2.1. Line-driven wind theory for single non-rotating stars

Here we briefly review the theory of line-driven winds for single non-rotating stars (see Chapter 8 of Lamers & Cassinelli 1999). In the atmospheres of hot luminous stars, the ions can have many narrow absorption lines. These lines can efficiently capture part of the ultraviolet radiation from the star and then symmetrically re-emit, so that the ions gain a net outwards momentum from the incoming photons. Due to the strong coupling between the ions and the surrounding gas, the atmosphere is accelerated outwards as a stellar wind. The acceleration in turn enables the lines to capture radiation over different frequencies via Doppler shifting: gas moves at greater velocities further away from the star, so can capture redder photons that passed through gas at smaller radii. Castor et al. (Reference Castor, Abbott and Klein1975) (hereafter CAK) established that the acceleration due to spectral lines can be approximated as a power lawFootnote b

(1) \begin{equation} g_\mathrm{L}=\frac{\sigma_eL}{4\pi r^2c}kt^{-\alpha},\end{equation}

where $\sigma_e$ is a reference value for the electron scattering opacity, L the luminosity of the star, r is the distance from the centre of the star, and c the speed of light. The dimensionless optical depth parameter t is given by

(2) \begin{equation} t\equiv\sigma_ev_\mathrm{th}\rho\left(\frac{dv}{dr}\right)^{-1},\end{equation}

where $v_\mathrm{th}$ is the mean thermal velocity of the protons in the wind, $\rho$ the density and v the velocity of the wind. The so-called force multiplier parameters k and $\alpha$ are tabulated for various types of stars (e.g. Abbott Reference Abbott1982; Shimada et al. Reference Shimada, Ito, Hirata, Horaguchi, Balona, Henrichs and Le Contel1994; Lattimer & Cranmer Reference Lattimer and Cranmer2021). For hot stars with effective temperatures $3.8\lesssim\log{(T_\mathrm{eff}/K)}\lesssim4.7$ , these (correlated) parameters fall in the ranges $0.1\lesssim k\lesssim1$ and $0.45\lesssim\alpha\lesssim0.7$ .

2.1.1. Point source limit

Given the expression for the line acceleration, the momentum equation can be written in the point source limit as

(3) \begin{equation} v\frac{dv}{dr}=-\frac{GM}{r^2}-\frac{1}{\rho}\frac{dp}{dr}+g_e+g_\mathrm{L},\end{equation}

where G is the gravitational constant, M the stellar mass, p is pressure and

(4) \begin{equation} g_e=\frac{GM}{r^2}\Gamma\end{equation}

is the continuum acceleration. Here we have defined $\Gamma\equiv\sigma_eL/(4\pi cGM)$ which is the Eddington factor. By using the mass continuity equation ( $\rho=|\dot{M}|/(4\pi r^2v)$ ) and some simple manipulation we get

(5) \begin{equation} \left(1-\frac{c_s^2}{v^2}\right)r^2v\frac{dv}{dr}=-GM(1-\Gamma)+2c_s^2r+C\left(r^2v \frac{dv}{dr}\right)^\alpha,\end{equation}

where $c_s\equiv\sqrt{dp/d\rho}$ is the isothermal sound speed and C is a constant that contains all the model parameters

(6) \begin{equation} C\equiv\frac{\sigma_eLk}{4\pi c}\left(\frac{\sigma_ev_\mathrm{th}|\dot{M}|}{4\pi}\right)^{-\alpha}.\end{equation}

We hereafter ignore the pressure gradient ( $c_s\rightarrow0$ ), which is known to only make a minor difference in the velocity distribution. Requiring that Equation (5) has a unique solution, the velocity can be solved for analytically

(7) \begin{equation} v(r)=\sqrt{\frac{\alpha}{1-\alpha}}v_\mathrm{esc}\left(1-\frac{R}{r}\right)^{0.5},\end{equation}

where $v_\mathrm{esc}\equiv\sqrt{2GM(1-\Gamma)/R}$ is the escape velocity of the star. The parameter C cannot be arbitrarily chosen but has to take a specific form

(8) \begin{equation} C=\frac{1}{\alpha}\left(\frac{\alpha}{1-\alpha}GM(1-\Gamma)\right)^{1-\alpha}.\end{equation}

By equating this with the definition of C (Equation (6)), the mass-loss rate $\dot{M}$ can be uniquely obtained given all the other stellar parameters. We can see from Equation (7) that the wind has a terminal velocity

(9) \begin{equation} v_\infty=\sqrt{\frac{\alpha}{1-\alpha}}v_\mathrm{esc},\end{equation}

which is very close to the escape velocity of the star in the typical range of $\alpha$ ( $\alpha \sim 0.45$ –0.7). Equation (7) corresponds to a so-called $\beta$ -law which takes the form

(10) \begin{equation}v=v_\infty\left(1-\frac{R}{r}\right)^\beta,\end{equation}

with $\beta=0.5$ .

2.1.2. Finite disc correction

In reality, stars are not point sources and have finite sizes. The wind material can feel radiative forces from a wide range of angles when it is still close to the surface of the star. The net outwards momentum can thus be reduced close to the surface, as many photons approach the wind material from a non-radial angle. This effect can be accounted for by simply multiplying the line-driven acceleration by

(11) \begin{equation} K_{FDCF}=\frac{(1+\sigma)^{\alpha+1}-(1+\sigma\mu_*^2)^{\alpha+1}}{(1-\mu_*^2)(\alpha+1)\sigma(1+\sigma)^\alpha},\end{equation}

where $\sigma\equiv d\ln v/d\ln r-1$ , and $\mu_*\equiv\sqrt{1-(R/r)^2}$ . This gives $K_{FDCF}=1/(1+\alpha)$ at $r=R$ and $K_{FDCF}\rightarrow1$ for $r\rightarrow\infty$ . So the momentum equation with the finite disc correction becomes

(12) \begin{equation} r^2v\frac{dv}{dr}=-GM(1-\Gamma)+CK_{FDCF}\left(r^2v \frac{dv}{dr}\right)^\alpha.\end{equation}

Although this cannot be solved analytically, it is known that the solution to this equation can be well fitted by a $\beta$ -law of

(13) \begin{equation} v(r)=2.5\cdot \frac{\alpha}{1-\alpha}v_\mathrm{esc}\left(1-\frac{R}{r}\right)^{0.7},\end{equation}

based on both analytical (Pauldrach et al. Reference Pauldrach, Puls and Kudritzki1986) and numerical (Müller & Vink Reference Müller and Vink2008) studies. Notice that the terminal velocity is a few times faster than that of the point source limit. The factor 2.5 is consistent with both observations (Lamers et al. 1995) and numerical simulations (Vink et al. Reference Vink, de Koter and Lamers1999) for stars with temperatures $T\gtrsim21\,000$ K. This velocity law can be approximately obtained by setting the constant C to

(14) \begin{equation} C=C_0\equiv\frac{1+\alpha}{\alpha}\left[\frac{\alpha}{1-\alpha}GM(1-\Gamma)\right]^{1-\alpha}\end{equation}

in Equation (12) and integrating from the surface of the star outwards.

2.2. Velocity distribution for rotating stars

When the star is rotating, an extra term accounting for the centrifugal force enters the momentum equation, based on the assumption that the material in the wind conserves the specific angular momentum of the surface of the star. This alters the velocity distribution of the wind. Numerical investigations show that the terminal velocity decreases with increasing rotational velocity without significantly affecting the power $\beta$ (Castor Reference Castor and De Loore1979; Abbott Reference Abbott1980; Friend & Abbott Reference Friend and Abbott1986; Curé & Rial Reference Curé and Rial2004; Owocki Reference Owocki, Kraus, Miroshnichenko and Society2006; Madura et al. Reference Madura, Owocki and Feldmeier2007).

The wind velocity decreases because the escape velocity is effectively reduced due to the centrifugal force. With this in mind, we apply an extra factor in the constant C to account for the reduction in effective gravity

(15) \begin{equation} C=C_\mathrm{rot}\equiv C_0\left[1-\left(\frac{v_\mathrm{rot}}{v_\mathrm{crit}}\right)^2\right]^{1-\alpha},\end{equation}

to see if we can reproduce the wind profile for rotating stars. Here $v_\mathrm{rot}$ is the equatorial rotational velocity of the star and $v_\mathrm{crit}\equiv\sqrt{GM(1-\Gamma)/R}$ is the critical rotational velocity. In Figure 1, we show the velocity distribution obtained by integrating Equation (12) with the constant C as in Equation (15). The non-rotating case is in good agreement with the theoretical prediction expressed by Equation (13). As the rotational velocity increases, the wind velocity decreases while preserving the overall $\beta$ -law shape, which is consistent with previous findings (Friend & Abbott Reference Friend and Abbott1986).

Figure 1. Wind velocity distribution for stars with different rotational velocities. Model parameters $\alpha=0.55$ and $\Gamma=0.1$ are used for this calculation. Solid curves are the numerically computed values for rotational velocities $v_\mathrm{rot}/v_\mathrm{crit}=0, 0.2, 0.4, 0.6, 0.8$ , and the red dashed curve shows the $\beta$ -law presented in Equation (13).

We plot the terminal wind velocity against the rotational velocity in Figure 2. Our numerically obtained values are in almost perfect agreement with the analytical values based on a simple ‘nozzle analysis’ presented in Owocki (Reference Owocki, Kraus, Miroshnichenko and Society2006). The nozzle analysis uses the fact that the parameter C is proportional to the mass flux ( $C\propto|\dot{M}|^{-\alpha}$ ). The maximum allowed mass flux is determined by the condition that the velocity profile should be monotonically increasing outwards. The integration fails in our models with rotation velocities faster than $v_\mathrm{rot}/v_\mathrm{crit}\gtrsim0.8$ , which is roughly consistent with the critical rotation derived from the nozzle analysis as indicated by the jump in the red dashed curve (see also Araya et al. Reference Araya and Curé2018). The perfect agreement between our model and the nozzle analysis at $v_\mathrm{rot}/v_\mathrm{crit}\lesssim0.75$ implies that setting the constant C to Equation (15) is equivalent to restricting the mass flux to the ‘nozzle’ value.

Figure 2. Terminal wind velocity as a function of rotational velocity. The black solid curve shows the numerical value while the red dashed curve shows the analytical value following the analysis of Owocki (Reference Owocki, Kraus, Miroshnichenko and Society2006). The acceleration parameter used for this figure is $\alpha=0.5$ .

2.3. Equation of motion in a binary system

When the wind-emitting star is in an X-ray binary system, the gravitational force of the compact object should also be included in the equation of motion of the wind. The full equation of motion in the corotating frame becomes

(16) \begin{align} \frac{d{\textbf{v}}}{dt}=&-\frac{GM_1(1-\Gamma)}{|{\textbf{r}}-{\textbf{r}}_1|^2}{\textbf{n}}_1-\frac{GM_2}{|{\textbf{r}}-{\textbf{r}}_2|^2}{\textbf{n}}_2\nonumber\\ &+CK_{FDCF}\left(v_1|{\textbf{r}}-{\textbf{r}}_1|^2 {\textbf{n}}_1\cdot\nabla\left[{\textbf{n}}_1\cdot{\textbf{v}}\right]\right)^\alpha \frac{{\textbf{n}}_1}{|{\textbf{r}}-{\textbf{r}}_1|^2}\nonumber\\ &-\boldsymbol\Omega\times(\boldsymbol\Omega\times{\textbf{r}})-2\boldsymbol\Omega\times{\textbf{v}},\end{align}

where $M_i, {\textbf{r}}_i$ , ${\textbf{n}}_i({\textbf{r}})\equiv({\textbf{r}}-{\textbf{r}}_i)/|{\textbf{r}}-{\textbf{r}}_i|$ $(i=1, 2)$ are respectively the mass, position vector, and normal vector pointing from the i-th star and $\boldsymbol\Omega=\sqrt{G(M_1+M_2)/a^3}$ is the orbital angular velocity where $a\equiv|{\textbf{r}}_1-{\textbf{r}}_2|$ is the orbital separation. The first term includes the gravitational and continuum acceleration from the donor (primary), the second term is the gravitational acceleration from the accretor (secondary), the third term expresses the line-driven acceleration from the primary star, the fourth and fifth terms correspond to the centrifugal and Coriolis force. We will discuss the value of the parameter C in the following section.

2.4. Tides and GD

Here we are focusing on the case where the secondary star is a compact object (BH or NS) and therefore the primary star is the only wind emitter. In a close binary system on a circular orbit, the donor star can feel a strong tidal force from the companion and becomes synchronised with the binary orbit. The star would then adjust its shape such that it fills the equipotential surface of the Roche potential. This distortion of the stellar shape along with the tidal force from the secondary causes the radial acceleration on the surface of the star to vary drastically depending on the position. Since the terminal velocity of winds are strongly related to the surface effective gravity (see Section 2.2), the wind velocity field could be highly asymmetric. We approximately account for this effect by choosing the value of C proportional to the surface effective gravity as

(17) \begin{equation} C=C_\mathrm{bin}\equiv C_0\left(\frac{|{\textbf{g}}_0|}{|{\textbf{g}}_\mathrm{pole}|}\right)^{1-\alpha},\end{equation}

where ${\textbf{g}}_0$ is the radial acceleration at the base of a wind streamline (surface of the star) and ${\textbf{g}}_\mathrm{pole}$ is the radial acceleration at the pole of the star. Here the acceleration includes gravity from both stars and the centrifugal force. This expression is equivalent to Equation (15) when applied to single rotating stars. In binaries, this will lead to a non-axisymmetrical wind distribution, with lower wind velocities in the direction towards and away from the accretor.

In addition to determining the wind velocity distribution, the effective surface gravity is expected to change the surface radiative flux of the star via GDFootnote c. Based on von Zeipel’s theorem (von Zeipel Reference von Zeipel1924), the radiative flux F at the surface of the star scales as

(18) \begin{equation} F\propto T_\mathrm{eff}^4\propto |{\textbf{g}}_0|^{\beta_1},\end{equation}

where $T_\mathrm{eff}$ is the effective temperature of the star and $\beta_1$ is the GD exponent. The exponent is $\beta_1=1$ in the original von Zeipel theorem which was derived for radiative stars, but can take much lower values for convective stars ( $\beta_1 \sim 0.3$ –0.4; Lucy Reference Lucy1967; Claret Reference Claret2012). In rotating stars, this leads to a darkening of the equator due to the lower effective gravity, which has actually been observed for the case of Altair (Domiciano de Souza et al. Reference Domiciano de Souza, Kervella, Jankov, Vakili, Ohishi, Nordgren and Abe2005).

Figure 3. Illustration of the effective gravity distribution over the surface of the star in a tidally locked equal mass binary with a Roche lobe filling factor of 1. Panels (a) and (b): 3D shape of the tidally distorted star from two different angles. The surface is coloured by the surface acceleration. The black dots indicate the position of the accretor. Panel (c): Mercator projection of the surface acceleration. The coordinate origin points in the direction of the accretor star. The acceleration is normalised by the surface acceleration of a spherical star with the same volume. The wind originating from the region surrounded by the white line intersects with the accretion radius of the companion when the velocity correction is on (see section 3.2).

Due to the tidal force, the surface flux distribution on a star in a close binary system can be even more asymmetric than rotating single stars. For example, for a fully Roche lobe filling star, the effective gravity at the first Lagrangian point is zero, meaning that the radiative flux should also be zero at that point. Overall, such a star will have a vanishing flux at the point facing towards the accretor, a greatly reduced flux away from the accretor, slightly darkened sides and bright poles (White et al. Reference White, Baumgarte and Shapiro2012; Hadrava & Čechura 2012; Čechura & Hadrava Reference Čechura and Hadrava2015). This effect appears as ‘ellipsoidal variations’ in the optical light curves of X-ray binary donors (e.g. Avni & Bahcall Reference Avni and Bahcall1975). We illustrate the surface effective gravity map in Figure 3. It is clear that there are two dark patches, but the patch facing towards the accretor is much darker than the opposite side. The acceleration is normalised by the surface acceleration of a spherical star with the same mass and volume as the distorted star ( ${\textbf{g}}_\mathrm{sph}$ ). Notice that the acceleration at the poles slightly exceeds that of a spherical star, as the tidal force acts to flatten the star in the vertical direction.

Since the stellar winds of massive stars are mainly driven by radiation, GD will cause the wind mass flux to vary over the surface of the star. This has been thoroughly studied theoretically for the case of rotating stars where the polar mass flux is expected to be enhanced over the equatorial mass flux (Cranmer & Owocki Reference Cranmer and Owocki1995; Owocki et al. Reference Owocki, Cranmer and Gayley1996; Owocki & Gayley Reference Owocki, Gayley, Nota, Lamers and Society1997; Owocki et al. Reference Owocki, Gayley, Cranmer, Howarth and Society1998; Gagnier et al. Reference Gagnier, Rieutord, Charbonnel, Putigny and Espinosa Lara2019). Although GD has been observed for some stars, observational evidence for its influence on the wind driving is still ambiguous. However, such polar-enhanced winds are known for some massive stars such as $\eta$ Carinae (Smith et al. Reference Smith, Davidson, Gull, Ishibashi and Hillier2003). The rotation of the star has not been measured, but given all the other indirect evidence supporting a stellar merger origin for $\eta$ Carinae (e.g. Smith et al. Reference Smith2018; Hirai et al. Reference Hirai and Podsiadlowski2021), it is natural to associate the latitudinal dependence with rapid rotation. We therefore expect that the asymmetric GD in close binaries could also lead to highly asymmetric mass flux. We take this into account by assuming

(19) \begin{equation} \frac{\dot{M}}{\dot{M}_\mathrm{pole}}=\left(\frac{|{\textbf{g}}_0|}{|{\textbf{g}}_\mathrm{pole}|}\right)^{\beta_1},\end{equation}

where $\dot{M}$ is the local mass flux and $\dot{M}_\mathrm{pole}$ is the mass flux at the pole.

We ignore any variations of $\alpha$ over the surface, although this could also be an important factor as suggested in the context of B[e] stars via the so-called bi-stability mechanism, which is caused by the change in ionisation below some threshold effective gravity (Lamers & Pauldrach Reference Lamers and Pauldrach1991).

Figure 4 displays the total mass flux from a gravity darkened star ( $\beta_1=1$ ) compared to that of a non-darkened case ( $\beta_1=0$ ). For an equal mass binary ( $q=1$ ), the total mass flux can be reduced by up to ${\sim}20\%$ when it is close to filling its Roche lobe. The reduction is slightly larger for unequal mass binaries, but the mass ratio dependence is minor.

Figure 4. Total reduction of mass flux due to gravity darkening in binaries. Different colours show results for different mass ratios, which is defined as $q\equiv M_2/M_2$ . All curves were computed assuming maximal gravity darkening ( $\beta_1=1$ ). The $q=0.5$ curve is almost identical to the $q=2.0$ curve and is therefore not visible.

2.5. Caveats

We made several simplifying assumptions in our formulation of the wind acceleration. First, we have ignored the pressure gradient in the wind acceleration, which can slightly raise the terminal velocity. Multiple scattering of photons have been ignored, which could lead to enhanced winds in optically thick cases (e.g. Friend & Castor Reference Friend and Castor1983; Abbott & Lucy Reference Abbott and Lucy1985). While multiple scattering can be important for stars like Wolf-Rayet stars, it is expected to be minor in the optically thin winds of O/B stars. We have assumed the radiative forces from the donor are always radial. This is not strictly true when the star suffers GD and there are flux variations over the surface (Gayley & Owocki Reference Gayley and Owocki2000). However, the non-radial component of the radiative force is at most ${\lesssim}10 \%$ and rapidly decreases with distance from the star (Čechura & Hadrava Reference Čechura and Hadrava2015). There also could be limb darkening effects as the stars do not have sharp surfaces (Cranmer & Owocki Reference Cranmer and Owocki1995).

Another key assumption is that we use the same Eddington factor and acceleration parameter $\alpha$ over the entire streamline, using the value at its launching point. This can be incorrect around the locations where there are significant flux and temperature variations due to GD, or when the streamlines are bent significantly by the Coriolis force. The former can reduce the effect of GD, because the wind material can see brighter parts of the star as it travels out, even if it was launched at much darker patches. Also, we have ignored any variations of $\alpha$ over the surface of the star. For example, the bi-stability mechanism may boost the mass flux at the darker areas, counteracting the GD effect we apply here (Lamers & Pauldrach Reference Lamers and Pauldrach1991; Owocki & Gayley Reference Owocki, Gayley, Nota, Lamers and Society1997; Owocki et al. Reference Owocki, Gayley, Cranmer, Howarth and Society1998; Gagnier et al. Reference Gagnier, Rieutord, Charbonnel, Putigny and Espinosa Lara2019).

Additionally, we have assumed a stationary wind velocity and density distribution. Line-driven winds are subject to instabilities, which may cause large variabilities leading to X-ray emissionFootnote d (see Owocki Reference Owocki1994 for a theoretical review on wind instability). As a result, the wind can become very clumpy, which can in principle cause stochastic accretion patterns (Oskinova et al. Reference Oskinova, Feldmeier and Kretschmar2012). However, the inhomogeneities on the orbital scale can be wiped out by hydrodynamical interactions before reaching the X-ray emitting region (El Mellah et al. Reference El Mellah, Sundqvist and Keppens2018). Indeed, the observed X-ray variability is much smaller than what is predicted by simply translating the orbital scale stochasticity to X-rays (Oskinova et al. Reference Oskinova, Feldmeier and Kretschmar2012). Also, the time-averaged wind profiles usually follow the stationary solution (e.g. Owocki et al. Reference Owocki, Castor and Rybicki1988), so the actual effect of wind clumping may be limited in the long term.

One major effect we have ignored in this study is the X-ray feedback from the accretor. The ionisation structure of the wind material can be altered by the irradiation, strongly influencing the wind driving (Hatchett & McCray Reference Hatchett and McCray1977; MacGregor & Vitello Reference MacGregor and Vitello1982; Masai Reference Masai1984; Stevens & Kallman Reference Stevens and Kallman1990; Sander et al. Reference Sander, Fürst, Kretschmar, Oskinova, Todt, Hainich, Shenar and Hamann2018). When the metals become photoionised by the X-rays, the UV absorption lines are significantly weakened. The radiative flux from the donor cannot push on the wind any more, reducing the wind terminal velocity. The consequences of diminishing acceleration strongly depend on how far the ionisation front reachesFootnote e, which in turn depends on the luminosity of the X-ray feedback. If the feedback is relatively weak, the photoionisation only occurs in a region where the wind has already accelerated sufficiently, and the effect of ionisation acts to decrease the terminal velocity. This will increase the accretion radius and therefore raises the total mass accretion rate (e.g. Karino Reference Karino2014; Čechura & Hadrava Reference Čechura and Hadrava2015). In the presence of strong feedback where the ionisation front reaches the surface of the donor, it can completely shut off the wind from the hemisphere facing the accretor (Blondin Reference Blondin1994; Krtička et al. Reference Krtička, Kubát and SkalickÝ2012; Čechura & Hadrava Reference Čechura and Hadrava2015; Krtička & Kubát Reference Krtička and Kubát2016; Krtička et al. Reference Krtička, Kubát and Krtičková2018). Such a situation cannot be sustained because the X-ray feedback will eventually die out as the accretor exhausts the matter that was powering the strong X-rays in the first place. This means that HMXBs may be in a self-regulated state where the X-ray luminosity varies around the critical value for wind inhibition (Krtička et al. Reference Krtička, Kubát and SkalickÝ2012; Krtička et al. Reference Krtička, Kubát and Krtičková2018). Krtička & Kubát (Reference Krtička and Kubát2016) show that many X-ray binaries including Cyg X-1 have X-ray luminosities close to the wind inhibition limit. This effect may be weakened due to wind clumping (Oskinova et al. Reference Oskinova, Feldmeier and Kretschmar2012). Although the X-ray irradiation is indeed a critical ingredient to understand the true wind accretion process, we shall leave quantitative investigation for future work and only briefly discuss the qualitative effects in this paper.

3.1. Method

Most of our methodology is similar to El Mellah & Casse (Reference El Mellah and Casse2017) except for the asymmetrical effects on the wind acceleration, which is the main focus of this paper.

To model the tidally distorted donor surface, we compute the equipotential surface within the Roche potential

(20) \begin{equation} \Phi({\textbf{r}})=-\frac{GM_1}{|{\textbf{r}}-{\textbf{r}}_1|}-\frac{GM_2}{|{\textbf{r}}-{\textbf{r}}_2|}-\frac12|\Omega\times{\textbf{r}}|^2,\end{equation}

assuming that the star is tidally locked to the orbit. Using the Roche potential implies point masses for the binary components, which is not a bad assumption given that massive main sequence stars have very centrally concentrated mass distributions. We characterise the size of the donor star with the Roche lobe filling factor defined as $f\equiv(V_*/V_\mathrm{RL})^{1/3}$ , where $V_*$ is the volume of the star, assumed to fill a region bounded by an equipotential surface, and $V_\mathrm{RL}$ is the volume of the Roche lobe. Varying f corresponds to either changing the stellar size with a fixed orbital separation, or changing the orbital separation for a fixed star.

From each point on the surface of the star, we integrate the equation of motion (16) to obtain the velocity distribution of the wind. In order to calculate the amount of mass and angular momentum that accretes onto the companion, we draw a sphere around the compact object with a radius equal to the BHL accretion radius (Hoyle & Lyttleton Reference Hoyle and Lyttleton1939; Bondi & Hoyle Reference Bondi and Hoyle1944) expressed as

(21) \begin{equation} r_\mathrm{acc}=\frac{2GM_2}{v_\bullet^2},\end{equation}

where $v_\bullet$ is the wind velocity in the vicinity of the accreting object. For the value of $v_\bullet$ , we use the vector sum of the orbital velocity and the radial wind velocity at a distance a from a single rotating star with the same volume and angular velocity as the tidally distorted star. At a distance $r_\mathrm{acc}$ from the accretor, the wind is already significantly focused and hydrodynamical effects could become important. We consider an extended accretion radius $r_\mathrm{ext}>r_\mathrm{acc}$ to compare results at different scales.

3.2. Models

We use units of $G=a=M_1=1$ for all of our calculations. The mass ratio is defined as $q\equiv M_2/M_1$ . We carry out calculations for a grid of binary models with different mass ratios $q=0.5, 1, 2$ and Roche lobe filling factors $f\in(0,1]$ . We also vary the force multiplier parameter $\alpha=0.5, 0.6$ and Eddington factor $\Gamma=0.1, 0.3$ to compare results for stars with different surface properties.

For each binary model, we compute two different wind velocity distributions with different expressions for the constant C when integrating the equation of motion: one with $C=C_0$ (Equation (14)) and one with $C=C_\mathrm{bin}$ (Equation (17)). The latter includes a reduction factor proportional to the lower effective surface gravity, and hereafter we will call this the velocity correction (VC). We also investigate the impact of GD, by setting $\beta_1=0$ or 1 in Equation (19). $\beta_1=1$ corresponds to a fully gravity darkened case and $\beta_1=0$ corresponds to a uniformly bright star. Note that in our formulation, the VC only influences the velocity distribution whereas GD only affects the mass flux along each streamline.

Figure 5. Wind streamlines in the equatorial plane (white curves). Model parameters are listed in the legend. The orbital angular momentum points out of the page. The background is coloured by the mass flux along each streamline normalised by the mass flux at the pole of the donor, assuming fully efficient GD ( $\beta_1=1$ ). Black dashed curves indicate the Roche lobe. Yellow curves show the locus of momentum cancellation in the equatorial plane. Insets show zoomed-in images of the accretion stream. Left and right panels compare the streamlines with and without the VC.