ABSTRACT

A huge optical luminosity of the supercritical accretion disc and powerful stellar wind in the high-mass X-ray binary SS433 make it difficult to reliably estimate the mass ratio of the binary components from spectroscopic observations. We analyse different indirect methods of the mass ratio estimate. We show that with an account of the possible Roche lobe overflow by the optical star, the analysis of X-ray eclipses in the standard and hard X-ray bands suggests the estimate q = Mx/Mv ≳ 0.3. We argue that the double-peak hydrogen Brackett lines in SS433 should form not in the accretion disc but in a circumbinary envelope, suggesting a total mass of |$M_\mathrm{v}+M_\mathrm{x}\gtrsim 40 \, \mathrm{M}_\odot$|⁠. The observed long-term stability of the orbital period in SS433 |$|\dot{P}_b/P_b|\le 1.793\times 10^{-14}$| s−1 over ∼28 yr period is used to place an independent constraint of q ≳ 0.6 in SS433, confirming its being a Galactic microquasar hosting a superaccreting black hole.

1 INTRODUCTION

The unique Galactic microquasar SS433 is a high-mass eclipsing X-ray binary with an orbital period of Pb ≃ 13|${^{\rm d}_{.}}$|08 at an advanced evolutionary stage with a precessing (Pprec ≃ 162|${^{\rm d}_{.}}$|3) supercritical optically bright accretion disc and relativistic jets (⁠|$v$|j/c ≃ 0.26) (Margon et al. 1979; Cherepashchuk 1981; Margon 1984; Cherepashchuk 1989). Over 40 yr of studies in the optical, X-ray, and radio, many features of SS433 have been studied in detail (see the review Fabrika 2004 and references therein). However, the masses of the binary components remain to be a highly debatable issue. Meanwhile, the component masses in SS433 are important to understand the evolutionary stage of this peculiar X-ray binary. The point is that SS433 provides a unique example of a high-mass X-ray binary at the stage of the secondary mass exchange when the more massive optical component overfills its Roche lobe avoiding the formation of a common envelope, and the system remains to be a semidetached binary (the formation of a common envelope at this stage is predicted by the modern theory of massive binary evolution, see e.g. Massevich & Tututkov 1988). At this stage, the removal of the mass and angular momentum from the system is mediated by the supercritical accretion disc around the relativistic component with the formation of powerful wind outflow from the disc (⁠|$v$||$\mathrm{ w}$| ≃ 2000 km s−1) and relativistic jets.

This peculiar feature of SS433 has recently been addressed by van den Heuvel, Portegies Zwart & de Mink (2017). These authors noted that when the donor star in a high-mass X-ray binary has a radiative envelope and starts filling the Roche lobe, and the mass ratio of the binary is q = Mx/Mv ≳ 0.29 (here Mx and Mv is the mass of the relativistic accretor and visual donor star, respectively), the formation of the common envelope can be avoided. The system evolves as a semidetached binary with rapid but stable Roche lobe overflow when the mass transfer through the inner Lagrangian point L1 is expelled from the vicinity of the relativistic star (the so-called isotropic re-emission mode, or 'SS433-like mode' of mass-loss from the system). If the mass ratio q ≲ 0.29, the high-mass X-ray binary at the secondary mass exchange stage inevitably passes through the common envelope stage, and in this case, depending on the initial angular momentum either a close WR+BH (Cyg X-3- like) binary forms or a Thorne–Zytkow object is produced (Thorne & Zytkow 1977). The authors (van den Heuvel et al. 2017) stress that in the case of a neutron star, the mass ratio q is much smaller than 0.29, and the binary system always evolves through the common envelope stage with the most likely formation of a Thorne–Zytkow object, unless the orbital period is longer than about 100 d.

Due to the huge optical luminosity of the supercritical accretion disc and powerful stellar wind in SS433, absorption lines in the optical companion spectrum are difficult to recognize, and the emission lines related to the precessing accretion disc are blended with emission lines from the powerful radial stellar wind. This makes it impossible to reliably estimate the dynamical mass ratio of the components in SS433.

Of the recent spectroscopic mass estimates in SS433, we note the paper by Hillwig & Gies (2008; |$m_x=4.3\pm 0.8 \, \mathrm{M}_\odot$|⁠, |$m_o=12.3\pm 3.3. \, \mathrm{M}_\odot$|⁠) and by Kubota et al. (2010; |$m_x=2.5^{+0.7}_{-0.6} \, \mathrm{M}_\odot$|⁠, |$m_o=10.4^{+2.3}_{-1.9} \, \mathrm{M}_\odot$|⁠), with the last study not excluding a neutron star in SS433. With all accuracy of spectroscopic observations carried out in these papers, the reliability of the mass estimates in SS433 can be put in doubt. For example, Bowler (2010, 2011a,b) showed that the absorption lines in the SS433 spectra that were used to estimate the component masses can also be formed not in the optical star photosphere but in a circumbinary shell whose inner region rotates with a velocity of ∼250 km s−1. This shell can also be found in the radio, where an image of the equatorial outflow perpendicular to relativistic jets was found (Blundell et al. 2001). Two-peaked emission hydrogen lines found by infrared (IR; Filippenko et al. 1988) and optical (Perez & Blundell 2010) spectroscopic observations also suggest the presence of a circumbinary gas shell.

In this paper, we analyse different possibilities to determine the component mass ratio in SS433 and discuss the reliability of the estimates of the mass ratio and total mass. In particular, we derive a new constraint on the component mass ratio from observational limits on the orbital period change with taking into account the possible mass-loss through the outer Lagrangian point L2.

2 THE MASS RATIO ESTIMATE FROM THE ANALYSIS OF X-RAY ECLIPSES

In the standard X-ray 2–10keV range, the width of the primary X-ray eclipse of SS433 is rather high, ∼2|${^{\rm d}_{.}}$|4 (Kawai et al. 1989; Brinkmann, Kawai & Matsuoka 1989; Kotani et al. 1996), with an orbital period of ∼13|${^{\rm d}_{.}}$|08. In this case, the model of the eclipse of a thin relativistic jet by the optical star filling its Roche lobe yields a small mass ratio q ≃ 0.15 (Kawai et al. 1989; Brinkmann et al. 1989; Kotani et al. 1996). This is significantly smaller than the critical mass ratio q = 0.29 (van den Heuvel et al. 2017), and in this case, it is not clear why SS433 does not enter the common envelope stage and remains a semidetached binary.

Recently, studies on the stability of mass transfer in massive close binaries appeared (Pavlovskii & Ivanova 2015; Pavlovskii et al. 2017). It is shown that nozzle-like mass transfer through the inner Lagrangian point L1 is restricted by gas-dynamic and thermodynamic effects, and the optical star can steadily overfill its Roche lobe and even expel the matter through the outer Lagrangian point L2. Thus, in the analysis of the 2–10keV X-ray eclipses, it should be taken into account that the optical star radius can exceed the mean Roche lobe radius. This suggests that the mass ratio estimate q = 0.15 obtained in Kawai et al. (1989), Brinkmann et al. (1989) and Kotani et al. (1996) is only a lower limit. Due to a weak dependence of the mean Roche lobe radius on the mass ratio (RRL/a ∼ 0.38q0.208, where a is the binary orbital separation), a ∼15 per cent increase in the optical star radius increases the mass ratio estimate by a factor of two. Thus, a ∼ 10−20 per cent Roche lobe overflow by the optical star enables the X-ray eclipse width in SS433 to be described for q ≃ 0.3.

The analysis of hard X-ray eclipses in SS433 (20–60keV) observed by INTEGRAL with an account for the precession flux variability suggests the component mass ratio q ≳ 0.3 (Cherepashchuk et al. 2013). In this analysis, the optical star was assumed to fill its Roche lobe and to eclipse an extended quasi-isothermal hot corona above the accretion disc. In addition, precession variability of the hard X-ray flux from SS433 limiting the corona height above the accretion disc was taken into account. Clearly, in this model, an account for a significant overfilling of the Roche lobe by the optical star will enhance the inequality q > 0.3.

Thus, the possibility of the Roche lobe overflow by the optical star in the analysis of standard and hard X-ray eclipses allows us to estimate the component mass ratio in SS433 q > 0.3, which exceeds the critical mass ratio q = 0.29 for the stable evolution of the binary system without the common envelope formation (van den Heuvel et al. 2017).

3 THE ESTIMATE OF THE TOTAL MASS OF SS433 FROM THE ANALYSIS OF CIRCUMBINARY ENVELOPE

The analysis of the stationary H α emission in the SS433 spectrum carried out in Blundell et al. (2001), Bowler (2010, 2011a,b), and Perez & Blundell (2010) by the line decomposition in the Gaussian components suggests the presence of a circumbinary shell rotating with a velocity of ∼250 km s−1 at the inner radius. These authors concluded that the total mass of the binary system in SS433 exceeds |${\sim } 40 \, \mathrm{M}_\odot$|⁠, and the relativistic object is undoubtedly a high-mass black hole. The spectroscopic appearances of such a shell are also consistent with radio observations that reveal traces of the equatorial gas outflow from SS433 in the plane perpendicular to the relativistic jets (Blundell et al. 2001).

The double-peak stationary hydrogen lines were also detected in the IR spectroscopic observations by Filippenko et al. (1988; Paschen series) and Robinson et al. (2017; Brackett series). Filippenko et al. (1988) interpret the double-peak hydrogen lines by two models: the model of a circumbinary shell (suggesting a high total mass of SS433), and the model in which these lines are produced in the accretion disc (and then the relativistic object can be a neutron star with a mass of |${\lesssim } 1.4 \, \mathrm{M}_\odot$|⁠).

On the other hand, Robinson et al. (2017) argue that the double-peak stationary hydrogen emissions are formed in the accretion disc around a 1.4 |$\, \mathrm{M}_\odot$| neutron star. Their arguments are based on high-resolution IR spectroscopic observations revealing broad line wings that can be due to, as these authors believe, the widening by a rapid rotation in the inner parts of the accretion disc. Robinson et al. (2017) carried out a numerical simulation of the emission line profiles from the accretion disc around a neutron star and obtained good agreement of theoretical profiles (double-peak profile with broad wings) with the IR spectroscopic observations of the Brackett hydrogen lines in SS433.

However, Robinson et al. (2017) have ignored an important observational fact: in the middle of X-ray eclipses in SS433, the centre of the accretion disc should be screened by the optical star. As the orbital inclination in SS433 is reliably known from the analysis of moving emission lines (i ≃ 79°), in the case of a circular orbit at the moment of the eclipse the distance between the accretion disc centre and the optical star centre is Δmin = cos i (in units of the orbital separation a). For i = 79°, we have Δmin ≈ 0.191, which is definitely smaller than the radius of the optical star that fills or even overfills its Roche lobe. Therefore, in the case of a circular orbit, in SS433 a significant part of the central part of the accretion disc around the relativistic object in the middle of the X-ray eclipse should be screened by the optical star. Therefore, the broad wings of the emission lines, if formed in the rapidly rotating central parts of the accretion disc, should disappear at the middle of the X-ray eclipse, which is not observed (Robinson et al. 2017). Additionally, the broad wings of the lines from the disc would strongly vary with the phase of the 162.3-d precession period.

In the case of an elliptical orbit, Δmin can be larger than cos i at the middle of the X-ray eclipse, and the central parts of the accretion disc can remain non-eclipsed. However, the latter possibility is untenable because X-ray observations show a monotonic increase of the eclipse depth with energy, and in the hard X-ray band (∼60 keV) the X-ray eclipse depth exceeds 80 per cent (Cherepashchuk et al. 2005, 2013). This suggests that irrespective of the form of the orbit, the optical star edge eclipses the central, hottest parts of the accretion disc in the middle of the X-ray eclipse. Thus, there is an additional argument suggesting that in the model of the rotational broadening of stationary hydrogen emission lines, the broad line wings should disappear in the middle of the eclipse. As this is not observed, there are grounds to reject the model of the double-peak line formation in the outer parts of the accretion disc.

A much more preferable seems to be the model that the Brackett hydrogen emission lines observed by Robinson et al. (2017) arise in a circumbinary shell and have composite profiles: the central double-peak component is related to the circumbinary shell rotation, and wide wings of these lines are produced in the radial stellar wind outflow from the supercritical accretion disc with a velocity of ∼2000 km s−1. As the stellar wind forms an extended region, the line formation region in the wind is not fully eclipsed by the optical star, and the wide wings of the Brackett hydrogen lines should not disappear during the eclipses. In this case, the double-peaked structure of the Brackett lines observed by Robinson et al. (2017) is formed in an extended rotating circumbinary shell, and the estimates made by Bowler (2010, 2011a), Blundell et al. (2001), Filippenko et al. (1988), and Perez & Blundell (2010) that the total mass of SS433 exceeds |${\sim } 40 \, \mathrm{M}_\odot$|obtain additional support.

Next, we show that the analysis of the long-term stability of the orbital binary period in SS433 independently confirms this conclusion.

4 THE MASS RATIO CONSTRAINTS FROM THE STABILITY OF THE ORBITAL PERIOD OF SS433

SS433 demonstrates a surprisingly stable orbital period: during almost 40 yr of observations, its period remains constant. According to the recent analysis of SS433 observations over a period of ∼28 yr carried out by V.P. Goranskij (Goranskij 2011), the ephemeris of the primary eclipse minima in SS433 reads  
\begin{eqnarray*} \mathrm{Min\, I} &=& \mathrm{JD}_{\mathrm{ hel}}\, 2450023.746\pm 0{^{\rm d}_{.}}030\nonumber\\ &&+\,(13{^{\rm d}_{.}}08223\pm 0{^{\rm d}_{.}}00007)\cdot E\, . \end{eqnarray*}
(1)
By assuming a 3σ rms error of the orbital period 0|${^{\rm d}_{.}}$|00021 to be an upper limit on the possible orbital period change over ∼28 yr, it is possible to constrain the mass ratio q. To do this, we will use a model of the mass-loss from SS433 due to isotropic re-emission by taking into account the evidence for the presence of a circumbinary envelope in SS433 which is formed because of the mass-loss through the outer Lagrangian point L2.
The model can be specified as follows. Consider a binary system with the component masses Mv and Mx = qMv, the mass ratio q ≤ 1, and the relative orbital separation a. The total mass of the system is M = Mv + Mx = Mv(1 + q). For simplicity, we will consider a circular orbit, which is justified by the high mass transfer rate in the binary. Assume a stable mass-loss rate from the optical star |$\dot{M}_\mathrm{v}\lt 0$|⁠, and the mass growth rate of the accretor is |$\dot{M}_\mathrm{x}=(\eta -1) \dot{M}_\mathrm{v}\ge 0$|⁠, 0 ≤ η ≤ 1. In this case, the conservation of angular momentum implies (see e.g. Soberman, Phinney & van den Heuvel 1997; Postnov & Yungelson 2006)  
\begin{eqnarray*} \frac{\dot{a}}{a}=-2\left(1+(\eta -1)\frac{M_\mathrm{v}}{M_\mathrm{x}}-\frac{\eta }{2}\frac{M_\mathrm{v}}{M}\right)\frac{\dot{M}_\mathrm{v}}{M_\mathrm{v}}+2\frac{\mathrm{ d}J/\mathrm{ d}t}{J}\, , \end{eqnarray*}
(2)
where J is the orbital angular momentum of the binary. Conservative mass transfer corresponds to η = 0 and dJ = 0, and in the non-conservative case, (dJ/dt)/J specifies the orbital angular momentum loss by the mass escaping the binary.
In the isotropic re-emission mode, the mass is transferred to the accretor Mx and then spherically symmetrically expelled carrying out a specific angular momentum of the accretor. For SS433, η ≈ 1 is a good approximation (most of the accreted matter is expelled by the supercritical accretion disc). In this mode, the angular momentum loss term reads  
\begin{eqnarray*} \left.\frac{\mathrm{ d}J/\mathrm{ d}t}{J}\right|_\mathrm{iso}=\beta \frac{\dot{M}_\mathrm{v}}{M} \frac{M_\mathrm{v}}{M_\mathrm{x}}. \end{eqnarray*}
(3)
Here, 0 ≤ β ≤ 1 characterizes the fraction of mass-loss in the isotropic re-emission mode.
We will also assume that a fraction (1 − β) of the mass-loss occurs through the outer Lagrangian point L2 behind the less-massive component (accretor) forming a circumbinary ring. In this case (see e.g. Soberman et al. 1997), the angular momentum from the system is lost through the L2 point and  
\begin{eqnarray*} \left.\frac{\mathrm{ d}J/\mathrm{ d}t}{J}\right|_\mathrm{cbr}=(1-\beta) \frac{\dot{M}_\mathrm{v}}{M} \frac{M^2}{M_\mathrm{x}M_\mathrm{v}}\sqrt{\frac{a_\mathrm{cbr}}{a}}, \end{eqnarray*}
(4)
where acbr is the circumbinary Keplerian ring radius. In the first approximation, we can assume |$a_\mathrm{cbr}=r_{L_2}$|⁠, where the distance from system’s barycentre to L2 point can be written as a power series in the modified mass ratio Mx/M = q/(1 + q) (see e.g. Deprit 1965; Emelyanov & Salyamov 1983):  
\begin{eqnarray*} \frac{r_{L_2}}{a} &\approx & \frac{1}{1+q}+\left(\frac{q}{3(1+q)}\right)^{1/3}+\frac{1}{3}\left(\frac{q}{3(1+q)}\right)^{2/3}\nonumber \\ &&-\,\frac{1}{9}\left(\frac{q}{3(1+q)}\right)+\frac{50}{81}\left(\frac{q}{3(1+q)}\right)^{4/3} \end{eqnarray*}
(5)
In the mass ratio range 0 ≤ q ≤ 1, this distance falls within the range from 1 to ≃ 1.27 (Fig.1).
Figure 1.

Distance to the L2 point from the binary system’s barycentre |$r_{L_2}$| in units of the relative orbital separation a for mass ratio 0 ≤ q ≤ 1 (the solid upper curve, equation 5). The dashed curve shows the value |$\sqrt{r_{L_2}/a}$| determining the angular momentum loss through L2.

Figure 1.

Distance to the L2 point from the binary system’s barycentre |$r_{L_2}$| in units of the relative orbital separation a for mass ratio 0 ≤ q ≤ 1 (the solid upper curve, equation 5). The dashed curve shows the value |$\sqrt{r_{L_2}/a}$| determining the angular momentum loss through L2.

Summarizing, in the case η = 1 (no accretor mass growth), we find  
\begin{eqnarray*} \frac{\dot{a}}{a}=-2\left(1-\frac{1}{2}\frac{M_\mathrm{v}}{M}\right)\frac{\dot{M}_\mathrm{v}}{M_\mathrm{v}}+2\beta \frac{M_\mathrm{v}}{M_\mathrm{x}}\frac{\dot{M}_\mathrm{v}}{M}+2(1{-}\beta)x\frac{M^2}{M_\mathrm{x}M_\mathrm{v}}\frac{\dot{M}_\mathrm{v}}{M}, \nonumber\\ \end{eqnarray*}
(6)
where |$x\equiv \sqrt{r_{L_2}/a}$|⁠. Differentiating the third Kepler’s law, |$2\dot{P}_b/P_b=3(\dot{a}/a)-\dot{M}/M$|⁠, and noticing that |$\dot{M}/M=\dot{M}_\mathrm{v}/M$|(η = 1), we can recast this equation to the form  
\begin{eqnarray*} \frac{\dot{P}_b}{P_b}=-\frac{\dot{M}_\mathrm{v}}{M_\mathrm{v}}\frac{3(1+2q)q-6\beta -6x(1-\beta)(1+q)^2+q}{2q(1+q)}\, . \end{eqnarray*}
(7)
Therefore, the observational constraint |$|\dot{P}_b/P_b|\lt \epsilon$| can be written in the form of two inequalities:  
\begin{eqnarray*} A_-(x,\alpha ,\beta)q^2+B_-(x,\alpha ,\beta)q+C(x,\alpha)\le 0, \end{eqnarray*}
(8)
 
\begin{eqnarray*} A_+(x,\alpha ,\beta)q^2+B_+(x,\alpha ,\beta)q+C(x,\alpha)\ge 0, \end{eqnarray*}
(9)
with coefficients  
\begin{eqnarray*} &A_-(x,\alpha ,\beta)=1-\frac{1}{3}\alpha -x(1-\beta)\, , \end{eqnarray*}
(10)
 
\begin{eqnarray*} &B_-(x,\alpha ,\beta)=\frac{2}{3}(1-\frac{1}{2}\alpha -3x(1-\beta))\, , \end{eqnarray*}
(11)
 
\begin{eqnarray*} &A_+(x,\alpha ,\beta)=1+\frac{1}{3}\alpha -x(1-\beta)\, , \end{eqnarray*}
(12)
 
\begin{eqnarray*} &B_+(x,\alpha ,\beta)=\frac{2}{3}(1+\frac{1}{2}\alpha -3x(1-\beta))\, , \end{eqnarray*}
(13)
 
\begin{eqnarray*} &C(x,\alpha)=-\beta -x(1-\beta)\, . \end{eqnarray*}
(14)
Here, |$\alpha = \epsilon \left|\frac{M_\mathrm{v}}{\dot{M}_\mathrm{v}}\right|$| is the dimensionless coefficient that can be estimated from observations. Taking the uncertainty in the period ephemeris σ = 0|${^{\rm d}_{.}}$|00007 obtained over ∼28 yr of observations (Goranskij 2011) and assuming the optical star mass |$M_\mathrm{v}=15 \, \mathrm{M}_\odot$| with the mass-loss rate |$\dot{M}_\mathrm{v}=10^{-4} \, \mathrm{M}_\odot$| yr−1, we get the estimate  
\begin{eqnarray*} \alpha =\frac{3\sigma }{\Delta t}\frac{M_\mathrm{v}}{\dot{M}_\mathrm{v}}= \frac{0{^{\rm d}_{.}}00021}{13{^{\rm d}_{.}}082\times 28\mathrm{yrs}}\frac{15 \, \mathrm{M}_\odot }{10^{-4}\, \mathrm{M}_\odot \mathrm{yr}^{-1}}\approx 0.086. \end{eqnarray*}
(15)
Note here that in this estimate the precise value of the optical star mass is rather unimportant because the mass-loss rate is much more uncertain. The limitation on the mass ratio q can be readily found from inequalities (8), (9) and are shown in Fig. 2 for two values of the total mass-loss rate from the system: |$\dot{M}=10^{-4} \, \mathrm{M}_\odot$| yr−1 and |$10^{-5} \, \mathrm{M}_\odot$| yr−1 (left-hand and right-hand panels, respectively). At given β, the interval of q for each value of the parameter β corresponds to the uncertainty α in the orbital period change measurement (15). It is seen that in the limit β → 1 (i.e. in the absence of the mass-loss rate through the L2 point), there is a lower limit on the mass ratio, qmin ≈ 0.7 and qmin ≈ 0.6 for the two assumed mass-loss rates, respectively. In this limit, the inequalities (8) and (9) yield the allowed intervals of the mass ratio 0.7 ≲ q ≲ 0.74 and 0.59 ≲ q ≲ 0.94 for 10−4 and |$10^{-5} \, \mathrm{M}_\odot$| yr−1, respectively.
Figure 2.

Constraints on the mass ratio of SS433 for different values of parameters x = 1.0, 1.1, and α = 0.086 (left-hand panel) and 0.86 (right-hand panel) corresponding to the mass-loss rate |$\dot{M}_\mathrm{v}=10^{-4}$| and |$10^{-5} \, \mathrm{M}_\odot$| yr−1, respectively.

Figure 2.

Constraints on the mass ratio of SS433 for different values of parameters x = 1.0, 1.1, and α = 0.086 (left-hand panel) and 0.86 (right-hand panel) corresponding to the mass-loss rate |$\dot{M}_\mathrm{v}=10^{-4}$| and |$10^{-5} \, \mathrm{M}_\odot$| yr−1, respectively.

5 DISCUSSION AND CONCLUSIONS

In this paper, we have analysed different indirect estimates of the mass ratio and the total mass in SS433. Due to a huge optical luminosity of the supercritical accretion disc and powerful radiation-driven wind outflow, dynamical estimates of the mass ratio of the binary components cannot be fully reliable.

We note that if the possible overfilling of the Roche lobe by the optical star is taken into account, the analysis of the primary X-ray eclipse both in the standard (2–10keV) and hard (20–60keV) X-ray bands suggests a high mass ratio q = Mx/Mv ≳ 0.3.

We show that the stationary double-peak hydrogen Brackett emission lines observed by Robinson et al. (2017) are unlikely to be formed in the outer parts of the accretion disc around the compact star and should be produced in a circumbinary envelope. The existence of such an envelope in SS433 has been repeatedly put forward from radio, optical, and IR observations (Blundell et al. 2001; Bowler 2010, 2011a,b; Perez & Blundell 2010). In this model, the total mass of the components in SS433 |$M_\mathrm{v}+M_\mathrm{v}\gtrsim 40 \, \mathrm{M}_\odot$|⁠, given the radius of the stationary circumbinary disc being at ∼1.5a (a is the binary orbit separation), see e.g. table 1 in Bowler (201,0). The smaller radius of the disc would decrease the total mass estimate. However, in this case, a significant variability of the double-peak hydrogen emissions would be expected caused by non-stationary gas flow in the circumbinary shell immediately close to the L2 point, which, apparently, is not observed.

We assume that the emission hydrogen line profiles have double structure for two reasons (see Section 3): the central double-peak component is produced by the rotating circumbinary shell, and the wide wings of these lines are formed in a strong wind outflow from the bright supercritical accretion disc around the compact object.

We have also analysed the mass-loss from the system in the model of isotropic re-emission mode together with a mass-loss from the external Lagrangian point L2. In this model, the observed long-term stability of the binary orbital period of SS433 over ∼28 yr sets a lower limit on the mass ratio q ≳ 0.6.

The dynamical analysis given above assumed a particular model for mass-loss: an isotropic re-emission stellar wind from the supercritical accretion disc and possible mass-loss rate through the outer Lagrangian point L2 with the formation of a circumbinary ring: |$\dot{M}= \dot{M}_\mathrm{v}=\beta \dot{M}_\mathrm{v}|_\mathrm{iso} +(1-\beta)\dot{M}_\mathrm{v}|_\mathrm{cbr}$|⁠.

We have used the usual assumption that the specific angular momentum carried out by the disc wind is equal to the specific orbital angular momentum of the relativistic component. This is justified in so far as the supercritical disc spherization radius (Shakura & Sunyaev 1973) is well within the Roche lobe of Mx: |$R_\mathrm{sph}=3R_g(\dot{M}/\dot{M}_\mathrm{Edd})\simeq 10^{10}(\mathrm{cm})\dot{M}/(10^{-4} \, \mathrm{M}_\odot \mathrm{yr}^{-1})\ll R_L(M_\mathrm{x})$| (here, |$R_g\approx 3\times 10^5 (M_\mathrm{x}/\, \mathrm{M}_\odot)$| cm is the Schwarzschild radius of the relativistic star, |$\dot{M}_\mathrm{Edd}\approx 10^{-8}\, \mathrm{M}_\odot (M_\mathrm{x}/\, \mathrm{M}_\odot)$| yr−1 is the critical mass rate corresponding to the Eddington luminosity), which is justified for SS433.

Much more uncertain is the specific angular momentum loss from the L2 point. We parametrize it with the parameter |$x=\sqrt{a_\mathrm{cbr}/a}$|⁠, assuming a Keplerian circumbinary ring at distance acbr from the system’s barycentre. This parameter can be equivalently written as x = |$v$|x/|$v$|cbr(M/Mv) = |$v$|x/|$v$|cbr(1 + q), where |$v$|x/|$v$|cbr is the ratio of the orbital velocity of the relativistic star to the Keplerian velocity at the ring distance. Infrared observations of the Paschen and Brackett hydrogen emission line splitting ∼250 km s−1 (Robinson et al. 2017) can be interpreted in terms of the emission from the circumbinary ring, thus |$v$|cbr ∼ 250 km s−1. The spectroscopy of He ii emission lines (Hillwig et al. 2004) suggests |$v$|x = 168 ± 18 km s−1. Therefore, the expected value of x falls within the range ∼1.16−1.36 for mass ratios q lying in the interval 0.7−1. This estimate shows that the interpretation of the observations does not contradict to the presence of a circumbinary disc not far beyond the L2 point in SS433.

Thus, we conclude that the estimation of the component mass ratio in SS433 by independent methods yields q ≳ 0.3 ÷ 0.6, and the total mass of the system is most likely |${\gtrsim } 40 \, \mathrm{M}_\odot$|⁠, suggesting that a stellar-mass black hole powers this unique Galactic microquasar.

ACKNOWLEDGEMENTS

We thank the anonymous referee for careful reading of the paper and useful remarks. The authors thank Dr. N.V. Emelyanov for discussions. The work of AMCH and AAB is supported by the RSF grant 17-12-01241 (analysis of constraints from X-ray eclipses and circumbinary shell). The work of KAP (analysis of constraints from mass-loss modes) is supported by the RSF grant 16-12-10519.

REFERENCES

Blundell
K. M.
,
Mioduszewski
A. J.
,
Muxlow
T. W. B.
,
Podsiadlowski
P.
,
Rupen
M. P.
,
2001
,
ApJ
 ,
562
,
L79
Bowler
M. G.
,
2011a
,
A&A
 ,
531
,
A107
Bowler
M. G.
,
2011b
,
A&A
 ,
534
,
A112
Brinkmann
W.
,
Kawai
N.
,
Matsuoka
M.
,
1989
,
A&A
 ,
218
,
L13
Cherepashchuk
A. M.
,
1981
,
MNRAS
 ,
194
,
761
Cherepashchuk
A. M.
,
1989
,
Astrophys. Space Phys. Rev.
 ,
7
,
185
Cherepashchuk
A. M.
et al.  
,
2005
,
A&A
 ,
437
,
561
Cherepashchuk
A. M.
,
Sunyaev
R. A.
,
Molkov
S. V.
,
Antokhina
E. A.
,
Postnov
K. A.
,
Bogomazov
A. I.
,
2013
,
MNRAS
 ,
436
,
2004
Emelyanov
N. V.
,
Salyamov
V. N.
,
1983
,
Soviet Astron.
 ,
27
,
442
Fabrika
S.
,
2004
,
Astrophys. Space Phys. Rev.
 ,
12
,
1
Filippenko
A. V.
,
Romani
R. W.
,
Sargent
W. L. W.
,
Blandford
R. D.
,
1988
,
AJ
 ,
96
,
242
Goranskij
V.
,
2011
,
Perem. Zvezdy
 ,
31
:
Hillwig
T. C.
,
Gies
D. R.
,
2008
,
ApJ
 ,
676
,
L37
Hillwig
T. C.
,
Gies
D. R.
,
Huang
W.
,
McSwain
M. V.
,
Stark
M. A.
,
van der Meer
A.
,
Kaper
L.
,
2004
,
ApJ
 ,
615
,
422
Kawai
N.
,
Matsuoka
M.
,
Pan
H.-C.
,
Stewart
G. C.
,
1989
,
PASJ
 ,
41
,
491
Kotani
T.
,
Kawai
N.
,
Matsuoka
M.
,
Brinkmann
W.
,
1996
,
PASJ
 ,
48
,
619
Kubota
K.
,
Ueda
Y.
,
Fabrika
S.
,
Medvedev
A.
,
Barsukova
E. A.
,
Sholukhova
O.
,
Goranskij
V. P.
,
2010
,
ApJ
 ,
709
,
1374
Margon
B.
,
Ford
H. C.
,
Grandi
S. A.
,
Stone
R. P. S.
,
1979
,
ApJ
 ,
233
,
L63
Massevich
A. G.
,
Tututkov
A. V.
,
1988
,
Evolution of Stars: Theory and Observations
 .
Nauka
,
Moscow
, p.
280
Pavlovskii
K.
,
Ivanova
N.
,
2015
,
MNRAS
 ,
449
,
4415
Pavlovskii
K.
,
Ivanova
N.
,
Belczynski
K.
,
Van
K. X.
,
2017
,
MNRAS
 ,
465
,
2092
Perez
M. S.
,
Blundell
K. M.
,
2010
,
MNRAS
 ,
408
,
2
Postnov
K. A.
,
Yungelson
L. R.
,
2006
,
Living Rev. Relativ.
 ,
9
,
6
Robinson
E. L.
,
Froning
C. S.
,
Jaffe
D. T.
,
Kaplan
K. F.
,
Kim
H.
,
Mace
G. N.
,
Sokal
K. R.
,
Lee
J.-J.
,
2017
,
ApJ
 ,
841
,
79
Shakura
N. I.
,
Sunyaev
R. A.
,
1973
,
A&A
 ,
24
,
337
Soberman
G. E.
,
Phinney
E. S.
,
van den Heuvel
E. P. J.
,
1997
,
A&A
 ,
327
,
620
Thorne
K. S.
,
Zytkow
A. N.
,
1977
,
ApJ
 ,
212
,
832
van den Heuvel
E. P. J.
,
Portegies Zwart
S. F.
,
de Mink
S. E.
,
2017
,
MNRAS
 ,
471
,
4256
This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)