Modeling Luminous-Blue-Variable Isolation

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by  Mojgan Aghakhanlootakanloo

MNRAS000,1–12(2017) Preprint23January2017 CompiledusingMNRASLATEXstylefilev3.0
Modeling Luminous-Blue-Variable Isolation
1⋆ 1 2
Mojgan Aghakhanlootakanloo, Jeremiah W. Murphy † and Nathan Smith ‡
1Physics,FloridaStateUniversity,77ChieftanWay,Tallahassee,FL32306,USA
2StewardObservatory,933N.CherryAve,Tucson,AZ85719,USA
7 23January2017
1
0
2 ABSTRACT
Observationsshowthatluminousbluevariables(LBVs)arefarmoredispersedthanmassive
n
O-type stars, and Smith&Tombleson (2015) suggested that these large separations are in-
a
J consistentwithasingle-starevolutionmodelofLBVs.Instead,theysuggestedthatthelarge
distancesare mostconsistent with binaryevolutionscenarios. To test these suggestions,we
9
1 modeled young stellar clusters and their passive dissolution, and we find that, indeed, the
standardsingle-starevolutionmodelisinconsistentwiththeobservedLBVenvironments.As

] single stars, the lifetimes of LBVs indicatedby their luminositieswouldbe far shorterthan
R
the ages required by their observed isolation. The resolution for this isolation is either that
S LBVs are much less massive than their luminosity would suggest, or they are the result of
. binary evolution.For the binaryscenarios, our crude models suggestthat LBVs are rejuve-
h
natedstars.Theyareeithertheresultofmergers,ortheyaremassgainersandreceivedakick
p
- whentheprimarystarexploded.Inthemergerscenario,LBVshavemoretimetodispersebe-
o causetheyarethemergeroftwolessermassstars,inwhichtheprimaryhasanaveragemass
r
of about18 solar masses. In the mass-gainer and kick scenario, we find that LBV isolation
t
s isconsistentwitha wide rangeof kickvelocities,anywherefrom0 to∼200km/s.Ineither
a
scenario,binarityseemstoplayamajorroleintheisolationofLBVs.
[

1 Keywords: binaries:general–stars:evolution–stars:massive–stars:variables:general
v
6
2
6
1 INTRODUCTION has been that most stars above 25-30 solar masses pass through
5
0 an LBV phase and experience eruptions as a means to transi-
Initial mass is one of the primary characteristics that determines
. tion from a hydrogen rich star to a H-poor Wolf-Rayet (WR)
1 astar’sevolutionandfate(Woosley&Heger2015);thereforeun-
star (Humphreys&Davidson 1994). In this scenario, LBVs
0 derstanding mass-loss isimportant indeveloping acomplete the-
experience high mass loss due to an unknown instability, which
7 ory of stellar evolution. Yet, understanding the physics and rela-
may be driven by a high L/M ratio, near the Eddington limit
1 tiveimportanceofsteadyanderuptivemass-lossinthemostmas-
(Humphreys&Davidson 1994; Vink 2012). On the other hand,
v: sive stars remains a major challenge in stellar evolution theory. the LBV phenomenon might only be associated with rare cir-
i There has been substantial progress in understanding mass-loss
X cumstances if it is due to binary interaction (Smith&Tombleson
via steady line-driven winds of hot stars (Kudritzki&Puls2000;
2015).
r Pulsetal. 2008), and this effect is included in stellar evolution
a Smith&Tombleson (2015) present new data that suggests
models (Vinketal. 2001; Woosleyetal. 2002; Meynet&Maeder
the latter. If LBVs mark a brief transitional phase at the end of
2005; Martins&Palacios 2013). However, the mass-loss rates of
the main-sequence and before core-He burning WR stars, then
redsupergiantsandtheroleoferuptivemasslossremainunclear,
they should be found near other massive O-type stars. However,
and the influence on stellar evolution remains uncertain (Smith
Smith&Tombleson (2015) found that LBVs are quite isolated
2014;Smith&Owocki2006).TheLuminousbluevariable(LBV)
fromO-typestars,andevenfartherawayfromOstarsthantheWR
phenomenonisonesuchpoorlyconstrainedclassoferuptivestars.
stars are. Given their isolation, Smith&Tombleson (2015) con-
LBVs are luminous, unstable massive stars that suf-
cludethattheLBVphenomenon isinconsistentwithasingle-star
fer irregular variability and major mass-loss eruptions
scenario and ismost consistent with binary scenarios. Inthis pa-
(Humphreys&Davidson 1994). The mechanism of these
per, we develop simple models for the dispersal of massive stars
eruptions and the demographics of which stars experience these
onthesky,inordertobetterconstrainwhethertheO-starandLBV
is poorly constrained (Smith 2011, 2014). The traditional view
distributionsrequirebinaryscenarios.
Part of the reason that LBVs are poorly understood is be-
cause there are very few examples. There are only 10 unob-
⋆ [e-mail]
† [e-mail] scured in our galaxy and 19 known in the nearest galaxies, the
‡ [e-mail] LMCandSMC(Smith&Tombleson2015).Eventhissmallsam-
(cid:13)c 2017TheAuthors
2 Aghakhanlootakanlooet al.
ple includes “candidate” LBVs (see below). Classifying vari- O stars, LBVsare isolatedin the Milky Wayand the Magellanic
ous stars as LBVs or candidates can be somewhat controversial Clouds. Moreover, they find that LBVs appear to have a much
(Humphreys&Davidson 1994; Weis 2003; Vink 2012); here we largerseparationthanevenWRstars,whicharethoughttobethe
summarize their basic characteristics. LBVs are luminous, blue descendantsofLBVs.Theyconcludethattheirspatialdistribution
massive stars with irregular or eruptive photometric variability. isinconsistentwithsingle-starevolutionandmostconsistentwith
Stars that resemble LBVs in their physical properties and spec- binary models. Theysuggested a binary scenario in which LBVs
tra,butlackthetell-talevariability,areusuallycalled“LBVcandi- arethekickedmassgainersormergerproducts.
dates”.Thereasontheyaresometimesgroupedtogetherisbecause Humphreysetal.(2016)putforthadifferentinterpretationof
itissuspectedthattheLBVinstabilitymaybeintermittent,sothat LBVlocations,suggestingthattheydonotruleoutthesingle-star
candidatesaretemporarilydormantLBVsSmith(2011,2014). scenario.TheynotethatthesampleinSmith&Tombleson(2015)
AlthoughthesignatureeruptivevariabilityofLBVswasiden- isamixtureoflessluminousLBVs,moreluminousclassicalLBVs,
tified long ago, the physical theory of LBV eruptions is not yet andunconfirmedLBVs,andtheyproposethatseparatingthemal-
clear.Forthemostpart,LBVsseemtoexperiencetwoclassesof leviates the conflict with single star models. From their point of
eruptions; S Doradus (or S Dor) eruptions (1-2 mag) and Giant view, the single-star hypothesis still works because (1) the three
eruptions(>2mag). most luminous stars of the sample that are classical LBVs (with
SDoradusvariablestaketheirnamesakefromtheprototypi- initial masses greater than 50M ) have a distribution similar to
⊙
calLBVSDoradus(Hubble&Sandage1953).DuringSDorout- lateO-typestars,and(2)thelessluminousLBVs(withinitialmass
bursts, LBVs make transitions in the HR Diagram (HRD) from ∼25−40M )arenotassociatedwithanyOstars,buthaveadistri-
⊙
theirnormal,hotquiescentstatetolowertemperatures(goingfrom butionsimilartoredsupergiants(RSGs),whichHumphreysetal.
blue to red). In its quiescent state, an LBV has the spectrum of (2016)suggestisconsistentwiththembeingsingle-starsonapost-
a B-type supergiant or a late Of-type/WN star (Walborn 1977; RSGphase.Humphreysetal.(2016)stronglysuggestedtodistin-
Bohannan&Walborn 1989). In this state, LBVs are fainter (at guishbetweenthesetwotypesofLBVsinstatisticaltests.More-
visual wavelengths) and blue with temperatures in the range of over,Humphreysetal.(2016)criticizedthatfiveoftheLMCstars
12,000to30,000K(Humphreys&Davidson1994).Intheirmax- (R81,R126,R84,Sk-69271andR99)areneitherLBVsnorcandi-
imumvisiblestate,theirspectrumresemblesanF-typesupergiant dates.
witharelativelyconstanttemperatureof∼ 8000K.SDorevents However, Smith (2016) showed that even using the LBVs
were originally proposed to occur at constant bolometric lumi- sample subdivided as Humphreysetal. (2016) prefer does not
nosity (Humphreys&Davidson 1994). So a change in tempera- changetheresultthatLBVsaretooisolatedforsinglestarevolution
tureimpliesachangeinthephotosphericradius,L = 4πσR2T4. (overlookingthelackofstatisticalsignificance).Themostmassive
Humphreys&Davidson (1994) suggested that the eruption is so LBVsappear tobeassociatedontheskywithlateO-typedwarfs
optically thick that a pseudo photosphere forms in the wind or (point1above),which,however,haveinitialmasseslessthanhalf
eruption.However,quantitativeestimatesofmass-lossratesshow ofthepresumedinitialmassesoftheclassicalLBVs.Similarly,the
that theyare toolow toform alargeenough pseudo photosphere lower-luminosity LBVs have a similar distribution to RSGs, but
(deKoteretal.1996;Grohetal.2009).Similarstudiesalsoimply theseRSGsaredominatedbystarsof10-15M (Smith2016).
⊙
that the bolometric luminosityisnot strictlyconstant (Grohetal. Humphreysetal. (2016) also stated that the observed LBV
2009). Instead, it has been suggested that the observed radius velocities seems to be too small to be consistent with the kicked
changeofthephotospherecanbeapulsationorenvelopeinflation mass-gainerscenario,butSmith(2016)pointedoutthatwithouta
drivenbytheFeopacitybump(Gra¨feneretal.2012). quantitativemodelforthevelocitydistributions,itwouldbediffi-
Theotherdistinguishingtypeofvariabilityisintheformofgi- culttoruleanythinginorout.Inthispaper,wewillshowbothhigh
anteruptionslikethe19thcenturyeruptionofηCar(Smith2011). andlowluminositiesLBVandLBVcandidateshavelargersepara-
ThebasicdifferencefromSDoreventsisthatgianteruptionsshow tionsthanonewouldexpect,andin§4.4andFigure10weshow
astrongincreaseinthebolometricluminosityandaremajorerup- that a wide range of kick velocities are consistent with the large
tive mass loss events, whereas S Dor eruptions occur at roughly separations.
constantluminosityandarenotmajormass-lossevents.Themass The main goal of this paper is to quantitatively constrain
lossrateatSDormaximumisoforder10−4M yr−1orless(Wolf whethertherelativeisolationofLBVsisinconsistentwithasingle-
⊙
1989; Grohetal. 2009). On the other hand, giant eruption mass star evolution model. We begin by reproducing and verifying
lossrateistheorder of 10−1 to1M yr−1 (Owockietal.2004; Smith&Tombleson (2015) results (§ 2). In § 3, we introduce a
⊙
Smith&Owocki 2006; Smith 2014). It is unlikely that a normal simplemodel for young stellarclustersand their passive dissolu-
line-driven stellar wind is responsible for the giant eruptions be- tion.Totestthismodel,wealsocomparetheseparationsbetween
causethematerialishighlydenseandopticallythick(Owockietal. Ostarsforthemodelandobservations;wefindthatthemodelre-
2004; Smith&Owocki 2006). Instead, giant eruptions must be producessomegeneralpropertiesofthespatialdistributionofmas-
continuum-drivensuper-Eddingtonwindsorhydrodynamicexplo- sivestars,butitislackinginotherways.Sinceweconstructedthe
sions(Smith&Owocki2006).Bothoftheselackanexplanationof simplestmodelpossible,thisimpliesthatwemayimprovethedis-
theunderlyingtrigger;thesuper-Eddingtonwindreliesuponanun- persion model and learn even more about the evolution of mas-
explainedincreaseinthestar’sbolometricluminosity,whereasthe sive stars. Then in § 4, we present the primary consideration of
explosivenatureofgianteruptionswouldrequiresignificantenergy thispaper;wecomparesingle-starevolutionandbinaryevolution
deposition. There is much additional discussion about the nature inthecontextofclusterdissolution,andwefindthatthesingle-star
of LBV giant eruptions in the literature (Humphreys&Davidson evolutionscenarioisinconsistentforinitialmassesappropriatefor
1994; Owockietal. 2004; Smith 2011, 2014; Smith&Owocki LBVluminosities.Wediscusstwobinaryevolutionchannelsthat
2006). are consistent with the relative isolation of LBVs. We then sum-
Smith&Tombleson(2015)highlightaresultthatchangesthe marize,andwediscussfutureobservationstofurtherconstrainthe
emphasis on the most likely models. They find that compared to binarymodels(§5).
MNRAS000,1–12(2017)
ModelingLBV Isolation 3
2 OBSERVATIONS
Table 1. List of LBVs and LBV candidates adapted from
Smith&Tombleson (2015). For the stars in the SMC, we rescale
Inthefollowingsections,weexplorewhichtheoreticalmodelsare
theirangularseparationby1.2asiftheyarelocatedatthedistanceofthe
mostconsistentwiththedata,butbeforethat,weclearlydefine,an-
LMC.ParenthesesinthenamerepresentLBVcandidatesandparentheses
alyze,andcharacterizethedatainthissection.First,wereproduce
in mass column specify the LBVs with relatively poorly constrained
and verify the results of Smith&Tombleson (2015). Second, we luminosityandmass.
furthercharacterizethedata,notingthatthedistributionsofnearest
neighborsarelognormal.Sincelog-normaldistributionshavevery LBV(name) Galaxy(name) S(deg) Meff(M )
fewparameters,thisrestrictsthecomplexityandparametersofour ⊙
R143 LMC 0.00519 60
modelsin§3.
R127 LMC 0.00475 90
Smith&Tombleson(2015)foundthatLBVsaremuchmore
SDor LMC 0.0138 55
isolatedthanO-typeorWRstars,suggestingthatLBVsarenotan
R81 LMC 0.1236 (40)
intermediarystagebetweenthesetwoevolutionarystages.Inpar- R110 LMC 0.2805 30
ticular,theyfoundthatonaverage,thedistancefromLBVstothe R71 LMC 0.4448 29
nearestOstarisquitelarge(0.05deg).Forcomparison,theaverage MWC112 LMC 0.0892 (60)
distancefromearlyOstarstothenearestOstaris0.002degrees, R85 LMC 0.0252 28
andfrommidandlateOstarsare0.008and0.010degreesrespec- (R84) LMC 0.1575 30
tively. If single early- and mid-type O stars are indeed the main (R99) LMC 0.0412 30
(R126) LMC 0.0358 (40)
sequence progenitorsofLBVs,thenonewouldexpect thespatial
(S61) LMC 0.1432 90
separations betweenLBVsandother O starstobenot toodiffer-
(S119) LMC 0.3467 50
entthantheseparationbetweenearly-andmid-typeOstars.How-
(Sk-69142a) LMC 0.0522 60
ever, the LBVseparations arean order of magnitude farther than
(Sk-69279) LMC 0.0685 52
theearly- andmid-type separations. In fact,the LBVseparations (Sk-69271) LMC 0.040 50
are5timeslargerthaneventhelate-typeOstars,whichlivelonger HD5980 SMC 0.0191 150
andcaninprinciplemigratefarther. R40 SMC 0.1112 32
Smith&Tombleson (2015) quantified the difference in the (R4) SMC 0.0160 (30)
distributions of separations by using the Kolmogorov-Smirnov
(KS)test.ComparingthedistributionsofLBVstoearly-,mid-and
late-type give P-values of 5.5e-9, 1.4e-4 and 4.4e-6 respectively.
These values imply that O-stars and LBV distributions are quite
different. If true, then these results have profound consequences
andinthethird,theLBVdistributiononlycontainsthecandidates.
forourunderstandingofLBVsandtheirplaceinmassivestarevo-
When we include both confirmed and candidate LBVs, the LBV
lution. In fact, Smith&Tombleson (2015) suggest that the most
andO-stardistributionsareclearlynotdrawnfromthesamepar-
natural explanation is that LBVsare the result of extreme binary ent distribution. However, omitting LBV candidates reduces the
encounters.Laterwewilltestthisassertion,butfornowwerepro-
distinctionsbetweenthedistributions.Onemightarguethatsince
duceandverifytheirresults.
LBVsrepresentalaterevolutionarystage,thenthespatialsepara-
To verify the results of Smith&Tombleson (2015), we first tionsshouldbelarger,andtherefore,thedistributionsofearly-type
definethedata.Thedataconsistsoftwomainparts;LBVsandO OstarsandLBVsshouldnotrepresentthesamedistribution.How-
stars.TheirsampleincludesWRstars,sgB[e]stars,andRSGstoo, ever,wewillshowinsection4thatthelifetimesofmassivestarsare
butwedonotdiscussthemherebecauseatthemoment,wewantto fartooshorttoexplaintheselargediscrepancies.Inourinitialas-
keepourmodelsin§3&4simpleandwewillfocusjustonLBVs sessment,weagreewithSmith&Tombleson(2015),thelargesep-
andOstars.TheirLBVsamplesinclude16starsintheLMC,and arations present a challenge to the single-star evolution scenario.
three stars in the SMC. They did not consider Milky Way LBVs Inthenextsections,wewillpresenttheoreticalmodelstoquantify
because it’s difficult to develop a homogeneous catalog. In their thisinconsistency.
study,theyincludeLBVcandidateswithamassiveCSMshellthat Beforeweconstrainthemodels,notethattheseparationdis-
likelyindicatesapreviousLBV-likegianteruption.LBVsandtheir tributions are log-normal (see Figure 1). In fact, this simple ob-
importantparametersaresummarizedinTable1. servation greatly restricts the complexity of the models that we
Smith&Tombleson (2015) gathered thepositions of O-type may explore in the next sections. If a variable, such as the sep-
stars within 10◦ projected radius of 30 Dor from SIMBAD aration between stars shows a normal distribution, then there are
database. They also used the revised Galactic O-star Catalog only two free parameters that describe the distribution, the mean
(Ma´ızApella´nizetal. 2013) to check their O-star samples (not andthevariance.Inaddition,iftheseparationdependsuponother
shownintheirpaper),butastheyclaimedthisdidnotchangetheir variablessuchasavelocitydistribution,thenthankstothecentral-
overallresults.WecollectthesameO-starsamplesfromSIMBAD limittheorem,theseparationdistributionwillonlydependuponthe
database. meanofthevarianceofthesecondaryvariablessuchastheveloc-
After gathering the data, we find the distance from one star itydistribution.Thismeansthatwecannotproposeoverlycomplex
tothenearest Ostar.Thebottompanel ofFigure3showsthere- modelsforthevelocitydistribution.Wewouldonlybeabletoinfer
sultingcumulativedistributions;notethatthedistributionsforeach themeanand thevariance anyway. Fortunately, wemaymeasure
typeofstarisquitedistinct.Infact,KStestsshowthatthesedis- theseparationfordifferenttypesofOstarsandotherevolutionary
tributions very distinct. Our KS-test P values are listed in Table stages.Thismeansthatwemayinferthetemporalevolutioninad-
2.WeconsiderthreeKStests.Inone,wecomparetheseparation ditionthetothemeanandvariance.Whatevermodelswepropose,
forbothconfirmedandcandidateLBVswithO-stardistribution.In theycannotbetooelaborate;wewillonlybeabletoinferthemean
the second, the LBV distribution only includes confirmed LBVs, andvarianceofonequantityasafunctionoftime.
MNRAS000,1–12(2017)
4 Aghakhanlootakanlooet al.
Table2.Broadly,wereproducetheresultsofSmith&Tombleson(2015)
whofoundthatthedistributionofseparationsbetweenLBVsandthenear-
estOstarisquitedifferentthanthedistributionsfortheseparationsbetween
OstarsandthenearestOstar.ThesecondrowshowstheresultsofourKS
5
tests between the LBV separations and the early-, mid-,and late-type O
stars.OurresultsaresimilartothoseofSmith&Tombleson(2015)(first
p ∼ 0.5 O2-O5(Early)
row).LikeSmith&Tombleson(2015)weobtainthepositions ofOstars 4
andtheirspectraltypesfromSimbad.Smith&Tombleson(2015)updated
thespectral types withtheGalactic O-starCatalog (Ma´ızApella´nizetal.
2013);however,wedidnot.Thisslightdifferenceinspectraltypingiswhat 3
causesthemodestdifferenceinPvalues.Ineithercase,theLBVseparations
areinconsistentwithanyO-typeseparations.IfweexcludetheLBVcan-
2
didates(thirdrow),theconclusionsremainthesame,butthesignificanceis
greatlyreduced.
1
DataSet EarlyO MidO LateO
LBV+LBVc(Smith&Tombleson2015) 5.5e-9 1.4e-4 6.4e-06
LBV+LBVc(thiswork) 8.2e-08 2.8e-05 8.4e-05
LBV(thiswork) 9.2e-04 2.3e-02 5.7e-02 10 p ∼ 0.5 O6+O7(Mid)
LBVc(thiswork) 6.4e-06 5.1e-05 2e-04
8
y
c6
3 AGENERICMODELFORTHESPATIAL n
DISTRIBUTIONOFTHESTARSINAPASSIVE ue4
DISPERSALCLUSTER q
e
r
InordertomodeltherelativeisolationofLBVs,weneedtomodel F2
the dissolution of clusters and associations of massive stars. For
severalreasons,wemodelthedissolutionofyoungstellarclusters
withaminimumsetofparameters.Forone,theO-stardistributions
p ∼ 0.4 O8+O9(Late)
arelognormal.Therefore,thereareonlyafewparametersthatde- 20
scribethedatawhichonemayfit.Theonlydatathatwecanreliably
fitarethemean,variance,andtimeevolutionoftheseparations.So
15
whatevermodelswedevelop,theyshouldnotbeoverlycomplex.
Also, as far as we know, there are no simple self-consistent and
testedmodelsforthedissolutionofclusters.Therefore,wepropose 10
asimplemodelofclusterdissolutionandadaptittoconsidertwo
scenarios:clusterdissolutioninthecontextofsingle-starevolution, 5
and clusterdissolution withclosebinary interactions. Inthissec-
tion,wepresentaclusterdispersalmodelconsideringonlysingle-
0
star evolution. Whileour dissolution models represent thespatial
10−4 10−3 10−2 10−1 100
distributionsreasonably wellincertainrespects, wenotethat our
modelfailstomatchthedatainotherways.Thisimpliesthatour S [deg]
modelismissingsomething.Inotherwords,wemaybeabletoin-
fermorephysicsaboutthedissolutionofclustersfromthesimple
spatialdistributionofOstars.Inthenextsection,wecontrastthe
Figure1.Normality test. Thedistributions ofseparations forearly, mid,
single-starmodelwithamodelthatconsidersbinarity. andlateOstarsareconsistentwithalognormaldistribution.Ineachplot,
Ourmaingoalistointroduceamodelforyoungstellarclus- weshowtheprobability,p,thattheparentdistributionisalognormaldis-
terswhichpredictsthespatial distributionof massivestars, espe- tribution.
ciallyO stars.Westartbyconsidering thesimplest model. Inthe
following,wemodeltheaveragedistancetothenearestOstarby
nothingmorethanthepassivedispersalofacluster. willdriftawaywithaspeedroughlyequaltothevelocitydispersion
Beforewediveintothedetailsofthemodel,itisworthchar- whentheclusterwasbound.Hence,vd ∼σv.Allthatislefttodois
acterizingthescalesof atypical cluster.Webeginrightafter star estimateMT andR.AtypicalclusterhasR∼4pcandabout40O
formation ends and consider a system of gas and stars that is in stars;ifonly∼1%ofthegasingiantmolecularcloudsformstars
virialequilibrium.Inthiscase,wehave2T +U = 0,whereTis (Krumholz&Tan2007),thetotalmassofthemolecularcloud,MT
thetotalthermalpluskineticenergy,andUisthegravitationalpo- isoforder2×105 M .Giventheseapproximations,weestimate
tRenitsiabloeunnedrg,ya.nIdnitthiaellsyta,rtshehasvyesteamvewloicthitytodtaislpmerassisonMtThaatnsdcarlaedsiuass thattNheexdtr,ifwteveplroecsietyntis⊙aomfoorredesprevcdifi∼cd1i3s.s5o(lu2t×io1M0n5TmMo⊙de4Rplcto)12co.nvert
1
thegravitationalpotentialoftheentiresystemσv ∼(GMT/R)2. thisdispersalvelocityintoadistributionofseparationsasafunction
Thenthesystemlosesgasmassbysomeformofstellarfeedback oftime.Ratherthanusingthisestimateforthedispersalvelocity,
(UV radiation, stellar winds, etc.) and likely makes the stars un- wewillusethedataandourmodeltoinferthedispersalvelocities.
bound. If the system looses all of the gas quickly, then the stars WeproposeaMonte-Carlomodelforthedissolutionoftheclusters.
MNRAS000,1–12(2017)
ModelingLBV Isolation 5
First,werandomlysampleN clustersuniformlyintimebetween
cl
0Myrsand6Myrs.Foreachcluster,wedrawaclustermassfroma 80
60 1.0Myrs
distributionofclustermasses.Then,weestimatethetotalnumber 40 hSi ∼ vdt
of the stars (N∗), and for each cluster, we draw a distribution of 200 [NO(t)]1/2
stellarmasses(M )fromtheSalpeterdistribution. −20
First, we ran∗domly select a total number of O stars , N , −40
∗ −60
e1foov9fre9er7e,a)acw,chehddMcNaclrcuclellusmtse∝teorrs.MitsTinct−hhlt2eeer,edSwsictshehtedrericbeihnuMttteihorcenlfnuifsunrmocthtmbieoenmrwoa(hEsfiscOlmhofsewtgtaherreesedfcnrolaur&wsetaeEtcrhf.hreHecmsoluiwozsve-- y[pc]−8246800000−803.−260M−y4r0s−20 0 20 40 60 80 24680000 6.3Myrs
0 0
ter,soourfirstorderofbusinessistoexpresstheSchechterfunc- −20 −20
tionintermsofthenumber ofOstars.Themassoftheclusteris −40 −40
Mandclm=axAimuMMm∗∗12mMas∗s−O1.3s5ta,rwthhaetrewMec∗1onasniddeMr. ∗In2taerremthseofmtihniism,uthme −−8600−80−60−40−20 0 20 40 60 80−−8600−80−60−40−20 0 20 40 60 80
totalnumbRerofOstarsbecomes x[pc]
Figure 2. We Monte-Carlo model the separations between O stars and
N = Mcl M∗2M −2.35dM . (1) LBVs by considering a random sample of dissolving clusters at random
∗ MM∗∗12M∗−1.35dM∗ ZM∗1 ∗ ∗ awgieths.iHtseorwenweagseh.oNwottheethOatsttahresaovfetrhargeeesreapnadroamtiolyngbeentwereaetendthceluOstesrtsa,resaicnh-
R creaseswithagefortworeasons.First,theseparationsincreaseastheclus-
Therefore,thetotalnumberofstarsintheclusterisproportionalto terdisperseswithadriftvelocityv overtimet.Second,Ostarsdisappear
d
themassofthecluster(N ∝Mcl),andwecaneasilytranslatethe astheyevolve.
∗
distributioninmasstoadistributioninthenumberofstarsforeach
bcleutwsteere,ndd0NNca∗lnd∝1N,t∗h−e2n.tIhfeRt∗otiasldnruamwbnefrroomfstthaersuinniftohremclduisstterribiustion
3.1 COMPARINGTHEPASSIVESINGLE-STAR
DISSOLUTIONMODELWITHTHEDATA
1
N = , (2)
∗ R (N max−1−N min−1)+N min−1 Next, we compare the passive dissolution of single stars to the
∗ ∗ ∗ ∗ LMCandSMCnearest-neighbordistributions.Figure3showsthe
cumulative distribution for the separations for our simple disso-
whereN andN arethemaximumandminimumnumber
∗max ∗min lutionmodel (toppanel) andfor theobservations (bottompanel).
ofthestarsinthecluster.
Forillustrationpurposes,wesetv to14.5km/s,makingthemod-
d
Foreachstar,wedrawthemassfromtheSalpeterInitialMass
eleddistributionhaveaboutthesamemeanasthedata.Sofar,our
Function;
passivedissolutionmodelisingoodagreementwithobservations.
Both the model and observations show a log-normal distribution
M =( 1 )0.74 inseparations,andtheaverageseparationincreaseswithspectral-
∗ [Rm(Mmax−1.35−Mmin−1.35)]+Mmin−1.35 type,whichisexpectedsincelaterOstarsstarslivelongerandhave
(3) moretimetodisperse.
,whereRmisrandomnumberbetween0and1. TheMean and Std.deviation of theseparations aretheonly
Havingestablishedtheinitialconditions,wenowdescribethe informationwecangleanfromaGaussiandistribution,sonext,we
evolution. The average separation between O stars depends upon investigatetheirbehaviors.Theprimaryparameterinourmodelis
howmuchtheclusterhasdispersedandhowmanyOstarsareleft. v ,soinFigure4weplotthemean(bottompanel)andstd.devia-
d
SoweneedtomodelthedispersionoftheOstarsandtheirdisap- tion(toppanel)asafunctionofv .Thedashedlinesrepresentthe
d
pearance.Forthelifetimes,weusethePadovastellarevolutionary modeledmeanandstd.deviation,andthesolidbandsindicatethe
models (Bertellietal. 2009). Figure 2 shows the spatial distribu- observedvalues.Theverticalaxes(µ andσ )inFigure4,arethe
S S
tionofanexamplemodel.ThespatialdistributionisGaussianwith meanandstandarddeviationinthelog;µ(logS)andσ(logS)to
σ = vdt. Given the assumption that stars are coasting then the poweroftenrespectively.Thesolidbandsprovidesomeestimate
individualvelocitiesarer/t.Withtheseassumptions,thenthedis- ofuncertaintyinourinferreddriftvelocity,webootstraptheobser-
tributionofvelocitiesisGaussiantoo;p(v)= √2πtσ2e−r2/2σ2 . vations,givingavarianceforboththemeanandstd.deviation.
Withourmodeldefined,ourfirsttaskistoconstrainwhether We draw three main conclusions from Figure 4. For one,
theaveragedistancesbetweenLBVsandOstarsisconsistentwith the drift velocities that we infer by comparing our simple model
the passive dissolution of a cluster with single-star evolution. To withthedataareroughlywhatwewouldexpect;seeourorder-of-
compare ourmodels tothedata,wecalculatetheangular separa- magnitudeestimatein§3.Second,weinferlargerdriftvelocities
tions, assuming that the clusters are at the distance of the LMC. forthelatertypeOstars.Thismayhintattheimportanceofbinary
Furthermore,tobeconsistentwithSmith&Tombleson(2015),we evolutionandkicksamongtheO-typestarsthemselves.Third,our
subdividethemodeledOstarsintoearly,mid,andlatetypesbased simplemodelisnotabletoreproducethevarianceinthedistribu-
upontheirmasses.Toconvertfrommasstospectraltype,weused tions. This implies that something ismissing from our model. In
Martinsetal.(2005)data.Early-typeOstarshavemassesgreater otherwords,thereismorethatwecanlearnabouttheevolutionof
than34.17 M , late-typeOstarshavemasses6 24.15 M , and massivestarsinclustersfromtheirspatialdistributions.
⊙ ⊙
mid-typeOstarshavemassesinbetween. Inthenextsubsection, Wecan use theresultsin FigureMeanVariance to alsoinfer
wetestwhetherourpassivesingle-stardissolutionmodelisconsis- that LBVisolation putsinteresting constraints on their evolution.
tentwiththedata. The average separation for late-type O stars is 0.01 degrees. For
MNRAS000,1–12(2017)
6 Aghakhanlootakanlooet al.
1.0 10
O2-O5(Early) Model
O6+O7(Mid) 9
0.8 O8+O9(Late)
8
0.6 7
S
σ
6
0.4
5
4
0.2
al
t
to 3
of 5 10 15 20 25
n
tio OBS 0.020
c
a O2-O5(Early) OBS
Fr0.8 O6+O7(Mid) Model
O8+O9(Late)
0.015
0.6 VLate∼17-20km/s
0.4 µS 0.010
VMid∼23-25km/s
0.2 0.005
LLBBVV
0.0 0.000 VEarly∼10-13km/s
10−3 10−2 10−1 100 5 10 15 20 25
ProjectedseparationtonearestO-typestar[deg] Vdrift [km/s]
Figure3.Cumulativedistributionsfortheprojectedseparationtothenear- Figure4.Themean(bottompanel)andstd.deviation(toppanel)distanceto
estOstar.Thetoppanelrepresentsthemodeleddistributionandthebottom thenearestneighborvs.driftvelocity.Inbothpanels,dashedlinesrepresent
panelrepresentsthedata.Broadly,themodelreproducestheobservations; the passive dissolution model and solid lines represent the observational
bothshowalog-normaldistribution,andtheaverageseparationincreases data(Smith&Tombleson2015).Wehighlightthreemainconclusions.1)
withspectral-typebecausethelaterspectraltypeslastlonger. Thedriftvelocitiesthatweinferbycomparingoursimplemodelwiththe
dataareroughlywhatweestimatedin§3.2)Weinferlargerdriftvelocities
for the later type O stars, implying that binary evolution and kicks may
LBVs,theaverageseparationisroughlyfivetimesbigger.Dimen- be important. 3) The passive dissolution model is not able to reproduce
sionally,theaverageseparationshouldbeproportional tothedis- thevarianceinthedistributions, whichimpliesmissingphysicsfromour
model. In another words, there is room to improve our model and learn
persionvelocityandtheage,S ∼ vdtage.IfanLBVcomesfrom moreabouttheinterplaybetweenO-starevolutionandclusterdissolution.
the most massive stars, then one would not expect them to have
ageslargerthanthelate-typeOstars.Therefore,asaconservative
estimate, let us assume that an LBV is an evolved very massive thatitisindeedinconsistentwithobservations.Inaddition,wecon-
star which has about the same age as a low mass main sequence sidermodelsthatinvolvebinaryinteractionsinadispersingclus-
Ostar.Underthisassumption,sincetheseparationsforLBVsare ter.Ouraimistodevelopmodelstoseewhetherbinaryscenarios
5timesbigger thanlate-typeOstars,thisimpliesthatthedisper- areconsistentwiththeLBVobservedseparations.Atthemoment,
salvelocityis5timesbiggerthanthelate-typeOstarwhichisof there is very littleinformation other than the separations, so it is
order 100km/s.Tobemorequantitative,inthenext sections,we notworthdevelopinganoverlycomplexmodelforbinaryinterac-
extendthepassivemodeltoinfertheactualdispersal velocityfor tion.Wewouldnotbeabletoconstraintheextraparametersofthe
LBVs.Alternatively,weconsiderbinaryscenarioswhichmaygive model.Therefore,wedevelop thesimplestbinarymodelstocon-
aexplanationfortherelativelylargeisolationforLBVs. strainthedata. Inparticular, weconsider twosimplemodels that
involvebinaryevolutioninadispersingcluster.Inthefirstmodel,
weconsiderthataLBVistheproductofamergerandisarejuve-
natedstar;inthesecondmodel,weconsiderthataLBVisamass
4 CLUSTERDISSOLUTIONWITHCLOSEBINARY
gainerandreceivesakickwhenitsprimarycompanionexplodes.
INTERACTIONS
In§4.1,wefirstputtogetherananalyticmodelfortheaverage
Intheprevioussection,wesuggestedthatthesingle-stardispersal separationbetweentwostarsversustime.Thenin§4.2weusethis
model is inconsistent withthe isolation of LBVs.In this section, modeltoshowtheinconsistencyinthestandardsingle-starmodel,
weputthepassivesingle-stardispersalmodeltothetest,andshow andweshowthatLBVsareeitheroverluminousgiventheirmass,
MNRAS000,1–12(2017)
ModelingLBV Isolation 7
stars,whichmayhaveadifferentdensitydistributionthanthefirst.
Obviously,later’L’willrepresentLBVs.Inthiscase,thecombined
densityis
σO =vOt
3 nOL(r)= 2NπσO2 e−r2/2σO2 + 2NπσL2e−r2/2σL2 . (7)
c O L
p
/ Withthistwo-component expression for the density, we can
ars σL =vLt evaluate the local separation, Sˆ via eq. (5) and then we can cal-
culatetheaverageseparationfromeq.(4).Calculatingtheaverage
t
s
# separationisnumericallystraightforward.However,withasmall
butuseful assumption, wecanderive ananalyticestimatefor the
averageseparation.Tomakeiteasiertocalculatetheintegralana-
lytically,wemaketwoassumptions.First,weassumethattheaver-
r[pc] ageseparationisroughlygivenbythescaleofoneoverthesquare
rootoftheaveragedensity.Therefore,
1
Figure5.Twosimplespatial-distribution modelsforthederivationofour hS i≈ . (8)
analyticscalings. OL hnOLi1/2
Second,becauseLBVsareextraordinarilyrarecomparedtotheO
stars we assume that N ≫ N . By considering these two as-
ortheyaretheproductofamergerandarearejuvenatedstar.Al- O L
sumptionstheaveragedensityis
ternatively, in§4.3, weuse theanalytic model to develop a kick
model, and in § 4.4, we use this model to infer a potential kick N
hn i≈ O . (9)
velocityforLBVs.Insummary, weillustratethattheisolationof OL 2π(σ2 +σ2)
O L
LBVsisconsistentwithbinaryscenarioandisinconsistentwiththe
Onceweplugthisintotheequationforhn i,eq.(8)leadstothe
single-starmodel. OL
averagedistancefromLBVstothenearestOstar:
Toconstrainthemodels, wefirstderiveanalyticscalingsfor
theaverageseparations,andthenweexplorewhetherthesescalings 2π(σ2 +σ2) 1/2
areconsistentwithsimplebinarymodels.First,weconsidersimple hSOLi≈ NO(t) L (10)
models for the spatial distribution of two groups of stars, type O (cid:18) O (cid:19)
andtypeL.EachhasaGaussianspatialdistributionwithitsown SoonwewillusetheseparationbetweenOstarstohelpcon-
velocity dispersion vd ,which we label as vO and vL. Later, we strainthemodels for LBVs,so wenow derive an analyticmodel
will consider two scenarios; one in which these average velocity forhSOi.TheaveragedensityforOstarsis
dispersionsarethesame,andoneinwhichtheyaredifferent.For N (t)
avisualrepresentationofthesesimplemodels,seeFigure5.Given hnOi= 4πOσ2 , (11)
thesedistributions,wecalculatetheaverageseparationbetweena O
andsotheaverageseparationbetweenOstarsisroughly
starandtheneareststarinthesamegroup.Thenwecalculatethe
averageseparationbetweenastaringroupOtoastaringroupL. 1 4π 1/2
Theaverageseparationbetweenstarsinthesamepopulationis hSOi≈ hn i1/2 = N (t) σO. (12)
O (cid:18) O (cid:19)
hSi=2π ∞Sp(r)rdr, (4) Tomakeuseoftheseexpressionsfortheaverageseparation,
Z0 we need to compare the separations between two different tracer
whereSistheseparation,andp(r)istheprobabilitydensityfunc- populations. Because the masses of mid-type O stars correspond
tionp(r) = 2π1σ2e−r2/2σ2 whereσ = vt.Tocalculatethemean roughlytoinferredminimummassofLBVs,weusethemid-type
valueoftheseparation,weneedtofindtheseparation(S).Oneway Ostaraverageseparationasareference;
toestimatethedistancetothenearestneighboristousethespatial hS i 2 1 σ2 N (t )
densityofstars.Ingeneral,anestimateforthedistancetothenear- OL = 1+ L O O , (13)
estneighbor is,Sˆ ≈ 1/n1/d,wherenisthenumber densityand (cid:18) hSOi (cid:19) 2(cid:18) σO2 (cid:19) NO(tL)
d are the number of dimensions that we consider (ZeljkoIvezic wherewearecarefultoconsiderhowthenumberofOstarschanges
2014).Whenviewingclustersprojectedontothesky,d=2. withtimeandweevaluatethisfunctionattheageoftheLBVpopu-
lationandthereferenceOstarpopulation.Thisequationrepresents
Sˆ≈1/n1/2. (5)
thegeneralexpressionrelatingage,theaverageseparations,andthe
If we consider a simple density distribution, n(r) = drift(orkick)velocityofeachtracerpopulation.
N exp−r2/2σ2, then the average separation in two di- Intheexpressionsfortheaverageseparations,theseparations
2πσ2
mensionalspaceis growduetotwoeffects:adriftvelocityandthedeathofOstars.
Thedrift part is simplyproportional tot. Next, weexplicitly de-
2π 1/2
hSˆi=2 σ, (6) rive the number of O stars as a function of time, NO(t).Given
(cid:18)N (cid:19) a mass function dN/dM, the total number of O stars is NO =
whereN isthetotalnumberofstarsinthecluster. MM12 ddMNdM = αA+1(M2−α+1−M1−α+1),whereM1isthemin-
Nowweconsidertwopopulationsofstars.Onewerepresent imummassforan−Ostar(∼16M ),M2 isthemaximummassO
R ⊙
with’O’,whichrepresentsthelargestnumber oftracerstars.As star,whichisafunctionoftheageofthecluster,αistheslope(we
thelabelsuggests,wewilllaterconsiderOstarsasalargenumber useSalpeter,2.35),andAisanormalizationconstant.Ifweassume
oftracer stars.Theother, ’L’,representsamoreraresetoftracer apower-lawrelationshipbetweenmassofanOstaranditslifetime
MNRAS000,1–12(2017)
8 Aghakhanlootakanlooet al.
as an O star, then we can relate the age of an M2 O star to the ItisclearthatmostoftheLBVshavelargerseparationscompared
ageofanM1Ostar:M2 =M1(tt1)−1/β.FromthePadovastellar towhatthepassivemodelpredicted.
evolutionarymodels(Bertellietal.2009),wefindthatthevalueof Moreover, in the passive model, LBVs do not have enough
β ∼1.5.Combiningtheseexpressions,wegetanequationforthe timetogettotheobservedaverageseparation.Figure7showsthe
numberofOstarsasafunctionoftheageofthecluster,t, samepassivemodel,theobservedLBVseparation,butthistimewe
simplifythepossibleagesofLBVsbyshowingtheagesfortheav-
NO(t)= 1−Aα(1−(tt1)αβ−1)M11−α. (14) esirnaggleemstaasrs,tohfeonutrheLyBdVosnaomthpalev.eCelneoaurlgyh,itfimLBeVtosreevaochlvtehaeslaarngoersmepa-l
WithanexplicitfunctionforthenumberofOstars,wemay arations.Instead,let’sconsiderhowoldanLBVwouldhavetobe
nowderivetheequationrelatingseparation, age,andvelocity,in- inordertopassivelydispersetotheobservedseparations.Figure7
cluding explicitly all of the dependence on time. Substitutingthe showsthattheagewouldneedtobeabout7.3Myrs.Yetthisage
expressionforNO(t),eq.(14)intothegeneralanalyticexpression, correspondstothemainsequenceturnofftimeforan18M staror
eq.13,wefinallyarriveatthegeneralanalyticformula,explicitly thedeathof a27M star.Bothof thesevalues arebelow⊙theav-
relatingseparation,age,andvelocity: eragemassof theLB⊙Vs,50M (§2).It isclear that considering
⊙
LBVs in the context of a standard single-star evolution is incon-
α−1
hS i 2 1 v x 2 (1−x β ) sistentwiththeisolation.Ifwestickwiththeassumptionthatthe
OL = 1+ L O , (15)
(cid:18) hSOi (cid:19) 2 (cid:18)vOxO(cid:19) !(1−xαβ−1) driftvelocitiesoftheLBVsandtheOstarsarethesame,thenthere
aretwopossiblesolutions.EitherLBVsareoverluminousorthey
wherexO = ttO1 andx = ttL1.t1 (8Myrs)istheageofthemin- aretheresultofamergerandarerejuvenatedstars.SeeFigure8to
imum mass and t (2.5 Myrs) is a reference age. We estimate visualizethemergermodel.
O
thesevaluesfromPadovastellarevolutionarymodels(Bertellietal.
2009).
In the next subsections, we use our general analytic result, 4.3 KICKMODEL
eq. (15), to explore what average separations one would expect
AnotherbinarymodelthatisconsistentwiththeisolationofLBVs
when we consider the passive dissolution in three scenarios, a
isthekickmodel.Tovisualizethekickmodel,considerthebinary
single-star evolution scenario, a binary scenario that involves a
scenarioinFigure9.Inthismodel,theprimarystar,themoremas-
merger,orabinaryscenariothatinvolvesakick.
sivestar,evolvesfirstandtransfersmasstothesecondarystar.Ifthe
moremassivestarismassiveenoughtoexplodeasacore-collapse
supernova,thenthecompanionmayreceiveakick.Thiskickmay
4.1 PASSIVEMODEL
beimpartedbyeitheranasymmetricexplosion,theBlaauwmech-
Usingouranalyticestimatesfortheaverageseparation,weassume anism(Blaauw1961),oracombinationofboth.Inthismanuscript,
thatthedispersalvelocityforLBVsandOstarsarethesameand wedonotmodelthebinaryevolutionandkickvelocities.Ratherwe
estimatethe average separation for the passive single-star model. justassumethattherearetwopopulations,onemorenumerousand
Comparingthismodeltotheobservations,wefindthatthepassive doesn’t receivekicks(theOstars),andonethat islessnumerous
single-starmodelisinconsistentwiththeobservations.IfLBVsdo andwhosevelocitydistributionisdominatedbykicks.
passivelydispersewiththesamevelocityastherestoftheOstars, Onceagain,wemayuseourgeneral analyticexpression, re-
then we propose that LBVs are the product of a merger and are latingtheseparations,age,andvelocities,eq.(15),butthistimewe
rejuvenatedstars. expressvLintermsoftheage,x=tL/t1,andthemeasuredvalues
Inthiscase,weneedtoconsidertheaverageseparationwhen oftheseparations,
the dispersal velocity for LBVsand O starsare the same. Inthis
scenario,ourgeneralanalyticexpression,eq.(15),reducesto vL = xO 2 hSOLi 2 (1−xαβ−1) −1 1/2 . (17)
v x  hS i α−1 
SL = 1 1+ x 2 (1−xOαβ−1) 1/2 . (16) O  (cid:18) O (cid:19) (1−xOβ ) 
SO "2 (cid:18)xO(cid:19) ! (1−xαβ−1) # Smith&Tombleson(2015)showedtheaveragedistancefrom
LBVsto the nearest O star is ∼ 6.5 larger than the average dis-
Equation(16)representsthepassivemodel. tancefromOstartothenearestOstar.IftheageofLBVsaresim-
ilartotheaverage mid-typeOstar, thenintheassumption of the
kickmodel,thisimmediatelyimpliesthatvLisroughlyninetimes
4.2 INCONSISTENCYINTHEPASSIVEMODEL; largerthanvO.Inthenextsubsection,weestimatetheLBVs’drift
IMPLIESMERGERANDREJUVENATION velocitygiventhismodel.
Next, we use the passively dissolving solution, eq. (16), to show
thattheisolationofLBVsisinconsistentwiththesingle-starsce-
4.4 ESTIMATIONANDINTERPRETATIONOFTHE
nario. If LBVsaremassive starsabove 27 M and evolve asiso-
⊙ KICKVELOCITY
latedstars,thenFigures7&8demonstratethatthemaximumages
oftheseLBVsiswhollyinconsistentwiththelargeseparationsob- Figure10showstheinferredkickvelocityasafunctionoftheLBV
servedformassivestars. age.Ifthemassgainerwhicheventually becomes theLBVgains
Thepassive model predictsmuch lowerseparations thanthe littlemass, the there is littlediscrepancy between the zero main-
observational data. See Figure 6 for an illustration. The orange sequence mass and the final mass. In thiscase, there is littledif-
curve line represents the passive model, S in eq. (16). The ference between itsapparent age and itstruemain-sequence age.
LBV
solidbrownlineillustratestheLBVs’averageseparationobtained Thenitstrueageisrelativelyshortandtheonlywaytogetalarge
fromthedatacomparedtothereferenceaverageseparation, SLBV . separationwithalargekickvelocity.Inthisscenario,wefindthat
S0
MNRAS000,1–12(2017)
ModelingLBV Isolation 9
Mdeath [M⊙]
050505 0 5 0 5
766554 4 3 3 2
MTO [M⊙]
00 5050 5 0 5 0
1100 5544 3 3 2 2
−−11
1100
]]
gg
ee
dd −−22
[[ 1100
SS
gg
oo SLBV
ll
LBVc
−−33
1100
LBV
passive model
TO
death
−−44
1100
00 11 22 33 44 55 66 77 88
tt [[MMyyrrss]]
Figure6.LBVisolation isinconsistent withthesingle-star modelandpassive dissolution ofthecluster. Thesolidbrownline showstheLBVs’average
separationobtainedfromthedata.Theorangecurveshowsouranalyticdescription,eq.(16),fortheaverageseparationinthecontextofpassivedissolution.
Thismodelrequiresareference;weusedthemid-typeOobservationsasthereference.Thelinesegmentsshowthepurportedmassesandallowableagesfor
theLBVs(solidsegments)andLBVcandidates(dashedsegments).IfLBVmassestimatesarecorrect,thenevenwhenoneconsidersthemaximumagefor
LBVs,theseparationsaremuchlargerthanwhatthesingle-starpassivemodelpredicts.
thekickcanbeashighas200km/s.Ontheotherhand,ifthemass wepresentthescaleoftheproblem;inasubsequentpaper,wewill
gainishigh,thenthetruemain-sequenceagewouldbemucholder modelthedistributionofobservedvelocitiesonewouldexpect.
than the current mass imply. With a much older age the velocity
require to get a large separation is much lower. It might even be
zero,inwhichcase,theLBVistheproductofamerger.Thehor-
izontalblacksolidlinerepresentstheaverageobservedseparation 5 SUMMARY
formid-typeOstars.Thesolidbluelinelinerepresentsourmodel
Smith&Tombleson(2015)foundthatLBVsaresurprisinglyiso-
toinferthekickvelocity,eq.(17).
latedfromotherOstars.Theysuggestedthattherelativeisolationis
inconsistentwithasingle-starscenarioinwhichthemostmassive
Thoughwepredictthatthekickvelocitiesmaybeashighas starsundergo an LBVphase on their way toevolving into aWR
∼200km/s,wenotethatthekickmaybequitelow,evennearzero. star. Instead, they suggested that abinary scenario is likely more
Humphreysetal.(2016)arguedthatnoneoftheLBVsintheLarge consistentwiththerelativeisolationofLBVs.Inthispaper,wetest
MagellanicCloudhavehighvelocities.Theysuggestthatmostof theseassertionsbydevelopingcrudemodelsforsingle-starandbi-
theLBVvelocities(listedinTable3of Humphreysetal.(2016)) naryscenariosinthecontextofclusterdissolution.Evenwiththese
areconsistentwiththesystemicvelocitiesofthelargeMagellanic crudemodels,wefindthattheLBVs’isolationisinconsistentwith
Cloud,concludingthattheobservedvelocitiesareinconsistentwith thestandardpassivesingle-starevolutionmodel.EitherLBVsare
thekick.InFigure10,weshowthatthekickvelocitymaybeany- overluminousandtheyaremuchlessmassivethanstandardmod-
wherefrom0to∼200km/sdependingontheorbitalparametersat els would infer, or they are the result of binary interactions. We
thetimeoftheSN,andhowmuchmasswastransferred.Tofurther findthattheLBVisolationismostconsistentwithtwobinarysce-
constrainthemassgainerandkickmodel,onewillneedtoproperly narios;eitherLBVsaremassgainersandreceiveakickanywhere
model binary evolution including explosions and kicks. For now, from0to∼200km/s,or theyaretheproduct ofmergersandare
MNRAS000,1–12(2017)
10 Aghakhanlootakanlooet al.
Mdeath [M⊙]
050505 0 5 0 5
766554 4 3 3 2
MTO [M⊙]
5050 5 0 5 0
00..1144 5544 3 3 2 2
00..1122
00..1100 MLBV(TO) ∼ 18 M⊙
MLBV(death) ∼ 27 M⊙
]] 00..0088 MLBV∼ 50 M⊙ tLBV ∼ 7.3 Myrs
gg
ee
dd
[[ 00..0066 OBS
SS
00..0044
Passive Model
00..0022
00..0000
00 11 22 33 44 55 66 77 88
t [Myrs]
Figure7.TherelativeisolationofLBVsisconsistentwithabinarymergerinwhichtheLBVisarejuvenatedstar.Thesolidbrownlineshowstheobserved
LBVaverageseparation. IfLBVspassively dissolvewiththerestofthecluster(orange line), thenweinfer anaverage ageforLBVsof7.3Myrs.This
correspondstothemain-sequence-turn-offtimefora18M starandthedeathtimefora27M star.However,theaveragemassforLBVsestimatedfrom
theirluminositiesisroughly50M .Starsthismassivedo⊙notlivelongenoughtopassivelydis⊙persetolargedistances.Ontheotherhand,ifLBVsarethe
productsofamerger,andtheprima⊙ryhasamassbetweenabout18M and27M ,thentherejuvenatedstarcouldhaveahighluminosity,highmass,and
oldageallowingittodispersetolargerdistances. ⊙ ⊙
rejuvenatedstars.Ofcourse,LBVsmayactuallyrepresentacom- km/s.Forthemid-andlate-type,werequiredriftvelocitiesonthe
bination of thesetwoscenarios. Itisquitepossible that someare order of 24 and 18 km/s, respectively. The higher drift velocities
massgainersandsomearetheproductofmergers. for later-typeO starssuggests that binarity and kicks mayplay a
prominentroleinclusterdissolution.Infact,somefractionoflater-
Inordertoconstrainthesemodels,wefirstreproducethere-
type O stars may be mass gainers or the product of mergers. In
sultsofSmith&Tombleson(2015).Similarly,weobtaintheposi-
a future paper, we will investigate whether one can constrain the
tionof allO starswithin10◦ projected radiusof 30Dor, andwe
fractionofkicksandstrongbinaryinteraction.
construct distributionsof distancestothenearest O star.Ourdis-
tributionsareverysimilartotheirs.Wefindthat thedistributions
UsingtheresultsoftheMonte-Carlosimulationsasaguide,
for theLBVsand Ostarsareveryunlikely tobedrawnfromthe
we develop an analytical model for the average separation as a
sameparentdistributions.Inparticular,theaveragedistancetothe
functionofdriftvelocityandtime.Theseanalyticscalingsstrongly
nearestOstaris∼6.5timeslargerforLBVsthanOstars.Tobet-
suggest that LBV isolation is inconsistent with single-star stellar
terinformourmodels,wefurthercharacterizethedistributionsand
evolution.Moreover,thesescalingssuggestthatLBVisolationsug-
find that all of the nearest neighbor distributions are log normal.
gests that either LBV stars have lower initial masses (and hence,
Thefactthatthedistributionsaresimpleandlognormaldemands
longer lifetimes) than one would infer from luminosities, or the
thatourmodelsforclusterdissolutionarealsosimple.
isolation is most consistent with some sort of binary interaction;
WeproposesimpleMonte-Carloandanalyticmodelsforthe eitheramerger orkick.IfLBVshavethesamedispersion veloc-
dispersalofopenclustersofOstars.Inthismodel,wesamplefrom itythatweinferfrommid-typeOstars,thenthetimetogettothe
distributionsofclustersizes,theSalpeterIMFforstarsandrandom relatively large isolation is 7.3 Myrs. However, the average mass
ages.TomatchtheobservedseparationsforearlyOstars,wefind ofLBVsis50M whichhasamaximumtimeof∼4.7Myrs.This
⊙
thattheearlytypeclustersneedadriftvelocityontheorderof11 isclearlyinconsistent. Ontheother hand, binaryinteractions can
MNRAS000,1–12(2017)