The Efficiency of Mass Accretion and Disc Structure from the Stellar Wind Mass Transfer in Binary Systems

Abstract

There have been many research works involving mass transfer in stellar binaries, all of which are limited to certain systems with specific binary parameters. In this work, we use three-dimensional smoothed particle (3D-SPH) simulations to explore the impact of binary mass ratio and wind speed on the fraction of mass transferred to the accreting companion and the structure of accretion discs. We examine all possible cases of binary mass ratios as well as different conditions of wind speed in the vicinity of the accretor. We adhere to thermally driven winds, with sound speed being the main parameter, in which transonic stellar winds expand in the binary medium. We find that mass accretion fraction is close to unity for slow winds. However, fast winds lead to mass accretion fraction of thousandths which agree very well with the Bondi–Hoyle estimates. Mass accretion fraction is found to be the largest when the mass ratio is unity. Our results show that an increase in either sound speed or binary mass ratio leads to decrease in accretion disc size. In most cases, the disc shifts from being circular. These results would allow us to estimate the rate of mass accretion and the structure of accretion discs in any type of stellar binaries.

1. Introduction

Many stars exist in binary or multiple systems, in which two or more stellar objects are gravitationally bound and orbit around their common Centre of Mass (CM). Observations suggest that the majority of stars are in binaries [1,2,3]. One of the most important modes of interaction between the binary components is mass transfer. This happens when the accreting component captures part or all of the mass lost by the mass-losing companion in the binary. The consequences of this process may include, for example, orbital dynamics evolution through angular momentum transfer [4] as well as a significant influence on stellar structure and the evolutionary path of the stellar components [5]. Researches suggest that angular momentum transfer associated with mass accretion contributes to the rapid rotation in massive stars [6]. Moreover, mass transfer in binaries may cause astrophysical events such as recurrent novae, supersoft X-ray emission, or type Ia supernovae if a White Dwarf (WD) is to be an accretor in the binary [7,8,9]. Mass transfer and accretion onto the surface of the accretor occur via accretion disc as demanded by the law of conservation of momentum in the captured material [10,11,12,13]. The size of the accretion disc is believed to be significant for outbursts to happen due to a dwarf nova driven by the thermal viscous instability [14,15,16]. It is also confirmed that accretion disc size plays an important role in the possibility of the formation of the second-generation planets [17,18,19]. Mass transfer in binary interactions occurs through two main mechanisms. The first is Roche Lobe Overflow (RLOF) in close binary stars [20], which happens when one of the stars fills its Roche lobe under any condition, leading to mass transfer between stars, and nearly all the transferred material is accreted by the companion star. The second mechanism is the transfer of mass via stellar wind. The second mechanism is less efficient compared to the RLOF because a large fraction of the lost mass escapes the system rather than being captured by the accretor [2,21]. Knowing the vitality of mass transfer in binaries, there have been studies conducted to explore various aspects of the process. Theuns et al. [22] investigated mass accretion rates in binaries and found that the efficiency of wind capture strongly depends on the wind velocity relative to the orbital motion. When the wind velocity $v_w$ is much higher than the orbital velocity $v_{orb}$, the accretion is inefficient because the wind escapes before being gravitationally captured. Conversely, accretion efficiency increases due to stronger gravitational concentration when $v_w$ is lower than $v_{orb}$. Nagae et al. [23] performed three-dimensional hydrodynamic simulations to calculate mass accretion fraction for a binary system of mass ratio $q=1.00$. They characterised the value of radial velocity of the wind at the Roche surface of the mass-losing star to determine the mechanisms of mass transfer and mass accretion fraction. Davis et al. [24] used a binary evolution code to numerically calculate mass transfer through Roche Lobe Overflow in a nearly solar-type star and studied the variation of mass transfer in a period of an eccentric orbit. They found that rotation causes deformation in the donor star and determines whether mass transfer occurs. Dosopoulou et al. [25,26,27] studied binary orbital evolution analytically due to mass transfer. Church et al. [28] used Smoothed Particle Hydrodynamics (SPH) to model a Cataclysmic variable system of a $0.6M\odot$ donor and a $1.0M\odot$ WD and studied modes of mass transfer in eccentric orbits. Hogg et al. [29] used binary population synthesis code based on Milky Way properties to investigate the formation of discs and found that around 26% of binaries can host second- and 13% can host third-generation discs. Lajoie and Sills [30,31] used numerical methods of SPH to model RLOF and study mass accretion in binaries with mass ratios of 0.5 and 1.0 for orbits with low eccentricity. Their results showed that the enhancement in mass transfer during the periastron passage varies in other parts of the orbital phase. Liu et al. [2] studied mass transfer with numerical simulations of binaries of mass ratios equal to or greater than one with a wind model applicable to the Asymptotic Giant Branch (AGB) donors with an initial wind particle velocity comparable to that of the orbital one. Their result shows a decrease in mass and angular momentum accretion as the mass ratio increases. Maes et al. [32] also used SPH to model mass transfer from an AGB wind in a binary; they looked more into the morphology of the mass transferred for a certain range of wind speed and binary separation. Saladino and Pols [33] investigated the morphology of mass transferred and rate of mass and angular momentum accretion for a range of orbital eccentricity with a mass ratio of $2.00$. Their mass accretion efficiency results were comparable to those of Bondi–Hoyle–Lyttleton (BHL) model estimates. The work of Saladino et al. [34] focused on modelling winds from rotating and non-rotating AGBs and cases of mass ratio equal or greater than unity and binary separations from $4AU$ to $20AU$. They found that for relatively low wind initial velocity, the morphology of the mass transfer is RLOF-like. In addition to their indication of whether accretion discs form for each model. In addition to studying angular momentum accretion, they calculated accretion efficiency from thousandths for fast winds to tenths for relatively slow winds. The research conducted by de Val-Borro et al. [35] investigated mass transfer and accretion efficiency in symbiotic binaries containing an AGB donor. In this work, the density profile and rotational velocity curve of the accretion discs are examined for systems with binary mass ratio $q=M_d/M_a=2.00$.

The semi-analytic modelling by Vathachira and Hillman [36] involves a four-dimensional parameter space of WD mass, donor mass, donor radius, and binary separation in symbiotic binaries to identify the regime boundaries between Wind Roche Lobe Overflow (WRLOF) and the wind that follows BHL prescription. Hydrodynamic simulations by Lee et al. [37] for an S-type symbiotic system with mass ratio $q=2.5$ suggest BHL estimate for mass accretion fraction is only consistent for fast winds whose speed is higher than the binary orbital speed. They also found azimuthal asymmetry of the gas density distribution in the accretion disc formed around the WD. Subsequent research by Tejeda and Toalá [38] suggested a geometrical correction for BHL estimates for the low-speed wind regimes, which are mainly the case in symbiotic binaries.

In this work, we use 3D-SPH to simulate mass transfer from a mass-losing component to an accreting companion in a binary system. To our knowledge, the mass accretion efficiency has been studied for binaries with mass ratio $q\ge 1.00$. We examine the dependence of the mass accretion efficiency on q in a broader range to include the cases where the accretor is greater than the donor, that is, $(0.25\le q\le 4.00)$, in addition to investigating the relation between the mass accretion fraction and the wind speed. Moreover, we focus on the dependence of the formation and size of the accretion disc on the wind speed and the binary mass ratio.

2. Theory of Binary Systems and Mass Transfer

In a binary system, two stars orbit around their CM with a binary orbital speed

$$v_{orb}={\displaystyle \frac{2a\pi }{P}}$$

(1)

The binary mass ratio is defined as

$${\displaystyle \frac{M_d}{M_a}}=q$$

(2)

with $M_d$ and $M_a$ being the mass of the donor and the accretor, respectively.

One of the equipotential surfaces produced due to the gravitational effect of the stars represents the Roche lobes. The Roche lobe radius is [39]

$$R_L\approx {\displaystyle \frac{0.49q^{2/3}}{0.6q^{2/3}+ln\left(1+q^{1/3}\right)}}a$$

(3)

where a is the distance between the stars (binary separation).

The size of the Roche lobe determines the mode of mass transfer or the region where mass transfer occurs [35]. Mass exchange happens when one star loses mass while the other gains mass, leading to accretion. The RLOF is achieved when the size of the mass-losing star exceeds the size of its Roche lobe; therefore, gas from its surface spills into the Roche lobe of its accreting companion. Otherwise, mass can only be transferred to the companion through the spherical outflow of material on the donor surface in the form of stellar wind. Studies including Mohamed and Podsiadlowski [40] and Abate et al. [41] revealed an intermediate case of mass transfer caused by the gravitational effect of the accretor on the wind of the donor, producing a gravitationally focused stream of wind gas into the Roche lobe of the accretor. This is called WRLOF. In the cases of RLOF and WRLOF, mass accretion fraction is efficient and could reach 100%, whereas capture of the stellar wind gas is much less efficient. The mass accretion rate $\dot{M}$ according to the BHL model is given by

$${\dot{M}}_{acc}=\pi R_{acc}^2\rho v_{rel}$$

(4)

where $R_{acc}={\displaystyle \frac{2G{M}_a}{{v_{rel}}^2}}$ is the accretion radius, $\rho$ is the density of the gas particle, and $v_{rel}$ is the relative velocity of the wind to the accretor. This equation assumes that a body moves at a given speed in fast winds and that the wind is supersonic [42,43]. By incorporating the orbital motion of the stars and introducing a constant mass loss rate $\dot{M}$ for a spherically symmetric expansion of the donor wind, the BHL accretion fraction takes the form

$$f_{BHL}={\displaystyle \frac{v_{orb}^4}{{\left(1+q\right)}^2{v}_w{\left(v_w^2+v_{orb}^2\right)}^{3/2}}}$$

(5)

where $v_w$ is the wind velocity at the accretor’s location. The value of $v_w$ depends strongly on the mechanism behind the gas outflow from the star. Furthermore, the value of $v_w$ depends on the stellar properties of the accretor, which in turn affects the fraction of the mass accreted. Although not considered in this research, if the accretor has its own wind, then wind–wind collision and shock fronts occur, which might slow down the incoming gas from the donor. If the accretor is considerably luminous, then the radiation pressure would block a significant fraction of the wind to accrete through radiation braking [44].

The nature of the gas outflow in stars is different and complicated. The simplest model would be an isothermal wind in which the gas is accelerated outward as a result of only two opposing forces considered: the outward force due to the gradient of gas pressure and the inward gravitational force (toward the centre of the star). The momentum equation for this model will then be (assuming constant mass loss rate in time) [45]

$$v{\displaystyle \frac{dv}{dr}}+{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{dP}{dr}}+{\displaystyle \frac{G{M}_d}{r^2}}=0.$$

(6)

where G is the universal gravitational constant. Solving for radial velocity v (see reference [45] for more details),

$$vexp\left(-{\displaystyle \frac{v^2}{2c_s^2}}\right)=c_s{\left({\displaystyle \frac{r_c}{r}}\right)}^{2}exp\left(-{\displaystyle \frac{2r_c}{r}}+{\displaystyle \frac{3}{2}}\right)$$

(7)

where $c_s={\left(RT/\mu \right)}^{1/2}$ is the constant isothermal speed of sound in the gas and $r_c=G{M}_d/2{c_s}^2$ is the critical distance or the critical radius. The equation of motion above has many solutions but only one critical solution, which represents a transonic solution that is the wind to start subsonic at the surface of the star $(v<c_s)$ at $(r<r_c)$, pass through the sonic point $(v=c_s)$ at $(r=r_c)$, and end up supersonic at large radii $(v>c_s)$ at $(r>r_c)$.

3. Method

For our models, we use a modified version of the SPH code originally developed by Nixon et al. [46] to simulate mass transfer in binary stars via stellar wind [47]. It is a three-dimensional conservative Lagrangian framework in which the self-gravity of the gas is neglected so the particles interact through pressure and artificial viscous forces. Throughout this work, artificial viscosity is implemented with SPH linear viscosity $\alpha =1.0$, and the quadratic viscosity $\beta =2.0$. The smoothing length parameter is $\eta =1.10$, which yields approximately 45 neighbouring particles. The mass of each SPH particle is $m_{pSPH}={10}^{-15}M$, with M being the total binary mass. In our simulations, binary components evolve through mass transfer. Each star is represented by a point mass orbiting around their CM on a circular orbit relative to an inertial frame of reference. SPH particles are injected in a spherically symmetric manner at a certain radius $r_{inj}$ from the point representing the mass-losing component, and the surface of the sphere with a radius $r_{inj}=R_d$ would then be considered as the surface of the star. The simulation space is initially empty of gas particles, but as time passes, continuous particle injection leads to a steady state in the simulation environment. The wind gas particles in our models behave as an isothermal wind with a constant adiabatic index $\gamma =1.0$. In the meantime, a fraction of the particles become gravitationally bound to the component representing the accretor to form an accretion disc. The gas particles then spiral into a surface that represents the spherical surface of the accreting component at a radial distance $R_a$ from the point representing the centre of the accretor. Particles are removed from the simulation if they

(a) Reach a distance relative to the donor less than $R_d$ assumed to fall back onto the donor’s surface;

(b) Reach a relative distance to the accretor smaller than $R_a$ assumed to be accreted onto the companion;

(c) Reach a radial distance from the CM three times the binary separation $r=3a$, as these particles are considered to be energetic and easily escape the binary’s gravitation.

Different numbers of SPH particles are injected in different simulations based on the binary and wind properties. For example, models with low $c_s$ and high q demand a relatively higher number of particle injections per orbit than those with high $c_s$ and low q. In the WRLOF model of $c_{s}0.15q0.25$ (top panels in Figure 1), a total number of 16,000,000 particles are injected per orbit into the simulation from the donor’s surface; this results in nearly 318,000 active particles at any instant after the steady state of the system. On the other hand, in the supersonic wind model of $c_{s}0.80q0.25$ (bottom panels of Figure 1), the total number of particles injected per orbit is 6,400,000 to yield about 760,000 active particles in steady state. This is because in steady state, the conservation of the SPH particle number mandates that

$$N_{inj}=N_d+N_a+N_{out}$$

(8)

where $N_{inj}$ is the number of SPH particles injected at any timestep from the injection surface, $N_d$ is the number of particles taken out of the simulation as they fall back into the donor’s surface, $N_a$ is the number of particles removed due to accretion onto the companion, and $N_{out}$ is the number of particles escaping the binary system.

It is worth mentioning that we have tested the convergence of our results by comparing the calculated numerical accretion fractions for simulations with the same binary and wind parameters but different resolutions. The comparison showed that the simulation results of $f_{Nacc}$ differ by around $0.3\%$ at the highest when the number of active particles increased by a factor of about 6.

Basically, SPH particles possess thermal, kinetic, and gravitational potential energy according to the equation,

$$e={\displaystyle \frac{1}{2}}{v}^2-{\displaystyle \frac{GM}{r}}+u$$

(9)

where e and u are the specific total and thermal energy, respectively, v is the particle’s velocity relative to the centre of CM, M is the total binary mass, and r is the radial distance of the particle relative to the CM. We arbitrarily chose the rejection radius to be $r=3a$ to reduce unnecessary computation because we have investigated that any particles reaching that radial distance are too energetic to be bound to the system.

Furthermore, the SPH code used in this work is set up to make measurements in dimensionless code units rather than physical units to reduce computational time consumption. The values of the binary parameters are listed in

Table 1. Physical quantities in the SPH code.

Physical Parameter	Code Value	Common Physical Unit
Binary separation	$a=1$	$AU$
Binary total mass	$M=1$	$M_{\odot }$
Gravitational constant	$G=1$	$6.674\times {10}^{-11} \mathrm{N}{\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-2}$
Orbital period	$P=2\pi$	Days

.

Throughout this work, distances are measured in units of binary separation; sound speed is in units of orbital speed (Equation (1)). Mass is measured in units of binary mass, and the time is in units of orbital period.

4. Results

We investigate the characteristics of mass transfer by focusing on the effects of binary mass ratio (see Equation (2)) $q<1.00$, $q=1.00$, and $q>1.00$ and the values of the sound speed $c_s$. We aim to acquire a deeper understanding of mass accretion fraction $f_{Nacc}$, modes of mass transfer, and the morphology of accretion discs through 54 simulation models presented in

Table 2. Simulation models studied in this research work. Column 1 shows the model ID; for instance, $c_{s}0.15q0.25$ indicates a model in which $c_s=0.15v_{orb}$ and binary mass ratio $q=0.25$, and so forth. Column 2 shows the wind speed at a binary separation distance from the donor (at the accretor’s location). The third column shows the wind Mach number at the position of the accretor. The numerical results of the mass accretion fraction are listed in column 4.

Model ID	${\mathit{v}}_{\mathit{w}}$	$\mathit{Ma}={\mathit{v}}_{\mathit{w}}/{\mathit{c}}_{\mathit{s}}$	${\mathit{f}}_{\mathit{Nacc}}$
$c_{s}0.15q0.25$	0.0016	0.0106	0.9000
$c_{s}0.20q0.25$	0.0385	0.1925	0.1360
$c_{s}0.20q0.50$	0.0037	0.0185	0.6500
$c_{s}0.20q1.00$	0.0001	0.0005	0.5300
$c_{s}0.26q0.25$	0.1576	0.6061	0.0110
$c_{s}0.26q0.50$	0.0612	0.2353	0.0540
$c_{s}0.26q1.00$	0.0127	0.0488	0.2170
$c_{s}0.26q2.00$	0.0021	0.0081	0.1150
$c_{s}0.26q4.00$	0.0005	0.0019	0.0380
$c_{s}0.33q0.25$	0.3676	1.1139	0.0060
$c_{s}0.33q0.50$	0.1907	0.5779	0.0210
$c_{s}0.33q1.00$	0.0845	0.2561	0.1300
$c_{s}0.33q2.00$	0.0318	0.0964	0.0570
$c_{s}0.33q4.00$	0.0142	0.0433	0.0680
$c_{s}0.40q0.25$	0.5813	1.4532	0.0600
$c_{s}0.40q0.50$	0.3836	0.9590	0.0850
$c_{s}0.40q1.00$	0.2219	0.5548	0.1020
$c_{s}0.40q2.00$	0.1268	0.3170	0.0560
$c_{s}0.40q4.00$	0.0767	0.1917	0.0440
$c_{s}0.45q0.25$	0.7578	1.6840	0.1930
$c_{s}0.45q0.50$	0.5355	1.1900	0.0840
$c_{s}0.45q1.00$	0.3555	0.7900	0.0850
$c_{s}0.45q2.00$	0.2337	0.5193	0.0610
$c_{s}0.45q4.00$	0.1606	0.3569	0.0560
$c_{s}0.50q0.25$	0.9390	1.8780	0.1510
$c_{s}0.50q0.50$	0.6975	1.3950	0.2000
$c_{s}0.50q1.00$	0.5000	1.0000	0.0710
$c_{s}0.50q2.00$	0.3573	0.7146	0.0560
$c_{s}0.50q4.00$	0.2707	0.5414	0.0580
$c_{s}0.55q0.25$	1.1236	2.0429	0.1020
$c_{s}0.55q0.50$	0.8695	1.5809	0.1100
$c_{s}0.55q1.00$	0.6545	1.1900	0.0780
$c_{s}0.55q2.00$	0.4981	0.9056	0.0710
$c_{s}0.55q4.00$	0.3966	0.7211	0.0520
$c_{s}0.60q0.25$	1.3152	2.1920	0.0710
$c_{s}0.60q0.50$	1.0470	1.7450	0.0730
$c_{s}0.60q1.00$	0.8166	1.3610	0.0650
$c_{s}0.60q2.00$	0.6462	1.0770	0.0450
$c_{s}0.60q4.00$	0.5384	0.8973	0.0190
$c_{s}0.66q0.25$	1.5687	2.3768	0.0420
$c_{s}0.66q0.50$	1.2546	1.9009	0.0430
$c_{s}0.66q1.00$	1.0374	1.5718	0.0350
$c_{s}0.66q2.00$	0.8375	1.2689	0.0220
$c_{s}0.66q4.00$	0.7281	1.1031	0.0090
$c_{s}0.80q0.25$	2.1160	2.6450	0.0170
$c_{s}0.80q0.50$	1.7960	2.2450	0.0180
$c_{s}0.80q1.00$	1.5192	1.8990	0.0130
$c_{s}0.80q2.00$	1.3088	1.6360	0.0080
$c_{s}0.80q4.00$	1.1704	1.4630	0.0033
$c_{s}1.00q0.25$	2.9640	2.9640	0.0059
$c_{s}1.00q0.50$	2.5970	3.5970	0.0056
$c_{s}1.00q1.00$	2.2790	2.2790	0.0041
$c_{s}1.00q2.00$	2.0370	2.0370	0.0025
$c_{s}1.00q4.00$	1.8780	1.8780	0.0016

. In most of these models, the binary orbital speed is supersonic; in other words, $c_s\le v_{orb}$ for all simulations. Running models with $c_s>v_{orb}$ would not lead to any significant change in the mass transfer; therefore, we do not investigate models with higher sound speed. The values of sound speed in units of $v_{orb}$ include $c_s\le 0.20$ to represent cool and slow winds, which results in the so-called WRLOF.

The values of sound speed in the range $0.20<c_s<0.40$ give a certain mode of mass transfer that is neither WRLOF nor isotropic (spherical) wind. Finally, the cases where sound speed is comparable to orbital speed $0.40\le c_s$ are cases in which the outflow of the simulated gas is a spherically symmetric and fast wind to model the situations where BHL estimate applies. The wind properties follow the Parker model for all simulations so that the wind isothermally expands due to the resultant effect of only two opposing forces: the inward gravitational force from the mass-losing star and the outward force produced by the gradient of pressure in the gas. Figure 2 shows the radial expansion of the winds for two examples of our models. The solution of the isothermal Parker wind is also plotted. This radial velocity profile is tested for a stationary donor star, and the wind expands to a distance from the star many times the binary separation. It is clear that the radial speed of the gas particles agrees very well with that of the Parker model.

Since the mass and size of stars are related (see, e.g., [48,49]), it is convenient not to keep the stars’ radii at a fixed value for different stellar masses. We would rather set the donor’s radius to be related to its mass in the binary and hence to its Roche radius. Therefore, the radial distance of the surface from which the SPH particles are injected is chosen to be equal to $0.9R_L$ to ensure that the star’s size is within the Roche sphere of the donor (Equation (3)).

The efficiency of accretion or accretion fraction in such a state of the systems is defined as the amount of mass accreted onto the companion to the amount of mass lost by the donor per unit time. In steady state, the mass loss rate is the combination of mass accretion and mass escaping the binary system per unit of time.

$${\dot{M}}_d={\dot{M}}_a+{\dot{M}}_{out}$$

(10)

Therefore, according to the definition, the accretion fraction will be

$$f_{Nacc}={\displaystyle \frac{{\dot{M}}_a}{{\dot{M}}_a+{\dot{M}}_{out}}}$$

(11)

Figure 1 and Figure 3 show the column density of the gas transferred from the donor to the accretor in binary surroundings. These visualisations are made using SPLASH tools to render SPH particles [50]. It can be seen that the binary components orbit on the x,y plane and the $+z$ axis is toward, while the $-z$ axis is away from the reader. All panels in a row represent the time evolution of a model with a certain q and $c_s$. At the very beginning, each simulation starts with a binary space empty of gas. Continuous injection of the SPH particle (from the donor’s surface) brings the simulation to steady state, i.e., the shape and rate of mass transfer remain unchanged afterwards. The first columns in Figure 1 and Figure 3 show the start of each simulation, that is, $t=0.00$. The second columns show the system in their stable phase of mass transfer, and the state of the system remains the same thereafter. It should be mentioned that the time to reach this steady state is not the same for all simulations, as shown in Figure 4. Models with low sound speed need several orbits to settle, whereas models with high sound speed (fast winds) require a small fraction of an orbit to achieve steady mass transfer.

It can be inferred from Figure 1 that for cool and slow winds with sound speed as small as $0.15$, for instance, the mode of mass transfer is WRLOF. In such a case, the wind particles are too slow to expand spherically, and only have a chance to spill into the Roche sphere of the companion via the first Lagrange point $L_1$, producing a large accretion disc around the accretor. Moderate sound speed in the gas produces a particular situation in which some of the gas particles manage to leave the donor through the second as well as the first Lagrange point ($L_1$ and $L_2$) along with an enhanced mass escape from the accretion disc via the third Lagrange point $L_3$. The combined effect of mass loss from $L_2$ and $L_3$ causes relatively high mass loss from the binary through two spiral arms extended outward. An example of such situations is shown in the second row in Figure 1. Higher sound speed cases (e.g., $c_s\sim 0.40$ shown in Figure 3) lead to a nearly spherical outflow with sustained enhancement of the stream through $L_1$.

Dense shells in front of the accretor are formed as depicted in Figure 3 for the models $c_{s}0.4$. These shock streams leading the accretor are formed due to the ram pressure produced as the expanding gas from the donor collides with the outer edge of the existing accretion disc with a supersonic relative speed. These shocks extend to trail the accretor. Fast winds with $c_s\sim v_{orb}$ create a stream of bound gas particles following the accretor without forming any resolvable accretion disc. This stream is the wake of the accretion as shown in the bottom panel of Figure 1. These are the situations where the BHL estimates apply. The videos of the simulations are available at https://www.youtube.com/channel/UCJCHVwNqzNOBjM1ohqXoZiw (accessed on 7 August 2025).

4.1. Mass Accretion Fractions

We present the results involving numerical calculations of mass accretion efficiency for mass ratios $q=0.25-4.00$ (except for models having low $c_s$ and high q. Such models are computationally costly, in a way that some of these models would need months to run using the available computer facility). The mass accretion fraction for mass ratios greater than $4.0$ would be extrapolated from our results, and therefore, running more simulations would be unnecessary in this regard. The temperature of the gas plays an important role in determining the energy associated with the gas particles, and hence its speed at the position of the accretor. The sound speed in the gas is an indicator of the temperature of the gas; thus, a low sound speed in any of the simulated models represents a cool (slow) wind, whereas hot and fast winds are characterised by a high sound speed. In order to investigate the fraction of mass accretion under different wind conditions, we investigated cases involving sound speeds $c_s=0.15-1.00$. The mass accretion fraction is calculated after steady state is reached using Equation (11) in this work.

The relation of the mass accretion fraction to the speed of sound and the mass ratio is shown in Figure 5. In general, one can see that for a certain value of the mass ratio q, the mass accretion fraction $f_{Nacc}$ is high when $c_s$ is low. Slow winds with a sound speed as low as $c_s=0.15$ yield $f_{Nacc}=90\%$. The intermediate values of the sound speed correspond to $f_{Nacc}$ of tenths. Likewise, the efficiency of mass accretion is reduced as the sound speed increases so that $f_{Nacc}$ is a few thousandths when $c_s\sim 1.00$.

Figure 5 also shows the exotic variation of $f_{Nacc}$ with mass ratio. Keeping $c_s$ constant, one can notice that, predominantly, $f_{Nacc}$ has a maximum at around $q=1$; that is, the accretion fraction increases with increasing q, peaks between $q=0.50-1.00$, then starts to decrease with further increase in mass ratio. What is interesting about this is that $f_{BHL}$ estimate shows a similar behaviour of mass accretion efficiency variation with mass ratio (see Figure 6).

4.2. Accretion Disc Structure

The shape and size of accretion discs determine the occurrence of astrophysical events like dwarf novae [14,15,16]. Along with other physical conditions, disc size is one of the main constraints for any potential formation of second- and third-generation planets [17,18,19]. Here, we demonstrate the shape and size of accretion discs characterised by the value of the wind sound speed and binary mass ratio. Our results show that the accretion discs formed are not perfectly circular, especially for the relatively large discs, which are rather manifested in semi-circular to eccentric shapes. After stability is reached and disc size takes its final, unchanged shape and size (this is when accretion fraction reaches its steady state simultaneously), these discs are formed around the accreting star in the orbital plane. Our results show that the formation of the discs is limited to the slow and moderate speed winds. Simulations of fast winds associated with sound speed values $c_s\sim v_{orb}$ do not show any evidence of resolvable accretion discs. To estimate disc size, we investigate the radial profile of the gas density at different azimuthal angles around the accretor in the orbital plane; this is shown in Figure 7. As mentioned above, the radial density distribution is not the same in every azimuth. Thus, we find the minimum and maximum radial extent from the accretor at which the density drops to the background level, then we assume that these represent upper and lower values of the truncation radii of the accretion discs. For each calculation of the disc radius, we make sure that the tidal tail that forms one of the spiral arms is not included in the radial density estimate. We introduce the average accretion disc radius as $R_D=(R_{min}+R_{max})/2$. To study the relation of accretion disc size to the sound speed, we keep the binary mass ratio constant at $q=1.00$ and vary the sound speed. Through $c_s=(0.15–0.60)$, simulations with higher sound speeds do not produce clear accretion discs but rather BHL accretion lines trailing the accretor.

Figure 8a shows that the accretion disc can extend to half the binary separation for the models with slow winds; for the model of $c_s=0.15$, the average accretion disc radius can be as high as $R_D=0.6a$. When the wind speed increases through transonic values, the average disc radius shows a monotonic decrease. In the simulation with $c_s=0.60$, in which the wind has already become supersonic, the smallest resolvable disc is calculated to be $R_D=0.08a$. Figure 8b illustrates the effect of binary mass ratio on the average disc radius while sound speed is kept constant ($c_s=0.40$). The outflow in these simulations displays a spherically symmetric wind with a mass stream in the line connecting the two components through $L_1$. It is clearly seen that average disc size has an inverse relationship to the mass ratio q. For the model with $q=0.25$, that is, the accretor is way more massive, the accretion disc extends beyond halfway the binary separation a; whereas the disc formed in the model with $q=4.00$ has an average radius of $R_D<0.2a$. The error bars in both Figure 8a,b represent the azimuthal extremes of the disc size; the upper limit of the error bar represents $R_{max}$ and the lower limit corresponds to $R_{min}$.

5. Discussion

In this work, the mass accretion fraction and accretion disc morphology are studied by simulation of mass transfer in binary systems. We consider the isothermal $\gamma =1.0$ Parker wind model to describe the outflow of gas from the mass-losing component. This wind solution is computationally simple and inexpensive. It can be a good approximation to the cool winds, including those of low-mass main sequence (MS) and evolved post-MS stars.

With this model, each SPH particle follows a transonic profile as it starts to expand from the star’s surface with a subsonic initial speed. The initial speed is determined by the analytic solution of Parker wind for this injection radius. The gas particles pass through a sonic (critical) point during their expansion and continue to become supersonic thereafter. Although it is a numerical simulation, our radial wind profile matches exactly with the analytical model (see Figure 2). We have not implemented any additional “inverse-square” parameters to reduce the gravitational effect of the mass-losing star. We have not set the particles to inject with an initial escape speed from the star’s surface, as can be seen in previous studies. In addition to the previous studies of mass accretion efficiency for systems having $q\ge 1.0$, we investigated an extensive range of binary mass ratios covering ($q<1.00$ and $q\ge 1.00$) along with a wide range of wind profiles from slow to fast winds characterised by the value of sound speed. Mass transfer occurs in binaries by means of either stellar wind or RLOF. Stars may undergo RLOF for a certain time of their lives and switch to stellar wind mass transfer based on the physical processes that occur inside the stars [6]. Furthermore, slow winds that cannot achieve escape velocity before reaching the Roche sphere of the star manage gravitational focusing through $L_1$ to go through WRLOF. If the physical condition of the star or the wind changes, then the star may switch to the wind mass transfer regime or the other way around [36]. It can be seen from our model simulations that fast winds result in spherically symmetric expansion of gas particles while slow winds initiate the WRLOF. This result aligns with those in Abate et al. [41], Saladino and pols [33], de Val-Borro et al. [35], and Mohamed & Podsiadlowski [40].

Our results suggest that maximum possible accretion occurs at binaries with components having comparable or similar masses. This result may be attributed to the mutual gravitational effect of both stellar components on the gas particles. A small donor mass compared to the accretor would cause the gas particles to accelerate strongly away from the donor and become too fast at the position of the accretor to be captured. On the other hand, a massive donor compared to the accretor would give up gas particles energetic enough to escape its gravity; such particles, again, are too energetic to become bound by a weak gravitational force from a relatively less massive accreting companion. These two effects may be compromised in situations where both stars have about the same mass to offer the highest possible accretion fraction. We may refer to Equation (9) (with M being replaced by $M_a$ and $M_d$ at a time), in which the gravitational and kinetic energy terms determine which particle is able to escape a massive donor and what particle becomes captured by the accretor. This study confirms and extends the results of Saladino et al. [34] and Liu et al. [2], in that the value of $f_{Nacc}$ increases as the mass of the accretor increases in the range of mass ratio they studied ($q\ge 1.0$). Numerical mass accretion fraction is further studied in this work; the models of slow winds yield higher $f_{Nacc}$ than those models with high wind speeds. It is obvious that speed determines the specific kinetic energy of the gas particle (Equation (9)). Slow winds means relatively less energetic particles that easily become gravitationally captured and accrete. Conversely, fast winds are energetic to escape the gravitational potential of the accretor and therefore hardly accrete. Our findings regarding $f_{Nacc}$-$v_w$ relation are consistent with the research results of Theuns et al. [22], Nagae et al. [23], and Saladino et al. [34]. Similar to Lee et al. [37], our calculations of $f_{Nacc}$ converge to the $f_{BHL}$ when $v_w>v_{orb}$.

Accretion discs are formed in most of the models as a result of mass transfer in the binary system. The study of the shape and size of the accretion discs is vital for many astrophysical subjects, and in particular, the possibility of planet formation in such accretion discs [51,52].

Here, we investigate the condition of the accretion discs in our models. It appears form our analysis that the disc size increases by decreasing either mass ratio q, sound speed $c_s$, or both. This is consistent with the findings of Saladino et al. [34] and Lee et al. [37]. Different values of q correspond to different Roche spheres around the binary components. The size of each of the components’ spheres represents the extent of its gravitational dominance in the binary space. Thus, a larger Roche radius of the accretor would make a larger room for the accretion disc to form. Thinking the other way, the massive donor component produces greater tidal torque, which truncates the accretion disc around the companion at smaller radii.

It is clear that the conservation of angular momentum requires the presence of accretion discs around the accretors. It is shown in Table 1 that low sound speeds generate low wind speeds which correspond to large accretion radii (Section 2). In such cases, the gas particles acquire large specific angular momenta. Conservation of angular momentum demands large circularisation radii for the accreted material in such cases. Conversely, high-sound speed models are associated with high-speed gas particles at the position of the accretor with small specific angular momenta and accretion radii, consequently small circularisation radii. Even higher values of wind speed result in negligible disc size or direct accretion. Our results in Figure 8a show that the radius of the accretion discs diminish as the value of sound speed rises. Simulations with high $c_s$ show no resolvable discs, which means that the gas particles undergo direct accretion (these cases are characterised by very low accretion fractions).

The distortion of the accretion disc is also produced by the tidal effect of the donor. The right panel of Figure 9 illustrates that in the cases where $q<1.00$, the disc sizes are even larger than the Roche radius of the accretor; when $q=1.00$, the accretion disc is about the same size as the Roche sphere. The radius of the accretion disc becomes smaller compared to the Roche radius when $q>1.00$. The left panel of Figure 9 indicates a monotonic relationship between the average size of the accretion disc and the accretion radius. The accretion disc always forms within the accretion radius with ${\displaystyle \frac{R_D}{R_{acc}}}\sim 0.5$.

6. Conclusions

In light of our numerical results of mass transfer in binary stars, we conclude the following: The simple thermally driven wind can be a good approximation model to interpret the gas outflow from stellar objects, in which the gas temperature is a key parameter to determine whether the pressure gradient overcomes the gravitation of the star.

Cool gases have a small pressure force; therefore, they result in slow winds, which then lead to mass transfer via WRLOF in a binary system. Hot gases, on the other hand, correspond to fast winds, which result in spherically symmetric gas expansion from the mass-losing star. Gas with moderate temperature leads to a complicated mass transfer regime that resembles a transitional mode from WRLOF to spherical wind.

Our results suggest that the fraction of mass transfer is strongly dependent on the binary mass ratio and the sound speed. The mass accretion fraction decreases as the sound speed of the wind gas increases and vice versa. However, the mass accretion fraction has a peculiar relationship with the binary mass ratio; accretion rate is relatively small for both low and high binary mass ratios, while it is the highest when the mass ratio is about unity.

Likewise, the size of the accretion disc depends on the values of both the binary mass ratio and sound speed. The radius of the accretion disc is large with a low sound speed. When the sound speed is much smaller than the binary orbital velocity, the disc radius can be larger than the Roche radius of the accretor. Smaller-sized discs are produced when the sound speed increases until the sound speed becomes comparable to the orbital speed of the binary; in such situations, accretion discs with finite sizes are not formed. There is also a clear connection between disc size and mass ratio, as disc radius is inversely related to binary mass ratio. Limitations in our models mainly include radiative effects (e.g., radiative braking and dust-driven winds) and winds from massive, hot stars. Such radiative driven winds are highlighted in Marchant & Bodensteiner [6] as critical for hot winds from spectral OB or Wolf–Rayet stars.