Issue |
A&A
Volume 690, October 2024
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Article Number | A279 | |
Number of page(s) | 12 | |
Section | Astrophysical processes | |
DOI | https://doi.org/10.1051/0004-6361/202450012 | |
Published online | 17 October 2024 |
The correlation between dip width and peak flux in Cir X-1
1
Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 100049 Beijing, People’s Republic of China
2
Dongguan Neutron Science Center, 1 Zhongziyuan Road, Dongguan 523808, People’s Republic of China
3
School of Astronomy and Space Science and Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, 210023 Nanjing, People’s Republic of China
4
University of Chinese Academy of Sciences, Chinese Academy of Sciences, 100049 Beijing, People’s Republic of China
5
Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, People’s Republic of China
6
Institut für Astronomie und Astrophysik, Kepler Center for Astro and Particle Physics, Eberhard Karls, Universität, Sand 1, D-72076 Tübingen, Germany
Received:
18
March
2024
Accepted:
6
August
2024
In this work we analyzed the long-term X-ray variation of Cir X-1 and its NICER spectra. The dip width and the peak flux after periastron are obtained. Our findings show a negative correlation between the dip width and the peak flux. A new scenario is introduced to explain the correlation. The disk height is higher at a larger radius in the standard model. When the neutron star (NS) moves away from periastron, the contraction of the equipotential surface causes the height of the outer disk to increase significantly. It blocks the X-ray emission near the NS. Subsequently, the height of the outer disk gradually decreases due to the viscosity. The viscous timescale is inversely proportional to the height of the disk, and the height of the outer disk is proportional to the mass transferred from the companion near periastron, thus the peak flux. Consequently, the duration of the obscuring, corresponding to the dip width, is inversely proportional to the peak flux. Moreover, we introduce a new ephemeris MJD(N) = 43075.0 + 16.5843N − 4.778 × 10−5N2 based on the dip-in phases (ingress phase of the dip).
Key words: accretion / accretion disks / binaries: general / stars: individual: Cir X-1
© The Authors 2024
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
Cir X-1 is an unusual X-ray binary (XRB). The presence of type I bursts (Tennant et al. 1986a,b; Linares et al. 2010) confirms its nature as a neutron star (NS) with a weak magnetic field. Nevertheless, different from the confirmation of its compact star, the nature of the companion in Cir X-1 remains uncertain. Cir X-1 exhibits a complete Z track and shares some properties with atoll sources (Oosterbroek et al. 1995; Shirey et al. 1999a; Ding et al. 2003), supporting its classification as a low mass X-ray binary (LMXB), which is also consistent with a weak magnetic field NS. However, some recent studies have offered different views on Cir X-1. Jonker et al. (2007) proposed that its companion is a B5–A0 spectral type supergiant with an orbital eccentricity of 0.45. The studies of Johnston et al. (2016) and Schulz et al. (2020) also support the results. Heinz et al. (2013) discovered the natal supernova remnant of Cir X-1, limiting its age to younger than 4600 yr, making Cir X-1 the youngest known XRBs. These properties suggest that Cir X-1 is a high mass X-ray binary (HMXB). The conflict in classification suggests that Cir X-1 has unique properties that require further investigation.
The X-ray flux of Cir X-1 has been monitored for several decades since its discovery. It has changed dramatically by several orders of magnitude (see, e.g., Fig. 1 in Tominaga et al. 2023). The long-term monitoring provides opportunities to investigate its nature, and the wide flux range allows us to investigate accretion scenarios at different accretion rates. Parkinson et al. (2003) analyzed the long-term X-ray variation of Cir X-1 and derived its orbital evolution. Clarkson et al. (2004) examined X-ray dips present in every orbit and suggested that they are indicators of the periastron passage. Shirey et al. (1999b) and Ding et al. (2006) also investigated the X-ray spectral evolution and concluded that the X-ray dips are ascribed to the heavy absorption.
Currently, there are some mechanisms to explain the formation of the dips in Cir X-1. Johnston et al. (1999) introduced a simple model, where the accretion disk disrupted by the tidal interaction near periastron leads to a decrease in the accretion rate. This model, originally proposed for an eccentric LMXB system, could also be applied to an HMXB system. Asai et al. (2014) analyzed three different possible mechanisms for the formation of dips: the end of the outburst, the propeller effect, and the stripping effect from the stellar wind of the companion. They proposed that the stripping effect from the stellar wind causes a sudden decline in the accretion rate. Tominaga et al. (2023) analyzed the X-ray spectral properties of an orbital cycle through NICER and proposed a model for the formation of dips. The block caused by a bulge on the edge of the accretion disk leads to the formation of dips.
In this work we explore the correlation between the width of X-ray dips and the peak flux after periastron in Cir X-1, and introduce a new mechanism to describe its X-ray variation. Section 2 introduces the instruments and observations used in this work. In Section 3 X-ray light curves are analyzed to obtain the dip width and the peak flux after periastron. Section 4 investigates the relationship between the dips and the absorption using the X-ray spectroscopy. The correlation between the X-ray dip width and the peak flux, and a possible scenario are presented in Section 5. Finally, a brief summary of this paper is provided in Section 6.
2. Instruments and observations
The Rossi X-ray Timing Explorer (RXTE; Bradt et al. 1993) was launched in 1995. Its all-sky monitor (ASM; Levine et al. 1996), containing three Scanning Shadow Cameras (SSCs), monitored Cir X-1 from 1996–2011. The Monitor of All-sky X-ray Image (MAXI; Matsuoka et al. 2009) was launched in 2008, and has been monitoring Cir X-1 for over ten years since 2009. These observatories have monitored the long-term X-ray variations of Cir X-1 for many years, which could provide an insight into the accretion mechanism in Cir X-1. We retrieved the long-term RXTE and MAXI light curves and converted them to fluxes, where the flux in different SSCs of RXTE at the same time is averaged. The selected light curves that span from 1996–2023 are shown in Figure 1. This figure shows that Cir X-1 maintained a high X-ray flux (> 1 Crab) from 1996–2000 and then gradually decreased to a quiescent state (≲0.1 Crab) with occasional outbursts. Recently, it has resumed a higher X-ray luminosity (∼0.5 Crab). To investigate the long-term X-ray variation in each orbit, we folded the light curve with the ephemeris (Nicolson 2007)
where MJD0 and N represent the MJD at periastron and the cycle number, respectively. In this work we define Orbit N as the Nth orbit derived from the ephemeris. We only consider the situation at high flux for the evolution of the folded X-ray light curves. The RXTE ASM light curve was folded every 500 days before MJD 53500, and the MAXI light curve was folded from MJD 59000–60000. The related folded light curves are shown in Figs. 2 and 3. The short and long dips are present in these folded light curves, and are represented by the red and blue horizontal bands, respectively.
Fig. 1. Light curves of Cir X-1 from 1996–2023. The black and red data points are from MAXI and RXTE, respectively. |
Fig. 2. Folded light curves for RXTE ASM. The red and blue horizontal bands represent the short and long dips, respectively. |
Fig. 3. Folded light curves for MAXI. The blue horizontal band represents the long dips. |
To investigate the production mechanism of the dip, the X-ray spectra from the Neutron Star Interior Composition Explorer (NICER; Gendreau et al. 2016) were analyzed. The NICER consists of 56 focal plane modules (FPMs) that cover the 0.2–12 keV energy band and provide a large effective area in the soft X-ray energy band, which helps to constrain the absorption component well. Observations during the pre-dip and dip states (ϕ ∼ 0.75–0.9) were selected, and are listed in Table 1. These observations are divided into five groups based on different orbits.
Observation log for NICER observations.
3. RXTE and MAXI analysis
3.1. Folded X-ray light curves
As shown in Figure 2, Cir X-1 exhibited higher X-ray luminosity during MJD 50000–50500, with a short dip near periastron. The outbursts occurred from phase 0–0.4 and remained relatively steady thereafter. The phase of the short dip was not steady during MJD 50500–51000, varying between phase 0 and 0.2.
During MJD 51000–51500, two dips are observed: a short dip near periastron and a long dip near phase 0.1. The X-ray luminosity increased rapidly after the short dip and entered the long dip quickly. Subsequently, the X-ray luminosity gradually returned to a relatively stable state accompanied by occasional bursts.
After MJD 51500, the short dip disappeared and outbursts occurred following the long dip. It is observed that the duration of the long dip increases as the outburst X-ray peak flux decreases, indicating a correlation between the dip width and the peak flux after periastron. Moreover, Figure 3 shows that the X-ray variation during MJD 59000–60000 has a similar profile to that during MJD 52000–53000. RXTE, together with MAXI, provides long-term monitoring for Cir X-1, with the peak X-ray flux ranging from 0–4 Crab, providing an excellent opportunity to investigate the correlation mentioned above.
3.2. Dip width
The Bayesian blocks technique (Scargle et al. 2013) is a method that could be used to find the optimal segmentation of the observational data and has been applied in numerous studies (e.g., Ponti et al. 2017; Tominaga et al. 2023). It could help retrieve the orbital phase of dip-in and dip-out in every orbit1. First, the dates of the light curves are converted to orbital phases based on the ephemeris. The observational data are then divided into a series of light curves. Each light curve includes the data from the apastron of one orbit to the next. The Bayesian blocks technique implemented by AstroPy (Astropy Collaboration 2018) is applied to the light curve in every orbit to obtain the orbital phases of dip-in and dip-out. The parameters of the routine bayesian_blocks are adjusted to make the positions of dip-in and dip-out distinct in the optimal segmentation of the light curve in every orbit. Figure 4 presents some examples where the vertical lines represent the edges of the optimal segmentation for the observational data. The dip-in and dip-out phases were manually selected according to the X-ray flux and are marked with red vertical lines. For example, the edge is regarded as the dip-in phase if the flux drops to a low level and, conversely, the edge would be considered as the dip-out phase. The flux in the dip state is typically below 0.1–0.2 Crab, whereas it is considerably higher in the non-dip state. Therefore, the dip-in and dip-out phases could be easily identified. Furthermore, the sampling in RXTE and MAXI is not continuous, which introduces uncertainties on the phases of the edges obtained through the Bayesian blocks technique. The half interval of the nearest sampling points on the two sides of the edge are selected as the error associated with the phase, and the center of the two nearest sampling points is re-selected as the dip-in or dip-out phase. Figure 5 illustrates how the dip-in and dip-out phases vary with the orbits. The dip width in every orbit is computed by the difference between the phases of dip-in and dip-out, and is shown in Figure 62. Sometimes two dips are observed in some orbits before Orbit 500, corresponding to the structure of two dips during MJD 51000–51500. The obtained dip-in and dip-out phases, as well as the dip widths, are archived on Zenodo3.
Fig. 4. Some examples of the optimal segmentation in one orbit. The data points represent the light curves of Cir X-1 from RXTE and MAXI. The vertical lines show the edges of the optimal segmentation, where red lines are selected as the phases of dip-in and dip-out. |
Fig. 5. Orbital phases for dip-in and dip-out. The horizontal solid line represents the orbital phase of periastron. The vertical dashed line represents the Orbit 500. Multiple dips are present in some orbits prior to Orbit 500, while only one dip is present after Orbit 500. |
3.3. Peak flux
From Figs. 2 and 3, it can be seen that the highest X-ray flux typically occurs during the phase between 0 and 0.4 in the majority of orbits. Therefore, we define the average flux of the top three fluxes during the orbital phase 0–0.4 as the peak flux in every orbit in this work. Additionally, if the maximum observational gap during the orbital phase 0–0.4 exceeds a phase width of 0.05 in one orbit, its peak flux is considered unreliable and is discarded. Finally, the peak flux of Cir X-1 is obtained for every orbit. The variation in the peak flux over the orbits is shown in Figure 7. As shown in Fig. 7, the variation in the peak flux follows the long-term variation of Cir X-1, indicating the reliability of our peak flux estimate.
Fig. 6. Dip width in every orbit. The vertical dashed line indicates the location of Orbit 500. Multiple dips are present in some orbits before Orbit 500, while only one dip is present after Orbit 500. |
Fig. 7. Peak flux in every orbit, obtained from the average flux of the top three fluxes during the orbital phase 0–0.4. |
4. NICER analysis
4.1. Data reduction
The data processing was carried out using the NICER Data Analysis Software (NICERDAS 2023-08-22_V011a) with the CALDB xti20221001. First, the standard pipeline processing tool nicerl2 was used to process the data with overonly_expr set to 1.0 to mitigate the background influence. Light curves were then generated using nicerl3-lc to determine the state of Cir X-1. The flux decline, corresponding to the transition from the pre-dip to the dip state, is present in the early dip phase and was removed in the following spectral analysis. Finally, the spectra with SCORPEON File as the background and responses were obtained with the routine nicerl3-spec. In our analysis, we excluded FPMs 14 and 34 to avoid possible detector noises.
4.2. Spectral analysis and results
For this work we performed the spectral fitting with Xspec 12.13.1 (Arnaud 1996) and set the abundance table to aspl (Asplund et al. 2009). The distance of Cir X-1 adopted is 9.4 kpc (Heinz et al. 2015). Due to the lack of hard X-ray spectra, a Comptonized blackbody component is difficult to constrain. Therefore, a blackbody component with a disk blackbody component was selected as the model for the X-ray emission in this work. The blackbody emission is from the surface of the NS and/or the boundary layer near the NS, and the disk blackbody emission originates from the accretion disk. To account for the absorption, we chose tbabs and pcfabs as the interstellar and the intrinsic absorption components, respectively. The column density of the interstellar absorption component was fixed at 1.8 × 1022 cm−2 (Heinz et al. 2013). Our model consists of the following parameters: Tbb and Rbb, the temperature of the surface of the NS and/or the boundary layer near the NS and its emission radius; Tin and Rincos1/2θ, the temperature at the inner disk radius and its projected radius on the sky plane; and nH and f, the column density of intrinsic absorption and its covering fraction. Furthermore, in partial observations, especially during the dip state, the X-ray spectra exhibit multiple emission lines, including Si XIV-Kα, S XV-Kα, S XVI-Kα, Ar XVII-Kα, Ar XVIII-Kα, Ca XIX-Kα, Ca XX-Kα, Fe I-Kα, Fe XXV-Kα, Fe XXVI-Kα, Fe XXV-Kβ, Fe XXVI-Kβ, and Fe XXVI-Kγ. To describe these structures, we used Gaussian components in the spectral analysis. Figure 8 shows examples of the best-fit NICER spectra; the main parameters are listed in Table A.1.
Fig. 8. Some examples of the best-fit NICER spectra. |
As shown in Table A.1, the parameters in the absorption component during the pre-dip state are consistent with those during the dip state. It implies that the dip may be not caused by the absorption. Therefore, a decrease in the accretion rate may lead to the formation of the dip. Nevertheless, the possibility that the X-ray emission in the vicinity of the NS is blocked by an additional obscuring medium could not be ruled out. Our observations cover five groups, all of which show similar absorption properties, indicating that the results are robust.
5. Discussion
5.1. Comparison with previous works
The dips have been observed in the X-ray variation of many XRBs, for example (e.g., GX 13+1, D’Aì et al. 2014, and XTE J1710-281, Raman et al. 2018), including Cir X-1. Shirey et al. (1999b) and Ding et al. (2006) investigated the X-ray spectral properties near periastron during MJD 50000–50500. Both results show that the column density of absorption increases significantly to the order of 1024 cm−2 during the dip state, while it is ∼1022 cm−2 during the non-dip state. Tominaga et al. (2023) analyzed the X-ray spectral properties during the dip state after MJD 59000. They consider that a local medium blocks the X-ray emission in the dip state, although the column density remains unconstrained. The spectra from Group 3 in this work were also included in their analysis. Different from our model, Tominaga et al. (2023) used a weak power-law component instead of the blackbody component. The weak power-law component could be disregarded during the pre-dip state. For purposes of comparison, we removed the blackbody component and performed the best fit for the X-ray spectra during the pre-dip state, which gives χ2 = 189, 157, 174, 136, and 172 for the five groups. Furthermore, an F-test was conducted on the two models, giving p-values of 1.3 × 10−28, 9.7 × 10−8, 1.3 × 10−14, 1.5 × 10−9, and 8.0 × 10−10. Thus, the inclusion of an additional blackbody component improves the spectral fit significantly. The results of the spectral analysis in this work are not statistically consistent with theirs.
5.2. Correlation between the dip width and the peak flux
Figure 9 shows the correlation between the dip width and the peak flux after periastron. The dip width decreases as the peak flux increases. A logarithmic or inverse proportion correlation may be present between the dip width and the peak flux. Consequently, we attempted to fit the two correlations separately. In this process, we performed best fits with the weighted least-squares method, where the inverse squares of the errors of the dip widths are regarded as the weights. First, for a logarithmic equation,
Fig. 9. Correlation between the dip width and the peak flux. The blue and red lines represent the best fits of a logarithmic equation and an inverse proportion equation, respectively. |
where fpeak and w represent the peak flux and the dip width, respectively. The fpeak is in units of Crab. The best fit shows A = −0.0989 ± 0.0006 and B = 0.0950 ± 0.0004, which gives R2 = 0.70. Second, for an inverse proportion equation,
The best fit shows A = 0.0693 ± 0.0002, which gives R2 = 0.72. Both best fits could yield satisfactory results.
5.3. Correction for the dip-in phases
Figure 5 shows a negative correlation between the dip-in phase and the orbit after Orbit 500. This correlation indicates that the current ephemeris may require correction. Thus, we first attempt to provide a simple modified ephemeris to correct the correlation:
In this procedure, we fit x to minimize the variance of the corrected phases of dip-in4, where the inverse squares of the errors of the dip widths are regarded as the weights. The best fit shows x = 4.399, which is not consistent with the Nicolson (2007) ephemeris at the 95% confidence level (x = 3.86–4.16). The derived dip-in phase from the corrected ephemeris is presented in Figure 10, and the corrected mean dip-in phase is about 0.07. This implies that the ephemeris needs to be updated. Based on the obtained dip-in phases, we could provide a new ephemeris:
Fig. 10. Derived dip-in phases in every orbit from the simple modified ephemeris. The horizontal dashed line represents the average dip-in phase with the simple modified ephemeris. The vertical dashed lines represent Orbits 500 and 570. |
Here MJD(0), P, and represent the dip-in phase of Orbit 0, the orbital period of Orbit 0, and the orbital period derivative, respectively. In the simple modified ephemeris, we find that some dip-in phases deviate to a great extent from the mean dip-in phase of ∼0.07. The corrected dip-in phase of Orbit 761 is less than −0.1, and the corrected dip-in phases between Orbit 500 and 570 are significantly later than the mean dip-in phase of ∼0.07. With these dip-in phases considered as outliers, we fit the new ephemeris. The best fit shows MJD(0) = 43075.0 ± 0.3, P = 16.5843 ± 0.0008 and , where the errors show a 3σ confidence level. It is clear that the new ephemeris is not consistent with that of Nicolson (2007), which implies that the orbital precession may be present in Cir X-1. The posterior probability distribution for parameters and the dip-in phases under the new ephemeris are presented in Figs. 11 and 12.
Fig. 11. Posterior probability distribution for parameters in the new ephemeris. |
Fig. 12. Derived dip-in phases in every orbit from the new ephemeris. The vertical dashed lines represent Orbits 500 and 570. |
5.4. A possible scenario for Cir X-1
Figures 2 and 3 show that the X-ray variation of every orbit contains one or more dips, followed by an outburst. We introduce a possible scenario for such an X-ray variation in this section.
5.4.1. Mechanism of the formation for dips
Although the classification of Cir X-1 is not currently definitive, we adopt an HMXB classification in this work. It remains unclear whether the formation of the dip after MJD 59000 is ascribed to an obscuring medium, so two different scenarios need to be discussed: the decrease in the accretion rate leads to the dips; the obscuring medium causes the dips.
For simplification, we consider the accretion disk to be composed of numerous test particles. In this case, their motion in binary systems could be determined by the potential. The potential in an eccentric orbit could be described by (Sepinsky et al. 2007; Davis et al. 2013)
where q, e, and ϕ represent the mass ratio of the binary, the eccentricity, and the true anomaly, respectively. The accretor and the donor are located at (1, 0, 0) and at the origin of coordinates, respectively. The quantity f is the ratio of the spin angular velocity of the donor to the orbital angular velocity at periastron. The potential and the coordinates are in units of and D, where D is the separation of the two stars and can be described by
The semimajor axis a can be obtained from Kepler’s third law. We ignore the spin of the donor here, and rewrite the above equation in the frame of the NS reference
where the NS and the companion are located at the origin of coordinates and at (1, 0, 0), respectively.
In the calculation we adopt the following system: the masses of the NS and the companion are 1.4 M⊙ and 10 M⊙; the eccentricity of the binary is 0.45; the orbital period is 16.5 d. In addition, we disregard any mass transfer from the companion here. In polar coordinates the relation between the true anomaly ϕ and the time t follows (Keeton 2014):
In our analysis the viscous dissipation is ignored, and thus the potential is conservative. The discussion is limited on the plane of orbital motion. Figure 13 presents a diagram of a binary system. The NS, marked with a black point, is located at the origin of coordinates. The companion star, marked with a purple point, is located on the positive X-axis. The X coordinate of the companion star varies with the orbital motion of the binary. We select three points, A, B, and C at apastron and record their potentials. The points are located on the negative X-, positive Y-, and positive X-axis, respectively. Their coordinates are (−d, 0), (0, +d), and (+d, 0), where d represents the distance from the selected point to the NS. In this work we only consider two situations, d = 105 km and d = 106 km. We calculate the variations of the distances along the entire orbit with the potentials fixed. Figure 14 shows the variations of the distances for the three points. The X-axis and the Y-axis represent the time and the distance of the three points from the NS, respectively. The NS is located at periastron when t = 0 day and apastron when t = −8.25 and t = 8.25 day. It can be seen that the distances of the three points increase gradually as the NS moves from apastron to periastron. As the viscous dissipation is not considered, the gas in the accretion disk will move far away from the NS due to the conservation of energy. The radial velocities of three points along the entire orbit were obtained by calculating the difference in the distances, and are presented in Figure 15. It can be seen that the maximum radial velocity occurs at t ∼ ±1 day, with values of v ∼ 6–7 m s−1 and v ∼ 700–1000 m s−1 at d = 105 km and d = 106 km, respectively. The accretion disk expands when the NS moves from apastron to periastron. Moreover, the expansion velocity increases with the distance.
Fig. 13. Diagram of the binary system. The NS (black point) is located at the origin of coordinates. The companion (purple point) is located on the positive X-axis. The three selected points A, B, and C are located on the negative X-, positive Y-, and positive X-axis, respectively. The yellow circle represents the accretion disk around the NS. |
Fig. 14. Movement of three selected points throughout the entire orbit with the potential fixed. The left and right panels represent the movement of the selected points at distances of 105 and 106 km from the NS, respectively. |
Fig. 15. Radial velocity of three selected points throughout the entire orbit. The left and right panels represent the radial velocity of the selected points at distances of 105 and 106 km from the NS, respectively. |
In the astrophysical environment, the viscous dissipation cannot be ignored. Therefore, we compare the expanding velocity of the accretion disk with the radial velocity caused by the viscous dissipation. It is assumed that the accretion disk satisfies the standard model (Shakura & Sunyaev 1973), and thus the radial velocity of the accretion disk is (Frank et al. 2002)
where Ṁ, m, R, and R* represent the accretion rate in units of 1016 g s−1, the NS mass in units of 1 M⊙, the radius in units of 1010 cm, and the inner radius of the accretion disk in units of 1010 cm, respectively. The parameter α is related to the viscosity and is commonly less than 1. Given the Eddington accretion rate and α = 0.1, the radial velocity due to the viscosity is ∼150 m s−1 and ∼90 m s−1 at d = 105 km and d = 106 km, respectively. Comparing these values with the expanding velocities, it can be seen that the expanding velocity exceeds the radial velocity due to the viscosity when the distance is greater than several 105 km. Nevertheless, it is hard to cause a decrease in the accretion rate through such a mechanism, due to the viscous timescale tvisc ∼ d/vR = 7.7 d at d = 105 km.
Figure 15 shows that the equipotential surface contracts rapidly near t ∼ 1 d. Moreover, the velocity of contraction increases with distance. In the standard model the scale height of the disk H ∝ R9/8 (Frank et al. 2002). It means that the disk is thicker at the outer disk. Accordingly, the rapid contraction of the equipotential surface could result in an increase in the scale height at a certain radius of the outer disk. Additionally, the higher velocity at the larger radius means that the separation of the equipotential surface decreases, which will cause the gas to accumulate. It may further increase the scale height. The thickened outer disk could block the emission near the NS, resulting in a dip. Subsequently, the contraction of the equipotential surface becomes weak, and the viscosity then dominates the subsequent evolution. It is anticipated that the obscuring interval (dip width) is proportional to the viscous timescale. The viscous timescale , where H and cs are the disk scale height and the sound speed, respectively. The increased amount of gas transferred near periastron leads to a higher peak flux and a thicker outer disk, and thus the height of the outer disk is in proportion to the peak flux H ∼ fpeak. Given that α and cs are fixed, the viscous timescale at a certain radius R. In particular, the dip width is inversely proportional to the peak flux, which is consistent with the correlation discussed in Section 5.2.
5.4.2. Mechanism of the outburst in Cir X-1
Figures 2 and 3 show that the X-ray variation exhibits a rapid rise followed by a slow decline after MJD 52000, which is a typical characteristic of an outburst induced by thermal instability. Such outbursts have been observed in many XRBs and cataclysmic variables (CVs). There are two local critical accretion rates and (Lasota et al. 2008) between which an outburst induced by the thermal instability will occur. The corresponding critical surface densities Σmax and Σmin could be determined by the local critical accretion rates. If the local surface density exceeds Σmax, the thermal instability will be triggered, causing a rapid increase in the temperature of the disk. The increase in temperature leads to a higher viscosity, resulting in a higher accretion rate and eventually an outburst. Additionally, different accretion rates will produce different X-ray variations during the outburst, depending on the location of the initial trigger of instability (see Fig. 5.8 in Hellier 2001). If the accretion rate is relatively low, the surface density at the inner disk will exceed the local critical surface density Σmax initially. A heating wave spreads from the inside to the outside in the accretion disk, and results in a slow increase in luminosity. On the contrary, if the accretion rate is relatively high, the surface density at the outer disk will exceed the local critical surface density Σmax initially. A heating wave spreads from the outside to the inside in the accretion disk, leading to a rapid increase in the luminosity. Subsequently, the gas in the accretion disk flows into the NS. When the outburst ends, a cooling wave spreads from the outside to the inside. Typically, the timescale of the increase is about 0.1 times that of the decrease during the outside-in outburst, while they are similar to each other during the inside-out outburst (Frank et al. 2002). The flux evolution in Cir X-1 after MJD 52000 has, in principle, a profile characterized by a rapid rise and a slow decay, and therefore resembles an outside-in outburst.
5.4.3. The complete scenario of Cir X-1
In this paragraph we present the possible complete scenario of Cir X-1. The gas is transferred from the companion near periastron through the stellar wind and/or Roche-lobe overflow, which cannot flow directly onto the NS due to its extra angular momentum, and instead accumulates on the outside of the disk. In the standard model, the scale height is greater at a larger radius. As the NS moves away from periastron, the rapid contraction of the equipotential surface causes the scale height of the outer disk to increase. Consequently, it blocks the X-ray emission near the NS, leading to a dip. Subsequently, the height on the outer disk gradually decreases due to the viscosity. Next, the gas flows toward the NS, and the surface density at a specific radius of the outer disk exceeds Σmax first, triggering the thermal instability and leading to an outside-in outburst. The outburst results in a rapid increase in the accretion rate, and thus in the luminosity. The timescales of rising and decline also support an outside-in outburst.
This scenario provides a comprehensive description of the X-ray variation, but there are some remarkable discrepancies that require further clarification. The following discussion is based on the new ephemeris. The dip-in phase in every orbit is expected to be relatively stable according to our scenario, but there are some discrepancies prior to Orbit 570 and to Orbit 761. First, we present the X-ray variation for Orbit 761 in Figure 16, which reveals that the flux of Orbit 761 is notably low (≲0.1 Crab). The derived dip-in phase of Orbit 761 may be attributed to the fluctuation in the accretion rate. Second, the dip-in phases between Orbits 500 and 570, corresponding to MJD 51350–52500, are significantly later than those after Orbit 570. From Figs. 2 and 7 we find that the peak fluxes between 500 and 570 are significantly higher, which indicates that a greater quantity of gas is transferred from the companion near periastron. Moreover, the fluxes increase rapidly near periastron, which implies an outburst occurs before the entry into the dip. The outburst would consume some gas on the outside of the disk. The contraction of the equipotential surface causing the obscuring requires that the height of the outer disk reaches a certain level. Therefore, the gas loss on the outside of the disk leads to a delay of the dip-in phase. In addition, the X-ray fluxes exhibited a gradual increase following the dip phases during MJD 51000–51500. This implies that the X-ray emission near the NS is gradually exposed, which is also in agreement with our scenario. Third, the fluxes prior to Orbit 500 are even higher, indicating a greater quantity of gas transferred from the companion. Given such a high accretion rate, it is possible that the entire disk maintains a high viscous state throughout the whole orbit. The accumulated gas on the outside of the disk would dissipate rapidly, leading to a short dip. Owing to a higher mass transfer rate, the dip-in phases are located at ∼0 again in our new ephemeris. For the other dips (ϕ > 0.03 in the new ephemeris) prior to Orbit 500, it may be attributed to an alternative instability mechanism in the disk, which requires further investigation. Finally, the dip-in phases at the low mass transfer rate are located at t ∼ 1 d in this scenario. Therefore, periastron should be located at ϕ ∼ −0.06 in our new ephemeris.
Fig. 16. X-ray variation for Orbit 761. The red vertical line represents the dip-in phase derived from the Bayesian blocks technique. |
5.4.4. Application to other binary systems
Our scenario could be applied to other binary systems, indicating that there may be other binary systems that exhibit a similar X-ray variation throughout their orbit. Nevertheless, few XRBs exhibit X-ray variations similar to that of Cir X-1. To investigate this issue, we attempted to modify the orbital parameters and repeated the analysis steps outlined in Section 5.4.1. When the eccentricity decreases, the mass of the companion decreases, the mass of the NS increases and/or the period increases, and the maximum radial velocity of the equipotential surface will be slower. In addition, we investigated the impact of these parameters by varying one of the orbital parameters while keeping the others fixed. We analyzed the resulting changes in the radial velocity of Point A (d = 105 km) throughout the entire orbit, and present them in Figure 17. The blue and red curves correspond to a 10% decrease in the orbital eccentricity and the companion mass, respectively. The green and purple curves represent a 10% increase in the NS mass and the orbital period, respectively. The formation of the dip is primarily affected by the eccentricity, followed by the orbital period, while changes in the masses of the binary have the least effect. Furthermore, the occurrence of the outburst means that the accretion disk could not be affected by the magnetic field, suggesting the magnetic field of the NS should be weak. The formation of the obscuring requires a sufficiently high orbital inclination. To satisfy these conditions, the binary system must have a higher eccentricity, a weak magnetic field of the NS, a sufficiently high orbital inclination, a companion of higher mass, and a shorter orbital period. In most NS XRBs, the NS commonly has a stronger magnetic field and a higher eccentricity orbit in a younger binary (e.g., most HMXBs), but the opposite is true in an older system (e.g., most LMXBs). Thus, few systems could satisfy a weak magnetic field and a higher eccentricity orbit. In most BH XRBs, the presence of a more massive compact star weakens the effect. A higher eccentricity, a shorter orbital period, and/or a more massive companion are required in this situation. Moreover, a sufficiently high orbital inclination is necessary, which further limits the number of such binaries. Searching for these objects could help us examine our scenario in the future.
Fig. 17. Impact of various orbital parameters on the radial velocity. The black dashed curve represents the radial velocity of A in the orbital parameters mentioned in Section 5.4.1. The blue and red curves respectively represent the orbital eccentricity and the companion mass decreasing by 10%. The green and purple curves respectively represent the NS mass and the orbital period increasing by 10%. |
6. Conclusions
In this work we analyzed the variation of X-ray flux in Cir X-1 from 1996–2023 with the Bayesian blocks technique. The dip width in every orbit and the peak flux after periastron were obtained. We find that the dip width is negatively correlated with the peak flux in Cir X-1. The X-ray spectroscopy was also performed using the NICER spectra during the entry into the dip state. We find that it remains unclear whether the obscuring is responsible for the dips observed after MJD 59000.
A scenario is introduced to explain the negative correlation between the dip width and the peak flux in Cir X-1. In the standard model, the disk height is higher at a larger radius. As the NS moves away from periastron, the contraction of the equipotential surface causes the disk height on the outer disk increasing significantly, which leads to blocking the X-ray emission near the NS. Subsequently, the height on the outer disk gradually decreases due to the viscosity. Next, the thermal instability is triggered, leading to an outburst. The outburst results in a rapid increase in the accretion rate and thus the luminosity. The viscous timescale is inversely proportional to the height of the disk, and the height is proportional to the mass transferred from the companion near periastron, which is related to the peak flux. Consequently, the duration of the obscuring, corresponding to the dip width, is inversely proportional to the peak flux.
In addition, we introduce a new ephemeris MJD(N) = 43075.0 + 16.5843N − 4.778 × 10−5N2 based on the dip-in phases. According to our scenario, periastron is located at ϕ ∼ −0.06 in the new ephemeris.
Data availability
A supplementary table is available at https://zenodo.org/records/13271411
Acknowledgments
We thank the anonymous referee for the helpful comments that greatly improve this work. This research has made use of data and/or software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA/GSFC. This research has made use of the MAXI data provided by RIKEN, JAXA and the MAXI team. This work is supported by the National Key R&D Program of China (2021YFA0718500) and the National Natural Science Foundation of China under grants 12173103, 12333007, U2038101, and U1938103. This work is partially supported by the International Partnership Program of the Chinese Academy of Sciences (grant No. 113111KYSB20190020). L. D. Kong is grateful for the financial support provided by the Sino-German (CSC-DAAD) Postdoc Scholarship Program (91839752).
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Appendix A: Results of spectral fitting
Best-fit parameters for NICER observations
All Tables
All Figures
Fig. 1. Light curves of Cir X-1 from 1996–2023. The black and red data points are from MAXI and RXTE, respectively. |
|
In the text |
Fig. 2. Folded light curves for RXTE ASM. The red and blue horizontal bands represent the short and long dips, respectively. |
|
In the text |
Fig. 3. Folded light curves for MAXI. The blue horizontal band represents the long dips. |
|
In the text |
Fig. 4. Some examples of the optimal segmentation in one orbit. The data points represent the light curves of Cir X-1 from RXTE and MAXI. The vertical lines show the edges of the optimal segmentation, where red lines are selected as the phases of dip-in and dip-out. |
|
In the text |
Fig. 5. Orbital phases for dip-in and dip-out. The horizontal solid line represents the orbital phase of periastron. The vertical dashed line represents the Orbit 500. Multiple dips are present in some orbits prior to Orbit 500, while only one dip is present after Orbit 500. |
|
In the text |
Fig. 6. Dip width in every orbit. The vertical dashed line indicates the location of Orbit 500. Multiple dips are present in some orbits before Orbit 500, while only one dip is present after Orbit 500. |
|
In the text |
Fig. 7. Peak flux in every orbit, obtained from the average flux of the top three fluxes during the orbital phase 0–0.4. |
|
In the text |
Fig. 8. Some examples of the best-fit NICER spectra. |
|
In the text |
Fig. 9. Correlation between the dip width and the peak flux. The blue and red lines represent the best fits of a logarithmic equation and an inverse proportion equation, respectively. |
|
In the text |
Fig. 10. Derived dip-in phases in every orbit from the simple modified ephemeris. The horizontal dashed line represents the average dip-in phase with the simple modified ephemeris. The vertical dashed lines represent Orbits 500 and 570. |
|
In the text |
Fig. 11. Posterior probability distribution for parameters in the new ephemeris. |
|
In the text |
Fig. 12. Derived dip-in phases in every orbit from the new ephemeris. The vertical dashed lines represent Orbits 500 and 570. |
|
In the text |
Fig. 13. Diagram of the binary system. The NS (black point) is located at the origin of coordinates. The companion (purple point) is located on the positive X-axis. The three selected points A, B, and C are located on the negative X-, positive Y-, and positive X-axis, respectively. The yellow circle represents the accretion disk around the NS. |
|
In the text |
Fig. 14. Movement of three selected points throughout the entire orbit with the potential fixed. The left and right panels represent the movement of the selected points at distances of 105 and 106 km from the NS, respectively. |
|
In the text |
Fig. 15. Radial velocity of three selected points throughout the entire orbit. The left and right panels represent the radial velocity of the selected points at distances of 105 and 106 km from the NS, respectively. |
|
In the text |
Fig. 16. X-ray variation for Orbit 761. The red vertical line represents the dip-in phase derived from the Bayesian blocks technique. |
|
In the text |
Fig. 17. Impact of various orbital parameters on the radial velocity. The black dashed curve represents the radial velocity of A in the orbital parameters mentioned in Section 5.4.1. The blue and red curves respectively represent the orbital eccentricity and the companion mass decreasing by 10%. The green and purple curves respectively represent the NS mass and the orbital period increasing by 10%. |
|
In the text |
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