# Delayed Development of Cool Plasmas in X-ray Flares from $\kappa^{1}$ Ceti

Kenji Hamaguchi CRESST II and X-ray Astrophysics Laboratory NASA/GSFC,
Greenbelt, MD 20771, USA  Department of Physics, University of Maryland,
Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA  Jeffrey W.
Reep Space Science Division, US Naval Research Laboratory, Washington, DC
20375, USA  Vladimir Airapetian American University, 4400 Massachusetts Avenue
NW, Washington, DC 20016 USA  NASA/GSFC/SEEC, Greenbelt, MD, 20771, USA  Shin
Toriumi Institute of Space and Astronautical Science, Japan Aerospace
Exploration Agency, 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa, 252-5210,
Japan  Keith C. Gendreau X-ray Astrophysics Laboratory, NASA/GSFC, Greenbelt,
MD 20771, USA  Zaven Arzoumanian  X-ray Astrophysics Laboratory, NASA/GSFC,
Greenbelt, MD 20771, USA

(Received 2022 July 18; Accepted 2022 December 21)

###### Abstract

The Neutron star Interior Composition ExploreR (NICER) X-ray observatory
observed two powerful X-ray flares equivalent to superflares from the nearby
young solar-like star, $\kappa^{1}$ Ceti, in 2019. NICER follows each flare
from the onset through the early decay, collecting over 30 cnts s-1 near the
peak, enabling a detailed spectral variation study of the flare rise. The
flare in September varies quickly in $\sim$800 sec, while the flare in
December has a few times longer timescale. In both flares, the hard band
(2$-$4 keV) light curves show typical stellar X-ray flare variations with a
rapid rise and slow decay, while the soft X-ray light curves, especially of
the September flare, have prolonged flat peaks. The time-resolved spectra
require two temperature plasma components at kT $\sim$0.3$-$1 keV and
$\sim$2$-$4 keV. Both components vary similarly, but the cool component lags
by $\sim$200 sec with a 4$-$6 times smaller emission measure (EM) compared to
the hot component. A comparison with hydrodynamic flare loop simulations
indicates that the cool component originates from X-ray plasma near the
magnetic loop footpoints, which mainly cools via thermal conduction. The time
lag represents the travel time of the evaporated gas through the entire flare
loop. The cool component has several times smaller EM than its simulated
counterpart, suggesting a suppression of conductive cooling possibly by the
expansion of the loop cross-sectional area or turbulent fluctuations. The cool
component’s time lag and small EM ratio provide important constraints on the
flare loop geometry.

Main sequence stars (1000), Solar analogs (1941), Stellar flares (1603),
Stellar x-ray flares (1637)

††facilities: NICER(XTI), XMM(RGS)††software:  HEASoft (Nasa High Energy
Astrophysics Science Archive Research Center (2014), Heasarc), xspec (Arnaud,
1996), scipy (Virtanen et al., 2020), astropy (Astropy Collaboration et al.,
2013; Scargle et al., 2013), SAS (v19.0; Gabriel et al., 2004), HYDRAD
(Bradshaw & Mason, 2003)

##  1 Introduction

Solar and stellar flares are the most energetic events on low-mass stars
(Haisch et al., 1991; Güdel & Nazé, 2009). They represent the rapid conversion
of magnetic energy of active regions into kinetic and thermal energies,
radiating from radio to gamma-rays and ejecting high-energy nuclei and
electrons. Powerful solar flares have disrupted the Earth’s magnetosphere and
human activity, yet flares of young Sun-like stars can far surpass their solar
counterparts in energy and frequency, with their enhanced magnetic dynamos
driven by rapid rotations and deep convections. Their intense radiation could
impact the exoplanetary environment and habitability (e.g., Airapetian et al.,
2020).

These flares, even with substantial energy variations, share similar behavior
and characteristics and arise from the universal magnetic reconnection
mechanism. Magnetic reconnection efficiently accelerates particles to high
energies ($\gtrsim$10 keV), which bombards the footpoints of the loops with
high-energy particles and heats the chromosphere; the evaporated gas fills the
magnetic loop and gradually cools down via radiation. The evaporated gas at
$\approx$107 K radiates primarily in soft X-rays between 0.1$-$10 keV
($\approx$1$-$100 Å). Therefore, soft X-ray observations are crucial in
understanding the flare geometry and heating mechanisms.

During a typical flare, soft X-ray emission rises quickly as the evaporated
gas fills the magnetic loop and decays quasi-exponentially as it gradually
cools down radiatively. Earlier studies have focused on the peak and decay
phase of flares (White et al., 1986; van den Oord & Mewe, 1989; Reale &
Micela, 1998; Tsuboi et al., 1998; Favata et al., 2000; Sasaki et al., 2021).
They suggested that powerful flares tend to decay slowly and originate from
larger flare loops, which exceed the stellar radius in extreme cases. Direct
solar flare imagings, stellar flare occultation observations, or theoretical
models support this idea, but the models can significantly overestimate the
flare size due to continuous heating, multiple loop structures or subsequent
flares during the decay phase (e.g., Toriumi et al., 2017; Schmitt & Favata,
1999; Güdel et al., 2004; Reep & Toriumi, 2017).

The rising phase holds crucial information on the flare geometry and heating
mechanism (e.g., Reale, 2007) as it goes through initial heating, evaporation,
and loop filling. However, the rising phase is often shorter than the decaying
phase (e.g., Reep & Knizhnik, 2019; Getman et al., 2021), and so has been
mostly limited to duration or crude hardness ratio studies in the soft X-ray
band. An exception is an XMM-Newton observation of Proxima Centauri, which
caught a bright flare from the onset to the middle of the decay, recording
$\gtrsim$100 cnts s-1 near the peak (Güdel et al., 2002, 2004; Reale et al.,
2004). The X-ray hardness ratio reached its maximum in the middle of the rise
and started to decline near the flux peak. The timing of maximum hardness
coincides with the U band (3000$-$3900Å) flux peak measured with the onboard
Optical Monitor, suggesting a connection between the coronal and chromospheric
heating.

The NICER (Neutron star Interior Composition ExploreR) X-ray observatory
onboard the International Space Station (ISS) (Gendreau & Arzoumanian, 2017)
observed two bright X-ray flares from the nearby solar-like star
$\kappa$(kappa)1 Ceti (a.k.a. HD 20630, HIP 15457, $d=$9.16 pc, mass: 1.04 M⊙,
radius: 0.95$\pm$0.10 R⊙, effective temperature: 5665 K, Ribas et al., 2010;
Rucinski et al., 2004) during a monitoring program for the Sellers Exoplanet
Environments Collaboration (SEEC)111https://seec.gsfc.nasa.gov in 2019. This
star shows intense magnetic activity due to its fast stellar rotation ($P$
=9.2 days), emitting two orders of magnitudes higher coronal X-rays and
chromospheric UV light than the Sun. In 1986, the star showed a signature of a
superflare event in the He I D3 ($\lambda$5875.6 Å) optical line, with an
estimated total flare energy of $E\approx$2$\times$1034 ergs (Schaefer et al.,
2000). Still, the the radiation from tranition region and coronal plasma
satisfies a solar magnetic flux scaling law similar to other Sun-like stars
(Toriumi & Airapetian, 2022). These characteristics suggest that $\kappa^{1}$
Ceti is a young solar analog at 0.4$-$0.8 Gyrs old with an enhanced solar-type
coronal and chromospheric heating rates (Airapetian et al., 2021). Its global-
scale magnetic shear may cause superflares that eject huge masses of coronal
material (Lynch et al., 2019).

The NICER X-ray observatory primarily aims at studying rapidly rotating
neutron stars with very high timing resolution, but its superb soft X-ray
collecting power, wide dynamic range, high throughput and moderate background,
decent energy resolution, tolerance to optical light, and rapid maneuvering
capability make it a powerful tool for observing nearby solar-type stars with
sporadic bright X-ray flares. This manuscript describes analysis of NICER
observations of the two powerful X-ray flares from $\kappa^{1}$ Ceti and
performs hydrodynamic simulations of single loop flares to interpret the
observations. The result provides how X-ray plasmas develop during the flare
rising phase in detail.

Figure 1: Background subtracted light curves of $\kappa^{1}$ Ceti between
0.6$-$1.2 keV on 2019 September 16$-$17 (top) and 2019 December 10 (bottom).
NICER opportunely caught the rising phase of a bright X-ray flare during each
observation. The grey and yellow lines show the background level estimated
with the NICER tool nibackgen3C50 and an average count of each Bayesian block
derived with the astropy tool bayesian_blocks. The time on the top shows the
light curve origin, an estimated flare onset. The color bars present quiescent
(magenta), preflare (red), flare rise & peak (blue) and flare decay end (cyan)
intervals for the spectral analysis. Each data bin is 50 sec.

##  2 Observation

The NICER X-ray Timing Instrument (XTI) is an array of aligned 56 X-ray
modules, each of which consists of an X-ray concentrator (XRC, Okajima et al.,
2016) and a silicon drift detector (SDD, Prigozhin et al., 2016). Each XRC
concentrates X-rays within a $\sim$3′ radius field of view to the paired SDD,
which detects each photon with accuracy at $\sim$84 ns. The XTI as a whole has
one of the largest collecting areas among X-ray instruments between 0.2$-$12
keV ($\sim$1900 cm-2 at 1.5 keV). We use 50 XTI modules as the remaining six
(ID: 11, 14, 20, 22, 34, 60) are inactive or noisy.

NICER can continuously observe a target up to $\sim$2.5 ksec in every ISS
orbit ($\sim$5.5 ksec). However target visibility can be limited by Earth or
ISS structure occultation, or proximity to high particle regions such as the
South Atlantic Anomaly. NICER can quickly slew the telescope and so observes
multiple targets in each ISS orbit to maximize the observing efficiency. This
capability enables NICER to visit a target frequently, but it can also cause
scheduling conflicts with other timely targets.

NICER started monitoring $\kappa^{1}$ Ceti on 2019 September 16; it has
observed the star for $\sim$180 ksec between 2019$-$2021\. During these
monitoring observations, NICER detected two prominent X-ray flares on 2019
September 17 and December 10. Earlier X-ray imaging observations did not
detect any X-ray sources at significant X-ray brightness within 3′ from
$\kappa^{1}$ Ceti (e.g., Telleschi et al., 2005), indicating that the flares
originate from $\kappa^{1}$ Ceti.

We analyze the NICER observations ID: 2300020101, 2300020102, and 2300020114.
We reprocess the datasets with the NICER calibration ver. CALDB XTI(20210707),
using nicerl2 in HEASoft ver. 6.29c and NICERDAS ver. V008c. We evaluate
particle background using nibackgen3C50 ver. v7b with the parameters
dtmin=10.0, dtmax=60.0, hbgcut=0.1, s0cut=30.0 (Remillard et al., 2021). We
use python ver. 3.7, numpy ver. 1.20.3, scipy ver. 1.1.0 and astropy ver. 3.1.

##  3 Observation Results

Figure 2: 1st$-$5th rows: band-sliced light curves of $\kappa^{1}$ Ceti around
the first flare on 2019 September 17 (left) and the second flare on December
10 (right). These light curves have the same horizontal scales, so the
variation timescales are directly comparable. The soft band light curves on
the upper panels have delayed peaks compared with the 2$-$4 keV light curve.
The grey line shows the instrumental background level. Each light curve has 50
sec bins. Bottom row: time-series of the hardness ratio defined by
$(H-S)/(H+S)$ where $H$ and $S$ are the net count rates between 1.2$-$4 keV
and 0.3$-$1.2 keV, respectively. The color bars present preflare (red), flare
rise & peak (blue) and flare decay end (cyan) intervals for the spectral
analysis. The first bin of the September flare is below $-$1 as nibackgen3C50
overestimates the hard band background. In each flare, the hardness ratio
peaks before the total (0.3$-$4 keV) band count rate maximum.

###  3.1 Light Curves

The first flare occurred on 2019 September 17, during the last snapshot of the
one-day observation of $\kappa^{1}$ Ceti from September 16 (190917, Figure 1
top). The snapshot only lasts for $\sim$800 sec, but it covers the rise and
beginning of the decay of the flare as the flare varies very quickly. The
Bayesian block analysis tool, bayesian_blocks in the astropy package (Scargle
et al., 2013) — nonparametric statistical analysis to detect significant flux
change in time-series data — does not suggest that the 0.6$-$1.2 keV count
rate changes from the previous snapshot to the first 100 sec of this snapshot.
This result suggests that the flare begins around the boundary of the last
snapshot’s second and third time bins, 2019 September 17 at 9h 36m 52s UT.

Figure 2 left shows band-sliced light curves of the last snapshot. The 2$-$4
keV light curve is typical of solar and stellar X-ray flares (Benz & Güdel,
2010, and references therein), showing a sharp rise in $\sim$200 sec and a
steady decline by a factor of 3 in $\sim$500 sec after the peak. However, the
softer band light curves rise more slowly, peak later, and decay more
gradually. The light curves below 1.2 keV may not even show a decay during the
snapshot. The soft band light curves significantly deviate from the hard band
light curves. The hardness ratio in the bottom panel rises quickly, peaks
before the total (0.3$-$4 keV) light curve, and declines gradually. This
behavior is similar to the giant flare seen from Proxima Centauri with XMM-
Newton (Güdel et al., 2004).

The second flare occurred during the second snapshot on 2019 December 10
(191210, Figure 1 bottom). This observation (for $\sim$1.6 ksec) is longer
than for the first flare, but it similarly covers the rise and beginning of
the decay as the second flare develops more slowly. NICER misses the middle of
the decay, but the third snapshot covers the end of the decay as the light
curve connects smoothly from the second snapshot. On the other hand, the first
snapshot shows a slight elevation in the middle, but both intervals appear to
be in a quiescent state without significant variation. The light curve before
the first flare also shows similar count rate variations (Figure 1 top). A
Bayesian block analysis shows that the 0.6$-$1.2 keV count rate stays at
$\sim$6.6 cnts s-1 from the latter half of the first snapshot to the first
$\sim$150 sec of the second snapshot, suggesting that the flare begins during
the second snapshot at around 19h 14m 40s UT. Meanwhile, the last Bayesian
block of the third snapshot has almost the same count rate ($\sim$6.8 cnts
s-1), suggesting that the quiescent emission does not vary significantly
during the flare.

Besides the slow variation, the second flare has similar energy-dependent
variations to the first flare. The 2$-$4 keV light curve reaches its peak
before other energy bands. The softer band peaks are delayed: the 0.3$-$0.6
keV light curve does not show a clear peak during the second snapshot. This
delay continues to the third snapshot. The softer band light curves gradually
decline, while the 2$-$4 keV light curve is almost flat. The hardness ratio
reaches a maximum early during the rise, but otherwise the variation is
similar to the first flare.

Figure 3: Quiescent spectra of the first flare for an exposure of 870 sec from
$-$16921 sec to $-$16051 sec (left) and the second flare for 789 sec from
$-$4700 sec to $-$3911 sec (right). The solid yellow lines are the best-fit
models, and the dotted lines are the individual components. The upper right
corner of each panel shows the best-fit parameters (the units are keV for kT
and cm-3 for EM). The best-fit models reproduce the observed spectra well.
Table 1: Best-fit Values of the Quiescent Spectra

Component | 190917 | 191210
---|---|---
| kT | EM | kT | EM
| (keV) | (1051 cm-3) | (keV) | (1051 cm-3)
1 | 0.27${}_{-0.03}^{+0.03}$ | 2.4${}_{-0.4}^{+0.4}$ | 0.34${}_{-0.03}^{+0.02}$ | 3.9${}_{-0.6}^{+0.4}$
2 | 0.62${}_{-0.03}^{+0.04}$ | 1.9${}_{-0.4}^{+0.3}$ | 0.73${}_{-0.06}^{+0.06}$ | 1.9${}_{-0.4}^{+0.5}$
3 |  |  | 1.98${}_{-0.38}^{+0.87}$ | 1.2${}_{-0.5}^{+0.2}$
4 |  |  | 1.04${}_{-0.20}^{+0.29}$ | 0.5${}_{-0.2}^{+1.1}$
$\Delta\chi^{2}$/d.o.f | 55.44/64 | 162.06/175

Note. — The errors show 90% confidence ranges. The 4th component is required
for the latter interval spectrum of the 191210 observation.

###  3.2 Time Resolved Spectra

To understand the energy-dependent time variations, we analyze the time-
resolved spectra. We first produce a quiescent spectrum from the snapshot
directly preceding each flare (the magenta bar in Figure 1). The snapshot of
the September flare shows no significant variation in count rates, while that
in December does show a small but significant count rate increase in the
middle at $-$4.7 ksec (Figure 1 bottom). Therefore, we produce two spectra
separated at the time with a statistically significant count rate change
(change point) in the Bayesian analysis (Figure 3). The quiescent spectra show
a prominent hump between 0.7$-$0.9 keV, consistent with emission lines of Fe
XVII-XX, Ne IX-X, and OVIII ions seen in the XMM-Newton RGS spectra of
$\kappa^{1}$ Ceti in 2002 (Telleschi et al., 2005). The spectrum has a steep
hard slope, with negligible emission above $\sim$2 keV, but no absorption cut-
off in the soft band down to 0.3 keV. We, thus, apply a thermal plasma
emission model (apec) without absorption (Table 1). Assuming the best-fit
elemental abundances of the XMM/RGS spectrum (Appendix A), we find that a
2-temperature (2$T$) model with kT = 0.27 & 0.62 keV provides an acceptable
fit to the quiescent spectrum for the September flare. The quiescent spectra
of the December flare require a 3$T$ model for the former interval and an
additional 1$T$ component for the latter interval to account for the flux
increase. We use this 4$T$ model as the fixed quiescent component for the
December flare spectra.

We produce time-resolved spectra during the flares, every 50 sec with a
minimum 25 counts per bin for the September flare to track the fast variation
and every 100 sec with a minimum of 50 counts per bin for the December flare
to get good photon statistics (Figures 4, 5). For each flare, we make one
spectrum before the flare onset in the same snapshot (the red bars in Figure 2
between $-$95—$-$35 sec for the first flare and $-$80—0 sec for the second
flare), which we call the preflare spectrum. In the third snapshot of the
December flare, due to a decreased count rate, we increase the time interval
of each bin to 400$-$600 sec. We also apply longer time intervals for the
third snapshot of the December flare near the decay end. The preflare spectrum
at the top left panel of each figure matches well the corresponding quiescent
spectrum in the solid yellow line, consistent with the Bayesian block analysis
of the flare onset timings.

During the first flare, the flux in the entire energy band increases for the
first 300 sec. The 0.7$-$0.9 keV hump becomes broader to the high energy side
as the flux at $\sim$1.1 keV, probably originating from Fe XXII-XXIV or Ne X
emission lines, is enhanced. After that, the hard band emission gradually
declines, while the flux at $\sim$0.9 keV, probably produced by Fe XVII-XX
lines, strengthens. The second flare evolves more slowly but similarly to the
first flare. The whole band increases until $\sim$800 sec, and then the hump
at $\sim$0.9 keV begins to strengthen. In the third snapshot, the flux
declines nearly to the preflare level with some residual in the soft and hard
bands in the prior $\sim$1 ksec.

Figure 4: Time-resolved spectra of $\kappa^{1}$ Ceti during the first flare
(190917). The red/blue line depicts the best-fit cool/hot component of the
flare spectrum, and the yellow line does the fixed quiescent component. The
solid black line is their sum. The hot component soars in the rising phase,
dominating most energy bands, while the cool component is more significant at
$\sim$0.9 keV with Fe L emission lines after $\sim$350 sec. The top right of
each panel shows each spectrum’s time interval in seconds.  Figure 5: Time-
sliced spectra of $\kappa^{1}$ Ceti during the second flare (191210). The
flare spectral components behave similarly to those of the first flare. The
cool component exceeds the hot component at $\sim$0.9 keV after $\sim$1 ksec
but not so much as during the first flare.

###  3.3 kT and EM Variations during the Flares

The time-resolved spectra do not suggest any significant variation of the
quiescent component during the flares. We, therefore, reproduce each time-
resolved spectrum by a model with variable flare components plus the fixed
quiescent component. The hard slopes of most spectra require a kT $\sim$3 keV
plasma. However, collisional equilibrium plasma at that temperature does not
emit Fe XVII-XX emission lines at $\sim$0.9 keV, which are enhanced near the
flare peaks. Non-equilibrium ionization plasmas emit these lines at
$\tau\sim$2$\times$1010 s cm-3, but they do not reproduce the emission lines
at $\sim$1.1 keV observed during the rising phase. This result suggests that
the flare spectra need at least two plasma components. We, thus, apply a 2$T$
apec plasma model for the flare emission, with elemental abundances fixed to
the best-fit XMM/RGS values. We find that a model with kT $\sim$2$-$4 keV and
0.3$-$1 keV components reproduces each spectrum well and that these
temperatures vary monotonically over time. However, the spectral parameters
are poorly constrained near the flare onset and end due to weak flare
emission. We, therefore, fit all spectra simultaneously, assuming that each
component’s plasma temperature varies linearly with time, and find reasonable
results with $\chi^{2}/d.o.f.$ at 685.96/616 for the first flare and
1118.41/919 for the second flare. Figures 4 & 5 plot the best-fit model for
individual spectra and Figure 6 shows the best-fit kT and EM values. The third
and fourth columns in Table 2 show the best-fit kT slopes.

The cool component EMs are mainly determined by fits of the Fe L emission line
complex to the $\sim$0.9 keV excess. We examine whether the line intensity
constraints in the applied model are adequate for this analysis. First, the
model fixes the elemental abundances during the flares at the best-fit XMM/RGS
quiescent spectrum values. However, some solar or stellar X-ray flares show
apparent elemental abundance changes from the pre or post-flare states (e.g.,
Osten et al., 2000; Audard et al., 2001; Mondal et al., 2021). The best-fit
spectral models also fit well the $\sim$1.1 keV bump with the Fe L and Ne K
lines in the hot component. Since the hard spectral slope determines the hot
component’s EM, the hot component’s Fe and Ne abundances are consistent with
the assumed abundances, i.e., the coronal abundances are not observed to
significantly change during the $\kappa^{1}$ Ceti flares. Second, suppose the
cool component did not reach equilibrium at ionization timescales of
$\lesssim$1010 s cm-3 as opposed to the model assumption. In that case, the
plasma should emit weaker Fe L lines than the equilibrium case and require a
larger EM to account for the observed 1.1 keV bump. However, no observed
spectra show strong emission below $\sim$0.7 keV expected from low ionized
oxygen and carbon emission lines from such non-equilibrium plasmas. In
addition, preflare loops probably have densities over $\sim$1011 cm-3 (e.g.,
Osten et al., 2006), suggesting that the Fe L line complex develops within
$\approx$0.1 sec. These results suggest that the cool component EM
measurements are robust.

The hot component explains most of the initial flux rise in each flare. The
component in the second flare is slightly hotter and cools down significantly
slower than the first flare, but the component stays kT $\gtrsim$2 keV
throughout both observations. On the other hand, the cool component develops
more slowly than the hot component. The plasma temperature does not vary
strongly at $\sim$1 keV around the flare peak, but it declines to $\sim$0.3
keV by the end of the second flare.

EM time series in Figure 6 bottom panels confirms the similarity of the two
flares: i) the hot EM varies with a linear rise and a slow decay, ii) the cool
EM varies similarly to the hot EM but with a delay. To quantitatively evaluate
their variations, we fit the EM time series with the following conventional
formula for stellar flares.

$\displaystyle EM(t)$ $\displaystyle=0$ $\displaystyle t<t_{\rm onset}$
$\displaystyle=EM_{\rm peak}\frac{t-t_{\rm onset}}{\Delta t_{\rm rise}}$
$\displaystyle t_{\rm onset}\leq t<t_{\rm peak}$ $\displaystyle(1)$
$\displaystyle=EM_{\rm peak}\exp(-\frac{t-t_{\rm peak}}{\tau_{\rm decay}})$
$\displaystyle t_{\rm peak}\leq t$

where $t_{\rm onset}$, $t_{\rm peak}$, $EM_{\rm peak}$ and $\tau_{\rm decay}$
are free parameters and $\Delta t_{\rm rise}=t_{\rm peak}-t_{\rm onset}$. For
the fittings, we use curve_fit in the scipy package. We fix $\Delta t_{\rm
rise}$ and $\tau_{\rm decay}$ of the 190917 flare’s cool component at the
best-fit values of the hot component as the cool component does not show a
clear EM peak. Table 2 shows the best-fit result. The hot component’s $t_{\rm
onset}$ is close to zero, again consistent with the Bayesian blocks
measurement of the flare onset in each flare. In contrast, the cool
component’s $t_{\rm onset}$ is significantly delayed from the hot component’s
$t_{\rm onset}$ (hereafter $\Delta t_{\rm delay}=t_{\rm onset}({\rm
cool})-t_{\rm onset}({\rm hot})$). In the second flare, the cool component has
similar $\Delta t_{\rm rise}$ to, but a factor of two longer $\tau_{\rm
decay}$ than, the hot component. The second flare has longer durations in
$\Delta t_{\rm delay}$, $\Delta t_{\rm rise}$, and $\tau_{\rm decay}$ than the
first flare.

These behaviors explain the energy-dependent variations of the light curves.
The 2$-$4 keV band light curve is dominated by the hot component’s behavior,
showing a conventional stellar flare variation. The softer bands add the cool
component’s behavior with a conventional flare variation but a time delay
compared to the hot component. The 0.6$-$1.2 keV light curve deviates most
with the strong 0.9 keV hump from the cool component. The deviation is
stronger in the first flare with a larger relative time delay ($\Delta t_{\rm
delay}/\Delta t_{\rm rise}$) and a larger EMpeak ratio (EM${}_{\rm peak}({\rm
cool})$/EM${}_{\rm peak}({\rm hot})$).

Figure 6: Best-fit kT (top) and EM (bottom) values of the time-resolved flare
spectra by the 2$T$ apec models (left: 190917, right: 191210). The red/blue
color shows the cool/hot plasma component. In the kT plots, the solid lines
and filled areas are the best-fit kT linear models and the 90% confidence
areas. In the EM plots, the data points show the best-fit values and their 90%
confidence ranges of the combined spectral fits. The solid lines show the
best-fit linear rise plus exponential decay model to these EM measurements. In
each flare, the cool component rises similarly to the hot component but with a
time delay.  Table 2: Flare Parameters

Flare | Comp. |  kT($t=0$) | kTslope | $t_{\rm onset}$ | $\Delta t_{\rm rise}$ | $\tau_{\rm decay}$ | EMpeak | LXpeak | $E_{\rm X}$ | $E_{\rm bol}$
---|---|---|---|---|---|---|---|---|---|---
|  | (keV) | (keV/ksec) | (sec) | (sec) | (sec) | (1052 cm-3) | (1029 ergs s-1) | (1032 ergs) | (1033 ergs)
190917 | cool | 1.00${}_{-0.13}^{+0.11}$ |  $-$0.18${}_{-0.24}^{+0.25}$ | 156 (20) | 218 (fix) | 572 (fix) | 0.71 (0.061) | 1.2 | 0.79 | 2.0/3.2
| hot | 2.86${}_{-0.46}^{+0.51}$ |  $-$1.6${}_{-0.95}^{+0.96}$ | 14 (7) | 218 (15) | 572 (86) | 2.6 (0.13) | 3.2 | 2.2 |
191210 | cool | 1.02${}_{-0.04}^{+0.03}$ |  $-$0.13${}_{-0.01}^{+0.02}$ | 227 (35) | 923 (99) | 2591 (324) | 0.45 (0.032) | 0.75 | 2.3 | 6.6/4.0
| hot | 3.65${}_{-0.25}^{+0.24}$ |  $-$0.32${}_{-0.09}^{+0.16}$ | 33 (37) | 1003 (59) | 1135 (111) | 2.8 (0.12) | 3.9 | 6.4 |

Note. — kT($t=0$), kTslope: Best-fit kT linear time variation model of the
combined spectral fits. The errors show 90% confidence ranges. $t_{\rm
onset}$, $\Delta t_{\rm rise}$, ${\tau}_{\rm decay}$, EMpeak: Best-fit linear
rise plus exponential decay model of the EM time series. The parentheses show
1$\sigma$ confidence ranges. LXpeak: Peak X-ray luminosity between 0.3$-$10
keV. $E_{\rm X}$: Total X-ray flare energy between 0.3$-$10 keV. $E_{\rm
bol}$: Total bolometric flare energy. The left values use a relation to the
GOES band (1.55-12.4 keV) flare-radiated energy for active stars (Table 2 in
Osten & Wolk, 2015). The right values use a relation to the GOES band solar
flare peak flux (equation (13) in Aschwanden et al., 2017).

Figure 7: Left: Whole loop EM variations of the $10\times 10^{9}$ cm loop
simulation. The EMs are divided by logarithmic temperature buckets and binned
every 30 seconds. The plot shows two EM components — the evaporated plasma
that envelopes individual temperature peaks standing up from the hot side,
getting a maximum at $\sim$600 sec, and the footpoint plasma, a group of low-
temperature buckets that rises and falls similarly at one-third of the
evaporated plasma component. Right: Whole loop EM variations for the first 500
sec, summed over the temperature ranges log $T=$7.3$-$8.0 K (blue) and
6.6$-$6.9 K (red). The number that labels each line is the loop length in 109
cm. The evaporated component starts to rise at $\sim$50 sec, while the
footpoint component is significantly delayed from the evaporated component,
more with longer loops. The overlapping triangular base in the red plot up to
$\sim$250 sec originates from the initial heating.

##  4 Hydrodynamic Simulations of Single Loop Flares

The hot component constitutes the major part of the flare emission. As
discussed in numerous studies, it should originate from radiatively-cooling
plasma inside the flare magnetic loops. Then, what is the cool component?
Flare spectral fits often require two temperatures or more (e.g., Sasaki et
al., 2021; Paudel et al., 2021), but the nature of the cool component is
poorly known. Our NICER study provides this component’s time variation through
the flare rise. We run hydrodynamic simulations of single magnetic loop flares
to understand the cool component.

We employ a field-aligned hydrodynamic code, the HYDrodynamics and RADiation
code (HYDRAD222https://github.com/rice-solar-physics/HYDRAD; Bradshaw & Mason,
2003), used to study heating in the solar corona and solar flares. The code
solves the equations for the conservation of mass, momentum, and energy for
plasma confined to a magnetic flux tube (Bradshaw & Cargill, 2013). The loops
are assumed to be heated by non-thermal electrons, accelerated by magnetic
reconnection near the loop’s apex. As the electrons propagate, they deposit
their energy through Coulomb collisions with the ambient plasma. The majority
of the heat is deposited in the upper chromosphere, causing a rapid increase
in temperature and pressure. It then drives an expansion of material
(chromospheric evaporation), carrying hot and dense plasma into the corona.
The assumed form of the heating function that we use was derived by Emslie
(1978), with modification for non-uniform ionization in the chromosphere
(Hawley & Fisher, 1994). As the loop evolves, the plasma cools through thermal
conduction and radiation, which we calculated using the atomic database
CHIANTI (Dere et al., 1997), version 10 (Del Zanna et al., 2021). We use the
elemental abundances derived from the XMM/RGS spectra (Table 5), but our
preliminary study using solar elemental abundances provides a similar result.
Our simulations assume a magnetic loop with a uniform cross-section and
injected particles with a power-law energy distribution for 200 sec with an
energy flux peaking at 1011.5 ergs cm-2 s-1 at 100 sec. Since the two NICER
flares have different flare decay timescales and plausibly different magnetic
loop lengths (e.g., Toriumi et al., 2017; Reep & Toriumi, 2017), the
simulations consider loop lengths at 5, 10, 15, 20, and 25$\times$109 cm. The
derived EM normalization can be adjusted by changing the cross-section of the
magnetic loops (equivalently, the total volume of the loops).

Table 3: HYDRAD Simulation Result

Loop Length |  $t_{\rm onset}$(eva) |  $t_{\rm onset}$(foot) | $\Delta t_{\rm delay}$ |  $\Delta t_{\rm rise}$(eva) |  $\tau_{\rm decay}$(eva) |  EMpeak ratio |  kTpeak(cool/hot)
---|---|---|---|---|---|---|---
(109 cm) | (sec) | (sec) | (sec) | (sec) | (sec) |  | (keV/keV)
5 | 63 | 77 | 15 | 330 | 760 | 1.21 | 0.32/1.40
10 | 64 | 153 | 89 | 491 | 1207 | 1.04 | 0.31/1.67
15 | 40 | 205 | 164 | 496 | 2268 | 0.78 | 0.32/1.66
20 | 25 | 248 | 223 | 639 | 3013 | 0.65 | 0.34/1.57
25 | 17 | 294 | 276 | 752 | 3601 | 0.60 | 0.35/1.57
190917 |  |  | 142 | 218 | 572 | 0.27 | 0.96/2.50
191210 |  |  | 194 | 1003 | 1135 | 0.16 | 0.88/3.31

Note. — In each simulation, the time origin is the particle injection start.
$t_{\rm onset}$(eva)/$t_{\rm onset}$(foot): onset time of the
evaporated/footpoint component derived from one/two linear fits to the
EM[7.3$-$8.0]/EM[6.6$-$6.9] time series. $\Delta t_{\rm delay}$ = $t_{\rm
onset}$(foot) $-$ $t_{\rm onset}$(eva). $\Delta t_{\rm rise}$(eva)/$\tau_{\rm
decay}$(eva): rise/decay time of the evaporated component derived from a fit
to the peak EMs of individual temperature buckets by a linear rise plus
exponential decay model. EMpeak ratio, kTpeak(cool/hot): EMpeak ratio and
plasma temperatures at the cool/hot EM peaks, derived from fits to the
synthetic flare spectra with 100 sec bins by a 2$T$ apec model.

The left panel of Figure 7 shows the EM evolution of the 10$\times$109 cm cm
flare loop simulation. The EM is dominated by the hottest plasma emission that
peaks near $\sim$600 sec. This component represents a radiative-cooling,
evaporated plasma that fills the magnetic loop, corresponding to the hot
component of the observing flares. Since the evaporated plasma cools down
gradually under thermal equilibrium, a single temperature bucket dominates
near its peak and drops to zero quickly once the plasma cools. A secondary
component is a group of low-temperature buckets that rises and falls similarly
to the main component at one-third the EM of the evaporated plasma component.
Each temperature bucket stays in this group until the evaporated plasma cools
down to its temperature range. This secondary component represents plasmas at
transition regions near the magnetic footpoints. Because the conductive heat
flows from the looptop to the footpoints, the plasma has a strong temperature
gradient and responds to the evaporated plasma’s variation. The other loop
length simulations show similar EM variations with different time scales.

In the first 500 sec, the log $T\geq$7.3 (K) buckets only reflect the
evaporated plasma component, while the log $T<$7.0 (K) buckets reflect the
footpoint plasma component. We, therefore, define two temperature ranges, log
$T=$7.3$-$8.0 (K) and 6.6$-$6.9 (K), and sum up EMs within each range to
understand their behaviors near the rising phase (Figure 7 right). First, the
EM[6.6$-$6.9] time series of various loop lengths vary similarly for $\sim$200
sec from the beginning. This EM base originates from the initial heating of
the plasma in the upper chromosphere by the injected particles, and peaks at
$\sim$100 sec in response to the assumed particle injection flux. The
EM[7.3$-$8.0] does not show this component clearly, but the slow rise in the
first $\sim$50 sec originates from the initial plasma heating.

All EM[6.6$-$6.9] plots except the 5$\times$109 cm simulation show the
footpoint components’ onsets as clear kinks (e.g., at $\sim$150 sec for the
10$\times$109 cm simulation). We measure the timing of each kink from a two
linear fit ($t_{\rm onset}$(foot) in Table 3). The onset ranges between
$\sim$80$-$300 sec, and longer loops have later onsets. The evaporated
component does not have a clear onset signature, so we measure the onset
timing from a fit to the first 200 sec of EM[7.3$-$8.0] by a linear function
starting at $t_{\rm onset}$(eva). The onset ranges between 17$-$64 sec and
does not appear to correlate with the loop length. The time lags $\Delta
t_{\rm delay}$ (=$t_{\rm onset}$(foot) $-$ $t_{\rm onset}$(eva)) clearly
increase with longer loops.

Figure 8: Density (top) and temperature (bottom) spatial distribution of the
$10\times 10^{9}$ cm loop simulation at 40, 60, 140 and 220 seconds from the
particle injection start. The horizontal axis shows the distance from a
footpoint along the loop: the loop top is at 5$\times$109 cm and the two
footpoints are at 0, 10$\times$109 cm. From left, i) at 40 sec, the particle
injection heats the footpoint chromosphere, and the evaporated gas soars into
the magnetic loop. The evaporated spectral component starts to increase. ii)
at 60 sec, the upward evaporation flows collide at the loop top, heating the
gas further. iii) at 140 sec, the shock propagate down the other leg,
smoothing the corona’s density. iv) at 220 sec, by the time the shock reaches
the footpoints, the loop has enough high density that thermal conduction
becomes extremely efficient, and the footpoint spectral component emerges. The
red and blue lines are for electrons and hydrogen, respectively. The black and
white double arrows point to the locations of the evaporation or shock fronts
from either side.  Figure 9: Electron and hydrogen density product (top) and
electron temperature (bottom) distributions of the $10\times 10^{9}$ cm loop
simulation near a footpoint region at $t=$40 sec (dotted), 60 sec (dash-
dotted), 140 sec (dashed), and 220 sec (solid). The horizontal axis is the
same as Figure 8 but on a logarithmic scale. The filled grey area in the
bottom panel shows the log $T=$6.6$-$6.9 K bucket. The boxes in the top panel
show the one-dimensional volumes and density product ranges of this
temperature bucket. Both the density product and the volume significantly
increase between $t=$140 sec and 220 sec.  Figure 10: EM value (blue) at each
temperature bucket peak and the corresponding plasma temperature (grey) in the
$10\times 10^{9}$ cm loop simulation. The solid green line is the best-fit
model of the EM values by a linear plus exponential decay model. The model
reproduces the EM variation well.  Figure 11: Left: Synthetic NICER spectrum
of the $10\times 10^{9}$ cm simulation at the flare peak (600$-$700 sec). The
flare spectrum is normalized to have the 0.3$-$2 keV flux at 2.2$\times$10-11
ergs cm-2 s-1 and combined with the quiescent spectrum of the September flare.
Right: The same spectrum but with EMs below the evaporated plasma temperature
(log $T<$7.3) reduced to 10%. The left spectrum has strong emission at
$\sim$0.8 keV, while the right spectrum is close to the observed flare peak
spectra. The upper right corner of each panel shows the best-fit parameters of
a 2$T$ apec model (the units are keV for kT and cm-3 for EM). Each plot uses
the same color scheme as of Figures 4 and 5.

Figure 8 shows why the footpoint component is delayed. In the plots, the
evaporated component is located in the middle part where temperature and
density quickly increase after the particle injection. The footpoint component
is located near both ends, whose temperature and density do not increase until
the shocks produced by the evaporated gas’s collision at the looptop propagate
down to the footpoints. Figure 9 displays the hydrogen and electron density
product and the electron temperature, magnifying the left end on a logarithmic
scale. Both the log $T=$6.6$-$6.9 K depth and density product increase between
140 sec and 220 sec. The time delay corresponds to the travel time of the
flare loop by the evaporated flows and the collisional shocks, which is
approximately the sound crossing time. It is, therefore, roughly proportional
to the loop length.

We also measure the decay timescales of the simulated flares. As described
above, the evaporated plasma is in a single temperature bucket at each
temperature peak. We, thus, take the peak EM of each temperature bracket and
fit them by a linear plus exponential decay model from equation (1). The fits
reproduce the EM variations well except around the peak (Figure 10 for the
10$\times$109 cm loop simulation). Longer loop flares have longer decay
timescales (${\tau}_{\rm decay}$(eva) in Table 3), as suggested in earlier
studies (e.g., van den Oord & Mewe, 1989; Toriumi et al., 2017).

We make synthetic NICER spectra of the simulated EM distributions to compare
the spectral properties. For each simulation, we produce spectral models with
100 sec bins, assuming an apec plasma model for each temperature bucket. We
normalize them to a peak 0.3$-$2 keV flux at 2.2$\times$10-11 ergs cm-2 s-1 to
match the observed two NICER flares. We then generate a synthetic spectrum for
each spectral model with the xspec fakeit tool, by convolving the model with
the NICER on-axis responses, nixtiref20170601v003.rmf and
nixtiaveonaxis20170601v005.arf. We increase photon statistics by a factor of
10 to reduce statistical uncertainty, equivalent to a 1 ksec exposure. We,
then, bin each synthesized spectrum to have $\geq$50 counts per bin, and fit
each spectrum by a 2$T$ apec model. Figure 11 left shows a synthetic spectrum
of the 10$\times$109 cm loop between 600$-$700 sec, adding the quiescent
component of the September flare for a comparison. Table 3 shows the EMpeak
ratios and plasma temperatures at the EM peaks. The peak plasma temperatures
$\sim$0.3$-$0.35 keV and 1.4$-$1.67 keV are similar among the simulations and
significantly lower than the observed values. The EMpeak ratio is the highest
with the 5$\times$109 cm loop simulation at 1.21 and smaller with longer loop
simulations. This result is naturally understood since the footpoint plasma
volume does not change with the loop length.

The numerical simulations demonstrate that the footpoint component is delayed
to rise by 100$-$300 sec from the evaporated component. This result indicates
that the cool component in the NICER spectra originates from the footpoint
plasma. The simulations also suggest that longer flare loops have longer
delays of the footpoint component rise and smaller EM peak ratios, as well as
longer decay timescales. All these properties are consistent with the
properties of the two NICER flares, suggesting that the December flare
originates from a longer flare loop than the September flare.

##  5 Discussion

The $\kappa^{1}$ Ceti flares in 2019 are an order of magnitude more powerful
than the most powerful solar flare ever seen, the Carrington Event in 1859 (LX
$\sim$1028 ergs s-1, Cliver & Dietrich, 2013; Sakurai, 2022). Their X-ray
luminosities are near the upper end of the flare luminosity ranges of solar-
type G stars (LX $\lesssim$1030 ergs s-1, Schaefer et al., 2000; Tsuboi et
al., 2016). Their bolometric flare-radiated energies 3$-$8$\times$1033 ergs,
evaluated from two independent empirical relations to the X-ray radiations
among solar and active stellar flares (Aschwanden et al., 2017; Osten & Wolk,
2015, see Table 2), qualify them as superflares ($>$1033 ergs, e.g., Maehara
et al., 2012) and are comparable to the $\kappa^{1}$ Ceti superflare recorded
in 1986 ($\sim$2$\times$1034 ergs, Schaefer et al., 2000). Nonetheless, their
X-ray luminosities and released X-ray energies are modest among active or
young stellar flares. (LX $\lesssim$1032-33 ergs s-1, e.g., Benz & Güdel,
2010, and references therein). The other X-ray characteristics — the hot
plasma temperature, hard band light curve, and hardness ratio variation — are
similar to solar and stellar X-ray flares (e.g., Pye et al., 2015). We
conclude that the $\kappa^{1}$ Ceti flares in 2019 are conventional magnetic
reconnection events.

The $\kappa^{1}$ Ceti flare spectra require an additional cool (kT $\lesssim$1
keV) temperature component. Although such a component has not received much
attention, well exposed stellar X-ray flare spectra usually require one or
more components with kT $\sim$0.3$-$1 keV (e.g., GT Mus: Sasaki et al. 2021,
EV Lac: Paudel et al. 2021). The high-resolution XMM/RGS spectra of the
Proxima Centauri flare suggested that the flare EM distribution was broad with
a peak at $\sim$30MK and a low-temperature tail during the rise and steadily
moved to low-temperatures as the flare developed (Güdel et al., 2004; Reale et
al., 2004). In solar flares, the low-temperature EM ($<$16.5 MK) peaks later
than the high-temperature EM (McTiernan et al., 1999), perhaps suggesting the
presence of a similar cool component. The cool component is probably
ubiquitous in solar and stellar flares and represents an average of the low-
temperature tail in the EM distribution.

The cool component’s EMs of the $\kappa^{1}$ Ceti flares increase steadily
during the rising phase, but the footpoint plasma’s EMs in the HYDRAD
simulations rapidly increase to half the maximum at the onsets. The September
flare may be statistics limited due to its quick rise, but the December flare
clearly shows that the cool component’s EM steadily increases with a possible
step-wise increase in the middle of the rise. This may suggest that each flare
is an assembly of multiple loops, which is well known from e.g. spatially-
resolved UV and optical imaging of solar flares (e.g., Aschwanden & Alexander,
2001). Multiple loop models can reproduce energy-dependent X-ray time
variations of solar flares (Reep & Toriumi, 2017).

If the observed flares are multiple loop events, we should ideally convolve
the HYDRAD simulations with single loop event rates. Very hard X-rays ($>$20
keV) or microwave emission can trace the flux variation of the injected non-
thermal reconnection particles (e.g., Benz, 2017), but we do not have
simultaneous data in these bands, unfortunately. Earlier flare observations
suggest that these emissions drop before the soft X-ray peaks (Lin et al.,
2003; Asai et al., 2004; Veronig et al., 2005), which are $\sim$200 sec in the
September flare and $\sim$1 ksec in the December flare. A convolution in each
timescale may change the EMpeak ratio and the flare decay timescale. It should
not change the cool component delay timescales.

We compare the derived $\kappa^{1}$ Ceti flare parameters, $\Delta t_{\rm
delay}$, $\tau_{\rm decay}$, and the EMpeak ratio with the simulation (Table
3). These values vary monotonically with the loop length in the simulation, so
we estimate the loop length for each parameter by linearly interpolating or
extrapolating the two neighboring values (Table 4). We also list three other
estimates from the literature. The first estimate is an empirical relation of
the ribbon distance with the decay timescale in solar flares (Toriumi et al.,
2017, equation 4). We approximate the decay timescale of the 1$-$8Å energy
flux with ${\tau}_{\rm decay}$ of the hot component in Table 2 and assume a
semi-circular flare loop shape to derive the loop length. The second estimate
is a quasi-statistic cooling model for a constant radiative and conductive
timescale ratio (van den Oord & Mewe, 1989; Tsuboi et al., 2000, equation A5).
A problem with this estimate is that flares never truly cool statically (e.g.,
Cargill et al., 1995). The third estimate is a magnetic reconnection model,
assuming that the gas pressure of a flare loop is comparable to the magnetic
pressure (Shibata & Yokoyama, 2002, SY02). A caveat is that the model requires
the unmeasurable preflare proton density. All estimates but the EMpeak ratio
are consistent with $\approx$1010 cm loop lengths. All estimates but SY02
suggest that the December flare has a longer flare loop than the September
flare.

Table 4: Flare Loop Length Estimate

Flare | HYDRAD | Sun | QS | SY02
---|---|---|---|---
| $\Delta t_{\rm delay}$ |  EMpeak ratio | ${\tau}_{\rm decay}$ |  |  |
190917 | 13.5 | 60.3 | 2.9 | 6.2 | 8.9 | 12.9
191210 | 17.6 | 72.3 | 9.2 | 13.6 | 22.5 | 8.7

Note. — Unit in 109 cm. HYDRAD: linear interpolation or extrapolation of the
nearest two values of the HYDRAD simulation in Table 3. Sun: Solar flare
ribbon distance relation in equation 4 of Toriumi et al. (2017). The derived
$d_{\rm ribbon}$ values are multiplied by $\pi$/2. QS: Quasi-Static cooling
model in Tsuboi et al. (2000), equation A5. SY02: Equation 7b in Shibata &
Yokoyama (2002) for the preflare proton density at 1011 cm-3.

The derived loop length of $\approx$1010 cm is near the upper end but still
within the range of solar flare loops (e.g., Toriumi et al., 2017). Since
$\kappa^{1}$ Ceti has about the same stellar radius as the Sun, we can safely
assume that the observed $\kappa^{1}$ Ceti flares have similar magnetic field
geometries to moderately large solar flare loops. However, the peak EMs
$\sim$3$\times$1052 cm-3 are about two orders of magnitudes larger than the
EMs of solar flares with similar loop lengths. One solution is that the
$\kappa^{1}$ Ceti flares have an order of magnitude higher flare plasma
density. Such high-density plasma radiatively cools with an order of magnitude
shorter timescales, but the $\kappa^{1}$ Ceti’s flare decay timescales are
consistent with the solar flare’s decay time relation (Table 4). The other
solution is that the $\kappa^{1}$ Ceti flares have two orders of magnitude
larger widths and/or have thicker magnetic loops.

The EMpeak ratio derives inconsistent loop lengths, possibly because the
HYDRAD simulation systematically overestimates the footpoint component. The
footpoint component is comprised of all temperature buckets below the
evaporated plasma temperature (see Figure 7 left). We therefore reduce the
footpoint component’s EMs of Figure 11 left simulation — the EMs below the
evaporated plasma temperature, log $T$ =7.3 (K) — to 10%, as a trial. Then,
the synthetic spectrum looks more similar to the observed spectra near the
flare peaks (Figure 11 right), and the best-fit 2$T$ apec model has a smaller
EMpeak ratio at $\sim$0.18. As observed, this model also derives a higher hot
component temperature at 2.0 keV.

The footpoint plasma at the height of $\sim$1.2$-$1.6$\times$108 cm is in the
transition region (Figures 8, 9). The line of sight should have more
intervening material than the evaporated plasma in the corona. Still,
attenuating $\sim$0.9 keV X-rays by $\sim$80% requires a hydrogen column
density at NH $\sim$1022 cm-2, corresponding to a physical depth of
$\sim$1010-11 cm for the density of the transition region ($n\sim$1011-12
cm-3). Flare loops need to be viewed almost edge-on to have this depth, but
realizing such geometries for both loops are less likely. Therefore, the
observed flares should have less footpoint plasma to the evaporated plasma
than the HYDRAD simulations. The RHESSI observatory found in the hard X-ray
band ($>$10 keV) that solar flares have several times higher electron rates at
the looptop than at the footpoints during the impulsive phase333 During the
impulsive phase, the magnetic reconnection accelerates charged particles,
which emit hard non-thermal X-rays (e.g., Benz, 2017). This phase occurs
mostly before the cool component rises. , implying the electrons accumulate in
the looptop (Simões & Kontar, 2013). The $\kappa^{1}$ Ceti flares may also
have a mechanism to suppress electron transportation to the footpoints and to
reduce thermal conduction. Such a mechanism may also explain the slower
cooling of the evaporated plasma compared to the HYDRAD simulations.

One possible mechanism to suppress electron transport is that the flare
magnetic loop expands toward the looptop, trapping charged particles in a
magnetic mirror. Solar coronal loops, whether quiescent or flaring, do not
necessarily show an expansion of the loop width along their lengths (Klimchuk
et al., 1992; Klimchuk, 2000; Klimchuk & DeForest, 2020, and references
therein). However, the magnetic field strength falls off with height in the
corona, implying that there should be an expansion of the cross-sectional area
of loops (e.g., Dudík et al., 2014), and models are unable to reproduce both
hot and cool emission simultaneously without an area expansion (Warren et al.,
2010; Reep et al., 2022a). The loop expansion reduces the thermal conductivity
near the footpoints. A preliminary 10$\times 10^{9}$ cm loop simulation with
the expansion geometry in Reep et al. (2022b) does not produce a small EMpeak
ratio, but the cool component EM peaks significantly later than the constant
loop simulation. The other possible mechanism is that the flare loops have
turbulent magnetic fluctuations, which would increase the frequency of Coulomb
collisions, suppressing the energy transport and reducing the thermal
conductivity (e.g., Bian et al., 2016; Allred et al., 2022). This mechanism
increases coronal temperatures compared to those in a model with collisionally
dominated transport.

##  6 Conclusion

NICER observed two moderately strong X-ray flares from $\kappa^{1}$ Ceti, a
nearby young solar analog, in 2019. NICER’s excellent soft X-ray sensitivity,
good energy resolution, and large collecting area bring rare details of bright
X-ray flares from the onsets through the peaks. Both flares show conventional
stellar flare variations above 2 keV with a rapid rise and decay, having
similar X-ray fluxes at $\sim$2.2$\times$10-11 ergs cm-2 s-1 between 0.3$-$2
keV and high plasma temperatures at $\sim$3 keV near the peaks. Their
bolometric energies estimated from the X-ray radiated energies,
$\sim$3$-$9$\times$1032 ergs, are comparable to superflares. The flare on
September 17 varies in several hundred seconds in X-rays, with an interesting
flat soft X-ray flux peak. The flare on December 10 varies 2$-$4 times more
slowly, showing a similar but less extreme variation in the soft band.

The time-resolved spectra show that, in the rising phase, the hard band slope
increases first, and a hump at $\sim$0.9 keV, originating from the Fe L line
complex, follows. Most spectra require two temperature optically-thin thermal
plasma components at kT $\sim$1 keV and $\sim$3 keV on top of the quiescent
component. The hot component mainly reproduces the hard band slope, and the
cool component does the 0.9 keV hump. Both components’ EMs rise linearly on
similar timescales, but the cool component is delayed by 100$-$200 sec. The
September flare has a longer delay time relative to the flare rise duration
and a more substantial cool component than the December flare, producing a
heavily rounded flare peak.

The HYDRAD field-aligned numerical simulations demonstrate that the cooler
footpoint plasmas start to increase a few hundred seconds after the hot
evaporated plasmas increase — longer flare loops have longer time delays and
weaker cool components. This result indicates that the cool components in the
$\kappa^{1}$ Ceti flares originate primarily from the footpoint plasma and
that the September flare stems from a shorter flare loop than the December
flare. The estimated loop lengths of $\approx$1010 cm are large but still
within the range of solar flare loops. Since the $\kappa^{1}$ Ceti flares have
more than two orders of magnitudes larger EMs than the solar flares, they need
significantly higher loop plasma densities or thicknesses. A significant
discrepancy in the EMpeak ratio may suggest that the HYDRAD simulations
overestimate the footpoint EMs and require a mechanism to suppress electron
transport, such as expanded magnetic loops or turbulent magnetic fluctuations.
A difference in the cool component’s EM rise may suggest that both flares are
multiple loop events, as seen in solar flares.

The NICER’s $\kappa^{1}$ Ceti observations and the HYDRAD simulations
demonstrate that the time delay of the cool component and the peak EM ratio of
the two temperature plasma components can be used as new, effective parameters
for estimating the flare loop length. We should confirm the derived relations
with more flare samples of various luminosities, durations, peak temperature,
and stellar types with existing or future NICER observations. Simultaneous
multi-wavelength observations will also greatly help constrain the flare
parameters. In particular, UV and optical observations with Hubble Space
Telescope or the TESS observatory trace hot chromospheric gas, helping
understand the whole chromospheric and coronal heating process. The HYDRAD
numerical simulations still have discrepancies with observations. We should
decipher the cause with further studies and improve the model to explain the
observations.

The material is based upon work supported by NASA under award number
80GSFC21M0002. JWR was supported by the Office of Naval Research 6.1 Support
Program. VSA acknowledges the funds from NICER GO Cycle 2 project award number
80NSSC21K0101. This work is supported by JSPS KAKENHI Grant Nos. JP20KK0072,
JP21H01124, and JP21H04492, and by NINS Grant Nos. 01321802 and 01311904. This
research has made use of data and/or software provided by the High Energy
Astrophysics Science Archive Research Center (HEASARC), which is a service of
the Astrophysics Science Division at NASA/GSFC. We thank Mr. Craig Gordon for
helping resolve a PYXSPEC problem. We thank Dr. Andrew Pollock for suggestions
of XMM-Newton RGS data analysis. We thank Drs. Stephen Drake, Yuta Notsu,
Michael F. Corcoran and Konstantin V. Getman for discussions about stellar
flare physics.

##  Appendix A Elemental Abundance Measurement

Telleschi et al. (2005) extensively studied the coronal elemental abundance of
$\kappa^{1}$ Ceti using XMM/RGS data in 2002. However, the XMM-Newton
instrumental calibration444https://www.cosmos.esa.int/web/xmm-
newton/calibration-documentation and the plasma emission codes (e.g.,
ATOMDB555http://www.atomdb.org) have significantly improved since then. The
elemental abundance of the star might also have changed in 17 years. We, thus,
independently measure the coronal elemental abundance of $\kappa^{1}$ Ceti
using the XMM/RGS data obtained on 2018 July 30 and 2019 January 29 (ObsID:
0822790901, 0822791001, PI: Wargelin).

We reprocess these datasets with SAS version
19.0666https://www.cosmos.esa.int/web/xmm-newton/sas. EPIC/MOS2 turns off
during these observations, while the EPIC-pn uses the timing mode with
relatively poor spectral resolution. We thus analyze EPIC/MOS1 and RGS data.
For MOS1, we take a 15′′ radius circular source region centered at the X-ray
peak position. The MOS1 on-axis CCD operates with the small window mode so
that we take background data from a source-free region from the surrounding
CCDs. The MOS1 light curves of these observations do not show significant time
variations. EPIC/MOS1 measures the 0.6$-$1.2 keV flux during the second
observation at $\sim$3.5$\times$10-12 ergs cm-2 s-1, which is $\sim$16% lower
than the first observation. This flux is nearly the lowest among the NICER
monitoring observations of $\kappa^{1}$ Ceti.

We produce the MOS1 spectra using the same source and background regions. For
RGS, we run rgsproc for the target position measured from the MOS1 image and
produce the source and background spectra (Figure 12). We only use the first-
order RGS spectra as the second-order RGS spectra do not have enough photon
counts to identify emission lines. We fit the unbinned MOS1 and RGS spectra
simultaneously using Cash statistic (c-stat) built in xspec (Cash, 1979). The
Cash statistic needs to add background as an additive model component so that
we simultaneously fit background spectra by an empirical model (power-law + 4
Gaussians), convolved with the source response (rmf) weighted with the
background areal scale (backscal). For the source spectra, we assume a 2$T$
thermal plasma model with various abundance values (vapec) and fit all
MOS1/RGS source/background spectra of the two observations simultaneously. We
allow for varying spectral normalization between MOS1 and RGS to account for
calibration uncertainty and kT and normalization between the two observations
for time variation. Table 5 lists the derived elemental abundance. We use
these values for the NICER data analysis and numerical simulation.

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Table 5: Applied Elemental Abundance

Element |  $\kappa^{1}$ Ceti | Sun
---|---|---
| Relative to Sun | Number | Number
H | 1.00f | 1.00E+00 | 1.00E+00
He | 1.00f | 8.51E-02 | 8.51E-02
Li | 1.00f | 1.12E-11 | 1.12E-11
Be | 1.00f | 2.40E-11 | 2.40E-11
B | 1.00f | 5.01E-10 | 5.01E-10
C | 0.42 | 1.12E-04 | 2.69E-04
N | 0.44 | 2.99E-05 | 6.76E-05
O | 0.40 | 1.95E-04 | 4.90E-04
F | 1.00f | 3.63E-08 | 3.63E-08
Ne | 0.62 | 5.29E-05 | 8.51E-05
Na | 1.00f | 1.74E-06 | 1.74E-06
Mg | 0.68 | 2.70E-05 | 3.98E-05
Al | 1.00f | 2.82E-06 | 2.82E-06
Si | 0.55 | 1.77E-05 | 3.24E-05
P | 1.00f | 2.57E-07 | 2.57E-07
S | 0.19 | 2.51E-06 | 1.32E-05
Cl | 1.00f | 3.16E-07 | 3.16E-07
Ar | 0.19 | 4.83E-07 | 2.51E-06
K | 1.00f | 1.07E-07 | 1.07E-07
Ca | 0.58 | 1.28E-06 | 2.19E-06
Sc | 1.00f | 1.41E-09 | 1.41E-09
Ti | 1.00f | 8.91E-08 | 8.91E-08
V | 1.00f | 8.51E-09 | 8.51E-09
Cr | 1.00f | 4.37E-07 | 4.37E-07
Mn | 1.00f | 2.69E-07 | 2.69E-07
Fe | 0.64 | 2.03E-05 | 3.16E-05
Co | 1.00f | 9.77E-08 | 9.77E-08
Ni | 1.59 | 2.64E-06 | 1.66E-06
Cu | 1.00f | 1.55E-08 | 1.55E-08
Zn | 1.00f | 3.63E-08 | 3.63E-08

Note. — Abundance numbers relative to H. Solar abundance reference: Asplund et
al. (2009). ffixed at the solar values in the XMM-Newton spectral fits.

Figure 12:  XMM-Newton RGS1$+$2 grating spectrum of $\kappa^{1}$ Ceti combined
from the 2018 and 2019 observations. The spectrum includes both source and
background data (black). The red line shows the best-fit 2$T$ apec model, and
the blue line does the corresponding background model. The black line is the
sum of these models. The prominent emission lines are labeled.