Spectroscopic modelling of two high-mass X-ray binaries, Cyg X–3 and 4U 1538–522

The companion stars of Cyg X–3 and 4U 1538-522 are a Wolf-Rayet (WR; Kerkwijk et al., 1996) and a B0Iab star (Reynolds et al., 1992), respectively. The wind velocity profile from a super-giant (SG) star is described by $\beta$-velocity law (Castor et al., 1975),

Here $\beta$ is velocity gradient and $v_{\infty}$ denotes the terminal velocity. In general, $v_{\infty}$ ranges from 1000 to 3000 km s^-1. For a steady-state wind, density profile n_H(r) depends on the mass-loss rate $\dot{\text{M}}$. The hydrogen density profile n_H(r) is given by the equation,

where $\mu$ and m_H represent the mean molecular weight of the gas and mass of hydrogen atom, respectively. For simplicity, we assume $\beta=0$ and a constant mass loss rate. As a result, the density decreases with distance in a power-law with a power $-2$,

To be noted here that CLOUDY measures distance from the ionizing source which is fixed in the code. In our models, the compact star is the main source of X-ray ionization. Hence, we use the wind density distribution along the line of sight from the X-ray source calculated for a wind spherically symmetric with respect to the compact star. We consider the elemental abundances of the wind in our calculation. Earlier Szostek & Zdziarski (2008) also assumed similar density profile for Cyg–X3. Hence, we use the following radius dependent power-law density profile

for all our models presented here. Here $n_{H}(r_{0})$ is the density at the illuminated face at $r_{0}$ and the radius $r$ is centered on the compact star. For hydrogen density $n_{H}(r)$ (cm^-3), we use the total hydrogen density. In general, physical conditions vary at different phases of the orbit. In our model we incorporate this by using a covering factor, $f$, where $f$ is the fraction of 4$\pi$ $\it{sr}$ covered by gas, as viewed from the central source of radiation. The value of $f$ lies between 0 to 1.

In our models presented here, we assume that the stellar wind is irradiated by radiation field coming from both the stars of the binary system. A blackbody is considered to explain the X-ray emission from and/or near the compact star. As a result, our considered radiation is composed of three components: i) a black body source appropriate to the companion stars’ surface temperature (T_BB), ii) another blackbody continuum with temperature equivalent to a few tenth of one keV arising from the thermal emission near the compact star’s surface (van der Meer et al., 2005), and iii) a Comptonised power-law continuum $\nu$^-α. This power-law continuum behaves as $\nu$^5/2 at lower energy to account for self-absorbed synchrotron (Rybicki & Lightman, 1979), $\nu$^-2 at higher energies and $\nu$^-α between 10 microns and 50 KeV. The photon index, $\alpha$, is a free parameter. It is to be noted that, without considering the power-law continuum, we cannot predict the observed fluxes of Fe XXV and Fe XXVI lines.

To study the effect of magnetic field in our models, we use a tangled magnetic field (in units of G) following the equation

where, $B_{0}$ is a free parameter and the term in parenthesis is the ratio of the density at a distance $r$ to the density at the illuminated face at $r_{0}$. The standard value of the adiabatic index $\gamma$ for a tangled magnetic field is 4/3. Lack of information on the angle between the radiation field of the central object and the magnetic field restrains us from using an ordered magnetic field. Here we propose that with a prior knowledge of other physical conditions, the strengths of highly ionised X-ray lines originating in the stellar wind of the companion star can be utilised to estimate an upper limit of the local magnetic field. Earlier, other modellers, e.g. Szostek & Zdziarski (2008), used CLOUDY to study the effect of X-ray on the stellar wind. The current updated c-17.02 version of CLOUDY uses improved energy levels of K${\alpha}$ transitions (Chakraborty et al., 2020a) which improves the model predictions. In addition to that, we study the effects of magnetic field on highly ionised X-ray lines.

Cloudy has a built-in optimization process based on phymir algorithm (, 1997) which calculates a non-standard goodness-of-fit estimator $\chi$² and minimises it by varying input parameters. The $\chi^{2}$ is determined by the following relation,

Here, $n$ is the number of emission lines used in the model, $O_{i}$ is the observed value for the i-th observable, $M_{i}$ is the modelled value for this observable, and $\sigma_{i}$ is the relative error in the observed value. When the number of observed lines are greater than the number of input parameters, we use this method (, 2006, 2016). Otherwise, we perform a grid of models in the parameter space (within an acceptable range) (Mondal et al., 2019; , 2018). We then pick the parameter values which predict better match to the observed values and finally fine-tune relevant input parameters so that the observed data can be matched maximally.

Here we consider a sample model for HMXBs and show the effects of various input parameters on Fe XXVI and Fe XXV line strengths as both the HMXBs considered here show Fe XXV and Fe XXVI lines. We assume a solar metallicity (Grevesse et al., 2010) stellar wind irradiated by radiation field coming from both the stars of the binary system consisting of (i) a blackbody source at 160,000 K with luminosity 10^38.65 erg s^-1 mimicking the radiation from a WR companion star, (ii) another blackbody continuum with temperature 0.7 keV and luminosity 10^39.4 erg s^-1 presenting the thermal emission from the compact star, and (iii) a Comptonised power-law continuum with a photon index 1.3 and luminosity 10^38.3 erg s^-1. The temperature, at which Fe XXV and Fe XXVI lines form, is much higher than the sublimation temperatures of graphite and silicate dust grains. Hence we do not include dust in our calculation. Further, we assume $n_{H}(r_{0})$ = 10^12.3 cm^-3 and the wind velocity to be 1500 km s^-1. We do not resolve the binary stars to keep the uncertainties of the model low. The inner radius is 10¹¹ cm away from the compact star and the extension of the ionized gas is set by a given column density of Hydrogen. For the sample model, this value is 10^22.5 cm^-2. The sample model parameters are shown in Table 1. Fig.1 shows the total incident continuum and its constituting individual parts. Fig.2 shows the transmitted continuum of the sample model as a function of wavelength. The transmitted continuum consists of the attenuated incident continuum and the outward component of the diffuse continuum and line emission. The resolving power of High Energy Transmission Grating (HETG) instrument on-board $\it{Chandra}$ varies from $\sim$ 800 at 1.5 keV to $\sim$ 200 at 6 keV. We thus set the coarse continuum mesh resolution E/$\Delta$E = 660.

The surface temperature of the companion star is much less than the excitation temperatures of Fe XXV and Fe XXVI. Hence, the luminosities of Fe XXV and Fe XXVI lines do not depend on the surface temperature of the companion star. So, among all the parameters, the temperature of the blackbody continuum from the companion star surface has no effect on the fluxes of Fe XXV and Fe XXVI lines of the sample model. Therefore, we do not consider the blackbody radiation from the companion star while modelling Cyg X–3 and 4U 1538–522.

To gauge the effects of a parameter on the line fluxes, we only vary that parameter while keeping all others fixed. Table 2 shows how the line fluxes change with changing parameters.

We notice that the wind velocity has very negligible effect on the Fe XXV and Fe XXVI line fluxes. To check the effect of changing hydrogen number density, we change the density by $\pm$0.3 dex from the default value, 10^12.3 cm^-3. It is clear from the plot that the line fluxes are density dependent (below the critical density) and Fe XXV line is more sensitive as discussed in the introduction. Individual line fluxes as well as Fe XXV / Fe XXVI line ratio increase with increasing density. In the following, we discuss the effects of other parameters.

We vary the inner radius by $\pm$0.3 dex from the default value to check its effect. For a given source luminosity, the radiation flux decreases as $r^{-2}$, and hence the temperature decreases with increasing inner radius. The collisional ionisation is temperature dependent, and its value decreases with decreasing temperature. Hence, individual line fluxes as well as Fe XXV / Fe XXVI line ratio increase with increasing inner radius. For the sample model, the effect of inner radius is more profound for Fe XXV.

The extension of the ionised gas is defined by N(H). So, a higher N(H) means a greater extension of the ionised cloud. As a result, the Fe XXV and Fe XXVI line fluxes increase with increasing N(H). However, Fe XXV line flux increases more than the Fe XXVI line. Hence, Fe XXV/Fe XXVI ratio increases with increasing N(H).

To check the effect of changing metallicity, the abundance of elements heavier than He, we change the metallicity by a factor of 2 from the default value. The density considered in the sample model are lower than the critical densities of Fe XXV and Fe XXVI. Hence, the luminosity of both the lines increases with increased metallicity. In addition to this, metallicity plays an important role in gas cooling. An Increased metallicity decreases the gas temperature and affects physical processes which are temperature dependent. As discussed earlier, the collisional ionisation is temperature dependent. Hence, increased metallicity increases Fe XXV/Fe XXVI.

To check the effect of changing photon index of the power-law spectrum we run two models with photon index 1.0 and 1.6. The number of incident photon flux ($\propto$ $\nu^{-photon-index}$) increases as photon index decreases, and it is more for smaller frequency. Hence, the line fluxes as well as Fe XXV/Fe XXVI increases as photon index decreases.

To check the effect of the compact star’s blackbody temperature, we run two cases with surface temperature 0.8 keV and 0.6 keV. A blackbody with a few tenth of keV temperature peaks in X-rays. Lowering this temperature reduces the height of the peak and shifts the peak to higher wavelengths. Hence, the continuum luminosity is affected by changing the blackbody temperature. The gas becomes less ionised and temperature also decreases with lower surface temperature. As a result, luminosity/intensity of the X-ray lines also decreases with decreasing surface temperature. With a higher surface temperature, the Fe XXV/Fe XXVI ratio decreases.

Cyg X–3 (20^h 32^m 25.78^s, +40^∘ 57’ 27.9”) (Cutri et al., 2003) is a well-observed HMXB in the constellation Cygnus, 7.4 kpc (McCollough et al., 2016) away from us. The nature of the compact star is not well understood, it can either be a neutron star or a black hole. Whereas, the companion star of this binary system is a Wolf-Rayet star (Kerkwijk et al., 1996) with an orbital period of 4.8 hours (Liu et al., 2007). Koljonen & Maccarone (2017) had estimated the mass of the WR star to be 8-14 M_⊙. Whereas, the mass of the compact object is 2.4${}^{+2.1}_{-1.1}$ M_⊙ (Zdziarski et al., 2013).

For a binary stellar system, the average distance between the two stars with known masses and known orbital period can be calculated using the simplistic expression of Kepler’s 3rd law, and using that one can calculate the average distance between the compact star and the companion star for Cyg X–3 to be $\approx$ 2-3 $\times$ 10¹¹ cm. Assuming a stellar wind with a velocity of the order of 10³ km s^-1, the Bondi-Hoyle wind accretion radius is found to be $\approx$ 3-12 $\times$ 10¹⁰ cm.

The compact object is very close to the Wolf-Rayet star in this case. Hence, the materials of the stellar wind get pulled away from the Wolf-Rayet star by the compact object and become hotter and shine in X-rays. This system exhibits many interesting properties like jets (Dubus et al., 2010) and radio flare associated with $\gamma$ ray emission etc (Corbel et al., 2012; Pahari et al., 2018). But here we focus only on the spectroscopic study of highly ionised X-ray lines.

Spectroscopic observation of Cyg X–3 has been done by many groups (Fender et al., 1999; Paerels et al., 2000; Vilhu et al., 2009; Kallman et al., 2019). Fender et al. (1999) performed IR observations, whereas Paerels et al. (2000) and Vilhu et al. (2009) used Chandra to study Cyg X–3 in X-rays. They detected Si XIV (2.005 keV), S XVI (2.62 keV), Fe XXV (6.7 keV) and Fe XXVI (6.966 keV) lines using Chandra. In this work, we model the X-ray line intensities observed by Vilhu et al. (2009) as they provided the line flux values in a tabular form which eases modelling. Vilhu et al. (2009) used HETG instrument on-board $\it{Chandra}$. The spectral resolutions are 0.023 Å and 0.012 Å for Medium energy Grating (MEG) and High energy Grating (HEG), respectively. The observed Fe XXV line is a composite of 1.869 Å, 1.859Å, 1.852 Å lines arising partially in the wind with intensity ratios 0.35:0.45:0.19, and partially in the region where the Fe XXVI line originates. Earlier Szostek & Zdziarski (2008) used CLOUDY (v6.02) to study the effect of stellar wind on the X-ray lines for the same source observed using BeppoSAX. Our current model uses Chandra data with higher resolution and we use a more advanced version of Cloudy, c-17-02, which is capable to predict the line energies and line intensities of highly excited ions more accurately (Chakraborty et al., 2020a). In addition, we also study the effect of magnetic field on highly ionised lines.

In our model, we assume that the stellar wind is irradiated by the compact object with a blackbody temperatures 0.7 KeV. The corresponding luminosity is 10^39.40 erg s^-1 (Vilhu et al., 2009). In addition to this, there is a Comptonised continuum having a power-law with the photon index $\alpha$. It is to be noted that the blackbody radiation from the companion star does not effect the X-ray line intensities. However, the metallicities of the companion star is important. We consider the typical Wolf-Rayet abundances assuming a WN(4-7) star: H=0.1, He=10, C=0.56, N=40, O=0.27 (Vilhu et al., 2009). The abundances of other elements are fixed at their solar values. In general, the stellar wind velocity of HMXBs range from 1000 to 3000 km s^-1 (Kaper, 1995). In this model, we choose a stellar wind with velocity 2000 km s^-1. Vilhu et al. (2009) noticed that the line intensities are maximum around phase 0.5, when the compact star is in front. Here the geometry is complex, and as a first approximation we fix the value of the covering factor to be 0.25 for our simple model. From earlier literature we find that the value of photon index for this source is around 2 for soft X-ray observations (Koljonen et al., 2018). We choose 2 as the value of photon index for our model. We fix the elemental abundances with respect to hydrogen (in log scale) as, He/H = -0.071, N/H = -2.568, C/H = -3.822, O/H = -3.878, Fe/H = -4.500, S/H = -4.733, Si/H = -4.448, similar to a WN(4-7) star. Vilhu et al. (2009) performed an XSTAR simulation (Bautista & Kallman, 2001) and predicted the hydrogen density to be $10^{11}$ cm^-3 and photon index to be 2. Our predicted hydrogen density at the illuminated face is close to this value. Our predicted hydrogen column density also matches to the value derived by Vilhu et al. (2009). The electron density (Fig. 4) and the electron temperature (Fig. 3) vary across the ionized gas. The value of electron temperature averaged over the extension of the cloud is 9.05$\times$10⁷ K. Fig. 5 show the predicted transmitted spectra. In Table 3 we list the model parameters without considering any magnetic field.

In Table 4 we compare the observed and predicted luminosity of HMXB Cyg X–3. Our predicted luminosities for lines Fe XXVI, Si XIV, S XVI matches very well with the observation. Our predicted total luminosity for Fe XXV also matches with the observation, though the ratios among the Fe XXV lines do not match. It is to be noted that the ionising wind model of Vilhu et al. (2009) could not reproduce the observed line fluxes of Fe XXV and Fe XXVI. In addition to the observed lines mentioned in Vilhu et al. (2009), we also predict Ca XX and Ar XVIII with luminosity above 10^34.5 erg s^-1. These lines were not reported in Vilhu et al. (2009). This can be due to interstellar absorption and/or subsolar abundances of Ca and Ar. To check this, we run a model with Ca and Ar abundances lowered by a factor of -1. and -0.7 (in log scale), respectively. The predicted luminosities decrease by a factor of 10 and 5, respectively. Hence, we conclude that either the abundances of these two elements are subsolar ($\sim$ 1 dex lower) and/or there is absorption by interstellar medium.

Next, we include magnetic field in our model to study its effect on line intensities. Very hot ionised gas cools via cyclotron radiation, and we find that with a strength of the magnetic field $\geq$ 10³ G, all the Fe XXV and Fe XXVI line fluxes exceed the observed ranges. At this point, the cyclotron cooling becomes the dominant cooling mechanism.

So, we reoptimise all the parameters including magnetic field. Table 5 and Table 6 list the predicted model parameters and the line luminosities, respectively. The variation of electron density and electron temperature have features similar to the previous case without magnetic field. However, the electron temperature averaged over the extension of the cloud is 8.66$\times$10⁷ K, less than the previous case without magnetic field. Our model predicts a magnetic field with strength 10^2.5 Gauss. The predicted spectra is similar to that of without magnetic field.

Previously Hubrig et al. (2016) had observed several Wolf-Rayet stars using FORS2 spectropolarimetry and estimated the highest value for the average longitudinal magnetic field to be 327$\pm$141 G. However, de la Chevrotière et al. (2014) have concluded 0 < B_wind $\leq$ 1700 G based on spectropolarimetric study of a sample of 11 bright WR stars. Our predicted magnetic field ($\approx$ 10^2.5 G) is consistent with the observations of Hubrig et al. (2016).

The HMXB 4U 1538–522 is an interesting eclipsing HMXB. It was first observed using UHURU satellite by Giacconi et al. (1974) and later followed by many observers. It is located at a distance of 6.4 kpc (Reynolds et al., 1992; Clark, 2004; Bailer et al., 2018) with an orbital period of 3.73 days (Clark, 2000; Baykal et al., 2006; Mukherjee et al., 2006; Falanga et al., 2015). This HMXB system is powered by the strong stellar wind from a B0Iab star QV Nor (Reynolds et al., 1992), and highly ionised Fe XXV and Fe XXVI lines are observed in emission (Aftab et al., 2019; Mukherjee et al., 2006; Rodes-Roca et al., 2011). Unlike the previous case, here the compact star is a pulsar. Reynolds et al. (1992) estimated the mass of the companion star to be 19.9 $\pm$3.4 M_⊙, and the mass of the compact object is 1.06 $\pm$ 0.27 M_⊙ (Falanga et al., 2015). Clark (2000) and few more observers suggest that the orbit is probably eccentric, with an eccentricity approximately 0.18. This source shows many interesting properties, like significant evolution in the orbital period and cyclotron lines (Hemphill et al., 2019) but as stated earlier, our main interest is to do spectroscopic study of the highly ionised X-ray lines.

In this work, we study the physical properties of HMXB 4U 1538–522 by modelling its X-ray line luminosities observed by Aftab et al. (2019) using XMM- Newton. The line fluxes of these lines were given in units of 10^-4 photons cm^-2 s^-1 for the eclipse phase. For calculation purpose, we multiply it with corresponding line centre energy and convert into ergs cm^-2 s^-1 unit. Here we consider $f=0.6$ for eclipse position. Furthermore, we use 0.2 as the photon index as estimated by Aftab et al. (2019) for the eclipse position.

Similar to the earlier system (Cyg X–3), we assume that the stellar wind is irradiated by the compact object with a blackbody temperatures of a few tenth of KeV and a power-law continuum. The blackbody radiation from the companion star has no effect on the X-ray lines, as expected. We consider typical solar abundances for this source (Grevesse et al., 2010). Here also we do not consider dust grains, as the temperature of the region where Fe XXV and Fe XXVI lines form is much higher than the dust grain sublimation temperature. In this model, we choose a stellar wind with velocity 1500 km s^-1 (Abbott et al., 1982). In Table 7 we list our predicted model parameters. Our predicted hydrogen density at the illuminated face of the wind is $10^{11.99}$ cm^-3 and the illuminated face is 10^10.43 cm away from the compact star.

Table 8 compares the predicted and observed fluxes for eclipse phase of HMXB 4U 1538–522. We predict observed fluxes of both the Fe XXV and Fe XXVI lines simultaneously within the observed range. It is to be noted that the model parameters could be constrained better for this source if we have more observed lines of different species. The electron density (Fig. 7) and the electron temperature (Fig. 6) vary across the ionized gas. The value of electron temperature averaged over the extension of the cloud is 2.865$\times$10⁶ K. Fig. 8 show the predicted transmitted spectra without any magnetic field.

Like the previous source, in the final stage, we switch-on the magnetic field in our model to check its effects on the X-ray lines. We find that above a magnetic field $>$ 10^3.5G, the Fe XXVI and Fe XXVI line fluxes exceed the observed ranges. We believe this is caused due to cyclotron cooling. Next, like the earlier source, we reoptimise all the parameters including magnetic field. The predicted electron density, electron temperature, and the predicted spectra are similar to the case without magnetic field. Table 9 lists the predicted model parameters. The predicted magnetic field is 10^2.5 Gauss. Table 10 compares the predicted and observed fluxes for eclipse phase of HMXB 4U 1538–522. Many groups estimated magnetic fields for OB stars (Przybilla et al., 2016; Castro, N et al., 2017) and found it to be of order of 1kG.