Draftversion February5,2008 PreprinttypesetusingLATEXstyleemulateapjv.10/09/06 A NOZZLE ANALYSIS OF SLOW-ACCELERATION SOLUTIONS IN ONE-DIMENSIONAL MODELS OF ROTATING HOT-STAR WINDS Thomas I. Madura and Stanley P. Owocki BartolResearchInstitute, DepartmentofPhysics&Astronomy,UniversityofDelaware,Newark,DE19716 Achim Feldmeier Astrophysik,Institut fu¨rPhysik,Universita¨tPotsdam,AmNeuenPalais10,14469Potsdam,Germany Draft versionFebruary 5, 2008 ABSTRACT 7 0 One-dimensional(1D)stellarwindmodelsforhotstarsrotatingat&75%ofthecriticalrateshowa 0 sudden shift to a slow-accelerationmode, implying a slower, denser equatorial outflow that might be 2 associatedwiththe dense equatorialregionsinferredfor B[e]supergiants. Herewe analyzethe steady 1D flow equations for a rotating stellar wind based on a “nozzle” analogy for terms that constrain n the local mass flux. For low rotation, we find the nozzle minimum occurs near the stellar surface, a J allowing a transition to a standard, CAK-type steep-acceleration solution; but for rotations & 75% of the critical rate, this inner nozzle minimum exceeds the global minimum, implying near-surface 1 3 supercritical solutions would have an overloaded mass loss rate. In steady, analytic models in which the acceleration is assumed to be monotonically positive, this leads the solution to switch to a slow 1 accelerationmode. However,time-dependentsimulationsusinganumericalhydrodynamicscodeshow v that, for rotation rates 75 - 85% of critical, the flow can develop abrupt “kink” transitions from a 7 steep acceleration to a decelerating solution. For rotations above 85% of critical, the hydrodynamic 0 simulations confirm the slow acceleration, with the lower flow speed implying densities 5 - 30 times 0 higher than the polar (or a nonrotating) wind. Still, when gravity darkening and 2D flow effects are 2 accountedfor,it seems unlikely that rotationallymodified equatorialwind outflowscould accountfor 0 the very large densities inferred for the equatorial regions around B[e] supergiants. 7 Subject headings: hydrodynamics — stars: early-type — stars: mass loss — stars: winds, outflow — 0 stars: emission-line, Be — stars: rotation / h p - 1. INTRODUCTION sociated with the reduced radiation temperature near o the equator can lead to a factor several enhancement r Hot,luminous,massivestarsofspectraltypesOandB t are generally quite rapid rotators, with inferred surface in the radial mass flux. By itself, this seems inade- s quate to explain equatorial densities estimated to be a rotationspeedstypicallyintherangeofseveral100km/s, : orasubstantialfractionofthecriticalrotationspeed(ca. hundreds or even thousands of times the densities of v the polar winds in these stars (Zickgraf et al. 1985; 400-600km/s) atwhich materialat the rotatingequato- i X rial surface would be in Keplerian orbit. Such luminous Kraus & Miroshnichenko 2006). However, a recent se- ries of papers by M. Cur´e and colleagues (Cur´e 2004; r stars are also characterized by strong stellar wind out- a flows, driven by the line-scattering of the star’s contin- Cur´e & Rial 2004; Cur´e et al. 2005) proposes that, for veryhigh,near-criticalrotation,aswitchofthewindout- uumradiationflux (Castor et al.1975, hereafterCAK). flowtoaslower,shallow-accelerationsolutioncanleadto A longstanding question is how such wind outflows are afurtherenhancementindensitythat,togetherwiththe affectedbythestar’srotation,andinparticular,whether bi-stability effect, might reach the equatorial densities this mightplayaroleinthe enhancedequatorialdensity inferred in B[e] supergiants. outflows and/or disks inferred in certain classes of par- Thepresentpaperaimstounderstandbetterthephys- ticularly rapid rotators,e.g., Be and B[e] stars. ical origin of these shallow wind acceleration solutions For classical Be stars, there is now substantial obser- for high rotation rates, and to examine critically their vational evidence (see, e.g., papers in Ignace & Gayley likely relevance for explaining dense equatorial disks or 2005) that the disks are Keplerian in nature, with very outflows. limited radial outflow; they are thus probably not a di- Modelling rotating, hot-star winds began with the rect result of feeding by a steady stellar wind (Owocki studiesbyFriend & Abbott(1986, hereafterFA)andby 2005). However, for the disk and/or enhanced equa- Pauldrachet al. (1986, hereafter PPK), who extended torial outflows inferred in supergiant B[e] stars, wind the CAK formalism by adding the effect of an outward mechanisms seem still a viable option. For example, centrifugal acceleration to one-dimensional (1D) models Lamers & Pauldrach (1991) and Pelupessy et al. (2000) of the wind outflow in the equatorial plane. Both FA have noted that the “bi-stability” enhancement in opac- and PPK independently derived a modified CAK model ity which occurs for some value of the local surface ef- (“mCAK”) that relaxes the CAK “point-star” approx- fective temperature (i.e. for B-type stars at 22,000 ∼ imation and properly accounts for the finite cone-angle K, see Pauldrach& Puls (1990); Vink et al. (1999)) as- subtended by the star. They then each found that the Electronicaddress: [email protected]@[email protected]ffeective gravity by the outward cen- 2 trifugal force tends to increase the mass-loss rate and account of the multi-dimensional effects noted above. decrease the wind speed. However, for the models com- Nonetheless, to provide a solid basis for such multi- puted,uptoabout75%ofthecriticalrotationrate,both dimensional models, it is important to have a clearer, changes are limited to only a factor of a few and are dynamicalunderstanding of these novel1D slow outflow thusinsufficienttoproducethelargeequatorialdensities solutions. Combininganalyticstudieswithnumericalhy- and low velocities inferred in B[e] supergiants. More- drodynamic simulations, this paper examines the reality over, for still faster rotation, above about 75% criti- and physical origin of the shallow wind acceleration so- cal, FA found that the equations for outward accelera- lutions for high rotation rates, with emphasis on their tion could no longer be integrated beyond some finite possiblerelevancetodiskoutflowsfromB[e]supergiants. radius, and thus they were unable to derive any com- We begin( 2)withabasicreviewofthe generaltime- § plete flow solutions for such rapid rotation speeds. Sub- dependent wind equations, together with their CAK- sequent1Dmodelshaveinvestigatedtheroleofmagnetic type, steady-state solutions in a non-rotating wind. To forces (Friend & MacGregor 1984; Poe & Friend 1986) provideaphysicalbasisfor extending these steady mod- and sound-waves (Koninx & Hearn 1992), but as sum- elstoincluderotation,wefirst( 3)applyasimple“noz- § marizedbyBjorkman & Cassinelli(1993)neithermecha- zle” analysis (see Holzer 1977; Abbott 1980), originally nismseemsfavorabletoproducingslow,denseequatorial developed to study winds driven from luminous accre- outflows. tion disks (Pereyra et al. 2004). With a few judicious, Morefundamentally,thephysicalrelevanceofanysuch yetquite reasonableapproximations(e.g.,neglectinggas 1D models may be limited since accounting for latitu- pressure terms by taking the zero-sound-speedlimit; us- dinal flows toward or away from the equator requires ing a beta velocity law to evaluate the finite-disk correc- at least a two-dimensional (2D) treatment. A vivid tion as an explicit function of radius), it is possible to example comes from the 2D “Wind Compressed Disk” obtainsimple integrationsofthe equationofmotionand (WCD) model of Bjorkman & Cassinelli (1993), which to study the scalings of the mass loss rate with rotation, argues that conservation of angular momentum should as well as the switch from steep to shallow acceleration tend to channel material from higher latitudes toward solutions beyond a threshold rotation rate. To test the an equatorially compressed disk flow. If one assumes a validityofthese simpleanalyticsolutions,wenextapply purely radial driving force, 2D hydrodynamical simula- a numerical hydrodynamics code ( 4) to evolve 1D ro- § tions(Owocki et al.1994)confirmthebasicWCDeffect, tating wind models to asymptotic steady states ( 5.1). § but show that, depending on whether material reaches Resultsconfirmatransitiontosloweraccelerationatvery the equator above or below some “stagnation point”, it highrotation(aboveabout85%ofcritical),butalsoshow either drifts outward or falls back toward the star. Such anewclassofnon-monotonic“kink”solutionsthatapply simultaneous infall-plus-outflow behavior is not possible formoderately fastrotation(ca. 75-85%of critical). We inasteady1Dmodel,butisaperfectlynaturaloutcome thenexamine( 5.2-5.3)thetimeevolutionofsolutionsin § in a 2D simulation. Furthermore, Cranmer & Owocki various rotation domains, with emphasis on the kink so- (1995) showed that, when computed from proper angle lutions,andonapeculiartransitioncase(85%ofcritical integration of intensity from the rotationally distorted rotation), characterized by an initial wind overloading stellar surface, the line-force also has nonzero, nonradial followed by a flow stagnation and eventual reaccretion componentsinbothazimuthandlatitude. Owocki et al. of material onto the star. We conclude with a summary (1996, seealsoPetrenz & Puls(2000))showedthatsuch and outlook for future work ( 6). § nonradiallineforcescaninhibit theformationofaWCD. 2. GENERALFORMALISMFORLINE-DRIVENMASS-LOSS Finally, equatorialgravitydarkeningcan actuallyreduce the wind mass flux from the equator, and so lead to an 2.1. 1D Time-Dependent Equations of Motion equatorial wind density that is lower, not higher, than In this paper, we examine 1D radiatively driven out- near the poles. flow in the equatorial plane of a rotating star. For the Despite this likely importance of such 2D effects, general time-dependent simulations discussed in 3, the several recent analyses (Cur´e 2004; Cur´e & Rial 2004; relevant equations for conservation of mass and§radial Cur´e et al.2005)havereexaminedthe1Dequatorialflow component of momentum have the form models of FA, with a particular focus on the failure to ∂ρ 1 ∂(r2ρv) obtain monotonically accelerating wind solutions for ro- + =0 (1) tation above about 75% of the critical speed. In partic- ∂t r2 ∂r ular, these papers argue that for such very high, near- ∂v ∂v vφ2 1∂P GM∗(1 Γe) critical rotation, the wind solution can switch to an al- +v = − +g , (2) ∂t ∂r r − ρ ∂r − r2 lines ternative mode, characterized by a much slower out- wardacceleration. Togetherwithamoderatelyenhanced whererandtaretheradiusandtime,andρandvarethe mass flux, the resulting lower-speed outflow then im- mass density and radial component of the velocity. The pliesasubstantialenhancementindensity,relativetothe body forces here include the outward radiative accelera- standard CAK, steep acceleration applicable at higher tion from line-scattering, g , and an effective inward lines latitudes. When combined with parameterizations in- gravitational acceleration GM∗(1 Γe)/r2, reduced by − tended to mimic a “bi-stability” enhancement in the the outward continuum radiative force from scattering line-driving (Lamers & Pauldrach 1991; Pelupessy et al. by free electrons, as accounted for by the Eddington pa- 2000), Cur´e et al.(2005) predict equator-to-poledensity rameter Γe = κeL∗/4πGM∗c. For the centrifugal term, contrasts of the order of 102 104. v2/r, we avoid explicit treatment of an azimuthal mo- − φ Assessing the physical relevance of such claims for mentum equation by assuming simple angular momen- understanding B[e] stars will eventually require proper tum conservation(which is a good approximationin the Nozzle Analysis of Rotating Hot-Star Winds 3 supersonic flow domain considered here), yielding then where α is the CAK power index. Here we have elimi- for the azimuthal speed nated aninverse dependence on density ρ in favorof the mass loss rate M˙ =4πr2ρv, with the line force constant R∗ v =v , (3) thus defined by φ rot r where vrot is the rotation speed at the star’s equatorial C 1 L∗ α Q¯Γe 1−α , (8) surfaceradiusR∗. Forsimplicity,wealsoavoidadetailed ≡ 1−α(cid:20)M˙ c2(cid:21) (cid:20)1−Γe(cid:21) treatment of the wind energy balance by assuming an with L∗ the stellar luminosity, and Γe the Eddington isothermal outflow, for which the pressure is written as parameter for the gravitationally scaled radiative ac- P = ρa2, where a is the (constant, isothermal) sound celeration from electron scattering opacity, κ (cm2/g) e speed. (Lamers & Cassinelli 1999; Owocki 2004). We have also usedtheGayley(1995)Q¯notationfortheoverallnormal- 2.2. Steady-State Equations with Rotation izationofthelineopacity. Notethat,forfixedsetsofstel- For the simplified case of a steady state, the time- lar parameters (L∗,M∗,Γe) and line-opacity (α,Q¯), the dependent terms vanish (∂/∂t = 0), yielding for the constantC scales with the mass loss rate as C 1/M˙ α. ∝ steady acceleration As already noted, the smallness of the dimension- less sound-speed parameter w implies that gas pressure vdv = GM∗(1−Γe) + vr2otR∗2 +g a2dρ. (4) playslittle roleinthe dynamicss ofany line drivenstellar dr − r2 r3 lines− ρ dr wind. Hence, to a good approximation, we can obtain accurate solutions by analyzing the much simpler limit The steady form for mass conservation implies a con- of vanishing sound speed a √w 0, for which the stancyfortheoverallmasslossrate,M˙ 4πρvr2. Using ∝ s → line-driven-wind equation of motion reduces to ≡ this to eliminate the density ρ, the equation of motion w′ = 1+ω2(1 x)+Cw′α. (9) (4) takes the form − − 2.3. Classical CAK Solution for a Point-Star (cid:20)1− av22(cid:21)vddvr =−GM∗(r12−Γe)+vr2ort3R∗2+glines+2ra2 . Letusfirstreviewthe standardCAK solutionwithout rotation, setting ω = 0. Note then that since the pa- (5) rametersΓ andC are spatiallyconstant,the solutionis Thesquare-bracketfactorontheleft-hand-sideallowsfor e independent of radius. For high M˙ and small C there a smooth mapping of the wind base onto a hydrostatic atmosphere below the sonic point, where v < a. How- arenosolutions,while for smallM˙ andhighC there are ever, in radiatively driven winds the pressure terms on two solutions. The CAK critical solution (denoted by the right-hand-side are generally negligible since, com- thesubscriptc)correspondstoamaximal masslossrate, pared to the gravitational acceleration term that must whichrequiresatangential intersectionbetweentheline- be overcome to drive a wind, these are of order w forceCw′α andthe combinedinertiaplus gravity1+w′, s (a/vesc)2 ≈ 0.001, where vesc ≡ 2GM∗(1−Γe)/R∗ ≡is for which αC w′α−1 =1, (10) the effective escape speed from thpe stellar surface radius c c R∗. and thus, together with the equation of motion (9), we Sincethekeytoastellarwindistoovercomegravity,it have α isconvenienttodefineadimensionlessequationofmotion w′ = , (11) that scales all accelerations by the effective gravity, c 1 α − with (1−ws/w)w′ =−1+ω2(1−x)+Γlines+ (14wsx), (6) Cc = αα(1 1α)1−α . (12) − − whereΓlines glinesr2/GM∗(1 Γe),andthegravitation- Usingeq. (8),this thenyieldsthe standardCAKscaling allyscaledin≡ertiaisw′ dw/d−x=r2v(dv/dr)/GM∗(1 for the mass loss rate, ≡ − Γcoeo).rdTinhaeteinxdepe1ndeRn∗t/vra,rwiahbilleehtheerediesptehnedeinnvtevraseriarabdleiuiss M˙ = L∗ α Q¯Γe (1−α)/α . (13) the ratio of th≡e r−adial kinetic energy to effective surface CAK c2 1 α(cid:20)1 Γe(cid:21) − − escape energy, w v2/v2 . Gas pressure effects are ac- Moreover,sincethe scaledequationofmotion(9)hasno tcroiufungteadl efoffrecbtystferro≡mmsrcootnaettsaicoinninagrewcsh≡araa2c/tevre2iszce,dwihniletecremns- etixopnliwci′tasppaptliieasl tdherpoeungdheonucte,ththeewsincadl.edThcrisitcicaanl tahcecreeleforrae- c of the ratio of the equatorial rotation speed to critical be trivially integrated to yield speed, ω vrot/vcrit = √2vrot/vesc, under the assump- w(x)=w(1)x, (14) ≡ tionthatthewindmaterialconservesits surfacevalueof wherew(1)=α/(1 α)istheterminalvalueofthescaled specific angular momentum, rvφ(r)=vrotR∗. flowenergy. Interm−sofdimensionalquantities,this rep- WithintheCAKformalismfordrivingbyscatteringof resentsaspecific caseofthe general“beta”-velocity-law, a point-source of radiation by an ensemble of lines, the β deshadowingofoptically thick lines by the Doppler shift R∗ associated with the wind acceleration gives the scaled v(r)=v∞(cid:18)1− r (cid:19) , (15) radiative acceleration Γ a power-law dependence on lines where in this case β =1/2,andthe wind terminal speed the flow acceleration w′, scales with the effective escape speed from the stellar Γlines =Cw′α, (7) surface, v∞ =vesc α/(1 α). − p 4 1.1 wind. For a rotating star and wind, the fdcf can be- b=1 f 1 come even more complicated, modified by the rotational shear of the wind outflow, and by the oblateness of the 5 0.9 =0. star, and possibly also by the equatorial gravity dark- b 0.8 b= 5 eGnaiynlgeyof&thOewsoocukrice20r0a0d)ia.tiHoonw(eCvrearn,mfoerrs&imOplwicoitcyk,il1e9t9u5s; nonetheless base our analysis on the spatially explicit 0.7 form obtained by assuming a canonical β = 1 velocity law (15) within the finite disk factor (16) for a simple 0.6 spherical expansion. In the zero sound-speed limit, the scaled equation of motion (9) can now be written in the 0.2 0.4 0.6 0.8 x 1 form, Fig. 1.— Spatialvariationofthefinite-disk-correctionfactorf, w′ α plottedvs. scaledinverseradiusx=1 R∗/r,forCAKexponent w′ = 1+ω2(1 x)+fCc , (17) α = 1/2 and various velocity-law expo−nents, β = 0.5, 1, 2, and − − (cid:18)m˙ (cid:19) 5. The horizontal dashed curves denote the unit correction that wherewe have normalizedthe line force interms related 1ap=pl0ie)s,aatndthaetploairngteodfiisstoatnrcoepsicwehxepraentshioenst(awrhaeprperσoa=chdeslntvh/edplonirn−t- to the “point-star” CAK model, with m˙ M˙ /M˙CAK, ≡ sourceformassumedintheoriginalCAKmodel. the ratio of the mass-loss rate to the point-star CAK value. Note then that for the non-rotating (ω = 0), 2.4. Finite-Disk Form for the CAK Line-Force point-star (f = 1) case of the classical CAK model, the critical solution (with maximal mass loss) is given by Theaboveanalysisisbasedontheidealizationofradi- m˙ = 1 and w(x) = α/(1 α)x. As noted in 2.3, this allystreamingradiation,asifthestarwereapointsource implies a CAK mass loss−rate M˙ = M˙ a§nd a ve- at the origin. This was the basis of the original CAK CAK wind solutions, although they did already identify (but locity law, v(r) = v∞ 1−R∗/r, with terminal speed did not implement) the appropriate “finite-disk correc- v∞ = α/(1 α)vesc.p − tion factor” (fdcf) to account for the full angular extent To apnalyze models with rotation, a particularly con- of the star (see CAK eq. [50 ]), venient case is to take α = 1/2, for which the equation of motion (17) (using eq. (12) for C ) becomes a simple c f(r)= (1+σ)1+α−(1+σµ2∗)1+α , (16) quadratic in √w′, (1+α)σ(1+σ)α(1 µ2) − ∗ w′ 2f w′/m˙ +g(x)=0, (18) − with µ∗ ≡ 1−R∗2/r2 the cosine of the finite-cone an- whereforconveniencep,we havedefinedarotationallyre- gle of the sptellar disk, and σ ≡ dlnv/dlnr−1. When ducedgravityasg(x) 1 ω2(1 x). Wecanthensolve thisfactorisincludedtomodify the point-starCAKline ≡ − − for a shallow (–) and steep (+) acceleration solution, acceleration (from eq. [7]), its complex dependence on radius, velocity, and velocity gradient complicates the g(x)n(x) 2 w′ (x)= 1 1 m˙/n(x) , (19) solution of the full equation of motion. Full solutions ± m˙ h ±p − i derived independently by FA and PPK yield a some- with the “nozzle function”, what reduced mass-loss rate M˙ M˙ /(1+ α)1/α fd CAK and higher terminal speed v∞ ≈3v≈esc. n(x) f(x)2 = f(x)2 . (20) But if we approximate the wind velocity law by the ≡ g(x) 1 ω2(1 x) simple “beta-law” form of eq. (15), then the fdcf can − − The significance of this nozzle function stems from its be evaluated as an explicit spatial function. Figure 1 appearancewiththe masslossratem˙ within the square- illustratestheresultingvariationoff withthescaledco- root discriminant(cf. Laval nozzle; Abbott (1980)). In ordinatexforα=1/2andvariousvaluesofβ. Notethat particular,we canreadilysee that maintaining a numer- the overall form is quite similar for all cases, increasing ically real flow acceleration requires1 a mass loss rate fromasurfacevaluef∗ f(R∗)=1/(1+α)topastunity ≡ m˙ min[n(x)]. As such, the locationofthe globalmini- at the isotropic expansion radius (where dv/dr = v/r), ≤ mum of this function (the smallest nozzle “throat”)rep- r/R∗ = (1+β) [corresponding to x = β/(1+β)], and resents the critical point that sets the maximal allowed eventually returning asymptotically to unity from above value of the mass loss rate, m˙ = min[n(x)], that is con- at large radii (x 1). → sistent with a monotonically accelerating outflow. Inthesteadywindanalysisinthenextsection,wethus Figure 2 plots n(x) vs. x for various rotation rates choosethe canonicalvalue β =1 to representthe fdcf as ω, using a β = 1 velocity law to obtain a spatially ex- an explicit spatial function (PPK). plicit approximation to the fdcf. Note that for no or 3. STEADY-STATESOLUTIONSFOR1DMODELSOF low rotation (about ω <0.75), the minimum of the noz- ROTATING,LINE-DRIVENSTELLARWINDS zle function is less than unity, and occurs at the stellar 3.1. Nozzle Analysis for Steady Wind Acceleration 1 Actually,thisrestrictionreallystemsfromourCAKscalingof Let us now examine how the combined effects of the theline-forcewithw′α(inthiscase√w′),whichrequiresastrictly fdcf and rotation alter the classical CAK result. Note positive acceleration, w′ > 0. But if we provide a backup scaling that we are ignoring here gravity darkening and oblate- fornegative accelerations, then“overloaded” situations,forwhich thesquare-rootdiscriminantineq. (19)becomes negative, simply ness effects, as well as any “bi-stability” in the line- lead to an abrupt switch, a so-called “kink” (Cranmer&Owocki driving parameters between the polar and equatorial 1996),toadecelerating solution. See 4and6forfurtherdetails. § Nozzle Analysis of Rotating Hot-Star Winds 5 3 pastthethresholdrateatω 0.75,thesolidcurvesshow ≈ an abrupt shift from steep acceleration to shallow accel- n w=0 eration, with the mass loss saturating to the point-star 2 .95 CAK value, m˙ = 1. The dashed curves show extrapo- w=0.8 lated results if the local nozzle minimum at the surface is instead used to set flow conditions; the mass loss in this case is set by the scaling m˙ in eq. (21), and the 0 1 terminal speed is derived by assuming a pure gravita- w=0 tional coasting for all radii with n(x) < m˙ . The data 0 points again compare corresponding results for the full 0 dynamical simulation, as described further in the next 0.2 0.4 0.6 0.8 1 sections. x 4. SPECIFICATIONSFORNUMERICALHYDRODYNAMICS Fig. 2.— Nozzlefunctionn(x)plottedvs. scaledinverseradius SIMULATIONS x=1 R∗/r,forvariousrotationratesω=0,0.5,0.6,0.7,0.8,0.9, 0.95,r−angingfromlowermosttouppermost. Allcurvesuseaβ=1 The above nozzle analysis provides a helpful frame- velocity law in evaluating the fdcf. The horizontal dashed line at workforunderstandingthe natureofflowsolutionsfrom unit value represents the nozzle function for the CAK point-star a rotating wind model, but is based on some key simpli- model,withn=f =g=1. fications, e.g. neglect of gas pressure (inclusion of which would lead to a wind that is not super-critical directly surface, x = 0. This allows the flow to transition to from the static surface boundary) and an approximate, a super-critical outflow directly from the static surface spatiallyexplicitformforthefinite-diskcorrectionfactor. boundaryconditionw(0)=0,followingthesteeper,plus Moreover,itimplicitlyassumesthatthederivedsolutions (+) root for the acceleration in equation (19), but with are the only relevant stable, attracting steady states for a mass loss rate less than the point-star CAK value, the rotating wind outflow. To test the validity of these f2(x=0) 4/9 simplifications and assumptions, let us now examine the m˙ =m˙0 ≡n(x=0)≡ 1 ω2 = (1 ω2). (21) time evolution of analogous 1D flow models, including − − also both the finite gas pressure and the dynamically Note that the factor 4/9 in the numerator is just the computed finite disk correction factor. Our specific ap- α=1/2 value for the zero-rotation,finite-disk-corrected proach here is to use a numerical hydrodynamics code mass loss scaling derived by FA and PPK, to evolve a 1D time-dependent model of the equatorial plane a line-driven stellar wind from a rotating star to- m˙fd ≡ M˙M˙fd =f∗1/α = (1+1α)1/α . (22) wTahredreasnuletqsuoiflitbhreiusemsismteualadtyi-osntsatceanfotrhtehnebreecsuomltipnagrefldowto. CAK those predicted by the steady-state nozzle analysis pre- By contrast,for large rotationrates (about ω >0.75), sented above. this nozzle minimum is unity, and occurs at large radii, The numerical models presented here were com- x = 1; satisfying the static surface boundary condition puted using a piecewise parabolic method (PPM; nowimpliesthattheflowatallfiniteradiishouldremain Collela & Woodward 1984) hydrodynamics code called sub-critical, following the shallower, minus (–) root for VH-1, originally developed at the University of Vir- the acceleration in equation (19), now with a mass loss ginia (J. Blondin). The basic code was modified for the rate just equal to the point-star CAK value, m˙ =1. present study to include radiative driving terms, solv- This thus provides the basic explanation for the ing the time-dependent equations for 1D spherical out- switch from steep to shallow accelerations inferred by flow (1) and (2). The spatial mesh uses n = 600 ra- r Cur´e et al. (2005). dialzonesextending fromthe base atthe stellar surface, 3.2. Nozzle SolutioRnsotfaotrintgheWVienldoscity Law in 1D rw1it=h tRhmeinzo=neRs∗p,actoinag smtaarxtiimngumatr∆60r01==R6.m1a8x×=1100−05RR∗∗, andthenincreasingby2.5%perzoneouttor350 =15R∗, The associated wind velocity laws can be obtained after which it remains constant at ∆r =0.333R∗ to the by simple numerical integration of equation (19) from outerboundary. Testswithdoublethegridresolutionfor a static boundary w(0) = 0, following either the steep selected cases give similar results to the above standard or shallow solution, depending on whether the rotation values. rateishighenoughtoshiftthecriticalpoint(wheren(x) The parabolic method requires flow variables to be has its absolute minimum) from the surface (x = 0) to specified in a phantom zone beyond each boundary. At largeradii(x=1). Figure3aplotstheresultingvelocity the outer radius, we assume supercritical outflow, with laws for selected slow vs. rapid rotation rates, yielding boundary conditions set by simple flow extrapolation as- respectively the steep vs. shallow types of flow solution. suming constant gradients. This is justified because, ThedashedcurveinFigure3aplotstheescapespeedv as discussed in the Appendix, when finite sound-speed esc as a function of x, showing that these winds are capable termsareincluded,thecriticalpointforeventheshallow of escaping the star. The right panel compares results accelerationsolutionsshouldbewellwithinourassumed from full dynamical simulations described below. outer boundary radius of Rmax =100R∗. Figure 4 illustrates the associated terminal speed and At the inner boundary, the velocity in the two radial masslossratesforthesesolutions(solidlines),plottedas zones below i is set by constant-slope extrapolation, min a function of rotation rate ω. As the rotation increases thus allowing the base velocity to adjust to whatever is 6 = 0 = 0 w = 0.5 w v/vesc2 ww = 0.6 v/vesc2 w == 00..65 1.5 ww ==00.7.74 1.5 ww = 0.7 w =0.75 1 1 w =0.80 w =0.82 0.5 ww ==00..7955 0.5 w =0.84 ww ==00.8.965 w =0.85 0 0 0.00 0.25 0.50 0.75 x 0.00 0.25 0.50 0.75 x Fig. 3.— (a) Flow speed over escape speed, v/vesc = √w, plotted vs. scaled inverse radius x = 1 R∗/r, as derived from a nozzle analysis, using a steep acceleration for no or modest rotation, ω = 0, 0.5, 0.6, 0.7, 0.74 and shallow ac−celeration for rapid, near-critical rotation, ω=0.75, 0.8,0.85, 0.9,0.95. Thedashed curveshowsthe escapespeed asafunctionofx. b)Sameplotasin(a), but asfound using asymptotic states of full hydrodynamical simulations. Note again the steep, supercritical accelerations for no or modest rotation, ω = 0, 0.5, 0.7, and shallow, subcritical accelerations for rapid, near-critical rotation, ω =0.86, 0.9, 0.95. However, note also the “kink” solutionspresentforω=0.75,0.8,0.82,0.84andthecollapsedsolutionforω=0.85. entire spacialmeshat some starting time t=0. For this we generally use a standard, finite-disk-corrected CAK 3 wind, computed by relaxing a 1D, non-rotating simu- lation to a steady state; however, for selected models 2.5 with moderately rapid rotation, we also explore using v¥ /v a slow-acceleration initial condition (see 5.3). From 2 esc § the assumed initial condition, the models are advanced forward in time steps set to a fixed fraction 0.25 of the 1.5 Courant time. OurversionofVH-1issetuptooperateinCGSunits, 1 requiring specification of physical values for the basic m parameters for both the star (e.g., mass, radius, lumi- 0.5 nosity) and wind (e.g., CAK k, α, and δ). Building upon our earlier studies of Be stars, the specific pa- 00 0.25 0.5 0.75 w 1 rameters chosen here are for a main sequence B star with mass M∗ = 7.5M⊙, radius R∗ = 4R⊙, luminosity Fig. 4.— Upper solid curve: Terminal flow speed over escape L∗ = 2310L⊙, and temperature T = 2 104 K; but we speed, v∞/vesc = pw(1), plotted vs. rotation rate ω, showing have also explored models with parame×ters appropriate the shiftfrom fast to slow wind as rotation rate is increased past forsupergiantB[e]stars. Thesestellarparametersimply ca. ω =0.75. Lower solidcurve: Mass loss rateinunits of point- star CAK value, again plotted vs. rotation rate ω, showing the an Eddington parameter of Γe = 0.008, an isothermal saturationattheCAKmasslossrateforrapidrotation,ω>0.75. sound speed of a = 16.6 km/s, and escape and critical Dashedcurvesshowthecontinuedsteadydecreaseinv∞/vesc and speedsofv =845km/sandv =597km/s. We also increase in m˙ if the local nozzle minimum at the stellar surface esc crit is used to set flow conditions ( 3.2). The circles (v∞/vesc) and assume a CAK power-law index of α = 1/2 and cumu- triangles(m˙)showcorrespondin§gresultsfromfullhydrodynamical lative line-strength parameter Q = 1533 (Gayley 1995). simulations. The value of δ has been set to zero in all simulations. In any case, for a given choice of the CAK power-law index α, we find that results are largely independent of appropriate for the overlying flow (Owocki et al. 1994). the specific physical parameters when cast in appropri- This usually corresponds to a subsonic wind outflow, al- atelyscaledunits,normalizingforexampleradiusbythe though inflow at up to the sound speed is also allowed. The base density is fixed at ρ =8.709 10−13 g cm−3, stellar radius R∗, velocity by the wind terminal velocity 0 × v∞ (which in turn scales with the stellar surface escape a value chosen because, for the characteristic wind mass speed), time by the characteristic flow time R∗/v∞, and fluxes of these models, it yields a steady base outflow mass loss rate in terms of the classical(point-star)CAK that is moderately subsonic. A lower-boundary density value given in eqn. (13). To faciliate comparison with much smaller than this produces a base outflow that is the analytic nozzle analysis in 2-3, we again choose supersonic, and thus is unable to adjust properly to the §§ α= 1/2, and plot all simulation results using the above mass flux appropriate to the overlying line-driven wind. scalings. Ontheotherhand,amuchlargerbasedensitymakesthe We further note that essentially all the key VH- lower boundary too “stiff”, leading to persistent oscilla- 1 results reported here were very well reproduced tions in the base velocity (Owocki et al. 1994). by a completely independent, simple dimensionless These time-dependent simulations also require setting hydrodynamics code developed by one of us (AF; an initial condition for the density and velocity over the Nozzle Analysis of Rotating Hot-Star Winds 7 Feldmeier & Nikutta 2006). Figure 4 shows that the fully dynamical results for Finally, in such time-dependent simulations of line- the scaled ratio of right boundary speed (circles) and driven winds, one must also supply a generalized scal- CAK scaled mass loss rate (triangles) are generally in ing for the line-force that applies in the case of non- goodagreementwiththepredictionsofthesimplenozzle monotonicflowacceleration. Ingeneralthisrequirestak- analysis (solid curves), with the modest, ca. 10% differ- ing into account non-local couplings of the line-transfer ences likely attributable to inclusion in the simulations (see, e.g., Feldmeier & Nikutta 2006), but we wish here of a small but finite sound speed (Owocki and ud-Doula to retain the substantial advantages of using a purely 2004). However, for rotation rates 0.75 < ω < 0.85, the local form for the line-driving. Noting that a nega- dynamical results tend to follow the dashed curves of tive velocity gradient implies a prior line resonance that the extended nozzle analysis, representing extended fast shadows radial photons from the star, a lower limit solutions rather than the abrupt shift to slow solutions would be just to set g = 0 whenever dv/dr < 0. indicated by the solid curves. As discussed above, this line On the other hand, since forward scattering can sub- rangeofrotationratesischaracterizedbykinksolutions. stantially weaken any such shadowing by a prior reso- To understand better this development of fast vs. slow nance,anupperlimitwouldbetocomputethelocalline vs. kinksolutions,letusnowexaminethetimeevolution force using the absolute value of the velocity gradient, of the simulations toward asymptotic states. g dv/drα. As a simple compromise between these line ∝| | 5.2. Time Relaxation of 1D Rotation Models two extremes, we choose here a scaling that truncates the radial velocity gradient to zero whenever it is nega- For the specific rotation rates ω = 0.7, 0.8, 0.84, & tive,i.e. dv/dr max(dv/dr,0). Forapoint-starmodel 0.90, which span the parameter range between fast and → with radially streaming radiation, this would give a zero slowaccelerationsolutions,Figure5usesgray-scaleplots line-force (since dv/dr < 0), but when one accounts for of the mass-loss rate (in units of the point-star CAK the lateral expansion v/r within the finite-disk correc- value) to illustrate the time relaxation from the CAK tion factor, it leads to a line force in which the usual initial condition (set to a non-rotating, finite-disk cor- dependence on radial velocity gradient is replaced by a rectedCAKmodel)toanasymptoticsteady-state. Time dependence on the expansion gradient, isgiveninunitsoftheflowtimet=R∗/v∞,andthefinal g (v/r)α. (23) valuesofthescaledmass-lossrateareindicatedabovethe line ∝ This leads to a line accelerationthat is intermediate be- associated plot. Note that there are distinct differences tween the underestimate and overestimate of the two inthe timeevolutionofeachmodel,withthe morerapid more extreme scalings. rotationcases characterizedby a longer relaxationtime, butthe temporalandspatialconstancyofthe finalmass 5. COMPARISONWITHTIME-DEPENDENT loss rates illustrate the steady nature of the asymptotic HYDRODYNAMICALSIMULATIONS solutions. 5.1. Asymptotic Steady-States of Time-Dependent Despitethistemporalandradialconstancyinthemass Simulations flux, the asymptotic states of the models with ω = 0.8 In our basic parameter study, each simulation is run and 0.84 include a kink, or abrupt discontinuity in their using the same initial condition, spatial mesh, bound- velocity gradient. To illustrate the formation of these ary conditions, base density, etc., with the only varia- kinks, the gray-scales in figure 6 show the evolution of tion being the rotation rate ω, which is set to specific the scaled velocity gradient. In both cases, the abrupt values ranging from 0.1 to 0.97. With only one excep- switchfromasteepaccelerationsolutiontoadecelerating tion (for the ω = 0.85 case, which turns to be a rather solution is apparent from the sharp transition from pos- pathological value; see 5.2), all simulations asymptoti- itive to negative velocity gradient, with the kink radius, § cally relax to a well-defined steady state. Moreover, for r ,indicatedaboveeachplot. Since,despitetheradial kink both moderate rotation (ω < 0.75) and high rotation discontinuity, the velocity gradient contours all become (ω > 0.85), these asymptotic states agree remarkably constant in time, we see clearly that the kink solutions well with the predictions of the above nozzle analysis. are indeed perfectly valid steady-states. Figure 3b shows the velocity laws for these final states, Note,however,thatthekinkradiusshiftsfromr = kink scaled in the same form used in figure 3a for the nozzle- 4.18R∗ for the ω = 0.8 model to rkink = 2.12R∗ for the analysisresults. Notethatbothfigures3aand3bshowa more rapid-rotation, ω =0.84 model. From the velocity steepaccelerationfornoormodestrotation(ω =0–0.7), law plots in figure 3b, this also implies that the more and shallow acceleration for rapid, near-critical rotation rapidrotating case has a more extended decelerationre- (ω = 0.86, 0.9 and 0.95). gion,andthusendsupwithamuchlowerfinalspeed. We However, for the moderately high rotation-rate cases canthus anticipate that a somewhatfaster rotationrate ω =0.75,0.8,0.82 and0.84,notealsotheappearanceof shouldgive anevenlowerkink radius,with the moreex- anewclassofkink solutions,characterizedbyanabrupt tended decelerating regionleading to an even lowerfinal shift to a decelerating or “coasting” flow beyond a well- speed, or perhaps even to a flow stagnation (zero veloc- defined “kink radius” r . These kink solutions thus ity) at a finite radius. In particular, note from figure 4 kink represent a kind of intermediary final state of the time- thatthedashedextrapolationcurvesuggeststhattheon- dependent simulations in the parameter ranges 0.75 < set of such flow stagnation should occur near a rotation ω < 0.85, effectively smoothing the abrupt jump from rate of ω 0.85. ≈ fastto slowaccelerationsolutions expected fromsteady- Infact,ournumericalsimulationsdoshowthatmodels state analyses. The formation of such kinks, and their near this rate have a quite pathological behavior. This underlying physical cause, are discussed further in the is illustrated in figure 7, which presents gray-scale plots section below ( 5.2) on time evolution. of the time-evolution for (a) the mass-loss rate and (b) § 8 w=0.8, m=1.39 300 250 m ) 1.60 R/v¥*200 1.40 ( 1.20 e 150 1.00 m Ti 0.80 100 0.60 0.40 50 0.20 0 25 50 75 DistanceinStellarRadii w=0.84, m=1.68 w=0.9, m=1.04 800 800 700 700 m m ) 600 1.95 )600 1.30 R/v¥*500 1.72 R/v¥*500 1.16 ( 1.48 ( 1.01 e 400 1.25 e 400 0.87 m m Ti 300 1.02 Ti 300 0.73 0.79 0.59 200 0.55 200 0.44 0.32 0.30 100 100 0 0 25 50 75 25 50 75 DistanceinStellarRadii DistanceinStellarRadii Fig. 5.— Gray-scale plots showing time evolution of mass-lossrate inunits of the point-star CAK value. The corresponding rotation rateandfinalmass-lossratearegivenaboveeachplot. Timeisinunitsoftheflowtimet=R∗/v∞. Noteineachtheeventual constancy ofthemass-lossratewithboth radiusandtime,indicatingrelaxationtoasteady-state solution. Notethattheranges forbothgray-scale andtimedifferforeachpanel. w=0.8, r =4.18R w=0.84, r =2.12R 300 kink * 300 kink * dv/dr 250 dv/dr 5.5E-03 3.3E-02 v)¥200 3.9E-03 v)¥200 2.4E-02 R/* 2.4E-03 R/* 1.4E-02 ( 7.9E-04 ( 4.7E-03 e e150 m -7.9E-04 m -4.7E-03 Ti -2.4E-03 Ti -1.4E-02 100 100 -3.9E-03 -2.4E-02 -5.5E-03 -3.3E-02 50 0 2 4 6 8 10 0 2 4 6 8 10 DistanceinStellarRadii DistanceinStellarRadii Fig. 6.— Gray-scaleplotsshowingtimeevolutionofthescaledvelocitygradientincgsunits. Thecorrespondingrotationrateandkink locationrkink aregivenaboveeachplot. Nozzle Analysis of Rotating Hot-Star Winds 9 non-rotating model used for the initial conditions in the TABLE 1 simulation models discussed above. For each of the spe- Comparison of Kink cific rotation cases in this transitional range, ω = 0.75, Locations 0.80,0.82,and0.84,wethus recompute simulationsthat ω rkink,hydro rkink,nozzle instead use an initial condition set to the slow accelera- (R∗) (R∗) tion steady-state found for the faster, near-critical rota- tion case ω = 0.9. 0.80 4.18 4.54 0.82 2.95 3.03 For the ω = 0.75 case, we find that the model again 0.83 2.52 2.54 relaxes to a fast solution with an outer-wind kink at 00..8845 21..1727 21..1866 rkink ≈ 10R∗. Steep acceleration solutions are also re- coveredin all the slower rotation models as well. However, as shown in figure 8, for the ω = 0.80, 0.82, Note. — rkink,hydro and rkink,nozzle are the kink loca- and 0.84 cases the final states now do approachslow ac- tionsfoundrespectivelyfromnu- celerationsolutions,apartfroma persistentpeculiar up- merical hydrodynamical models ward kink near the outer boundary (i.e. for x > 0.9 in and from the semi-analytic noz- figure 8). Such models also show persistent small-scale zleanalysis. fluctuations with an amplitude of ca. 10% in the mass the scaled velocity gradient in this ω = 0.85 case. An flux,apparentlyreflectingsomedifficultyforthenumeri- overloadingofthewindandeventualcollapseofthetime- calsolutiontorelaxtoasubcriticalflowsolutioninthese dependent solution are shown in these figures. In the cases. The upward kink may stem from using a super- first 300 flow times, there is an initial kink formation, criticaloutflow boundary conditionforthis slowacceler- butatitslowerradiusofrkink =1.77R∗ thekinkoutflow ationsolution,whichdoesnotbecomesupercriticaluntil speedisquitelow,v 370km/s,wellbelowthelocal kink ≈ far from the star. escape speed v (r ) 640 km/s. With the reduced esc kink ≈ Finally,evenforthisslowaccelerationinitialcondition, line-driving in the decelerating region, the wind outflow the peculiar case ω=0.85 still forms an overloaded con- now stagnates at a finite radius, rstag ≈ 10R∗. There dition with flow stagnation and reaccretion, after which material accumulates until it is eventually pulled back it never fully recovers a steady outflow result. by the stellar gravity into a reaccretion onto the star, effectively quenching both the kink and base outflow. 5.4. 1D Results for Equatorial Density For this particular case, the outflow never fully recov- ers fromthis quenching, but for only slightly more rapid These results also allow us to identify the radial vari- rotation, ω >0.86, any flow stagnation occurs relatively ation of the relative density enhancement in the slow close to the∼star, with a correspondingly faster and less equatorialwind of a rotating star, comparedto the non- massivereaccretion,followedby a recoveryto a slow ac- rotating solution that applies to the polar wind. We are celerationsolution,asindicatedinfigures3and4. Inthe ignoringheregravitydarkeningandoblatenesseffects,as very rapid, near-critical rotation case ω = 0.90, no kink well as any “bi-stability” in the line-driving parameters or flow stagnation forms, and the solution relaxes more betweenthe polar and equatorialwind. Fromthe analy- directly to the slow solution, as illustrated in the lower sis of 3, the relative density enhancements are givenby right panel of figure 5. the ra§tios of the quantity m˙/√w between the rotating Finally,table1comparesthe radiusofthekinksfound and non-rotating models. in our numerical simulations with the radius at which The dashed curves in figure 9 show the spatial vari- thenozzleanalysisindicatesasteepaccelerationsolution ation of this density enhancement for rotating models can no longer be maintained. For cases in which the with ω = 0.8, 0.86, 0.9, and 0.95. Note that the en- mass flux set at the base m˙ = m˙0 = n(x = 0) is above hancements are a few factors of ten, not insignificant, the CAK value, i.e. m˙ = n(x = 0) > 1 , this occurs but not sufficient to reproduce the inferred densities at a radius xk where a declining nozzle function falls of B[e] disks, which are factors of order 104 or more backton(xk)=m˙. Thecorrespondingradiusrkink,nozzle denser than a typical polar wind outflow (Zickgraf et al. agrees quite well with the kink locations found from the 1985; Kraus & Miroshnichenko 2006). Inclusion of bi- hydrodynamical simulations, rkink,hydro. Note also that stability effects could give about another factor of a the requirement n(x = 0) = (4/9)/(1 ω2) > 1 implies few by increasing the equatorial mass loss around the − rotation rates of ω > √5/3 0.745, representing the cooler, gravity-darkened equator, thus yielding an over- onset for either possible kink≈solutions or a switch to a allenhancementoforder102(Lamers & Pauldrach1991; slow acceleration solution. Pelupessy et al.2000). Significantlyhigherenhancement wouldrequireanunrealisticallylowαtoincreasefurther 5.3. Results for Slow Acceleration Initial Condition the equatorial mass flux, and/or assuming an equato- rial surface rotation within a sound speed of the critical A central finding of the dynamical simulations is that (orbital) speed. In the latter case, minor disturbances moderately fast rotation models 0.75 < ω < 0.85 form (e.g. pulsations) in the stellar envelope or photosphere fast-accelerationkink solutionsinsteadofthe slowaccel- could instead eject material into an orbiting, Keplerian eration solutions predicted from steady-state analyses. disk (Lee et al. 1991; Owocki 2005), obviating the need But since the non-linear character of the flow equations to invoke any central role for radiatively driven outflow allowsmorethanonesolution,thisraisesthequestionof solutions. whether slow accelerationsolutions in this regime might also be stable attractors, perhaps for initial conditions thatareclosertotheirsloweroutflowformthanthefast, 6. SUMMARYANDCONCLUSIONS 10 w=0.85, m=0 w=0.85, r =1.77R 600 600 kink * 500 500 dv/dr m 1.7E-01 me(R/v)¥*340000 2111....07410112 me(R/v)¥*340000 1185....4102EEEE----00001122 Ti 0.83 Ti 2.4E-02 200 0.54 200 -4.7E-03 0.24 -3.3E-02 100 -0.05 100 0 0 2 4 6 8 10 2 4 6 8 10 DistanceinStellarRadii DistanceinStellarRadii Fig. 7.— (a) Gray-scaleplotof the local mass-lossrate inunits of the point-star CAK valueforω=0.85, showingthe collapse ofthe solution. Notethepile-upofmaterialatadistanceofr 10R∗,andtheeventualstagnationofthewindandreaccreationofmaterialback ontothestellarsurface. (b) Gray-scaleplotofthescale≈dvelocitygradientincgs unitsforω=0.85. Inthetimeinterval between t=150 saonldut3io0n0bfleocwomtiemseosv,earlokaindkedinantdheuvnesltoacbiltey,gevraedntieunatllyisccollelaarplsyinvgi.sible, with the kink location at rkink ≈ 1.77R∗. After t ≈ 300, this kink Using an analytic approach combined with numerical hydrodynamicsimulations,weinvestigatethereasonsfor v/vesc the switch from a steep to shallow acceleration in 1D 0.4 line-driven stellar wind models as stellar rotation rates w=0.82 w=0.8 areincreasedbeyondathresholdvalueofω 0.745. The w=0.84 results indicate that the cause of this switc≈h is the over- 0.3 loading of the base mass-loss rate beyond the point-star CAK value. The latter represents the maximal allowed 0.2 mass loss for which there can be a monotonically accel- Initial Condition erating flow speed throughout the entire wind. Further- 0.1 more, the finite-disk correction-factor (fdcf) reduces the drivingeffectivenessnearthestellarsurface,andthusre- duces the maximalmass lossthat canbe initiated there. 0.00 0.25 0.50 0.75 x This reduction allows the outer wind to maintain a pos- itive acceleration even as other effects (e.g. centrifugal Fig. 8.— Finalwindvelocitylaw(flowspeedoverescapespeed, reduction in effective gravity near the surface) allow for v/vesc, plotted vs. scaled inverseradius x) for ω =0.8, 0.82, and 0.84 simulations that used the ω = 0.9 final state (dashed line) an increase of the base mass loss from its fdcf value. as an initial condition. The solid lines now approximate the ex- This problem of wind overloading at large rotation pectedslowaccelerationsolution,exceptforanunexplaineduptick rates was first noticed by Friend & Abbott (1986), who invelocitygradientneartherightboundary, x>0.9. found indeed that beyond some threshold rotation rate, the super-critical, steep acceleration solutions they were deriving could not be followed beyond some finite ra- dius. The contributions of M. Cure´ and collaborators have since shown that this termination can be avoided r by switching to a shallow accelerationsolution. As rota- 35 r w tion increases beyond the threshold value of ω 0.745, w=0 30 the base mass-loss becomes greater than the po≈int-star CAKvalue,andsotheonlyglobally accelerating solution 25 w= possible is a shallow one with a subcritical outflow. 0.95 20 w =0.9 caHlsoimweuvlaert,ioannsiimspthoarttatnhtelflesoswondoleeasrnnoetdnfreocmessoaurrilnyufmolleorwi- 15 w= 0.86 the solution with a globally monotonic acceleration. We w =0.8 10 seethatthe steady1Dsolutionsforrotatingwindsactu- ally fall into four domains. First, for no or low rotation 5 rates(ω <0.745),the minimum ofthe nozzlefunction is less than unity and occurs at the stellar surface (x = 0). 0.25 0.5 0.75 x 1 This allows the flow to transitionto a super-criticalout- flowdirectlyfromthestaticsurfaceboundaryandfollow Fig. 9.— Density enhancement of slow wind solutions with rapid rotation rates, ω = 0.8, 0.86, 0.9, and 0.95, relative to a thesteepaccelerationsolutionwithamass-lossratethat non-rotatingwindwiththesamewindparameters. Dashedcurves is less than the point-star CAK value. areforanalyticresults,whilesolidcurvesarethoseobtainedfrom Next, there exits a “gray zone” for rotation rates be- numericalsimulationsusingVH-1. tweenω =0.745and 0.86wheretwodifferentsolutions ∼ arepossible. Ifwerestrictourselvesto astrictlypositive