SELF-REGULATED SHOCKS IN MASSIVE STAR BINARY SYSTEMS

Using the procedure outlined above, we can calculate M(t) accounting for the effects of excess ionization (as controlled by ξ). As an example, we show results for a star with effective temperature T_eff = 38, 500 K, surface gravity log g = 3.92, and solar metallicity log (Z/Z_☉) = 0 in Figure 2.

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As expected, our results are in good agreement with Stevens & Kallman (1990). For each value of ξ, M is largest (and constant) when t is sufficiently small that all lines are optically thin. At large t, M decreases as lines become optically thick and saturate. For calculations with low ionization parameter (log ξ < 0), M remains significant (M ≳ 1) up to around t ∼ 1 (M is essentially independent of ionization parameter for log ξ < −2). As found by Stevens & Kallman (1990), we also see that as log ξ is increased beyond zero, M drops and the regime in which M(t) is well described by the optically thin limit extends to higher t values. For log ξ > 3, the force multiplier is always small.

For comparison, we also show in Figure 2 the M values reported by Abbott (1982) from calculations for a star with similar parameters (T_eff = 40, 000 K, log g = 4.00, and n_e/W = 1.8 × 10¹¹ cm⁻³). Since no excess ionization radiation was included by Abbott (1982), his calculations should be compared to our results for the lowest ionization parameter shown (log ξ = −2). In general, the agreement is very good—the biggest discrepancy occurs around log t = −2.5 and is at worst a factor of two.

Although the force multiplier M can be directly used to specify the line force, it is convenient to parameterize its dependence on t for use in wind calculations. Although this approach means that the full complexity of M(t) is not captured, it is widely used because of the relative ease of manipulating simply parameterized forms for M(t) when deriving wind solutions.

The basic ansatz under the CAK approximation is to fit a power law to the run of M(t) with t,

$$\begin{equation} M(t) = k t^{-\alpha } , \end{equation} \tag{ 6 }$$

where α defines the slope and k the amplitude of M at t = 1 (i.e., k = M(1)). To capture the flattening of M(t) for small t, we follow Owocki et al. (1988) and modify Equation (6) such that the force multiplier becomes constant at low t (as it must be in the optically thin limit),

$$\begin{equation} M(t,\xi) = k(\xi) t^{- \alpha } \left[\frac{(1 + \tau _{\rm max})^{1-\alpha }-1}{\tau _{\rm max}^{1 - \alpha }}\right] , \end{equation} \tag{ 7 }$$

where τ_max = η_max(ξ)t. In Equation (7) we now explicitly indicate that M depends on both t and ξ. Throughout this paper, we will choose to describe the influence of ξ on M via the CAK parameters k(ξ) and η_max(ξ). Allowing for ξ-dependence in these quantities captures the two systematic changes in M(t, ξ) with ξ: decreasing η_max(ξ) with increasing ξ allows the turnover in M(t) to shift to higher t with increasing ξ, while a reduction in k(ξ) at large ξ describes the overall decrease in M for larger ionization parameters.

We follow Stevens & Kallman (1990) in choosing that α does not vary with ξ. Although it is certainly possible to allow α to vary, this mostly just adds unwarranted complexity to the parameterization. As is clear from Figure 2, the slope of log M versus log t is not constant, meaning that a best-fit α is in any case a function of the range across which it is fit. Therefore, in all of our calculations, we do not fit α but rather fix it to the value that is required in order to reproduce the correct observed terminal velocity for a single star of the appropriate spectral type in a standard CAK theory.³ For the example star discussed in Section 2.1, this is α = 0.57.

With α fixed, we derive values of k(ξ) and η_max(ξ) by fitting our computed M(t) curves to the functional form given by Equation (7). We restrict this fitting to log t < 0, the regime in which M(t) is expected to be dynamically significant. To illustrate the accuracy of this approach, the derived fits from our example calculation are overplotted in Figure 2. As expected, the fits are always very good in the optically thin limit and generally agree to within a few tens of percent across the range of interest (i.e., when M ≳ 1). However, there are clear imperfections, particularly in cases where the slope of M(t) deviates from a constant power law (e.g., in our log ξ = 2 case). Nevertheless, the parameterized form provides a convenient description and reproduces the force multiplier to within a factor of two, which is adequate precision for the purpose of this investigation. We provide tabulated values of k(ξ) and η_max(ξ) in the Appendix.

As mentioned in the previous section, in fitting k(ξ) and η_max(ξ) we specified the value of α a priori with the aim that the resulting M(t, ξ) produced a terminal wind velocity in agreement with observed values. We now also rescale the force multiplier M(t, ξ) to produce a wind mass-loss rate in agreement with observed values, which equates to multiplying k(ξ) by a correction factor.⁴ At the cost of some subjective rescaling, this approach has the advantage of ensuring that the line force used in the colliding wind model in the following section will produce sensible wind parameters while also allowing the influence of X-ray irradiation to be explored. We note that this modification is of smaller magnitude than the current uncertainties in mass-loss rates and wind acceleration in massive stars (see Puls et al. 2008 for a recent review).

Line force calculations were performed for two massive stars: O6V and O4III (see Table 1 for full sets of stellar parameters). In Figure 3 we show the resulting k(ξ) and η_max(ξ). Clearly, for log (ξ) > 0 the line force is effectively suppressed, whereas for log (ξ) < 0 ionization effects are negligible and radiative acceleration is largely unaffected.

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Table 1. Parameters Used for Stellar Model Atmospheres

Model	Teff	M*	R*	log (L/L☉)	log (g)	Z	$\dot{M}$	v∞	k(ξ = 0)	α
	(K)	(M☉)	(R☉)				( M☉ yr−1 )	(km s−1)
O6V	38500	31.7	10.2	5.3	3.92	1	2 × 10−7	2530	0.12	0.57
O4III	41500	48.8	15.8	5.8	3.73	1	5 × 10−6	2750	0.18	0.63

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To compute the wind acceleration we follow Stevens et al. (1992). Alterations have been made to couple the model with a means of estimating the X-ray luminosity from the wind–wind collision shocks, and then allow for an ionization parameter (ξ) dependence of the line force. The calculation proceeds by solving for the wind of one of the stars, with the influence of the companion star appearing in the effective gravitational potential and as a contribution to the total radiative flux in the line force. Subsequently, we change to the frame of reference of the companion star and solve for its wind in an equivalent manner. In the following we describe the solution procedure for each wind. In Section 3.2, we discuss how to infer the shock properties from the two wind solutions.

To simplify the problem we consider steady-state solutions for the flow along the line of centers between the stars with the forces arising due to orbital motion ignored.⁵ We assume that the flow is symmetric about the line of centers and that the wind flows purely radially from the star. We also assume that the wind is isothermal with temperature, T = 0.8T_eff, where T_eff is the effective stellar surface temperature. Lastly, we do not consider stellar radiation reflected from the opposing star's photosphere. Some approximate expressions are used in the model to keep the calculations straightforward, while achieving an accuracy at the order unity level, and in Section 7 we discuss possible alternatives. The equations for mass and momentum conservation in the wind are

$$\begin{eqnarray} \oint \rho {\bf v}\cdot d{\rm S} &=& 0 \rightarrow \dot{M}_{\Omega } = r^2 \rho v, \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} F(r, v, dv/dr) &=& \left(1 - \frac{a^2}{v^2} \right)v\frac{dv}{dr} + \frac{d \Phi }{dr} - \frac{2 a^2}{r} - g_{\rm rad}, \quad\\ F(r, v, dv/dr) &=& 0, \nonumber \end{eqnarray} \tag{ 9 }$$

where r is the distance along the line of centers measured from the center of star 1, v is the wind velocity along the line of centers, ρ is the density, $\dot{M}_{\Omega }$ is the mass-loss rate per steradian, and a is the isothermal speed of sound. The gravitational potential due to both stars is

$$\begin{equation} \Phi = -\frac{{\it G M}_{\ast 1}(1 - \Gamma _{1})}{r} - \frac{{\it G M}_{\ast 2}(1 - \Gamma _{2})}{d_{\rm sep} - r}, \end{equation} \tag{ 10 }$$

where M_*1 and M_*2 are the respective masses of stars 1 and 2, Γ₁ and Γ₂ are the respective Eddington ratio for each star ($\Gamma _{\rm i} = \sigma _{\rm e}L_{\ast \rm i}/4 \pi G M_{\ast \rm i}c$), d_sep is the separation of the stars (measured between their centers), and G is the gravitational constant. The combined radiative line force from both stars, g_rad takes the form,

$$\begin{equation} g_{\rm rad} = \frac{\sigma _{\rm e}M(t,\xi)}{c}(F_1 K_1 - F_2 K_2), \end{equation} \tag{ 11 }$$

where F₁, F₂ are the radiative fluxes and K₁, K₂ are the finite disk correction factors (FDCFs) for stars 1 and 2, respectively. M(t, ξ) is the line force multiplier (Equation (7)), which in our formulation has a dependence on both optical depth, t (Equation (2)) and the ionization parameter, ξ (Equation (5)).

The FDCF is a multiplicative factor used to correct the point-source approximation for the finite size of the stellar disk (Castor 1974; CAK; Pauldrach et al. 1986). For our adopted geometry and assumptions about the flow along the line of centers, we have

$$\begin{eqnarray} K_{\rm i}(r,v,dv/dr) & = & \frac{(1 + \sigma _{\rm i})^{1 + \alpha } - \left(1 + \sigma _{\rm i}\mu _{\ast i}^2\right)^{1 + \alpha }}{\sigma _{\rm i}(1 + \alpha)(1 + \sigma _{\rm i})^{\alpha }\left(1 - \mu _{\ast i}^{2}\right)}, \end{eqnarray} \tag{ 12 }$$

where μ_i = cos θ_i with θ_i being the angle subtended by the respective stellar disk viewed from a point in the wind, $\mu _{\ast 1}^2 = 1 - R_{\ast 1}^2/r^2$ and $\mu _{\ast 2}^2 = 1 - R_{\ast 2}^2/(d_{\rm sep} - r)^2$, and σ₁ = (r/v)(dv/dr) − 1 and σ₂ = (d_sep − r)/v)(dv/dr) − 1. Following Stevens & Pollock (1994) and Pauldrach et al. (1986), we approximate the FDCFs as purely radial functions and neglect any velocity or velocity gradient terms. In this limit,

$$\begin{eqnarray} &&K_{\rm i}(r,v,dv/dr) \rightarrow K_{\rm i}(r) = \frac{1 - [1 - \mu _{\ast \rm i}^2]^{1 + \alpha }}{(1 + \alpha)(1 - \mu _{\ast \rm i}^2)}. \end{eqnarray} \tag{ 13 }$$

In the model considered here, we enforce monotonicity in the flow by ensuring dv/dr = max (dv/dr, 0), which ensures that the optical depth parameter, t, is a positive-valued scalar variable. It follows that our models do not permit radiative braking (which requires an inflection in the velocity gradient). Gayley et al. (1997) comment that correctly accounting for non-monotonicity in the FDCF allows radiative braking. Another way of viewing this is that radiative braking involves a bridging between two monotonic flows, which is not facilitated by standard CAK theory.

To proceed, we make a coordinate transform using the substitution of variables (Abbott 1980),

$$\begin{eqnarray} u &=& \frac{-2 {\it GM}_1(1 - \Gamma _1)}{r a^2}, \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} w &=& \frac{v^2}{a^2}, \end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray} w^{\prime } &=& r^2 v \frac{dv}{dr} [{\it G M}_1 (1 -\Gamma _1)]^{-1}, \end{eqnarray} \tag{ 16 }$$

leading to

$$\begin{equation} F(u, w, w^{\prime }) = \left(1 - \frac{1}{w} \right)w^{\prime } + h(u) - g E(w^{\prime },\xi) B(u)w^{\prime \alpha }, \end{equation} \tag{ 17 }$$

where

$$\begin{equation} g = \Gamma _{1}/(1 - \Gamma _{1}), \end{equation} \tag{ 18 }$$

$$\begin{eqnarray} &&E(w^{\prime },\xi) = k(\xi) C^{-\alpha } w^{\prime } \left[\left(\frac{1}{\eta _{\rm max} C} +\frac{1}{w^{\prime }}\right)^{1-\alpha } - (\eta _{\rm max} C)^{\alpha -1}\right], \nonumber\\ \end{eqnarray} \tag{ 19 }$$

$$\begin{equation} C = \frac{\sigma _{\rm e} \dot{M}_{\Omega } v_{\rm th}}{G M_{\ast 1}(1 - \Gamma _{1})}, \end{equation} \tag{ 20 }$$

$$\begin{equation} B(u) = K_1(u) - \left(\frac{M_{\ast 2} \Gamma _2}{M_{\ast 1} \Gamma _1} \right)K_{2}(u)A(u), \end{equation} \tag{ 21 }$$

$$\begin{equation} h(u) = 1 + \frac{4}{u} - \frac{M_{\ast 2}(1 - \Gamma _2)}{M_{\ast 1}(1 - \Gamma _1)}A(u), \end{equation} \tag{ 22 }$$

and

$$\begin{equation} A(u) = \left(\frac{u_{\rm d}}{u - u_{\rm d}} \right)^2. \end{equation} \tag{ 23 }$$

(Note that our definition of C in Equation (20) differs from Stevens & Pollock's (1994) Equation (13)). The FDCFs

$$\begin{equation} K_{1}(u) = \frac{1 - [1 - (u/u_{\ast 1})^2]^{1 + \alpha }}{(1 + \alpha) (u/u_{\ast 1})^{2}} \end{equation} \tag{ 24 }$$

and

$$\begin{equation} K_{2}(u) = \frac{1 -[1 - (u/u_{\ast 2})^2A(u)]^{1 + \alpha }}{(1 + \alpha) (u/u_{\ast 2})^2 A(u)}, \end{equation} \tag{ 25 }$$

where u_d = u(d_sep) and $u_{\rm \ast \rm i} = u(R_{\ast \rm i})$. The strategy for finding a consistent wind solution is centered around the use of the critical point conditions,

$$\begin{eqnarray} f_{1}(w_{\rm c},w^{\prime }_{\rm c},C_{\rm c}) &=& F(u,w,w^{\prime }) = 0, \end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray} f_{2}(w_{\rm c},w^{\prime }_{\rm c},C_{\rm c}) &=& \frac{\partial F}{\partial w^{\prime }} = 0, \end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray} f_{3}(w_{\rm c},w^{\prime }_{\rm c},C_{\rm c}) &=& \frac{\partial F}{ \partial u} + w^{\prime }\frac{\partial F}{\partial w} = 0. \end{eqnarray} \tag{ 28 }$$

Equations (26)–(28) are, respectively, the equation of motion, the singularity condition, and the regularity condition. The subscript "c" denotes the value of the given parameter at the critical point. Our set of equations differs slightly from those of Stevens & Pollock (1994). Specifically, we lack the singular presence of the eigenvalue of the problem (namely the mass-loss rate). Therefore, we cannot use the equations for f₁, f₂, and f₃ to derive closed form relations for w_c and $w^{\prime }_{\rm c}$. Instead, noting that Equations (26)–(28) compose three equations in three unknowns, we solve for w_c, $w^{\prime }_{\rm c}$, and C_c using a multi-dimensional root finder (see, e.g., Press et al. 1986), where the critical point conditions derived by Stevens & Pollock (1994) are used as the initial guess.

Once the wind profiles have been calculated we proceed to estimate the X-ray luminosity from the individual wind collision shocks. The separate values are then combined to evaluate the total shocked-wind X-ray luminosity. The final step is to use the estimate of the intrinsic X-ray luminosity from the shocked winds (Equation (29)) to evaluate the ionization parameter, ξ (Equation (5)). (Note that when we swap from the frame of reference of one of the stars to its companion's, we interchange the indices in Equations (29) and (32)).

We approximate each shock as a thin shell, and take the pre-shock wind velocity and density to be v_sh = v(r_bal) and ρ_sh = ρ(r_bal), respectively, where r_bal is the ram pressure balance point. The mean post-shock gas temperature (i.e., averaged over the bow shock), T_ps, can be estimated from the Rankine–Hugoniot shock jump conditions, kT ≃ (1/2) × 1.17(v_sh/10⁸ cm s⁻¹)² keV, where the factor of a half is a correction to account for shock obliquity. The 0.01–10 keV X-ray luminosity from the respective wind–wind collision shocks is then estimated using the simple relation:

$$\begin{eqnarray} &&L_{\rm X i} = \frac{1}{2}\dot{M_{\rm i}}v_{\rm i}^{2} \Xi _{\rm i}\left(\frac{1}{1+\chi }\right), \end{eqnarray} \tag{ 29 }$$

where the total mass-loss rate is approximated as $\dot{M}\approx 4 \pi \dot{M}_{\Omega }$. (Note that the X-ray-emitting region of the shocks is taken to be a point source situated at the ram pressure balance point along the line of centers). The parameter Ξ approximates the thermalization of wind kinetic power. In Section 4 we consider models of wind–wind collision and wind-photosphere collision (as a result of the stronger wind overwhelming the weaker wind). In the latter circumstance we take Ξ to be the fractional solid angle subtended by the disk of the companion star,

$$\begin{equation} \Xi = \frac{1}{2}(1 -\epsilon _{\ast }), \end{equation} \tag{ 30 }$$

where $\epsilon _{\ast }^2=1- (R_{\ast 2}/d_{\rm sep})^2$. For a wind–wind collision we take Ξ to be the fractional wind kinetic power normal to the contact discontinuity (Zabalza et al. 2011),

$$\begin{equation} \Xi = \frac{1}{4}\left(\frac{\pi \zeta _{\rm eff}}{1+\zeta _{\rm eff}} \right)^2, \end{equation} \tag{ 31 }$$

where the effective wind momentum ratio of the system is

$$\begin{equation} \zeta _{\rm eff}=\frac{\dot{M}_{2} v_{\rm sh 2}}{\dot{M}_{1} v_{\rm sh 1}}. \end{equation} \tag{ 32 }$$

The cooling parameter, χ, appearing in Equation (29) derives from the ratio of the characteristic flow time to the cooling time (Stevens et al. 1992),

$$\begin{equation} {\chi} = \left(\frac{v_{\rm sh}}{10^8 {{{\rm \,cm}{\rm \,s}^{-1}\,}}}\right)^4 \left(\frac{d_{\rm sep}}{10^{12}\;{\rm cm}}\right) \left(\frac{\dot{M}}{10^{-7}{{{{\,M_{\odot }}}{\rm \,yr}^{-1}\,}}}\right)^{-1}. \end{equation} \tag{ 33 }$$

If χ ≲ 1 the post-shock gas is radiative, whereas if χ ≫ 1 the post-shock gas is adiabatic. It is useful to note the different scalings of L_X as the importance of cooling changes. For adiabatic shocks we have $\hbox{$L_{\rm X}$}\propto (\dot{M}/v_{\rm sh})^2$; therefore, a decrease in v_sh leads to an increase in L_X. In contrast, when the shocks are radiative $\hbox{$L_{\rm X}$}\propto \dot{M}v_{\rm sh}^2$, and a decrease in v_sh reduces L_X.

In Section 4 we consider calculations with either an attenuated or unattenuated X-ray flux. For the latter we merely have F_X(r) = (L_X1 + L_X2)/4π(r_bal − r)². For the former, an attenuated flux is calculated by first scaling a 0.01–10 keV X-ray spectrum—derived from the MEKAL plasma code (Kaastra 1992; Mewe et al. 1995)—such that its total luminosity matches the value from Equation (29). The spectra from both winds are then combined, and the column density of gas upstream of the shock, $N_{\rm H} = \int ^{r_{\rm bal}}_r \rho (r) dr$, is used to attenuate the spectrum. (Absorption due to the post-shock layers is neglected.) Finally, the resulting spectrum is integrated to acquire the attenuated luminosity, L_Xatt, from which the ionization parameter can be determined. To this end we use version c08.00 of Cloudy (Ferland 2000; see also Ferland et al. 1998) to calculate the opacity.

To test the accuracy of our model, we made a calculation for an O6V+O6V binary at a separation of 30 R_☉ and compared the estimated X-ray luminosity to model cwb1 from Pittard (2009) and Pittard & Parkin (2010); a 3D hydrodynamical model with radiatively driven winds. We found that our model overpredicted the intrinsic 0.1–10 keV X-ray luminosity by a factor of roughly two. Therefore, for all models examined in this paper we multiply Equation (29) by a factor of 1/2. Furthermore, when computing log (L_X/L_bol) from our models we define L_X as the 0.5–10 keV X-ray luminosity to be consistent with observational studies (see, for example, Nazé 2009; Nazé et al. 2011; Gagné et al. 2011).

To summarize, the steps in the calculation are as follows.

• 1.  
  Begin by setting ξ = 0 everywhere.
• 2.  
  Compute v(r) and ρ(r) using the wind acceleration model described in Section 3.1 for the respective stars.
• 3.  
  Determine the ram pressure balance point between the winds and use this to find L_X for both post-shock winds. For a wind-photosphere collision, the balance point is taken to be at the surface of the companion star.
• 4.  
  Calculate the X-ray ionization parameter, ξ(r) (see Section 3.2).
• 5.  
  Calculate the change in v_sh relative to the last iteration and if convergence is not achieved⁶ then repeat steps (2)–(4).

Figure 4 shows the number of iterations required to reach convergence. A larger number of iterations are required for smaller separations where the effect of SRSs is greatest.

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We begin by examining the O6V+O6V binary at a separation of d_sep = 60 R_☉. Figure 5 shows the dramatic influence of SRSs on the wind acceleration compared to a calculation without this effect included. Three results are immediately apparent from this plot: (1) the pre-shock velocity is considerably reduced, (2) the wind acceleration is inhibited, and (3) the acceleration region is smaller.

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What causes such a significant difference between wind calculations with and without SRSs? Figure 6 shows the variation of ξ with radius from the star. Close to the shocks (which reside at r = 30 R_☉ in this example), log (ξ) ≫ 0 which is sufficient to strongly suppress the line force (Section 2). In fact, throughout most of the wind the value of ξ is large enough that the wind acceleration, g_rad∝k(ξ), will be inhibited somewhat (see Figure 3), and this becomes clear when one examines the run of k(ξ) against radius. We note, however, that the sharp rise in ξ close to the shocks is an unphysical consequence of concentrating all of the X-ray emission from post-shock winds at the stagnation point. SRSs are important when ξ reaches relatively high values in the wind acceleration region (i.e., well away from the shocks) and, therefore, the spike seen in the top panel of Figure 6 does not affect the main conclusions of this work.

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For comparison, we also show in Figure 6 (top panel) a calculation performed with an unattenuated X-ray flux, which differs only very slightly from the calculation with an attenuated X-ray flux. Although the total accrued column density, N_Htot, steadily increases when tracking back from the shocks toward the star (lower panel of Figure 6), it never reaches a sufficiently high value to impact the X-ray flux from the wind–wind collision shocks. Therefore, the decrease in ξ as one moves away from the shocks results from an increase in the wind density and geometrical dilution of the X-rays. This result holds true for all of the models considered in this work. However, the influence of attenuation may become more important for higher mass-loss rates and/or when the winds contain optically thick clumps.

The wind mass-loss rate is largely unaffected by SRSs. This is because the mass-loss rate is set very close to the star and, as is evident from Figure 6, the ionization parameter is low enough in this region to have little effect on the line force. It is interesting to note that the influence of SRSs inhibits the wind acceleration which causes the inner wind density to increase, thus reducing the influence of X-ray ionization (ξ∝ρ⁻¹).

To better understand the region of parameter space in which SRSs will influence the dynamics of the flow and the observable properties of a binary system, we have performed further model calculations for the O6V+O6V binary at a range of binary separations, the results of which are shown in Figures 7 and 8. As anticipated from Section 4.1, and illustrated by the calculations with and without SRS, the mass-loss rate does not vary greatly due to SRSs. The decrease in mass-loss rate with decreasing binary separation occurs due to the inhibition of wind acceleration by the opposing star's radiation field (Stevens & Pollock 1994).⁷

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Also evident from Figure 7 is that SRSs reduce the pre-shock wind velocity for a large range of binary separations—for the O6V+O6V binary the relative difference in v_sh between calculations with and without SRSs is 45% at d_sep = 40 R_☉ and steadily decreases to 6% at d_sep = 1000 R_☉. SRSs also cause the downturn in v_sh to occur at larger separations than without SRSs.

A secondary effect of a lower v_sh is that the importance of radiative cooling increases (lower χ). As Figure 7 illustrates, without SRSs we would not expect radiative shocks (χ ≲ 1) until d_sep ≲ 40 R_☉ (P_orb ≲ 4 days). In contrast, with SRSs this range increases out to d_sep ≲ 70 R_☉ (P_orb ≲ 10 days). Therefore, we expect that systems will have radiative shocks for larger separations than previously anticipated.

The inclusion of SRSs causes a reduction in plasma temperature, kT, at all separations but only significantly affects the X-ray luminosity, L_X, for a limited range of separations (Figure 8). At large separations (d_sep ≳ 200 R_☉) we do not predict a considerable difference in L_X due to SRSs. Within our model, this stems from the scaling $L_{\rm X} \propto (\dot{M}/v_{\rm sh})^2$ for adiabatic shocks (see Equations (29) and (33)), and the differences in $\dot{M}$ and v_sh between models with and without SRSs (Figure 7). These competing effects effectively cancel to produce almost identical X-ray luminosities. For example, for separations greater than 200 R_☉ there are uniform offsets of roughly 7% for $\dot{M}$ and v_sh between calculations with or without SRSs.

There are, however, a range of separations where SRSs are predicted to make the system intrinsically brighter. For the O6V+O6V binary, this range is 70 < d_sep < 200 R_☉ and arises because v_sh decreases more rapidly with decreasing binary separation with SRSs than without (and because $\hbox{$L_{\rm X}$}\propto (\dot{M}/v_{\rm sh})^2$ at the relevant values of χ). With SRSs, and at d_sep < 70 R_☉, χ < 1 therefore $L_{\rm X} \propto \dot{M} v_{\rm sh}^2$ and the decrease in v_sh caused by SRSs reduces the intrinsic brightness of the wind–wind collision shocks. The abrupt flattening of L_X as the separation is reduced below d ∼ 70 R_☉ in the model with SRSs (Figure 8) is a consequence of the transition from χ > 1 to χ < 1, which alters the dependence of L_X on v_sh.

We now consider an O4III+O6V binary with unequal wind momenta. Based on values for $\dot{M}$ and v_∞ from Table 1 (which are calculated using isolated single-star wind models) the wind–wind momentum ratio, ζ = 0.04 (in favor of the O4III star). Assuming terminal velocity winds (i.e., neglecting radiative inhibition and SRSs), the distance of the wind–wind momentum balance point from the star with the weaker wind is

$$\begin{equation} r_{2} = d_{\rm sep} \frac{1+\zeta ^{1/2}}{\zeta ^{1/2}}. \end{equation} \tag{ 34 }$$

Setting r₂ = R_*2 = 10.2 R_☉, one estimates the wind–wind collision to remain away from the surface of the O6V star for separations greater than 61 R_☉. If we improve on this estimate using our wind model without the inclusion of SRSs we find a stable wind–wind collision down to separations of 80 R_☉ (dashed vertical lines in Figures 9 and 10). However, when SRSs are included a stable wind–wind collision is not predicted to occur for separations smaller than 300 R_☉ (dotted vertical lines in Figures 9 and 10). The reason for this drastic increase is that SRSs tend to make the weaker wind even weaker as it is closer to the source of the X-rays at the wind–wind collision. The tendency for SRSs to reduce the strength of the weaker wind, therefore, becomes more pronounced as the separation of the stars is reduced. Consequently, even for comparatively large separations, wind launching fails. This general result states that SRSs will cause a wind-photosphere collision in unequal wind systems up to larger separations than otherwise expected.

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For sufficiently large separations (d_sep  >  300 R_☉) a wind–wind collision is predicted to occur, and in this regime SRSs introduce a minor reduction in $\dot{M}$'s of roughly 4% for both stars (Figure 9). SRSs also cause a reduction in v_sh for both winds, most notably at smaller separations, where a sharp downturn arises in v_sh for the O6V wind, signifying the sudden failing of the wind–wind collision as the O6V's wind becomes increasingly weakened by the X-rays from the shocks. Somewhat surprisingly, although we anticipate that a wind-photosphere collision will ensue for relatively large separations, the post-shock gas is expected to be adiabatic (χ ≫ 1). This differs from previous models in which a wind-photosphere collision is typically accompanied by highly radiative shocks from the weaker wind (Pittard 1998; Parkin & Gosset 2011).

Similar to the O6V+O6V binary, L_X is largely unaffected by SRSs for the O4III+O6V binary (Figure 10). A noticeable reduction in kT values is, however, introduced particularly for smaller separations. In this case we find that SRSs introduce an offset in kT values for separations larger than 700 R_☉ (P_orb > 300 days), and that for closer separations SRSs cause a sharp downturn in kT, reflecting the behavior of v_sh for the O6V's wind—Figure 9. L_X remains similar between the models, with SRSs causing an upturn in L_X for d_sep ≃ 300–600 R_☉.

We note that although the O4III star has the stronger wind, the X-ray emission is greater from the shocked O6V's wind because a larger fraction of its wind is shocked at an angle close to the shock normal (thus converting a larger fraction of its kinetic energy into thermal energy—Pittard & Stevens 2002). It follows that the observed kT will also be predominantly weighted by the weaker wind. For the specific parameters used in our model, and at separations less than 1000 R_☉ (P_orb < 400 days), the ratio of X-ray luminosity from the winds is 2:1 in favor of the O6V.

As mentioned in the preceding section, the inclusion of SRSs (and radiative inhibition—Stevens & Pollock 1994) considerably increases the range of binary separations where a wind-photosphere collision will occur in a massive star binary system. For our O4III+O6V model we found that a ram pressure balance between the winds could not be achieved for separations less than 300 R_☉. In this section we consider the wind-photosphere collision occurring at d_sep < 300 R_☉. For this purpose we use the model described in Section 3 with the difference that the O6V's wind is not included and the shock is assumed to occur at the surface of the companion star, r = d_sep − R_*2. The fractional wind kinetic power that is thermalized, Ξ, is approximated by the solid angle subtended by the O6V star as viewed by the O4III star (see Section 3.2).

As is clear from the plots of $\dot{M}$, v_sh, and χ in Figure 11 and kT and log (L_X/L_bol) in Figure 12, SRSs have very little effect on the wind-photosphere collision. This is because log (ξ) < 0 in the inner wind acceleration region which allows the wind to accelerate to a similar velocity to the case with no SRSs. log (ξ) > 0 is only reached in regions beyond the acceleration zone, meaning that the driving is unaffected. To illustrate this, in Figure 13 we show radial profiles of wind velocity, log (ξ), and k(ξ) computed for binary separations of 50 and 200 R_☉. Clearly, the line force is only suppressed by SRSs in regions where log (ξ) > 0, which reflects the almost step function like behavior of k(ξ) at log (ξ) ≃ 0–1 (Figure 3).

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In the equal winds case we find reasonably good agreement between our O6V+O6V model and the observed kT and log (L_X/L_bol) (Figure 8). Including SRSs improves the match to the observed kT values. The O6V+O6V results (Figure 8) show that for binary separations less than ∼200 R_☉ (P_orb < 30 days) we expect roughly equal wind systems to be brighter than the expected luminosity from embedded wind shocks in the respective stars (log (L_X/L_bol) ≃ −7; e.g., Sana et al. 2006; Nazé et al. 2011). Roughly equal wind systems with separations larger than ∼200 R_☉ will not, therefore, be identifiable as CWB systems from their X-ray luminosity but instead they may be identifiable by kT > 0.6 keV (i.e., hotter plasma temperature than anticipated from a single massive star—Owocki & Cohen 1999). Indeed, it may be the case that only early-type O+O binaries with intermediate orbits and strong winds will be prolific X-ray emitters (e.g., Cyg OB#9—P_orb = 858 days—Nazé et al. 2012), with the majority of later-type massive binaries only being identifiable as CWBs (in X-rays) via plasma temperatures above 0.6 keV.

There is significant scatter in the observed kT values and log (L_X/L_bol) for orbital periods less than six days. We do not attempt to compare our model against these systems because, when the separation of the stars becomes comparable to their stellar radii, one expects additional effects that we have not considered to become important, for example, tidal deformation, gravity darkening, photospheric reflection, and the possibility of mass transfer (Gayley et al. 1999; Dessart et al. 2003; Owocki 2007; Dermine et al. 2009).

At the separations of the observed unequal wind binaries, a wind–wind collision is predicted from models without SRSs while a wind-photosphere collision is expected based on our SRS calculations. Comparing Figures 10 and 12 one sees that both cases do arguably similarly well at matching the observations—although the wind–wind collision with no SRSs does appear to overpredict the observed log (L_X/L_bol). Interpreting this comparison is complicated, and it may simply be indicating that reality lies between these two different cases, i.e., a wind–wind collision prevailing to smaller separations but with some wind suppression due to SRSs.

Examining the wind-photosphere collision in more detail, the models systematically overpredict kT values by roughly a factor of two, irrespective of whether SRSs are considered or not (Figure 12). However, the agreement between the model and observations is reasonably good for log (L_X/L_bol), with the exception of the two systems with orbital periods of roughly 6 days: HD93205 (Townsley et al. 2011; Nazé et al. 2011) and HD101190 (Chlebowski et al. 1989; Sana et al. 2011; Gagne et al. 2013). We remind the reader that orbital periods have been converted to binary separations under the basic assumption of circular orbits, which is accurate for the majority of the systems in the sample. Considering the two outliers with orbital periods of roughly 6 days, the former, HD93205, has an orbital eccentricity of 0.37 (Morrell et al. 2001; Rauw et al. 2009). Using the ephemeris from Morrell et al. (2001) and the date of the Chandra observation of HD93205, we estimate an orbital phase of ∼0.2. As this is relatively close to periastron, we cannot appeal to the larger separation that will occur at apastron to improve the match against our model results. Adopting the recently derived orbital solution for HD101190 with an eccentricity of ∼0.3 (Sana et al. 2011) does not help the agreement between our model and its $\log (\hbox{$L_{\rm X}$}/L_{\rm bol})$ data point either. A more detailed hydrodynamical model of a wind-photosphere collision is warranted to investigate the systematic discrepancy in kT values and $\log (\hbox{$L_{\rm X}$}/L_{\rm bol})$.