Issue 
A&A
Volume 544, August 2012



Article Number  A120  
Number of page(s)  9  
Section  Stellar atmospheres  
DOI  https://doi.org/10.1051/00046361/201118753  
Published online  10 August 2012 
Coronal winds powered by radiative driving
Astrophysics Group, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, UK
email: l.lucy@imperial.ac.uk
Received: 28 December 2011
Accepted: 10 July 2012
A twocomponent phenomenological model developed originally for ζ Puppis is revised in order to model the outflows of latetype O dwarfs that exhibit the weakwind phenomenon. With the theory’s standard parameters for a generic weakwind star, the ambient gas is heated to coronal temperatures ≈3 × 10^{6} K at radii ≳ 1.4 R, with cool radiationdriven gas being then confined to dense clumps with filling factor ≈0.02. Radiative driving ceases at radius ≈2.1 R when the clumps are finally destroyed by heat conduction from the coronal gas. Thereafter, the outflow is a pure coronal wind, which cools and decelerates reaching ∞ with terminal velocity ≈1000 km s^{1}.
Key words: stars: earlytype / stars: massloss / stars: winds, outflows
© ESO, 2012
1. Introduction
The Xray emission from O stars (Harnden et al. 1979) is now generally agreed to arise from numerous shock fronts distributed throughout their winds. An early theory of such Xray emitting winds (Lucy & White 1980, LW) was based on a twocomponent phenomenological model for the finite amplitude state reached by unstable linedriven winds. Subsequently, the fundamental approach of computing the growth of the instability using the equations of radiation gas dynamics was pioneered by Owocki et al. (1988) and Feldmeier (1995), albeit with the then necessary restrictions to 1D flow and simplified radiative transfer.
A question meriting further research is how and where this windshock model fails. According to LW, failure occurs at the low massloss rate (Φ) of a main sequence B0 star, because the assumption of rapid radiative cooling of shocked ambient gas then breaks down for blob velocities v_{b} ≳ 10^{3} km s^{1}, resulting in the heating of the blobs and consequent loss of linedriving. They conjecture that “thereafter, the relative motions of the two components dissipate and the smoothed wind coasts out to infinity”. In effect, LW suggest that a wind that is initially radiatively driven converts into one that relies on thermal pressure to reach ∞ – i.e., a coronal wind.
In addition to this question’s intrinsic interest, it is notable that the locus of this expected failure coincides with that of stars exhibiting the weakwind phenomenon (e.g., Marcolino et al. 2009, M09). Accordingly, this paper elaborates LW’s conjectures for the outflow from a weakwind star.
2. A multizone wind: zone 1
The twocomponent model must be generalized to remove the assumption of instantaneous cooling of shocked gas and to incorporate blob destruction at finite radius. To achieve these aims, a multizone model is adopted, with each zone corresponding to different physical circumstances.
Zone 1 starts just beyond the sonic point and ends when the isothermalshock assumption is no longer justified. We assume instability has grown to full amplitide and adopt the LW description in which radiationdriven blobs (b) interact dynamically with a low density ambient medium (a). Apart from the infinitesimallythin cooling zones at shock fronts, both components are in thermal equilibrium with the photospheric radiation field, so that T_{a,b} = T_{eq}.
2.1. Blobs
The blobs can be identified with the clumps that are now a standard and spectroscopicallyrequired feature of diagnostic codes for Ostar winds (e.g., Bouret et al. 2005). In such codes, the clumps are assumed to obey the βvelocity law (1)where R is the photospheric radius and v_{∞}, the terminal velocity, is determined from the violet edges of P Cygni absorption troughs.
Given the wide use of the βlaw, this now replaces LW’s Eq. (10). But here, since Eq. (1) ceases to apply when r > r_{S}, the blobs’ destruction radius, v_{∞} is not an observable. The highest velocity at which UV absorption is detected is a measure not of v_{∞} but of v_{b}(r_{S}).
For given β, diagnostic modellers choose the clumps’ filling factor f_{b} and massloss rate Φ_{b} so that absorption troughs have their observed strengths. For a strong line at frequency ν_{0}, this typically requires that, despite clumpiness, a continuum photon emitted between ν_{0} and ν_{0}(1 + v_{∞}/c) has small probability of escaping to ∞, and so most of the photon momentum in this interval is transferred to the clumps. In an LW wind, essentially the same requirement arises as a consistency criterion: since the ambient gas is assumed not to be radiatively driven, it must be shadowed by the blobs. The optical depth criterion adopted by LW is that τ_{1}(r) > 1.5 for all r, where τ_{1} is given in Eq. (11) of LW. For the βvelocity law, the function χ in their τ_{1} formula becomes (2)where x = R/r.
2.2. Dynamics
In zone 1, the dynamical interaction of the blob and ambient components is treated exactly as in LW. For more recent treatments and applications, see Howk et al. (2000) and Guo (2010).
The equation of motion obeyed by the blobs is (3)where g_{R},g_{D}, and g are the forces per gram due to radiation, drag, and gravity, respectively. The drag force g_{D} retards the blobs but accelerates the ambient gas and is the means by which photon momentum is transferred to this component. The resulting equation of motion of the ambient gas is (4)where are the smoothed densities, and with .
With the LW assumption of no mass exchange between the components, the two equations of continuity integrate to give (5)where Φ_{a,b} are constants whose sum is the star’s massloss rate Φ.
The drag force mg_{D} on a blob of mass m is computed using De Young and Axford’s (1967) theory of inertiallyconfined plasma clouds – see Sect. II b) in LW. The resulting formula is (6)where is the blob’s mean cross section, U = v_{b} − v_{a} is the blob’s velocity relative to the ambient gas, and the drag coefficient C_{D} = 1.519.
2.3. Filling factors
At a point (v_{b},v_{a},r) in an outward integration with specified Φ_{a,b}, the smoothed densities are given by Eq. (5). The ambient density is then , where f_{a} is the filling factor of the ambient gas. Correspondingly, the mean density of the stratified De YoungAxford blob is . Now, in the absence of a void component, f_{a} + f_{b} = 1, and so only one of f_{a} and f_{b} is independent. To determine f_{b}, say, we must iterate. Solution by repeated bisection is adopted, starting with upper and lower limits f_{U} = 1 and f_{L} = 0. Then, with the estimate , the blob’s volume V_{b} is computed from LW’s Eq. (5). The mean density of the blob is then ρ_{b} = m/V_{b}, corresponding to . If , the new upper limit is . On the other hand, if , the new lower limit is . The iterations continue until f_{U} − f_{L} < 10^{7}. Then, with the resulting converged value of f_{b}, all quantities required to continue the integration can be evaluated.
2.4. Switch criterion
The assumption of instantaneous cooling breaks down at low densities because the cooling rate per unit volume . If the cooling time scale t_{c} increases to the extent that a parcel of shockheated gas encounters another shock before cooling back to T_{eq}, then shockheating raises the mean temperature of the ambient medium. An approximate criterion for this transition to zone 2 is derived as follows:
First, consider radiativelycooled flow of monatomic gas (γ = 5/3) emerging from a steady shock (see Fig. 1 in Draine & McKee 1993). Since this flow is subsonic, the pressure gradient may be neglected in comparison to that of temperature. Thus, in the shock’s frame, (7)the cooling timescale is therefore .
Now consider flow into the bow shocks. The entire mass ρ_{a} of ambient gas in unit volume is shocked in time interval , where is the number density of blobs, and is the mass flow rate through each bow shock. Hence t_{i} ≈ 4/3f_{b} × σ/U.
The criterion for switching from zone 1 to zone 2 is then simply t_{c} > t_{i}.
3. A multizone wind: zone 2
The outward integration of the wind continues in zone 2 with the same basic model except that T_{a,b} ≠ T_{eq}. The ambient gas is now heated by being repeatedly shocked, and the blobs in turn gain heat by conduction from the ambient gas. Zone 2 ends when the blobs can no longer achieve thermal equilibrium.
Fig. 1 Mach number M as a function of u_{p}/u_{e} along the supersonic (M_{ + }) and subsonic (M_{ − }) solution branches. When u_{p}/u_{e} = 5/4, M_{ + } = ∞ and M_{ − } = 1/^{√}5 = 0.447. 
3.1. Blob survival
The survival of blobs (clumps) in stellar winds has similarities to that of clouds in the interstellar medium. In that context, Cowie & McKee (1977) studied the evaporation of a spherical cloud embedded in a hot tenuous medium. Importantly, they treated the saturation of heat conduction when the electron mean free path in the surrounding medium is ≳ the cloud’s radius and estimated, under the assumption of steady outflow, the reduced evaporation rate. In a companion paper (McKee & Cowie 1977; see also Graham & Langer 1973), they consider the effects of radiative losses, finding that evaporation is replaced by condensation if the losses exceed the heat input from the hot gas. However, numerical calculations by Vieser & Hensler (2008) cast doubt on the assumption of steady outflow. In the case of saturated conduction, they find a further reduction of evaporation rate by a factor ~40 due to changes of the cloud’s environment caused by the outflow.
Given the evident difficulty of reliably predicting when cool gas is eliminated by its interaction with surrounding hot gas, a simple prescriptive approach is adopted here: as in zone 1, the blobs retain their fixed mass m throughout zone 2. However, when the heat input from the ambient gas exceeds their maximum cooling rate, the blobs are assumed to merge instantly with the ambient gas.
3.2. Heating and cooling of blobs
In zone 2, the blobs are surrounded by shockheated gas and so will be heated by thermal conduction. But if the ambient gas reaches coronal temperatures, heat conduction is fluxlimited. Moreover, conductivity may be suppressed by magnetic fields. An approximate formula interpolating between the classical and saturated limits and incorporating a suppression factor φ is derived in Appendix A.
If ℒ_{in} is the rate of heat flow from the ambient gas, a blob will achieve thermal equilibrium at T_{b} > T_{eq} if the enhanced radiative cooling rate (8)where Λ(T) is the opticallythin cooling function, and the blob is treated as isothermal and of uniform density. (But note that LW’s definition of ρ_{b} is such that Δℒ_{b} is exact for the density stratification of an isothermal De YoungAxford blob.)
Because Λ(T) reaches a maximum at T_{†}(K) = 5.35 dex (Dere et al. 2009), the solution of Eq. (8) with T_{b} < T_{†} is appropriate as the blobs are heated to above T_{eq}. When T_{b} reaches T_{†}, the corresponding ℒ_{in} is the maximum value consistent with thermal equilibrium. Any further increase in ℒ_{in} cannot be matched by increased cooling. Accordingly, we take this as the point beyond which the blobs cannot survive.
Note that if conduction is completely suppressed (φ = 0), then ℒ_{in} = 0 and the solution of Eq. (8) is T_{b} = T_{eq}. The blobs therefore survive, and zone 2 extends to ∞.
3.3. Heating and cooling of ambient gas
According to LW, the rate at which energy is being dissipated per unit volume is (9)Dividing by , we find that the rate per blob is (10)showing that in unit time each blob’s bow shock dissipates the kinetic energy in a column of inflowing gas of length U and cross section . In zone 1, this dissipated energy is radiated by a thin cooling layer, and so determines the sum of these layers’ frequencyintegrated emissivities – Eq. (7) in LW. But in zone 2 where t_{c} > t_{i}, we jump to the opposite limit, treating dissipation as a heat source distributed uniformly throughout the ambient gas, and similarly for cooling. Accordingly, the energy equation for stationary flow of the monatomic ambient gas is (11)Note that since f_{a} ≈ 1 terms arising from radial changes in f_{a} have been neglected.
The total cooling rate per unit volume of ambient gas, , is the sum of the losses due to radiative cooling and to conduction into the blobs. Thus, (12)Integration of Eqs. (4), (5) and (11) continues until T_{b} = T_{†}, at which point the blobs are deemed to merge instantly with the ambient gas (Sect. 3.1). Accordingly, the transition from zone 2 to zone 3 occurs at S, a surface of discontinuity (e.g., Landau & Lifshitz 1959), across which the fluxes of mass (13)momentum (14)and energy (15)are continuous. Zone 2 thus ends at r_{S} with the evaluation of J,Π and F.
4. A multizone wind: zone 3
The outward integration continues in zone 3, but now the blobs have disappeared, leaving a single fluid component with no driving force (g_{D} = 0) and no heat input . The initial conditions for the resulting ODE’s are obtained from the continuity across S of J,Π and F. Thus, v,ρ, and T for the flow emerging from S^{ + } are given by and (18)where J,Π, and F are given by Eqs. (13)–(15).
4.1. Solution branches
From Eqs. (16)–(18), we readily derive the quadratic equation (19)where u_{p} = 5Π/8J and u_{e} = ^{√}(F/2J). The two solutions are (20)The corresponding temperatures T_{ ± } are derived from the isothermal sound speeds given by (21)and the densities are ρ_{ ± } = J/v_{ ± }.
If u_{p} = u_{e}, the two solutions coincide. When this happens, v = u_{p} = ^{√}(5/3)a – i.e., the outflow at S^{ + } is exactly sonic. If u_{p} > u_{e}, the solutions are real and distinct. The v_{ + } solution is supersonic (M_{ + } branch), and the v_{ − } solution is subsonic (M_{ − } branch). Note that the M_{ + } branch has a singularity at v_{p}/v_{e} = 5/4, at which point a_{ + } = 0. Mach numbers for the two branches are plotted against v_{p}/v_{e} in Fig. 1.
The M^{ − } branch corresponds to S being the locus not only of merging but also of a stationary shock front. This branch would perhaps be appropriate if there were a preexisting slower wind (cf. Macfarlane & Cassinelli 1989), but this is not the circumstance evisaged here. Instead, therefore, the M^{ + } branch is selected since this corresponds to a high speed twocomponent flow at S^{ − } emerging at S^{ + } as a singlecomponent supersonic flow.
4.2. Dissipation at S
In addition to the roots of Eq. (19) being real, a further condition is mandatory: the transition from S^{ − } to S^{ + } must be such that kinetic energy is dissipated (entropy production) and not the reverse. For the two branches, kinetic energy is thermalized at the rates (22)whose positivity must be checked.
Note that the kinetic energy dissipated at S is not radiated away by a thin cooling zone. Instead, this energy contributes to the flow’s enthalpy at S^{ + }, which then does PdV work in the subsequent expansion.
4.3. Outward integration
The solution for the singlecomponent gas in zone 3 is obtained by integrating the equations of motion (23)continuity (24)and energy (25)the initial conditions at r_{S} are v_{ + }, ρ_{ + } and T_{ + } derived in Sect. 4.1.
This integration continues to r = ∞. However, this is only possible if the energy density at S^{ + } is sufficient to overcome both the remaining potential barrier and the cooling losses. If not, a stationary, sphericallysymmetric wind solution of this type does not exist.
5. An example
To illustrate the ideas presented in Sects. 2–4, the solution for a generic weakwind star is now described in detail.
5.1. Standard parameters
The model has several parameters, for which standard values are now adopted. Given their uncertainty, sensitivity to changes are reported in Sect. 6.
Because the theory does not predict Φ, this is derived from previouslytabulated mass fluxes (Lucy 2010b; L10b). The chosen model has T_{eff} = 32.5 kK and log g = 3.75, consistent with the weakwind stars ζ Oph and HD 216532 – see Table 3 in M09. The model’s mass flux J (g cm^{2} s^{1}) = −7.11 dex.
The star’s mass ℳ = 24.1 ℳ_{⊙} is determined by finding the point on the ZAMS from which the evolutionary track during core Hburning has log g = 3.75 when T_{eff} = 32.5 kK. This point is reached after 5.75 × 10^{6} yrs when R = 10.83 R_{⊙} and the luminosity L = 1.18 × 10^{5} L_{⊙} = 4.52 × 10^{38} erg s^{1}. The assumed composition is X = 0.70,Z = 0.02.
With R and L determined, Φ = 4πR^{2}J = 8.80 × 10^{9} ℳ_{⊙} yr^{1} = 1.10L/c^{2}. This theoretical Φ derives from the constraint of regularity at the sonic point (v = a) in the theory of moving reversing layers. In the weakwind domain, this theory’s predictions exceed the highly uncertain (±0.7 dex) observational estimates of M09 by ≈ 0.8 dex but are lower than the Vink et al. (2000) formula by ≈ 1.4 dex (Lucy 2010a, L10a).
For the parameters in Eq. (1), we adopt the observationallysupported Ostar values β = 1 and v_{∞} = 2.6v_{esc}(R) = 2394 km s^{1}.
The mass m of the blobs must also be specified. Recent modelling of Ostar spectra finds that “In most cases, clumping must start deep in the wind, just above the sonic point” (Bouret et al. 2008). We therefore retain LW’s assumption that blobs form at or near the sonic point and have diameters comparable to H_{ρ}, the local scale height. At v = a in model t325g375, ρ = 4.27 × 10^{14} g cm^{3} and H_{ρ} = 9.86 × 10^{8} cm, so that the crude LW estimate is m = 2 × 10^{13} g.
The ratio η = Φ_{b}/Φ must also be specified. Following LW, we determine η by imposing the constraint that τ_{m} = 1.5, where (26)Typically, the minimum occurs at the end of zone 1 where inertial confinement is greatest. In zone 2, shadowing rapidly becomes irrelevant since the rapid rise of T_{a} – see Fig. 3 – destroys driving ions.
Finally, the conductivity suppression factor φ introduced in Appendix A must be specified. As standard value, we set φ = −1.0 dex, a moderate degree of suppression compared to estimates for galaxy clusters (e.g., Ettori & Fabian 2000).
5.2. Zone 1
In this highdensity zone close to the photosphere, both components are assumed to be in thermal equilibrium with the star’s radiation field, a condition approximated by setting T_{eq} = 0.75 T_{eff}, as in L10a,b.
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 (27)the outward integration starts, as in LW, with v_{b} = 150 km s^{1} and v_{a} = 100 km s^{1}, a point sufficiently beyond the presumed onset of clumpiness that the twocomponent state may be regarded as established. The starting radius from Eq. (1) is r_{i} = 1.067 R.
Equation (27) has a singularity when v_{a} = a_{a}. Since the integration starts with v_{a} > a_{a}, this singularity only arises if insufficient drag g_{D} causes the flow to decelerate. A parameter set for which this happens does not admit a steady wind of this type.
As shown in Fig. 2, the standard parameters result in an outflow of ambient gas that accelerates throughout zone 1. This continues until the switch to zone 2 is triggered by the onset of the inequality t_{c} > t_{i} – see Sect. 2.4. This occurs at r/R = 1.28, with v_{b} = 528 km s^{1} and v_{a} = 325 km s^{1}. The relevant timescales are t_{c} = t_{i} = 2.0 × 10^{3} s, which are ≪ the local flow timescale, r/v_{b} = 1.8 × 10^{4} s.
The postshock cooling rate required in calculating t_{c} is given by n_{e}n_{H}Λ(T), where Λ(T) is the opticallythin cooling function for photospheric abundances tabulated by Dere et al. (2009). This rate is computed at the apex of the bow shock with n_{e} = 1.18n_{H}, corresponding to complete electronstripping.
At the end of zone 1, the postshock temperature has risen to 6.0 × 10^{5} K, so Xray emission from zone 1 is negligible.
Fig. 2 Velocities of blobs (v_{b}) and ambient gas (v_{a}) as functions of radius. Zone boundaries are indicated. The surface of discontinuity S where blobs merge with ambient gas occurs at r/R = 2.14. 
5.3. Zone 2
With the isothermal assumption dropped, the structure of zone 2 is determined by Eqs. (4) and (11). With dependent variables v_{a} and T_{a}, the ODE’s to be integrated are (28)and (29)Since all variables are continuous at this transition, the integration starts at the point (v_{a},v_{b},T_{a},T_{b},r) reached by the zone1 integration.
Equations (28) and (29) are a pair of algebraic equations for the two derivatives. The determinant of the coefficients’ matrix is zero when v_{a} = ^{√}(5/3)a_{a} – i.e., at the adiabatic sonic point. If this singularity is ecountered, the parameters are inconsistent with the conjectured wind structure.
Figure 2 shows that, with the standard parameters, the flow continues to accelerate throughout zone 2 reaching v_{a} = 940 km s^{1} at r_{S} = 2.14 R, at which point v_{b} = 1277 km s^{1}.
The corresponding temperature structure predicted for zone 2 is shown in Fig. 3. At the start, T_{a,b} = T_{eq} = 24.4 kK. Thereafter, shockheating of the ambient component overcomes radiative, conductive and adiabatic cooling to give a rapidly increasing T_{a}, reaching the coronal value 10^{6} K at r = 1.35 R and 3.7 × 10^{6} K at r_{S}.
The profile for T_{b} shows discontinuous jumps at the beginning and end of zone 2. These result from nonmonotonic variations of Λ(T). For example, Λ’s peak at T_{†}(K) = 5.35 dex is preceded by lower peak at 5.00 dex. Accordingly, after reaching T_{b}(K) = 5.00 dex, a slight increase in ℒ_{in} results in a discontinuous jump to T_{b}(K) = 5.18 dex, followed quickly by blob destruction when T_{b} = T_{†}. Because of these jumps, the radiative driving of the blobs, which is ultimately reponsible for T_{a}’s increase to coronal values, occurs mostly between T_{b} = 40 and 90 kK.
Blob temperatures are derived algebraically from Eq. (8) on the assumption that blobs adjust instantaneously to thermal equilibrium. At r_{S}, the heating time scale 1.5nkT_{†} × V_{b}/ℒ_{in} = 0.9 × 10^{2} s compared to the flow time scale r/v_{b} = 1.3 × 10^{4} s.
In computing cooling rates for blobs, we set n_{e} = 1.12n_{H}, corresponding to metals being stripped of ~2–3 electrons.
Fig. 3 Temperatures of blobs (b) and ambient gas (a) as functions of radius. Zone boundaries are indicated. 
5.4. Surface of discontinuity S
At S^{ − }, the blobs have filling factor f_{b} = 0.024, velocity v_{b} = 1282 km s^{1} and temperature T_{b} = 2.24 × 10^{5} K. The corresponding values for the ambient component are f_{a} = 0.976, v_{a} = 944 km s^{1} and T_{a} = 3.66 × 10^{6} K. After merging, the flow at S^{ + } has two possible solutions (Sect. 4.1). For the rejected M_{ − } solution, the flow emerges with v_{ − } = 313 km s^{1}, T_{ − } = 1.90 × 10^{7} K, corresponding to Mach 0.48, and the implied rate at which kinetic energy is dissipated erg s^{1} or 6.9 × 10^{6}L.
For the selected M_{ + } solution, the flow emerges with v_{ + } = 1094 km s^{1}, T_{ + } = 2.58 × 10^{6} K, corresponding to Mach 4.6, and the implied dissipation rate erg s^{1} or 2.0 × 10^{7} L.
Notice that v_{ + } ∈ (v_{a},v_{b}), as expected if S is the locus only of merging. In contrast, v_{ − } < v_{a}, so there is a coincident shock, as also indicated by the far greater dissipation rate .
5.5. Zone 3
The single component flow emerging from S is a pure coronal wind: the only outward force is the gradient of thermal pressure.
The structure of zone 3 is obtained by continuing the integration of Eqs. (28) and (29), but now with and . The initial conditions at r_{S} are v_{ + } and T_{ + } given in Sect. 5.4.
A short segment of this outflow is plotted in Figs. 2 and 3, showing that the flow decelerates and (inevitably) cools. For these standard parameters, the energy density at S suffices to overcome cooling and power escape to ∞. At r_{f}/R = 100, the flow has slowed to 984 km s^{1}, way beyond the local v_{esc} = 92 km s^{1}
The temperature drops below the coronal value 10^{6} K at r/R = 4.24 and to 10^{5} K at r/R = 13.2.
5.6. Emission measure
With standard parameters, our generic weakwind star is predicted to have a corona (T > 10^{6}) that extends from r_{1} = 1.35 R to r_{2} = 4.24 R and so will be an Xray emitter. As a crude guide to detectability, we compute the emission measure of coronal gas (30)and its hardness parameter (31)The results are ε(cm^{3}) = 53.51 dex and ⟨ kT ⟩ = 0.20 keV.
5.7. Energy budget
The global energy budget of this multizone wind is of interest. The input is the rate of working in zones 1 and 2 of g_{R}, the force per unit mass acting on the blobs. This rate L_{wrk} = 5.4 × 10^{33} erg s^{1}.
The balancing output is L_{M} + L_{W}, where L_{M} is the rate at which matter gains kinetic and potential energy, and L_{W} is the wind’s radiative luminosity. For the interval (r_{i},r_{f}), L_{M} = 4.8 × 10^{33} erg s^{1} or 88.5% of L_{wrk}. The remaining 11.5% is accounted for by L_{W}, which comprises radiative losses from shock fronts in zone 1, cooling radiation from blobs and ambient gas in zone 2, and cooling radiation from the coronal flow in zone 3.
For an idealized linedriven wind in which gas remains (by assumption) at T_{eq}, PdV work is negligble so that L_{M} = L_{wrk}. In contrast, for a pure coronal wind, L_{wrk} = 0, so that L_{M} is entirely due to the PdV work of the hot gas. The relative contributions of these two mechanisms in this hybrid case is of interest.
In answering this, we must first integrate from Eq. (9) over zones 1 and 2 to obtain the total dissipation rate L_{D} = 1.1 × 10^{33} erg s^{1}. The quantity L_{wrk} − L_{D} = 4.3 × 10^{33} erg s^{1} is then the contribution to L_{M} due directly to radiative driving. On the other hand, the contribution of PdV work by hot gas is L_{D} − L_{W} = 0.4 × 10^{33} erg s^{1}.
A measure of the proximity of a hybrid to a pure coronal wind is the ratio (32)which = 0 for a conventional linedriven wind and =1 for a coronal wind. With standard parameters, the multizone wind has θ = 0.08, so direct radiative driving still dominates in accounting for L_{M}.
A further quantity of interest is the integrated cooling rate of gas with T_{e} > 10^{6} K, since this is approximately the wind’s Xray luminosity. For zones 2 and 3, this gives L_{X} ≈ 3.4 × 10^{31} erg s^{1}, so that L_{X}/L ≈ 0.76 × 10^{7} L, similar to the ratio found for earlytype O stars.
6. Nonstandard parameters
The theory developed in Sects. 2–4 has several parameters, each of which would either be predicted or rendered unnecessary if calculations could be carried out from first principles. Sensitivity of the results to these currently unavoidable parameters must therefore be investigated. Accordingly, sequences of solutions are now reported in which a single parameter is varied while keeping others at the standard values of Sect. 5.
Key properties of the models are given in Table 1. The quantities reported are as follows:

Column 1: Sequence identifier.

Column 2: Exponent in Eq. (1), the velocity law.

Column 3: Log of total massloss rate in ℳ_{⊙} yr^{1}.

Column 4: Log of blobs’ mass in g.

Column 5: Log of conductivity suppression factor – see Eq. (A.9).

Column 6: Fraction of massloss in blobs =Φ_{b}/Φ.

Column 7: Shadowing optical depth – see Eq. (26).

Column 8: v_{b} in km s^{1} at the destruction radius r_{S}.

Column 9: Maximum ambient gas temperature in 10^{6} K.

Column 10: Log of emission measure in cm^{3} – see Eq. (30).

Column 11: Hardness parameter in keV – see Eq. (31).
Solutions with nonstandard parameters.
Fig. 4 Sensitivity of coronal winds to φ, the magnetic suppression factor – see Eq. (A.9). Values of log φ are shown. The vertical segments are the surfaces of discontinuity S. 
6.1. Sequence I
In this sequence, the conductivity suppression factor varies from φ = 0.001, an extreme value but with observational support for galaxy clusters (Ettori & Fabian 2000), to φ = 1, the value for a nonmagnetized plasma.
Not surprisingly, the predictions are highly sensitive to φ, and this might eventually be exploited diagnostically. With φ = 1, conductive heating of the blobs destroys them already at r_{S} = 1.57 R where v_{b} = 873 km s^{1}. However, with φ = 0.001, blobs survive out to r_{S} = 22.0 R where v_{b} = 2286 km s^{1}.
Table 1 also shows that coronal temperature and the hardness parameter increase as φ → 0. However, the emission measure ε remains ~53.4–53.5 dex after an initial sharp rise from 53.18 dex for φ = 1. The predicted temperature profiles of the coronae as φ varies are plotted in Fig. 4.
This sequence demonstrates the diagnostic potential of UV and Xray data in constraining magnetic suppression of conductivity. The UV data measures the highest velocity at which wind matter transfers photon momentum to the gas and the Xray data measures the hardness of coronal emission.
A further diagnostic test provided by P Cygni lines is the weakness of emission components. As r_{S} decreases with increasing φ, the fraction of scattered photons occulted by the star increases and the emission component weakens. 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 weakwind stars investigated in M09 all have C iv resonance doublets with weak or absent emission components. Diagnostic modelling of these stars would improve if UV scattering were truncated at finite radius.
6.2. Sequence II
As noted in Sect. 5.1, the Φ of weakwind stars is poorly determined. This sequence explores sensitivity to this uncertain parameter.
When Φ is increased above the standard value from L10b, the blobs survive to higher velocities, and the higher coronal densities give the approximate scaling law ε ∝ Φ^{1.3}. Interestingly, the quantities T_{max} and ⟨ kT ⟩ are insensitive to Φ.
The attempt to continue this sequence to lower Φ’s failed at − 8.36 dex because the singularity in zone 2 discussed in Sect. 5.3 is encountered. This arises as follows: the sharp initial rise of T_{a} in zone 2 causes M to decrease despite increasing v_{a} – see Figs. 2 and 3. But as T_{a} levels off M reaches a minimum and then rises again. Sequence II terminates for Φ between − 8.36 and − 8.26 dex when this minimum falls to M = 1. For the solution plotted in Figs. 2 and 3, this zone2 minimum is M = 1.88 at r = 1.47 R.
6.3. Sequences III–V
In sequence III, the velocitylaw exponent varies from β = 0.5 – rapid acceleration – to β = 2.5 – slow acceleration. 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_{b}(r_{S}) is moderately sensitive.
In sequence IV, solution sensitivity to the highly uncertain blob mass is explored. Fortunately, coronal properties are only moderately sensitive, with ε ∝ m^{0.22} and ⟨ kT ⟩ ∝ m^{0.03}. In regard to blob destruction, this occurs as expected at low velocities for small m. In consequence, sequence IV terminates for m(gm) between 11.8 and 11.9 dex because the outflow in zone 3 is then unable to reach ∞ on account of negative energy density – see Sect. 4.3.
Finally, sensitivity to the shadowing parameter τ_{m} is investigated with sequence V. Again only moderate sensitivity is found.
7. Conclusion
The aim of this paper has been to investigate the structural changes of Ostar winds when Φ decreases to the extent found for the weakwind stars. To this end, the twocomponent phenomenological model developed originally for ζ Puppis is modified to incorporate LW’s conjectures following the breakdown of that model’s assumptions for τ Sco. When applied to a generic weakwind star, the revised model predicts that shockheating of the ambient gas gives rise to coronal temperatures, that conductiveheating eventually destroys the blobs, and that the resulting singlecomponent flow coasts to ∞ as a pure coronal wind. Thus, in broad outline, the volumetric roles of hot and cool gas in Ostar winds are reversed. In the now standard picture for a star such as ζ Puppis the Xray emitting gas occupies a tiny fraction of the wind’s volume, with the bulk of the volume being highlyclumped cool gas with T ~ T_{eq}. In contrast, in the picture suggested here for the weakwind stars, Xray emitting gas fills most of the volume for r ≳ 1.3 R, with surviving cool gas in the form of dense clumps with f_{b} ~ 0.01–0.03.
As is common elsewhere in astrophysics, the approach adopted in this paper is phenomenological modelling. A simplified picture of the phenomenon is combined with approximate treatments of the expected physical effects to create an “endtoend” tractable theory that obeys conservation laws and makes testable predictions (e.g., Xray spectra and UV line profiles). Such theories are of course always an interim measure, to be discarded when the obstacles to calculation from first principles are overcome. Unfortunately, in this case, these obstacles are formidable: 3D timedependent gas dynamics, radiative transfer, and heat conduction including saturation and possibly magnetic suppression.
Evidently, the fundamental approach is unlikely to yield results anytime soon. Accordingly, possible improvements of the crude modelling described herein should be investigated. Also diagnostic codes should incorporate features of such models to extract more reliable parameters from observational data.
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Appendix A: Heat conduction into a spherical blob
In zone 2, the blobs are surrounded by gas whose temperature is rising to coronal values. Conduction will therefore transfer heat into the blobs, and this constitutes a loss term in the energy equation for the ambient gas.
Given that f_{b} ≪ 1, it suffices to consider a single spherical blob with temperature T_{b} and radius σ located at r = 0 in an infinite medium with T → T_{a} as r → ∞. If is the cooling rate per unit volume and κ is the conductivity, the equilibrium temperature profile for r > σ is given by the equations (A.1)and (A.2)with boundary conditions (A.3)In the blob’s absence, the gas is isothermal and has cooling rate , which is subtracted in Eq. (A.1). Accordingly, ℒ(∞) is the additional cooling due to the blob’s presence. Of this, Δℒ = ℒ(∞) − ℒ(σ) represents emission from ambient gas cooled below T_{a} by the blob, and ℒ(σ) is the rate of heat conduction into the blob.
In thermal equilibrium, ℒ(σ) is balanced by emission from within the blob. Now, since radiative cooling is ∝ ρ^{2} and ρ_{b} ≫ ρ_{a}, we expect that Δℒ ≪ ℒ(σ). Therefore, to a first approximation, ℒ(r > σ) = ℒ(σ), a constant, and this allows Eq. (A.2) to be solved analytically when κ ∝ T^{5/2} (Spitzer 1962). The resulting temperature profile is given by
(A.4)where t = T(r)/T_{a}, and the corresponding rate of heat conduction into the blob is (A.5)since κT ∝ T^{7/2}, ℒ_{cl} is insensitive to T_{b} when T_{a} ≫ T_{b}. For the solutions reported in Sects. 5 and 6, we take κ = 1.0 × 10^{6} T^{5/2}, corresponding to Coulomb logarithm lnΛ = 17.
The above discussion treats conduction in the diffusion limit – i.e., where the mean free path of the electrons is ≪ macroscopic length scales. In the opposite limit, heat conduction into the blob is fluxlimited and saturates at (A.6)where q_{sat} is estimated by Cowie & McKee (1977) to be (A.7)and is here evaluated at T_{a},(n_{e})_{a}. Interpolating between these limits (cf. Balbus & McKee 1982), we take the rate of heat conduction into the blob to be ℒ_{cond}, where (A.8)at high temperatures, this gives ℒ_{cond} ∝ T^{3/2} in place of ℒ_{cl} ∝ T^{7/2}.
The above formula is for a nonmagnetized plasma. But since stars in and near the weakwind domain have detected magnetic fields (e.g., Oskinova et al. 2011), we include the possibility of magnetic suppression of heat conduction by writing (A.9)in this investigation, φ is varied to explore its impact on the solutions. In future, it may be determined or constrained by fitting observational data.
The suppression of thermal conductivity in astrophysical plasmas has been strikingly confirmed by the discovery of cold fronts in Xray maps of clusters of galaxies
(e.g., Carilli & Taylor 2002). For the cluster Abell 2142, Ettori & Fabian (2000) estimate a reduction factor of between 250 and 2500. They speculate that, as a result of cluster merging, different magnetic structures are in contact and so remain to high degree thermally isolated. The displacements of wind clumps from their nascent ambient surroundings might well lead similarly to substantial reduction factors.
All Tables
All Figures
Fig. 1 Mach number M as a function of u_{p}/u_{e} along the supersonic (M_{ + }) and subsonic (M_{ − }) solution branches. When u_{p}/u_{e} = 5/4, M_{ + } = ∞ and M_{ − } = 1/^{√}5 = 0.447. 

In the text 
Fig. 2 Velocities of blobs (v_{b}) and ambient gas (v_{a}) as functions of radius. Zone boundaries are indicated. The surface of discontinuity S where blobs merge with ambient gas occurs at r/R = 2.14. 

In the text 
Fig. 3 Temperatures of blobs (b) and ambient gas (a) as functions of radius. Zone boundaries are indicated. 

In the text 
Fig. 4 Sensitivity of coronal winds to φ, the magnetic suppression factor – see Eq. (A.9). Values of log φ are shown. The vertical segments are the surfaces of discontinuity S. 

In the text 
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