Astronomy & Astrophysics manuscript no. lblucy c ESO 2012 (cid:13) January 4, 2012 Coronal winds powered by radiative driving L.B.Lucy AstrophysicsGroup, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW72AZ, UK Received ; Accepted 2 1 ABSTRACT 0 2 Atwo-componentphenomenologicalmodeldevelopedoriginallyforζ Puppisisrevisedinordertomodeltheoutflowsof n late-typeOdwarfsthatexhibittheweak-wind phenomenon.Withthetheory’sstandardparametersforagenericweak- a wind star, theambient gas is heated to coronal temperatures 3 106K at radii >1.4R, with cool radiatively-driven J gas beingthenconfinedtodenseclumpswith fillingfactor 0≈.02.×Radiativedrivin∼gceases at radius 2.1R when the clumps are finally destroyed by heat conduction from the c≈oronal gas. Thereafter, the outflow is a pu≈re coronal wind, 2 which cools and decelerates reaching with terminal velocity 980 km s−1. ∞ ≈ ] Key words.Stars: early-type- Stars: mass-loss - Stars: winds, outflows R S . h 1. Introduction Zone1startsjustbeyondthesonicpointandendswhen p the isothermal-shock assumption is no longer justified. We - The X-ray emission from O stars (Harnden et al. 1979) is assume instability has grown to full amplitide and adopt o now generally agreed to arise from numerous shock fronts r the LW description in which radiative-driven blobs (b) in- t distributedthroughouttheirwinds.Anearlytheoryofsuch teractdynamicallywithalowdensityambient(a)medium. s X-rayemitting winds (Lucy&White 1980;LW)wasbased a Apart from the infinitesimally-thin cooling zones at shock [ on a two-component phenomenological model for the fi- fronts, both components are in thermal equilibrium with niteamplitudestatereachedbyunstableline-drivenwinds. the photospheric radiation field, so that T =T . 1 Subsequently, the fundamental approach of computing the a,b eq v growth of the instability using the equations of radiation 3 gas dynamics was pioneered by Owocki et al. (1988) and 2.1. Blobs 8 Feldmeier (1995), albeit with the then necessary restric- 4 The blobs can be identified with the clumps that are now tions to 1-D flow and simplified radiative transfer. 0 a standard and spectroscopically-required feature of diag- . A question meriting further research is how and where 1 nostic codes for O-star winds (e.g., Bouret et al. 2005). In this wind-shock model fails. According to LW, failure oc- 0 suchcodes,theclumpsareassumedtoobeytheβ velocity 2 curs at the low mass-loss rate (Φ) of a main sequence B0 law − star, because the assumption of rapid radiative cooling of 1 β shocked ambient gas then breaks down for blob velocities R : v =v 1 (1) v v > 103km s−1, resulting in the heating of the blobs and b ∞(cid:18) − r(cid:19) b i con∼sequentlossofline-driving.Theyconjecturethat’there- X where R is the photospheric radius and v , the terminal after,the relativemotions ofthe two components dissipate ∞ r velocity, is determined from the violet edges of P Cygni a andthesmoothedwindcoastsouttoinfinity.’Ineffect,LW absorption troughs. suggestthat a wind that is initially radiatively driven con- Giventhewideuseoftheβ-law,thisnowreplacesLW’s verts into one that relies on thermal pressure to reach - ∞ Eq. (10). But here, since Eq. (1) ceases to apply when r > i.e., a coronal wind. r , the blobs’ destruction radius, v is not an observable. In addition to this question’s intrinsic interest, it is no- S ∞ The highest velocity at which UV absorptionis detected is table that the locus of this expected failure coincides with a measure not of v but of v (r ). that of stars exhibiting the weak-wind phenomenon. (e.g., ∞ b S For given β, diagnostic modellers choose the clumps’ Marcolino et al. 2009; M09). Accordingly, this paper elab- filling factor f and mass-loss rate Φ so that absorp- orates LW’s conjectures for the outflow from a weak-wind b b tion troughs have their observed strengths. For a strong star. line at frequency ν0, this typically requires that, despite clumpiness, a continuum photon emitted between ν0 and 2. A Multi-zone wind: Zone 1 ν0(1+v∞/c) has small probability of escaping to , and ∞ so most of the photon momentum in this interval is trans- The two-component model must be generalized to remove ferred to the clumps. In an LW wind, essentially the same theassumptionofinstantaneouscoolingofshockedgasand requirementarisesas a consistency criterion:since the am- to incorporate blob destruction at finite radius. To achieve bientgasisassumednottoberadiativelydriven,itmustbe these aims, a multizone model is adopted, with each zone shadowedbytheblobs.Theopticaldepthcriterionadopted corresponding to different physical circumstances. by LW is that τ1(r) > 1.5 for all r, where τ1 is given in Send offprint requests to: L.B.Lucy Eq.(11) of LW. For the β velocity law, the function χ in − 1 Lucy:Coronal winds their τ1 formula becomes 2.4. Switch criterion 1 χ(x;β)= (1 x)1−2β /[2β+x (1+β)x2] (2) The assumption of instantaneous cooling breaks down at 2 − − low densities because the cooling rate per unit volume where x=R/r. ˙ ρ2. If the cooling time scale t increases to the ex- c C ∝ tent that a parcel of shock-heated gas encounters another shock before cooling back to T , then shock-heatingraises 2.2. Dynamics eq the meantemperatureofthe ambientmedium.Anapprox- In zone 1, the dynamical interaction of the blob and am- imate criterion for this transition to zone 2 is derived as bient components is treated exactly as in LW. For more follows: recent treatments and applications, see Howk et al. (2000) First,considerradiatively-cooledflowofmonatomicgas and Guo (2010). (γ =5/3)emergingfromasteadyshock(seeFig.1inDraine The equation of motion obeyed by the blobs is & McKee 1993). Since this flow is subsonic, the pressure gradient may be neglected in comparison to that of tem- dv v b = g g g (3) perature. Thus, in the shock’s frame, b R D dr − − where gR,gD, and g are the forces per gram due to radi- dlnT 2 ˙ 1 ation, drag, and gravity, respectively. The drag force g C = (7) D dr ≈ −5 Pv ℓ retards the blobs but accelerates the ambient gas and is c the means by which photon momentum is transferred to The cooling timescale is therefore t =ℓ /v=5/2 P/˙. this component. The resulting equation of motion of the c c × C Nowconsiderflowintothebowshocks.Theentiremass ambient gas is ρ ofambientgasinunitvolumeisshockedintimeinterval a dv 1 dP¯ ρ¯ t = ρ / j , where = f /V is the number density of v a = a + b g g (4) i a Nb b Nb b b a dr −ρ¯a dr ρ¯a D− blobs, and jb ≈ ρaUAb is the mass flow rate through each bow shock. Hence t 4/3f σ/U. Here ρ¯ are the smoothed densities, and P¯ = a2ρ¯ with i ≈ b× a,b a a a Thecriterionforswitchingfromzone1tozone2isthen a2a =kTa/µmH. simply tc >ti. With the LW assumptionof no mass exchangebetween the components, the two equations of continuity integrate to give 3. A Multi-zone wind: Zone 2 Φ =4πr2ρ¯ v (5) a,b a,b a,b The outward integration of the wind continues in zone 2 where Φ are constants whose sum is the star’s mass-loss a,b with the same basic model except that T = T . The rate Φ. a,b 6 eq ambient gas is now heated by being repeatedly shocked, The drag force mg on a blob of mass m is computed D and the blobs in turn gain heat by conduction from the using De Young and Axford’s (1967) theory of inertially- ambient gas. Zone 2 ends when the blobs can no longer confinedplasmaclouds-seeSect.IIb)inLW.Theresulting achieve thermal equilibrium. formula is 1 mg = C ρ U2 (6) D D a b 2 A 3.1. Blob survival where = πσ2 is the blob’s mean cross section, U = b A The survivalof blobs (clumps) in stellar winds has similar- v v is the blob’s velocity relative to the ambient gas, b a − ities to that of clouds in the interstellar medium. In that and the drag coefficient C =1.519. D context, Cowie & McKee (1977) studied the evaporation of a spherical cloud embedded in a hot tenuous medium. 2.3. Filling factors Importantly, they treated the saturation of heat conduction when the electron mean free path in the surround- At a point (v ,v ,r) in an outward integration with spec- b a ing medium is > the cloud’s radius and estimated, under ified Φ , the smoothed densities ρ¯ are given by Eq. a,b a,b the assumption∼of steady outflow, the reduced evaporation (5). The ambient density is then ρ = ρ¯ /f , where f a a a a rate.Inacompanionpaper(McKee&Cowie1977;seealso is the filling factor of the ambient gas. Correspondingly, Graham & Langer 1973), they consider the effects of ra- the mean density of the stratified De Young - Axford blob diative losses, finding that evaporation is replaced by con- is ρ = ρ¯ /f . Now, in the absence of a void component, b b b densation if the losses exceed the heat input from the hot f + f = 1, and so only one of f and f is indepen- a b a b gas. However, numerical calculations by Vieser & Hensler dent. To determine f , say, we must iterate. Solution by b (2008) cast doubt on the assumption of steady outflow. In repeated bisection is adopted, starting with upper and thecaseofsaturatedconduction,theyfindafurtherreduc- lower limits f = 1 and f = 0. Then, with the esti- U L tion of evaporationrate by a factor 40 due to changes of mate f˜ = (f +f )/2, the blob’s volume V is computed ∼ b U L b the cloud’s environment caused by the outflow. from LW’s Eq.(5). The mean density of the blob is then Given the evident difficulty of reliably predicting when ρ = m/V , corresponding to f = ρ¯ /ρ . If f < f˜, the b b b b b b b cool gas is eliminated by its interaction with surrounding new upper limit is fU = f˜b. On the other hand, if fb f˜b, hotgas,asimple prescriptiveapproachisadoptedhere:As thenewlowerlimitisf =f˜.Theiterationscontinue≥until in zone 1, the blobs retain their fixed mass m throughout L b f f < 10−7. Then, with the resulting converged value zone2.However,whentheheatinputfromtheambientgas U L − off ,allquantitiesrequiredtocontinuetheintegrationcan exceedstheirmaximumcoolingrate,theblobsareassumed b be evaluated. to merge instantly with the ambient gas. 2 Lucy:Coronal winds 3.2. Heating and cooling of blobs IntegrationofEqs.(4),(5)and(11)continuesuntilT =T , b † atwhichpointtheblobsaredeemedtomergeinstantlywith Inzone2,theblobsaresurroundedbyshock-heatedgasand theambientgas(Sect.3.1).Accordingly,thetransitionfrom so will be heated by thermal conduction. But if the ambi- zone2tozone3occursatS,asurfaceofdiscontinuity(e.g., ent gas reaches coronal temperatures, heat conduction is Landau & Lifshitz 1959), across which the fluxes of mass flux-limited. Moreover,conductivity may be suppressed by magnetic fields. An approximate formula interpolating be- J =Ja+Jb =faρava+fbρbvb (13) tween the classical and saturated limits and incorporating momentum a suppression factor φ is derived in Appendix A. Π=f (P +ρ v2)+f (P +ρ v2) (14) If is the rate of heat flow from the ambient gas, a a a a a b b b b in blob wLill achieve thermal equilibrium at T > T if the and energy b eq enhanced radiative cooling rate 1 5 1 5 F =J v2+ a2 +J v2+ a2 (15) a (cid:18)2 a 2 a(cid:19) b (cid:18)2 b 2 b(cid:19) ∆ = (n n ) (Λ(T ) Λ(T )) V = (8) b e H b b eq b in L − × L are continuous.Zone 2 thus ends at r with the evaluation S Here Λ(T) is the optically-thin cooling function, and the of J, Π and F. blob is treated as isothermal and of uniform density. (But notethatLW’sdefinitionofρ issuchthat∆ isexactfor b Lb 4. A Multi-zone wind: Zone 3 thedensitystratificationofanisothermalDeYoung-Axford blob.) The outward integration continues in zone 3, but now the Because Λ(T)reaches a maximum atT†(K)=5.35 dex blobs have disappeared, leaving a single fluid component (Dere et al. 2009), the solution of Eq.(8) with Tb < T† is with no driving force (gD =0) and no heat input (Q˙ =0). appropriate as the blobs are heated to above Teq. When The initialconditions for the resulting ODE’s are obtained Tb reaches T†, the corresponding in is the maximum fromthecontinuityacrossS ofJ,ΠandF.Thus,v,ρ,and L value consistent with thermal equilibrium. Any further in- T for the flow emerging from S+ are given by crease in cannot be matched by increased cooling. Lin ρv =J (16) Accordingly, we take this as the point beyond which the blobs cannot survive. P +ρv2 =Π (17) Note that if conduction is completely suppressed (φ = and 0), then Lin = 0 and the solution of Eq.(8) is Tb = Teq. J 1v2+ 5a2 =F (18) The blobs therefore survive, and zone 2 extends to . (cid:18)2 2 (cid:19) ∞ where J,Π, and F are given by Eqs. (13)-(15). 3.3. Heating and cooling of ambient gas According to LW, the rate at which energy is being dissi- 4.1. Solution branches pated per unit volume is From Eqs. (16)-(18), we readily derive the quadratic equa- Q˙ = g ρ¯ U (9) tion D b v2 2u v+u2 =0 (19) − p e Dividing by Nb, we find that the rate per blob is where up = 5Π/8J and ue = √(F/2J). The two solutions are 1 Q˙b = 2ρaU2×U ×CDAb (10) v± =up±qu2p−u2e (20) The corresponding temperatures T are derived from the ± showing that in unit time each blob’s bow shock dissipates isothermal sound speeds given by the kinetic energy in a column of inflowing gas of length kT 2 UergayndiscrraodsisatseedctbioynaCtDhAinb.coInoliznognela1y,ert,hiasnddisssoipQa˙teddeteenr-- a2± = µm±H = u2e− 5upv± (21) mines the sum of these layers’ frequency-integrated emis- and the densities are ρ =J/v . ± ± sivities - Eq.(7) in LW. But in zone 2 where t > t , we If u = u , the two solutions coincide. When this hap- c i p e jump to the opposite limit, treating dissipation as a heat pens, v =u =√(5/3)a - i.e., the outflow at S+ is exactly p source distributed uniformly throughout the ambient gas, sonic. If u > u , the solutions are real and distinct. The p e and similarly for cooling. Accordingly,the energy equation v+ solutionissupersonic(M+ branch),andthev− solution for stationary flow of the monatomic ambient gas is is subsonic (M− branch). Note that the M+ branch has a singularity at vp/ve = 5/4, at which point a+ = 0. Mach v dPa 5 a2dρa = 2 (Q˙ ˙ ) (11) numbers for the two branches are plotted against vp/ve in a(cid:18) dr − 3 a dr (cid:19) 3 −Ctot Fig. 1. The M− branch corresponds to S being the locus not Notethatsincefa 1termsarisingfromradialchangesin only of merging but also of a stationary shock front. This ≈ fa have been neglected. branch would perhaps be appropriate if there were a pre- The total cooling rate per unit volume of ambient gas, existingslowerwind(cf.Macfarlane&Cassinelli1989),but ˙ , is the sum of the losses due to radiative cooling and this is not the circumstance evisaged here. Instead, there- tot tCo conduction into the blobs. Thus, fore, the M+ branch is selected since this correspondsto a highspeed two-componentflow atS− emergingatS+ as a ˙ = (n n ) Λ(T )+ (12) single-component supersonic flow. tot e H a a b in C N L 3 Lucy:Coronal winds 5. An example To illustrate the ideas presented in Sects. 2-4, the solution for a generic weak-wind star is now described in detail. 1 5.1. Standard parameters The model has severalparameters,for which standardval- uesarenowadopted.Giventheiruncertainty,sensitivityto 0.5 changes are reported in Sect. 6. Because the theory does not predict Φ, this is derived frompreviously-tabulatedmassfluxes(Lucy2010b;L10b). The chosen model has Teff = 32.5kK and logg = 3.75, consistentwith the weak-windstarsζ OphandHD216532 0 - see Table 3 in M09. The model’s mass flux J(gm cm−2 s−1) = -7.11 dex. The star’s mass = 24.1 is determined by finding the point on thMe ZAMS Mfro⊙m which the evolution- -0.5 ary track during core H-burning has log g = 3.75 when Teff = 32.5kK. This point is reached after 5.75 106 yrs 1 1.1 1.2 whenR=10.83R andtheluminosityL=1.18 ×105L = 4.52 1038 erg s⊙−1. The assumed compositio×n is X⊙ = Fig.1. Mach number M as a function of up/ue along the su- 0.70,×Z =0.02. personic (M+) and subsonic (M−) solution branches. When With R and L determined, Φ = 4πR2J = 8.80 up/ue =5/4, M+ =∞ and M− =1/√5=0.447. 10−9 yr−1 =1.10L/c2.ThistheoreticalΦderivesfrom× the cMon⊙straint of regularity at the sonic point (v = a) in 4.2. Dissipation at S the theory of moving reversing layers. In the weak-wind domain, this theory’s predictions exceed the highly uncer- InadditiontotherootsofEq.(19)beingreal,afurthercon- tain ( 0.7 dex) observational estimates of M09 by 0.8 ditionis mandatory:the transitionfromS− to S+ mustbe dex bu±t are lower than the Vink et al. (2000) formu≈la by suchthat kinetic energyis dissipated (entropyproduction) 1.4 dex (Lucy 2010a;L10a). and not the reverse. For the two branches, kinetic energy ≈ For the parameters in Eq.(1), we adopt the is thermalized at the rates observationally-supported O-star values β = 1 and v =2.6v (R)=2394 km s−1. 1 ∞ esc L±S = 2(Φava2+Φbvb2−Φv±2) (22) The mass m ofthe blobs must alsobe specified.Recent modelling of O-star spectra finds that ’In most cases, whose positivity must be checked. clumping must startdeep inthe wind, just abovethe sonic Note that the kinetic energy dissipated at S is not ra- point’ (Bouret et al 2008). We therefore retain LW’s as- diated away by a thin cooling zone. Instead, this energy sumption that blobs form at or near the sonic point and contributes to the flow’s enthalpy at S+, which then does have diameters comparable to H , the local scale height. ρ PdV work in the subsequent expansion. At v = a in model t325g375, ρ = 4.27 10−14 gm cm−3 and H =9.86 108 cm, so that the cru×de LW estimate is ρ m=2 1013 g×m. 4.3. Outward integration × The ratio η = Φ /Φ must also be specified. Following b The solution for the single-component gas in zone 3 is ob- LW, we determine η by imposing the constraint that τ = m tained by integrating the equations of motion 1.5, where dv 1dP τm =min[τ1(r)] inzone1 (26) v = g (23) dr −ρ dr − Typically, the minimum occurs at the end of zone 1 whereinertialconfinementisgreatest.Inzone2,shadowing continuity rapidly becomes irrelevant since the rapid rise of T - see a Φ=4πr2ρv (24) Fig.3 - destroys driving ions. Finally, the conductivity suppression factor φ intro- and energy ducedinAppendixAmustbespecified.Asstandardvalue, dP 5 dρ 2 we set φ = 1.0 dex, a moderate degree of suppression v a2 = nenHΛ(T) (25) compared to−estimates for galaxy clusters (e.g., Ettori & (cid:18)dr − 3 dr(cid:19) −3 Fabian 2000). The initial conditions at rS are v+,ρ+ and T+ derived in Sect.4.1. 5.2. Zone 1 This integration continues to r = . However, this is onlypossibleiftheenergydensityatS+∞issufficienttoover- In this high-density zone close to the photosphere, both come both the remaining potential barrier and the cooling componentsareassumedtobe inthermalequilibriumwith losses. If not, a stationary, spherically-symmetric wind so- the star’s radiation field, a condition approximatedby set- lution of this type does not exist. ting Teq =0.75Teff, as in L10a,b. 4 Lucy:Coronal winds With the assumptions of isothermal flow, specified v , b and no mass exchange between components, the structure of zone 1 is obtained by integrating the ODE dlnv r 2a2 ρ¯ a = a + b g g (27) dlnr v2 a2 (cid:20) r ρ¯ D− (cid:21) a− a a 1000 The outward integration starts, as in LW, with v = 150 b km s−1 and v = 100 km s−1, a point sufficiently beyond a the presumed onset of clumpiness that the two-component state may be regarded as established. The starting radius from Eq.(1) is r =1.067R. i Eq.(27) has a singularity when v = a . Since the in- a a 500 tegration starts with v > a , this singularity only arises a a if insufficient drag g causes the flow to decelerate. A pa- D rametersetforwhichthishappensdoesnotadmitasteady wind of this type. AsshowninFig.2,thestandardparametersresultinan outflow of ambient gas that accelerates throughout zone 1. Thiscontinuesuntiltheswitchtozone2istriggeredbythe 0 0 0.1 0.2 0.3 0.4 0.5 onsetof the inequality t >t - see Sect.2.4.This occurs at c i r/R = 1.28, with v = 528 km s−1 and v = 325 km s−1. b a The relevant time-scales are tc = ti = 2.0 103 s, which Fig.2. Velocities of blobs (vb) and ambient gas (va) as func- are the local flow time-scale, r/v =1.8 ×104 s. tions of radius. Zone boundaries are indicated. The surface of b T≪he post-shockcoolingrate ˙ requiredi×ncalculating t discontinuity S where blobs merge with ambient gas occurs at C c r/R=2.14. is given by n n Λ(T), where Λ(T) is the optically- thin e H cooling function for photospheric abundances tabulated by Dereetal.(2009).Thisrateiscomputedattheapexofthe The profilefor T showsdiscontinuousjumps atthe be- bow shock with n = 1.18n , corresponding to complete b e H ginningandendofzone2.Theseresultfromnon-monotonic electron-stripping. variations of Λ(T). For example, Λ’s peak at T (K)=5.35 At the end of zone 1, the post-shock temperature has † risen to 6.0 105K, so X-ray emission from zone 1 is neg- dex is preceded by lower peak at 5.00 dex. Accordingly, × after reaching T (K) = 5.00 dex, a slight increase in ligible. b Lin results in a discontinuous jump to T (K) = 5.18 dex, fol- b lowed quickly by blob destruction when T = T . Because b † 5.3. Zone 2 of these jumps, the radiative driving of the blobs, which is ultimately reponsible for T ’s increase to coronal values, With the isothermal assumption dropped, the structure of a occurs mostly between T =40 and 90kK. zone2isdeterminedbyEqs.(4)and(11).Withdependent b BlobtemperaturesarederivedalgebraicallyfromEq.(8) variables v and T , the ODE’s to be integrated are a a on the assumption that blobs adjust instantaneously (va2−a2a)ddllnnvra + a2addlnlnTra = 2a2a+r(cid:18)ρρ¯¯abgD−g(cid:19) (28) s1toc.5alntehkreT/r†mv×a=lVbe1/q.L3uiinlib=1r0i4u0.ms9..×A10t2rsSc,omthpearheedattiongthetiflmoewstcimalee b × and In computing cooling rates for blobs, we set ne = 1.12n , corresponding to metals being stripped of 2 3 2 dlnva + dlnTa = 4 + 2 r (Q˙ ˙ ) (29) electroHns. ∼ − tot 3 dlnr dlnr −3 3P v −C a a Since all variables are continuous at this transition, the in- 5.4. Surface of discontinuity S tegration starts at the point (v ,v ,T ,T ,r) reached by a b a b the zone-1 integration. At S−, the blobs have filling factor fb = 0.024, velocity Eqs.(28) and (29) are a pair of algebraic equations for vb = 1282km s−1 and temperature Tb = 2.24 105K. × the two derivatives. The determinant of the coefficients’ The corresponding values for the ambient component are smoantircixpoiisnzt.erIfotwhihsesninvgaul=ar√ity(5is/3e)caoaun-tie.ree.,d,atthtehpeaardaimabetaetrics fmaer=gi0n.g9,7t6h,evaflo=w9a4t4kSm+ sh−a1satnwdoTpaos=sib3l.e66so×lu1t0io6nKs.(ASeftcetr. are inconsistent with the conjectured wind structure. 4.1). For the rejected M− solution, the flow emerges with Fig.2showsthat,withthestandardparameters,theflow v− =313kms−1,T− =1.90 107K,correspondingtoMach × continuestoacceleratethroughoutzone2reachingv =940 0.48, and the implied rate at which kinetic energy is dissi- km s−1 at rS =2.14R, at which point vb =1277 kma s−1. pated L−S =0.31×1034 erg s−1 or 6.9×10−6L. The corresponding temperature structure predicted for For the selected M+ solution, the flow emerges with zTohneere2aifstesrh,oswhnocikn-hFeiga.t3in.gAtofththeestaamrtb,iTena,tbc=omTepqo=nen2t4.o4vkeKr-. vM+ac=h41.60,94aknmdths−e1i,mTp+lied=d2is.s5i8pa×tio1n06rKat,ecLo+rre=sp0o.n92ding10t3o2 comesradiative,conductiveandadiabaticcoolingtogivea erg s−1 or 2.0 10−7L. S × rapidly increasing Ta, reaching the coronal value 106K at Notice that×v+ (va,vb), as expected if S is the locus r =1.35R and 3.7 106K at r . only of merging. In∈contrast, v < v , so there is a coinci- S − a × 5 Lucy:Coronal winds 5.7. Energy budget 7 The global energy budget of this multi-zone wind is of interest. The input is the rate of workingin zones 1 and 2 of 6.5 gR, the force per unit mass acting on the blobs. This rate L =5.4 1033 erg s−1. wrk × ThebalancingoutputisL +L ,whereL istherate M W M atwhichmattergainskineticandpotentialenergy,andL 6 W is the wind’s radiative luminosity. For the interval (r ,r ), i f L =4.8 1033 erg s−1 or 88.5% of L . The remaining M wrk × 11.5% is accounted for by L , which comprises radiative W 5.5 losses from shock fronts in zone 1, cooling radiation from blobsandambientgasinzone2,andcoolingradiationfrom the coronal flow in zone 3. 5 For an idealized line-driven wind in which gas remains (by assumption) at T , PdV work is negligble so that eq L =L .Incontrast,forapurecoronalwind,L =0, M wrk wrk 4.5 so thatLM is entirelydue to the PdV workofthe hot gas. The relative contributions of these two mechanisms in this hybrid case is of interest. 0 0.1 0.2 0.3 0.4 0.5 Inansweringthis,wemustfirstintegrateQ˙ fromEq.(9) overzones1and2toobtainthetotaldissipationrateL = D Fig.3. Temperatures of blobs (b) and ambient gas (a) as func- 1.1 1033ergs−1.ThequantityL L =4.3 1033erg wrk D tions of radius. Zone boundaries are indicated. s−1×isthenthecontributiontoL du−edirectlyto×radiative M driving. On the other hand, the contribution of PdV work by hot gas is L L =0.4 1033 erg s−1. dent shock, as also indicated by the far greater dissipation D W − × rate L−. Ameasureof the proximity ofa hybrid-to a pure coro- S nal wind is the ratio 5.5. Zone 3 θ =(LD LW)/LM (32) − ThesinglecomponentflowemergingfromS isapurecoro- which = 0 for a conventional line-driven wind and = 1 nalwind: the only outwardforce is the gradientof thermal for a coronal wind. With standard parameters, the multi- pressure. zone wind has θ = 0.08, so direct radiative driving still The structure of zone 3 is obtained by continuing the dominates in accounting for LM. integrationofEqs.(28)and(29),butnowwithg =0, Q˙ = A further quantity of interest is the integrated cooling 0 and ˙ = n n Λ(T). The initial conditionDs at r are rate of gas with Te > 106K, since this is approximately Ctot e H S the wind’s X-ray luminosity. For zones 2 and 3, this gives v+ and T+ given in Sect. 5.4. L 3.4 1031 erg s−1, so that L /L 0.76 10−7L, A short segment of this outflow is plotted in Figs. 2 X ≈ × X ≈ × similar to the ratio found for early-type O stars. and 3, showing that the flow decelerates and (inevitably) cools.Forthesestandardparameters,theenergydensityat S suffices to overcome cooling and power escape to . At 6. Non-standard parameters r /R=100,theflowhasslowedto984kms−1,wayb∞eyond f the local v =92km s−1 The theory developed in Sects. 2-4 has several param- esc Thetemperaturedropsbelowthecoronalvalue106Kat eters, each of which would either be predicted or ren- r/R=4.24 and to 105K at r/R=13.2. deredunnecessaryif calculationscouldbe carriedoutfrom first principles. Sensitivity of the results to these currently unavoidable parameters must therefore be investigated. 5.6. Emission measure Accordingly, sequences of solutions are now reported in which a single parameter is varied while keeping others at With standard parameters, our generic weak-wind star is the standard values of Sect. 5. predicted to have a corona (T > 106) that extends from Key properties of the models are given in Table 1. The r1 =1.35R to r2 =4.24R and so will be an X-ray emitter. quantities reported are as follows: As a crude guide to detectability, we compute the emission measure of coronal gas Col. 1: Sequence identifier. r2 ε = 4π nenH r2dr (30) Col. 2: Exponent in Eq.(1), the velocity law. Z r1 and its hardness parameter Col. 3: Log of total mass-loss rate in yr−1. M⊙ = 4πε−1 r2kT n n r2dr (31) Col. 4: Log of blobs’ mass in gm. e H Z r1 Col. 5: Log of conductivity suppression factor - see Eq. The results are ε(cm−3) = 53.51 dex and < kT >= 0.20 (A.9). keV. 6 Lucy:Coronal winds A further diagnostic test provided by P Cygni lines is theweaknessofemissioncomponents.Asr decreaseswith S increasing φ, the fraction of scattered photons occulted 7 by the star increases and the emission component weak- ens. This effect was invoked for τ Sco by LW in arguing that ’outflowing gas loses its ability to scatter UV radiation while still close to the star’s surface.’ Note that the weak-wind stars investigated in M09 all have C iv reso- nance doublets with weak or absent emission components. Diagnostic modelling of these stars would improve if UV 6.5 scattering were truncated at finite radius. 6.2. Sequence II As noted in Sect.5.1, the Φ of weak-wind stars is poorly determined. This sequence explores sensitivity to this un- 6 certain parameter. When Φ is increased above the standard value from L10b,the blobs surviveto highervelocities,andthe higher 0 0.5 1 1.5 coronaldensitiesgivetheapproximatescalinglawε Φ1.3. ∝ Interestingly, the quantities T and are insensi- max tive to Φ. Fig.4.Sensitivityofcoronal windstoφ,themagneticsuppres- The attempt to continue this sequence to lower Φ’s sionfactor-seeEq.(A.9).Valuesoflogφareshown.Thevertical segments are the surfaces of discontinuity S. failed at 8.36 dex because the singularity in zone 2 dis- − cussedinSect.5.3isencountered.Thisarisesasfollows:the sharp initial rise of T in zone 2 causes M to decrease de- a Col. 6: Fraction of mass-loss in blobs = Φb/Φ . spite increasing va - see Figs.2 and 3. But as Ta levels off M reaches a minimum and then rises again. Sequence II Col. 7: Shadowing optical depth - see Eq.(26). terminates for Φ between 8.36 and 8.26 dex when this − − minimum falls to M = 1. For the solution plotted in Figs. Col. 8: v in km s−1 at the destruction radius r . 2 and 3, this zone-2 minimum is M =1.88 at r =1.47R. b S Col. 9: Maximum ambient gas temperate in 106K. 6.3. Sequences III-V Col. 10:Logofemissionmeasureincm−3 -seeEq.(30). In sequence III, the velocity-law exponent varies from β = 0.5 - rapid acceleration - to β = 2.5 - slow acceleration. Col. 11: Hardness parameter in keV - see Eq.(31). The standard value β = 1.0 is approximately a stationary point as regards the coronal properties ε and < kT >, so these are insensitive to β. However, v (r ) is moderately b S sensitive. 6.1. Sequence I In sequence IV, solution sensitivity to the highly un- certain blob mass is explored. Fortunately, coronal prop- Inthis sequence,the conductivitysuppressionfactorvaries erties are only moderately sensitive, with ε m−0.22 and from φ = 0.001, an extreme value but with observational m0.03. In regardto blob destructio∝n, this occurs support for galaxy clusters (Ettori & Fabian 2000),to φ= ∝ as expected at low velocities for small m. In consequence, 1, the value for a non-magnetized plasma. sequence IV terminates for m(gm) between 11.8 and 11.9 Not surprisingly, the predictions are highly sensitive to dex because the outflow in zone 3 is then unable to reach φ, and this might eventually be exploited diagnostically. on account of negative energy density - see Sect.4.3. With φ=1, conductive heating ofthe blobs destroysthem ∞ already at r = 1.57R where v = 873km s−1. However, Finally, sensitivity to the shadowing parameter τm is S b investigated with sequence V. Again only moderate sensi- with φ = 0.001, blobs survive out to r = 22.0R where S tivity is found. v =2286km s−1. b Table 1 also shows that coronal temperature and the hardness parameter increase as φ 0. However, the emis- 7. Conclusion → sion measure ε remains 53.4 53.5 dex after an initial ∼ − sharp rise from 53.18 dex for φ = 1. The predicted tem- The aim of this paper has been to investigate the struc- perature profiles of the coronae as φ varies are plotted in tural changes of O-star winds when Φ decreases to the ex- Fig.4. tent found for the weak-wind stars. To this end, the two- This sequence demonstrates the diagnostic potential of component phenomenological model developed originally UV and X-ray data in constraining magnetic suppression for ζ Puppis is modified to incorporate LW’s conjectures ofconductivity.TheUVdatameasuresthehighestvelocity following the breakdown of that model’s assumptions for at which wind matter transfers photon momentum to the τ Sco. When applied to a generic weak-wind star, the re- gas and the X-ray data measures the hardness of coronal visedmodelpredicts thatshock-heatingofthe ambientgas emission. givesrisetocoronaltemperatures,thatconductive-heating 7 Lucy:Coronal winds Table 1. Solutions with non-standard parameters. Seq. β logΦ logm logφ η τm vb(rS) Tmax logε I 1.0 -8.06 13.3 -3.0 0.45 1.5 2286 8.4 53.43 0.40 1.0 -8.06 13.3 -2.5 0.45 1.5 2180 8.2 53.44 0.39 1.0 -8.06 13.3 -2.0 0.45 1.5 1991 7.7 53.44 0.36 1.0 -8.06 13.3 -1.5 0.45 1.5 1678 6.2 53.47 0.30 1.0 -8.06 13.3 -1.0 0.45 1.5 1277 3.7 53.51 0.20 1.0 -8.06 13.3 -0.5 0.45 1.5 1015 2.1 53.47 0.13 1.0 -8.06 13.3 0.0 0.45 1.5 873 1.4 53.18 0.11 II 1.0 -8.26 13.3 -1.0 0.47 1.5 1109 3.5 53.27 0.19 1.0 -8.06 13.3 -1.0 0.45 1.5 1277 3.7 53.51 0.20 1.0 -7.76 13.3 -1.0 0.41 1.5 1509 3.7 53.92 0.21 1.0 -7.46 13.3 -1.0 0.37 1.5 1719 3.6 54.31 0.21 1.0 -7.16 13.3 -1.0 0.32 1.5 1918 3.3 54.70 0.21 1.0 -6.86 13.3 -1.0 0.27 1.5 2121 2.9 55.06 0.19 III 0.5 -8.06 13.3 -1.0 0.68 1.5 1348 3.6 53.14 0.17 0.6 -8.06 13.3 -1.0 0.59 1.5 1380 4.0 53.37 0.20 0.8 -8.06 13.3 -1.0 0.50 1.5 1341 3.9 53.49 0.20 1.0 -8.06 13.3 -1.0 0.45 1.5 1277 3.7 53.51 0.20 1.5 -8.06 13.3 -1.0 0.37 1.5 1135 3.2 53.50 0.17 2.0 -8.06 13.3 -1.0 0.32 1.5 1033 2.8 53.45 0.15 2.5 -8.06 13.3 -1.0 0.29 1.5 962 2.5 53.40 0.14 IV 1.0 -8.06 11.9 -1.0 0.20 1.5 889 2.6 53.81 0.14 1.0 -8.06 12.3 -1.0 0.26 1.5 989 3.0 53.74 0.16 1.0 -8.06 12.8 -1.0 0.35 1.5 1135 3.4 53.63 0.18 1.0 -8.06 13.3 -1.0 0.45 1.5 1277 3.7 53.51 0.20 1.0 -8.06 13.8 -1.0 0.55 1.5 1400 4.0 53.38 0.21 1.0 -8.06 14.3 -1.0 0.66 1.5 1503 4.2 53.23 0.22 1.0 -8.06 14.6 -1.0 0.72 1.5 1555 4.4 53.12 0.22 V 1.0 -8.06 13.3 -1.0 0.41 1.0 1322 4.2 53.67 0.24 1.0 -8.06 13.3 -1.0 0.45 1.5 1277 3.7 53.51 0.20 1.0 -8.06 13.3 -1.0 0.48 2.0 1246 3.3 53.40 0.17 1.0 -8.06 13.3 -1.0 0.50 2.5 1225 2.9 53.31 0.16 eventuallydestroystheblobs,andthattheresultingsingle- Appendix A: Heat conduction into a spherical blob component flow coasts to as a pure coronalwind. Thus, ∞ In zone 2, the blobs are surrounded by gas whose temper- in broad outline, the volumetric roles of hot and cool gas ature is rising to coronal values. Conduction will therefore in O-star winds are reversed. In the now standard picture transferheatintotheblobs,andthisconstitutesalossterm forastarsuchasζ Puppis the X-rayemitting gasoccupies in the energy equation for the ambient gas. a tiny fraction of the wind’s volume, with the bulk of the Giventhatf 1,itsufficestoconsiderasinglespher- vtroalustm,ienbtehinegpihcitguhrley-sculgugmepsteeddchoeorlegfaosrwthitehwTea∼k-Tweiqn.dInstcaorns-, icalblobwithtemb ≪peratureTbandradiusσ locatedatr =0 X-ray emitting gas fills most of the volume for r > 1.3R, in an infinite medium with T Ta as r . If ˙ is the → → ∞ C with surviving cool gas in the form of dense clum∼ps with cooling rate per unit volume and κ is the conductivity, the equilibrium temperature profile for r > σ is given by the f 0.01 0.03. b ∼ − equations As is common elsewhere in astrophysics, the approach dL =4πr2[ ˙(T) ˙(T )] (A.1) a adopted in this paper is phenomenological modelling. A dr C −C simplified picture of the phenomenon is combined with ap- and proximate treatments of the expected physical effects to dT = L (A.2) create an ’end-to-end’ tractable theory that obeys conser- dr 4πr2κ vation laws and makes testable predictions. Such theories with boundary conditions are of course always an interim measure, to be discarded when the obstacles to calculation from first principles are T(σ)=T and T( )=T (A.3) b a ∞ overcome. Unfortunately, in this case, these obstacles are Inthe blob’sabsence,thegasisisothermalandhascooling formidable: 3-D time-dependent gas dynamics, radiative transfer,andheatconductionincludingsaturationandpos- rate ˙(Ta), which is subtracted in Eq. (A.1). Accordingly, C sibly magnetic suppression. ( ) is the additional cooling due to the blob’s presence. L ∞ Of this, ∆ = ( ) (σ) represents emission from am- L L ∞ −L Evidently,thefundamentalapproachisunlikelytoyield bientgascooledbelowT bythe blob,and (σ) is the rate a L results anytime soon. Accordingly, possible improvements of heat conduction into the blob. of the crude modelling described herein should be inves- In thermal equilibrium, (σ) is balanced by emission tigated. Also diagnostic codes should incorporate features from within the blob. Now, sLince radiative cooling is ρ2 ∝ of such models to extract more reliable parameters from and ρ ρ , we expect that ∆ (σ). Therefore, to a b a ≫ L ≪ L observational data. first approximation, (r >σ)= (σ), a constant, and this L L 8 Lucy:Coronal winds allows Eq. (A.2) to be solved analytically when κ T5/2 Draine,B.T.,&McKee,C.F.1993,ARA&A,31,373 ∝ (Spitzer 1962). The resulting temperature profile is given Ettori,S.,&Fabian,A.C.2000,MNRAS,317,57 by Feldmeier,A.,1995,A&A,299,523 Graham,R.,&Langer,W.D.,1977,ApJ,179,469 t7/2 =t7/2+(1 t7/2)(1 σ) (A.4) GHauron,dJe.nH,.F,.20R1.0,,JAr&.,AB,r5a1n2d,u5a0rdi, G., Gorenstein, P., Grindlay, J., b − b − r Rosner, R., Topka, K., Elvis, M., Pye, J. P., & Vaiana, G. S. where t = T(r)/T , and the corresponding rate of heat 1979ApJ,234L,51 a Howk, J.C., Cassinelli, J.P., Bjorkman, J.E., Lamers, H.J.G.L.M. conduction into the blob is 2000,ApJ,534,348 8 Landau, L.D., & Lifshitz, E. M. 1959, Fluid Mechanics (Pergamon cl = πσ[(κT)a (κT)b] (A.5) Press) L 7 − Lucy,L.B.2010a,A&A,512,33(L10a) SinceκT T7/2, isinsensitivetoT whenT T .For Lucy,L.B.2010b,A&A,524,41(L10b) cl b a b Lucy,L.B.,&White, R.L.1980,ApJ,241,300(LW) ∝ L ≫ the solutions reported in Sects. 5 and 6, we take κ=1.0 Macfarlane,J.J.,&Cassinelli,J.P.1989,ApJ,347,1090 10−6T5/2, corresponding to Coulomb logarithm lnΛ=17×. McKee,C.F.,&Cowie,L.L.1977,ApJ,215,213 The above discussion treats conduction in the diffusion Marcolino, W. L. F.,Bouret, J.-C.,Martins, F., Hillier,D. J., Lanz, T.,&Escolano,C.2009,A&A,498,837(M09) limit - i.e., where the mean free path of the electrons is Oskinova,L.M.,Todt,H.,Ignace,R.,Brown,J.C.,Cassinelli,J.P., macroscopic length scales. In the opposite limit, heat &Hamann,W.-R.2011, MNRAS,416,1456 ≪ conduction into the blob is flux-limited and saturates at Owocki,S.P.,Castor,J.I.,&Rybicki,G.B.1988,ApJ,335,914 Spitzer, L. 1962, Physics of Fully Ionized Gases (New York: =4πσ2q (A.6) Interscience) sat sat L Vieser,W.,&Hensler,G.2007,A&A,475,251 where q is estimated by Cowie & McKee (1977) to be Vink,J.S.,deKoter,A.,&Lamers,H.J.G.L.M.2000,A&A,362,295 sat 2kT 1/2 e q =0.4 n kT (A.7) sat e e (cid:18)πm (cid:19) e and is here evaluated at T ,(n ) . Interpolating between a e a these limits (cf. Balbus & McKee 1982), we take the rate of heat conduction into the blob to be , where cond L −1 = −1+ −1 (A.8) Lcond Lcl Lsat At high temperatures, this gives T3/2 in place of cond T7/2. L ∝ cl L ∝ Theaboveformulaisforanon-magnetizedplasma.But sincestarsinandneartheweak-winddomainhavedetected magnetic fields (e.g., Oskinova et al. 2011), we include the possibility of magnetic suppression of heat conduction by writing =φ (A.9) in cond L L Inthisinvestigation,φisvariedtoexploreitsimpactonthe solutions. In future, it may be determined or constrained by fitting observational data. The suppression of thermal conductivity in astrophysi- cal plasmas has been strikingly confirmed by the discovery of cold fronts in X-ray maps of clusters of galaxies (e.g., Carilli&Taylor2002).FortheclusterAbell2142,Ettori& Fabian (2000) estimate a reduction factor of between 250 and 2500. They speculate that, as a result of merging, different magnetic structures are in contact and so remain to high degree thermally isolated. The displacements of wind clumpsfromtheirnascentambientsurroundingsmightwell lead similarly to substantial reduction factors. References Balbus,S.A.,&McKee,C.F.1982, ApJ,252,529 Bouret,J.-C.,Lanz,T.,&Hillier,D.J.2005,A&A,438,301 Bouret, J.-C., Lanz, T., Hillier, D. J., & Foellmi, C. 2008, in Clumping in Hot Star Winds, W.-R. Hamann, A.Feldmeier, & L.M.Oskinova,eds.(Potsdam:Univ.-Verl.) Carilli,C.L.,&Taylor,G.B.2002, ARA&A,40,319 Cowie,L.L.,&McKee,C.F.1977,ApJ,211,135 DeYoung,D.S.,&AxfordW.I.1967, Nature,216,129 Dere, K.P., Landi, E.,Young, P. R.,Del Zanna, G., Landini, M.,& Mason,H.E.2009,A&A,498,915 9

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