#  AB Dor: Coronal imaging and activity cycles

Gurpreet Singh Aryabhatta Research Institute of Observational Sciences
(ARIES), Manora Peak, Nainital 263001, India  Department of Physics, Deen
Dayal Upadhyaya Gorakhpur University, Gorakhpur 273009, India  J. C. Pandey
Aryabhatta Research Institute of Observational Sciences (ARIES), Manora Peak,
Nainital 263001, India

###### Abstract

Using long-term X-ray observations, we present the short-term and long-term
X-ray variability analysis of the ultra-fast rotating active star AB Dor.
Flaring events are common in X-ray observations of AB Dor and occupy a
substantial portion of the total observation time, averaging around
57$\pm$23%. The flare-free X-ray light curves show rotational modulation,
indicating the presence of highly active regions in its corona. We have
developed a light curve inversion code to image the corona of active fast
rotating stars. The results of coronal imaging reveal the presence of two
active regions of different brightness that are separated by $\sim$180∘in
longitude. These active regions are also found to migrate along the longitude
and also show variation in their brightness. Our analysis of long-term X-ray
data spanning from 1979 to 2022 shows multiple periodicities. The existence of
a $\sim$19.2 yr cycle and its first harmonic indicates the presence of a
Solar-like, long-term pattern. In comparison, the periodicities of $\sim$3.6
and $\sim$5.4 yr are possibly due to the presence of a flip-flop cycle in
X-rays, which is also supported by similar periods findings from the optical
data in the earlier studies. Further confirmation of the existence of the
X-ray flip-flop cycle requires long-term observations at regular intervals in
the quiescent state.

Stellar activity (1580) — Stellar Coronae (305) — X-ray star (1823) — X-ray
astronomy (1810) — Stellar imaging (1145)

##  1 Introduction

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

The STVs due to flares in X-rays have been studied and modelled in the past
for a long and helped 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; Johnstone et al., 2010; Cohen et al., 2010, etc),
and light curve inversion techniques (Siarkowski, 1992; Siarkowski et al.,
1996; Drake et al., 2014; Singh & Pandey, 2022) are main techniques to explain
such type of STVs in X-rays. These techniques have their own limitations.
Doppler imaging of X-ray data requires high spectral resolution, which is
inadequate for most of the 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. The 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 the LTVs are studied in detail, and
theoretical models are constructed to explain it. The LTVs for the
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 of Mt. 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) found that stars with cyclic variations in Ca
ii H&K flux tend to show less X-ray activity than the stars, which show
irregular variability in Ca ii emission. These activity cycles serve as
proxies for the stellar dynamo. However, 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 for which 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), $\alpha$ Cen A and $\alpha$ Cen B (Robrade et
al., 2012; Wargelin et al., 2017), $\iota$ Horologii (Sanz-Forcada et al.,
2019), and $\epsilon$ 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. 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 ultra-fast 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, 2020). AB Dor A being a fast rotator
(P${}_{rot}\sim$0.51 d), shows violent magnetic activity with average X-ray
luminosity in the range of $\sim 10^{30}$ 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. The 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 is also
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 is observed.
Based on a longer X-ray data set, Lalitha & Schmitt (2013) have found a
probable activity cycle of about 17 years, with X-ray amplitude fluctuation
substantially smaller than the Sun.

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

Our paper is organized as follows: In section 2, we explain the coronal
imaging method. The 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 Coronal Imaging

###  2.1 The method

Lucy (1974) and Withbroe (1975) developed an iterative technique that
maximizes the likelihood. We call this method as LW method throughout the
paper. This imaging technique has been used extensively in medical science and
astronomy (for example Siarkowski, 1992; Siarkowski et al., 1996; Shepp &
Vardi, 1982; 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) corona is optically thin,
(b) it rotates rigidly,
(c) active region lasts for at least one complete cycle,
(d) rotational modulation is due to those regions in the corona which are
being eclipsed by the photosphere of the star, and
(e) 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). Firstly, 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$\times$0.05$\times$0.05 $R_{\odot}^{3}$ and assign a uniform
emission density, $f_{em}(x,y,z)$ to each cubical bin in corona.
Step (ii). An occultation matrix (M), which is dependent on the angle of
inclination, is calculated for each observed phase. It assigns 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 ($\phi$) is the sum of contributions from all the cubical bins which
are visible at the given phase, i.e.

$F(\phi)=\sum_{x,y,z}f_{em}(x,y,z)\textbf{M}(\phi,x,y,z)dxdydz$ (1)

Step (v). The discrepancy between the model and observed light curve is
calculated using standard $\chi^{2}$ statistics and updated the emission
density of each bin until the reduced $\chi^{2}$ converges to $\sim$1 using
the following equation

$f_{em}^{n+1}(x,y,z)=f_{em}^{n}(x,y,z)\frac{\sum_{i}\frac{F_{o}(\phi_{i})}{F_{c}(\phi_{i})}\textbf{M}(\phi_{i},x,y,z)}{\sum_{i}\textbf{M}(\phi_{i},x,y,z)}$
(2)

where $F_{c}(\phi_{i})$ and $F_{o}(\phi_{i})$ are modeled and observed flux at
phase $\phi_{i}$.

To optimise the adequate grid size, the model was run on the 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$\times$0.05$\times$0.05
$R_{\odot}^{3}$, does not improve the standard deviation of residual but
requires more 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 modelled
coronal images. Therefore, the grid size of 0.05$\times$0.05$\times$0.05
$R_{\odot}^{3}$ was found to be the optimal grid size for the adequate
solution.

###  2.2 Validation of 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 modelled using the above
method to construct the coronal image. During each iteration, we computed the
conditional probability (Pi) of observing counts $F_{o}(\phi_{i})$ at a phase
$\phi_{i}$. 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 $M(b,\phi_{i})$ as

$\displaystyle
P_{i}(F_{o}(\phi_{i})|f,M(\phi_{i}))=e^{-\lambda}\frac{\lambda^{F_{o}(\phi_{i})}}{F_{o}(\phi_{i})!};$
(3) $\displaystyle\lambda=\sum_{b}f(b)M(b,\phi_{i})db$

The corresponding log-likelihood function is defined as

$\displaystyle
log(L)=\sum_{i}log(P_{i})=\sum_{i}-\lambda+F_{o}(\phi_{i})log(\lambda)$ (4)
$\displaystyle-log(\Gamma(F_{o}(\phi_{i})+1))$

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

Figure 1: (a) Coronal image with injected active regions, (b) Simulated and
modelled light curves: Blue dots represent simulated data, while the black
solid curve depicts the modelled light curve, (c) Coronal image returned by
model, and (d) $\Delta log(L)$vs iteration plot with $\Delta log(L)$=
$log(L_{n})$-$log(L_{n-1})$ and $n$ is the number of iterations.

##  3 Application to AB Dor

###  3.1 X-ray Observations

AB Dor has been observed by many X-ray missions since its discovery as an
active star. It is being monitored regularly by XMM-Newton as it serves as a
calibration source. The XMM-Newton has five detectors dedicated to X-ray
observations: three European photon imaging cameras (EPIC; one PN (Strüder et
al., 2001) and two MOS (Turner et al., 2001)), two reflection grating
spectrometers (RGS; den Herder et al., 2001). RGS provides a spectral
resolution of 200-800 in the energy range of 0.3 – 2.5 keV. AB Dor was
regularly observed by the RGS detector; therefore, we have used the RGS
observations for further analysis. The XMM-Newton has observed AB Dor for 42
epochs from the year 2000 to 2022. The data were reduced using standard XMM-
Newton Science Analysis System (SAS) v19.0.0 software. The task rgsproc was
used to reprocess the data. The data was further screened for background
flaring events and extraction mask size. The background-subtracted light
curves were then extracted using rgslccorr task in SAS. AB Dor’s background-
subtracted X-ray light curves from RGS2 are shown in Figure A.1 in the
appendix A.

(a) Obs ID: 0123720201 (2000-05-01)

(b) Obs ID: 0134521301 (2001-10-13)

(c) Obs ID: 0134521501 (2002-04-12)

(d) Obs ID: 0134522201 (2003-01-23)

Figure 2: Left panels: Phase folded X-ray light curves along with the modelled
X-ray light curves. Right panels: Coronal images as obtained from the model
with phase folded X-ray light curves.

(e) Obs ID: 0412580701 (2011-01-02)

(f) Obs ID: 0791980101 (2016-10-07)

(g) Obs ID: 0791980401 (2017-10-10)

(h) Obs ID: 0810850501 (2019-09-30)

Figure 2: Continued…

(i) Obs ID: 0810850601 (2020-09-29) Figure 2: Continued…

###  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 clip to the
2$\sigma$ value of the mean, while the lower sigma clip is set free. By doing
so, large flares are removed automatically from the data, which was then
screened visually to remove further flaring events left unchecked by the
algorithm above. The flaring regions are shown by red open circles in Figure
A.1.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$\pm$0.23, where the error in the flare duty cycle is the
standard deviation. This indicates that AB Dor remains in the flaring states
for 57$\pm$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.

$MJD=44296.075+0.51479E$

Where MJD corresponding to the ”0” phase is the phase when the dominating
active region (say spot A) is in the centre 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 are 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.

###  3.3 Coronal imaging of AB Dor

The above-discussed model (Section 2.1) was applied to model the X-ray
rotational modulation of AB Dor. We have taken the angle of inclination of
60∘, the radius of 0.96 $R_{\odot}$ (Guirado et al., 2011), and $h_{cor}$ of
0.4 $R_{*}$(Hussain et al., 2007).

Figure 2 shows the model results, with the right panels showing the image of
the corona and the left panels showing the best fit modelled phase folded
light curves along with the observed light curves. The corona of AB Dor was
found to have two active longitudes that were located approximately 180∘apart.
Two active regions were located near the same longitudes during the epochs
2000-05-01 (see Figure 2 a) and 2001-10-13 (see Figure 2 b). The brightest
active region was located near the longitude of 0∘, whereas the less active
longitude was near -180∘. During the observation of the epoch 2002-04-12 (see
Figure 2 c), both active regions have shifted by 50∘ towards the decreasing
longitude. After nine months during the epoch 2003-01-23 (see Figure 2 d), the
corona of AB Dor became dominated by a single active region, spreading over
more than half a longitude from -90∘ to +90∘. On the epoch 2011-01-02 (see
Figure 2 e), AB Dor again showed two active regions near the longitude of -75∘
and +105∘. The corona of AB Dor in the yr 2016 (Figure 2 f), 2017 (Figure 2
g), and 2019 (Figure 2 h) appears to be similar to the corona during the
periods of 2000-2002. It appears that after $\sim$ 17 yr of the observations,
AB dor shows a similar activity level, which is close to the photometric
activity cycle of $\sim$16.96 years. In 2020, the corona of AB Dor exhibits a
similar pattern, with one active longitude spreading from +60∘to -90∘, but
with a 180∘shift from the active longitude observed in 2003. It appears that
the corona of AB Dor also shows a flip-flop like cycle and displays long-term
similarities that repeat roughly every $\sim$ 17 years.

##  4 Long-term X-ray activity

We have compiled a comprehensive dataset of X-ray observations for AB Dor from
1979 to 2022 to analyse the LTV. This dataset was sourced from both X-ray
archives and literature. The Einstein observatory data was 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
database111https://heasarc.gsfc.nasa.gov/W3Browse/exosat/me.html. The on-
source background-subtracted light curves from EXOSAT were obtained in 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.
Figure A.1 ll and A.1 mm show the resulting light curves. We have averaged the
quiescent state count rates over yearly bins to investigate long-term
variability. The count rates were then calibrated in the energy range of
0.3–10.0 keV using the WebPIMMS222https://heasarc.gsfc.nasa.gov/cgi-
bin/Tools/w3pimms/w3pimms.pl 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 B.1 and is plotted in Figure 3.

We have performed the Lomb-Scargle periodogram analysis on the long-term
quiescent binned data, revealing significant peaks at periods of 3.6$\pm$0.1,
4.3$\pm$0.1, 5.4$\pm$0.2, 9.5$\pm$0.2, and 19.2$\pm$2.1 yrs. The periods of
3.6 and 5.4 yrs 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 LX model to the data to further investigate the
long-term variability and computed the $\chi^{2}$ as 976. This $\chi^{2}$ was
then compared with the critical value of $\chi^{2}$ of 67.9 for the dof of 36
and the confidence limit of 99.9%. The derived $\chi^{2}$ was found to be much
higher than the critical value of $\chi^{2}$, indicating that the LX is
undoubtedly variable. As mentioned above, we fit the sine curves with all four
periods. This fit gives better $\chi^{2}$ (= 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 $\chi^{2}$ of 55.3. Adding the period of 19.2
yr does not improve the fit significantly. However, sinusoidal behaviours are
unreliable when there are fewer than three cycles, as the available dataset
was only for 44 years. The best-fit curve is overplotted in the upper panel of
Figure 3 as a solid black line with grey shaded regions corresponding to
1$\sigma$ confidence level.

Figure 3: Long-term evolution of corona of AB Dor. The solid black line in the
upper panel shows the best-fit line along with 1$\sigma$ confidence limits as
grey-shaded regions. The lower panel shows the Lomb-Scargle periodogram of
long-term X-ray data of AB Dor.

##  5 Discussion

We have carried out the STVs and LTVs observed in the ultra-fast 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 $\sim$ 60 % of its observing time, which is high
among other active stars (e.g. Pandey & Singh, 2012; Stelzer et al., 2000).

(a) Evolution of eastern and western hemisphere with time

(b) Eastern vs Western hemisphere

(c) LS power spectra of Eastern and western hemispheres

Figure 4: (a) Plot of average count rate of two opposite hemispheres with time
as obtained by phase folded quiescent state light curves. (b) The plot between
average X-ray count rate from the eastern hemisphere and average X-ray count
rates from the western hemisphere,(c) LS power spectra Eastern and Western
hemispheres’ light curves.

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). For 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 X-ray light curve modelling 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 modelled. 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 LS
analysis of the data reveals four periods at 3.6, 4.3, 5.4 and 9.5 yr in the
power spectra of X-ray light curves. In the earlier studies, the periods at
3.4 and 5.4 ys were also obtained and explained in terms of the flip-flop
cycle. The phenomenon of a flip-flop is also evidenced by the coronal images
obtained from LCIT, as explained in Section 2. For example, the coronal image
of the epoch 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 does not allow us to 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 $\pm$90∘(or $\pm$0.25 phase)
of spot A, whereas the western hemisphere is around the $\pm$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
years for every 100th rotation, with the brightness of the active regions
varying according to a 5.5-year 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-year 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 anti-correlated 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 $3\times 10^{-7}$. This suggests that the hemispheres
interchange their activity with time.

We conducted periodogram analysis using the LS method for both the flux of the
eastern and western hemispheres. The results, depicted in Figure 4(c), reveal
prominent peaks at periods of 3.6$\pm$0.2 yr and 5.6$\pm$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 optical data has been
previously attributed to the flip-flop cycle (Järvinen et al., 2005). Thus,
the presence of common periodicities of $\sim$ 5.5 yr in 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, 1993), the 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 analogues (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 analogues
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 non-axisymmetric 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 signatures for the
longer period is seen at 19.2 yr and its first harmonic at period 9.5 yr,
which suggests that the non-axisymmetric mode coexist with the axisymmetric
mode.

##  6 Conclusions

The present study indicates that the AB Dor exhibits frequent flare events,
accounting for an average of 57 $\pm$ 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,
the coronal imaging of AB Dor has shown the presence of two distinct active
longitudes, each situated 180 degrees 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
lightcurve exhibits multiple periodic signals. Notably, periods of
approximately $\sim$3.6, $\sim$5.4, and $\sim$19.2 yr manifest, resembling
findings from previous studies of optical observations. It is plausible that
the $\sim$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.

##  7 Acknowledgements

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 is 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.

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##  Appendix A X-ray light curves

(a) Obs ID: 0123720201 (2000-05-01) (b) Obs ID: 0126130201 (2000-06-07) (c)
Obs ID: 0123720301 (2000-10-27) (d) Obs ID:0133120701+0133120101+ 0133120201
(2000-12-11) (e) Obs ID: 0134520301 (2001-01-20) (f) Obs ID: 0134520701
(2001-05-22) (g) Obs ID: 0134521301 (2001-10-13) (h) Obs ID: 0134521401
(2001-12-26) (i) Obs ID: 0134521501 (2002-04-12) (j) Obs ID:
0155150101+0134521601 (2002-06-18) (k) Obs ID: 0134521701 (2002-11-15) (l) Obs
ID: 0134521801 (2002-11-05)

Figure A.1: XMM-Newton RGS2 X-ray lightcurves of AB Dor with a 300-sec bin
size. The red open circles represent the flaring state, while the blue-filled
circles show the quiescent state. The blue solid line and shaded region
correspond to the mean quiescent state and standard deviation, respectively.

(m) Obs ID: 0134522001 (2002-12-03) (n) Obs ID: 0134522101 (2002-12-30) (o)
Obs ID: 0134522201 (2003-01-23) (p) Obs ID: 0134522301 (2003-03-30) (q) Obs
ID: 0134522401 (2003-05-31) (r) Obs ID: 0160362501+0160362601 (2003-08-02) (s)
Obs ID: 0160362701 (2003-10-23) (t) Obs ID: 0160362801 (2003-12-08) (u) Obs
ID: 0160362901 (2004-11-27) (v) Obs ID: 0160363001 (2005-04-18) (w) Obs ID:
0160363201 (2005-10-16) (x) Obs ID: 0412580201 (2007-07-19) (y) Obs ID:
0412580301 (2008-01-03) (z) Obs ID: 0412580401 (2009-01-04 (aa) Obs ID:
0602240201 (2009-11-25)

Figure A.1: Continued

(ab) Obs ID: 0412580601 (2010-01-11) (ac) Obs ID: 0412580701 (2011-01-02) (ad)
Obs ID: 0412580801 (2011-12-31) (ae) Obs ID: 0791980101 (2016-10-07) (af) Obs
ID: 0791980401 (2017-10-10) (ag) Obs ID: 0810850101 (2018-10-02) (ah) Obs ID:
0810850501 (2019-09-30) (ai) Obs ID: 0810850601 (2020-09-29) (aj) Obs ID:
0810850701 (2021-12-04) (ak) Obs ID: 0810850901 (2022-10-19) (al) EXOSAT
(1984-11-12) (am) EXOSAT (1986-01-21)

Figure A.1: continued

##  Appendix B Long-term quiescent X-ray data

Table B.1: Yearly averaged X-ray data of AB Dor as observed from various X-ray
missions from 1979 to 2022. Here, $\sigma_{CR}$ and $SE_{CR}$ correspond to
standard deviation and standard error in count rate, whereas the $\sigma_{L}$
and $SE_{L}$ are standard deviation and standard error in Luminosity,
respectively.

Telescope | Date | MJD | Count rate ($CR$) | $\sigma_{CR}/SE_{CR}$ | Luminosity ($L$) | $\sigma_{L}/SE_{L}$
---|---|---|---|---|---|---
| YYYY-MM-DD |  | count/s | counts/s | ($\times$1030 erg/s) | ($\times$1030 erg/s)
Einstein1 | 1979-06-01 | 44025.00 | 2.17 | 0.18 / 0.18 | 1.26 | 0.01 / 0.01
EXOSat2 | 1984-12-11 | 46045.74 | 0.98 | 0.39 / 0.03 | 2.11 | 0.83 / 0.07
| 1986-01-21 | 46451.36 | 0.95 | 0.35 / 0.03 | 2.06 | 0.76 / 0.07
Ginga3 | 1990-10-01 | 48165.00 | 2.02 | 0.57 / 0.18 | 1.37 | 0.38 / 0.12
ROSAT4 | 1990-07-02 | 48074.4 | 1.62 | 0.08 / 0.08 | 1.18 | 0.06 / 0.06
| 1991-10-26 | 48555.47 | 1.45 | 0.03 / 0.03 | 1.05 | 0.02 / 0.02
| 1992-03-14 | 48695.88 | 1.72 | 0.02 / 0.02 | 1.26 | 0.01 / 0.01
| 1994-07-06 | 49539.53 | 1.97 | 0.01 / 0.01 | 1.43 | 0.01 / 0.01
| 1995-07-07 | 49905.25 | 1.84 | 0.01 / 0.01 | 1.34 | 0.01 / 0.01
| 1996-07-28 | 50292.34 | 1.85 | 0.01 / 0.01 | 1.35 | 0.01 / 0.01
| 1997-07-12 | 50641.49 | 2.11 | 0.01 / 0.01 | 1.54 | 0.01 / 0.01
| 1998-03-02 | 50874.26 | 1.82 | 0.02 / 0.02 | 1.33 | 0.02 / 0.02
BeppoSAX5 | 1997-11-10 | 50762.23 | 0.24 | 0.03 / 0.01 | 2.38 | 0.32 / 0.09
| 1997-11-29 | 50781.92 | 0.28 | 0.05 / 0.01 | 2.80 | 0.49 / 0.1
| 1999-12-09 | 51521.83 | 0.14 | 0.02 / 0.0 | 1.40 | 0.17 / 0.05
| 2000-06-04 | 51699.74 | 0.13 | 0.02 / 0.0 | 1.30 | 0.23 / 0.04
Suzaku6 | 2006-12-15 | 54084.50 | – | – / – | 1.48 | 0.02 / 0.02
XMM-Newton7 | 2000-08-19 | 51775.82 | 1.43 | 0.05 / 0.003 | 1.39 | 0.05 / 0.003
| 2001-05-30 | 52059.21 | 1.61 | 0.07 / 0.006 | 1.56 | 0.07 / 0.006
| 2002-09-24 | 52541.19 | 1.78 | 0.05 / 0.005 | 1.73 | 0.05 / 0.005
| 2003-07-05 | 52825.46 | 1.55 | 0.04 / 0.004 | 1.5 | 0.04 / 0.004
| 2004-11-28 | 53337.09 | 1.32 | 0.09 / 0.007 | 1.28 | 0.08 / 0.007
| 2005-07-18 | 53569.47 | 1.42 | 0.04 / 0.006 | 1.38 | 0.04 / 0.006
| 2007-07-19 | 54300.44 | 1.61 | 0.12 / 0.009 | 1.56 | 0.12 / 0.009
| 2008-01-04 | 54469.09 | 1.46 | 0.17 / 0.014 | 1.42 | 0.17 / 0.013
| 2009-06-16 | 54998.66 | 1.22 | 0.08 / 0.006 | 1.18 | 0.08 / 0.006
| 2010-01-11 | 55207.87 | 1.19 | 0.09 / 0.007 | 1.15 | 0.09 / 0.007
| 2011-01-03 | 55564.0 | 1.37 | 0.12 / 0.009 | 1.34 | 0.12 / 0.008
| 2012-01-01 | 55927.01 | 1.37 | 0.08 / 0.006 | 1.34 | 0.08 / 0.006
| 2016-10-07 | 57668.63 | 1.32 | 0.14 / 0.008 | 1.28 | 0.13 / 0.007
| 2017-10-11 | 58037.25 | 1.48 | 0.12 / 0.007 | 1.43 | 0.12 / 0.007
| 2018-10-03 | 58394.13 | 1.19 | 0.13 / 0.007 | 1.15 | 0.13 / 0.007
| 2019-10-01 | 58757.51 | 1.25 | 0.09 / 0.005 | 1.22 | 0.08 / 0.005
| 2020-09-30 | 59122.11 | 1.15 | 0.09 / 0.005 | 1.12 | 0.09 / 0.005
| 2021-12-05 | 59553.24 | 1.86 | 0.14 / 0.008 | 1.81 | 0.14 / 0.008
| 2022-10-20 | 59872.32 | 1.35 | 0.08 / 0.004 | 1.31 | 0.07 / 0.004
AstroSat |  82016-01-23 | 57410.85 | – | –/– | 1.37 | 0.1/0.1
|  92018-02-26 | 58165.37 | 0.93 | 0.32 / 0.003 | 1.24 | 0.07 / 0.07

1 Elvis et al. (1992), 2 This paper, 3Vilhu et al. (1993), 4 ROSAT HRI
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Sairam et al. (2023), 9 This paper.