4.1. Orbital Phase Averaged Spectrum

We have extracted the orbital phase averaged spectrum of Cen X-3 (see Figure 7) with MAXI/GSC using the MAXI on-demand processing³, carefully excluding any contamination by nearby brighter sources. For spectral analysis we used the XSPEC fitting package, released as a part of XANADU in the HEASoft tools. We tested both phenomenological and physical models commonly applied to accreting X-ray pulsars and rebinned all extracted spectra to obtain spectral bins by Gaussian distribution.

Fig. 7 Orbital phase averaged spectrum of Cen X-3 in the 2.0-20.0 keV band. Top panel: data and best-fit model described by equation (2). Bottom panel: residuals between the spectrum and the model. The resulting fit parameters are reported in Table 2. The color figure can be viewed online. 

Absorbed Comptonisation models have been successfully applied to HMXBs in the 2.0-20.0 keV energy range covered by MAXI/GSC, such as Vela X-1 (^{Doroshenko et al. 2013}), 4U 1538-52 (^{Rodes-Roca et al. 2015}) and Cen X-3 (^{Rodes et al. 2017}). We started to describe the orbital phase-averaged spectrum using a simple partial absorbed Comptonisation model modified by a Gaussian absorption line at 5 keV to compensate the incompleteness of the MAXI/GSC response (Nakahira private communication). An iron fluorescence emission line present in the spectrum is modelled with a Gaussian component, Fe Kα at 6.4 keV. This model is described by equation (1).

where, in terms of XSPEC, pcfabs is a partial covering fraction absorption that affects only a fraction f of the model component multiplied by it, gabs is the Gaussian absorption line component, compST is the Comptonisation of cool photons on hot electrons (^{Sunyaev & Titarchuk 1980}), and the Gaussian line is added to describe the Fe Kα line. The absorption cross sections were taken from ^{Verner & Yakovlev (1995)} and the abundances were set to those of ^{Wilms et al. (2000)}.

Although this model describes the averaged spectra between 2.0-20.0 keV well (χ_r² = 1.09), it cannot offer a good statistical and/or physical solution for all orbital phase-resolved spectra. For example, fitting the eclipse spectrum, i.e. when the direct X-ray emission is totally blocked by the companion, we obtained that the covering fraction factor was f ≈ 0. Model parameters were also not well constrained. As the aim was to satisfactorily describe both the aver aged spectrum and the orbital phase-resolved spectra with the same model, it was decided to reject it.

Recently, ^{Aftab et al. (2019)} and ^{Sanjurjo-Ferrín et al. (2021)} carried out spectral analysis of XMM-Newton data in the eclipse and out-of-eclipse phases in the energy band (0.8-10.0 keV). They obtained best fits by combining blackbody and power-law components and used them to describe both eclipse and out-of-eclipse spectra. In addition, ^{Wojdowski et al. (2001)} argued that the stellar wind in the system is smooth and concluded that the wind is most likely driven by X-ray heating of the illuminated surface of the companion star as proposed by ^{Day & Stevens (1993)}. Compared to previous studies, thanks to MAXI/GSC’s observation strategy, a large number of complete orbits have been observed and divided into 10 orbital phase intervals. According to the previous discussion, finally, models including either a blackbody component (bbody in XSPEC) or a Comptonisation component (compST in XSPEC) have been combined to describe the average spectrum.

Interstellar medium (ISM) absorption and local absorption components have been included by means of a partial covering fraction defined through the parameter C. The tbnew⁴ component is a recent version of the Tübingen-Boulder absorption model which updates the absorption cross sections and abundances (^{Wilms et al. 2000}); the gabs factor has been described above; po is a simple photon power law consisting of a dimensionless photon index (Γ) and the normalisation constant (K), the spectral photons keV⁻¹ cm⁻² s⁻¹ at 1 keV; bbody corresponds to a blackbody model whose parameters include the temperature kT_bb in keV and the normalisation norm, defined as L₃₉/D², where L₃₉ is the source luminosity in units of 10³⁹ erg s⁻¹ and D² is the distance to the source in units of 10 kpc. This model is given by equation (2).

where GL represents the Gaussian functions added to account for the emission lines. Here, parameters derived from ASCA data by ^{Ebisawa et al. (1996)} were used to describe the Fe Kα complex.

Fitted parameters for the continuum model are listed in Table 2 where it is also included the equivalent width (EW) of the Gaussian emission lines. Figure 7 shows the data, the best-fit model described by equation (2), and residuals as the difference between observed flux and model flux divided by the uncertainty of the observed flux.

Since the surface luminosity of a blackbody only depends on its temperature, it is possible to calculate the radius of the emitting region, R_bb, by using the expression:

where D is the distance to the source in kpc, F_bb is the unabsorbed flux in erg s^-1 cm^-2 in the energy range 2.0-20.0 keV, T_bb the temperature in keV. Taking into account the distance to the source given by GAIA (E)DR3 d(kpc) = 6.8-0.5+0.6, and an unabsorbed flux of (1.7-0.4+0.5) × 10^-9 ergs^-1 cm^-2 in the MAXI/GSCenergy band, from equation (3) we found a radius of the emitting surface of R_bb = 0.71-0.16+0.19 km. If we assume thermal emission from the NS polar cap, this radius may be consistent with the expected size (also compatible with ^{Sanjurjo-Ferrín et al. 2021}).

The intrinsic bolometric X-ray luminosity is a key parameter to infer the stellar wind parameters and the details of the accretion processes. It is usually estimated from the measured X-ray flux of the source by using equation (4). Although the accretion flow is not expected to be as isotropic as the stellar wind, here, it is assumed that the system is emitting isotropically. A few cautions should be kept in mind when using this assumption: in accreting neutron stars, the bulk of the x-rays are produced in the accretion columns near the two magnetic poles; the emission profiles of these regions are not well known; the bolometric flux is measured on a small energy band and derived from phenomenological rather than physically justified models (see ^{Martínez-Núñez et al. (2017)} for a review of stellar winds from massive stars).

where L_X is the X-ray luminosity, D is the distance to the source and ƒ_{no_abs} is the unabsorbed flux in the 2.0-20.0 keV energy band. We obtained L_X = (1.9-0.8+1.0), × 10³⁷ ergs^-1, ƒ_{no_abs} = (3.5-0.9+1.1) × 10^-9 ergs^-1 cm^-2, indicating that the accretion mode is not only due to the stellar wind (< 10³⁶ erg s^-1) but should also be enhanced by disk accretion or gas stream accretion.

From the bbody normalization, L₃₉ / D²₁₀, a value of 0.04-0.08+0.09 is derived, which is consistent with the experimental result if we take into account the uncertainties of the parameters.

Another model has been also tested by replacing the blackbody by a Comptonisation model, maintaining the rest of the components unchanged. This model is described by equation (5).

Best-fit model parameters are listed in Table 3 and Figure 8 shows the averaged spectrum. From the model, the inferred unabsorbed flux was (3⁺⁶_-1) × 10⁻⁹ erg s⁻¹ cm⁻² in the MAXI/GSC energy band, corresponding to an X-ray luminosity of L_X = (2⁺³_-1) × 10³⁷ erg s⁻¹ which agrees completely with the previous result.

Fig. 8 Orbital phase averaged spectra of Cen X-3 in the 2.0-20.0 keV band. Top panel: data and best-fit model (equation (5)). Bottom panel: residuals between the spectrum and the model. The resulting fit parameters are reported in Table 3. The color figure can be viewed online. 

The Comptonisation parameter y = kTτ² / (m_e c²), where k is the Boltzmann constant, T is the temperature, τ is the optical depth, m_e is the electron mass and c is the light speed, determines the efficiency of the Comptonisation process (^{Titarchuk 1994}; ^{Prat et al. 2008}), and its value from the averaged spectrum y = 2.7-0.3+0.2 indicates an efficient process that corresponds to a moderate accretion rate.

The iron fluorescence emission line shows an interesting evolution along the orbital phase: it is centred at 6.4 keV out-of-eclipse but it is shifted to 6.7 keV at the ingress suggesting the coexistence of two iron lines at different energies (^{Rodes et al. 2017}). Sensitive X-ray observatories such as ASCA and XMM-Newton have detected and resolved the Fe complex in Cen X-3 (^{Ebisawa et al. 1996}; ^{Sanjurjo-Ferrín et al. 2021}, respectively). In contrast, MAXI/GSC is not able to resolve it and therefore the values obtained for these lines by ^{Ebisawa et al. (1996)} have been used to fit them. For this purpose, all their parameters except the line intensities were fixed (see Tables 2 and 3). The line flux ratio [Fe XXVI] / [Fe XXV] can be used to estimate the ionisation state of the emitting plasma.

The results obtained in the averaged spectrum for both models were 0.6 ± 0.3 (equation 2) and 1.1 ± 0.5 (equation 5), respectively. Assuming that this procedure is an approximation of the ionisation state of the system, these average values point to a highly ionised plasma with log ξ ≈ (3.4 - 3.8), according to the ionisation parameter calculated by ^{Ebisawa et al. (1996}, their Figure 8, upper panel).

Since the partial covering fraction modifies the continuum at low energies, two hydrogen column components were applied: one to describe the ISM towards the system, N²_H, and the other to describe the ISM plus local absorption, N¹_H. The mean value of the Galactic column density of hydrogen N_H,tot in the direction of Cen X-3 is 1.16 × 10²² cm⁻² (^{Willingale et al. 2013}). ^{Valencic & Smith (2015)} reported an ISM absorption towards this source of (1.6 ± 0.3) × 10²² cm⁻². The column N² derived from both models are compatible taking into account its uncertainties.

4.2. Orbital Phase Averaged, High and Low States Spectra with MAXI/GSC

As a consequence of the large variability of the light curve, high and low states have been defined. Thus, the good time intervals (GTIs) of both states have been identified in the energy range 2.0-20.0 keV and their respective averaged spectra have been extracted. In this analysis, we fitted the high and low states spectra with the same models as we used in the orbital phase-averaged spectrum, except that only a single here. Gaussian has been included The fits are shown in Figures 9 and 10; meanwhile, the best-fit model parameters are listed in Tables 4 (equation 2) and 5 (equation 5).

Fig. 9 Orbital phase averaged, high states and low states spectra of Cen X-3 in the 2.0-20.0 keV band. Top panel: Data and best-fit models defined by equation (2). Bottom panel: residuals for the model. The resulting fit parameters are reported in Table 4. The color figure can be viewed online. 

Fig. 10 Orbital phase averaged, high states and low states spectra of Cen X-3 in the 2.0-20.0 keV band. Top panel: Data and best-fit models defined by equation (5). Bottom panel: residuals for the model. The resulting fit parameters are reported in Table 5. The color figure can be viewed online. 

To determine the periods of high and low activity we established a number of counts greater than ≈ 0.5 photons s⁻¹ cm⁻² and less than ≈ 0.3 photons s⁻¹ cm⁻², respectively. Then, more than 100 bins (one bin ≈ 0.1 MJD) were accumulated to calculate the low-activity intervals and grouped 10 bins by 10 bins for the high-activity events.

The unabsorbed fluxes of the orbital phase-averaged, high and low activity spectra in units of erg s⁻¹cm⁻² are (1.72 ± 0.03) × 10⁻⁹, (4.1^+1.3) × 10⁻⁹ and (5.4^+1.5_−1.2) × 10^-10, respectively. For such fluxes, the radius of the blackbody emitting area is found R_bb = 0.71^+0.07_-0.06 km (averaged spectra), R_bb = 1.6 ± 0.5 km (high states) and R_bb = 0.36^+0.12_−0.10 km (low states). All values are on the order of 1 to 2 km which is compatible with a hot spot on the NS surface (^{Sanjurjo-Ferrín et al. 2021}).

Using the definition of the bbody normalization, L₃₉/D²₁₀, we have derived its value in the three spectra: 0.04^+0.08_−0.07 (averaged spectra), 0.1 ± 0.3 (high states) and 0.009^+0.019_−0.016 (low states). These values are consistent with the experimental ones, taking the uncertainties into account (see Table 4).

The Comptonisation parameter y (see Table 6) shows an efficient process that corresponds to a moderate accretion rate if we take into account the uncertainties.

During the high state, when L_X is close to 10³⁸ erg s⁻¹, the Fe Kα is not present in the spectrum. Moreover, the EW of the Fe Kα is three times higher during the low states than that in the averaged spectra. These facts suggest that the strong x-ray radiation may mask the possible presence of iron emission lines.

The unabsorbed flux is found to vary with a similar trend for both models. In fact, the X-ray luminosity is the same, taking the uncertainties into account (see Table 6, Column 2 corresponds to equation 2 and Column 3 corresponds to equation 5). On the one hand, it indicates (for averaged spectra and high states) that the accretion mode is not only due to the stellar wind (> 10³⁶ erg s⁻¹) but should also be enhanced by disk accretion or gas stream accretion. On the other hand, the luminosity indicates that the accretion mode in low states spectra is due to stellar wind. Therefore, the difference in X-ray luminosity between the high and low states can be attributed to a decrease in the accretion rate rather than an overall rise in absorption.

Tables 7 and 8 show the unabsorbed flux and the luminosity of each model component (equations 2 and 5, respectively) as well as the total unabsorbed fluxes and luminosities whose values agree with those given in Table 6.

4.3 Orbital Phase-Resolved Spectra with MAXI/GSC

Previous studies following this direction of analysis have been performed in one or two orbits (^{Nagase et al. 1992}; ^{Suchy et al. 2008}) or over shorter orbital phases (^{Ebisawa et al. 1996}; ^{Audley et al. 1996}; ^{Wojdowski et al. 2001}; ^{Aftab et al. 2019}; ^{Sanjurjo-Ferrín et al. 2021}). We have obtained orbital phase-resolved spectra of the HMXB pulsar Cen X-3, accumulating the 60 s duration scans into ten orbital phase bins covering entirely its orbit (^{Rodes-Roca et al. 2015}).

Based on the results from § 4.1, we fitted the orbital phase-resolved spectra with the same two models as we used in the orbital phase-averaged spectrum. Both models gave acceptable fits to observational data and the results are shown in Figures 11 and 12 for selected orbital phase-resolved spectra.

Fig. 11 Orbital phase-resolved spectra of Cen X-3 in the 2.0-20.0 keV band. Top panel: Selected spectra and best-fit models defined by equation (2). Bottom panel: residuals for the model. The color figure can be viewed online. 

Fig. 12 Orbital phase-resolved spectra of Cen X-3 in the 2.0-20.0 keV band. Top panel: Selected spectra and best-fit models (defined by equation (5)). Bottom panel: residuals for the model. The color figure can be viewed online. 

The degeneracy between the properties of the accretion rate and the physical parameters of the NS in all available models produces a certain degree of degeneracy between fit parameters. The behaviour of different parameters of the model described by equation (5) towards eclipse suggests degeneracies between them. To better explore the photon index variation, we constrained its value to the interval obtained between orbital phase 0.0 to 0.6, i.e. 1.90 < Γ < 2.85, and fitted the spectra of pre-ingress, ingress, eclipse and egress orbital phases again. The errors were obtained with the error task, provided by XSPEC, and with propagation of uncertainties. The new values obtained by this procedure differed only slightly from the previous values within errors. However, the photon index was not well constrained and exhibited relatively large errors. Our results are marked with open blue circles in Figures 14-18 and 20, where filled red squares represent the initial parameter values. The possible degeneracy between the spectral photon index Γ and absorption spectral parameters is known. Therefore, we produced χ² contour plots of Γ and N¹_H for both models (see Figure 13). It is clear from this figure that the value of N ¹ is moderately correlated to that of Γ as expected. However, the narrow energy band of MAXI/GSC as well as the observational mode makes it extremely difficult to remove this degeneracy through spectral fitting. In the following, we discuss the orbital phase spectral variation of the model parameters described by equations (2) and (5).

Fig. 13 The ellipses are χ²-contours for two parameters (N¹_H and Γ). The contours are 68%, 90% and 99% confidence levels for two interesting parameters. Top panel: averaged spectrum fitted by equation (2). Bottom panel: averaged spectrum fitted by equation (5). The color figure can be viewed online. 

In Figures 14-18 and 20-21, we show the evolution of the relevant parameters of both models throughout orbital phase. The temperature of the blackbody is almost constant, decreasing drastically in the eclipse and increasing in the eclipse-egress. That implies that the emission zone of this component should be large. The normalisation of the black-body component shows a smooth decrease, which could be compatible with a constant value, reaching the minimum value at eclipse (Figure 14). The radius of the emission zone is of the order of 1 to 3 km whereas in eclipse it is about 9 ± 3 km (Figure 14). This fact could suggest soft X-ray reflection from the inner accretion disk region and agree with that reported by ^{Sanjurjo-Ferrín et al. (2021)}.

Fig. 14 Evolution of some parameters of the model described by equation (2). Unabsorbed flux is in units of 10⁻⁹ erg cm⁻² s⁻¹. Bbody norm is in units of L₃₉/D²₁₀. Power-law normalization is expressed in units of keV⁻¹ s⁻¹ cm⁻². Filled red squares: initial values. Open blue circles: values obtained by constraining the range of values of the photon index (pre-ingress, ingress, eclipse and egress). The color figure can be viewed online. 

Fig. 15 Evolution of some parameters of the model described by equation (5). Unabsorbed flux is in units of 10⁻⁹ erg cm⁻² s⁻¹. Power-law normalisation is expressed in units of keV⁻¹ s⁻¹ cm⁻². Filled red squares: initial values. Open blue circles: values obtained by constraining the range of values of the photon index (preingress, ingress, eclipse and egress). The color figure can be viewed online. 

Fig. 16 Evolution of the Gaussian intensity of the Fe emission lines with orbital phase. Top, second and third panels: model described by equation (2). Fourth, fifth and bottom panels: model described by equation (5). The unit of the line flux I is 10⁻⁴ photons s⁻¹ cm⁻². Filled red squares: initial values. Open blue circles: values obtained by constraining the range of values of the photon index (pre-ingress, ingress, eclipse and egress). The color figure can be viewed online. 

Fig. 17 Evolution of the equivalent width (EW) of the Fe emission lines with orbital phase. Top, second and third panels: model described by equation (2). Fourth, fifth and bottom panels: model described by equation (5). EW is in units of keV. Filled red squares: initial values. Open blue circles: values obtained by constraining the range of values of the photon index (pre-ingress, ingress, eclipse and egress). The color figure can be viewed online. 

Fig. 18 Evolution of the partial covering fraction C and line intensity ratio Fe XXVI/Fe XXV. Top and second panels: model described by equation (2). Third and bottom panels: model described by equation (5). Filled red squares: initial values. Open blue circles: values obtained by constraining the range of values of the photon index (pre-ingress, ingress, eclipse and egress). The color figure can be viewed online. 

When the neutron star is embedded into the stellar wind of the donor, the hydrogen column density shows a modulation along the orbit. Therefore, besides the interstellar medium absorption component (consistent with a constant value), we also allowed for the presence of a local absorber, modulated by a partial covering fraction that acts as a proxy for some features of the stellar wind of the donor star or the surroundings of the compact object.

For the model defined by equation (2), the normalisation of the power law decreases smoothly during out-of-eclipse before reaching a minimum during pre-eclipse, eclipse and eclipse-egress (Figure 14). The power-law photon index is rather stable during all orbital phases (Γ ≈ 2.1), but it drops to ≈ 1.3 in the eclipse-egress ([0.96-1.0]) (Figure 20, top panel).

Fig. 19 Schematic view of the binary system. For simplicity we have assumed a circular orbit whose parameters r, s and θ are shown. We note that in this sketch the orbital phase 0 (1) is centred on the mid-time of eclipse. The color figure can be viewed online. 

Fig. 20 Evolution of the photon index Γ and absorption columns N¹_H and N²_H. Top, second and third panels: from equation (2). Fourth, fifth and bottom panels: from equation (5). Filled red squares: initial values. Open blue circles: values obtained by constraining the range of values of the photon index (pre-ingress, ingress, eclipse and egress). The color figure can be viewed online. 

For the model defined by equation (5), the normalisation of the power law (Figure 15) shows a similar pattern to that of the photon index (Figure 20, fourth panel), showing a flat behaviour outof-eclipse, increasing as ingress takes place and decreasing at egress. Apparently, the evolution along the orbit of the power-law parameters seems to be different. However, if the range of photon index is constrained, the values obtained with the new fits become consistent with each other taking into account the uncertainties. This would suggest that there is a certain degree of degeneracy between model parameters. The orbital variation of temperature and Comptonisation normalisation have two local peaks and the lowest value occurs at the eclipse.

The orbital evolution of unabsorbed flux for both models shows a very similar trend. In fact, the corresponding X-ray luminosities are the same taken into account uncertainties (see in Table 9, Column 2 corresponds to equation 2 and Column 3 corresponds to equation 5). The unabsorbed flux is consistent with a constant value, except for eclipse and eclipseegress. Although for an almost circular orbit and disc accretion no orbital modulation in the amount of material to accrete is expected, variations of the flux can be due to local absorption produced by an emerging accretion stream (eclipse-egress), probably corotating with the compact object, and/or a decrease in the accretion rate, most likely associated to instabilities at the inner edge of the disc interacting with the neutron star magnetosphere.

The X-ray continuum is modified by a partial covering fraction where the column N² _H corresponds to the ISM towards the source and N¹_H represents the ISM plus the circumstellar environment absorption (Figure 20). We have depicted the variation of the covering fraction measured from our models in Figure 18. It is seen that the covering fraction tends for most of the orbit to have values lower than 0.8, 0.4 < C < 0.9, which means that there are large inhomogeneities in the stellar wind of the giant star. Although both models seem to follow the same trend in the covering fraction, the orbital variation of C, given by equation (5), shows a certain modulation while the error bars obtained in equation (2) prevent us from confirming this orbital modulation. Nevertheless, taking into account uncertainties, most data points are consistent with 0.8 in both models. The covering factor variation from 0.76 to 0.9 is compatible with the compact object being deeply embedded into the stellar wind of the companion according to ^{Sanjurjo-Ferrín et al. (2021)}. From the long-term observations used here C > 0.8 at orbital phases [0.2-0.3] and eclipse.

4.4 Iron Line Complex

The spectral resolution of MAXI/GSC is not good enough to resolve the iron line complex, i.e. fluorescence emission lines (Fe Kα at ≈ 6.4 keV, Fe Kβ at ≈ 7.1 keV) and recombination emission lines (Fe XXV at ≈ 6.7 keV and Fe XXVI at ≈ 6.9 keV). From MAXI/GSC observations, the iron line feature is found to peak, in general, at ≈ 6.5 keV in the out-of-eclipse phase and peak at higher energies ≈ 6.7 keV in the ingress phase. This suggests the presence of both Fe Kα and recombination lines. The high spectral resolution provided by other missions such as ASCA, XMM-Newton or Chandra, proved to be instrumental in resolving these emission lines, if present (^{Ebisawa et al. 1996}; ^{Aftab et al. 2019}; ^{Sanjurjo-Ferrín et al. 2021}). Other sources where this dichotomy has been reported are Vela X - 1 (^{Martínez-Núñez et al. 2014}; ^{Doroshenko et al. 2013}; ^{Malacaria et al. 2016}), 4U 1538 - 52 (^{Rodes-Roca et al. 2011}, ²⁰¹⁵) and GX 301 - 2 (^{Fürst et al. 2011}; ^{Islam & Paul 2014}). Therefore, to describe the Fe complex, values obtained from ASCA observations were adopted and used to fit our orbital phase-resolved spectra (^{Ebisawa et al. 1996}), where the energies and line widths were fixed and the line intensities were left as free parameters in the fit.

Unfortunately, none of the iron emission lines could be detected by MAXI/GSC, neither in the eclipse nor in the egress phases (Figure 16).

All free parameters, line intensities, equivalent widths and line intensity ratio Fe XXVI/Fe XXV (Figures 16, 17 and 18, respectively) were found consistent with constant values within errors out-of-eclipse.

4.5 The Stellar Wind in Cen X-3

Thanks to our orbital phase-resolved spectroscopy, we can study the properties of the stellar wind in Cen X-3 by analysing the variation of the hydrogen column N¹_H along the orbit. As a first step, we have applied a simple spherically stellar wind model to describe our observational data. This simple model reproduced the observed shape of the absorption curve along the entire orbit in the wind-fed HMXB 4U 1538-52 (^{Rodes et al. 2008}) and it is consistent with the stellar wind description of Cen X-3 reported by ^{Wojdowski et al. (2001)}.

The radial flow velocity takes the form (^{Castor et al. 1975}; ^{Abbott 1982}):

where v_∞ is the terminal velocity of the wind, Rc is the radius of the companion star, r is the distance from the centre of the companion star and α is the velocity gradient.

Conservation of mass requires:

where Ṁ_C is the mass loss rate from the primary and n_H is the wind density. Combining equations (6) and (7) and integrating the wind density along the line of sight to the X-ray source, it is possible to find a model which properly describes the variation in N_H with orbital phase. Defining s as the distance through the stellar wind along the line from the compact object toward the observer (see Figure 19), we have:

The angle θ is related to the orbital phase and equation (8) can be rewritten as:

As we have mentioned before, N¹_H represents the ISM plus the environment that surrounds the star. The stellar wind model is described by means of the following equation:

where X_H, the hydrogen mass fraction which is equal to 0.76, m_H is the hydrogen atom mass, Ṁ_C the mass-loss rate, v_∞ is the terminal velocity of the wind in the range [1 000-3 000] km s⁻¹ (^{Falanga et al. 2015}) and r represents the binary separation. At-tempts were made to fit the local absorption N¹_H using equation (10). A smooth wind model could possibly describe the behaviour of our data, although the large uncertainties prevent us from drawing strong conclusions from this result.

The evolution of the absorption column for both models is shown in Figure 20, and it is consistent with a constant value. The variation of the N¹_H columns have been plotted together against the orbital phase in Figure 21. As can be seen from the graph, both values are consistent within errors and the fit has been represented by a dotted line. ^{Suchy et al. (2008)} studied Cen X-3 over two consecutive orbits with observations taken by the RXTE observatory. They found that N_H increased near eclipse-ingress and egress during both orbits. However, during the second orbit, N_H was relatively constant prior to mid-orbit and rose continuously afterward. They also interpreted the small increase in N_H seen in the orbital phase [0.3-0.4] as a bow shock in front of the accretion stream. The two-dimensional numerical simulations performed by ^{Blondin et al. (1991)} provided the variation of the absorbing column of material, N_H, as a function of orbital phase. The dependence on binary separation showed a double-peaked structure, with one peak occurring slightly before orbital phase 0.5 and the second somewhat after (see their Figure 2). On the other hand, the orbital phase dependence of N_H taking the tidal stream and accretion wake into account showed small peaks associated with clumps in the wind and a strong jump at orbital phase 0.6 (see their Figure 8). They also plotted the column density of the undisturbed wind model showing a concave shape below the full simulated models. ^{Blondin et al. (1990)} also showed in their simulations that the column density throughout the orbit changes between consecutive orbits due to variability in the accretion flow. We have not found this behaviour in our long-term MAXI/GSC observations, although a small enhanced in N¹_H could be present in the orbital phase [0.4-0.5] (see Figure 20). However, it cannot be directly probed within the context of our MAXI/GSC analysis, which only allows inferring the long-term properties of the source averaged over several orbital phase bins. To sum up, the models carried out by ^{Blondin et al. (1991)} do not describe well the observed behaviour for Cen X−3.

Fig. 21 Evolution of the absorption column N¹_H (ISM plus local absorption). Red filled squares: from equation (2). Blue open circles: from equation (5). The dotted line represents the fit to a constant. A better fit could be obtained by applying equation (10), but the large uncertainties prevent us from draw by firm conclusions. The color figure can be viewed online.