Multicomponent radiatively driven stellar winds

Transcription

1 A&A 369, DOI: 1.151/4-6361:21121 c ESO 21 Astronomy & Astrophyscs Multcomponnt radatvly vn stllar wnds I. Nonsothrmal thr-componnt wnd of hot B stars J. Krtčka 1,2 and J. Kubát 2 1 Kata tortcké fyzky a astrofyzky PřF MU, Kotlářská 2, Brno, Czch Rpublc 2 Astronomcký ústav, Akadm věd Čské rpublky, Ondřjov, Czch Rpublc Rcvd 12 Sptmbr 2 / Accptd 12 January 21 Abstract. W computd modls of a thr-componnt nonsothrmal radatvly vn stllar wnd for dffrnt spctral typs of hot B stars. W showd that frcton hats manly th outr parts of th wnd and th modfd tmpratur stratfcaton lads to a dcras of th outflow vlocty. Contrary to th sothrmal cas, vn th possblty of dcouplng of radatvly absorbng ons and major nonabsorbng componnt s xcludd for a ralstc form of th vng forc. Rgardlss of th actual form of th vng forc, w drvd gnral condtons undr whch dcouplng of a stllar wnd may occur. W dmonstratd that th possblty of dcouplng s closly rlatd to th functonal dpndnc of th vng forc and to th rato of th dnsts of ndvdual componnts. W dscuss svral consquncs of a multflud phnomnon n hot star wnds, partcularly mtallcty ffcts and th mplcatons of possbl hlum dcouplng on chmcally pcular stars. W propos an xplanaton of th low trmnal vlocty of τ Sco basd on frctonal hatng. Ky words. hyodynamcs stars: mass-loss stars: arly-typ stars: wnds stars: ndvdual: τ Sco 1. Introducton In fundamntal paprs, Lucy & Solomon 197 and Castor t al. 1975, hraftr CAK showd that hot stars should hav a wnd vn by radaton absorbd n spctral lns. Aftr som rfnmnts of th thory of ths wnd Abbott 1982; Frnd & Abbott 1986; Paulach t al t was possbl to xplan basc obsrvd faturs of th radatvly vn stllar wnd. Snc th formulaton of th basc thory t has bn frquntly assumd that th wnd was sothrmal to mak th problm as smpl as possbl. Consquntly, th thrmal structur of th radatvly vn wnd was studd only occasonally. Drw 1985 computd statstcal, thrmal, and onzaton qulbrum of P Cygn s wnd and latr xtndd th calculatons for many arly-typ stars Drw Sh showd that nrgy balanc of a hgh dnsty wnd s domnatd by th qulbrum btwn photoonzaton hatng and radatv coolng. For low dnsty wnds, addtonal ffcts bcom mportant, such as th so-calld Dopplr hatng and coolng ntroducd by Gayly & Owock Ths ffct s a natural consqunc of th dpndnc of th radatv forc on th vlocty gradnt va th Dopplr ffct and Snd offprnt rqusts to: J.Krtčka, -mal: krtcka@physcs.mun.cz physcally t arss from th dffrnc btwn absorpton and msson frquncy n th movng mdum. Th scond mchansm s th frctonal hatng Sprngmann & Paulach 1992 causd by nlastc collsons btwn dffrnt partcls. Th lattr ffct s also mportant for th ovrall structur of th wnd, bcaus for low-dnsty wnds, frcton may not b capabl of vng passv nonabsorbng plasma. Sprngmann & Paulach proposd that th so-calld on runaway may occur. Howvr, thy dd not solv th hyodynamc quaton tslf; thy manly ntndd to comput tmpratur structur of th wnd affctd by frctonal hatng. Thy xpland th low trmnal vlocty of τ Sco as a consqunc of dcouplng of absorbng ons. Babl 1995 computd a thr-componnt modl of th radatvly vn wnd and showd that for th cas of a man-squnc A star, frcton may lad to dcouplng of th passv plasma. H xpland th lmntal sparaton n th atmosphr of chmcally pcular stars on th bass of th multcomponnt natur of th wnd. Ths rsults stmulatd many authors to study th possbl ffcts of dcouplng, howvr, not by slf-consstnt solvng of th approprat hyodynamc quatons. Portr & Drw 1995 studd dcouplng n th outflow from B stars, Portr & Skouza 1999 usng th thory of th dcoupld wnd pontd out th possblty of th prsnc of Artcl publshd by EDP Scncs and avalabl at or

2 J. Krtčka and J. Kubát: Multcomponnt radatvly vn stllar wnds. I. 223 pulsatng shlls around stars wth low-dnsty radatvly vn wnd, and Hungr & Groot 1999 studd dcouplng of hlum n a stllar wnd. On th othr hand, Krtčka & Kubát 2, hraftr KK showd on th bass of a smpl thory an sothrmal two-componnt wnd that for lat B stars frcton dos not lad to dcouplng of th radatvly acclratd ons. Thy uncovrd a surprsng fact that frcton lads to an abrupt dcras n th outflow vlocty gradnt. Howvr, thy pontd out that for xtrmly low-dnsty wnds, on runaway may occur. Th basc shortfall of ths modl s th sothrmcty of th wnd and nglctng of lctrons. In ths papr, w ntnd to nclud both lctrons and th possbl ffct of th nonsothrmal wnd nto our modl. 2. Dscrpton of th modl 2.1. Basc assumptons W assum that th star has a statonary sphrcally symmtrc radatvly vn stllar wnd consstng of thr dal gas componnts, namly of a passv plasma hyogn ons wth mass qual to proton mass m p and charg qual to proton charg q p, of absorbng ons wth mass m = A m p and charg q, and of lctrons wth mass m and charg q. Fnally, w assum that th tmpratur s th sam for all thr componnts T = T = T p = T Contnuty and momntum quatons Each componnt of th wnd s adquatly dscrbd by th contnuty quaton d r 2 ρ a v ra = 1 and by th quaton of moton cf. Bragnskj 1963 dv ra v ra =grad a g 1 dp a ρ a + q a E + 1 m a ρ a b=,p,, b a R ab, 2 whr a =, p, and subscrpts, p, stand for mtallc ons, passv plasma, and lctrons, rspctvly. In ths quatons v ra and ρ a ar radal vlocty and dnsty of th corrspondng partcls, p a s partal prssur of ach componnt, p a = a 2 a ρ a whr sothrmal sound spd s a 2 a = kt/m a, g s gravtatonal acclraton, ga rad s radatv acclraton actng on absorbng ons and lctrons w assum no radatv forc actng on passv plasma, E s th charg sparaton lctrc fld and R ab s th frctonal forc xrtd by componnt b on componnt a. Gravtatonal acclraton actng on ach componnt of th flud has th form g = GM/r 2,whrM s th stllar mass and G s th gravtatonal constant. Th radatv acclraton actng on an lctron s gvn by th absorpton on th fr lctrons and has th form g rad = m p m GΓM r 2, 3 whr Γ= σ L 4πcGM s th rato of th actng radatv forc causd by absorpton of radaton by fr lctrons and gravtatonal forc L s th stllar lumnosty. W tak th radatv acclraton actng on absorbng ons n th form s Castor t al g rad = 1 δ α σ L Y 4πr 2 c f ρ / Wm Y dv r 1 11 cm 3 k, 4 σ v th ρ wth forc multplrs k, α, δ aftr Abbott Hr, α+1 f = 1 + σα+1 1+σµ 2 c α +11 µ 2 c σ 1 + σ α, 5 s th fnt dsk corrcton factor Frnd & Abbott 1986; Paulach t al. 1986, v th = 2kT/m p s th thrmal spd, and Y s th rato of th mtallc on dnsty to passv plasma dnsty w usd th sam valu as n KK, Y =.127. W ntroducd th lattr rato to nabl calculaton of th radatv forc not from th whol plasma dnsty as s usually adoptd n th standard CAK thory but from th absorbng on dnsty. Thr s an apparnt contradcton n usng th hyogn thrmal spd nstad of th on thrmal spd. Us of th on thrmal spd sms natural, snc only ons ar acclratd by radaton. As was found by Abbott 1982, th acclraton from a sngl ln n th sothrmal wnd s ndpndnt of th thrmal spd. Thrfor th choc of th formula for v th s compltly arbtrary, and h dcdd to choos th hyogn thrmal spd for th calculaton of th wnd paramtrs k, α, andδ. Snc w ar usng hs paramtrs for th calculaton of th ln forc, t s ncssary to us th sam thrmal spd for th calculaton of th ln forc as Abbott 1982 dd, namly th hyogn thrmal spd. Frctonal forc pr unt volum R ab actng btwn both componnts has th form Sprngmann & Paulach 1992: R ab = n a n b k ab Gx ab. 6 Th frcton coffcnt k ab s gvn by k ab = 4π ln Λq2 aq 2 b kt v ra v rb v ra v rb 7 Th Coulomb logarthm s of th form cf. Lang 1974 [ ] 3/2 24π kt ln Λ = ln n 4π 2, 8 whr n s th partcl dnsty n = n p + n + n, s th lmntary charg, Gx s so th calld Chanaskhar functon, whch s dfnd n trms of th rror functon Φx s Drcr 1959 Gx = 1 2x 2 Φx x dφx dx. 9

3 224 J. Krtčka and J. Kubát: Multcomponnt radatvly vn stllar wnds. I. Th Chanaskhar functon has th followng bhavor. For x 1sGx x, forx 1 rachs ts maxmum valu and for x 1sGx x 2. Th argumnt of th Chanaskhar functon n Eq. 6 s x ab = v ra v rb A ab, 1 v th whr A ab = A a A b / A a + A b s th rducd atomc mass. Th trm v r dv r / n th lctron momntum quaton can b nglctd compard to th lctron prssur trm and consquntly th lctron momntum quaton can b usd to dtrmn charg sparaton lctrc fld E E = g rad g + 1 R p + R 1 dp m ρ ρ 11 Elctron vlocty and dnsty ar calculatd from th condton of quas-nutralty and from th zro currnt condton n = n p + z n 12 n v r = n p v rp + z n v r Enrgy quaton W assum a nonsothrmal wnd, so th systm of contnuty and momntum quatons should b supplmntd by th nrgy hat transfr quaton whch n th cas of qual tmpratur of all componnts taks th form a=p,, 1 d r 2 r 2 u a v ra + a 2 a ρ a a=p,, = Q rad 1 2 a=p,, b=p,, b a 1 r 2 d r 2 v ra R ab v ra v rb. 14 Th ntrnal nrgy s chosn to b u a = 3/2 n a kt. Apparntly, usng contnuty Eq. 1 th hat transfr quaton can b rwrttn n a mor smpl form 3 2 k dt a=p,, v ra ρ a m a + = Q rad 1 2 a=p,, a=p,, b=p,, b a a 2 a ρ 1 a r 2 d r 2 v ra R ab v ra v rb. 15 Two trms on th lft-hand sd dscrb advcton and adabatc coolng, rspctvly. Th rght-hand sd trms stand for radatv hatng/coolng and for frctonal hatng, rspctvly Radatv nrgy trm For consstnt dtrmnaton of radatv hatng and coolng t would b ncssary to nclud all bound-fr and fr-fr transtons for all spcs consdrd and also nlastc collsons, and to solv a complt st of quatons of statstcal qulbrum as wll as th radatv transfr quaton. It would b computatonally vry costly. Th man sourcs of radatv hatng and coolng n th atmosphrs of hot stars ar th hyogn Lyman bound-fr and fr-fr transtons. Accordng to NLTE hyogn-hlum modl atmosphr calculatons Kubát 21 th contrbuton of nlastc collsons to hatng/coolng s much smallr n th outrmost parts of th atmosphr at last by a factor of 1. Not that radatv bound-bound transtons do not contrbut to th radatv hatng/coolng trm Q rad, snc thy transfr nrgy btwn radaton and ntrnal nrgy of atoms and th amount of nrgy transfrrd ctly to th onc moton s nglgbl. Thus, w dcdd to stmat th radatv hatng/coolng trm Q rad usng th formr two mchansms only hyogn Lyman bound-fr and frfr transtons. Th dtald form of hatng and coolng n th abov-mntond transtons s s Kubát t al Q H ff =4πn N H + Q C ff =4πn N H + Q H bf Q C bf =4π l =4π l α ff ν, T J ν dν, α ff ν, T J ν + 2hν3 c 2 16a hν/kt dν, 16b n l α bf,l νj ν 1 ν l dν, 16c ν n l α bf,l ν J ν + 2hν3 c 2 hν/kt 1 ν l ν dν, and th total nrgy transfrd va radaton s 16d Q rad = Q H ff + Q H bf Q C ff Q C bf. 17 Fr-fr and photoonzaton cross-sctons ar takn from Mhalas 1978, J ν s consdrd to b constant throughout th wnd and t s takn as mrgnt radaton from a sphrcally symmtrc statc hyogn modl atmosphr for a corrspondng stllar typ Kubát Crtcal ponts Gnrally, crtcal ponts ar ponts whr drvatvs of ndvdual varabls cannot b dtrmnd ctly from quatons. A thr-componnt stllar wnd gnrally should hav thr crtcal ponts. Howvr, nglctng th trm contanng th drvatv of th lctron vlocty n th lctron quaton of moton smplfs th problm, lavng only two crtcal ponts. Blow w wrt th quatons of moton and nrgy n a smplfd form. W nclud all trms wthout drvatvs nto th trms F = a for th quatons of moton, =3

4 J. Krtčka and J. Kubát: Multcomponnt radatvly vn stllar wnds. I. 225 for th nrgy quaton. Th quatons of moton thn rad v ra dv ra = a2 a dn a n a da2 a z a 2 a dn a n z da 2 a a + ga rad + F a r, v ra,n a,v rb,n b,t, 18 whr z a = q a /. Smlarly, th nrgy quaton can b rwrttn as 1 T dt = 2 3 dv rb n b b + F 3 v ra,n a,v rb,n b,t. 19 n b v rb b Exprssng drvatvs of tmpratur, dnsts of ach componnt and lctron vlocty wth th hlp of drvatvs of passv plasma and th vlocty of absorbng ons, w arrv at a matrx form of th quatons, namly D y = b, 2 whr y s th column vctor, dfnd as y = 1 dv rp v rp 1 dv r v r. 21 Th column vctor b dos not dpnd on any of th drvatvs and ncluds th trms F from Eqs. 18 and 19. Dagonal lmnts th of matrx D ar g rad a 2 D aa = v ra v ra dv ra / a 2 a 1+n a n z2 a z a n v ra + z a v r 1 + z b n b v 22 rb b whr a stands for or p, and off-dagonal lmnts rad D ac = a 2 n c a n z cz a z a n v rc + z c v r 1 + z b n b v, 23 rb b whr a, c stands for or p. In ths quatons w accountd for quasnutralty and zro-currnt condtons Eqs. 12, 13. At sngular ponts, th dtrmnant of matrx D quals zro. Gnral calculaton of sngular ponts for arbtrary valus of numbr dnsts would b rathr complcatd. On should look for two roots of an quaton D pp D D p D p =. 24 Thus, w confn our dscusson to a ralstc cas, whn th onc numbr dnsty s much lowr than th passv plasma numbr dnsty. In ths cas, th off-dagonal trms D p can b nglctd, and Eq. 24 can b splt nto two quatons that dtrmn th crtcal ponts, namly D pp = 25a D =. 25b Not that such sparaton of two crtcal ponts s possbl not only n th standard cas n n p, but also n th nonralstc cas, n n p. Howvr, whn th wnd s not dcoupld.. v r v rp, th scond condton 25b s not mt anywhr n th wnd, bcaus th non-zro trm contanng th drvatv of th radatv forc s substantally hghr than th othr trms s also KK. Thrfor, dffrnt condton must b usd to fx th mass-loss rat. Such condton can b drvd from momntum quatons 2 by multplyng th quatons for absorbng ons and passv plasma by th corrspondng dnsty and summng thm agan, only trms ncludng drvatvs of varabls ar xplctly wrttn, th rst s ncludd n th trm F, namly ρ p v rp a2 p dvrp v rp + ρ v r a2 v r a 2 p ρ + 1 T ρp a 2 p + ρ a 2 dt + ρ p ρ d dvr ρ g rad a 2 ρ ρ d + z ρ = F 4 r, ρ a,v ra,t. 26 Usng th quasnutralty Eq. 12 and rcallng Eq. 19, w obtan ρ p v rp a2 p dvrp v rp + ρ v r a2 dvr v r ρ g rad a 2 ρ p p ρ + z a T ρ m ρ 2ρp a 2 p +1+z ρ a 2 dt ρ p dv rp m p v rp + z m ρ dv r m v r = F 5 r, ρ a,v ra,t. 27 Fnally, f th condton ρ p v rp a2 p dvrp v rp + ρ v r a2 grad dvr v r dv r / + 1 2ρp a 2 p T +1+z ρ a 2 dt a 2 ρ p p + z a 2 ρ ρ ρ m ρ p dv rp m p v rp +z m ρ dv r =28 v r holds, thn th total quaton of moton 26 dos not dpnd on th drvatvs of any varabls. Ths condton s usd to fx th mass-loss rat s Sct Th fact that th onc crtcal pont s not mt anywhr n th wnd s vry ntrstng, partcularly n conncton wth th so-calld Abbott wavs Abbott 198. At frst glanc, on can conclud that such purly onc Abbott wavs can dssmnat from any pont n th wnd upstram of th flow. Howvr, our prlmnary smpl m

5 226 J. Krtčka and J. Kubát: Multcomponnt radatvly vn stllar wnds. I. lnar stablty analyss of th two-componnt cas ndcats that such purly onc Abbott wavs ar havly dampd whn th corrspondng Chanaskhar functon dos not rach ts maxmum valu. Thus, smlar to th on-componnt cas, th last pont n th wnd from whch upstram wavs can propagat s that dfnd by Eq. 28. Th dtald stablty analyss wll b prsntd n our nxt papr Boundary condtons Boundary condton for tmpratur W assum that at th nnr boundary, radatv qulbrum holds, thrfor th boundary condton for tmpratur s Q rad R = Boundary condton for vlocty To avod problms concrnng two crtcal ponts, w start to calculat our modls at th passv plasma crtcal pont; consquntly, th boundary condton for passv plasma vlocty s s Eq. 22, 25a 2 v rp R =a 2 p 1+n p n z2 p z p n vrp + z p v r 1 + z b n b v. rb b 3 Th boundary condton for absorbng ons can b drvd from th passv plasma quaton of moton 2, whch at th passv plasma crtcal pont may b rwrttn as R p ρ p + = GM r 2 1 n p n Rp ρ p 1 Γ+ m m p + m R m p ρ 4a2 p r, 31 whr w usd th boundary condton for tmpratur 29 and nglctd frctonal hatng. For stars wth solar abundancs, w may furthr nglct th numbr dnsty of absorbng ons, n n, and takng nto account that th lctron and passv plasma numbr dnsts ar narly qual n p n w wrt th boundary condton for absorbng ons n a smplr form, R p ρ p + m R = 4a2 p m p ρ r + GM r 2 1 Γ, 32 whch was usd n our modls. Furthr smplfcaton of th boundary condton for on vlocty may b obtand by takng nto account th fact that th ndvdual componnts ar fully coupld at th bas of th wnd, as s th cas n our modls. Th boundary condton thn taks th much smplr form v rp R =v r R. 33 Thus, our rsults do not chang sgnfcantly f w suppos that vlocts of all componnts ar qual at th bas of th wnd. W ncorporatd th possblty of dffrnt boundary vlocts only for th sak of physcal consstnc of our modls Boundary condton for dnsty Provdng that at som pont dp n th atmosphr wth radus r at <R, both passv plasma and absorbng on vlocts ar narly qual, thn ρ r at =Y ρ p r at. 34 Fnally, usng th contnuty Eq. 1, th boundary condton for th passv plasma dnsty may b wrttn n th form ρ p R = 1 ρ R v rr Y v rp R 35 Th boundary valu of onc dnsty s chosn n ordr to obtan th thr-componnt CAK soluton s Sct Boundary condtons for lctrons Th charg consrvaton Eq. 12 and zro currnt condton 13 can b usd ctly as boundary condtons for lctron dnsty and lctron vlocty, rspctvly Numrcal mthod Thr xst svral numrcal mthods that may b sutabl for ths problm. As an xampl, mthods basd on numrcal smulatons ar frquntly appld. Th systm s smply allowd to dvlop untl t rachs a statonary stat. Such a mthod was usd for th nvstgaton of stllar wnd structur, partcularly n 2D,.g. by Owock t al and Ptrnz & Puls 2. Anothr mthod ntgral mthod was dvlopd by Butlr 1979 and was appld to a solar wnd problm by Bürg Th Hnyy mthod Hnyy t al was usd to solv quatons of th radatvly vn stllar wnd by Nobl & Turolla Ths mthod a modfd Nwton- Raphson mthod. Th hyodynamc quatons ar lnarzd and also th CAK condton s ncludd nto th st of lnarzd quatons. Th lattr mthod was slctd by us. Equatons, whch w solv.. Eqs. 1, 2, 12, 13, 15 togthr wth th approprat boundary condtons Eqs. 12, 13, 29, 32, 35 may formally b wrttn as P ψ =, 36 whr ψ s a column vctor of varabls NR s th numbr of grd ponts, ψ =ρ 1,v r1,ρ p1,v rp1,ρ 1,v r1,t 1, ρ 2,v r2,ρ p2,v rp2,ρ 2,v r2,t 2,..., ρ NR,v rnr,ρ pnr,v rpnr,ρ NR,v rnr,t NR 37

6 J. Krtčka and J. Kubát: Multcomponnt radatvly vn stllar wnds. I. 227 and P s a non-lnar oprator. Th Nwton-Raphson mthod adoptd hr mploys th tratv schm Ĵ n δψ n+1 = P n ψ n, 38 whr ψ n s soluton n th nth traton and Ĵ n s th Jacoban n th nth traton, Ĵ n kl = P k ψ l To solv th systm of Eqs. 38 for δψ n+1 w usd th numrcal packag LAPACK, dvlopd at th Unvrsty of Tnnss. Although th orgnal Hnyy mthod handls crtcal ponts wll Nobl & Turolla 1988, du to som numrcal problms w dcdd to modfy th mthod as dscrbd n KK. Hr w only brfly summarz our approach. For ach modl w sarch for th boundary dnsty ρ = ρ p R that nabls us to pass smoothly through th pont smlar to th CAK pont dfnd by th Eq. 28. In ach stp w fx th valu of ρ, prform svral Nwton- Raphson tratons dscrbd abov and aftr nspcton of th obtand rsults w modfy ts valu n th followng mannr. If condton 28 s not mt, w ncras th valu of ρ, whras n th oppost cas or vn f th modl dos not convrg w dcras th valu of ρ.th stps ar rpatd untl a satsfactory valu of ρ s obtand.. th rror of our stmat of ρ s 1%. Not that ths a bt complcatd procdur affcts only th rsultng mass-loss rat and dos not chang th ovrall pctur of th downstram flow On-componnt modls For comparson purposs, w also calculatd oncomponnt modls of a stllar wnd. Thy ar dscrbd by contnuty Eq. 1, quaton of moton 2, and nrgy Eq. 15, whr ρ a and v ra ar rplacd by dnsty and vlocty of th whol flud ρ, v r, rspctvly. Frcton trms and lctrc charg sparaton lctrc fld ar not accountd for. Th on-componnt quatons ar d r 2 ρv r = 4a dv r v r grad + g + 2 d a 2 p ρ = 4b ρ 3k dt v ρ r +2a 2 p m ρ 1 d r 2 p r 2 v r Q rad =, 4c whr th radatv forc s compard to thr-componnt notaton g rad r, ρ, v r, dv r rad = Y g r, Y 1 ρ,v r, dv r CAK crtcal pont B B Fg. 1. Uppr panl: on-componnt dottd ln and thrcomponnt radatvly acclratd ons dashd ln, passv plasma and lctrons full ln radatvly vn stllar wnd modl of a B star. Notc that both curvs concd. In addton, th locaton of th CAK crtcal pont for both wnd modls s almost th sam and t s gvn by condton 28. Lowr panl: comparson of tmpratur stratfcaton of oncomponnt dottd ln and thr-componnt sold ln radatvly vn stllar wnd modls of a B star systm of quatons th wll known CAK crtcal pont n th form v r 1 a 2 p grad = v r dv r / To solv ths st of quatons, w usd th sam mthod as dscrbd n Sct. 2.6, howvr, wth ssntally lowr numbr of varabls and quatons. 3. Rsults of calculatons Insrtng drvatvs of dnsty from contnuty Eq. 4a and tmpratur from nrgy Eq. 4c nto th quaton of moton 4b w obtan th crtcal pont of th abov Paramtrs of modl stars ar gvn n Tabl 1. Man squnc stllar paramtrs ar takn from Harmanc 1988 and forc multplrs ar adoptd from Abbott 1982.

7 228 J. Krtčka and J. Kubát: Multcomponnt radatvly vn stllar wnds. I B2 CAK crtcal pont CAK crtcal pont B B B Fg. 2. Th sam as Fg. 1 for a B2 star Tabl 1. Adoptd paramtrs of modl stars. M s th stllar mass n unts of solar mass, R s th stllar radus n unts of solar radus, T ff s th star s ffctv tmpratur, k, α, and δ ar radatv forc multplrs, q /q p s th onc charg and A s th atomc numbr of a radatvly vn on Stllar Stllar paramtrs Wnd paramtrs Avrag on typ M R T ff star [M ] [R ] [K] k α δ A q /q p B B B B B B τ Sco Stllar wnds wth no sgnfcant ffct of frcton For stars wth hgh dnsty wnds whr th ft vlocty btwn componnts s low compard to th thrmal spd, no sgnfcant dffrncs btwn on-componnt and thr-componnt wnds occur. Ths bhavor s shown n Fgs. 1 and 2 for th cas of B and B2 stars, rspctvly. Th tmpratur profl of such wnd s controlld Fg. 3. Th sam as Fg. 1 for a B3 star. Notc th dffrncs both n vlocts and n tmpratur structur causd by frctonal hatng s th txt manly by qulbrum btwn radatv hatng and radatv coolng. Smlar modls hav bn prvously dscrbd by othr authors Sprngmann & Paulach 1992, modl of ζ Pup thrn, KK, modl of ɛ Or thrn, howvr, usng smplr assumptons Dcras of th outflow vlocty du to frctonal hatng For stars wth lowr dnsty, frcton bgns to play an mportant rol s Fgs. 3 and 5 for th cas of B3 and B4 stars, rspctvly. To mantan th common flow, a suffcnt amount of momntum must b transfrrd to th passv componnt. Nar th stllar photosphr, th wnd s rlatvly dns. From nspcton of th frctonal forc Eq. 6, t follows that th ft vlocty btwn absorbng ons and passv plasma can b low compard to th thrmal spd of hyogn. Consquntly, frctonal hatng can b nglctd n ths rgon and th tmpratur structur s st manly by th radatv procsss and adabatc coolng s Drw 1989 for mor dtald

8 J. Krtčka and J. Kubát: Multcomponnt radatvly vn stllar wnds. I B4 2 1 CAK crtcal pont B Fg. 4. Radatv hatng or coolng dashd ln and frctonal hatng sold ln rlatv to th adabatc coolng n th modl of a B3 star wnd. Not that n th rang 2 R 6 R th wnd s slghtly radatvly coold s th lowr panl for dtald plot of radatv coolng n ths rgon calculatons. Hnc, th thr-componnt tmpratur s qual to ts on-componnt valu at th bas of th wnd. Howvr, ths s not vald throughout th wnd. At th outr parts of th wnd th dnsty dcrass. Snc th frctonal forc dpnds on th product of dnsts, th ft vlocty should ncras to acclrat th passv plasma cf. Fg. 6 for th quantty x p. Consquntly, frctonal hatng ncrass, but t also dpnds on th product of th dnsts of th ntractng partcls. Snc th dnsty of partcls dcrass, frctonal hatng rachs ts maxmum valu and also dcrass n th outrmost parts of th wnd. Thrfor, at th outr parts of th wnd th tmpratur s gvn by th balanc btwn radatv hatng and coolng and th thr-componnt tmpratur s agan qual to ts on-componnt valu. Fnally, not that bcaus x p > 1, th maxmum of th Chanaskhar functon s not mt anywhr n th wnd. Thus, dcouplng of radatvly acclratd ons and major passv componnt s xcludd n such modls Fg. 5. Th sam as Fg. 1 for a B4 star. Hatng s vn gratr than for a B3 star cf. Fg. 3 x p Fg. 6. Th dpndnc of x p s Eq. 1 on radus for a B4 star Th bhavor of ndvdual hatng or coolng trms s shown n Fg. 4. Not that n a partcular cas of a B3 star, radatv hatng of a wnd changs to radatv coolng n th doman of th wnd whr frctonal hatng s mportant. Th most mportant componnt n frctonal

9 23 J. Krtčka and J. Kubát: Multcomponnt radatvly vn stllar wnds. I CAK crtcal pont B Fg. 7. Th dpndnc of vlocts on radus for a B5 star. Th manng of partcular lns s sam as n Fg. 1 hatng s causd by ncountrs of non-absorbng and absorbng ons. Encountrs of lctrons and absorbng ons supply about two magntuds lss to frctonal hatng. Fnally, frctonal hatng orgnatng from ncountrs of non-absorbng ons and lctrons s nglgbl du to th low ft vlocty btwn ths spcs. Bcaus th charg sparaton lctrc fld s not a domnant trm n momntum quatons of both passv and actv componnts, th lctron componnt s only mportant n th tmpratur quaton. Thus, th only ffct of th lctron componnt s a slght ncras of tmpratur n th rgon whr frctonal hatng s mportant. In most of our modls, th lctron vlocty s vry smlar to th passv plasma vlocty. Th vlocty profl s affctd by changs n th tmpratur structur through th dpndnc of radatv acclraton on th thrmal spd s Eq. 4. Th hghr th tmpratur, th gratr th thrmal spd and th lowr th radatv acclraton. Consquntly, th wnd trmnal vlocty dcrass. Smlarly Vnk t al concludd that lowrng th radatv acclraton n th suprsonc rgon lads to th lowrng of th outflow vlocty. To our knowldg, th ffct of lowrng of th outflow vlocty by frctonal hatng s not mntond anywhr n th ltratur Hyogn backfallng to th stllar surfac At th lowst dnsts, th onc componnt of th wnd s unabl to ag th passv plasma componnt out of th atmosphr, as dsplayd n Fg. 7. W s that th onc and passv plasma dcoupl wll blow th pont whr th scap vlocty s rachd and th passv plasma falls back onto th stllar surfac. Howvr, ths raccrton should b studd usng tm-dpndnt calculatons cf. Portr & Skouza Morovr, for such low dnsts, a dffrnt typ of soluton s vald. As was shown by Babl 1996, th hyostatc soluton for passv plasma and th wnd soluton for absorbng ons xst smultanously Fg. 8. Comparson of thr-componnt modls wth dashd ln and wthout sold ln Dopplr hatng/coolng. Only onc vlocty s shown for both modls Ths typ of soluton was also studd for th cas of a twocomponnt wnd KK, modl of B5 star wth q =1.1 q p. Only n ths cas dos th rsultng mass-loss rat of th thr-componnt modl dffr sgnfcantly from ts on-componnt valu. Nar th CAK pont, tmpratur s hghr almost by a factor of two than n th on-componnt cas. As wll b dscussd n Sct. 4.3, a hghr tmpratur blow th CAK pont lowrs th massloss rat. In our partcular cas, th mass-loss rat computd n th thr-componnt cas s M yr 1, contrary to ts on-componnt valu M yr 1. Morovr, a lowr tmpratur blow th CAK pont lowrs th flow vlocty and for ths cas th dpndnc of vlocty on radus n on-componnt and thr-componnt modls dffr sgnfcantly vn blow th crtcal pont. Ths ffct was nabld by ncludng th varatons n tmpratur and t s compltly mssng for sothrmal wnds n a two-componnt modl s KK and modl of B5 star wth q =1.1 q p thrn.

10 J. Krtčka and J. Kubát: Multcomponnt radatvly vn stllar wnds. I Th ffct of Dopplr hatng and coolng Howvr, thr s anothr mchansm that s abl to affct th thrmal qulbrum of th wnd, namly th Dopplr hatng and coolng. Ths mchansm was studd n dtal by Gayly & Owock Dopplr hatng/coolng transfrs nrgy ctly from radaton to th moton of th atoms wth th hlp of a dffrnc btwn th absorpton and msson frquncy n a movng mdum wth a vlocty gradnt wthn th sam ln. Incluson of ths ffct rqurs a dtald knowldg of th ln lst and, consquntly, a hug amount of computr tm, so w calculatd only approxmat valus of th Dopplr hatng/coolng basd on stmats from Gayly & Owock 1994 n ordr to qualtatvly stmat ts ffct. W addd th Dopplr hatng/coolng trm Q dh nto th radatv hatng/coolng trm Q rad.forq dh w usd th formula Q dh = ρ g rad 2kT m w w dff 42 gvn by Gayly & Owock 1994, Eq. 26 thrn, whr w s a hatng componnt of ct radaton and w dff s a hatng componnt of dffus radaton Not that thr hatng/coolng trm q dh s dfnd pr unt mass whras Q dh s dfnd pr unt volum, so w nsrtd th trm ρ. For th dffrnc w w dff w usd th rough stmat suggstd by th sam authors w w dff W computd a B3 star modl wth ncluson of Dopplr hatng/coolng s Fg. 8. Th ffct of Dopplr hatng/coolng s comparabl to th ffct of frctonal hatng and th total ffct approxmatly doubls our tmpratur maxmum. Howvr, th qualtatv pctur of our rsults dos not chang. Dtald ncluson of Dopplr hatng/coolng as wll as mprovd tratmnt of th ln lst wll b dalt wthn our nxt paprs. 4. Gnral consquncs of a multflud approach 4.1. Dpndnc of forc multplrs on mtallcty From th dpndnc of th radatv forc n th multcomponnt approach Eq. 4 on th mtallc dnsty w can ctly drv th dpndnc of th forc-multplrs k, α on th mtallcty. If th mtallcty changs, thn th actual mtallc dnsty s modfd as ρ = zρ, 44 whr ρ s th corrspondng solar valu of th on dnsty and z s th actual abundanc rlatv to ts solar valu. Consquntly, th radatv acclraton actng on absorbng ons vars accordng to s Eq. 4 g rad = z α rad g. 45 Snc th radatv forc actng on th flud n th oncomponnt approxmaton s gvn by ρ g rad, mtallcty ffct can b dscrbd by th dpndnc of th forcmultplr k on th mtallcty as k = z 1 α k, 46 whch s th sam rsult as obtand by Puls t al. 2, who usd th ln statstcs approach. Ths scalng law was also dscussd by Abbott Possbl dcouplng of ndvdual componnts Consdr a stuaton n whch th flow conssts of two spcs, on of whch s actv and rcvs momntum say, va radatv acclraton and th scond on s passv and s aggd va frcton. Lt us chang th dnsty n th wnd.g. changng th boundary condton. For hgh dnsts, th frctonal forc s abl to transfr suffcnt amount of momntum from th actv to th passv componnt. If th dnsty s lowr, thn, bcaus th vlocty law s narly th sam, th frctonal acclraton should b th sam too. Howvr, th corrspondng dnsty of th actv componnt s now lowr; thrfor, to mantan th sam frctonal acclraton, th ft vlocty btwn both componnts should b hghr s Eqs. 6, 7. Apparntly, ths pctur has ts lmtatons. If th maxmum of th Chanaskhar functon s rachd, th frctonal acclraton s not abl to mantan th twocomponnt flow. Th bhavor of th vlocty law out of ths pont dpnds gnrally on th dnsts of both componnts and on th forc actng on actv componnt. Ths wll b dscussd for two basc dffrnt cass blow Th major actv componnt If th dnsty of th actv componnt s sgnfcantly hghr than th dnsty of th passv componnt, thn dcouplng of th passv componnt from th major flow s possbl. Ths can b sn asly from th quaton of moton of th actv componnt Eq. 2. Th frcton trm can b nglctd bcaus th passv plasma dnsty s substantally lowr than th actv plasma dnsty and thrfor th passv componnt dos not nflunc th actv componnt. Contrary, lowrng of th frctonal trm n th quaton of moton of a passv componnt should b balancd by lowrng of th passv plasma vlocty gradnt,.. dcouplng s possbl. Clarly, f th passv componnt rachs th scap vlocty, thn t lavs th star sparatly; f ts vlocty s lowr than th actual scap vlocty thn th passv componnt falls back onto th star. Intrstngly, th pont whr th mnor componnt dcoupls from th man flow dos not dpnd on ts dnsty. Ths can b sn asly from th quaton of moton 2 of th mnor componnt. Th only trm whch dpnds on th mnor componnt dnsty s th gas prssur trm, whch can b nglctd. Thus, f th wnd s not sgnfcantly hatd durng dcouplng, th pont whr th mnor componnt dcoupls dos not dpnd on ts dnsty.

11 232 J. Krtčka and J. Kubát: Multcomponnt radatvly vn stllar wnds. I suffcnt amount of momntum to th passv plasma. Quanttatvly, th frctonal forc s lowr by R compard to th cas whn t s capabl of mantanng th common flow of both componnts. Thrfor, th vlocty gradnt of th passv componnt p should op by dv rp /. Th quaton of moton ylds v rp a p 2 dv rp v rp = R, ρ p whr th contnuty quaton was usd. Ths dcras of th frctonal forc also affcts th actv componnt. For th chang n th vlocty gradnt of actv plasma dv r / w can wrt from th quaton of moton of th actv plasma v r a 2 v r dv r = g R R, ρ whr g R s th chang of forc that vs actv plasma. Such a pctur s vald for both sothrmal and nonsothrmal wnd. Howvr, spcally n th lattr cas, t should b justfd by numrcal calculatons. Gnrally, thr ar two possbl cass of downstram flow, basd on whthr th vng forc dpnds on th vlocty gradnt or not. Constant vng forc. Frst, w assum that th vng forc g R dos not dpnd on th vlocty gradnt dv r /. Thn th vlocty gradnt should chang substantally, gvn that n ths rgon v r v rp and th chang of th actv componnt vlocty gradnt rads Fg. 9. Wnd modl wth major actv componnt dashd ln allowng for dcouplng of th passv componnt sold ln Th locaton of ths pont dpnds on th mass and charg of both componnts and on th major componnt dnsty. Howvr, n ths cas, w compltly nglctd th ffct of frctonal hatng. Thrfor, th stuaton dscrbd may b gnrally mor complcatd, spcally f th forc actng on th major componnt dpnds on tmpratur as th ln-forc actually dos. To manfst such thortcal rsults w computd a wnd modl of a B3 star usng paramtrs lstd n th Tabl 1 wth an artfcally nhancd on-tohyogn rato of Y = 1. Th numrcal modl obtand n Fg. 9 supports prcdng thortcal consdratons. Both componnts dcoupl. Morovr, th wnd s sgnfcantly hatd, attanng a maxmum tmpratur of about 23 K Th mnor actv componnt Th stuaton s compltly dffrnt n th cas whn th componnt rcvng th major amount of momntum s a mnor lmnt rlatv to th passv plasma. Ths s th common cas n th radatvly vn stllar wnd. At th pont whr th Chanaskhar functon rachs ts maxmum, frctonal acclraton s not capabl of transfrng dv r = ρ dv r ρ p In ths cas, dcouplng s possbl. To prov ths concluson, w computd nonsothrmal modls wth constant acclraton.. acclraton whch dos not dpnd on th vlocty gradnt. W slctd a modl of a B3 star and as th vng acclraton w chos th radatv forc n th CAK approxmaton computd for th bta-vlocty law wth β =.45 and v = 11 km s 1. Ths computd forc w hld fxd durng Nwton-Raphson tratv stps. For hgh valus of q for whch th Chanaskhar functon dos not rach ts maxmum, no dcouplng occurs. On th othr hand, for low valus of onc charg q for whch frcton s not capabl of vng th passv plasma flow, runaway of ons occurs, nabld by stp ncras n th onc vlocty gradnt, as was alrady dscussd s Fg. 1. Fnally, th frcton hats th flow, ncrasng th wnd tmpratur by of ordrs of magntud. Ths offrs a plausbl modl of how wnd tmpraturs suffcnt to produc X-rays can b attand. Drvng forc ncrasng wth th vlocty gradnt. On th othr hand, f th vng forc s an ncrasng functon of th vlocty gradnt, thn an abrupt ncras of th vlocty gradnt of th actv componnt contrary to th prvous cas nhancs th vng forc and furthr dsrupts th momntum balanc. Thrfor, n ths cas, th

12 J. Krtčka and J. Kubát: Multcomponnt radatvly vn stllar wnds. I ncrass wnd dnsty such that th Chanaskhar functon dos not rach ts maxmum valu andthusvnth possblty of dcouplng s xcludd s Sct Fg. 1. Modl of a B3 star wnd wth constant vng acclraton for q =.6q p s th txt. Th notaton s th sam as n Fg. 1 momntum balanc would not b achvd by an ncras of th vlocty gradnt, but rathr by a dcras. Th vlocty gradnt should op accordng to v r a 2 g R dv r v r dv r / = R ρ If th frst two lft-hand sd trms can b nglctd as n th cas of th radatv forc n th Sobolv approxmaton, s KK, th abov quaton can b rwrttn g R g R dv r / dv r = R ρ In th cas of th Sobolv approxmaton ths mans that th actv componnt vlocty gradnt ops and dcouplng s not possbl. Thus, n th suprsonc flow, th possblty of on runaway crucally dpnds on th vng forc. Howvr, ths concluson cannot b provd n th nonsothrmal cas, du to th dpndnc of radatv acclraton on tmpratur. Whn th vlocty dffrnc btwn actv and passv componnts rss, th wnd s sgnfcantly hatd by frcton and th radatv acclraton s lowr. Ths dcrass th outflow vlocty and Narly qual dnsts of both componnts Th most complcatd stuaton occurs whn th dnsts of both actv and passv componnts ar narly qual. In ths cas, th abov-mntond quatons cannot b usd n such a smpl form bcaus th possblty of dcouplng crucally dpnds on th functonal dpndnc of th vng forc,.. dpndnc on th vlocty gradnt, tmpratur tc. Th quston of whthr dcouplng s possbl or not should thus b solvd only usng carful numrcal computatons. Howvr, th lattr cas probably has lmtd astrophyscal rlvanc, bcaus hyogn s th most abundant lmnt n th majorty of stars. On th othr hand, thr xsts an ntrstng class of hyogn-dfcnt stars whr th cas of narly qual dnsts may occur. Howvr, such analyss, although ntrstng, gos far byond th scop of ths papr Mass-loss rat Snc condton 28, usd to fx th mass-loss rat n th cas whn both componnts ar coupld.. v rp v r and dv rp / dv r / s narly th sam as th CAK condton 41, th thr-componnt mass loss rat and th on-componnt mass loss rat showd n th Tabl 2 n ths lmt ar narly th sam. Morovr, a condton 28 lads to a maxmum mass-loss rat, for whch a stabl soluton xsts, smlar to th on-componnt cas s Po t al Tabl 2. Mass-loss rats of on-componnt 1C and thrcomponnt 3C modls. All valus ar M yr 1 Star B B1 B2 B3 B4 B5 τ Sco 1C C Ths s not tru n th cas whn th gas s sgnfcantly hatd by frcton nar th CAK pont. Hghr tmpratur ncrass thrmal spd and thrfor dcrass radatv acclraton n th Sobolv approxmaton Eq. 4. As was dscussd by Vnk t al. 1999, as th ln acclraton n th subsonc rgon dcrass, mass-loss rat dcrass. It s ntrstng to dscuss th nflunc of hatng on th obsrvd mass-loss rats. Rcnt dtald thortcal study of mass-loss rats of Vnk t al. 2 showd that thr s a qut good agrmnt btwn thortcal massloss rats for O stars and mass-loss rats dducd from rado msson and Hα profl. On th othr hand, thortcal mass-loss rats for B stars ar n good agrmnt wth mass-loss rats calculatd from rado msson

13 234 J. Krtčka and J. Kubát: Multcomponnt radatvly vn stllar wnds. I Fg. 11. Dcouplng of hlum dottd ln from common hyogn sold ln and absorbng on dashd ln flow n th modl of a B3 star four-componnt wnd. In th outrmost parts of th wnd th lctron vlocty dashd dottd ln dffrs slghtly from th hyogn vlocty Scudr t al but not wth mass-loss rats dducd from Hα profls Kutzk t al Snc th rado msson orgnats at svral hund stllar rad Lamrs & Lthrr 1993 whr th radatv qulbrum holds s Fg. 5, th rado mass-loss rats ar not affctd by frctonal hatng. Howvr, th Hα ln orgnats from layrs closr to th star and thrfor can b affctd by frctonal hatng. Thus, propr ncluson of th frctonal and possbly Dopplr hatng can hlp undrstandng of th dscrpancy btwn mass-loss dtrmnd by varous mthods for B stars. 5. Applcatons of thr-componnt modl 5.1. Dcouplng of hlum Th ntrstng papr of Hungr & Groot 1999 motvatd us to tst whthr w ar also abl to prdct dcouplng of hlum on th bass of our multcomponnt modl. Thrfor w ntroducd two nw varabls to dscrb th hlum flow, namly hlum dnsty ρ H and hlum radal vlocty v rh. W assumd hlum atoms to hav mass m H =4m p and charg q H. Th hlum vlocty and dnsty ar obtand from contnuty Eq. 1 and quaton of moton Eq. 2. W addd approprat hlum trms nto th nrgy Eq. 15 and nto th crtcal pont condton Eq. 28. W assumd a solar abundanc of hlum at th bas of th wnd and qual bas vlocts of hlum and hyogn. Th on vlocty s dtrmnd as dscrbd n Sct W strss that ths modls ar not fully slf-consstnt bcaus w dd not nclud th hlum crtcal pont. Ths s compltly byond th scop of th prsnt artcl; w am only to show that hlum dcouplng from th man absorbng ons hyogn flow s possbl. Dtald calculatons of a four-componnt modl ncludng hlum wll b publshd lswhr. Th modl of th four-componnt flow n a B3 star s gvn n Fg. 11 for q H =.3q p.atapontwhr th common flow of hyogn and absorbng ons s not abl to support hlum flow, th lattr dcoupls from th othr componnts and ts vlocty dcrass. Bcaus th hlum vlocty s lowr than th scap vlocty, hlum can raccrt to th star, cratng possbl surfac hlum ovrabundanc f magntc flds ar prsnt Hungr & Groot 1999 or possbl H shlls Portr & Skouza Not that such statonary modls cannot b xtndd to an arbtrary radus. Durng dclraton th hlum crtcal pont s agan rachd, and furthr opraton of th gravtatonal forc rvrss th cton of th hlum flow and causs ts backfallng to th stllar surfac. Such a stuaton cannot b corrctly dscrbd by our modl. Th wnd s sgnfcantly hatd by frcton btwn hlum and othr componnts byond th pont whr hlum dcoupls from th man flow. Hghr tmpraturs would lowr th argumnt x p of th Chanaskhar functon s Eq. 1 and thrfor lowr th frctonal forc btwn hyogn and mtallc componnts. Thus, to mantan a common flow of hyogn and absorbng ons, th ft vlocty should ncras. Hlum dcouplng s snstv to th hlum charg and thrfor to th tmpratur. For such low tmpraturs, whch ar commonly attand n th stllar wnd of B stars, dcouplng would b possbl. Howvr, f th hlum charg would rs du to hghr wnd tmpratur.g. to q H = q p no hlum dcouplng would occur. Thus, th dtald pctur of hlum dcouplng s much mor complcatd and nds furthr study. If th prsnt pctur of hlum dcouplng s corrct, thn Bp and Ap stars wth wnds could b X-ray sourcs. Rcntly ths was justfd by Dachs & Humml 1996 for th cas of th young opn clustr NGC Unfortunatly, t s not clar f th sourc of th X-rays ar Bp stars thmslvs or thr possbl cool companons wth hot corona τ Sco Ths wll-studd star s a puzzlng objct. Although t bcam a bnchmark star for thortcal and obsrvatonal

14 J. Krtčka and J. Kubát: Multcomponnt radatvly vn stllar wnds. I τsco τ Sco Fg. 12. Thr-componnt wnd modl of τ Sco. Th notaton s th sam as n Fg. 1. Not that ncluson of frctonal hatng lowrs th outflow vlocty towards obsrvd valus studs, ts natur s not yt wll known. Th obsrvd trmnal vlocty stmat about 2 km s 1 Abbott 1978; Lamrs & Rogrson 1978 contradcts th valu 385 km s 1 obtand from a dtald thortcal study by Paulach Sprngmann & Paulach 1992 wr abl to rduc th thortcal valu usng th twocomponnt modl of th stllar wnd, howvr, usng a lowr valu of a mass-loss rat. Our on-componnt modl wth much smplr atomc physcs than Paulach 1987 prdcts a trmnal vlocty 3 km s 1, wll abov th xprmntal rsult. Thrfor, w computd a thr-componnt modl of ths star to fnd out f our modls ar abl to xplan th low trmnal vlocty of ths star. Accordng to Klan 1994, w rducd th mtallcty to th valu z =.6. Unfortunatly, vn such a thr-componnt modl yldd narly th sam valu for trmnal vlocty as th on-componnt modl. Thrfor, w assumd slghtly rducd mtallcty contrary to Klan, z =.3, to allow for gratr frctonal hatng. Ths artfcal chang of th mtallcty dos not ndcat our mstrust of obsrvd mtallcty valus. Rathr t ndcats that our modl assumptons.g. th rprsntatv natur of our ons do not corrspond to th ralty of τ Sco. W blv mor dtald modls n th futur can xplan th low trmnal vlocty of τ Sco mor prcsly. Nvrthlss, basd on ths assumptons, w showd that frctonal hatng mayplay an mportant roln th modl ofτ Sco wnd bcaus t rducs ts trmnal vlocty to 24 km s 1,closr to th obsrvd valus. Morovr, th computd vlocty profl s Fg. 12 s vry smlar to that drvd from a dtald UV-ln ft by Hamann W would lk to mphasz that our computd low valu for th trmnal vlocty was not obtand du to th rducd valu of mtallcty, bcaus our wnd modl wth z =.3 wthout frctonal hatng ylds narly th sam trmnal vlocty as th on-componnt modl wth z =1.. Th low valu of trmnal vlocty was obtand du to th ncluson of frctonal hatng nto our modls. Although Sprngmann & Paulach 1992 also xpland th low trmnal vlocty of τ Sco on th bass of multcomponnt flow, our approach s substantally dffrnt, thr xplanaton bng on on runaway. Anothr problm arsng for othr low-dnsty wnd stars s th dtrmnaton of th mass-loss rat. Our obtand valu, M yr 1, s about a factor of 3 lowr than that obtand from th usual CAK approxmaton s.g. Cohn t al. 1997, who usd th cookng formula of Kutzk t al. 1989, du to th rducd mtallcty n our modls. Ths valu s slghtly hghr than a nw uppr-lmt of th mass-loss rat, M yr 1, constrand by Zaal t al from nfrard msson; howvr, ths rsult s snstv to a star s ffctv tmpratur. Fnally, Watrs t al rportd th prsnc of hyogn msson lns n th nfrard spctrum of τ Sco and concludd that τ Sco s a pol-on B star. Howvr, Murdoch t al suggstd that nfrard hyogn msson s causd by th prsnc of NLTE ffcts s also Zaal t al Our NLTE sphrcally-symmtrc statc modl of th photosphr of τ Sco prdcts nfrard msson both n hyogn and hlum lns. 6. Conclusons W computd thr-componnt modls of a radatvly vn wnd of hot B stars. W showd that frcton has a nglgbl ffct n hgh dnsty wnds,.. n stars wth a rlatvly hgh mass-loss rat. For ths stars, thrcomponnt calculatons yld th sam rsults as th common on-componnt ons. For low dnsty wnds th ncluson of frcton lads to hatng of th flow. W showd that n slf-consstnt hyodynamc modls, frcton can hat th gas to tmpraturs up to two tms th ffctv tmpratur. Ths hatng lads to lowrng of th radatv forc and, consquntly, to th lowrng of th outflow vlocty. Lowr outflow vlocty lads to hghr dnsty and ths mans, togthr wth hghr thrmal spd, that th ft vlocty of both componnts s too low to allow dcouplng of passv and absorbng componnts.

15 236 J. Krtčka and J. Kubát: Multcomponnt radatvly vn stllar wnds. I. Fnally, stars whr absorbng plasma s not abl to ag th passv plasma out from th atmosphr hav a purly mtallc wnd, as has alrady bn shown by Babl 1995, W drvd gnral condtons for dcouplng of a multcomponnt flow. W showd that f th dnsty of th acclratd componnt s hghr than th dnsty of th passv componnt, thn dcouplng s possbl, rgardlss of th actual form of vng forc. On th othr hand, f th dnsty of th acclratd componnt s lowr than th dnsty of th passv componnt, thn th possblty of dcouplng s gvn by th functonal dpndnc of th vng forc. If th vng forc dos not chang substantally n th rgon whr dcouplng s possbl, thn dcouplng of ndvdual componnts s agan possbl. If th vng forc scals wth vlocty gradnt as n th Sobolv approxmaton, thn dcouplng of ndvdual componnts s not possbl. Our calculatons ndcat that frcton s abl to hat th wnd to th tmpraturs of th ordr of 1 6 Ks Fg. 1. Such a hot mdum s abl to mt X-ray radaton. Howvr, quanttatv stmats of X-ray lumnosty ar byond th scop of ths papr. Thr ar many applcatons of th multflud modl of stllar wnd. Consdrng hlum as a fourth componnt, on can apply th modl of a multcomponnt wnd to th study of chmcally pcular stars. W ar also abl to ft th outflow vlocty of th star τ Scoclosrtoth obsrvd valu. Unfortunatly, our approach of handlng th problm s stll not satsfactory. Manly, th approxmaton of ontyp absorbng ons wth paramtrs whch ar not rgdly gvn may lad to som msundrstandng. Th approxmaton of constant onc charg s not slf-consstnt bcaus th dgr of onzaton changs wth tmpratur and dnsty. Morovr, allowng for a macroscopc lctrc fld may lad to som obsrvabl ffcts Portr 1999, prvat communcaton. Such studs wth th ncluson of nspcton of stablty of multcomponnt flow wll b th subjct of futur paprs. Howvr, w ar now abl to dscrb th gnral pctur of multflud flow, whch wll lkly not b strongly affctd by futur rfnmnts. Acknowldgmnts. Th authors would lk to thank Dr. John Portr for hs commnts on th manuscrpt. Ths rsarch mad us of NASA s Astrophyscs Data Systm Abstract Srvc Kurtz t al. 2; Echhorn t al. 2; Accomazz t al. 2; Grant t al. 2. Ths work was supportd by projcts K1-3-61/4 and K Appndx A: Dtals of numrcal mthod Th non-lnar oprator P has followng form: for =1 P 1 = ρ 1 ρ A.1a P 2 = GM r Γ 4a p 2 r 1 +R p ln Λ+ m m p R ln Λ A.1b P 3 = ρ p1 1 v r1 ρ 1, A.1c Y v rp1 P 4 = v rp1 R sa p, A.1d P 5 = ρ 1 m ρ p1 + z ρ 1, A.1 m p A P 6 = ρ 1 v r1 m ρ p1 v rp1 + z ρ 1 m p A v r1 A.1f P 7 = Q rad. A.1g for1<<nr [ ] d r 2 ρ v r P 7 6 = A.2a [ ] [ ] [ ] dvr P 7 5 = v r, + a2 dρ da 2 + q E ρ, m [ ] α 1 dvr x α +g+r p + R lnλa.2b ρ, [ ] d r 2 ρ p v rp P 7 4 = A.2c [ ] [ ] [ ] dvrp P 7 3 = v rp, + a2 p dρp da 2 p + q p E ρ p, m p + g + R p ln Λ + R p ln Λ A.2d P 7 2 = ρ, m ρ p, + z ρ,, A.2 m p A P 7 1 = ρ, v r m ρ p, v rp, + z ρ, v r,, A.2f m p A P 7 = 3 [ ] dt 2 k ρ a, v ra, m a=p,, a + [ ] a 2 a ρ 1 d r 2 v ra a, Q rad 1 2 a=p,, a=p,, r 2 b=p,, b a for =NR [ ] d r 2 ρ v r P 7 6 = P 7 5 P 7 4 = P 7 3 P 7 2 P 7 1 ρ a R ab ln Λ v ra, v rb,. = v r, v r, 1 v r, 1 v r, 2 r r 1 [ ] d r 2 ρ p v rp = v rp, v rp, 1 r = ρ, m ρ p, + z m p = ρ, v r m m p v rp, 1 v rp, 2 r 1 A.2g A.3a A.3b A.3c A.3d ρ,, A.3 A, A.3f ρ p, v rp, + z A ρ, v r,

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