AB Dor: Coronal Imaging and Activity Cycles

The variable nature of light curves of late-type active stars can be observed over the entire electromagnetic spectrum with a variability timescale ranging from a few minutes to a few decades. Both short-term variability (STV) and long-term variability (LTV) are found to present in solar-type stars and are due to the different manifestations of magnetic activities. STVs last from a few minutes to a few days and are generally attributed to flaring activity and rotational modulation due to inhomogeneities. LTVs last from a few years to a few decades and are linked to the stellar activity cycles.

The STVs due to flares in X-rays have been studied and modeled in the past for a long time and have helped us to understand the extreme physical condition of solar-type stars (e.g., Haisch et al. 1991; Reale 2007; Pandey & Singh 2008, 2012). Stellar coronae are spatially unresolved; therefore, different techniques have been developed to extract information from the periodic STVs due to the rotational modulations of active regions in the stellar atmosphere. Doppler imaging (Brickhouse et al. 2001), extrapolating the surface magnetic maps (Hussain et al. 2007; Cohen et al. 2010; Johnstone et al. 2010, etc.), and light-curve inversion techniques (Siarkowski 1992; Siarkowski et al. 1996; Drake et al. 2014; Singh & Pandey 2022) are the main techniques to explain such types of STVs in X-rays. These techniques have their own limitations. Doppler imaging of X-ray data requires a high spectral resolution, which is inadequate for most stars due to instrumental and observing limitations. Inferring the coronal structures on the basis of magnetic surface maps requires simultaneous observations in optical and X-ray bands. Light-curve inversion techniques (LCITs) are a mathematically ill-posed problem where 3D information is extracted from 1D time series data. However, these techniques have gained interest with time due to the easy availability of time series data.

The study of LTVs is useful to understand the underlying dynamo mechanisms. The Sun is the only star for which LTVs have been studied in detail, and theoretical models have been constructed to explain them. The LTVs for magnetically active stars have been studied extensively over the past several decades in optical bands (e.g., Messina & Guinan 2002; Lockwood et al. 2007). In a sample from the Mount Wilson HK program (Wilson 1968, 1978), about 60% of the stars have shown periodic and cyclic variations in their chromospheric activity (Baliunas et al. 1995, 1998). Radick et al. (1998) have shown that the photometric cycle is in phase with the chromospheric cycle for older active stars, whereas the opposite relation was found for younger active stars. Hempelmann et al. (1996) observed that stars exhibiting cyclic variations in Ca ii H&K flux tend to display lower levels of X-ray activity compared to those displaying irregular variations in  Ca ii emission. These activity cycles serve as proxies for the stellar dynamo. However, finding the X-ray activity cycle corresponding to these cycles in the stellar corona remains challenging due to their longer cyclic period and limited X-ray data. Only six stars have been reported to have X-ray activity cycles so far. These are 61 Cyg A (Hempelmann et al. 2006; Robrade et al. 2012), HD 81809 (Favata et al. 2008; Orlando et al. 2017), α Cen A and α Cen B (Robrade et al. 2012; Wargelin et al. 2017), ι Horologii (Sanz-Forcada et al. 2019), and Eridani (Coffaro et al. 2020). Apart from a longer activity cycle, the Sun and other Sun-like stars have been found to have an activity cycle whose period is nearly one-third of the longer activity cycle (Berdyugina 2005), which is linked with the phenomenon of periodic switching between two active longitudes, i.e., a flip-flop cycle. Moreover, for the Sun, it has been found that the flip-flop cycle is different in the northern and southern hemispheres, where it was shown that the northern hemisphere flip-flop period is 5% slower than that of the southern hemisphere (Berdyugina & Usoskin 2003).

For the present work, we have taken the ultrafast rotating active star AB Dor A due to the availability of sufficient observations in the X-ray band by the XMM-Newton satellite. AB Dor A is part of a quintuplet stellar system. It is a K0-type star that has recently reached the main sequence and is located at a distance of 14.85 pc (Gaia Collaboration 2021). Being a fast rotator (P_rot ∼ 0.51 day), AB Dor A shows violent magnetic activity with average X-ray luminosity in the range of ∼10³⁰ erg s⁻¹ (Lalitha & Schmitt 2013). In X-rays, the contribution from other components in the AB Dor system is negligible; thus, AB Dor A (hereafter AB Dor) can be regarded as a single X-ray-emitting star.

AB Dor has gained the attention of most X-ray missions because of its higher X-ray flux and location advantage of being distant from the Galactic plane. Since the first detection in X-rays by Pakull (1981), AB Dor is found to show frequent flaring episodes. STVs due to flares have been extensively studied in the past (e.g., Franciosini et al. 2002; Lalitha & Schmitt 2013; Didel et al. 2024, etc.). Moreover, STVs due to rotational modulation have also been reported by several authors in the past (Collier Cameron et al. 1988; Jetsu et al. 1993; Hussain et al. 2007; Hackman et al. 2013, etc.). Kuerster et al. (1997) reported the first LTV X-ray study, where they reported partial rotational modulation with no long-term X-ray activity trends during the 5.5 yr of X-ray observations, but a slight increase in X-ray flux was observed. Based on a longer X-ray dataset, Lalitha & Schmitt (2013) have found a probable activity cycle of about 17 yr, with X-ray amplitude fluctuation substantially smaller than the Sun.

A long-term photometric study revealed different types of activity cycles: one with a period of 5–7 yr in which activity switches between the two active longitudes, i.e., a flip-flop cycle, and another with a period of 19–22 yr, which is similar to the 11 yr solar cycle (Amado et al. 2001; Järvinen et al. 2005).

Our paper is organized as follows: in Section 2, we explain the coronal imaging method. Section 3 deals with the observation, light-curve analysis, and application of the coronal imaging model to the star AB Dor. The long-term X-ray activity of AB Dor is analyzed in Section 4. The results obtained are discussed in Section 5, whereas we conclude our findings in Section 6.

2.1. Method

Lucy (1974) and Withbroe (1975) developed an iterative technique known as the maximum likelihood approach. We call this method the LW method throughout this paper. This imaging technique has been used extensively in medical science and astronomy (e.g., Shepp & Vardi 1982; Siarkowski 1992; Siarkowski et al. 1996; Singh & Pandey 2022, etc.). We have applied the LW method to image the stellar coronae of single active stars. Thus, our LCIT is a method to invert light curves to get information on the geometry of emitting plasma. It is based on the following assumptions.

• (a)  
  the corona is optically thin,
• (b)  
  it rotates rigidly,
• (c)  
  the active region lasts for at least one complete cycle,
• (d)  
  rotational modulation is due to those regions in the corona that are being eclipsed by the photosphere of the star, and
• (e)  
  the distance to the star is much larger than its radius, so the shadows can be cylindrical shaped.

Here, we explain the algorithm for the method in the following different steps.

Step (i). First, we generate a uniform corona of the star from the photosphere to the coronal height (h_cor) with a resolution of a cubical bin of 0.05 × 0.05 × 0.05 ${R}_{\odot }^{3}$ and assign a uniform emission density, f_em(x, y, z), to each cubical bin in the corona.

Step (ii). An occultation matrix ( M ), which is dependent on the angle of inclination, is calculated for each observed phase. It assigns a weight to each cubical bin as 0 for occulted and 1 for visible.

Step (iii). Cubical bins with a constant weight throughout the observations are removed from the solution space as they do not contribute to the rotational modulation of the light curve.

Step (iv). Since the plasma is optically thin, so the total flux observed at any phase (ϕ) is the sum of contributions from all the cubical bins that are visible at the given phase, i.e.,

$$\begin{eqnarray}&&F(\phi )=\displaystyle \sum _{x,y,z}{f}_{\mathrm{em}}(x,y,z){\boldsymbol{M}}(\phi ,x,y,z){dxdydz}.\end{eqnarray} \tag{ 1 }$$

Step (v). The discrepancy between the model and observed light curve is calculated using standard χ² statistics and the emission density of each bin is updated until the reduced χ² converges to ∼1 using the following equation:

$$\begin{eqnarray}&&{f}_{\mathrm{em}}^{n+1}(x,y,z)={f}_{\mathrm{em}}^{n}(x,y,z)\displaystyle \frac{{\sum }_{i}\tfrac{{F}_{o}({\phi }_{i})}{{F}_{c}({\phi }_{i})}{\boldsymbol{M}}({\phi }_{i},x,y,z)}{{\sum }_{i}{\boldsymbol{M}}({\phi }_{i},x,y,z)},\end{eqnarray} \tag{ 2 }$$

where F_c (ϕ_i ) and F_o (ϕ_i ) are the modeled and observed flux at phase ϕ_i .

To optimize the adequate grid size, the model was run on a synthetic light curve (see Section 2.2) using different grid resolutions of 0.3, 0.2, 0.1, 0.05, 0.03, and 0.02. Subsequently, we examined the standard deviation of the residuals and the number of iterations as a function of different grid resolutions. Making the grid size less than 0.05 × 0.05 × 0.05 ${R}_{\odot }^{3}$ does not improve the standard deviation of the residual but requires a greater number of iterations to converge the solution, thus making computation expensive. However, increasing the grid size does not give the appropriate solutions as a large number of outliers were obtained in the modeled coronal images. Therefore, a grid size of 0.05 × 0.05 × 0.05 ${R}_{\odot }^{3}$ was found to be the optimal grid size for the adequate solution.

2.2. Validation of the Method

To validate our model, we initiated the process by generating simulated light curves, assuming a fully transparent corona with two distinct active regions located at longitudes ranging from 60° to 120° and 220° to 300°. In Figures 1(a) and (b), we show an artificial coronal image and the corresponding light curve, respectively. The synthetic light curve was modeled using the above method to construct the coronal image. During each iteration, we computed the conditional probability (_Pi ) of observing counts F_o (ϕ_i ) at a phase ϕ_i . This computation involved utilizing known emissions in each cubic bin denoted as f(b), their volume element db = dxdydz, and the occultation matrix denoted as M(b, ϕ_i ) as

$$\begin{eqnarray}\begin{array}{rcl}{P}_{i}({F}_{o}({\phi }_{i})| f,M({\phi }_{i})) & = & {e}^{-\lambda }\displaystyle \frac{{\lambda }^{{F}_{o}({\phi }_{i})}}{{F}_{o}({\phi }_{i})!};\\ \lambda & = & \displaystyle \sum _{b}f(b)M(b,{\phi }_{i}){db}.\end{array}\end{eqnarray} \tag{ 3 }$$

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The corresponding log-likelihood function is defined as

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}(L) & = & \displaystyle \sum _{i}\mathrm{log}({P}_{i})=\displaystyle \sum _{i}-\lambda +{F}_{o}({\phi }_{i})\mathrm{log}(\lambda )\\ & & -\mathrm{log}({\rm{\Gamma }}({F}_{o}({\phi }_{i})+1)).\end{array}\end{eqnarray} \tag{ 4 }$$

Figure 1(c) displays the coronal image generated through modeling the artificial light curve. Additionally, in Figure 1(b), we have overlaid the simulated light curve for comparison. Figure 1(d) shows ${\rm{\Delta }}\mathrm{log}(L)$ versus the iteration plot, where ${\rm{\Delta }}\mathrm{log}(L)=\mathrm{log}({L}_{n})-\mathrm{log}({L}_{n-1})$ and n is the number of iterations. Here, in Figure 1(d), ${\rm{\Delta }}\mathrm{log}(L)$ shows that the difference in likelihood keeps on decreasing, indicating convergence. Further, ${\rm{\Delta }}{log}(L)\geqslant 0$, which shows that the method increases the likelihood at each successive iteration.

3.1. X-Ray Observations

3.2. Flare Detection and Rotational Modulation

AB Dor showed frequent flaring episodes in almost all the observations. Therefore, it is necessary to remove the flaring events before looking for the rotational modulations. To remove flaring events from X-ray light curves, we have applied the sigma clipping method with the upper sigma clipped to the 2σ value of the mean, while the lower sigma clip is set free. By doing so, large flares are removed automatically from the data, which were then screened visually to remove further flaring events left unchecked by the algorithm above. The flaring regions are shown by red open circles in Appendix A. To quantify the flaring duration, we have calculated the flare duty cycle for each observation, which we define as the ratio of the flare duration to the total observing time of that epoch. AB Dor's average flare duty cycle was found to be 0.57 ± 0.23, where the error in the flare duty cycle is the standard deviation. This indicates that AB Dor remains in a flaring state for 57% ± 23% of the observation time. Further, there seems to be no preferred rotation phase for the occurrence of flares during all the light curves analyzed in this study. During two epochs, 2016 and 2019, AB Dor showed very strong flares where the count rates during flare peaks were found to be 10–35 times higher than the quiescent count rates. A detailed analysis of these flares is carried out in our other paper (Didel et al. 2024).

In order to study the rotational modulation, we removed all the flaring events from the original light curves. The flare-free light curves were then phase-folded using the following ephemeris:

$$\begin{eqnarray*}&&\mathrm{MJD}=44296.075+0.51479E,\end{eqnarray*}$$

where the MJD corresponding to the "0" phase is the phase when the dominating active region (say, spot A) is in the center of the visible disk, and another less active region (say, spot B) was 180° apart from spot A (see Vilhu et al. 1993). The quiescent state for at least one rotational phase was observed only for the nine epochs of observations and is shown in the left panels of Figure 2. These phase-folded light curves clearly show the X-ray rotational modulation for all nine epochs. Other light curves also show the signature of the rotational modulation, but complete phase coverage was not observed.

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3.3. Coronal Imaging of AB Dor

We have compiled a comprehensive dataset of X-ray observations for AB Dor from 1979 to 2022 to analyze the LTV. This dataset was sourced from both X-ray archives and the literature. The Einstein observatory data were taken from the Einstein Slew Survey (Elvis et al. 1992). Additionally, quiescent count rates were obtained from Vilhu et al. (1993) and Franciosini et al. (2002) for Ginga and BeppoSax observations, respectively. We have also included data from the ROSAT HRI observations and discarded those observations that were flagged as potentially variable. The EXOSAT observations were obtained from the HEASARC database. ³ The on-source background-subtracted light curves from EXOSAT were obtained in the 1–8 keV energy band, which were combined using the fmerge task of FTOOLS. The combined light curve was binned to 300 s, and the flare-detection method was applied. Figures A1(ll) and (mm) show the resulting light curves. We have averaged the quiescent state count rates over yearly bins to investigate LTV. The count rates were then calibrated in the energy range of 0.3–10.0 keV using the WebPIMMS ⁴ tool, with a single apec model and an assumed coronal temperature of 0.87 keV (Didel et al. 2024). The resulting fluxes were converted to luminosity using the distance of 14.85 pc. The quiescent fluxes in 0.3–10.0 keV using Suzaku data during the years 2006 and 2007 are taken from Slee et al. (2014). The evolution of quiescent luminosity is summarized in Table B1 in Appendix B and is plotted in Figure 3.

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We have performed Lomb–Scargle periodogram analysis on the long-term quiescent binned data, revealing significant peaks at periods of 3.6 ± 0.1, 4.3 ± 0.1, 5.4 ± 0.2, 9.5 ± 0.2, and 19.2 ± 2.1 yr. The periods of 3.6 and 5.4 yr closely align with the period identified in previous studies from optical data (Amado et al. 2001; Järvinen et al. 2005). The peak at 19.2 yr bears a strong resemblance to the periods of 16.96 and 20 yr found in studies by Lalitha & Schmitt (2013) and Järvinen et al. (2005), respectively. The 4.3 yr peak appears to be the beat period of 3.6 and 19.2 yr, indicating the presence of a long-term activity cycle. Furthermore, the 9.5 yr periodicity seems to be the first harmonic of the long-term activity cycle with a period of 19.2 yr.

We first fit the constant L_X model to the data to further investigate the LTV and computed the χ² as 976. This χ² was then compared with the critical value of χ² of 67.9 for the degrees of freedom of 36 and the confidence limit of 99.9%. The derived χ² was found to be much higher than the critical value of χ², indicating that the L_X is undoubtedly variable. As mentioned above, we then fit the sine curves with all four periods. This fit gives better χ² of 197 than that from the constant model fitting. Excluding the outliers at epochs 1997.53, 2002.73, and 2021.93, we found a much better fit with χ² of 55.3. Adding the period of 19.2 yr does not improve the fit significantly. However, sinusoidal behaviors are unreliable when there are fewer than three cycles, as the available dataset was only for 44 yr. The best-fit curve is overplotted in the upper panel of Figure 3 as a solid black line with gray-shaded regions corresponding to the 1σ confidence level.

We have carried out a detailed analysis of the STVs and LTVs observed in the ultrafast rotating star AB Dor. The X-ray light curves of AB Dor showed a very dynamic nature with at least one flaring episode per epoch and the presence of rotational modulation. We found that AB Dor flares ∼60% of its observing time, which is high among other active stars (e.g., Stelzer et al. 2000; Pandey & Singh 2012).

The X-ray emission from AB Dor is found to be rotationally modulated. The X-ray rotational modulation in AB Dor has been reported many times in the past (e.g., Vilhu et al. 1993; Kuerster et al. 1997; McIvor et al. 2003; Lalitha & Schmitt 2013, etc.). A total of nine epochs of observations show complete rotational modulation, whereas, for other epochs of observations, rotational modulation could not be seen for a complete cycle due to a higher flare duty cycle. In order to analyze rotational modulation in AB Dor, we have developed an LCIT to image stellar coronae of single active fast rotators. The coronal images obtained from the X-ray light-curve modeling show the bimodal distribution of active regions across the longitudes for most of the epochs. Due to the geometrical constraints of the inclination angle of the star, only a part of the latitudes can be modeled. This fact can be seen in Figure 2, where we could see active regions between −30° to +60° of latitude.

We have also carried out a long-term X-ray study of AB Dor using the X-ray data obtained from various X-ray missions from the year 1979 to 2022. The Lomb-Scargle (LS) analysis of the data reveals four periods at 3.6, 4.3, 5.4 and 9.5 yr in the power spectra of the X-ray light curves. In earlier studies, the periods at 3.4 and 5.4 yr were also obtained and explained in terms of a flip-flop cycle. The phenomenon of a flip-flop is also evidenced by the coronal images obtained from the LCIT, as explained in Section 2. For example, the coronal images of the epochs 2017 and 2019 show opposite coronal brightness. However, the limited number of coronal images does not allow us to establish firmly the presence of a flip-flop cycle.

As mentioned above, most of the X-ray light curves suffer from flaring events, and we do not have a complete quiescent state for a single rotation to know the flux of active regions. Further, the unavailability of simultaneous/quasi-simultaneous optical and X-ray images means we cannot compare the active longitudes. Therefore, we have compared the average X-ray count rates of the first half of the folded light curve with that of the next half for all the quiescent data. This fixed hemisphere method is equivalent to comparing the X-ray flux from one hemisphere (say, eastern) to that of the next hemisphere (say, western). The eastern hemisphere is around ±90° (or ±0.25 phase) of spot A, whereas the western hemisphere is around ±90° of spot B. This method may be susceptible to the migration of active regions in and out of the fixed hemispheres.

To explore the impact of migration on the light curves of a star with two active regions in its corona, we simulated synthetic light curves over 20 yr for every 100th rotation, with the brightness of the active regions varying according to a 5.5 yr flip-flop cycle, as found in AB Dor from observations in the optical band. Employing the fixed hemisphere method, which calculates the normalized flux of eastern and western hemispheres for each rotation and a Lomb–Scargle periodogram analysis, we first confirmed that our method accurately recovers the 5.5 yr flip-flop period without migration. We tested various migration rates by introducing synchronous linear migration in the longitude of both active regions. We found that a flip-flop cycle can be detected with synchronous migration of active regions below 0.1 degrees per rotation. However, the method did not recover peaks in the periodogram for migration rates exceeding or equal to 1 degree per rotation. Even at a relatively high migration rate of 0.1 degrees per rotation, our method still captured signatures of the flip-flop cycle. This method gains a significant advantage in the case of AB Dor as active regions steadily migrate throughout a considerable portion of the starspot cycle.

We have taken only those light curves for which at least 25% of the phased light curve is available for each hemisphere. The resulting evolution of X-ray emission from both the eastern and western hemispheres is depicted in Figure 4(a). Here, the error in each data point is the weighted standard deviation of X-ray counts of that hemisphere. Figure 4(b) shows the plot of count rates from the eastern hemisphere to count rates from the western hemisphere. The count rates from both hemispheres are found to be anticorrelated with each other. The Spearman correlation coefficient between the count rates of the eastern and western hemispheres is found to be −0.78 with a null hypothesis probability of 3 × 10⁻⁷. This suggests that the hemispheres interchange their activity with time.

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We conducted periodogram analysis using the LS method for the fluxes of both the eastern and western hemispheres. The results, depicted in Figure 4(c), reveal prominent peaks at periods of 3.6 ± 0.2 yr and 5.6 ± 0.4 yr. These frequencies are also present in the power spectra of the long-term X-ray light curve. The existence of the 3.4 and 5.4 yr periods in the optical data has been previously attributed to the flip-flop cycle (Järvinen et al. 2005). Thus, the presence of common periodicities of ∼5.5 yr in the optical and X-ray data may also suggest the presence of the flip-flop cycle in X-rays.

Since the first discovery of flip-flops on a G-type giant FK Com (Jetsu et al. 1991; Jetsu et al. 1993), flip-flop cycles have been reported for many stars in single as well as binary systems (RS CVn (Berdyugina & Tuominen 1998; Buccino & Mauas 2009), young solar analogs (Messina & Guinan 2002; Berdyugina & Järvinen 2005; Patel et al. 2013), active M dwarfs (Vida et al. 2010), short-period active binaries (Oláh et al. 2013), W UMa systems (Wang et al. 2015; Luo et al. 2017; Mitnyan et al. 2018), and CABS (Jetsu et al. 2017)). However, no flip-flops have been reported in the X-ray band for any active star in the past.

Previous studies, such as those by Berdyugina et al. (2002), have put forth theoretical models to explain the flip-flop cycles observed in solar analogs and the Sun. They suggest that flip-flop cycles are due to the existence of two distinct magnetic dynamo modes. One of these modes is an oscillating axisymmetric mode that corresponds to sunspot-like cycles, while the other is a nonaxisymmetric mode linked to active longitudes. Moss (2004) and Fluri & Berdyugina (2004) further explored the possible combinations of these modes resulting in flip-flops and active longitudes. Comparisons between the Sun and young dwarf stars reveal that the relative strength of these modes can change over time. Currently, the Sun is primarily dominated by an axisymmetric dipole-like mode, while both modes are prominent in younger active stars. AB Dor shows periodicities at 3.6 and 5.4 yr in long-term X-ray data that were linked to a flip-flop cycle in previous studies, while the signature for the longer period is seen at 19.2 yr and its first harmonic at period 9.5 yr, which suggests that the nonaxisymmetric mode coexists with the axisymmetric mode.

The present study indicates that AB Dor exhibits frequent flare events, accounting for an average of 57% ± 23% of the total observation time occupied by these flares. Analysis of quiescent light curves revealed the presence of rotational modulation in most observational epochs. In addition, coronal imaging of AB Dor has shown the presence of two distinct active longitudes, each situated 180° apart. These active longitudes exhibit migration and variations in their relative brightness, with one active longitude dominating the other. The periodogram analysis of long-term X-ray light curves exhibits multiple periodic signals. Notably, periods of approximately ∼3.6, ∼5.4, and ∼19.2 yr period manifest, resembling findings from previous studies of optical observations. It is plausible that the ∼5.4 yr period is linked to an X-ray flip-flop cycle, whereas the 19.2 yr may correspond to a long-term cycle. The X-ray flip-flop cycle needs to be confirmed using long-term X-ray observations with adequate cadence.

We wish to extend our sincere appreciation to the designated referee and scientific editor of this paper for their insightful comments and valuable suggestions. The EXOSAT, ROSAT, and XMM-Newton data have been retrieved from the HEASARC archive database. The majority of this work is based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. XMM-Newton data are reduced using Science Analysis System (SAS), available at https://www.cosmos.esa.int/web/xmm-newton/sas-download. HEASOFT used for light-curve analysis is available at https://heasarc.gsfc.nasa.gov/lheasoft/download.html.

Figure A1 shows the X-ray light curves of the AB Dor as observed from the XMM-Newton RGS2 instrument. The blue and red open circles show the X-ray count rate for the quiescent and flaring states. The shaded blue regions in each subfigure show the mean quiescent count rate.

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Table B1 shows the yearly averaged quiescent X-ray count rate and luminosity from 1979 to 2022. The data has been tabulated based on the analysis of data from the current study and previous studies.