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Mon.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.hydrogen 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 selected 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 composition).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 ≈ ⊙