Table Of ContentMon.Not.R.Astron.Soc.000,000–000(0000) Printed5February2008 (MNLATEXstylefilev2.2)
NLTE models of line-driven stellar winds
II. O stars in SMC
⋆
Jiˇr´ı Krticˇka
U´stavteoreticke´fyzikyaastrofyzikyPrˇFMU,CZ-61137Brno,CzechRepublic
6
0
0 Received
2
n
ABSTRACT
a
J We calculate NLTE line-drivenwindmodelsof selected O stars in the spectralrangeof O4
to O9 in the Small Magellanic Cloud (SMC). We comparepredictedbasic wind properties,
6
i.e. the terminal velocity and the mass-loss rate with values derived from observation. We
1 foundrelativelygoodagreementbetweentheoreticalandobservedterminalvelocities.Onthe
v other hand, predicted mass-loss rates and mass-loss rates derivedfrom observationare in a
9 good agreementonly for higher mass-loss rates. Theoreticalmass-loss rates lower than ap-
2 proximately 10−7M year−1 are significantly higher than those derived from observation.
⊙
1 These results confirm the previously reported problem of weak winds, since our calculated
1 mass-lossratesareinafairagreementwithpredictionsofVinketal.(2001).Westudymulti-
0
componentmodelsforthesewinds.Forthispurposewedevelopamoredetaileddescription
6
ofwinddecoupling.Weshowthattheinstabilityconnectedwiththedecouplingofindividual
0
windelementsmayoccurforlow-densitywinds.Inthecaseofwindswithverylowobserved
/
h mass-lossratethemulticomponenteffectsareimportantforthewindstructure,howeverthis
p isnotabletoconsistentlyexplainthedifferencebetweenpredictedmass-lossrateandmass-
- loss rate derivedfromobservationfor these stars. Similar to previousstudies, we foundthe
o
dependenceofwindparametersonthemetallicity.Weconcludethatthewindmass-lossrate
r
t significantlyincreaseswithmetallicityasM˙ ∼Z0.67,whereasthewindterminalvelocityon
s
a averagedependsonmetallicityonlyslightly,namelyv∞ ∼Z0.06(forstudiedstars).
:
v
Keywords: stars:winds,outflows–stars:mass-loss–stars:early-type–hydrodynamics–
i
X instabilities–galaxies:MagellanicClouds
r
a
1 INTRODUCTION verse,hydrogenandheliummostlycontributetothewinddensity,
their contribution to the radiative acceleration is very small (e.g.
Inrecentfewyears8mclasstelescopesbecameroutinelyavailable
Abbott 1982). Heavier elements like carbon, nitrogen, oxygen or
forthestellarresearch.Thisenableddetailedstudyofmanystars
ironaremuchmoreimportantforthewindaccelerationofpresent
intheLocalGroup.Clearly,manyastrophysicallyimportanttypes
hot stars.Thisisbecause heavierelementshaveeffectivelymuch
ofstarscanbestudiedfromanotherperspective.Thisisespecially
morelinesthatareavailabletoabsorbthestellarradiationandto
trueforstarsfromtheSmallMagellanicCloud(SMC).Theirlower
acceleratethestellarwind.Clearly,inmostcasestheradiativeforce
metallicitycomparedtothesolarvalue(theaveragemetallicityof
shallbehigherforhighermetallicity.Consequently,thebasicwind
individual elements is Z/Z = 0.2, e.g. Venn 1999, Bouret at
⊙ propertiesshalldependonthemetallicity.
al.2003)enablestostudythestellarpropertiesandevolutionwith
Althoughrelativelysimpleexpressionsforthedependenceof
respecttometallicity.
themass-lossrateandtheterminalvelocityonthemetallicitycan
Oneofthemostimportantpropertiesofhotstars,thatcanin-
beobtainedevenonthebasisofthestandardCAK(Castor,Abbott
fluenceboththeobservedspectrumandthestellarevolution,isthe
& Klein 1975) theory (e.g. Puls et al. 1998), the detailed depen-
presence of the stellar wind. The existence of significant depen-
denceisprobablymorecomplicated(Pulsetal.1998,Vinketal.
denceofthebasicwindproperties(i.e.theamountofmassexpelled
2001).Itisclearthatmoreprecise(NLTE)windmodelsareneces-
fromthestarperunitoftime,themass-lossrateandthewindve-
sarytostudythisproblemindetail.
locityatlargedistancesfromthestar,theterminalvelocity)onthe
metallicityhasbeenanticipatedalreadyattheverybeginningofthe There are several independent NLTE wind models that can
theoretical study of hot star winds (Abbott 1982, Kudritzki et al. be used for the study of the metallicity dependence of the wind
1987).Althoughthemostabundantelementsintheobserveduni- properties.Thesecodesdifferinthelevelof sophisticationofthe
treatmentofthewindproblem.Oneofthemostimportantaspects
that influences the reliabilityof individual NLTE wind models is
⋆ E-mail:[email protected] theinclusionofproperbound-freeandbound-boundblanketingin
2 J. Krticˇka
theUVdomain.WhilemodelsofVinketal.(2001,hereafterVKL)
Table1.AtomsandionsincludedintheNLTEcalculations.Inthistable,
useMonte Carlomethod for thesolution of theradiativetransfer levelmeanseitheranindividuallevelorasetoflevelsmergedintoasuper-
equation,modelsofPauldrachetal.(2001)solvethisequationsin level.
detail.WindmodelsofGra¨fener&Hamann(2005),thatweresuc-
cessfullyusedforthecalculationofwindmodelsofWRstars,use Ion Levels Ion Levels Ion Levels
comoving-frameformulationoftheradiativetransferequation.All
thesemodelstosomeextentproperlytakeintoaccountblanketing HI 9 NeIV 12 SIV 18
HII 1 NeV 17 SV 14
effectsintheUVdomainthatareimportantforthecorrectcalcu-
HeI 14 NeVI 1 SVI 16
lationoftheionizationbalanceandoftheradiativeforce.Hence,
HeII 14 NaII 13 SVII 1
thesemodelsareabletoreliablypredictthemostimportantwind
HeIII 1 NaIII 14 ArIII 25
parameter–themass-lossrateanditsmetallicitydependence. CII 14 NaIV 18 ArIV 19
However, alsothe windvelocity fieldand especially theter- CIII 23 NaV 16 ArV 16
minal velocity can beinfluenced bythe metallicity.To studythis CIV 25 NaVI 1 ArVI 1
effect,thesolutionofhydrodynamicequationsisnecessary.Since CV 1 MgII 14 CaII 16
modelsofVKL,thatarewidelyusedinhotstarevolutionarycal- NII 14 MgIII 14 CaIII 14
culations,donotsolvetheseequations,thesemodelsassumeapre- NIII 32 MgIV 14 CaIV 20
specifiedvelocitylawandcannotbeusedforthepredictionsofthe NIV 23 MgV 13 CaV 22
NV 13 MgVI 1 CaVI 1
velocitystructure.Thisisanimportantdifferencebetweenmodels
NVI 1 AlIII 14 FeIII 29
of VKL and e.g. Pauldrach et al. (2001) or Gra¨fener & Hamann
OII 50 AlIV 14 FeIV 32
(2005). Hydrodynamical wind models for different metallicities
OIII 29 AlV 1 FeV 30
were calculated by Kudritzki (2002). According to these models OIV 39 SiIII 12 FeVI 27
thewindterminalvelocitydecreaseswithdecreasingmetallicity,in OV 14 SiIV 13 FeVII 1
agreementwithSMCobservations(seeanextensivecompilationof OVI 20 SiV 15 NiIII 36
Kudritzki&Puls2000). OVII 1 SiVI 1 NiIV 38
However,thewindmetallicitydoesnotvarytheradiativeforce NeII 15 SII 14 NiV 48
only.Themomentumacquiredbytheheavierelementsfromthera- NeIII 14 SIII 10 NiVI 1
diativefieldistransferredtothebulkwindcomponent(i.e.hydro-
gen and helium) via the Coulomb collisions. Since the frictional
force caused by these collisions depends on number densities of equationstogetherwiththeradiativetransferequation.Theradia-
heavierelementsandhydrogen-helium component, theeffectsin- tivetransferequationinlinesissolvedintheSobolevapproxima-
duced bythecollisions aremore important inthelow-metallicity tion(Sobolev1947,Castor1974),whereasthecontinuumradiative
environment (Kudritzki 2002,Krticˇkaetal. 2003).NLTEmodels transferequationissolvedbyFeautriermethodinthesphericalco-
availabletostudythisproblemwerepresentedbyKrticˇka&Kuba´t ordinates(seeMihalas&Hummer1974orKuba´t1993).Derived
(2004,hereafterKK1). occupationnumbersareusedtocalculatetheradiativeforce(inthe
Somehot starswithlow luminosities(andconsequently low Sobolevapproximation)andtheradiativecooling/heatingterm(us-
masslossrates)showsignaturesofveryweakwinds,muchweaker ingthethermalbalanceofelectronsmethod,Kuba´tetal.(1999)).
thanthatdeduced fromstandardwindtheory(Bouretetal.2003; This enables to solve the hydrodynamic equations. These proce-
Martinsetal.2004;2005).Sincethisdiscrepancyisbasedmostly duresareiteratedtoobtainconsistentmodelstructure.Finally,wind
on wind models of Vink et al. (2001), it would be interesting to mass-lossratesandterminalvelocitiesofstudiedstarscanbeob-
testthisresultalsowithindependentNLTEwindmodelsandtotest tainedfromourmodelsandcomparedwithobservations.
whetherthisdiscrepancycanbecausedbymulticomponenteffects. Although our models involve some simplifying assumptions
Thestudyofthemetallicitydependenceofstellarwindprop- (especiallythesimplifiedtreatmentoftheradiativetransfer(split-
ertiesisnotimportantonlyforourunderstandingofstellarwinds tingoftheradiativetransferincontinuumandinthelines,neglect
ormassivestarsitself.Sincestellarwindsplayaroleinthechem- oflineoverlaps)comparedtomoreadvancedmodelsofe.g.Paul-
icalenrichmentofgalaxiesandareresponsibleforinputofalarge drachetal.(2001),VKLorGra¨fener&Hamann(2005)),theywere
amount of momentum and energy in the interstellar medium, the abletocorrectlypredictbasicwindparametersoflateOstars(see
detailed knowledge of variations of basic wind parameters with KK1)andofAsupergiant(seeKrticˇka&Kuba´t2004).Moreover,
metallicityisimportantalsoforotherfieldsofastrophysics. our models have some advantages compared tosome other mod-
Aswehave discussed, windproperties of hot starsshall de- elsavailableintheliterature,forexamplethedirectcalculationof
pend on the stellar metallicity. Low-metallicity environment of theradiativeforcewithoutusingofforcemultipliersormulticom-
SMCoffersauniquepossibilitytotesttheseavailablepredictions. ponent treatment of model equations (seeKrticˇka&Kuba´t 2001,
Todoso,wepresentherewindmodelsforselectedSMChotstars. hereafterKKII).
Presentedmodelswereonlyslightlymodifiedwithrespectto
thestatusdescribedbyKK1.First,weincludedacceleratedlambda
iterationsincontinuum(or,moreprecisely,approximateNewton-
2 MODELDESCRIPTION
Raphsoniterations)basedonRybicki&Hummer(1992)paperand
ThemodelsappliedinthispaperwereindetaildescribedbyKK1. Ngacceleration(Ng1974)toacceleratetheconvergenceofthesys-
Herewe only summarise the basic model properties and refer an temof statisticalequilibriumequations (seealsoHillier&Miller
interestedreadertothispaper. (1998),orHubeny(2003)forareview).Thiswillbedescribedin
Models assume spherically symmetric and stationary stellar afollowingpaper (Krticˇka&Kuba´t, inpreparation). Second, our
wind. Occupation numbers of selected atoms and ions (see Ta- previous set of included atomic models based on TLUSTY files
ble1)areobtainedbythesolutionofstatisticalequilibrium(NLTE) (Hubeny1988,Hubeny&Lanz1992,Hubeny&Lanz1995,Lanz
NLTE modelsof line-drivenstellarwinds II. 3
& Hubeny 2003) or Opacity Project (Seaton 1987, Luo & Prad- butforasignificantpartofstarstheevolutionarymassisroughly
han1989,Sawey&Berrington1992,Seatonetal.1992,Butleret 1.5 higherthanthespectroscopicone.Theuseofspectroscopic
×
al.1993,Nahar&Pradhan1993)andIronProject(Hummeretal. massesinsteadoftheevolutionaryoneswouldhelptoobtainabet-
1993,Bautista1996,Nahar&Pradhan1996,Zhang1996,Bautista teragreementbetweenobservationandtheoryforsomestars,how-
&Pradhan1997,Zhang&Pradhan1997,Chen&Pradhan1999) everitwouldcausedifferencesforsomeothers.
datawasslightlyextended(seeTable1).Thesementionedchanges Whenavailable,theabundancesof individual elementswere
donotsignificantlyinfluencethederivedresults.Tocompletethe takenfromtheliterature,however inother cases weassumed av-
listofatomicdatabasesused,theoscillatorstrengthsnecessaryfor eragevalueZ/Z = 0.2derivedforSMCstars(e.g.Venn1999,
⊙
thecalculationoftheradiativeforceareextractedfromtheVALD M04).WeuseGalacticheliumabundance.
database(Piskunovetal.1995,Kupkaetal.1999). ThereareindicationsthatstellarwindsofOstarsareclumped
The boundary radiative flux is taken from grid of line- (e.g.B03).Fromtheobservationalpointofview,thepossiblewind
blanketedplane-parallelmodelatmospheresOSTAR2002(Lanz& clumpingdecreasesthewindmass-lossrateinferredfromtheob-
Hubeny2003)insteadoffromH-Hesphericallysymmetricmodels servation because the line profiles of clumped wind mimic those
ofKuba´t(2003).Sinceline-blanketedfluxeshavegenerallylower withhighermass-lossrate.Accordingtothenumericalsimulations
fluxintheUV region(wherearemany linesimportant for radia- ofwindinstability(e.g.Feldmeieretal.1997,Runacres&Owocki
tivedriving), obtainedmass-lossratesareslightlylower (roughly 2002) the theoretical mean mass-loss rate is nearly the same for
1.4 ) than that derived using H-He fluxes. However, this differ- smoothandstructuredwinds.Thus,iftheobservationsreallyshow
×
enceislowerthanthedispersionofmass-lossratesforGalacticO signaturesofclumpingatallspectralregionswherethestellarwind
stars(seeKK1)anddoesnotsignificantlyinfluencepredictedwind is observed, then, according to our present knowledge, the theo-
terminalvelocities(thedifferencebetweentheterminalvelocities reticallypredicted values of mass-loss rates should basically cor-
isabout100kms−1). respond to the values derived from observation with account of
clumping.Forourstudyweadoptedthevaluesderivedfromobser-
vationsassuming”smooth”windspartlybecausethesevalueswere
forlargerSMCsample(toourknowledge)derivedonlybyB03and
3 CALCULATEDWINDMODELS partlybecausethemodelsofwindclumpingarestillschematic.
Windparametersadoptedforthecomparisonwiththeoretical
3.1 ParametersofstudiedSMCstars
values will be discussed individually for those stars, for which it
Sinceourmainintentionforfuturestudiesistostudyweakwinds isnecessary. Notethatinthetextwewilltermthemass-lossrate
of B stars, for the present study we selected only those cooler estimatedfromobservationas”observedmass-lossrate”,although
(i.e. with effective temperatures T . 42000K) O SMC stars, one should keep inmind that thisquantity cannot be directlyde-
eff
forwhichatleastreliableestimateoftheirmass-lossrateisavail- rivedfromspectraandthatitismodel-dependent.
ableinthe literature.Wetriedtoomit those starsfor which their
windparameters are uncertain and which arebinaries. Moreover, NGC 346 WB 1 Thisisamultiplesystem (Heydari-Malayeri &
wealsoaimtobasethelistonbroadersurveyswithlargernumber Hutseme´kers1991).Theterminalvelocityderivedforthissystem
of individual stars studied. Stellar parameters and wind parame- by P96 v = 2650kms−1 is marked as uncertain due to the
tersofthesestarsaregiveninTab.2.Stellareffectivetemperatures complexit∞yoftheabsorptionprofile.P96notethatanother possi-
andradiiaretakenfromPulsetal.(1996,hereafterP96),Bouretet blevalueoftheterminalvelocityisv = 2250kms−1.Prinja&
al.(2003,hereafterB03)andMasseyetal.(2004,hereafterM04). Crowther(1998)obtainedvalueofedg∞evelocity(velocityatwhich
WhereasparametersgivenbyP96wereobtainedbymodelswithout thelineprofilemeetsthecontinuumlevel)asvedge =2830kms−1
wind-blanketing,parametersgivenbyB03andM04werederived for NV lines. Using their derived approximate relation v =
usingmodelswithwind-blanketing. Tostudythethin-windprob- 0.8vedge we obtain terminal velocity v = 2260kms−1.∞Thus,
lemindetailweaddedalsostarsfromMartinsetal.(2004,hereafter weadoptedv =2250kms−1asaval∞ueoftheterminalvelocity.
Mr04)sample.ThesestarsaresuspectedVzstars,i.e.starscloseto ∞
theZAMS.Theyexhibitmuchlowerwindspectralsignaturethan
NGC 346 WB 4 The observed terminal velocity by P96 v =
that predicted from standard theory. On the other hand there are
onlyupperlimitsoftheirobservedmass-lossratesandlowerlimits 1550kms−1 is quoted as uncertain. The typical terminal v∞eloc-
ity for stars of similar spectral type is higher, the terminal ve-
oftheirobservedterminalvelocitiesavailable.
locity calculated as twice the escape velocity (roughly suitable
SMCstarsareinsomesensemoresuitableforthetestofthe-
oretical models than starsfrom our Galaxy. Since thedistance to forSMC stars,B03) is1900kms−1. Usingapproximate relation
v = 0.8v of Prinja& Crowther (1998) and their measure-
SMC is known with relatively high precission, the stellar radius edge
andmass-lossratemaybealsoderivedmorereliably. m∞entofedgevelocityofCIVlinesweobtainv =1950kms−1.
∞
Hence,weadoptthisvalueofterminalvelocity.
For our study we adopted evolutionary stellar masses either
derived by B03 or by us using evolutionary tracks of stars with
initial metallicity Z/Z = 0.2 calculated by Charbonnel et al. NGC346WB6 TheedgevelocityobtainedforthisstarbyPrinja
(1993).Theuseofthee⊙volutionarymassesisarelativelyimportant & Crowther (1998) vedge = 1925kms−1 is lower than the ter-
assumptionthatcansignificantlyinfluencetheresultsderiveddue minal velocity v = 2250kms−1 derived by P96 or v =
tothediscrepancybetweenstellarmassesofhotstarsderivedfrom 2300kms−1 by∞B03.Thisdifferenceprobablyillustratesth∞efact
evolutionary tracks and from spectroscopy (Herrero et al. 1992). thatthecorrectdeterminationofwindvelocitiesforSMCisdiffi-
SinceKK1intheiranalysisusedevolutionarymasses,wealsouse cultduetotheirlowwinddensity.
evolutionarymasseshere,butinfactformanystarsfromKK1sam- This star was independently studied by P96, who obtained
plethesemassesarenearlyequal.OurSMCsampleisnothomo- slightlylowervalueoftheeffectivetemperaturethanB03(T =
eff
geneousinthatsense,sinceforsomestarsthesemassesareequal, 40000K).
4 J. Krticˇka
Table2.StellarandwindparametersofselectedSMCstars.Stellarparameters(radiusR ,theeffectivetemperatureTeff andthemetallicityrelativetothe
∗
solarvalueZ/Z )wereadoptedeitherfromPulsetal.(1996,hereafterP96),Bouretetal.(2003,hereafterB03),Masseyetal.(2004,hereafterM04)and
Martinsetal.(20⊙04,hereafterMr04).ForthestellarmassMweassumevaluesderivedusingevolutionarytracks.ForstarswithmetallicitiesdenotedasB03
andMr04weadopteddetailedabundancedeterminationsfromB03andMr04forC,N,O,Si,SandFeandZ/Z =0.2forotherheavierelements.Observed
windparameters(i.e.themass-lossratesM˙ andthewindterminalvelocitiesv )werealsomostlytakenfrom⊙P96,B03,M04andMr04(whenavailable),
∞
howeverseethediscussionforindividualstars.Predictedvaluesofwindparameterswerederivedbyourcode.Forstarsforwhichv valueisnotgivenin
∞
thetableauthorsprovideonlylowerlimitsthatisinagreementwithourpredictedvalueofv .
∞
Star Stellarparameters MasslossratesM˙ Terminalvelocitiesv Source
∞
Sp. R M Teff Z/Z observed predicted observed predicted
[R∗] [M ] [K] ⊙ [M yr−1] [M yr−1] [kms−1] [kms−1]
⊙ ⊙ ⊙ ⊙
NGC346WB1 O4III 23.3 95 42000 0.2 4.810−6 3.510−6 2250 2240 P96
NGC346WB4 O5.5V 14.2 53 42000 0.2 ≤110−7 7.810−7 1950 2200 P96
NGC346WB6 O4V 11.2 40 41500 B03 2.710−7 3.510−7 2300 2180 B03
NGC346MPG368 O5.5V 10.6 38 40000 B03 1.510−7 1.810−7 2100 2580 B03
NGC346MPG113 O6V 7.8 33 40000 B03 310−9 4.810−8 3370 B03
N81#2 O6.5 7.9 31 40000 Mr04 .10−8 410−8 2880 Mr04
AzV75 O5III 25.4 92 40000 0.2 3.510−6 3.810−6 2100 2120 M04
N81#1 O7 10.3 34 38500 Mr04 .10−8 910−8 2530 Mr04
AzV26 O6I 27.5 86 38000 0.2 2.510−6 3.810−6 2150 1850 M04
AzV232 O7Ia 29.3 93 37500 0.2 5.510−6 4.110−6 1400 1880 P96
AzV207 O7V 11.0 33 37000 0.2 110−7 1.010−7 2000 2060 M04
N81#11 O7.5 6.9 23 37000 Mr04 .10−9 210−8 2270 Mr04
N81#3 O8.5 5.0 19 36000 Mr04 .310−9 310−9 2260 Mr04
AzV238 O9III 15.5 37 35000 0.2 1.310−7 2.810−7 1200 1830 P96
NGC346MPG487 O6.5V 10.2 25 35000 B03 310−9 3.010−8 2300 B03
AzV469 O8.5II 21.2 38 32000 0.2 1.810−6 4.010−7 2000 2010 M04
NGC346MPG12 O9.7V 10.1 21 31000 B03 110−10 1.710−8 1820 B03
AzV 232 Crowther etal. (2002) applied models withwindblan-
3000
keting to study the stellar parameters of this star and obtained
much lower value of the effective temperature than P96 (T = 2800
eff
32000K).However,wecalculatedwindparameterswiththisnew d) 2600
dsteetlelramrminaastiso,nraodfiussteallnadrpaabruanmdeatnecress(io.ef.Cth,eNe,fOfe)ctaivnedtwemepoebrtaatiunreed, serve 2400
ltoowoelorwthvaanlutheeofobthseermveadssvlaolsuse.raAtels(oM˙m=ode9l1s0o−f7VMK⊙Lyprr−ed1i)c,tmwuicthh 1] (ob 22020000
−
thisnewparametersaboutfivetimeslowermass-lossratethanthe m s 1800
observedvalue.Probably,thismaybecausedbythefactthatthis k
[ 1600
star has a very peculiar chemical composition (Z(C)/Z (C) = v¥
⊙ 1400
0.07, Z(O)/Z (O) = 0.1, whereas Z(N)/Z (N) = 2.0,
Crowtheretal.⊙2002)andabundancesofotherelem⊙entswhichare 1200
alsoimportantfordrivingofthestellarwindwerenotdetermined.
1500 2000 2500 3000
Thus,tokeepoursamplemorehomogeneous, weusedthestellar v¥ [km s−1] (calculated)
parametersderivedbyP96.
Figure1.Comparisonofcalculatedandobservedterminalvelocities.Stars
forwhichthereisnoreliableestimateoftheirterminalvelocities (infact
3.2 Comparisonofcalculatedwindparameterswith
thosestarsthatexhibitveryweakwinds)wereexcludedfromtheplot.Line
observation
denotesonetoonerelation.
In Fig. 1 we compare calculated wind terminal velocities for se-
lected stars with observed values. While for some stars there is
averygoodagreement betweenobservedandcalculatedterminal NGC346 MPG368, AzV26, AzV 232 andAzV 238 therest of
velocities, for some starsthe agreement isworse. Before wewill terminalvelocityestimationsliewithinthementioneduncertainty
discussthisprobleminanextsectionletusconclude,thatderived interval.
scatterbetweenobservedandcalculatedwindterminalvelocitiesis According tothepredictions of stellarwindtheorythewind
slightlyhigherthanthatforGalacticstars(seeKK1). terminal velocity v iscorrelated withthe surface escape veloc-
∞
Generally, it isdifficultto assessthe accuracy of determina- ityv .ForGalacticOstarsthevalueofv /v doesnotdepend
esc esc
∞
tionofwindparameters.P96statethatduetotheedgevariability on their effective temperature and has a mean value of about 2.5
andcontaminationoftheedgesofwindabsorptionlinesbyunder- (Lamerset al. 1995). Thus, we also plot the ratio of v /v for
esc
∞
lyingphotosphericlinestheterminalvelocitiesshouldberegarded studied stars (see Fig. 2 where we compare ratio of v /v de-
esc
∞
asaccurateupto 10%.Takingthisasatypicalerrorofterminal rivedusingtheoreticalandobservedvaluesofv ).Themeanvalue
± ∞
velocity determination we conclude, that with exception of stars ofv /v 2.3isslightlylowerthanthatobtainedbyKK1for
esc
∞ ∼
NLTE modelsof line-drivenstellarwinds II. 5
al.2000 andalsoSect.4.3)and alsowithuncertain valuesof the
3.5 theoretical terminalvelocities.
observed
The comparison of calculated mass-loss rates and mass-
3 loss rates derived from observations in Fig. 3 shows relatively
good agreement for stars with higher mass-loss rates (M˙ &
esc 2.5 1(K0−K71)M.⊙Thyisr−is1)p,rionbmabalnyydcuaesetosbtheettefracthtathnaftotrhsetadrisstiannocuertoGaSlMaxCy
v
/ isknownwitharelativelyhighprecisionand,thus,thebasicstel-
v¥ 2 larparametersareknownalsowithahighprecision(probablywith
exception of some systematic effects like wind blanketing effect
1.5 whichmayinfluenceparametersofbothstellargroups).However,
there is a significant disagreement between theoretical and ob-
1 served values for lower mass-loss rates (M˙ . 10−7M⊙yr−1).
30000 35000 40000 45000 Inthiscasethepredictedmass-lossratesaremorethantentimes
higher than that derived from spectral analysis of observed data.
T [K]
eff
This is not only a problem of our models, since also predictions
Figure2.Comparisonoftheratioofthewindterminalvelocityv tothe of VKL show the same behaviour (see B03; Martins et al. 2004;
∞
surfaceescapevelocityvesc derivedusingtheoreticalandobservedvalues 2005). To demonstrate this conclusion, we have added the mass-
ofv .Notethattheobservedvaluesforstarswithonlylowerlimitofthe lossratepredictionsofstudiedstarscalculatedusingVKLrecipe
∞
terminalvelocity available (missingv inTab.2)arenotplotted inthis
intoFig.3.Thepossibleoriginofthisdiscrepancywillbediscussed
∞
graph.
inSect.5.
OurresultsforpossibleyoungVzstarswiththinwinds(M˙ .
10−7M yr−1) fromMr04sampledo not show that thesystem-
⊙
10−5 aticdisagreementbetweenpredictedmass-lossratesandmass-loss
rates derived from observation is higher than for generally older
starswiththinwindsfromB03sample(seeFig.3).
10−6 For stars with very small mass-loss rates (M˙ .
d) 10−7M yr−1) only upper limitsof their terminal velocities are
ve10−7 available⊙in the literature. Predicted terminal velocities of these
ser starsareconsistentwiththeseupperlimits.Notehoweverthatthis
b
1] (o10−8 idsicntostthaacthtehcekteorfmthineamlvoedleolcsitbieescaaunsdemthaesss-tlaonsdsarradtewsianrdetrheelaotreydp–rei-f
−
yr themass-lossrateofthesestarsisreallymuchlowerthanthepre-
.M [M .°10−9 dmicutcihonhidgehreivr.edinthispaper,thentheirterminalvelocitiesmaybe
10−10 this work Themass-lossratesderivedbyVKLrecipeareslightlyhigher
thanthatofusalthoughKK1foundrelativelygoodagreementbe-
VKL
upper limit tweenthesetheoreticalrates.Thisiscausedbyadifferentboundary
10−1110−9 10−8 10−7 10−6 10−5 fluxused.Theline-blanketedfluxesfromOSTAR2002gridusedin
. this work have lower flux in the UV domain (this domain is im-
M [ M . yr−1] (calculated)
° portantfortheline-driving)thanH-HemodelsusedbyKK1,con-
sequentlyderivedmass-lossratesareslightlylower.However,the
Figure 3. Comparison of calculated mass-loss rates (either by us or by effectofdifferentboundaryfluxesonterminalvelocitiesissmall.
VKL)andmass-lossratesderivedfromobservation.Pointscorresponding
totheupperlimitswereobtained usingourmodelpredictions andupper
limitsderivedfromobservations.Linedenotesonetoonerelation.
3.3 Windmomentumluminosityrelationship
Wind momentum-luminosity relationship (see Kudritzki & Puls
Galacticstars.Ourcalculatedv /v ratioisslightlylowerthan (2000)andreferencestherein)mayprovideanindependentmethod
esc
recentobservationalfindingofE∞vansetal.(2004),whoderivedthe forthedeterminationofstellarandconsequentlyalsogalacticdis-
medianv /v =2.63forSMCstarswitheffectivetemperatures tances. However, to achieve this, detailed calibration of this re-
esc
highertha∞n24000K.SinceB03obtainedfortheir(however lim- lationship is necessary, especially with respect to the metallicity.
ited)samplev /v 2.3weconcludethatourcalculationsare Low-metallicity environment of SMC provides an ideal tool for
esc
inagreement w∞ithoth∼er studies and that our resultsmay indicate suchacalibration.
thatSMChotstarsterminalvelocitiesareslightlylowerthanthat Wecomparetheoreticalmodifiedwindmomentum-luminosity
ofGalactichotstars. relationshipM˙v R /R 1/2 obtainedforconsideredstarsus-
SimilarlytoGalacticO stars,there isa largescatter of both ing our NLTE w∞ind(cid:0) m∗ode⊙ls(cid:1)with a relationship derived from ob-
observed and theoretical v /v values (see Fig. 2). The scatter servations of these stars (see Fig. 4). We excluded observed val-
esc
∞
of theoretical values of v /v is slightly higher for SMC stars ues for stars for which only upper limits of their terminal veloc-
esc
∞
than for Galactic stars (KK1). The origin of this scatter is prob- ities were derived from observation. We conclude that there is a
ably partly connected with high sensitivity of terminal velocities relativelygoodagreementbetweencalculatedvaluesandthosede-
ondetailedwindparametersintheouterwindregions(seePulset rived from observation. Note however that the agreement for ex-
6 J. Krticˇka
and to a lesser extent by a downward revision of mass-loss rates
31
this work duetotheuseofblanketedmodelatmospheres.
S) observation
G 30 linear regression
C Galactic (KK1)
2] ( 4 INFLUENCEOFTHEUNCERTAINTIESOFSTELLAR
1/ 29
)⊙ PARAMETERSDETERMINATION
R
R/* 28 Individual stellarparameters(liketheeffectivetemperature, mass
( orradius)arenotinsomecasesdeterminedwithahighdegreeof
v¥
.g[M 27 ptarienctiiseisoonf.Tdherisiviesdesspteelclaiarllryadtiriue(ffoorrGhaoltascttaircssftoarrswdhuicehtolaurgneceurntacienr--
o
l tiesofthedeterminationoftheirdistance)andstellarmasses(see
26 Herreroetal.1992,Lanzetal.1996)mayexist.
4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2
Theuncertaintiesofdeterminationofthestellarparametersin-
log(L / L )
⊙ fluencealsothepredictedwindparameters(i.e.themass-lossrates
andtheterminalvelocities).Todescribethisindetail,weperform
Figure4.Comparisonofcalculated(crosses)andobserved(circles)mod- studyofvariationsofpredictedwindparameterswithsmallchange
ifiedwindmomentumforconsideredstars.Notethatobservedvaluesfor of stellar parameters. From the scaling of modified CAK theory
starswithunknownterminalvelocitywereexcludedfromthisplotsinceit (Kudritzkietal.1989)
ispossibletoobtainatmostonlylowerlimitstotheirmodifiedwindmo-
mentum(forstarswithknownobservedmass-lossrates).Linearregression M˙ ∼N01/α′L1/α′[M(1−Γ)]1−1/α′, (2)
ofcalculateddataofSMCandGalactic(KK1)starsisalsoplotted.
whereN0 isconnectedwithnumberoflinesthateffectivelydrive
thestellarwind,ΓistheEddingtonparameterand
Table3.Comparisonofmodifiedwindmomentum-luminosityrelationship α′ =α δ, (3)
−
(seeEq.(1))takenfromdifferentsources.SMCdenotessampleconsidered
inthispaper,eithercalculated fromtheoretical valuesorfromvalues de- whereαandδareusualCAKparameters,soα′istypicallyequal
rivedfromobservation(againexcludingtheobservedvaluesforstarswith toabout 0.5. Hotstarwindmass-lossratemostlydepends onthe
veryweakwinds). stellar luminosity L. With decreasing metallicity parameter N0
decreases causing a decrease of mass loss rate. Finally, with de-
Sample logD0 x creasingeffectivemassM(1−Γ)masslossrateincreases.Since
Eq.(2)doesnottakeintoaccountthevariationsofα′withe.g.the
Galactic(Vinketal.2000) 18.68±0.26 1.826±0.044 effective temperature, we also include the scaling of Vink et al.
Galactic(KK1) 18.7±2.3 1.83±0.40
(2000)obtainedforGalacticOBstars
SMC(theoretical) 16.6±0.2 2.08±0.04
SMC(observed) 16.6±2 2.1±0.4 M˙ L2.2M−1.3Teff(v /vesc)−1.3, (4)
∼ ∞
withresultofmetallicitydependencederivedbyVKLas
cludedstarswithunknown terminalvelocity(forstarswithM˙ < M˙ Z0.69. (5)
10−7M year−1) is very poor (using either lower limits of their ∼
⊙ FromtheclassicalCAKtheoryitisknownthattheterminalveloc-
terminalvelocitiesorterminalvelocitiesderivedfromourcalcula-
ityv dependsmostlyontheescapevelocityv (seeKudritzki&
tions), since their predicted mass-loss ratesare much higher than esc
∞
Puls(2000)forareview).
theobservedones.Ontheotherhandstandardtheorypredictsthat
Here we do not aim to fit expressions like Eq. (2) for large
terminalvelocitiesandmass-loss ratesarerelated.Insuchacase
amountofstars,howeverjusttostudyvariationsofwindparame-
the terminal velocities of these stars could be much higher and
terswithvaryingstellarparameters.
theagreement betweenpredicted andobserved windmomentum-
luminosityrelationshipcouldbeimproved.
Wehavealsocalculatedthelinearregressionofboththeoret-
4.1 Effectivetemperatureandstellarmass
ical and observed modified wind momentum-luminosity relation-
shipforconsideredstars Comparisonofmass-lossratesandterminalvelocitiesforstudied
starscalculatedwithoriginalsetofparametersandwithstellaref-
loghM˙v∞(cid:0)R∗/R⊙(cid:1)1/2i=xlog(L/L⊙)+logD0, (CGS()1) efenctltyiv,efotremhipgehreartuerfefebctyiv1e0t0e0mKperhaitguhreerthisegmivaessn-lionssFriagt.e5i.sAhipgphaerr-,
and compared it with the theoretical values derived for Galactic inagreementwithothertheoreticalpredictions(seeEqs.(2),(4)).
stars(Tab.3).First,duetorelativelygoodagreementofmass-loss Fromourcalculationsitispossibletoderivethatthemass-lossrate
ratesthereisagoodagreement betweentheoretical andobserved scaleswiththeeffectivetemperatureas
modified wind momentum-luminosity relationship for SMC stars M˙ T10.64, (6)
(exludingstarswiththinwind).Moreover,duetolowermass-loss ∼ eff
rates of SMC stars the D0 value is significantly lower than that fromwhichtheeffectiveparameter α′ = 0.38. Thisscalingisin
forGalacticstars.Alsotheslopeisslightlydifferent.Roughly5 arelativelygoodagreement withscalingofVinketal.(2000,see
6 lower modifiedwindmomentum derived inthepresent stud−y Eq.(4)),whichimpliesM˙ T9.8.Thedependenceoftheterminal
× ∼ eff
comparedtothatderivedbyKK1forGalacticstarsismostlydue velocityontheeffectivetemperature(Fig.5)ismorescattered.
to lower wind mass-loss rates caused by low metallicity of SMC Thevariationsofthemass-lossrateandtheterminalvelocity
NLTE modelsof line-drivenstellarwinds II. 7
10−5
K)
+1000 10−6 0 K) 4000
Teff,0 +100 3500
−1 yr] (T=eff⊙1100−−87 1] (T=Teffeff,0 23500000
M -s
.M [ km 2000
[
v¥
10−9 1500
10−9 10−8 10−7 10−6 10−5 1500 2000 2500 3000 3500 4000
M . [M⊙ yr−1] (Teff=Teff,0) v¥ [km s-1] (Teff=Teff,0)
Figure5.Comparisonofthemass-lossrates(leftpanel)andtheterminalvelocities(rightpanel)ofstudiedstarscalculatedwithoriginaleffectivetemperature
andeffectivetemperaturehigherby1000K.Solidlinedenotesonetoonerelationanddashedlinedenoteslinearfittotherelationbetweentwogroupsof
windparameters.
10−5
M)010−6 4000
1.05 M)0 3500
1] (M= 10−7 =1.05 3000
−M yr⊙ -1s] (M 2500
.M [ 10−8 km
[ 2000
v¥
10−9 1500
10−9 10−8 10−7 10−6 10−5 1500 2000 2500 3000 3500 4000
M . [M⊙ yr−1] (M=M0) v¥ [km s-1] (M=M0)
Figure6.Comparisonofthemass-lossrates(leftpanel)andtheterminalvelocities(rightpanel)ofstudiedstarscalculatedwithoriginalmassandmass1.05
timeshigher.Solidlinedenotesonetoonerelationanddashedlinedenoteslinearfittotherelationbetweentwogroupsofwindparameters.
withstellarmassaremorestringent(Fig.6).Withincreasingstellar theuncertaintiesofdeterminationofstellarmassandeffectivetem-
massthemasslossratedecreases,onaverage perature,thensimilareffectshouldbealsopresentinKK1,where
slightlybetteragreementbetweentheoreticalandobservedtermi-
M˙ M−1.60, (7)
∼ nalvelocitieswasderived.
withfairagreement withEqs.(2),(4)forthederivedvalueα′ =
0.38. Theterminalvelocityisproportional totheescapevelocity.
4.2 Abundances
Fromourcalculationswederivetheaveragerelation
v M0.52, (8) To study the variations of wind parameters with metallicity we
∞ ∼ recalculated wind models with the same stellar parameters as in
whichclearlyreflectsthisproportionality. Tab. 2, however with abundance of heavier elements 1.5 times
Itisclearthatsomepartofthediscrepancybetweenobserved higher.Comparisonofthemass-lossratesandtheterminalveloci-
andtheoreticalwindparametersmaybeattributedtotheuncertain- tiescalculatedwithdifferentmetallicitiesisgiveninFig.7.
tiesinthedeterminationofthestellarparameters(massandtheef- Forhighermetallicitytheradiativeforceishigherandconse-
fectivetemperature).Especiallythedownwardrevisionofthestel- quently also the mass loss rate is higher. We have found that for
larmassbythefactorof1.5 (e.g.duetothedifferencebetween studiedstarstherelation
×
spectroscopicandevolutionarymasses)wouldcausedownwardre-
M˙ Z0.67 (9)
visionoftheterminalvelocitiesbythefactorofabout1.3andthe ∼
increaseofthemass-lossrateby2 .Ontheotherhand,sincethe holds.ThisisinagoodagreementwithVKL(seeEq.(5)).
×
distancetotheSMCisknownwitharelativelyhighdegreeofpreci- Thesituationwiththevariationsoftheterminalvelocitywith
sionweconcludethatprobablyalsoanothersourceofdiscrepancy themetallicityismorecomplicated.InSect.3.2weconcludedthat
betweenobservedandpredictedwindterminalvelocitiesispresent. theratioofthewindterminalvelocitytothestellarescapeveloc-
Moreover, if higher terminal velocities scatter is given purely by ityv /v maybeslightlyhigherforGalacticstarsthanforSMC
esc
∞
8 J. Krticˇka
10−5
)MC10−6 4000
S
1.5 Z )SMC 3500
= Z
.−1M [M yr] (Z⊙1100−−87 -1km s] (Z=1.5 23500000
[ 2000
v¥
10−9 1500
10−9 10−8 10−7 10−6 10−5 1500 2000 2500 3000 3500 4000
M . [M⊙ yr−1] (Z=ZSMC) v¥ [km s-1] (Z=ZSMC)
Figure7.Comparisonofthemass-lossrates (leftpanel)andtheterminal velocities (rightpanel)ofstudiedstarscalculated withoriginal metallicity and
metallicity1.5timeshigher.Solidlinedenotesonetoonerelationanddashedlinedenoteslinearfittotherelationbetweentwogroupsofwindparameters.
O,seePauldrach1987,Vinketal.1999andalsoFigs.8and9).In
0.7
such a case, the line acceleration is very sensitive to the detailed
AzV 238
0.6 O wind structure (i.e. temperature, density and chemical composi-
tion).Partlyduetothis,largevariationsofv /v (bothobserved
esc
0.5 andtheoretical,seeLamersetal.1995and∞Pauldrachatal.1990)
occurforindividualstars.Thisisalsolikelycauseforthehighscat-
rad 0.4 Fe terbetweenterminalvelocitiesderivedusingdifferentmetallicities
ad/f 0.3 inFig.7.Thisscatteralsooccurswhenvariationsofterminalve-
rfh N locitywithtemperature(seeFig.5)arestudied.Toalesserextent
0.2 thisisalsotrueforthevariationsofstellarmass,seeFig.6.
S
C The metallicity of studied stars is not known with a high
0.1 Ar precision. For many stars we used just scaled Galactic chemical
composition,althoughsignificantdeviationsfromthisscalingexist
0
0.01 0.1 1 10 (e.g.B03;theyarealsoprobablymanifestedbyadifferentnumber
r/R -1 ratioofWNtoWCstarsintheClouds,seeMikula´sˇek1969).More-
*
over,wedonothaveanyinformationontheabundanceofmanyel-
Figure8.Theradialvariationofthecontributionoftheradiativeforcefrad ementswhichareimportantfortheradiativedrivingatall(e.g.Ne,
h
actingonelementhtothetotalradiativeforcefradforstarAzV238(plotted Ar).Evenworse,therearelargedifferencesinthechemicalcom-
aswhrad = fhrad/frad).Ironisthemostimportantelementfortheradiative positionof individual stars(e.g. B03). Toconclude, some part of
drivingclosetothestar,whileoxygendominatesintheouterregions.
higher scatterbetweenobserved andpredictedterminalvelocities
canbeattributedtopoorlyknownchemicalcompositionofmostof
stars. With varying metallicity the wind density changes, conse- thestudiedstars.Uncertaintiesofthedeterminationofotherstellar
quentlywindionizationandexcitationstatevaries,andhencealso parameters,i.e.thestellarmassandtheeffectivetemperaturemay
terminalvelocitiesofstudiedstarsvary.However,thesevariations (besidestheapproximations involvedinourcode) alsocontribute
arenotmonotonic,ascanbeseenfromFig.7.Forsomestarsthe tothisscatter.
terminalvelocitydoesnotsignificantlychangewithmetallicity,for
someofthemincreases,forsomeofthemdecreases.Onaverage,
theterminalvelocityslightlyincreaseswithincreasingmetallicity
5 MULTICOMPONENTEFFECTS
forstudiedSMCstars,
v Z0.06. (10) Stellarwindsofhotstarsareacceleratedmainlybytheabsorption
∞ ∼ ofradiationintheresonancelinesofheavierelements.Theradia-
FromcalculationsofKudritzki(2002)wecaninfermoresteeppro- tive acceleration acting on individual elements is however differ-
portionality,roughlyv Z0.12. ent,andconsequentlyindividualelementshavedifferentvelocities.
∞ ∼
Consequently,stellarwindsofhotstarshaveamulticomponentna-
ture (e.g. Springmann & Pauldrach 1992, KKII). For many stars
4.3 Sensitivityofv ontheconditionsintheouterwind
∞ the velocity differences are small, thus the wind multicomponent
Althoughonaveragethevalueofv dependsonmetallicityonly naturecanbeneglectedinthiscase.Krticˇkaetal.(2003)showed
∞
slightly,thehighscatterbetweenterminalvelocitiesderivedusing thatinthelow-metallicityenvironmentthevelocitydifferencesbe-
differentmetallicitiesinFig.7mayseemsurprising.Similareffect tweenwindcomponentsarelarger,thusmulticomponenteffectsare
washowever,inadifferentcontext,reportedalsobyotherauthors moreimportant.However,aswasnotede.g.byMr04,evenforrela-
(e.g.Pulsetal.2000).TheOstarwindsintheouterregionsareac- tivelylow-metallicityenvironmentofstellarwindofSMCstarsthe
celeratedmainlybyfewdozenlinesoflighterelements(likeC,N, multicomponenteffectscanbeneglected.
NLTE modelsof line-drivenstellarwinds II. 9
1 5000 1 5000
AzV 238, r ≈1.05R AzV 238, r ≈18.5R
* *
0.8 4000 0.8 4000
s s
0.6 3000 ne 0.6 3000 ne
radradf/fline,tot 0.4 2000 number of li radradf/fline,tot 0.4 2000 number of li
0.2 1000 0.2 1000
0 0 0 0
10-7-10-610-6-10-510-5-10-410-4-10-310-3-10-210-2-10-1 10-7-10-610-6-10-510-5-10-410-4-10-310-3-10-210-2-10-1
frad/frad frad/frad
line line
Figure9.Contribution oflines withdifferent strengths totheradiative forceatthecritical point(r ≈ 1.05R ,left panel)andintheouterwindregion
(r ≈ 18.5R ,rightpanel)forstarAzV238.Thetotalrelative contribution totheradiative forcefrad /frad∗summedoverlines withgiven individual
∗ line,tot
contributiontotheradiativeforcefrad/fradisplottedusingboxes.Solidlinedenotesnumberofspectrallineswhoserelativecontributiontotheradiativeforce
line
liesinagiveninterval.Radiativeaccelerationclosetothestarismainlyduetohundredsoflineswhoseflriande/fradliesintheinterval(10−3−10−2),whereas
theaccelerationintheouterregionsisgivenjustbyfewdozenslineswith10−2<flriande/frad<10−1.
Ontheother hand, theseconsiderations arebased ontheas- (T and T are temperatures of wind components a and b) and
a b
sumptionthattheheavierelementscanbedescribedbyonecom- lnΛistheCoulomblogarithm.TheargumentoftheChandrasekhar
ponentonly.Byotherwords,usuallyitisassumedthatatomicmass functionG(x )is
ab
ofheavierionsisthesame,theradiativeaccelerationonheavierel-
v v
ementsisthesameandconsequentlytheseelementshavethesame xab= | rbα− ra|, (14)
mean velocity. However, in reality individual elements have dif- ab
where
ferentmass,theradiativeaccelerationactingonthemisalsodiffer-
entandconsequentlytheseelementshavedifferentmeanvelocities. α2 = 2k(maTb+mbTa). (15)
Hence,multicomponenteffectsmayoccurforhigherdensitiesand ab m m
a b
metallicitiesthanpreviouslyassumed.
For low velocity differences (x . 1) the flow iswell coupled.
ab
However, for higher velocity differences (x & 1) the Chan-
ab
drasekharfunctionisdecreasingandthisbehaviourmayenabledy-
5.1 Velocitydifferencesinthestellarwind namicaldecouplingofwindcomponents(seeSpringmann&Paul-
drach1992,KKII).
Toestimatetheimportanceofmulticomponenteffectsinthestellar
Inthemomentum equationof heavier ionsEq.(11)theleft-
windofSMCstarswecalculateapproximatevelocitydifferences
hand side term and the pressure term can be neglected. Also the
betweenindividualheavierelementsh(acceleratedbythelineab-
gravitational acceleration and the electric polarisation field term
sorption) and passive wind component p (hydrogen and helium)
can be neglected. Finally, due to high number density of passive
usingthemomentumequationofindividualwindelements.Inthe
component (hydrogen and helium) compared to the number den-
caseofstationarysphericallysymmetricstellarwindthisequation
sityof heavier elementsthe only important frictional termisthat
reads(cf.Burgers1969,KKII)
betweenpassiveandothercomponents.Consequently,theapprox-
dv 1 d q imativemomentumequationofheavierelementhis
v ra =grad g a2ρ + a E+
ra dr a − − ρadr (cid:0) a a(cid:1) ma grad = 1 K G(x ). (16)
+ ρ1a Xb=aKabG(xab)|vvrrbb−−vvrraa|, (11) Thisequationstatesthhatwholρehmohmpentuhmpobtainedbyindividual
6 heavierelementsduetotheline-absorptionistransferredbyfriction
wherev andρ arethevelocityandthedensityofcomponenta
ra a tothepassivecomponent.UsingtheTaylorexpansionoftheChan-
(a=pora=h),aaistheisothermalsoundspeed,Eisthecharge drasekharfunction(appropriateforx < 1)G(x ) 2xhp the
separationelectricfield,gisthegravitationalacceleration,grad is hp hp ≈ 3√π
a relativevelocitydifferencebetweenpassivecomponentandagiven
the radiative acceleration and q and m are charge and mass of
a a heavierionhis
particlea.Frictionalparameterhasthefollowingform:
Kab=nanb4kπTqa2qb2 lnΛ, (12) xhp = |vrpα−hpvrh| ≈ghradmnph8√πq3h2kqTp2lnΛ, (17)
ab wherewehaveassumedT T T.Forourdiscussiontherel-
p h
where na and nb are number densities of individual components evantquantityisnottheradi≈ativea≈ccelerationghraditself,however
andmeantemperatureofbothcomponentsis theradiativeforcefrad =ρ grad(perunitvolume).Thus,
h h h
Tab= mamTab++mmbbTa (13) xhp ≈fhradnh1np 8√πq3h2kqTp2lnΛ. (18)
10 J. Krticˇka
Theworkdonebytheradiativeaccelerationisusedtoliftthewind Eq.(24)providesonlyaveryapproximateexpressionforthe
material from the stellar gravitational well. Thus, for stars with velocity difference, mainly due to a very simplified velocity gra-
the same mass-loss rates (and the same velocity field) the radia- dientassumedinEq.(21).Foramorereliableestimateitismuch
tiveforce(atcorrespondingradii)shallbesimilar.Thesestarswith bettertouseEq.(18).
lower metallicities have lower n and consequently higher x . Notethatthispictureofdecouplingissomehowdifferentfrom
h hp
Inreality,starswithlowermetallicitieshavealsolowermass-loss thatpresentedbyotherstudies,i.e.Springmann&Pauldrach(1992)
rates, lower n and consequently even higher x . Due to these andKKIIanddiscussed byMr04. Thesestudiesassumed that all
p hp
two effects (see also Krticˇka et al. 2003) stars with lower metal- heavier ions which are accelerated by line-transitions decouple
licitieshavehighervelocitydifferencesbetweenwindcomponents. fromthemean(passive)flow.Howevernowwediscussamorere-
Thecrucialpointis,however,thatheavierelementswhicharenot alisticdescriptionthatindividualionsdecouplefromthemeanflow
abundantinthestellarwind(i.e.n n )andwhichsignificantly separately.
h p
contributetotheradiativeacceleratio≪n(i.e.theirfradislarge)may
h
havelargerelativevelocitydifferencesx ,inmanycasesx 1.
hp hp
≈
Insuchacase,instabilityconnectedwiththedecouplingofconsid- 5.2 VelocitydifferencesinthewindsofstudiedSMCstars
eredelementmayoccur.
WeusedourNLTEwindcodetocalculateapproximativevelocity
Letusfirstroughlyestimateinwhichsituationthismayoccur.
differencesinstudiedSMCstellarwinds.Forthispurposeweap-
Let be wrad the relative contribution of agiven element h to the
h pliedEq. (18), whereboth thecontribution of a given element to
totalradiativeforcefrad,i.e.
theradiativeforcewh andchargesofwindcomponentsqh andqp
frad =wradfrad. (19) arecalculatedusingourNLTEwindcode.
h h
Calculated values of x for individual stars and individual
hp
Neglecting the gravity, the pressure term and the electric polari-
elementsaregiveninFig.10.Generally,therelativevelocitydif-
sationfieldthetotalradiativeforcecanbeapproximatedfromthe
ferences are smallest close to the star, where the stellar wind is
momentumequationEq.(11)ofpassivecomponentas
relativelydense.Asthestellarwindaccelerates,thewinddensityis
dv lowerandvelocitydifferencesarehigher.Atsomepointtheveloc-
frad ≈ρpvr drr, (20) itydifferenceshaveitsmaximumandforlargerradiidecreaseout-
wherev isthemeanwindvelocity,wemayassumev v .The wardsduetodecreasingvelocitygradient.Thisgeneralbehaviour
velocityrgradientcanbeestimatedas r ≈ rp ofrelativevelocitydifferenceswasdescribedelsewhere(KKII).In
ourcasethebehaviourismorecomplicatedmainlyduetothepro-
vrddvrr ≈ vR∞2 , (21) cesseFsoorfmioonsitzaotfiotnheanstdarrsecaonmdbfionramtioons.toftheelementstheveloc-
∗
fornhwecanwritefromtheapproximatecontinuityequation itydifferences arerather low, xhp ≪ 1,thus thereoccursno de-
couplinginthiscase.However,insomecases,especiallyinthose
M˙
n Z , (22) wherethewinddensityislow,therelativevelocitydifferencebe-
h ≈ h4πR2v mh tweenargon (or sulphur) and passive component becomes higher
∗ ∞
whereZ isthedensityofagivenelementrelativetothebulkden- andclosetoone.Thisiscausedbythefactthatargonandsulphur
h
sEiqtys.ρ(1i8n)–th(e22s)tewlleardeartmiveosphere(ρh = Zhρ).Consequently using choavnetrivbeurtyetloowthemreatdalilaitciviteyf(oZrcAer,S(MwCrad≈ 301.10)−.5T)haenvdelsoicgintyifidciafnfetlry-
Ar ≈
encesaresohigh,thattheymaycauselargefrictionalheating.
x wradv3 R mpmh 3√πkT . (23)
hp ≈ h ∞ ∗ ZhM˙ 2qh2qp2lnΛ
Inthecasewhentheheavierionsareapproximatedbyonecompo- 5.3 Multicomponentwindmodels
nentonly, wehavewrad = 1andwearriveatEq.(6)ofKrticˇka
h Totestthepossibilityoffrictionalheating,wehavecalculatedfour-
et al. (2003). However, in many cases the abundance of individ-
componentwindmodels,wherethefourthcomponentiseitherar-
ual element Z is much lower than Y (theratio of total density
h i gon, sulphur or carbon, depending on which element for a given
of heavier ”absorbing” ions and passive component in the atmo-
star has a maximum velocity difference. The other three compo-
sphere)andthus,predictedvelocitydifferencesarelarger.Eq.(23)
nentsareheavierions,passivecomponent (hydrogenandhelium)
canbealsorewritteninamoreconvenientformas
andfreeelectrons.
xhp ≈0.015whradv83R12ZhAM˙h 11zTh2z4p2, (24) reliabFliyrsptr,ewdeicttetshteedvwelhoectihtyerdtihffeerseimncpelse.eWqueaptiloonttEhqe.r(e1s8u)ltisonablylefotor
−
thecaseofstarNGC346WB1(seeFig.11),howevertheresultthat
where we have assumed mp = mH, qp = ezp, qh = ezh and
Eq. (18) is able to very reliably predict the approximate velocity
m =A m ,whereeistheelementarychargeandm isthepro-
h h H H
tonmass,andscaledquantitiesareM˙ 11 M˙/(10−11M yr−1), differencesissimilaralsoforotherstars.
v8 v /(108cms−1), R12 −R ≡/(1012cm), and⊙T4 Forstarswithhighestvelocitydifferencesthefrictionalheat-
T/(1≡04K)∞. From Eq. (24) it foll≡ows t∗hat for heavier elemen≡ts ing occurs1. This is shown in Figs. 12, 13, where we compare
(Ah 10) with low abundance (Zh 10−5) which signifi-
≈ ≈
cantly contribute to the radiative force (i.e. w 0.1) the de-
h ≈ 1 Inthecommoncasethatoccursinthestellaratmosphereandwinditis
coupling (x & 1) canoccur for relativelylargemass-lossrates
hp usuallynotnecessarytodifferentiatebetweenkinetictemperaturesofindi-
(oforder10−8M yr−1).Themass-lossrateforwhichthedecou- vidualparticlessincethesetemperaturesarenearlyequal.However,inthe
⊙
plingoccursinthewindsofSMCstarsisapproximatelyfivetimes presentcasethetemperaturesofindividualwindcomponentssignificantly
higherthanthemass-lossrateforwhichthedecouplingoccursin differ.Sincethecollisionaltermsthatenterintothestatisticalequilibrium
thewindsofGalacticstars,sinceZh,SMC 0.2Zh, . equations arecausedbythecollisions withfreeelectrons thatmovewith
≈ ⊙
