2. Cyclotron lines

Since the physical conditions are expected to vary over the emission region, the X-ray spectrum is expected to change with the viewing angle and therefore with pulse phase. This variation can be because during one rotation phase different parts of the surface are exposed and also due to change in local field structure due to accretion dynamics, e.g. change in accretion rate, as is seen for sources like Her X-1 and V0332+53. The difference in time scales of variation for cyclotron line energy $E_{\mathrm{cyc}}$ and luminosity will allow researchers to distinguish between these two distinct cases. (Nagase et al. Reference Nagase1991; Wilson et al. Reference Wilson, Finger and Camero-Arranz2008). In this work, we have selected 11 persistent sources with SG companions known to have at least one cyclotron line (see Table 1). Here, our analysis of all pointing observations provides an opportunity to infer the values of magnetic field strength according to their spectra. The gravitational redshift z that at the NS surface is approximately

(1) \begin{equation}z\simeq\frac{1}{\sqrt{1-\frac{2GM_{\mathrm{NS}}}{R_{\mathrm{NS}}c^{2}}}}-1\simeq0.3,\end{equation}

where $M_{\mathrm{NS}}, R_{\mathrm{NS}}$ , G, and c are the NS mass, radius, gravitational constant and speed of light, respectively. Assuming canonical values for $M_{\mathrm{NS}}$ and $R_{\mathrm{NS}}$ of 1.4 $\mathrm{M}_{\odot}$ and 10 km, thus z = 0.3. As such, the line energy changes the fundamental cyclotron line (keV) to become related to the magnetic field strength and its configuration (Trüemper et al. Reference Trüemper1978) by the equation

(2) \begin{equation}\begin{array}{rcl} E_{\mathrm{cyc}}=11.6 keV B_{12} (1+z)^{-1} .\end{array}\end{equation}

Here B $_{12}$ is the magnetic field strength in units of 1012 G, and the higher harmonics have an energy n times the fundamental energy $E_{\mathrm{cyc}}$ (see, e.g. Wilson et al. Reference Wilson, Finger and Camero-Arranz2008, and references therein). The effect of NS rotation on the variation in the shape of the cyclotron line features is considered. This could be correlated to the accretion geometry and other mechanisms (Bachetti et al. Reference Bachetti2014).

In general, the magnetic field of NSs spans a range from $10^{8}\mathrm{G}$ or less (LMXBs) to $10^{15} \mathrm{G}$ (magnetars). We list the computed magnetic strengths along with other system parameters for selected SG-HMXBs in Table 2. These values show no correlation with the spin period of wind-fed systems (see Figure 1).

Table 2.

List of derived parameters for SG HMXBs.

The data on mass and radius of donor are taken from Falanga et al. (Reference Falanga2015), Reig et al. (Reference Reig2016), Chaty et al. (Reference Chaty2008), Rawls et al. (Reference Rawls2011), Cusumano et al. (Reference Cusumano2010), Mason et al. (Reference Mason2012). The luminosity and effective temperature are computed by their approximated stellar evolution track given by Hurley et al. (Reference Hurley, Pols and Tout2000). From these donor data, the terminal velocity of the wind and the wind mass-loss rate from SG stars are derived by polynomial approximation given by Vink et al. (Reference Vink, de Koter and Lamers2001).

Figure 1.

The magnetic field as a function of the spin period of SG-HMXBs.

It should be noticed that, of course, we need to consider some observational biases, such as: too strong of a magnetic field prevents accretion onto the NS and we could not observe such systems as bright X-ray sources (Taani et al. Reference Taani2020; Karino Reference Karino2020). Besides such possibility of biases, this concentration around $B \approx 10^{12} \mathrm{G}$ will draw a lot of interest and promote further studies on the NS magnetic field (Taani et al. Reference Taani2019a, b). The fundamental energy covers a wide range, starting at 10 keV for Swift J1626.6-5156 (DeCesar, Pottschmidt, & Wilms Reference DeCesar, Pottschmidt and Wilms2009) to 100 keV for LMC X-4 (la Barbera et al. Reference la Barbera2001).

The strong energy variation of the cyclotron lines (for example, in V0332+53, GRO J1008-57 and GX301-2) can be used to argue that during different phases of the X-ray pulses, regions with different magnetic fields are observed.

It is noteworthy to mention here that 4U 0115+634 is one of the pulsars whose CRSFs have been studied in great detail (see, e.g., Wheaton et al. Reference Wheaton1979; Nagase et al. Reference Nagase1991; Nishimura Reference Nishimura2005). In previous outbursts, CRSFs have been detected up to the fifth harmonic (Heindl et al. Reference Heindl2003; Ferrigno et al. Reference Ferrigno2011). This high number of detected CRSFs in 4U 0115+634 makes this system an outstanding laboratory to study the physics of cyclotron lines in X-ray pulsars.

3. Investigating wind parameters in SG HMXB systems

Under the assumption that the spin period of a NS is nearly in torque equilibrium at a steady mass accretion rate, the magnetic radius (where the magnetic pressure balances with the accretion ram pressure) and corotation radius $r_{\mathrm{co}}$ (the radius where the Keplerian angular velocity equals the NS spin angular velocity) have similar values (Donati et al. Reference Donati2011), and enough amount of angular momentum could be received via wind accretion (Karino et al. Reference Karino, Nakamura and Taani2019), one estimates the magnetic field strength as

(3) \begin{equation}B_{\mathrm{NS}} = 2.2 \times 10^{12} \mathrm{G} \times \zeta^{1/2}\left( \frac{\dot{\textit{M}}}{10^{18} \mathrm{g\, s}^{-1}} \right)^{1/2}\left( \frac{\textit{P}_{\mathrm{s}}}{1 \mathrm{s}} \right)^{7/6} ,\end{equation}

assuming that the mass of the NS is 1.4 M $_\odot $ and its radius is 10 km (Ghosh & Lamb Reference Ghosh and Lamb1979; Campana et al. Reference Campana2002; Tsygankov et al. Reference Tsygankov, Mushtukov, Suleimanov and Poutanen2016). The parameter $\zeta$ is the ratio of accretion velocity to the free-fall velocity, and hereafter we fix this value as 0.5. However, we should note here that, the effect of the accretion flow geometry (spherical, disk-like or planar) must be taken into account (Karino et al. Reference Karino, Nakamura and Taani2019).

Figure 2.

Plots of the wind velocity vs mass-loss rate, in various accretion regimes. The position of each source is shown by a black filled circle. As shown clearly, the parameter space can be categorised into (A) supersonic inhibition regime, (B) subsonic inhibition regime, (C) supersonic propeller regime, (D) subsonic propeller regime, and (E) direct accretion regime indicated by the shaded region, and the solution of the wind equation is represented by solid curves.

The mass accretion rate $\dot{M}$ can be obtained as the following, if we assume the Hoyle–Lyttleton accretion scenario using the local density at the position of the NS (see, e.g. Bondi & Hoyle Reference Bondi and Hoyle1944 and references therein).

(4) \begin{equation}\dot{M} = \rho_{\mathrm{w}} v_{\mathrm{rel}} \pi R_{\mathrm{acc}}^{2} .\end{equation}

Under the condition of the spherical stellar wind (Walter, Lutovinov, & Bozzo Reference Walter, Lutovinov and Bozzo2015), the local density of the wind matter becomes

(5) \begin{equation}\rho_{\mathrm{w}} = \frac{ \dot{m}_{\mathrm{w}} }{ 4 \pi a^{2} v_{\mathrm{w}} },\end{equation}

where $\dot{m}_{\mathrm{w}}$ is the wind mass-loss rate from the donor. The estimation depends on the empirical stellar-evolution model. Here, orbital radius a can be obtained from the orbital period of the system and donor mass. Note that we assume here that, the systems are in circular orbits. Combining them, we have

(6) \begin{equation}\dot{M} = \dot{m}_{\mathrm{w}} \times\frac{ G^{2} M^{2}_{\mathrm{NS}} }{ a^{2} v_{\mathrm{w}} v^{3}_{\mathrm{rel}} } .\end{equation}

We can deduce the mass accretion rate from the observed X-ray luminosity. The relative velocity of the wind to the NS is

(7) \begin{equation}v_{\mathrm{rel}} = \left( v^{2}_{\mathrm{orb}} + v^{2}_{\mathrm{w}} \right)^{1/2}\end{equation}

and the velocity of the line-driven wind is usually prescribed in the model by so-called $\beta-$ low:

(8) \begin{equation}v_{\mathrm{w}} = v_{\infty}\left( 1 - \frac{R_{\mathrm{d}}}{a} \right)^{\beta} .\end{equation}

In this study, we assume $\beta$ to be in the range of around 0.6–1.5 based on models of the donor (Puls, Vink, & Najarro Reference Puls, Vink and Najarro2008), $v_{\infty}$ denotes the terminal velocity of the wind.

On the other hand, unless the orbital radius is very small, the ratio of mass accretion rate to mass-loss due to stellar wind, can be approximated as the ratio of the accretion region $\left(4 \pi G^{2} M^{2}_{\mathrm{NS}} / v^{2}_{\mathrm{rel}}\right)$ to the sphere whose radius is the orbital radius ( $4 \pi a^{2}$ ). This is equivalent to replacing the stellar wind velocity by the relative velocity in Equation (6). Then, with the help of this equation, we get

(9) \begin{equation}v^{2}_{\mathrm{w}} = -v^{2}_{\mathrm{orb}} \pm \sqrt{ \frac{ G^{2} M^{2}_{\mathrm{NS}} }{ a^{2} \dot{M} } \dot{m}_{\mathrm{w}} }.\end{equation}

The wind parameters such as $\dot{m}_{\mathrm{w}}$ and $v_{\infty}$ contain large uncertainties. The observational biases should be considered, as the opacity reduction and the unknowns in the geometry and emission mechanisms of the system (see, e.g., Mushtukov et al. Reference Mushtukov, Suleimanov, Tsygankov and Poutanen2015). Combining above wind equations with CRSF data introduced in the previous section, a single relationship between $\dot{m}_{\mathrm{w}}$ and $v_{\infty}$ can be obtained as a solution. In addition, the CRSF data help us to make a concentration on the estimation of the stellar wind parameters. The orbital semi-major axis a is obtained from the orbital period if the mass of the donor is known. In Table 2, we show the values of donor mass and donor radius, which has appeared in Equation (8).

Figure 3.

The same as previous, for the wind mass-loss rate, accretion mass-loss rate and accretion regimes. The position of each source is shown by a black filled circle.

Figure 4.

The same initial conditions in terms of the stellar wind parameters as in Figures 2–3, but these sources show different features and different distributions. These systems show that, the accretion stage can go from wind accretion to partial RLOF.

If we choose the mass-loss rate from the donor $\dot{m}_{\mathrm{w}}$ as a fundamental variable, then Equation (9) becomes a biquadratic equation and has four solutions. Two of them are physically nonsense, so we consider the other two solutions since they have a clear correlation between wind velocity and the mass loss rate of donors in SG-HMXBs. These solutions of the wind velocity are shown in Figures 2–4 by solid curves, as functions of mass-loss rate in SG-HMXBs. Note that, the uncertainty caused by the strength of the NS magnetic field is negligible, since this uncertainty is determined fairly accurately from CRSF data, and is also much smaller than the uncertainty of all other considered parameters such as in $\dot{m}_{\mathrm{w}}$ and $v_{\infty}$ . In these figures, the wind parameters given by the frequently used wind model by Vink, de Koter, & Lamers (Reference Vink, de Koter and Lamers2001) are shown at the same time. The approximated wind mass-loss rates from SG stars are given by the complex functions of mass, radius (escape velocity) and luminosity (effective temperature) of SG stars. To derive the mass-loss rate of the HMXB donors, we use the mass and radius data which are derived and introduced in previous papers (Chaty et al. Reference Chaty2008; Cusumano et al. Reference Cusumano2010; Rawls et al. Reference Rawls2011; Mason et al. Reference Mason2012; Falanga et al. Reference Falanga2015; Reig et al. Reference Reig2016) based on the observations and the stellar-evolution model. The estimation depends on the empirical stellar-evolution model. From these data, we derive the effective temperature and corresponding luminosity using approximated stellar evolution scheme. We compile Equations (1)–(30) in Hurley, Pols, & Tout (Reference Hurley, Pols and Tout2000) and find the evolution stage of each donors in SG-HMXBs listed in Table 2. Then, we derive the effective temperature at this evolutionary stage.

The wind parameters ( $v_{\infty}$ and $\dot{m}_{\mathrm{w}}$ ) we obtained are reported in Table 2. From these results, we could confirm that terminal velocities of the wind from donors in SG HMXBs are rather slow. In seven systems, the terminal velocities of donors dip from the typical wind velocity of galactic single SG stars $\left(1000-2000\, \mathrm{km\,s}^{-1}\right)$ . This result is consistent with recent result given by Giménez-García et al. (2016) who argued that the stellar wind of donors in persistent HMXBs is systematically slow.Footnote a For instance, from recent observations, it is suggested that the wind velocity in persistent SG HMXBs, Vela X-1, is relatively slow $\left(v_{\infty} = 700\,\mathrm{km\,s}^{-1}\right)$ , this might be due to the ionisation of the stellar wind by the radiation from NS, and thus prevents the acceleration of the stellar wind (Kretschmar et al. Reference Kretschmar2021). In our samples, even for systems with higher $v_{\infty}$ , the wind velocities at the NS positions are typically $v_{\mathrm{w}} \approx 500\,\mathrm{km\,s}^{-1}$ , and still show very slow wind. However, in some tight binaries, the tidal effects could reduce stellar wind velocity towards the accretor (Hirai & Mandel Reference Hirai and Mandel2021).

It is broadly considered that in wind-fed HMXBs, the wind matter is captured by the NS magnetic field at a certain radius, and transported onto the polar regions of the NS. In this process, around the polar region, the accretion column is formed, and the potential energy of the accretion matter is converted into strong X-ray radiation.

However, it is believed that, when the NS (and consequently NS magnetic field lines) rotates rapidly, the accretion matter cannot fall onto the NS surface and in some conditions it could be expelled out (Pfahl et al. Reference Pfahl2002; Podsiadlowski et al. Reference Podsiadlowski2004). This rotational inhibition of the accretion matter is called the propeller effect (see Reig and Zezas Reference Reig and Zezas2018). The propeller/accretion limit could be defined by three typical radii (accretion radius $R_{\mathrm{acc}}$ , is the distance at which inflowing matter is gravitationally concentrated toward the NS. Magnetic radius $r_{\mathrm{m}}$ , is the distance at which the pressure of the NS magnetic field balances the ram pressure of inflowing materials. Corotation radius $r_{\mathrm{co}}$ is the radius where the Keplerian angular velocity equals the NS spin angular velocity.

The parameter space could divided into five accretion regimes based on their magnitude relation (Stella, White, & Rosner Reference Stella, White and Rosner1986; Bozzo, Falanga, & Stella Reference Bozzo, Falanga and Stella2008). That is, the parameter space can be categorised into

• (A) supersonic inhibition regime ( $r_{\mathrm{m}} > r_{\mathrm{a}}, r_{\mathrm{co}}$ ),

• (B) subsonic inhibition regime ( $r_{\mathrm{co}} > r_{\mathrm{m}} > r_{\mathrm{a}}$ ),

• (C) supersonic propeller regime ( $r_{\mathrm{a}} > r_{\mathrm{m}} > r_{\mathrm{co}}$ ),

• (D) subsonic propeller regime ( $r_{\mathrm{co}} , r_{\mathrm{a}} > r_{\mathrm{m}}$ , $\dot{M} < \dot{M}_{\mathrm{c}}$ ),

• (E) direct accretion regime ( $r_{\mathrm{m}} < r_{\mathrm{a}}, r_{\mathrm{co}}$ ).

Here, $\dot{M}_{\mathrm{c}}$ $\left(\sim 10^{-7} \mathrm{M}_{\odot} \mathrm{yr}^{-1}\right)$ denotes the critical limit where radiative cooling starts working (see Bozzo et al. Reference Bozzo, Falanga and Stella2008).

In the figures shown below (Figures 2–4), the different accretion regimes (A) to (E) are divided by dashed lines. The propeller regime is defined when the accretion radius of the accretion disk is larger than the magnetic radius. In contrast, in the supersonic inhibition regime, the magnetic radius is larger than the accretion radius and corotation radius, so it rotates more slowly than the inner regions of the disk (Frank, King, & Raine Reference Frank, King and Raine2002). The shaded region denotes the direct accretion regime such that only systems in this region can be observed as a persistent HMXB. The efficiency of the propeller depends weakly on the magnetic moment of the star (Ustyugova et al. Reference Ustyugova2006). Since if the angular velocity of the star is larger, then the efficiency of the propeller becomes higher (Tsygankov et al. Reference Tsygankov, Mushtukov, Suleimanov and Poutanen2016). In contrast, the magnetic gate (or magnetic barrier), which refers to the effect of magnetic pressure, will prevent the material from accreting in the inhibition regime. Thus, the centrifugal gate (or centrifugal barrier), which refers to the propeller effect, will also propel away material along the magnetospheric boundary of the NS. The two gates are closed, preventing accretion onto the NS. In addition, the subsonic propeller regime becomes clearer as the strength of the propeller increases. As the strength increases there is a sharp decrease in the accretion rate to the star.

It is noteworthy to mention here that the CRSF data allow us to better identify the propeller region and the accretion region as shown clearly in the Figures 2–4. In particular, the solid curves in these figures represent the wind equation solution for the theoretical relations between $\dot{m}_{\mathrm{w}}$ and $v_{\mathrm{w}}$ as given by Equation (9) with the CRSF data. In addition, the CRSF data make us concentrate on the estimation of the stellar wind parameters. We show that when these curves come into the direct accretion region (shaded region) created in the figures, the systems with corresponding parameters can lead to X-ray emission.

5. Summary and conclusions

The following conclusions and implications are obtained:

We have investigated some physical quantities for several HMXBs with supergiant companions. The magnetic field, which is generated by the cyclotron lines- gives the mass accretion rate, based on the assumption of torque balance. The mass accretion rate allows us to derive a relation between the terminal wind velocity and the mass-loss rate of the donor. Furthermore, for all systems, our analysis (direct accretion condition shown by shaded region, and wind equation solution with CRSF data shown by solid curves) indicates that the wind velocity must be systematically slow.

By adopting the accretion regime model by Bozzo et al. (Reference Bozzo, Falanga and Stella2008), we have explored the parameter space in several regimes to support the intrinsic variabilities of mass accretion rate and wind velocity. This may allow us to study an evolutionary path for several SG-HMXBs in these diagrams. Different regimes are sufficient to distinguish the bright X-ray sources spatially, and the magnetic field-wind velocity can be probed. As a result, the persistent SG HMXBs within the shaded region can be observed through the direct accretion regime. This interpretation is predicated on its emission of accretion in high-energy X-rays.

It is seen that the wind velocity causes a significant effect on the results of their X-ray features and it could be used to determine the ejection mechanism of mass. When the wind velocity is slow, the accretion disk is often formed even in systems with large orbital period. This will allow us to better characterise the HMXB of both types, SG and Be, hosting NS, by deriving accurate properties of those compact binaries.

From the updated measurement of HMXB cyclotron lines, the derived magnetic fields given by the CRSF data are all concentrated around ${\sim}10^{12} \mathrm{G}$ . Note that, the fundamental energy during X-ray observation, spin and other physical parameters property diverges and varies. The existence of a high magnetic field has the potential to regulate their formation and evolution.

The accretion mechanism for the fast spinning NSs ( $P_{\mathrm{spin}} \leq 40$ s) with a short orbital period ( $P_{\mathrm{orb}} \leq 10$ d), like in LMC X-4, Cen X-3 and OAO1657 (see Figure 4) can not be constrained by our model. In LMC X-4 and Cen X-3, these two binary systems are extremely tight systems. Thus, the accretion mechanism can therefore not be approximated by spherical wind, because in such tight systems the concentrated asymmetric wind or RLOF accretion should be considered (El Mellah et al. Reference El Mellah, Sander, Sundqvist and Keppens2019a, b). Although OAO 1657 is a further evolved star with a long orbit, the donor of this system can be detected throughout its evolution as a Wolf-Rayet star. On the contrary, it can be seen from Figures 2 and 3, that the results are good for systems with long orbital periods, and we can successfully apply the stellar evolution code directly to these systems.

Finally, one would hope that the results of this work will be improved with data from INTEGRAL, ${eRosita}$ and ${HXMT}$ , which can provide significant increase in the observational sensitivity of a few cyclotron sources as well as population synthesis studies.