t/t
logE =- ms - 5.51039, (25)
2 2.20489- 1.89579(t/t )
ms
for 15.85 ≤M2,zams ≤25.12 M⊙. Here, tms is the main se-
quencelifetime. We define the end of the main sequence as
thehydrogenabundanceatthecorebeinglessthan0.01.
To follow the secular changes of the orbital parameters
associated with emissions of gravitational waves, we adopt
Equations (35) and (36) in Junker&Schäfer (1992), which
arederivedupto3.5post-Newtonianorder.
The rates of change in A and e due to wind mass loss
andwindaccretionontotheBHaredeterminedbyfollowing
Figure6. Theorbital evolution ofaselected winning binary. Right after
Equations(15)and(16)in(Hurleyetal.2002), theformationoftheBH(t=3.8Myr),thisbinaryconsistsofa14.8M⊙BH
anda21.7M⊙mainsequencestar.Thetoppanelsshowthetimeevolutionof
dA M˙ 2- e2 1+e2 M˙ orbitalperiodandeccentricity.Thebottompanelsshowtherateofchangesof
= - A 2 + + acc thesemi-majoraxisAandeccentricityeduetotidaleffects(solidline),wind
(cid:18)dt (cid:19)wind (cid:20)MBH+M2 (cid:18)MBH MBH+M2(cid:19)1- e2(cid:21) masslossandwindaccretion ontotheBH(dashedline), andgravitational
(26) radiation(dottedline).
de = - eM˙ 1 + 1 . (27) Theelementspresentedintheprevioussessionscannowbe
dt acc M +M 2M
(cid:18) (cid:19)wind (cid:18) BH 2 BH(cid:19) combinedto establish a completepicture of the evolutionof
Cygnus X-1 and the dynamics involved in the core collapse
The mass loss via stellar wind also induces a loss in the
event that formed the BH. After finding the successful evo-
spinangularmomentumoftheBH companion. Hurleyetal.
lutionarysequencesthatsatisfyalltheobservedpropertiesof
(2000) showed that if all the mass is lost uniformly from a
the BH companion and the bolometric X-ray luminosity as
thinshellatthesurfaceoftheMSstar,
discussed in Section4, we trace the motionof the system in
J˙ = d (I Ω) = 2M˙ R2Ω, (28) theGalaxybackintimetotheformationoftheBH.Weadopt
2,spin dt 2 3 2 2 themethodologyofGualandrisetal.(2005)toaccountforthe
uncertainties in the measured distance and velocity compo-
where J2,spin is the spin angular momentum of the BH com- nentsofCygnusX-1. ThetimeofBH formationisdifferent
panion. foreachsuccessfulsequence. ItisestimatedbytheBHmass
For each of simulated binaries, we follow the secular of the sequence, which connects to an approximate lifetime
changes of its orbital properties due to all the mechanisms ofthecorrespondingBHprogenitor. Thisproceduregivesus
mentioned in this section. The properties of binary com- a constraint on the system’s peculiar velocity right after the
ponents are adopted from the corresponding successful se- BHformation. WethenperformMonteCarlosimulationson
quence. Unlike finding Vpec,postSN in Section5, the orbital the orbitaldynamicsat core collapse for each successfulse-
evolutionof the binarygoes forwardin time, fromtBH tot2. quence. Therearesevenfreeparameters: theBHimmediate
Withinthisperiodoftime, theBH companionhastoalways progenitormass, the pre-SN orbital semi-majoraxis and ec-
fitwithinitsRochelobeatperiapsis. Inotherwords,itscal- centricity, the mean anomaly, the magnitude of kick veloc-
culated radius is constrained to be less than the Roche lobe ity imparted to the BH, and the two angles specifying the
radius at periapsis given by Sepinskyetal. (2007). Again, direction of the kick velocity. The Monte Carlo simulations
we allow an uncertainty of ±2.5 R⊙, due to the differ- produceapopulationofsimulatedbinaries,whichsatisfythe
encein calculatedstellar radiiamongstellar evolutioncodes post-SNsystem’speculiarvelocityconstraintderivedalready.
(Valsecchietal. 2010). Furthermore, the rotational angular Last,weevolvetheorbitsofthesesimulatedbinariesforward
velocity of the BH companion has to be smaller than the intimetothecurrentepoch. Iftheorbitalperiodandeccen-
breakupangularvelocityΩc. Iftheorbitalperiodandeccen- tricityofthesimulatedbinaryatcurrentepochmatchthemea-
tricityofthesimulatedbinaryatt2matchthemeasuredvalues suredvaluesofCygnusX-1,weclassifythatsimulatedbinary
ofCygnusX-1,weclassifythatbinaryasa"winningbinary". asa"winningbinary". Theresultspresentedinwhatfollows
Figure6 illustrates the time evolution of orbital parameters are derived from the winning binaries of all successful se-
foroneselectedwinningbinary,andshowsthatthechangein quences.
thesemi-majoraxisismainlydeterminedbythestellarwind In Figure7, we present the probability distribution func-
massloss,whilethechangeintheeccentricityisoverwhelm- tions(PDFs)oftheBHimmediate(He-rich)progenitormass
inglydominatedbythetidaleffects. (M ) andnatal kickmagnitude(V ). We find M to be in
He k He
a range of 15.0- 20.0 M⊙, and Vk to be ≤77 kms- 1, both
8. PROGENITORCONSTRAINTS at95%confidence. Figure8illustratesthe2DjointV –M
k He
UnderstandingCompactObjectformation.III. 9
Figure7. TheprobabilitydistributionfunctionsoftheBHimmediate(He- Figure9. The2DjointApreSN–epreSNconfidencelevels:68.3%(red),95.4%
rich)progenitormass(MHe)andnatalkickmagnitude(Vk)impartedtothe (yellow),and99.7%(blue).
BH. bornwithanextremespin,wefirstestimatehowmuchmass
theBHcouldhaveaccretedfromitscompanion’sstellarwind
since thetime of BH formation. The winningbinariesof all
successfulsequencesshow that at maximumthe BH hasac-
creted∼2×10- 3M⊙.SinceitisimpossibletospintheBHup
toa∗>0.95byaccretingthatnegligibleamountofmass,the
BHneedstohaveanextremespinatbirth.Thishighspinhas
implications about BH formation and the role of rotation in
corecollapse. Axelssonetal. (2011) also concludedthatthe
observedspinconnectstoprocessesinvolvedincorecollapse,
andisnotlikelytooriginatefromthesynchronousrotationof
theBHprogenitor.
Besides the constraints on the BH formation, our results
alsoshedlightontheevolutionarypictureofCygnusX-1.We
findthatrightaftertheformationoftheBH,theBHcompan-
ionhasa massof 19.8- 22.6M⊙, in anorbitwith periodof
4.7- 5.2days.Sincethen,theorbitalseparationofCygnusX-
1hasbeenincreasingwith time, asthe rate ofchangein the
semi-majoraxisisdominatedbytheinfluenceofstellarwind
masslossfromthesystem. Ontheotherhand,theorbitalec-
centricityhasdecreasedslightlysincetheBHformation.This
isbecausethetidesexertedonthecompanionbythe BH, as
the dominant mechanism of circularizing the orbit, are not
Figure8. The2DjointVk–MHeconfidencelevels:68.3%(red),95.4%(yel- strongenoughtodecreasetheorbitaleccentricitysignificantly
low),and99.7%(blue).
withinthetimeperiodofseveralmillionyearssincethetime
confidencelevels, which shows that if M is less than ∼17 of BH formation. We find that e ranges from 0.015 to
He postSN
M ,theBHmighthavereceivedanon-zeronatalkickatthe 0.022. However, this does not suggest that e has to be
⊙ preSN
corecollapse event. For smallM , a minimumV of ∼55 small. An eccentric pre-SN orbital could become fairly cir-
He k
kms- 1isnecessaryforexplainingthecurrentobservedprop- cular if there is a natal kick imparted to the BH at the right
ertiesofCygnusX-1.Furthermore,boththeM PDFandthe direction. As illustrated in Figure9, there are winningbina-
He
2D joint Vk–MHe confidence levels show that the maximum rieswithepreSN beingashighas∼0.53.
M isconstrainedbyouradoptedupperlimitof20M . We
He ⊙
imposethislimitbasedonthephysicsinvolvedintheevolu- 9. SUMMARY&DISCUSSION
tionofmassivestars. Adiscussiononthislimitcanbefound In this paperwe constrainedthe progenitorpropertiesand
in Section9.1. Given our understanding of mass loss from the formation of the BH in the persistent XRB CygnusX-
Heliumstars, itseemsthattheBH haspotentiallyreceiveda 1. Our analysis accounts for the orbital evolution and mo-
small natal kick velocity of ≤77 km s- 1 (95% confidence) tionthroughthe Galactic potentialrightafterthe BH forma-
duringthecorecollapseevent. tion,andthebinaryorbitaldynamicsatthetimeofcorecol-
Based on the dynamical model of Oroszetal. (2011), lapse. WefindthatthemassoftheBHimmediateprogenitor
Gouetal.(2011)foundthattheBHinCygnusX-1hasaspin falls within a range of 15.0- 20.0 M⊙ at 95% confidence.
parametera∗>0.95at3σ.TodeterminewhethertheBHwas Wenotethatthemaximumprogenitormassisconstrainedby
10 T.-W.Wongetal.
our adopted upper limit, which is discussed in Section9.1.
The BH has potentially received a small natal kick velocity
of ≤77 km s- 1 at 95% confidence. In fact if the progenitor
mass is less than ∼17 M⊙, a non zero natal kick velocity
is necessary to explain the currently observed properties of
CygnusX-1. Since the BH has only accreted mass from its
companion’sstellar wind, the total amountof mass accreted
sincetheBHformationislessthan∼2×10- 3M⊙. Thisin-
dicatesthattheobservationallyinferredBHspinofa∗>0.95
(Gouetal.2011)cannotbeexplainedbymassaccretionand
hastobenatal.ThishighspinhasimplicationsaboutBHfor-
mationandtheroleofrotationincorecollapse.Rightafterthe
BH formation, the BH companionhas a mass of 19.8- 22.6
M⊙,inanorbitwithperiodof4.7- 5.2daysandeccentricity
of 0.015- 0.022. Although the post-SN orbital eccentricity
is small, the pre-SN orbitcan potentiallybe fairly eccentric.
ThisispossibleiftheBHreceivesanatalkickvelocityatthe
rightmagnitudeanddirection.
The formation of the BH in CygnusX-1 has been
previously studied by Nelemansetal. (1999) and
Mirabel&Rodrigues (2003). Both studies assumed
symmetric mass loss during the core collapse event, and Figure10. Thesameplotof2DjointVk–MHeconfidencelevelsasFigure8,
considered only the binary orbital dynamics at the time of calculatedwithVpec,postSN≤10kms- 1insteadoftheoriginalrangederived
Section5.
core collapse. Comparing with these two earlier studies,
entassociationofCygnusX-1.ThisinfersthatV due
we consider the possible asymmetries developed during the pec,postSN
to the core collapse eventhas to be small. If we changethe
core collapse eventand the evolutionof the binarysince the constraintonV to≤10kms- 1,wefindM tobein
BstuHdifeosrmdaotinoont. cItonissiidmerpothrteanmt tuoltintuodteethoafttthheesoebtswerovaetaiorlniearl a range of 13.9p-ec1,p6os.t9SNM⊙ and Vk to be ≤24 kmH/se, both at
95%confidence. Besidesthechangein95%limits,non-zero
constraints taken into account here and hence the suggested
BH natal kicks are not needed for progenitors of M ≤17
progenitors are not complete solutions for the evolutionary He
M inordertoexplaintheobservedpropertiesofCygnusX-
historyofCygnusX-1. ⊙
Finally,wediscusssomeoftheassumptionsintroducedin 1,butbecomenecessaryforMHe>17.5M⊙ (seeFigure10).
Also, we note that a relatively small change on the range of
ouranalysisinthefollowingsub-sections.
V affectsthederivedconstraintonV qualitatively.
pec,postSN k
9.1. MaximumBHProgenitorMass
9.3. Super-SynchronizedOrbit
Unlike the case of GRO J1655-40 studied in Paper I, the
After considering several previous measurements of
analysis of orbital dynamics during the core collapse event
the BH companion’s surface rotation speed (V sini),
doesnot givean upperlimit on M . Instead, we have con- rot
He Caballero-Nievesetal. (2009) adopted V sini=95±6 km
servativelyadoptedanupperlimitofMHe≤20M⊙,basedon s- 1. Oroszetal. (2011) found that the ratroiot of the BH com-
physicsinvolvedin the evolution of massive stars. As men-
panion’sspinningfrequencytotheorbitalfrequency(fΩ)was
tioned in Section 6.1 of Paper II, by evolving a ZAMS star
1.400±0.084,whichwasderivedbasedontheirresultsofthe
of ∼100 M⊙ at solar metallicity, the maximum Helium star inclination angle i=27◦.06±0◦.76 and the companion ra-
mass one can achieve is ∼15 M⊙ when including moderate diusR =16.5±0.84. ThisindicatesthattheBHcompanion
stellarrotation,and∼17.5M⊙ whenassumingnostellarro- is supe2r-synchronized. We note that with the analysis pre-
tation. Whenadoptingtheupperlimitof17.5M ,thelower
⊙ sentedhere,wefindnoneofourwinningbinarieshavesuper-
limitof M decreasesslightly to 14.6M and the rangeof
He ⊙ synchronizedBHcompanionsatthecurrentepoch. Theyare
V becomes14- 81kms- 1,bothwith95%confidence. This
k allsub-synchronizedwith fΩ reaching∼0.87atmaximum.
rangeofV stillsuggeststhattheBHinCygnusX-1received
k Inan effortto examinehowourstandardassumptionscan
alowkickduringthecorecollapseevent.
bemodifiedandinvestigatewhethersuper-synchronismisat
all allowed by the models as indicated by the observations,
9.2. AssociationwithCygOB3
wemaketwo modificationstoouranalysis. We firstremove
The center of Cyg OB3 locates at l = 72.8◦ and b = theassumptionthatthepre-SNorbitispseudo-synchronized,
2.0◦, and at a distance of 1.4- 2.7 kpc away from the andrandomlydistributethepre-SNspinoftheBHcompanion
Sun(Masseyetal.1995;Dambisetal. 2001;Mel’Niketal. betweenzeroanditsbreakupangularfrequencyΩ . Next,we
c
2001; Mel’Nik&Dambis 2009). When comparing that to reduce the secular changes of the orbital parameters due to
the location of CygnusX-1 (Table1), it is clear that not the influence of tides by multiplying the right hand side of
only their Galactic coordinates are close to each other, but Equation(21)–(23)byaconstant f .
tide
also their distance estimations overlapwith each other. Fur- As shown in Figure11, by allowing the pre-SN spin of
thermore,the measurementsof propermotionandradialve- companion to be greater than pseudo-synchronization and
locity show that Cygnus X-1 is moving as the members of keepingthetidalstrengthunchanged(i.e. f =1.0),themax-
tide
Cyg OB3 (Dambisetal. 2001; Mirabel&Rodrigues 2003; imum fΩofthewinningbinariesincreasesto∼1.2.Although
Mel’Nik&Dambis 2009). Based on these observations, it is getting close, this value is still below the observation-
Mirabel&Rodrigues(2003) arguethat Cyg OB3 is the par- allyinferredone. Togetherwithaweakenedtidalstrengthof