© The Authors 2024

1. Introduction

High-mass X-ray binaries (HMXBs) are binary systems with a massive main sequence companion star with a mass ≥10 M_⊙ and a compact object, which can be a neutron star (NS), white dwarf, or black hole. The optical companions in HMXBs are massive OB-type stars, which are known to have strong stellar winds driven by line absorption (Lucy & Solomon 1970; Castor et al. 1975). Some of the mass lost in this stellar wind is accreted onto the compact object after being captured by its gravity, thus giving rise to X-ray emission.

The main sequence stars in HMXB systems can have strong stellar winds with a mass-loss rate as high as $\sim {10}^{-7}-{10}^{-6}{M}_{\odot }{\mathrm{yr}}^{-1}$ (Blondin et al. 1990). The matter lost can cause photoelectric absorption of X-ray photons from the NS along our line of sight. Stellar wind models are used to probe the stellar wind from an OB-type star in a binary system with an accreting NS using pointed source observations (Wojdowski et al. 2001; Suchy et al. 2008) or long-term observations with an all-sky monitor (Doroshenko et al. 2013; Islam & Paul 2014; Rodes-Roca et al. 2015; Manikantan et al. 2023).

Centaurus X-3 (Cen X-3) is an HMXB pulsar system with an O6-8 III giant star (V779 Cen) (Krzeminski 1974; Hutchings et al. 1979) as its optical companion. The companion star has a mass of $20.2_{-1.5}^{+1.8}{M}_{\odot }$, while the NS has a mass of $1.{34}_{-0.14}^{+0.16}{M}_{\odot }$ (van der Meer et al. 2007). The NS has a spin period of ∼4.82 s and the orbital period of the system is ∼2.1 days (Nagase 1989).

In this paper, we present a long-term study of Cen X-3 using the Monitor of All-sky X-ray Image (MAXI) / Gas Slit Camera (GSC) all-sky monitor (Matsuoka et al. 2009), which has been in operation for over a decade. Such an instrument is ideal for an orbital-phase-resolved spectral analysis (Torregrosa et al. 2022) of the source. Also, the source exhibits different intensity levels in the MAXI/GSC data. Therefore, we investigated the intensity level variation in Cen X-3 (Raichur & Paul 2008) and the orbital phase dependence of the source spectrum at each intensity level. After this, we fitted the photoelectric absorption observed with a symmetric stellar wind model and inspected the deviation of the measurements from an isotropic wind model at different intensity levels.

Previous observations of several HMXBs (Jackson 1975; Kaper et al. 1994) have shown the presence of additional structures around the compact object. These structures could be accretion wakes, photoionization wakes, or tidal streams (Blondin et al. 1990, 1991; Manousakis et al. 2012). Such structures lead to asymmetry in the orbital phase dependence of the absorption column density. In another study, long-term intensity variations were proposed to be caused by the accretion disc obscuring the compact object (Raichur & Paul 2008). For this work, we conducted orbital-phase-resolved spectral analysis for multiple intensity levels to thoroughly analyse differences in absorption as a function of the orbital phase for each intensity level.

In Section 3, we present the timing analysis we carried out, for which we divided the orbit into different regions after calculating the variation in hardness ratio (HR) from the soft and hard energy bands. In Section 4, we provide our orbital-phase-averaged and orbital-phase-resolved spectral analysis performed on both the intensity-averaged and intensity-resolved data. We then fitted a symmetric wind model for both the intensity-averaged and intensity-resolved data (Sect. 5) to investigate the variation in absorption for different intensity levels. A toy wake model was then used to understand the orbital phase dependence of absorption found for the lowest intensity level in Cen X-3.

2. Observation and data

Data from MAXI (Matsuoka et al. 2009), which has observed the whole sky every day for a period of approximately 14 years, were used in this study. MAXI is installed on the Japanese Experimental Module Exposed Facility (Kibo-EF) on board the International Space Station (ISS). As the ISS orbits the Earth, completing an orbit every ∼90 minutes, MAXI surveys the sky. Such observations allow for the study of the long-term variability of X-ray sources, which can be observed due to accretion disk instability, varying mass accretion rate, absorption, etc. The light curves for this study are binned to 90 minutes.

The MAXI instrument has two kinds of X-ray cameras, the Gas Slit Camera (GSC) (Mihara et al. 2011) and the Solid-state Slit Camera (SSC) (Tomida et al. 2011). The GSC instrument is a large-area position-sensitive proportional counter with a total detection area of 5350 cm², operating in the energy range of 2−30 keV. SSC has a detection area of 200 cm² and operates in an energy range of 0.5−12 keV. For this study, we used only the GSC instrument in the energy range of 2−20 keV. SSC was not used because of its low detection area and lower available exposure, which lead to a very low overall count rate, even for a bright source such as Cen X-3.

The GSC light curve and spectrum were obtained from the MAXI on-demand process website¹. We used GSC data ranging from MJD 55075 to MJD 60034, a total duration of ∼13.5 years. The effective exposure time of Cen X-3 is approximately 11.39 Ms, or ∼132 days. The light curves were obtained with a time resolution of 90 minutes. Orbital-phase-resolved and intensity-resolved data were similarly downloaded using appropriate good time intervals.

3. Timing analysis

The log-term light curve of Cen X-3 is shown in Figure 1, with a bin size of 30 days and presenting count rate modulation over several orbits. Normalized folded light curves of Cen X-3 for different energy bands (2−20 keV, 2−4 keV, 4−8 keV, 8−14 keV, and 14−20 keV) were generated (see Figure 2). The normalization of the folded light curve was performed by dividing the count rates in each phase bin by the average source count rate. The orbital ephemeris for Cen X-3 light curve folding is taken from Klawin et al. (2023) (see Table 1). The orbital period calculated independently using efsearch² is 2.08697 ± 0.00002 days for the ∼13.5 years of data (∼2009–2023). This is in good agreement with Raichur & Paul (2010) and Klawin et al. (2023). The orbital ephemeris from Klawin et al. (2023) was used for folding in a further analysis.

Fig. 1. Long-term MAXI/GSC 2−20 keV light curve of Cen X-3 with a time binning of 30 days.

The energy-resolved normalized orbital profiles exhibit visually similar phase variations for all the energy bands considered. But, looking more closely at the orbital profiles for the 2−4 keV (red) and 14−20 keV (green) energy bands in Figure 2, we see that the spectrum becomes harder around an orbital phase ∼0.6 when the softer band (red) goes below the harder band (green).

Fig. 2. Energy-resolved normalized orbital profile in multiple energy bands of 2−20 keV, 2−4 keV, 4−8 keV, 8−14 keV, and 14−20 keV. The orbital ephemeris is taken from Klawin et al. (2023).

To quantitatively study the variation in orbital profile across multiple energy bands, and thus the change in the shape of the source spectrum, we used the HR, which is defined as

$$\begin{array}{c}\hfill \mathrm{HR}=\frac{H-S}{H+S},\end{array}$$(1)

where S and H are the source count rates in the soft (2−4 keV) and hard (10−20 keV) energy bands, respectively. Here we chose 10−20 keV for the harder band (instead of 14−20 keV used above) for better statistics, since the count rate from the hard band is closer to the soft band this way. We calculated the variation of the HR across 64 orbital phase bins for an orbit and plotted it in Figure 3. The HR sharply increases before and after the eclipse, indicating an increase in absorption from the stellar wind as the compact object gradually goes behind the companion star. So we defined different regions in orbit based on different values of HR, which may originate from a variable absorption column density.

Fig. 3. Variation of the HR with the orbital phase as calculated from the energy bands, S: 2−4 keV and H: 10−20 keV. The plot is overlaid with the S-band (blue), the H-band (green), and the 2−20 keV (red) folded light curve. The vertical dashed purple lines divide the orbit into four regions: out of eclipse (OOE), eclipse ingress, eclipse egress, and eclipse.

The HR thus serves as a proxy to differentiate between the multiple orbital regions, namely the eclipse ingress when the HR increases sharply, eclipse egress when the HR decreases sharply, out of eclipse (OOE) when the HR stays almost constant for Cen X-3, and eclipse. The eclipse-egress, OOE, and eclipse-ingress regions have orbital phases ranging from ϕ_orb ∼ 0.102 − 0.164, 0.164−0.836, and 0.836−0.898, respectively, based on the variation of the HR with the orbital phase as seen in Figure 3.

4. Spectral analysis

All the spectra from MAXI/GSC were analysed using XSPEC v12.13.0c (Arnaud 1996). The elemental abundances used are from Wilms et al. (2000), and the photoionization absorption cross-sections are taken from Verner et al. (1996).

4.1. Intensity-averaged and orbital-phase-averaged spectrum

The long-term orbital-phase-averaged 2−20 keV MAXI/GSC spectrum of Cen X-3 is fitted with a partially absorbed power law with a high-energy cutoff continuum model and a Gaussian feature for the iron Kα fluorescent emission line. We used the spectral model TBabs*pcfabs*(powerlaw*highecut + gaussian). Cen X-3 has a soft excess, similar to many X-ray pulsars, which is usually fitted with a blackbody component. But the phenomenological spectral model used in this paper does not require an additional blackbody component; the reason for this is elaborated on in the discussion below.

The absorption in X-ray has two components, one due to the circumstellar matter around the binary system and the other due to the Galactic interstellar medium (ISM). The local absorption due to circumstellar matter from stellar wind was modelled with pcfabs, a partial covering absorption model. The absorption due to the Galactic ISM was modelled with TBabs, and the value of Tbabs: nH, the equivalent hydrogen column density, was fixed to the Galactic line-of-sight value of 1.11 × 10²² atoms cm⁻² (HI4PI Collaboration 2016). The best-fit values of the spectral parameters for the model used are given in Table 2 for the intensity-averaged and orbital-phase-averaged spectrum. The best-fit spectrum is plotted in Figure 4, along with the residuals to the best-fit model. The 2−20 keV flux for the phase-averaged spectrum is 2.89 ± 0.01 × 10⁻⁹ ergs s⁻¹ cm⁻².

Fig. 4. Long-term intensity-averaged and orbital-phase-averaged 2−20 keV spectrum of Cen X-3 for the ∼13.5 years of MAXI/GSC data. Top panel: 2−20 keV MAXI/GSC intensity-averaged and orbital-phase-averaged spectrum and the best-fit spectral model of Cen X-3. Bottom panel: Residual of the best-fit spectral model.

4.2. Intensity-averaged and orbital-phase-resolved spectroscopy

The binary orbit of Cen X-3 is divided into four regions, namely eclipse-egress, OOE, eclipse-ingress, and eclipse, based on the HR variation. The orbital phase gives the relative positions of the NS and the companion star with respect to our line of sight.

Further, the OOE region is divided into ten phase bins, while the eclipse-ingress and eclipse-egress regions are divided into two and three phase bins, respectively. The counts of the new phase bins for each orbital phase range were kept as uniform as possible to maintain similar statistics.

Orbital-phase-resolved spectroscopy was performed with the same spectral model used for the orbital-phase-averaged spectrum on the spectra extracted for the different orbital phase bins. For the spectral fitting in the orbital-phase-resolved spectra, some parameters were frozen to the best-fit values of the spectral model in Table 2. The spectral parameters that were frozen are TBabs, nH; powerlaw,Γ, highecut,E_cutoff; highecut,E_fold, gaussian,E_Fe; and gaussian,σ_Fe.

Cen X-3 has a highly circular orbit with eccentricity < 1.6 × 10⁻³ (Bildsten et al. 1997; Raichur & Paul 2010); therefore, the emission from the NS is likely to be independent of the orbital phase, as matter density in the circumbinary region used for accretion does not change due to a constant orbital separation between the NS and the optical companion. The accretion rate may still change within or across orbit(s), depending on the behaviour of the circumbinary medium or local conditions near the NS, but it will not have any clear orbital phase dependence in the long term. So only photoelectric absorption of the emission from the NS may change and show an orbital phase dependence. Thus, the continuum parameter Γ is frozen, as seen above, to the phase-averaged spectral fit value for orbital-phase-resolved spectroscopy. After the spectral fitting of each phase bin spectrum, we can see the evolution of the free parameters in the binary orbit in Figure 5.

Fig. 5. Variations of free spectral parameters plotted from the intensity-averaged orbital-phase-resolved spectral analysis of the 2−20 keV MAXI/GSC data. The 2−20 keV unabsorbed flux for the spectral model used is calculated and plotted at the bottom. The errors are quoted at a 1σ confidence level. The shaded region indicates the eclipse phase of Cen X-3.

4.3. Intensity-resolved analysis

The orbit-averaged out-of-eclipse count rate from Cen X-3 shows significant orbit-to-orbit variation. From the normalized folded light curve in Figure 2, we obtained the orbital phase range for the OOE region, which is ϕ_orb ∼ 0.164 − 0.836. Good time intervals for the OOE region were generated according to the orbital ephemeris given in Table 1 and used to generate a 2−20 keV MAXI/GSC light curve of Cen X-3 from MJD 55075 to MJD 60034. Then the average count rate for each orbit during the OOE phase was calculated. The variation in this orbit-averaged OOE count rate shows no obvious long-term trend. A histogram for the total number of orbits against the average OOE count rate for each orbit is seen in Figure 6. For this histogram, we redistributed the ∼13.5 years of MAXI/GSC data for Cen X-3 into three intensity levels, as shown with the vertical dashed lines in Figure 6, where the low, medium, and high intensity levels have 1477, 467, and 279 orbits, respectively. This was done to keep the total photon counts similar across all intensity levels in order to maintain similar statistics.

Fig. 6. Histogram of total number of orbits vs average out-of-eclipse count rate per orbit, where we divide all the orbits into three intensity levels.

4.4. Intensity-resolved and orbital-phase-averaged spectral analysis

The normalized orbital profile for each intensity level is plotted in Figure 7. The normalized orbital profile for the three intensity levels is significantly different. For the lowest intensity level, the profile is more asymmetric with respect to orbital phase 0.5 compared to the medium and high intensity levels. There is a steep decrease in source count rate beyond the orbital phase ϕ_orb ∼ 0.5 for the lowest intensity level. The orbital-phase-averaged spectrum for each intensity level was fitted with the same spectral model used in Section 4.1 for the overall spectra. The best-fit parameters for the spectral model for all intensity levels are shown in Table 2. The local absorption is smaller for higher intensity levels.

Fig. 7. Normalized orbital profile for all three intensity levels.

Fig. 8. Intensity-resolved and orbital-phase-averaged 2−20 keV spectrum of Cen X-3 for the ∼13.5 years of MAXI/GSC data. Top panel: 2−20 keV MAXI/GSC orbital-phase-averaged spectrum for all three intensity levels of Cen X-3 with their best-fit spectral model. Bottom panel: Residual of the best-fit spectral model for all intensity levels.

4.5. Intensity-resolved and orbital-phase-resolved spectroscopy

For each intensity level, the orbital profile was divided into two eclipse-egress, ten OOE, and two eclipse-ingress phase bins. The spectrum for each phase bin was then fitted with the best-fit spectral model for the intensity-resolved and the orbital-phase-averaged spectrum for a given intensity level. Just as in Section 4.2, we fixed the absorption due to Galactic ISM (TBabs: nH), the photon index (Γ), the high-energy cutoff (E_cutoff and E_fold), the Gaussian line centre (E_Fe), and the line width (σ_Fe) for the Fe K_α fluorescent emission line. The orbital variation in the free spectral parameter for each intensity level is plotted in Figure 9.

Fig. 9. Intensity-resolved and orbital-phase-resolved spectroscopy for all three intensity levels plotted in red, green, and blue, respectively. The 2−20 keV unabsorbed flux for the spectral model used is calculated and plotted at the bottom. The errors are quoted at a 1σ confidence level. The shaded region indicates the eclipse phase of Cen X-3.

The local photoelectric absorption increases significantly after ϕ_orb ∼ 0.5 for the lowest intensity level. Even though there is some excess absorption present at the high and medium intensity levels with respect to a symmetric wind model, there exists a significant deviation in absorption column density for the lowest intensity level, indicating a significant increase in absorbing matter along the line of sight for the source.

5. Wind modelling

5.1. Intensity-averaged wind modelling

The variation in local absorption (pcfabs: nH) observed in Figure 5 is not a symmetric function of ϕ_orb. The local absorption would have been a symmetric function of ϕ_orb for spherically symmetric stellar wind. There is excess absorption after ϕ_orb ∼ 0.5, which may originate from an asymmetric structure present in the binary environment of Cen X-3. Such a feature after ϕ_orb = 0.5 is attributed to the presence of matter that trails behind the NS or is disrupted by it (Blondin et al. 1990; Doroshenko et al. 2013). Also, the photoelectric absorption is larger at pre-eclipse than at post-eclipse.

The variation in photoelectric absorption with orbital phase in HMXBs is usually modelled with a spherically symmetric, radiatively driven wind model (Suchy et al. 2008; Doroshenko et al. 2013; Rodes-Roca et al. 2015). The wind model used is from Castor et al. (1975) (CAK75). The CAK75 velocity profile of the stellar wind is given by

$$\begin{array}{c}\hfill v(r)=v_{\infty }{(1-\frac{R_{\star }}{r})}^{\beta },\end{array}$$(2)

where v_∞ is the terminal velocity of the wind, R_⋆ is the radius of the companion star, and β is the velocity gradient parameter. The mass-loss rate due to stellar wind from the companion star is $\dot{m}$, and the radial mass density profile of the stellar wind is given by

$$\begin{array}{c}\hfill \rho (r)=\frac{\dot{m}}{4\pi r^{2}v(r)}=\left(\frac{\dot{m}}{v_{\infty }}\right)\frac{1}{4\pi r^2}\frac{1}{{(1-R_{\star }/r)}^{\beta }}.\end{array}$$(3)

The hydrogen mass density is ρ_H(r) = x_Hρ(r), where x_H is the hydrogen mass fraction for abundance taken from Wilms et al. (2000) during the spectral analysis above. Finally, the equivalent hydrogen number column density, N_H, was calculated by integrating ρ_H(r) from the position of the NS to the observer along the line of sight as

$$\begin{array}{c}\hfill N_{\mathrm{H}}(\varphi \prime )=\kappa {\displaystyle {\int }_{x_{\varphi \prime }}^{\infty }}{\rho }_{\mathrm{H}}(r(x\prime ,y_{\varphi \prime },z_{\varphi \prime }))\mathrm{d}x\prime ,\end{array}$$(4)

where κ is the conversion factor from the mass to hydrogen particle number, the x-axis is parallel to the line of sight, and r(x_ϕ′, y_ϕ′, z_ϕ′) is the radial distance of the NS from the centre of the companion star at orbital phase ϕ′. The radius of the companion star in Cen X-3 is taken as R_⋆ = 12.1 ± 0.5 R_⊙, the orbital radius as $r_d=19.1_{-0.5}^{+0.6}{R}_{\odot }$ (van der Meer et al. 2007), and the orbital inclination angle as i = 79 ± 3° (Sanjurjo-Ferrín et al. 2021).

The wind model parameter β was fixed to 0.8 (Friend & Abbott 1986) before we fitted the model and calculated the ratio as $\dot{m}({10}^{-6}{M}_{\odot }{\mathrm{yr}}^{-1})/v_{\infty }(1000{\mathrm{km}\mathrm{s}}^{-1})=0.33\pm 0.04$. It is clear in Figure 10 that the spherically symmetric wind model is not adequate for explaining the asymmetry found with respect to ϕ_orb = 0.5. The photoelectric absorption due to local matter starts rising earlier, after ϕ_orb = 0.5, and deviates from a spherically symmetric wind model, as clearly seen from the ratio variation in the bottom panel of Figure 10. Such a rise in absorption could be due to matter trailing behind the NS, called a wake (Blondin et al. 1990; Manousakis et al. 2012).

Fig. 10. Local absorption evolution with the orbital phase. Top panel: Variation in local absorption across the orbit overlaid with the best-fit spherically symmetric wind model for the intensity-averaged data. Bottom panel: Ratio between the observed absorption and the best-fit model.

5.2. Wind model for different intensity levels

For this section, we checked if the orbital phase dependence of photoelectric absorption at each intensity level follows a similar trend as that seen in the intensity average case. We used the CAK75 wind model for all intensity levels. With β = 0.8 (Friend & Abbott 1986) fixed, we get the best-fit values of ṁ/v_∞ for all intensity levels. We determined that the ratio $\dot{m}({10}^{-6}{M}_{\odot }{\mathrm{yr}}^{-1})/v_{\infty }(1000{\mathrm{km}\mathrm{s}}^{-1})$ is 0.26 ± 0.06, 0.33 ± 0.05, and 0.45 ± 0.06 for the low, medium, and high intensity levels, respectively. The variation in photoelectric absorption for both the data and the best-fit wind model is plotted in Figure 11 for each intensity level.

Fig. 11. Intensity resolved local absorption evolution with orbital phase. Top panel: variation in local absorption with orbital phase for the observed data and the best-fit curve for a spherically symmetric wind model for each intensity level. Bottom panel: variation in the ratio of data and best-fit model curve for each intensity level.

5.3. Wake toy model for the low intensity level

The presence of a wake structure trailing the NS at the low intensity level is evident from the increased absorption after ϕ_orb ∼ 0.5. We used a toy model for the accretion wake in Cen X-3 (Jackson 1975; Suchy et al. 2008), using a hemisphere for the bow shock, followed by a wake that extends ∼40 R_⊙ (see Figure 12). This toy model has a density of ∼3 × 10⁻¹³  g cm⁻³ on the boundary and decreases to null towards the inside of the wake and bow shock as a cosine, though accretion wakes should have thin and dense boundaries (Fransson & Fabian 1980). Such an assumption works for this demonstration. Along with this, a photoionization wake (Fransson & Fabian 1980; Blondin et al. 1990; Manousakis & Walter 2011) trailing the NS or a tidal stream (Blondin et al. 1991) between the optical star and NS can explain the late-orbital-phase absorption seen in a binary system. A sphere of uniform density ∼3 × 10⁻¹²  g cm⁻³, an order higher than the accretion wake and lagging behind the NS in Figure 12, was used in place of a photoionization wake or tidal stream. This explains qualitatively the photoelectric absorption observed during the late orbital phase at the low intensity level of Cen X-3, which can be caused by a photoionization wake or tidal stream that lags behind the NS (Fransson & Fabian 1980; Blondin et al. 1991). The absorption due to stellar wind and an accretion wake gets an initial bump between ϕ_orb ∼ 0.5 − 0.8, followed by higher absorption before eclipse due to a photoionization wake or tidal stream that results in dense matter lagging behind the NS (see Figure 13). We note that this is mostly for illustrative purposes, to understand qualitatively the late-orbital-phase absorption at the low intensity level of Cen X-3.

Fig. 12. Illustration showing the top view of Cen X-3 with the optical companion in green at the centre along with the dashed line for the orbit of the NS. The structure in red is the accretion wake with a blue tip for the bow shock. The smaller sphere between the dashed line and optical companion is dense matter lagging behind the NS, as a substitute for a photoionization wake or tidal stream.

Fig. 13. Variation in the observed local absorption with an orbital phase for the low intensity level. Also, absorption due to a stellar wind and other structures, such as an accretion wake, photoionization wake, or tidal stream, calculated for a toy model, is overlaid.

6. Discussion

We conducted a long-term study of Cen X-3 and probed the variation in spectra across the orbit to better understand the X-ray absorption characteristics of Cen X-3.

We used a phenomenological spectral model for our spectral analysis, which is different from the ∼3 keV blackbody component used along with the continuum in other reports (Sanjurjo-Ferrín et al. 2021; Torregrosa et al. 2022). However, the broadband spectrum of Cen X-3 observed with the BeppoSAX (Burderi et al. 2000), XMM-Newton (Devasia et al. 2010), and Suzaku (Tomar et al. 2020) observatories, covering a lower energy range than MAXI/GSC, has a blackbody temperature of ∼0.1 keV and did not require a ∼3 keV blackbody component. A low-temperature blackbody component, such as those at ∼0.1 keV seen in many other X-ray pulsars, would not have significant emission in the 2−20 keV energy band of MAXI/GSC used in this work.

Instead, we used a partially absorbed power-law continuum with a high-energy cutoff and a Gaussian feature for the strong iron K-alpha fluorescent emission line. For this spectral model, we clearly observed a variation in hydrogen column density, and thus photoelectric absorption, across the binary orbit. Such a variation in absorption helps us understand the characteristics of the absorbing matter in the circumbinary environment of the Cen X-3 binary system.

The variation of photoelectric absorption and the evidence of a wake or matter trailing the NS is clear for the intensity-averaged and orbital-phase-resolved spectra from Figure 10. We used a simple spherically symmetric wind model to fit the variation in observed photoelectric absorption. Though such a model is not suited for modelling the asymmetry in absorption, we can see how far the data deviate from the model. A clear deviation of the observed absorption away from the symmetric wind model is seen in the evolution of the ratio in the bottom panel of Figure 10. However, we cannot tell if such a feature is maintained for all the orbits in the long-term data of Cen X-3. So, all the orbits of Cen X-3 were divided into three intensity levels to conduct a more thorough analysis of the asymmetry present in the photoelectric absorption around the binary orbit.

An orbital-phase-resolved spectral analysis was carried out for each intensity level. Clearly, an early rise after ϕ_orb ∼ 0.5 is observed, along with very high photoelectric absorption before eclipse for the lowest intensity levels compared to higher intensity levels (see Figure 9). When we used a symmetric wind model as in the intensity average case, the deviation between the wind model and data for each intensity level was clearly not similar (see Figure 11). The photoelectric absorption at the highest intensity level is the most symmetric with respect to ϕ_orb ∼ 0.5, as seen in the ratio in Figure 11. Meanwhile, the asymmetry in absorption with respect to ϕ_orb ∼ 0.5 seems to be more prevalent at the lowest intensity level. This suggests that the presence of a wake structure that trails behind the NS is expected for the lowest intensity level of Cen X-3.

A greater value of L_X/Ṁ_w is expected to lead to a situation where the wind is highly ionized, thus forming a photoionization wake that increases absorption after ϕ_orb ∼ 0.5 until the eclipse (Blondin et al. 1990, 1991), where L_X is the X-ray luminosity and Ṁ_w is the stellar wind mass-loss rate. Assuming that Ṁ_w is similar across all intensity levels and the late-orbital-phase absorption at the low intensity level of Cen X-3 is due to a photoionization wake, that would mean that a photoionization wake forms for the low intensity level with the lowest L_X or L_X/Ṁ_w value. This is in contrast with Blondin et al. (1990), as the high intensity level in Cen X-3 shows the most symmetric absorption with respect to mid-phase, suggesting that a photoionization wake is not as prominent at the high intensity level of Cen X-3.

A structure such as a tidal stream was described by Blondin et al. (1991) to be steady in time and produce absorption after ϕ_orb ∼ 0.5 until the eclipse. They also implied that a strong absorption for orbital phase ϕ_orb ∼ 0.6 − 0.7 is expected for a significant tidal stream in their model. But we lack such strong absorption for that orbital phase for all intensity levels for Cen X-3 (see Figure 11). A Suzaku observation of Cen X-3 at a low intensity level (Naik et al. 2011) has shown extended dips in the X-ray light curve before the eclipse, which indicates X-ray absorption due to dense matter along our line of sight. Dedicated observations of Cen X-3 at different intensity levels with pointed instruments can shed further light on the wake formation in this intriguing binary system.

Acknowledgments

This research has made use of MAXI data provided by RIKEN, JAXA and the MAXI team. We thank the referee for the valuable comments which helped improve the quality of this paper.

References

All Tables

All Figures

	Fig. 1. Long-term MAXI/GSC 2−20 keV light curve of Cen X-3 with a time binning of 30 days.
In the text

	Fig. 2. Energy-resolved normalized orbital profile in multiple energy bands of 2−20 keV, 2−4 keV, 4−8 keV, 8−14 keV, and 14−20 keV. The orbital ephemeris is taken from Klawin et al. (2023).
In the text

	Fig. 3. Variation of the HR with the orbital phase as calculated from the energy bands, S: 2−4 keV and H: 10−20 keV. The plot is overlaid with the S-band (blue), the H-band (green), and the 2−20 keV (red) folded light curve. The vertical dashed purple lines divide the orbit into four regions: out of eclipse (OOE), eclipse ingress, eclipse egress, and eclipse.
In the text

	Fig. 4. Long-term intensity-averaged and orbital-phase-averaged 2−20 keV spectrum of Cen X-3 for the ∼13.5 years of MAXI/GSC data. Top panel: 2−20 keV MAXI/GSC intensity-averaged and orbital-phase-averaged spectrum and the best-fit spectral model of Cen X-3. Bottom panel: Residual of the best-fit spectral model.
In the text

	Fig. 5. Variations of free spectral parameters plotted from the intensity-averaged orbital-phase-resolved spectral analysis of the 2−20 keV MAXI/GSC data. The 2−20 keV unabsorbed flux for the spectral model used is calculated and plotted at the bottom. The errors are quoted at a 1σ confidence level. The shaded region indicates the eclipse phase of Cen X-3.
In the text

	Fig. 6. Histogram of total number of orbits vs average out-of-eclipse count rate per orbit, where we divide all the orbits into three intensity levels.
In the text

	Fig. 7. Normalized orbital profile for all three intensity levels.
In the text

	Fig. 8. Intensity-resolved and orbital-phase-averaged 2−20 keV spectrum of Cen X-3 for the ∼13.5 years of MAXI/GSC data. Top panel: 2−20 keV MAXI/GSC orbital-phase-averaged spectrum for all three intensity levels of Cen X-3 with their best-fit spectral model. Bottom panel: Residual of the best-fit spectral model for all intensity levels.
In the text

	Fig. 9. Intensity-resolved and orbital-phase-resolved spectroscopy for all three intensity levels plotted in red, green, and blue, respectively. The 2−20 keV unabsorbed flux for the spectral model used is calculated and plotted at the bottom. The errors are quoted at a 1σ confidence level. The shaded region indicates the eclipse phase of Cen X-3.
In the text

	Fig. 10. Local absorption evolution with the orbital phase. Top panel: Variation in local absorption across the orbit overlaid with the best-fit spherically symmetric wind model for the intensity-averaged data. Bottom panel: Ratio between the observed absorption and the best-fit model.
In the text

	Fig. 11. Intensity resolved local absorption evolution with orbital phase. Top panel: variation in local absorption with orbital phase for the observed data and the best-fit curve for a spherically symmetric wind model for each intensity level. Bottom panel: variation in the ratio of data and best-fit model curve for each intensity level.
In the text

	Fig. 12. Illustration showing the top view of Cen X-3 with the optical companion in green at the centre along with the dashed line for the orbit of the NS. The structure in red is the accretion wake with a blue tip for the bow shock. The smaller sphere between the dashed line and optical companion is dense matter lagging behind the NS, as a substitute for a photoionization wake or tidal stream.
In the text

	Fig. 13. Variation in the observed local absorption with an orbital phase for the low intensity level. Also, absorption due to a stellar wind and other structures, such as an accretion wake, photoionization wake, or tidal stream, calculated for a toy model, is overlaid.
In the text