11institutetext: Institut für Astronomie und Astrophysik Tübingen, Kepler
Center for Astro and Particle Physics, Eberhard Karls Universität, Sand 1,
72076 Tübingen,Germany
11email<EMAIL_ADDRESS>22institutetext: INAF - Osservatorio
Astronomico di Palermo, Piazza del Parlamento 1, I-90134 Palermo, Italy

# The X-ray activity of the young solar-like star Kepler-63 and the structure
of its corona

M. Coffaro 11 B. Stelzer The X-ray activity of the young solar-like star
Kepler-63 and the structure of its coronaThe X-ray activity of the young
solar-like star Kepler-63 and the structure of its coronaThe X-ray activity of
the young solar-like star Kepler-63 and the structure of its coronaThe X-ray
activity of the young solar-like star Kepler-63 and the structure of its
corona S. Orlando 22

(Received 24 September 2021 / Accepted 11 February 2022)

The X-ray satellite XMM-Newton has so far revealed coronal cycles in seven
solar-like stars. In this sample, the youngest stars $\epsilon$ Eridani ($\sim
400$ Myr) and $\iota$ Horologii ($\sim 600$ Myr) display the shortest X-ray
cycles and the smallest amplitudes, defined as the variation of the X-ray
luminosity between the maximum and minimum of the cycle. The X-ray cycle of
$\epsilon$ Eridani was characterized by applying a novel technique that
allowed to model the corona of a solar-like star in terms of magnetic
structures observed on the Sun (active regions, cores of active regions and
flares) at varying filling factors. The high surface coverage of $\epsilon$
Eridani with magnetic structures ($65-95\%$) that emerged from that study was
held responsible for the low cycle amplitude in the X-ray band. It was also
hypothesized that the basal surface coverage with magnetic structures may be
higher on the corona of solar-like stars while they are young. To investigate
this hypothesis, we started in 2019 the X-ray monitoring campaign of
Kepler-63. The star had been observed once before, in 2014, with XMM-Newton.
With an age of $210\pm 45$ Myr and a photospheric cycle of $1.27$ yr,
Kepler-63 is so far the youngest star observed in X-rays with the aim of
revealing a coronal cycle. Our campaign comprised four X-ray observations of
Kepler-63, spanning 10 months (i.e. 3/5 of its photospheric cycle). In the
long-term X-ray lightcurve we did not reveal a periodic variation of the X-ray
luminosity, but a factor two change would be compatible with the considerable
uncertainties in the low signal-to-noise data of this relatively distant star.
As for the case of $\epsilon$ Eridani, we described the coronal emission
measure distribution (EMD) of Kepler-63 with magnetic structures observed on
the Sun. The best match with the observations is found for an EMD composed of
cores and flares of GOES Class C and M following the canonical flare frequency
distribution. More energetic flares can be occasionally present but they do
not contribute significantly to the quasi-stationary high-energy component of
the emission measure probed with our modelling. This model yields a coronal
filling factor of $100\%$. This complete coverage of the corona with X-ray
emitting magnetic structures is consistent with the absence of an X-ray cycle,
confirming the analogous results derived earlier for $\epsilon$ Eridani.
Finally, combining our results with the literature on stellar X-ray cycles we
establish an empirical relation between the cycle amplitude $L_{\rm
X,max}/L_{\rm X,min}$ and the X-ray surface flux, $F_{\rm X,surf}$. From the
absence of a coronal cycle in Kepler-63 we infer that stars with higher X-ray
flux than Kepler-63 must host an EMD that comprises a significant fraction of
higher-energetic flares than those necessary to model the corona of Kepler-63,
i.e. flares of Class X or higher. Our study opens new ground for studies of
the solar-stellar analogy and the joint exploration of resolved and unresolved
variability in stellar X-ray lightcurves.

###### Key Words.:

X-ray: stars - Stars: solar-type - Stars: activity - Stars: coronae - Stars:
individual: Kepler-63

††offprints: M. Coffaro

##  1 Introduction

The magnetic field in solar-like stars is maintained by the so-called
$\alpha-\Omega$ dynamo, that is responsible of changing the configuration of
the magnetic field lines periodically. The time scale on which these changes
take place defines the length of the activity cycle (Böhm-Vitense 2007). A
consequence of the magnetic field re-configuration is the formation of
magnetic structures that rise on the stellar surface, evolve and decay
periodically in the course of the magnetic cycle. The surface coverage with
these structures, thus, varies continuously between a minimum and a maximum.
Therefore, by tracing the evolution of the magnetic structures we can
characterize the activity cycles of solar-like stars. These structures are
manifest in all layers of the stellar atmosphere, from the photosphere to the
corona. Due to the temperature gradient that characterizes the solar and
stellar atmospheres, triggering different emission mechanisms for each layer
of the atmosphere, activity cycles are expected to be manifest at different
wavelengths, from the optical to the X-rays.

Photospheric and chromospheric cycles are well studied. In particular, $\sim
60\%$ of the main-sequence stars show chromospheric cycles with periods
lasting up to 20 yr (Wilson 1978). These cycles are characterized with Ca II
H&K emission lines emitted by plages in the chromosphere. The photospheric
cycles can instead be characterized by tracing the evolution of the starspots.
One way of gathering information on the starspot evolution is the so-called
spot transit method (Silva 2003), that infers the presence of starspots
through the brightness changes during a planet-transit.

Our knowledge about cycles in the outermost atmospheric layer, the corona, is
still scarce. Since the same dynamo process is responsible for the magnetic
cycle in all parts of the stellar atmosphere, the length of X-ray cycles is
expected to coincide with that of the chromospheric cycles. Since the latter
have been seen that last up to 20 yr, long X-ray monitoring campaigns would be
required to detect their coronal counterparts. Moreover, the search for X-ray
cycles relies on sparsely sampled lightcurves. This is clearly a disadvantage
for reliable coronal cycle detections. Therefore, studies of long-term X-ray
variability have focused on stars with known chromospheric cycle.

Up to date we are aware of the existence of only seven stars that show X-ray
activity cycles, all detected by the X-ray satellite XMM-Newton. For most of
them, the common denominator is their age and the period of their coronal
cycles. These stars are $\alpha$ Centauri A and B (Robrade et al. 2012), $61$
Cygnus A and B (Hempelmann et al. 2006; Robrade et al. 2012), HD 81809 (Favata
et al. 2008; Orlando et al. 2017), with ages between $3$ Gyr and $6$ Gyr and
with X-ray activity cycles that last from $7$ up to $20$ yr. The remaining two
stars with observed X-ray cycle, $\iota$ Horologii (Sanz-Forcada et al. 2013,
2019) and $\epsilon$ Eridani (Coffaro et al. 2020), are young stars, with age
of 600 and 400 Myr and with short coronal cycles that have periods of $1.6$
and $2.9$ yr, respectively. Moreover, these two young stars show another
intriguing characteristics: the amplitude of their X-ray cycles, i.e. the
variation of the X-ray luminosity between the minimum and maximum of their
coronal emission, is remarkably small.

The X-ray emission of the Sun throughout its cycle is related to magnetic
structures whose coronal coverage fraction changes along with the solar cycle.
Exploiting the solar-stellar analogy, the hypothesis is that, for solar-like
stars, we can link their X-ray variability to the same phenomenon. In this
context, Coffaro et al. (2020) applied a novel technique which enables to
reproduce the X-ray emission of a star in terms of time-variations in the
coverage of its corona with the same kind of magnetic structures that are
observed on the Sun. This technique was developed in the framework of the
study ”The Sun as an X-ray star” (Orlando et al. 2000, Peres et al. 2000,
Reale et al. 2001, Orlando et al. 2001; see 3.1) in which observations of the
solar corona by the X-ray satellite Yohkoh were converted into a format
virtually identical to that of the observations obtained with non-solar X-ray
satellites, such as XMM-Newton. The final data products of this method are
synthetic X-ray spectra that are distinguished by a particular fractional
coverage of magnetic structures on the corona, and can be directly compared to
the observed spectra of a coronally active star: from such a comparison a
percentage surface coverage with solar magnetic structures can be inferred for
each stellar X-ray observation.

This method was first applied to HD81809 (Orlando et al. 2017). A more
detailed study was carried out by Coffaro et al. (2020) for $\epsilon$
Eridani. In that work we found a massive presence of magnetic structures on
the corona throughout the X-ray activity cycle of $\epsilon$ Eridani. We
argued that the high percentage (from $\sim 70$ to $\sim 90\%$) of the
magnetic structures on the corona do not allow the X-ray luminosity to
significantly vary during the coronal cycle, explaining the small amplitude of
$\epsilon$ Eridani’s X-ray luminosity.

To better understand the origin of small cycle amplitudes and to investigate
if solar-like stars younger than $\epsilon$ Eridani show X-ray activity
cycles, we started the study of Kepler-63, a young G2V star at a distance of
200 pc. With its age of $210\pm 45$ Myr (Sanchis-Ojeda et al. 2013), it is
younger than the other stars that show short coronal activity cycles. It is
also a fast rotator, with a rotational period of $\sim 5.4$ days and it hosts
a Jupiter-like planet with a polar orbit of $9.43$ days (Sanchis-Ojeda et al.
2013).

Kepler-63 was extensively observed with the NASA satellite Kepler and a
photospheric cycle was detected by Estrela & Valio (2016) and Netto & Valio
(2020) from the analysis of the Kepler lightcurves, by applying the spot
transit method from which they found a significant periodicity at $1.27$ yr.
Its young age and the short activity cycle make Kepler-63 an ideal target for
an X-ray monitoring campaign in search of its coronal cycle.

In 2 we present our X-ray monitoring campaign and the analysis of the X-ray
observations. In 3, we apply the method developed by Orlando et al. (2017) and
refined by Coffaro et al. (2020) with the aim to model the corona of Kepler-63
in terms of the same type of magnetic structures observed on our Sun. In 4, we
discuss our results and we give our conclusions.

##  2 Observations and data analysis

The X-ray monitoring campaign of Kepler-63 with the satellite XMM-Newton
started in 2019 and it continued until 2020 (PI: M.Coffaro; Proposal ID:
084123). During this monitoring, four X-ray observations were acquired. The
time span between each X-ray observation was of few months and the full X-ray
campaign covered 3/5 of the photospheric cycle.

Prior to this X-ray campaign, Kepler-63 had already been observed with XMM-
Newton once in 2014 (Lalitha et al. 2018). This earlier observation was used
to define the observational strategy of the X-ray monitoring campaign. In 1,
the XMM-Newton observing log of Kepler-63 is presented, including all
available observations.

Kepler-63 was observed employing all instruments on board of XMM-Newton. In
this paper, only the analysis of the EPIC/pn data is presented.

Table 1: Observing log of the X-ray monitoring campaign of Kepler-63.

Obs. date | Obs ID | Exposure time111Time of acquisition of each observation of the EPIC/pn detector, prior to the GTI cuts. | Count rate222EPIC/pn count rate obtained from the source detection procedure in the energy band $0.2-2.0$ keV.
---|---|---|---
|  | [ksec] | [cnt/sec]
2014-09-28 | 0743460301333Archival observation | $27$ | $0.0197\pm 0.0011$
2019-05-05 | 0841230201 | $15.9$ | $0.0169\pm 0.0020$
2019-08-21 | 0841230301 | $15.7$ | $0.0170\pm 0.0014$
2019-10-31 | 0841230401 | $9.5$ | $0.0185\pm 0.0022$
2020-03-06 | 0841230501 | $5.7$ | $0.0197\pm 0.0047$

For some observations the lightcurves of the EPIC/pn background showed
irregularities and high count rates during the acquisition. Our visual
inspection showed that the standard values for identifying the Good Time
Intervals (GTIs), i.e. RATE $<0.4$ cnt/s as suggested by the threads of
Science Analysis Software (SAS; version 1.3; de la Calle 2019) for the
extraction of the EPIC/pn scientific data, did not remove all times of high
background. Instead, we defined the GTIs ad hoc, identifying visually those
intervals of time were the full-detector count rate was constant and low. In
11, the lightcurves of the full-detector background of the EPIC/pn are shown
for the five observations. For the observations of September 2014 and August
2019, the GTIs were chosen over the whole lightcurve of the detector
background, corresponding to RATE $<0.8$ cnt/s and RATE $<1.45$ cnt/s
respectively (red solid lines in 11). For the remaining observations, the GTIs
were chosen as specific time intervals (highlighted by the red ranges in the
plots). Because of this choice, these latter observations effectively result
shorter in time with respect to their original exposure time.

After applying the source detection procedure of SAS, Kepler-63 results
detected in all 5 EPIC/pn observations. The positions of Kepler-63 on the
detector identified by the source detection procedure were used to localize
the extraction regions for the lightcurves and spectra: a circular region
centered at the coordinates given by the source detection and with a radius of
$20$ arcsec was chosen. The background regions were chosen in a source-free
part of the CCD where Kepler-63 was detected (and nearby the source). In the
following subsections, the details on the lightcurve and spectral analysis are
given.

###  2.1 EPIC/pn lightcurves

The EPIC/pn lightcurves of Kepler-63 were extracted in the soft energy band of
XMM-Newton ($0.2-2.0$ keV), employing the tools of SAS, evselect and
epiclccorr. All lightcurves were binned with a time bin size of $1000$
seconds, with the exception of the observation of March 2020. In this case,
the GTIs have a duration that is less than $1000$ seconds and, thus, a smaller
bin size is more suitable. We chose $600$ seconds.

1 shows the background-subtracted lightcurves of Kepler-63. In these plots,
the solid blue lines represent the time-averaged count rates calculated with
the source detection procedure, and its uncertainties (dashed lines). These
values are reported in 1.

We checked for short-term variability in each ligthcurve, employing the
software R and its package changepoint (Killick & Eckley 2014). This package
allows to look for changes of mean and variance in time series data. The
application of this method did not yield any significant short-term
variability in our observations, i.e. no flare-like events occurred during the
X-ray monitoring of Kepler-63.

(a)  (b)

(c)  (d)

(e)

Figure 1: EPIC/pn background-subtracted lightcurves for all observations of
Kepler 63. The blue lines are the mean count rate (solid lines) obtained from
the source detection procedure, with the associated error (dashed lines).

###  2.2 EPIC/pn spectra

The spectra of Kepler-63 were extracted for the whole energy band of XMM-
Newton, i.e. $0.2-12.0$ keV, and grouped with a minimum of counts per bin
equal to 15, with the exception of the observation from March 2020. For this
last observation, a grouping size of minimum 15 counts per bin would result in
very few spectral bins, as it is the one with the lowest photon statistics,
and with a short exposure time. Therefore, only the spectrum of the
observation of March 2020 was grouped choosing 5 as minimum counts per bin.
However, even with the reduced number of counts per bin, the spectrum of this
observation has too few bins to be properly fitted and, thus, it was excluded
from the analysis.

Each of the other four spectra was analysed with the standard spectral
software xspec (version 12.10; Arnaud 1996) and was fitted with the simplest
thermal model, i.e. a one-temperature (1-T) APEC model, considering the energy
range $0.3-2.0$ keV 444Beyond $2.0$ keV the spectra are extremely noisy and
the high uncertainties would compromise the goodness of the fit.. The
abundances $Z$ are kept frozen during the fitting procedure, on a value equal
to $0.3Z_{\odot}$, which is typical for the corona of solar-like stars (Maggio
et al. 2007). Tests for other values of abundances are explained in 2.2.1. The
photoelectric absorption is not considered in the spectral model: although
Kepler-63 is at a distance for which this effect plays a relevant role, i.e.
at $200$ pc, below $0.3$ keV the uncertainties on the counts are too high to
make the spectrum physically reliable.

Table 2: Best-fitting parameters for the X-ray spectra of Kepler-63. The
spectral model used for all spectra is the 1-T APEC model, with the exceptions
of the observations of September 2014, August 2019 and the simultaneous
fitting of all observations obtained in 2019: in these latter cases a 2-T APEC
model was used as well. For both spectral models the metal abundances are kept
frozen on the value $0.3Z_{\odot}$. The associated uncertainties are obtained
with the error command of xspec. In the last column, $\overline{\chi}^{2}$
stands for the reduced $\chi^{2}$ and d.o.f are the degrees of freedom.

Date | Model | $kT_{1}$ | $kT_{2}$ | $logEM_{1}$ | $logEM_{2}$ | $kT_{\rm av}$ | $F_{\rm X}$ |  $\overline{\chi}^{2}$ (d.o.f)
---|---|---|---|---|---|---|---|---
|  | [keV] | [keV] | $[\rm cm^{-3}]$ | $[\rm cm^{-3}]$ | [keV] | $10^{-14}\ \rm[erg/cm^{2}/s]$ |
|  |  |  |  |  |  |  $0.3-2.$ keV |
2014-09-28 | 1-T APEC | $0.60\pm 0.12$ | $\smallsetminus$ | $51.92\pm 0.13$ | $\smallsetminus$ | $\smallsetminus$ | $2.27\pm 0.16$ | 0.74 (27)
| 2-T APEC | $0.50\pm 0.13$ | $1.08\pm 0.47$ | $51.84\pm 0.41$ | $51.45\pm 0.80$ | $0.67\pm 0.18$ | $2.43\pm 0.22$ | 0.64 (25)
2019-05-05 | 1-T APEC | $0.79\pm 0.29$ | $\smallsetminus$ | $51.83\pm 0.34$ | $\smallsetminus$ | $\smallsetminus$ | $1.91\pm 0.35$ | 0.65 (11)
2019-08-21 | 1-T APEC | $0.82\pm 0.18$ | $\smallsetminus$ | $51.80\pm 0.22$ | $\smallsetminus$ | $\smallsetminus$ | $1.75\pm 0.26$ | 1.39 (13)
| 2-T APEC | $0.27\pm 0.09$ | $1.04\pm 0.41$ | $51.79\pm 0.47$ | $51.66\pm 0.49$ | $0.61\pm 0.18$ | $2.29\pm 0.28$ | 0.43 (10)
2019-10-31 | 1-T APEC | $0.74\pm 0.20$ | $\smallsetminus$ | $51.92\pm 0.30$ | $\smallsetminus$ | $\smallsetminus$ | $2.38\pm 0.38$ | 0.81 (7)
Simultaneous fit (2019) | 2-T APEC | $0.31\pm 0.16$ | $0.94\pm 0.23$ | $51.59\pm 0.63$ | $51.72\pm 0.38$ | $0.67\pm 0.15$ | $1.93\pm 0.19$ |  $0.75$ ($32$)

The best-fitting values resulting from this procedure are reported in 2. The
uncertainties of each best-fitting parameter are obtained with the error
command of xspec, that calculates the confidence interval for the model
parameters within a confidence level of $90\%$. The spectra of each
observation, with the spectral model overplotted, are shown in 12.

The observation from September 2014 and the one from August 2019 show
sufficiently high statistics to be fitted with a 2-T APEC model. For these two
spectra the best-fitting parameters for the 2-T model are also shown in 2. The
abundances are again frozen to the value of $0.3Z_{\odot}$.

####  2.2.1 Metal abundances

We carried out tests with the aim of constraining the coronal abundances for
each observation. To this end, we adopted a 1-T APEC model in which the metal
abundances were left free to vary during the fitting procedure. However,
because of the low photon statistics that each spectrum shows, it was not
possible to have a statistically significant fit. Only the fit of the archival
observation yielded a constrained value for $Z$, equal to $0.24\pm 0.07$, with
a reduced $\chi^{2}$ of $1.25$ (with 25 d.o.f).

We additionally noticed that when considering a range of different values of
metal abundances, each of them kept frozen in the fitting procedure, the
statistics of the fit did not show significant differences. Thus, with the
quality of the available data, we can not constrain the coronal metal
abundances.

For the further analysis, we kept the metal abundances always frozen to
$0.3Z_{\odot}$, as mentioned above.

####  2.2.2 Simultaneous fitting

We performed a simultaneous fit of all usable observations obtained in 2019
(three observations in total), in order to increase the signal-to-noise ratio.
This allowed us to use a 2-T APEC model. The 2-T fit enables (1) a more
detailed comparison to the observation from September 2014 and (2) a more
meaningful investigation of Kepler-63 in the framework of the study ”The Sun
as an X-ray star” (see 3).

We carried out the simultaneous fit as described in 2.2. The results are shown
in the bottom right panel of 12 and reported in 2.

##  3 Modelling the corona of Kepler-63 in terms of solar magnetic structures

Here, we present our technique for indirectly identifying magnetic structures
on the corona of Kepler-63. As explained in 1, the method is based on the
study ”The Sun as an X-ray star” and was previously applied to the old binary
star HD81809 (Favata et al. 2008; Orlando et al. 2017) and the young solar-
like star $\epsilon$ Eridani (Coffaro et al. 2020).

In 3.1, a summary of the study ”The Sun as an X-ray star” is presented,
followed by the description of its application to the case of Kepler-63 (3.2).

###  3.1 The solar emission measure distributions

In the study ”The Sun as an X-ray star”, different types of magnetic
structures, observed with the Soft X-ray Telescope (SXT) on board the solar
satellite Yohkoh on the corona of the Sun during the 22nd solar activity
cycle, were spatially and temporally analysed. The three kinds of structures
that were identified in the X-ray images of the Sun are quiet regions (also
termed background corona, BKCs), active regions (ARs) and cores of active
regions (COs). For each of these structures, a distribution of emission
measures as a function of the temperature (EMD) was derived from the
observations. In this way, for each phase of the solar cycle it was possible
to identify the contribution of each type of magnetic structure to the total
EMD.

The Yohkoh satellite provided also several images of solar flares on the
corona, and eight flares were analysed in time, from their rise throughout
their decay (Reale et al. 2001). These flares range from Class C (relatively
low energy flares) to Class X (the most energetic ones observed on the Sun),
and the corresponding flare EMDs were retrieved. Moreover, Yohkoh collected
observations of flares with different filters, hard filters sensitive to
plasma temperatures around and above $10^{7}$ K and soft filters for lower
temperatures. The vast majority of the flares were observed with a pair of
hard filters only; a few of them were observed also with a pair of soft
filters, thus allowing a better reconstruction of the EMD at temperatures as
low as $3$ MK. One of these flares (a Class M flare, namely a flare of mid
energy) was observed with two pairs of filters (soft and hard) and analysed by
Reale et al. (2001). It was also possible to trace the temporal evolution of
one AR and its corresponding CO from the rise to the decay (see Fig. 2 of
Orlando et al. 2004).

In the final paper of ”The Sun as an X-ray star” (Orlando et al. 2001), for
each observed type of solar EMD, i.e. that of BKCs, ARs and COs, a
corresponding synthetic spectrum was extracted, using the response matrix of
X-ray satellites that can observe stars. This way, Orlando et al. (2001)
simulated solar coronal observations as if the Sun was observed by non-solar
X-ray instruments.

Figure 2: Average EMDs per unit surface area for each type of magnetic
structures observed with Yohkoh. The black lines are the two flaring EMDs
tested within this work: a time-averaged distribution derived from a sample of
eight flares from Class C to X observed with a pair of hard filters with
Yohkoh (solid black line; 3.2.4) and a time-averaged distribution derived from
the flares of Class C and M in the sample (dotted black line; 3.2.5), the M
flares observed also with a pair of soft filters. Figure 3: X-ray luminosity
as a function of emission measure weighted thermal energy $kT_{\rm av}$ (left
panel) and the two temperatures and emission measures (right panel). The
circles represent the synthetic spectra extracted from EMDs that are composed
of time-averaged COs varying from a coverage of $10$ to $100\%$ of the total
coronal surface (change of color shade) and by FLs, from Class C to X time-
averaged over their evolution, that vary from $0$ to $4\%$ of the area of the
COs (change of symbol size). The best-fitting parameters of the observations
of Kepler-63 are overplotted with the blue square (2014) and the magenta
circle (2019).In the background, the occurrences of each best-fitting
parameter, that is simulated 1000 times for each combination of magnetic
structures by introducing Poisson noise, are shown with the corresponding
color bar in log scale. Figure 4: X-ray luminosity as a function of emission
measure weighted thermal energy $kT_{\rm av}$ (left panel) and the $kT$s and
$EM$s (right panel). The EMDs used for these grids assume the same magnetic
structures and the same percentage coverage as in 3, with the difference that
each EMD is scaled to a larger extension of the corona of Kepler-63, i.e. the
stellar radius is set to be $30\%$ larger. The coding of the plots follows 3.

###  3.2 The case of Kepler-63

With the aim of investigating whether and how the corona of young active stars
can be described by magnetic structures similar to those observed in the solar
corona, we applied the method presented in 3.1 to Kepler-63. The Sun is,
indeed, the only star on which magnetic structures can be spatially and
temporally resolved with current technology, and thus the corona of the Sun is
the only template that can be used for such investigations.

For applying the above study, we considered that at least two spectral
components are needed in order to achieve a basic description of the
temperature structure of the corona. However, the individual observations of
the 2019 campaign are too poor for a 2-T fit, and thus we have fitted the
three spectra jointly (see 2.2.2). We, therefore, have two epochs with
spectral fits for Kepler-63 available: one representing the archival
observation (from 2014) and one representing the average corona observed in
2019.

####  3.2.1 Choice of the appropriate magnetic structures

In the first step we aimed at finding the appropriate types of magnetic
structures whose EMDs, scaled to the coronal metal abundance (i.e.
$0.3Z_{\odot}$) and the surface area of Kepler-63 ($R_{\star}=0.91R_{\odot}$;
Sanchis-Ojeda et al. 2013), are suited to reproduce the X-ray luminosity
($L_{\rm X}$) and emission measure weighted thermal energies ($kT_{\rm av}$)
observed during the XMM-Newton monitoring campaign of the star. The $kT_{\rm
av}$ is calculated as

$kT_{\rm av}=\dfrac{kT_{1}\cdot EM_{1}+kT_{2}\cdot EM_{2}}{EM_{1}+EM_{2}},$
(1)

where $kT_{i}$ and $EM_{i}$ ($i=1,2$) are the spectral components of the 2-T
APEC model.

In 2 we show the EMDs derived from Yohkoh data for the different types of
coronal structures: BKCs (yellow line), ARs (red line) and COs (green line).
The figure also shows two EMDs for flaring regions: one derived from a sample
of flares from Class C to X observed with Yohkoh/SXT using a pair of hard
filters (Reale et al. 2001; solid black line) and the other one derived from
flares of the sample from Class C to M (dashed black line), with the latter
analysed also using a pair of soft filters in addition to the hard ones.

From 2 we see that the EMD per unit surface area of the solar BKCs and ARs are
much lower than the one of the COs and, thus, any percentage of these two
structures would negligibly contribute to the total EMD. Covering part of the
corona with BKCs and/or ARs implies less space left available for other
magnetic structures that have a higher surface brightness and would better
reproduce the X-ray luminosity of Kepler-63, which is at least two orders of
magnitude higher than that of the Sun ($\log L_{\rm X\ \rm Kep63}\rm\
[erg/s]\sim 29$ versus $\log L_{\rm X\ \odot}\rm\ [erg/s]\sim 27.3$; Orlando
et al. 2001). We interpret this as a clear indication that Kepler-63 requires
a high surface coverage with COs, so that the contribution of BKCs and ARs to
the X-ray emission is negligible. Thus, we do not consider the contribution of
BKCs and ARs in the following.

For the case of $\epsilon$ Eridani, Coffaro et al. (2020) used the EMD
obtained from the CO observed in 1996 by Orlando et al. (2004), time-averaged
during its evolution (solid green line in 2), together with the contribution
of the EMDs of FLs. These EMDs of COs and FLs were able to reproduce the X-ray
luminosity of $\epsilon$ Eridani ($\log L_{\rm X\ \epsilon Eri}\rm\
[erg/s]\sim 28$). Due to the higher X-ray luminosity of Kepler-63 ($\log
L_{\rm X\ \rm Kep63}\rm\ [erg/s]\sim 29$), it is necessary to increase the
percentage of surface coverage of COs and FLs in order to reach the observed
X-ray emission level of the star. We, therefore, started to test the same
scenario adopted for the case of $\epsilon$ Eridani, where the synthetic
corona is covered with time-averaged COs and FLs, albeit assuming a greater
percentage of them than in the case of $\epsilon$ Eridani. For the flaring
contribution, we tested two different FL EMDs as we describe in 3.2.4 and
3.2.5. These two distributions are plotted in 2 as solid and dotted black
lines respectively.

We took into account that the different classes of flares have different
occurrence rates and, therefore, the flare filling factor depends on the flare
energy (here approximated by peak flux, alias GOES class). To determine the
average contribution of the different flare classes to the total EMD we
assumed that the differential frequency of the peak flux ($F_{\rm peak}$) of
solar flares is described as $dN(F_{\rm peak})\propto F_{\rm
peak}^{-\alpha}dF_{\rm peak}$ with index $\alpha=1.57$. This is the value
calculated by Aschwanden & Parnell (2002) from Yohkoh observations of solar
flares. The flare flux frequency distribution (FFD) is normalized to the peak
flux of the flare with the lowest flux (a Class C flare in the case of the
sample of flares considered here). The flare filling factors we derive with
our modelling thus apply to a Class C5.8 flare, and filling factors for flares
with other peak fluxes can be obtained by scaling from the power-law (see
4.4). We note that the normalization of the FFD is degenerate with the filling
factor of flares. If the normalization was made on a less energetic flare the
absolute values for the flare frequencies of the more energetic events would
be lower, and a higher filling factor would be required to compensate for
this.

After the appropriate classes of magnetic structures have been defined (here,
COs and FLs), a grid of EMDs was constructed where each grid point is
distinguished from the others by a different fractional surface coverage with
each class of magnetic structures.

####  3.2.2 Synthesis of the X-ray spectra

We extracted a synthetic XMM-Newton spectrum for each grid point, making use
of the EPIC/pn response matrix and taking into account the distance of
Kepler-63. In order to estimate uncertainties, 1000 spectra were generated for
each grid point, and Poisson statistics were introduced in the extraction
procedure for the synthesis of the spectra. Each of these 1000 spectra was
then fitted in xspec with a 2-T APEC model, and with metal abundances frozen
to $0.3Z_{\odot}$, i.e. the same spectral model used to represent the observed
spectra of Kepler-63. In this way, each grid point, i.e. each combination of
magnetic structures, is not unequivocally represented by only two couples of
$kT$s and $EM$s for the cold and hot components, but rather by a set of 1000
values for each of the four fit parameters drawn from a Poisson distribution.
The four parameters (two $kT$s and two $EM$s) for each point of the final grid
are calculated as the mean of the 1000 values obtained from the spectral
analysis.

Furthermore, from analyzing the synthetic spectra we obtained the X-ray
luminosity as a function of the EM weighted coronal temperature, calculated as
in Eq. 1, and we performed a first visual comparison of these values with
those calculated from the spectra of Kepler-63, in order to verify if the
chosen combination of magnetic structures can reproduce the observations. The
left panel of 3 is an example of $L_{\rm X}$ as a function of $kT_{\rm av}$,
where the grid and the observations are compared. We discuss the details of
such comparisons for different grids in 3.2.4 and 3.2.5.

As a final step, we performed a quantitative match between the temperatures
and the emission measures, obtained from the analysis of the synthetic spectra
with those retrieved from the observations, according to the approach
described by Coffaro et al. (2020) and summarized in the subsequent section.

####  3.2.3 Selection criterion

The match between synthetic and observed X-ray spectra of Kepler-63 was
performed by using a set of four equations, one for each of the four best-
fitting parameters ($kT_{i}$ and $EM_{i}$, with $i=1,2$), following Coffaro et
al. (2020):

$P_{i}^{\rm obs}-\Delta P_{i}^{\rm obs}\cdot\sigma\leq P_{i,j,k}^{\rm syn}\geq
P_{i}^{\rm obs}+\Delta P_{i}^{\rm obs}\cdot\sigma,$ (2)

where $P_{i}^{\rm obs}$ are the parameters obtained from the spectral analysis
of the observations of Kepler-63, and $\Delta P_{i}^{\rm obs}$ are the
corresponding errors. $P_{i,j,k}^{\rm syn}$ are the best-fitting parameters of
the synthetic spectra. The index $j$ denotes the 1000 spectra, randomized with
a Poisson distribution, generated for each $k$-th grid point 555The index $k$
changes according to the number of combinations of magnetic structures present
in a given grid. In this work, the final grids (i.e. colored circles in 3, 4
and 7) were composed from $\sim 400$ to $\sim 1500$ points..

In Eq. 2, the parameter $\sigma$ is the same for all four equations and
determines the global confidence range of the match between observed and
synthetic values. Thus, the criterion finds the set of best-fitting parameters
for each 1000 sets ($j$) of spectra among all grid-points ($k$) within the
smallest $\sigma$. At the end of the selection procedure there are in total
1000 best matches between one observation and the synthetic spectra, that
correspond to a range of best-fitting combinations of young COs and FLs. From
these 1000 representations, the unique best-matching combination of young COs
and FLs was found from the median of the $j$ selected values, associating as
error the $10\%$ and $90\%$ quantile of these values.

(a)  (b)

Figure 5: EMDs constructed from solar magnetic regions that best match the
observations of Kepler-63 (on the left: 2014; on the right: 2019). The
contributions to the total EMD (grey line) are from time-averaged COs (green
line) and FLs (red line) from Class C to X time-averaged over their evolution.
The confidence level $\sigma$ within which the models are selected is given on
the upper left corner of each plot. The EMDs are scaled to a larger size of
the corona of Kepler-63, i.e. $+30\%$ than its stellar radius. The red squares
are the best-fitting parameters of the X-ray spectra of Kepler-63. The black
circles are the best-fitting parameters retrieved from the best-matching
synthetic spectra. The filling factor for the best-matching combination of
magnetic structures is given in the legend.  Figure 6: Ratios between the
best-fitting parameters of the spectra of Kepler-63 and the $kT$ and $EM$
values of the best-matching synthetic spectra666The error bars are not taken
into account as the calculated uncertainties are greater than the
corresponding ratios.. The blue symbols are the ratios obtained from the
comparison with the observation of 2014, whereas the magenta ones are obtained
for the 2019 observations. The circles refer to those parameters calculated
from the best-matching EMDs that have as flaring contribution the FLs from
Class C to X time-averaged over their evolution (3.2.4). The magenta triangles
are instead calculated from the best-matching EMDs that include flares of
Class C and M time-averaged over their evolution. (3.2.5) Figure 7: X-ray
luminosity as a function of emission measure weighted thermal energy $kT_{\rm
av}$ (left panel) and the two temperatures and emission measures (right
panel). The EMDs used for generating these grids are composed of time-averaged
COs covering from $10$ to $100\%$ of the total corona and FLs of Class C and
Class M, time-averaged over their evolution, that vary from $0$ to $15\%$ of
the area of the COs. The Class M flares were also observed with Yohkoh with a
pair of soft filters in addition to the pair of hard filters (used for the
analysis of the Class C flare). The coding of the plots follows 3.

(a)  (b)

Figure 8: EMDs constructed from solar magnetic regions that best match the
observations of Kepler-63. The contributions to the total EMDs are from COs
and FLs of Class C and Class M, time-averaged over their evolution. The
confidence level $\sigma$ within which the models are selected is given on the
upper left corner of each plot. The coding of the plots follows 5.

####  3.2.4 Match with the standard flaring Sun

The first grid of EMDs, and corresponding spectra, that we tested is composed
of time-averaged COs, varying from $10\%$ to $100\%$ coverage of the corona,
and by a distribution of FLs, time-averaged over their evolution, that vary
from $0\%$ to $4\%$ of the COs. The FL distribution was computed by
considering a sample of flares from Class C to Class X observed with Yohkoh,
from their rise to their decay phase (the average EMD per unit surface is
shown in 2 as black solid line), and taking into account the power-law FFD as
explained in 3.2.1.

The plot of the X-ray luminosity as a function of the $kT_{\rm av}$ calculated
from this grid is shown in 3 on the left: each of the circles represents one
point of the synthetic grid, i.e. one specific combination of magnetic
structures. The percentage of COs is designated with the color of the circles,
whereas the percentage of FLs is represented by the size of the circles. It
can be noticed that, as the coverage fraction of COs increases on the corona,
the X-ray luminosity increases while the EM-weighted thermal energy is less
affected. Similarly, the greater the percentage of FLs is, the higher the EM-
weighted thermal energy results, whereas the variation of X-ray luminosity is
moderate 777This trend of X-ray luminosity and EM-weighted thermal energy,
induced by magnetic structures, was already noticed for the Sun by Orlando et
al. (2004).. This general behaviour of the grid is the same we found for
$\epsilon$ Eridani (Coffaro et al. 2020).

The best-fit parameters obtained from the spectral analysis of Kepler-63 (see
2) are plotted with the blue and magenta symbols. The right panel in 3 shows
the thermal energies and emission measures calculated from the grid and from
the spectra of Kepler-63. The color and size code of the plotting symbols are
the same as in the left panel. We can see that an increase of the percentage
of COs corresponds to an increase of $EM_{1}$ and $EM_{2}$ only, whereas the
FLs influence both thermal energies and the $EM_{2}$.

In the background of each plot, the number of occurrences for each pair of the
best-fitting parameters, obtained from the 1000 simulations of each $k$-th
grid point, are shown. The spread of these values provides an indication for
the range of synthetic parameters compatible with the observations.

In 3, the $L_{\rm X}-kT_{\rm av}$ plot shows that the chosen combination of
magnetic structures can reproduce the EM-weighted thermal energy of Kepler-63,
but the observed X-ray luminosity is not reached by the grid, i.e. even a full
surface coverage ($100\%$) with COs and FLs does not produce enough $L_{\rm
X}$ at the observed values of $kT_{\rm av}$ to explain the brightness of the
corona of Kepler-63.

We note that the simulations presented above do not consider any extension of
the corona above the chromosphere. In other words, the corona is assumed to
have a thickness close to zero. However, the observations of the Sun show that
the corona expands significantly above the chromosphere, with coronal magnetic
loops of ARs extending up to $20-30\%$ of the solar radius (DeForest 2007). It
is, therefore, reasonable to consider an extension of the corona of Kepler-63
above the limb.

By assuming the same types of magnetic structures (time-averaged COs and FLs)
with the same range of fractional coverage as before, we produced a grid of
EMDs adopting a $30\%$ larger radius for the corona of Kepler-63 to account
for its extension above the chromosphere. The left panel in 4 shows $L_{\rm
X}$ as a function of $kT_{\rm av}$ calculated from this grid. The coding of
the plot follows that of 3. Here, the X-ray luminosity reproduces the
observations of Kepler-63, although at the limit within the error bars. We,
therefore, examined this model in more detail.

First, by adopting the selection criterion of Eq. 2, we obtained the EMDs that
best match the observations, shown in 5. In these plots, the grey line is the
total EMD that best matches the observations of Kepler-63, and the green and
red lines are the EMDs of the COs and FLs that contribute to the total
distribution at a given fraction coverage which is reported in the legend of
the plots. The red and black symbols are the best-fitting parameters obtained
from the spectral analysis of the actual spectra of Kepler-63 and the
synthetic ones, respectively.

In 6 the ratio between the best-fitting parameters of the spectra of Kepler-63
and those of the best-matching synthetic spectra are plotted as circles. It
can be noticed that for the observations of 2019 (magenta circles) there is a
better agreement with the selected models than for the observation of 2014
(blue circles) that shows larger discrepancies in particular in the ratios of
the parameters of the first spectral component.

####  3.2.5 Match with the modified flaring Sun

To reduce the discrepancies seen in 4 and 5 between the best-fitting
parameters of the models and those of the observations of Kepler-63, we need
to change the components of the EMDs. As can be seen in 4 (right panel), a
grid with lower $kT_{2}$ and $EM_{2}$ would better represent the data
(especially for 2014). It is also clear in the plot that the COs reach the
high X-ray luminosity of Kepler-63, and, therefore, this kind of structure is
needed. We can thus modify only the FL component, and, as from 4 we see that
we need smaller values of $kT_{2}$, a FL distribution composed of mid-
energetic flares appears more suited.

Among the solar flaring observations acquired with Yohkoh and presented by
Reale et al. (2001), there are two flares of Class M that were observed by the
satellite with a pair of soft filters, in addition to the pair of hard filters
that are most frequently used for Yohkoh observations of flares. The soft
filters yield an EMD that extends to lower temperatures, thus providing a more
complete description of the EMD of flaring regions (see Reale et al. 2001, for
more details). This is seen in 2 where the EMD that includes the soft-filter
observations (dotted black line) extends to lower temperatures than the time-
averaged EMD of all (hard-filter) flares in the sample observed with Yohkoh.

We, therefore, changed the FL distribution using the EMDs of solar flares of
Class M, analysed with the two pairs of filters (soft and hard) and time-
averaged across their evolution. Together with the Class M flares, we also
considered the contribution of Reale et al. (2001)’s Class C flare to the EMD,
as from the canonical flare frequency distribution (see 3.2.1) the number of
Class C flares is higher than that of Class M events, and therefore the former
ones can not be excluded.

In this new simulation the grid range of the coverage fraction of COs remains
unchanged, that is varying from $10$ to $100\%$. The FLs are instead allowed
to vary in a broader range of CO filling factor than before, namely $0-15\%$.
This is motivated by the fact that the new FL component show an EMD per unit
surface area smaller than the flares from Class C to X used in the previous
grid (see the dotted black line versus the solid one in 2). Moreover, the EMD
of Class M flares covers a wider range of temperatures than the one of the
time-averaged flares at different energies. This is due to the fact that the
flaring distribution of Class M FLs was derived by using both soft and hard
filters, providing a contribution to the EMD at lower temperatures. Thus, to
reproduce the observed $kT_{\rm av}$, we needed to assume a higher percentage
of FLs than in the previous case.

Starting from these EMDs, we extracted the corresponding synthetic spectra and
we analysed them as we have done previously, using a 2-T APEC model. Then, we
proceeded with the selection criterion explained in 3.2.3.

In 7, the plot of $L_{\rm X}$ as a function of $kT_{\rm av}$ (on the left) and
the plots of the $kT$s and $EM$s (on the right) are shown. The coding of these
plots is the same as for 3. Such a high percentage of FLs reproduces the X-ray
luminosity without the need to account for the thickness of the corona of
Kepler-63888We note that neglecting the thickness of the corona corresponds to
deriving an upper limit to the surface coverage of COs and FLs; see discussion
in 4.3.. In 8, the EMDs that best match the observations are shown. In 6, the
ratios between the observed best-fitting parameters and the ones of the best-
matching synthetic spectra for this test are shown as triangles. Here, it can
be noticed that overall the ratios between the temperatures are closer to one
than in the previous case, indicating that the replacement of the flaring EMD
has led to a better agreement between the observations and synthetic spectra.
On the other hand, the $EM$ values, and in particular those of the 2014
observation, have not changed strongly with respect to the previous
combination of magnetic structures.

##  4 Discussion

Kepler-63, a solar-like star with an age of only $\sim 200$ Myr, is the
youngest star that so far was observed in X-rays with XMM-Newton with the goal
of searching for a coronal magnetic cycle. Our X-ray monitoring campaign
consisted in four snapshots of its corona, distributed between 2019 and 2020,
thus spanning 3/5 of the photospheric cycle ($P_{cycl}=1.27$ yr; Estrela &
Valio 2016).

One additional observation was available in the XMM-Newton archive, dated back
to 2014. Lalitha et al. (2018) analysed this observation, using a 2-T APEC
model, and they obtained an X-ray luminosity $\log L_{\rm X}\ [\rm
erg/s]=29.00\pm 0.17$ and a EM-weighted thermal energy $kT_{\rm av}=0.59\pm
0.07$ keV. These values are in agreement with our analysis.

The spectral analysis of all available XMM-Newton observations, comprising the
archival observation of 2014, yields a mean X-ray luminosity $\log L_{\rm X}\
[\rm erg/s]=29.03\pm 0.49$, that corresponds to a X-ray surface flux of $\log
F_{\rm X,surf}[\rm erg/cm^{2}/s]=6.32\pm 0.05$.

###  4.1 Search for an X-ray cycle

The photospheric cycle of Kepler-63 was revealed from observations with the
Kepler telescope across 4 years (from 2009 to 2013), during which the
variation of stellar spots had been monitored. The first observed minimum and
maximum happened on August 2009 and March 2010 respectively (see Fig. 10 in
Estrela & Valio 2016). We propagated across time the sinusoidal function,
representing the $1.27$-yr photospheric cycle of Kepler-63, that Estrela &
Valio (2016) obtained from the Lomb-Scargle analysis of the number of spots
seen in the photosphere 999The number of spots seen in the transit curve of
Kepler-63 was scaled to the running mean over a range of five data points
(i.e. five stellar spots) and to the residuals of a quadratic polynomial fit
to the data. This implied that the final number of spots had non-interger and
negative values (Estrela & Valio 2016).. We found that the maximum and minimum
closest to our X-ray campaign were expected for the end of January 2019 and
for mid September 2019. In 9 we show the long-term X-ray lightcurve of
Kepler-63 where the solid blue line is the folded sinusoidal function
calculated by Estrela & Valio (2016) and the dotted line is our extrapolation
of the photospheric cycle. The red symbols are the X-ray fluxes derived from
our monitoring of Kepler-63. Clearly, the uncertainties of the X-ray flux in
each XMM-Newton observation are larger than the changes of the flux between
the observations. Therefore, we can only state that any potential X-ray cycle
during our campaign in 2019-2020 had a minimum-to-maximum variation smaller
than a factor 2.

Figure 9: Long-term X-ray lightcurve of Kepler-63. The red triangles are the
X-ray fluxes, with the associated errors, calculated from the XMM-Newton
observations. The blue solid line is the sinuisodal function describing the
photospheric cycle (1.27 yr) obtained by Estrela & Valio (2016). The dotted
black line is our extrapolation of the sine curve.

###  4.2 Activity level and cycle amplitude of Kepler-63

Figure 10: X-ray surface flux of all stars with known X-ray cycles as a
function of the cycle amplitude i.e. the variation of $L_{\rm X}$ between the
maximum and the minimum of the activity cycle. In the plot, also Kepler-63 is
included although we detect no X-ray cycle. The color bar denotes the ages of
the stars. The dotted red line is the linear regression performed on the data
set. The result of the fit is given in the legend of the plot, while the
residuals of the procedure are reported in the bottom panel.

In 10 we put Kepler-63 together with stars with known coronal cycle. It can be
seen that Kepler-63 is the most active of them, i.e. it presents the highest
surface X-ray flux. 10 shows the logarithm of the X-ray surface fluxes
($F_{\rm X,surf}=L_{\rm X}/4\pi R_{\star}$) as a function of the X-ray cycle
amplitude ($A=L_{\rm max}/L_{\rm min}$) in log scale, defined as the variation
of the X-ray luminosity between the maximum and minimum of the coronal
cycle101010The X-ray surface fluxes and the cycle amplitude of each star in
the sample are calculated from the X-ray luminosity and the long-term
lightcurves published in the literature (the Sun from Orlando et al. 2001;
$\alpha$ Centauri A and B and 61 Cygnus A and B from Robrade et al. 2012;
HD81809 from Orlando et al. 2017; $\iota$ Horologii from Sanz-Forcada et al.
2019; $\epsilon$ Eridani from Coffaro et al. 2020).. We notice that there is a
linear correlation between the log of these two quantities: as the X-ray
activity level of the stars decreases (and, in turn, the stellar age
increases), the amplitude of their coronal cycles increases. We interpret this
result as due to a decrease in the coverage of the stellar surface with
magnetic structures as the star evolves. When the star is young the massive
presence of magnetic structures limits the variations of X-ray luminosity
during the stellar cycle; when the star is old, the lower coverage of the
stellar surface with coronal structures allows the latter to have large
changes in their coverage during the cycle, leading to large variations of the
X-ray luminosity. This scenario is consistent with the expectation from dynamo
theory where the alternation of the $\alpha$\- and $\Omega$-effect causes
periodic changes of the number of magnetic structures, and their number is
larger for faster rotators. As rotation rate drops with stellar age our color
code in 10 can be taken as a code for rotation rate and, in fact, with $P_{\rm
rot}=5$ d Kepler-63 is the fastest rotator in the sample.

According to the linear regression on the dataset in 10, the X-ray surface
flux of the stars reaches a maximum value $\log F_{\rm
X,surf}\rm{[erg/cm^{2}/s]}\sim 6.3$ when the stars have no activity cycle
(i.e. when the amplitude $\log L_{\rm max}/L_{\rm min}=0$). This flux,
however, can not be considered as an upper limit to the X-ray flux that active
stars can reach because X-ray observations have revealed a significant
fraction of stars with fluxes up to $\log F_{\rm
X,surf}\rm{[erg/cm^{2}/s]}\sim 7.5$ (Schmitt 1997). Since we have shown
through our modelling of Kepler-63 (summarized in 4.3) that the stellar corona
is fully covered with magnetically active structures already at one order of
magnitude smaller flux, such high values for $F_{\rm X,surf}$ indicate not
only that the coronae of these stars have a 100% filling factor but also that
they must have flares of higher energy than the ones we have inferred to be
present on Kepler-63. The presence of high energy flares is a reasonable
hypothesis as e.g. pre-main sequence stars have flares with much greater
energies (and X-ray fluxes) than solar flares (Getman & Feigelson 2021).

###  4.3 Magnetic structures on the corona of Kepler-63 and implications for
cycles

We quantitatively investigated if the undetected X-ray cycle amplitude is
linked to a massive presence of magnetic structures on the corona of
Kepler-63, by applying a novel technique based on the study ”The Sun as an
X-ray star” (3.1). This technique allows us to indirectly gain information on
the magnetic structures that determine the X-ray emission level of Kepler-63.
The method builds on the EMDs of the magnetic structures observed by the solar
X-ray satellite Yohkoh on the corona of the Sun: BKCs, ARs, COs and FLs. The
EMDs can be built assuming that different percentages of these magnetic
structures determine the total emission. From these EMDs, X-ray spectra of a
pseudo-Sun (in the present case, Kepler-63) can then be synthesized as if they
were collected with XMM-Newton.

In our previous application of this technique to $\epsilon$ Eridani (Coffaro
et al. 2020), the high signal-to-noise ratio of the observations enabled a
very detailed description of the coronal temperature structure of the star,
outlined by a 3-T model fit to the EPIC/pn spectra. On the other hand for the
case of Kepler-63, we can analyse most of its EPIC/pn observations employing a
one temperature spectral model, as seen in 2.2. To increase the signal-to-
noise ratio, and thus to use a 2-T spectral model for achieving a more
detailed description of the corona of Kepler-63, it was necessary to reduce
the X-ray monitoring campaign to only two observational epochs, one describing
the corona in 2014 and the other one describing its average behaviour during
2019. This was achieved by simultaneously analysing all observations acquired
in that year.

We found that the coronal temperature structure of Kepler-63 is reproduced
when we assume time-averaged COs together with a FL EMD. BKCs and ARs are
neglected as they have a low surface brightness. Excluding BKCs and ARs does
not imply that such regions are not present on the stellar corona. However,
the high surface coverage of COs that is necessary to produce an X-ray
luminosity as high as that observed leaves only a small percentage of the
stellar surface available for BKCs and ARs. The low filling factor of these
types of regions together with the fact that they are characterized by a much
lower surface brightness in comparison with that of COs and FLs make the
contribution of BKCs and ARs to the total X-ray emission of the star
negligible with respect to that of COs.

For the flaring contribution to the total EMD, we tested two scenarios: in the
first one, we assumed that a distribution of flares from (GOES) Class C to
Class X, from their rise to their decay, are present in the corona of
Kepler-63, re-conducting its behaviour to a standard flaring Sun. In the
second scenario, we assumed that only lower-energetic flares, i.e. from Class
C (the least energetic flaring events of our sample) to Class M, are present.
This was necessary because, as shown in 6, the first scenario can not
reproduce the X-ray temperatures observed in Kepler-63.

For the case of the standard flaring EMD, we examined the effect of scaling
the simulated EMDs to two different sizes of the corona: in a first approach
we adopted the literature value for the radius of Kepler-63, neglecting the
extension of its corona above the chromosphere. In a second test we considered
the possibility that the corona of Kepler-63 significantly extends above its
surface, and in particular that the extension is $30\%$ of the stellar radius
($R_{\star}=0.91R_{\odot}$). This hypothesis is motivated by solar
observations, from which it is evident that the corona of the Sun is spread
above the surface up to $20-30\%$ of its radius (DeForest 2007).

5 and 8 show that, due to the large errors associated to each best-fitting
parameter for both observations and synthetic spectra (black and red symbols),
both tested scenarios (i.e. the EMDs with different flaring components and
different sizes of the corona) can reproduce the spectra of Kepler-63. The
parameter $\sigma$, defined in Eq. 2, is $\sim 1$ when the selection criterion
is run over the first scenario, i.e. the EMDs composed of all classes of FLs
plus a $30\%$ coronal extension. In the second tested case, i.e. the EMDs with
Class C and M FLs and no extension of the corona, $\sigma$ results to be $\sim
0.6$. Thus, we choose this latter model as the preferred one. For this
scenario we find that the magnetic structures cover the whole corona, i.e. the
filling factor is $100\%$ for both X-ray epochs of Kepler-63 (8). The complete
coverage of the corona of Kepler-63 strengthens the hypothesis that the
amplitude of an eventual X-ray cycle is significantly reduced by the massive
presence of coronal structures in all phases of the cycle.

As with this simulation we can reproduce the observations of Kepler-63 without
considering the thickness of the corona and, thus, we underestimate the
extension of the corona above the stellar surface, the filling factors of the
magnetic structures can be taken for upper limits. If we consider the corona
to have an extent of $30\%$ of the stellar radius covered with the same types
of structures (i.e. Class C and M flares and COs), the surface coverage with
magnetic structures would be reduced by a factor $1.7$ and only $\sim 60\%$ of
the corona would result filled by COs and FLs. Most likely, the regions not
covered with COs and FLs may be filled with BKCs and ARs. This, however, has a
negligible impact on the EMD and X-ray emission as explained above. The lower
coverage fraction of the extended corona would imply that a change of the
X-ray luminosity by a factor of two is possible during the cycle of Kepler-63,
as almost half of the corona would be free for new magnetic structures to rise
and evolve during a potential cycle maximum. Although our data do not suggest
the presence of an X-ray cycle, a factor 2 variation can not be excluded given
the large uncertainties of the individual flux measurements (see 9 and 4.1).

###  4.4 Nature of the flares on Kepler-63

Another intriguing characteristic we face is related to the nature of the
magnetic structures we used for modelling the corona of Kepler-63. To
reproduce best the coronal structure of Kepler-63, the hottest component of
the total EMDs needs to be at lower energy than the typical solar flare EMD
(which indicatively has temperatures around $7-10$ MK). To represent such
energies, we made use of low- and mid-energy flares from Class C to M, but
omitting the more energetic Class X events. Since young solar-like stars show
lower metallicity than the Sun, the radiative cooling is less efficient. This
in turn implies that the evolution of flares is slowed down and the events are
on average seen at lower temperatures. Thus, it is plausible that overall low-
and mid-energetic flares are dominant on Kepler-63, and such a scenario is
supported by our models.

We stress that we are exploring a quasi-stationary component of the corona of
Kepler-63 resulting from the superposition of several unresolved flares. From
the canonical power-law distribution, the frequency of higher-energy flares
(such as, for instance, Class X flares) is expected to be lower than that of
low-to-mid energetic flares and, therefore, they would not contribute
significantly to the quasi-stationary component. High-energy flares may be
revealed as short-term variability in the X-ray lightcurves. As a rough
estimate, we can assume that a flare would have been detected in the
lightcurve if it was $2\delta$ above the mean count rate. Here $\delta$ is the
standard deviation, and for Kepler-63 we find $\delta=0.01\ \rm cts/s$. Using
the rate-to-flux conversion factor obtained from the EPIC/pn spectral analysis
($CF=1.2\cdot 10^{-12}\rm erg/cm^{2}/cts$) such a 2$\delta$ event corresponds
to a flux amplitude of $2.5\cdot 10^{-14}\rm erg/cm^{2}/s$, and the
corresponding peak flare flux at the surface of Kepler-63 is $F_{\rm
X,surf,2\delta}=2.3\cdot 10^{6}\rm erg/cm^{2}/s$. The total surface flux of
Kepler-63’s quiescent X-ray emission (that includes the quasi-stationary
flaring component) is $2.1\cdot 10^{6}\ \rm erg/cm^{2}/s$. This means that a
2$\delta$ event has $1.1\times$ the total surface flux of the star, i.e. only
events that provide an X-ray flux that is higher than that of the whole rest
of the star’s corona can be identified in the EPIC/pn lightcurve. Such events
have $45\times$ the surface flux of an X9 Class flare111111This estimate does
not consider the different energy bands of XMM-Newton/EPIC and the GOES flare
classification; see below for an estimate of the conversion.. No clear flare
signatures are seen in 1, but the strong variability between two subsequent
1000-bins could, in principle, be caused by such high-energy flares. Judging
from the lightcurves, the rate of such events would be one every 2-3 ksec with
a duration of $\sim$ 1 ksec (one bin) each. In this interpretation, the mean
count rates indicated in 1 would not be representing the quiescent activity
level. We defer a more quantitative analysis of flare detection thresholds to
a future work. In the following we estimate the number of Class X flares on
Kepler-63 on the basis of the frequency distribution of flare energies.

The coronal filling factor for Class X flares can be estimated from the FFD
power-law. In our analysis we have normalized the FFD to the smallest Yohkoh
flare which is of Class C5.8. With the power slope $\alpha=1.57$ we infer a
flare frequency for Class X9 flares121212We arbitrarily choose the X9 flare
for our calculation since this is the largest flare in the Yohkoh sample
(Reale et al. 2001); a more accurate treatment should take account of the
continuous flare distribution. by $7.2\cdot 10^{-4}$ smaller than that of
Class C5.8. Our modelling of the X-ray spectra of Kepler-63 in 3.2.5 has
yielded a filling factor with FL of $\sim 13.4\%$ of the stellar surface. This
number refers to Class C5.8 flares. Accordingly, the filling factor for Class
X9 flares is $ff_{\rm X9}\sim 9.7\cdot 10^{-5}$. With a reasonable assumption
on the coronal area covered by one such flare we can estimate the number of
Class X9 flares present on the surface of Kepler-63 at any time. We carry out
the calculation assuming that the area covered by a Class X9 flare on the
corona is $A_{\rm f}=3\cdot 10^{19}\ cm^{2}$. This value is directly derived
from the Yohkoh images for the X9 flare of the sample which is our reference
point (Reale et al. 2001). We then scale this result to the total surface area
of Kepler-63. The number of Class X9 flares on Kepler-63, $N_{\rm X9}$, is
then given by the ratio of the filling factor $ff_{\rm X9}$ and the fractional
surface area of one such event, $A_{\rm f}/4\pi R_{\star}^{2}$. We find
$N_{\rm X9}\sim 0.16$. This means bright flares are rare events on Kepler-63,
justifying a posteriori the omission of Class X flares in our EMD modelling on
timescales consistent with those of observations.

We note that several uncertainties contribute to this calculation of the flare
number and energetics. In particular, the number of X9 Class flares depends
strongly on the area covered by such an event. The Yohkoh X9 flare from Reale
et al. (2001) was a so-called complex flare, characterized by an arcade of
loops. The value for the flare area derived by Reale et al. (2001) for this
event is consistent with the areas inferred by Aschwanden & Aschwanden (2008)
for GOES Class M and Class X flares. Our above estimate for the flare number,
therefore, is likely to be representative for solar flares.

A second uncertainty in our modelling is the slope of the FFD distribution for
which we used the Yohkoh value, $\alpha=1.57$. If, instead, we assume a
steeper FFD ($\alpha=2.0$, based on observations of stellar flares in the
optical, see e.g. summary by Ilin et al. 2021), the conditions are even more
relaxed, yielding flare numbers that are lower by approximately a factor 10.

Finally, as mentioned in footnote 7, we are mixing numbers derived from
different instruments. The flare classification is based on GOES $1-8\ \AA$
flux while our analysis uses data from the Yohkoh/SXT. Argiroffi et al. (2008)
determined the scaling between flare emission measures observed with these two
instruments to be $EM_{\text{Yohkoh}}\sim 0.5\cdot EM_{\text{GOES}}$. The
factor two uncertainties that results if a similar relation holds for the flux
values of Yohkoh and GOES is clearly smaller than the uncertainties caused by
the other assumptions described above.

To summarize, a calculation based on the properties of a Class X9 flare
observed with Yohkoh shows that the number of such bright flares present in
the corona of Kepler-63 is negligible, consistent with the requirements from
our ”The Sun as an X-ray star” modelling of the X-ray spectrum and with the
non-detection of such events in the X-ray lightcurve of the star. However, for
other plausible flare parameters a higher number of X9-type events can be
obtained, and such a scenario can not be excluded by the X-ray lightcurve that
has too poor a signal-to-noise ratio to distinguish individual Class X flares.

##  5 Conclusions and outlook

With our simulations and their comparison to the observed XMM-Newton spectra
we have identified the types of solar magnetic structures that best reproduce
the coronal X-ray emission of Kepler-63. We found that modifications are
required as to the presence and relative importance of the different magnetic
features with respect to those that dominate the X-ray emission of the Sun. In
our earlier analogous study of another young solar-like star, $\epsilon$
Eridani (Coffaro et al. 2020), we also had to adapt the solar magnetic
structures to be able to fit the observations of the stellar corona, but in
different ways from what we require for Kepler-63. We conclude that stellar
coronae come in a variety of physical characteristics.

The non-detection of an X-ray cycle on Kepler-63 is compatible with the
complete coverage of its corona with magnetic structures as we have derived
from our modelling. The correlation between cycle amplitude and X-ray surface
flux that we find for the sample of stars with X-ray cycles suggests a
limiting X-ray brightness for stars that can sustain cycles of $\log{F_{\rm
X,surf}}\,{\rm[erg/cm^{2}/s]}\sim 6.3$, the value of Kepler-63. Stars with
much higher values of X-ray surface flux than Kepler-63 are known (see e.g.
Schmitt 1997), and we speculate that these stars must present higher energy
flares in order to reach their observed X-ray radiation output.

The modelling is likely also affected by the quality of the observations, e.g.
the signal-to-noise ratio of the stellar X-ray spectra and the range of flares
included in the Yohkoh EMDs. Future work should address these points by
investigating additional stellar X-ray observations with the technique of ”The
Sun as an X-ray star” and by closing holes in the solar flare statistics.

###### Acknowledgements.

We thank the anonymous referee for the useful input that improved our final
discussion. MC acknowledges financial support by the Bundesministerium für
Wirtschaft und Energie through the Deutsches Zentrum für Luft- und Raumfahrt
e.V. (DLR) under the grant FKZ 50 OR 2008. S.O. acknowledges financial
contribution from the agreement ASI-INAF n.2018-16-HH.0. 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.

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##  Appendix A EPIC/pn lightcurve of the background detector

Here, the lightcurve of the background detector of the EPIC/pn for each
observation of Kepler-63 are shown. In each plot, the red lines and red
intervals denote the GTIs chosen in each lightcurve as explained in 2.

(a)  (b)

(c)  (d)

(e)

Figure 11: Lightcurves of the full-detector background of EPIC/pn. The red
lines and the red intervals show where the GTIs were chosen.

##  Appendix B Spectral analysis

The EPIC/pn spectra of Kepler-63 are shown in this section. The solid red line
is the spectral model (APEC) used for fitting the observations. The bottom
panels in each plot are the residuals of the fitting. For the observations of
2014, of August 2019 and the simultaneous fit of 2019 observations, a two
temperature spectral model is adopted. In all other cases, only one thermal
component is used, with the exception of the observation of March 2020. As
explained in 2.2, this latter observation suffers from low photon statistics,
and it is not possible to carry out a meaningful fit. Values of the best-
fitting parameters are found in 2.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 12: EPIC/pn spectra of Kepler-63. The solid line is the best-fitting
spectral model. The bottom panels are the reduced $\chi^{2}$ residuals of the
fitting.