# Statistical Signatures of Nanoflare Activity. III.
Evidence of Enhanced Nanoflaring Rates in Fully Convective stars as Observed
by the NGTS

S. D. T. Grant Astrophysics Research Centre, School of Mathematics and
Physics, Queen’s University Belfast, Belfast, BT7 1NN, UK  D. B. Jess
Astrophysics Research Centre, School of Mathematics and Physics, Queen’s
University Belfast, Belfast, BT7 1NN, UK  Department of Physics and Astronomy,
California State University Northridge, Northridge, CA 91330, USA  C. J.
Dillon Astrophysics Research Centre, School of Mathematics and Physics,
Queen’s University Belfast, Belfast, BT7 1NN, UK  M. Mathioudakis Astrophysics
Research Centre, School of Mathematics and Physics, Queen’s University
Belfast, Belfast, BT7 1NN, UK  C. A. Watson Astrophysics Research Centre,
School of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7
1NN, UK  J. A. G. Jackman School of Earth and Space Exploration, Arizona State
University, Tempe, AZ 85287, USA  Department of Physics, University of
Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK  Centre for Exoplanets and
Habitability, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK
D. G. Jackson Astrophysics Research Centre, School of Mathematics and Physics,
Queen’s University Belfast, Belfast, BT7 1NN, UK  P. J. Wheatley Department of
Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK  M. R.
Goad  School of Physics and Astronomy, University of Leicester, University
Road, Leicester, LE1 7RH, UK  S. L. Casewell School of Physics and Astronomy,
University of Leicester, University Road, Leicester, LE1 7RH, UK  D. R.
Anderson Department of Physics, University of Warwick, Gibbet Hill Road,
Coventry CV4 7AL, UK  M. R. Burleigh School of Physics and Astronomy,
University of Leicester, University Road, Leicester, LE1 7RH, UK  R. G. West
Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4
7AL, UK  J. I. Vines Departamento de Astronomia, Universidad de Chile, Casilla
36-D, Santiago, Chile

###### Abstract

Previous examinations of fully-convective M-dwarf stars have highlighted
enhanced rates of nanoflare activity on these distant stellar sources.
However, the specific role the convective boundary, which is believed to be
present for spectral types earlier than M2.5V, plays on the observed nanoflare
rates is not yet known. Here, we utilize a combination of statistical and
Fourier techniques to examine M-dwarf stellar lightcurves that lie on either
side of the convective boundary. We find that fully convective M2.5V (and
later sub-types) stars have greatly enhanced nanoflare rates compared with
their pre-dynamo mode transition counterparts. Specifically, we derive a
flaring power-law index in the region of $3.00\pm 0.20$, alongside a decay
timescale of $200\pm 100$ s for M2.5V and M3V stars, matching those seen in
prior observations of similar stellar sub-types. Interestingly, M4V stars
exhibit longer decay timescales of $450\pm 50$ s, along with an increased
power-law index of $3.10\pm 0.18$, suggesting an interplay between the rate of
nanoflare occurrence and the intrinsic plasma parameters, for example, the
underlying Lundquist number. In contrast, partially convective (i.e., earlier
sub-types from M0V to M2V) M-dwarf stars exhibit very weak nanoflare activity,
which is not easily identifiable using statistical or Fourier techniques. This
suggests that fully convective stellar atmospheres favor small-scale magnetic
reconnection, leading to implications for the flare-energy budgets of these
stars. Understanding why small-scale reconnection is enhanced in fully
convective atmospheres may help solve questions relating to the dynamo
behavior of these stellar sources.

Computational methods (1965) — Optical flares (1166) — Stellar flares (1603) —
Flare stars (540)

††journal: ApJ††facilities: Next Generation Transit Survey (NGTS)

.

##  1 Introduction

Magnetic reconnection is a fundamental physical process in conducting plasmas
that allows for the conversion of magnetic energy through the rearrangement of
magnetic fields (e.g., Priest, 1986; Reale, 2007; Cargill et al., 2015).
Magnetic reconnection came to initial prominence in astronomy as a proposed
mechanism for observed solar flare activity. The first derived reconnection
model was presented by Sweet (1958) and Parker (1957) as a two-dimensional
magnetohydrodynamic (MHD) configuration, where there exists a long, thin
diffusion region enabling magnetic fields to reconnect. The reconnection rate
of this Sweet-Parker model was found to be inversely proportional to the
square root of the dimensionless Lundquist number, $S$, which is defined as;

$S=\frac{Lv_{A}}{\eta}\ ,$ (1)

where $L$ is the length of the diffusion region, $v_{A}$ is the Alfvén speed,
and $\eta$ is the plasma resistivity (a full derivation is presented by Priest
& Forbes, 2007). Predicted Sweet-Parker reconnection rates for solar
conditions could not recreate actual observations and the derived energetics
from Sweet-Parker events were orders-of-magnitude smaller than what was
observed in the solar corona (e.g., Crosby et al., 1993). An alternative model
was presented by Petschek (1964), which allowed for faster rates by permitting
reconnection across far shorter length scales of the diffusion region (Priest
& Forbes, 2007). This remedies an issue with the Sweet-Parker model
(reconnection rate $\propto 1/\sqrt{S}$) due to the Petschek reconnection rate
being inversely proportional to the logarithm of the Lundquist number, which
limits the influence of plasma conductivity and provides more robust
similarities to the characteristics of large flare events (e.g., Aschwanden,
2020).

Observationally, magnetic reconnection is characterised by an impulsive
brightening as magnetic energy is converted into localised plasma heating, and
is classically seen as stochastic, macroscopic events (see the reviews of
Cargill & Klimchuk, 2004; Benz & Güdel, 2010; Fletcher et al., 2011; Benz,
2017). Subsequently, the plasma gradually cools over an extended period, which
manifests as an exponentially decaying intensity from the time of maximum
brightness (e.g., Moffett, 1974; Moffett & Bopp, 1976; Kowalski et al., 2013;
Pitkin et al., 2014). The quantification of this decay process is measured
through the $e$-folding time, $\tau$, which is the time taken for the flare
luminosity to decrease by a factor $1/e$. The magnitude of this value is
dependent on the underlying local plasma conditions, such as the efficiencies
of evaporative, non-evaporative, conductive, and radiative cooling processes
(Antiochos & Sturrock, 1978). Solar and stellar flare energies are governed by
a power-law relationship (Aschwanden et al., 2000). Here, the power-law
exponent governs the frequency, $dN/dE$, of flaring events with an associated
energy, $E$, through the relationship,

$\frac{dN}{dE}~{}\propto~{}E^{-\alpha}\ ,$ (2)

where $\alpha$ represents the power-law index. The nature of a power-law
relation dictates that low-energy flares will be many times more frequent than
larger events, and that small-scale events become more energetically important
as the power-law index, $\alpha$, increases. A range of power-law indices have
been documented across varying solar and stellar flare energy windows, from
$1.35\leq\alpha\leq 2.90$ (Berghmans et al., 1998; Krucker & Benz, 1998;
Aschwanden, 1999; Parnell & Jupp, 2000; Benz & Krucker, 2002; Winebarger et
al., 2002; Aschwanden & Freeland, 2012; Aschwanden et al., 2014, 2015).

The stochastic nature of large solar flaring eventsindicates that they are too
infrequent to be viable heating mechanism for the extra-ordinary temperatures
of the solar corona, known as the coronal heating paradox. Instead,
nanoflares, with individual energies around $10^{9}$ times less than their
large-scale counterparts, were proposed as an alternative due to their higher
occurrence rates (Parker, 1988). In order to be considered as consequential to
the flare energy budget, and thus atmospheric heating, it has been established
that the minimum requirement is $\alpha\geq 2$ (Parker, 1988; Hudson, 1991).
Due to their individual low energies, nanoflares are typically embedded within
the noise floor of the measured intensity signals, leading to difficulties
identifying individual nanoflare events. However, their higher occurrence
frequency means they can be recovered from time series data using statistical
techniques that do not rely on the individual identification of macroscopic
intensity signals.

Building on the work of Terzo et al. (2011) and Jess et al. (2014), Jess et
al. (2019, henceforth referred to as Paper I) developed a robust method for
nanoflare investigation. Through Monte-Carlo simulations, realistic nanoflare
lightcurves were generated for a wide range of $\alpha$ and $\tau$ values,
coupled with precise modeling of the noise characteristics of solar
observables. Through this, Paper I was able to uncover nanoflare signatures in
solar coronal observations, manifesting as asymmetric contributions to the
intensity fluctuation distributions of coronal images, consistent with power-
law distributions on the order of $1.82\leq\alpha\leq 1.90$. Despite showing
that the solar active region under study did not appear to contain the
necessary nanoflare activity to influence coronal heating, Paper I provided a
comprehensive method for analysing small-scale flare activity in intensity
time series, and suggested the same techniques could be applied to stellar
observations.

Subsequently, Dillon et al. (2020, henceforth referred to as Paper II)
utilised the techniques of Paper I on stellar lightcurves from A, K and M-type
stars to investigate whether signals previously interpreted as $p$-mode
oscillations in dMe flare stars could in-fact be caused by nanoflares. Fourier
analysis of each spectral class showed no enhanced power around $p$-mode
frequencies (i.e., ${1-1000}$ s) in A-types, power enhancement at $p$-mode
frequencies in the K-type, and power enhancements across the entire frequency
spectrum in M-dwarfs. These enhancements were classically seen as evidence of
global wave activity generated in both the K and M-type stars, however Monte-
Carlo simulations revealed that the M-type stars produced the asymmetric
intensity fluctuation distribution effects consistent with nanoflares, as
opposed to the symmetrical effects observed in the K-type distributions that
are consistent with dominant oscillatory behaviour. The M-dwarf flare activity
produced a power-law index of $\alpha=3.25\pm 0.20$, greater than previously
reported values, and enabled Paper II to show that the flaring rate was high
enough for nanoflare signals to appear quasi-periodic when an entire stellar
disk is integrated into a single lightcurve, thus explaining their influence
on the resulting Fourier power spectra. The reason for the enhanced nanoflare
power-law indices for M dwarfs was not known, but it was theorized that the
fully convective nature of these stars may be responsible.

While solar-like stars have a combination of convective and radiative zones
bridging their core and visible surface, some stars operate in a fully
convective manner. The change from partially- to fully-convective interiors
has been related to the ‘convective boundary’, distinguished by a lack of
tachocline in later stars that are fully convective. The tachocline is a thin
region of the stellar interior at the boundary between the radiative and
convective zones that contains large radial shears due to the imbalance
between the rigid radiative zone and the differentially rotating outer
convective zone (Spiegel & Zahn, 1992; Browning, 2008). Wright & Drake (2016)
estimated that this transition occurs in M-dwarf stars around M3V and later,
with recent studies suggesting a more precise transition at approximately
M${2.1-2.3}$V (Mullan & Houdebine, 2020).

Convection is a primary driver of magnetic reconnection in stars (Pedersen et
al., 2017). As magnetic reconnection is the driving force behind flares,
changes to the convective nature of a star have important implications for the
resulting flare dynamics. The tachocline is thought to play a role in
strengthening magnetic fields in partially-convective stars such as the Sun,
as the shear forces across the region can convert poloidal fields into
stronger toroidal configurations (Parfrey & Menou, 2007). However, it is
important to note that the tachocline is not necessary for magnetic field
generation, since fully convective stars also exhibit magnetism, where the
dynamo is theorized to be driven by helical turbulence (Durney et al., 1993;
Browning, 2008; Pipin & Seehafer, 2009), but this change in dynamo is still
under debate. Indeed, Wright & Drake (2016) and Wright et al. (2018)
investigated the relationship between stellar rotation and activity levels for
fully-convective late M-type dwarf stars. They found that the
rotation/activity relationship for fully convective stars was almost
indistinguishable from partially convective stars, suggesting the solar-type
dynamo is independent of the presence of a tachocline.

Returning to the results of Durney et al. (1993), Browning (2008) and Pipin &
Seehafer (2009), it is possible to hypothesize that the enhancement of
nanoflaring rates is linked to the conversion of late M-type interiors to
fully-convective, and hence to the consequent changes induced in the helical
dynamo processes. Previous examinations of stellar flares on late-type MV
stars have found a range of power-law indices. Early space-based observations
produced values of $\alpha\sim 1.5$ (Collura et al., 1988), although a range
of partially- and fully-convective MV stars were included in the sample
studied. Subsequent studies of late-type stars produced indices $\alpha>2$ and
showed that there were no discrepancies between ground- and space-based
observatories in these calculations (Robinson et al., 1995, 1999). A trend
developed where reported power-law indices increased as the complexity of the
methods for isolating marginal flare signals developed (e.g., Güdel et al.,
2003; Güdel, 2004; Welsh et al., 2006; Hawley et al., 2014a), with values as
high as $\alpha=2.7$. However, this remains below the nanoflare power-law
index ($\alpha\approx 3.25$) presented in Paper II. This may be a result of
the novel detection techniques of Paper I providing unprecedented access to
the lower energy spectrum of flares, as it has been seen in solar observations
that the power-law index, $\alpha$, can change depending on the size and
energetics of the flares under consideration in the sample (e.g., Wang & Dai,
2013; Ryan et al., 2016; Milligan et al., 2020). Therefore, there may exist a
similar noticable shift in the gradient of the power-law index around the
transition between large and small-scale flares in MV stars. Hence, it is
important to examine the power-law indices associated with nanoflare activity
across a wide range of MV spectral sub-types with the techniques described in
Paper I and Paper II to better understand the influence of fully-convective
interiors in the generation of nanoflares.

As discussed in Paper II, an alternative source for the enhanced rate of
small-scale reconnection may be the fully-convective stars having plasma with
a higher resistivity value (Mohanty et al., 2002), which lowers the associated
plasma Lundquist numbers. Small-scale flaring has been shown to occur more
favorably via Sweet-Parker reconnection (Tsuneta & Katsukawa, 2004), thus
enhanced nanoflaring can be expected in stars with low plasma Lundquist
numbers. If nanoflare rates are enhanced in fully convective stars, then
investigating whether this is due to the change in dynamo, or down to the
plasma resistivity, could answer important questions regarding the dynamo
physics in operation in these stars.

##  2 Background to Previous Statistical Stellar Nanoflare Analysis

Figure 1: Sample lightcurves of NGTS J052346.3-361114 (M0V-type; top panel),
and NGTS J050423.8-373021 (M4V-type; lower panel) spanning 24 000 s, each
normalized by their respective standard deviations, $\sigma_{N}$. The region
highlighted in red in the lower panel denotes the intensity values that are
removed from consideration due to an excursion above 3$\sigma_{N}$, as is the
formality for macroscopic flare signatures.

Paper II used a combination of statistical and Fourier analyses to investigate
nanoflare populations in a variety of stellar sources. Through comparisons
between observational lightcurves and synthetic time series with simulated
nanoflare signals, enhanced nanoflare activity in late-type MV stars was
observed, through a larger power-law index when the nanoflare occurrence rates
were plotted as a function of their underlying energy. In order to determine
why the power-law indices in these stars were significantly larger than in
other stellar and solar studies, we re-apply the analysis techniques put
forward by Paper I and Paper II to M-dwarf stars across a range of different
spectral types. By determining the change in observed nanoflare properties
(i.e., the change in the statistical and Fourier parameters associated with
the embedded nanoflare conditions) either side of the fully-convective
boundary, we aim to identify the role a fully-convective atmosphere plays in
nanoflare occurrence rates. This work constitutes the third contribution to a
series of studies on nanoflare behavior in solar and stellar atmospheres.
Given the consistency of techniques applied across these studies, it is
prudent to provide contextual information on previous works. For detailed
discussions of the analysis procedures and modeling set-ups, it is advised to
consult Paper I and Paper II directly.

The initial detection method for nanoflare signatures involves the statistical
analyses of quiescent intensity fluctuations following a traditional Z-scores
approach (Sprinthall, 1990). A histogram of the fluctuations is generated,
with two distinct statistical deviations away from a standardized Gaussian
distribution providing evidence of nanoflares. These signatures, identified by
Terzo et al. (2011) and Jess et al. (2014), were diagnosed through the
modeling of the observed lightcurves by embedding the rapid rises and
exponential decays of intensities around the noise limit that are associated
with the energetics of nanoflares. The first detection characteristic is a
negative median offset of the intensity fluctuation distribution, whereby the
median value of the histogram is $<0~{}\sigma_{N}$, i.e., offset from the mean
of the distribution that is equal to $0~{}\sigma_{N}$ following the
application of polynomial detrending. This was shown to be associated with the
exponentially decaying nature of the nanoflare lightcurve. The decay phase
produces more significant negative fluctuations below the mean than the
relatively brief elevated signal of the flare event, thus providing an offset
between the median and mean that is directly measurable. The second signature
of nanoflare activity is an excess of fluctuations at $\sim 2~{}\sigma_{N}$,
which is caused by the associated energetics of nanoflares producing
consistent peak brightenings around $\sim 2~{}\sigma_{N}$ (see Jess et al.
2014 and Paper I for further details on flare energy modeling). This gives
rise to an asymmetric distribution with a slight excess of fluctuations
visible at $\sim 2~{}\sigma_{N}$ in the corresponding histogram, which is best
categorized through Fisher skewness coefficients. In addition to the primary
nanoflare indicators described above, benchmarks on the shapes and widths of
the intensity fluctuation distributions are provided through calculation of
the kurtosis and $\zeta$ values, where $\zeta$ is the ratio of the full-width
at eighth-maximum to that of the full-width at half-maximum (i.e.,
FW$\frac{1}{8}$M-to-FWHM ratio) of the resulting distribution. These
parameters allow for additional information on the nature of the distribution,
with kurtosis linked to the prevalence of outliers at high $\sigma_{N}$
values, and deviations from the established $\zeta$ value for a normal
distribution ($\zeta=1.73$) revealing non-linear distributions of measurements
around the mean.

As established in Paper I & Paper II, a quiescent lightcurve exhibiting both
of these statistical signals, identified through the median offset and Fisher
skewness, contains embedded nanoflare signatures. Terzo et al. (2011)
presented an observation of nanoflare activity that only induced a median
offset in the distribution, due to the weak nanoflare signal not producing
sufficient peak amplitudes to influence the positive wing of the distribution.
However, both effects presenting in observations, and corroborated by the four
chief diagnostics, cannot be explained by any alternative mechanism. In an
observation of purely ambient solar or stellar plasma, noise would be the only
artifact. However, noise fluctuations follow a standard Gaussian distribution
as a result of Poisson statistics tending to a Gaussian in the limit of large
number statistics (Terzo et al., 2011). Solar and stellar observations contain
a variety of oscillatory phenomena, ranging from the signatures of MHD waves
in their atmospheres (see the review of Jess et al., 2023) to modulation
caused by the rotation of the star and starspots. In the context of this
study, these signals have no influence over the asymmetric effects under
consideration, as the sinusoidal nature of linear oscillations produces a
symmetric distribution around the mean. Starspots, characterized by periodic
reductions in intensity for times related to rotation, will decrease both the
median and mean of the sample, providing a minimal effect in the offset
between the two. Their effect on the timeseries is also mitigated through the
application of polynomial detrending, as the periodicity of their effect is on
the order of multiple days. MHD waves are also capable of steepening into non-
linear shocks that produce notable intensity enhancements over a short period
(e.g., Carlsson & Stein, 1997; Grant et al., 2018). However, these shocks
display a ‘saw-tooth’ pattern to their intensity morphology, in contrast to
the exponential decay of flares, and thus would not reproduce the statistical
effects described above. Therefore, with these effects discounted, and any
macroscopic flare events robustly removed from the data (see Section 3), there
is a confidence that the statistical effects derived from the intensity
fluctuation distributions are positive signatures of nanoflare activity.

Through inspection of nanoflare observations with insight from simulations,
Terzo et al. (2011) and Jess et al. (2014) found the interval between
nanoflares in a single lightcurve to be $\sim 360$ s, a similar frequency to
ubiquitous $p$-mode signatures ($\sim 1-10$ mHz; Andrews, 1989; Rodríguez-
López et al., 2014; Rodríguez et al., 2016). Subsequently, Paper I were able
to derive the frequency of flaring events given their power-law index, showing
that the greater the power-law index, the higher the frequency of nanoflares
in a given sample (see Figure 4 of Paper I). With the low flaring intervals
and high frequencies found, it was suggested that nanoflare signals, when
integrated across a field-of-view, can no longer be viewed as stochastic
events like their macroscopic counterparts. Instead, they can be considered to
be a quasi-periodic phenomena, particularly given the exceptionally large
power law index of $\alpha\sim 3.25$ reported by Paper II.

The quasi-periodic nature of nanoflares was successfully utilized in Paper II
to further verify their detection. When inspecting the statistical
distributions of K- and M-type stars, there was the potential, but marginal,
signature of nanoflares in K-types, as opposed to clear nanoflare signals in
the M-types. Paper II subsequently employed Fourier techniques to distinguish
the quasi-periodicities of stellar nanoflares. Power spectral densities (PSDs)
were computed for the longest continuous time series common across all stars
in order to maximize the resulting frequency resolution. These PSDs revealed
power at $p$-mode frequencies in both types, however the nature of that power
differed, with M-types showing power enhancement across a wide range of
frequencies, synonymous with nanoflare activity from a range of interval
times/frequencies, as opposed to the strictly oscillatory nature of K-type
atmospheres (see Figure 3 of Paper II). Additionally, the PSDs in M-types
displayed a prominent spectral slope following the peak energy value, which
was also found to be due to the underlying nanoflare signal. Fourier analysis
is therefore a valuable tool in both confirming nanoflare signal in the
source, and differentiating between those atmospheres with wave and nanoflare
interplay (i.e., partially-convective interiors) and where nanoflares dominate
the energy landscape (i.e., fully-convective interiors).

Throughout these works, the observational findings have been substantiated and
interpreted through simulated nanoflare time series. Paper I devised the
methodology and analysis techniques, based on the Monte-Carlo modeling of
flare intensities with additionally added camera-specific noise signatures,
coupled with a range of typical nanoflare amplitudes and decay rates, and
provides a detailed description of the set-up. Paper II took these techniques
and tailored them for a strictly stellar scenario, including the remodeling of
the noise profiles and resizing the observed area to mimic a full stellar
disk, as opposed to a sub-section of the solar atmosphere used in Paper I.
Flare energies between $10^{22}-10^{25}$ erg, typical of nanoflares, were used
alongside a linear scaling relationship to reproduce the observed counts. Each
simulated lightcurve has two variables controlling the nanoflare input: the
power-law index, $\alpha$, and the $e$-folding time, $\tau$. Lightcurves were
then generated for a dense grid of parameter ranges, $1\leq\alpha\leq 4$ (in
steps of 0.05) & $5\leq\tau\leq 500$ (in steps of 5 s), consistent with
previous viable observation ranges (Terzo et al., 2011; Jess et al., 2014).
This synthesis produced 6100 lightcurves embedded with characteristic noise
and unique nanoflare configurations. Subsequent comparison between synthetic
and observed lightcurves is achieved through forming distributions of each
synthetic time-series, and identifying the configuration with matching median
offset, Fisher skewness, kurtosis and $\zeta$ values (see Figure 5 of Paper
I). Given the stringent requirements of this matching criteria, unique
solutions are found for the observables in both Paper I and Paper II. Given
the computational demands, the simulations presented in Paper II are used in
this study, thus further details on their reproducability can be found in the
original paper.

In this paper, we apply proven statistical nanoflare analysis techniques to a
wide range of M-type stars that lie on either side of the predicted fully-
convective boundary. We compare their statistical and Fourier properties to
the simulations generated in Paper II in order to determine the probable
underlying nanoflare conditions and the effect of the convective boundary on
the uncovered nanoflare properties.

##  3 Observations With NGTS

To remain consistent with Paper II, the Next Generation Transit Survey (NGTS;
Wheatley et al., 2018) was utilized to obtain the observations. The long time
series (each in excess of $10^{5}$ frames) and short cadence ($\sim 12$ s)
available for thousands of M-type stars allow for the accumulation of suitable
number statistics necessary for nanoflare analyses. The initial spectral
classification generated by the NGTS pipeline (which utilizes Spectral Energy
Distribution fitting, see section 5.1.1 in Wheatley et al., 2018) was combined
with stellar parameters from the TESS Input Catalog Version 8 (TIC V8; Stassun
et al., 2018), to ensure robust spectral sub-type identification.

Figure 2: Histograms of intensity fluctuations, each normalized by their
respective standard deviations, $\sigma_{N}$, for the NGTS J052346.3-361114
(M0V-type; top panel), and NGTS J050423.8-373021 (M4V-type; lower panel)
lightcurves. A standardized Gaussian profile is overplotted in each panel
using a dashed red line for reference. The M4V-type distribution has a
negative median offset with respect to the Gaussian, in addition to elevated
occurrences at $\sim 2~{}\sigma_{N}$, which is consistent with the statistical
signatures of nanoflare activity. On the other hand, the M0V-type intensity
fluctuations provide effectively zero negative median offset, and no elevated
occurrences at $\sim 2~{}\sigma_{N}$. This is inconsistent with clear
statistical signatures of nanoflare activity, with the resulting distribution
remaining more consistent with the presence of photon-based shot noise. Zoomed
insets highlight the ranges spanning $-0.4\leq\sigma_{N}\leq 0.0$ and
$1.7\leq\sigma_{N}\leq 2.2$, where negative median offsets and occurrence
excesses, respectively, are clearly visible for the M4V stellar source. For
improved clarity, the blue and gold lines display the corresponding
distributions in each zoomed panel.

To ensure the ideal sample of objects for study, a number of selection
criteria were applied to the catalog extracted from NGTS to remove unwanted
artifacts. Initially, the magnitude of the stars were constrained to ensure
similar photon noise characteristics for each object. Thus, only stars with
magnitudes matching the range of the previous study of Paper II (spanning NGTS
magnitudes of $\sim 12-14$) were progressed. This ensured that the magnitude
of the fluctuations in each spectral type were approximately equivalent, with
M0V and M4V stars exhibiting standard deviations of $1.6\pm 0.1$% and $2.0\pm
0.6$%, respectively. The average standard deviation of the time-series across
the full sample was $2.0\pm 0.8$% of the mean, thus an equivalent fluctuation
profile can be applied to the modeling. Next, in a similar manner to Jackson
et al. (2023), complementary photometric and astrometric data from Gaia DR3
(Gaia Collaboration et al., 2016, 2022) were utilized to exclude unwanted
candidates from the sample. In particular, astrometric excess noise analysis
was applied to exclude any binary systems (Evans, 2018), and the photometric
filtering processes of Arenou et al. (2018) identified any blended sources. A
final step was to consider the rotation rate of the stars in the sample. Given
the length of the NGTS time series, it is not possible to extract the periods
associated with slow rotating M-dwarfs (i.e., above 30 days). Therefore,
generalized Lomb-Scargle techniques (Lomb, 1976; Scargle, 1982) were applied
to the NGTS data to identify fast rotators and exclude them from the sample.
This is done to ensure the sample contains similarly ‘slow’ rotators, as
defined in previous studies (e.g., Mondrik et al., 2019), thus removing the
influence rotation rate has on increasing stellar activity (West et al., 2008;
Candelaresi et al., 2014). Only two candidates exhibited definable
periodicities and were thus excluded, an M2V & an M4V with periods of 21.1
days and 19.8 days, respectively.

After accounting for magnitude considerations, avoiding blended sources,
ensuring TIC matching and excluding fast rotators, we were able to find 5
stars for each spectral sub-type, consisting of M0V, M1V, M2V, M2.5V, M3V, and
M4V. The stellar properties (NGTS identifier, RA/Dec, magnitude, etc.) of
these candidates are provided in Table A1. Only one suitable M5V star with
TIC-derived stellar parameters could be identified, and no sub-types later
than this were found. The intrinsic brightness of M-dwarfs decreases with
increasing sub-type (Yang et al., 2017), leading to difficulty in identifying
suitable candidate stars with the desired brightness properties. Future
investigations of post-M4V stars may be fruitful, but identifying a suitable
number of candidates may prove difficult with existing instrumentation. Hence,
we limit our current study within the range of M0V – M4V, where we have
multiple candidates available for comparison. This range also overlaps well
with the predicted dynamo mode transition to fully convective(M${2.1-2.3}$V;
Mullan & Houdebine, 2020), making it suitable for the study of the role that
fully-convective starsnplay in the resulting nanoflare activity.

The lightcurves were background corrected and flat-fielded via the NGTS data
reduction pipeline described and visualized in Wheatley et al. (2018). This
pipeline calculates a relative error in the flux at each data point in the
time series. This error correlates with cloudy weather and/or high airmass
values. Any fluctuations in this error exceeding 1$\sigma$ above the mean
value were removed, resulting in $\sim$10% of each time series being omitted.
This removed any data that had statistically significant increases in its
associated flux uncertainties, therefore preventing any large flux errors
(largely due to poor seeing conditions) from contaminating the final time
series.

Table 1: Averaged characteristics of the statistical properties by each spectral type. Spectral type | Median offset ($\sigma_{N}$) | Fisher skewness |  $\zeta$ ratio | Kurtosis
---|---|---|---|---
M0V | $-0.040\pm 0.008$ | $0.004\pm 0.002$ | $1.740\pm 0.005$ | $0.128\pm 0.010$
M1V | $-0.040\pm 0.008$ | $0.008\pm 0.002$ | $1.746\pm 0.005$ | $0.180\pm 0.013$
M2V | $-0.030\pm 0.006$ | $0.003\pm 0.011$ | $1.766\pm 0.010$ | $0.196\pm 0.037$
M2.5V | $-0.050\pm 0.000$ | $0.019\pm 0.003$ | $1.739\pm 0.004$ | $0.227\pm 0.072$
M3V | $-0.050\pm 0.000$ | $0.025\pm 0.004$ | $1.750\pm 0.006$ | $0.180\pm 0.017$
M4V | $-0.050\pm 0.000$ | $0.051\pm 0.014$ | $1.754\pm 0.010$ | $0.267\pm 0.032$

To prepare the data for statistical analysis, each lightcurve was detrended by
a low-order polynomial so the mean value is zero. Then, the time series is
subsequently renormalized by its own standard deviation, $\sigma_{N}$. Next,
the lightcurves extracted for each observing sequence were examined for the
presence of non-Gaussian intensity enhancements such as macroscopic flare
signatures following the methodology described by Paper II. Emission
signatures exceeding 3$\sigma_{N}$ above the mean value, lasting continually
for a minimum of 1 minute (5 datapoints), were identified in each lightcurve.
Based on a normal distribution, the probability of these event presenting
through Gaussian-Poisson noise is $\lesssim 2\times 10^{-13}$, hence allowing
for robust detection of macroscopic flaring activity. Every star, apart from
the M2V candidate NGTS J062005.7-372555, demonstrated macroscopic flare
signatures, resulting in the removal of a further $\sim 0.2-2.5$% of the
remaining M-type time series. The degree of macroscopic flare emission varied
with the spectral sub-type, with M4V stars exhibiting approximately five times
more detected flares than the M0V stellar types, consistent with previous
studies (Hawley et al., 2014a; Yang et al., 2017). Once the larger-scale flare
signatures had been identified, they were subsequently removed from the time
series using an interval of $\pm$5 minutes ($\pm$25 datapoints) from the first
and last detection above the 3$\sigma_{N}$ threshold (see Figure 1). Removing
these signatures allows for the assumption of normality in the intensity
distribution, i.e. shot and readout noise combined with ambient stellar
intensity (Terrell, 1977; Delouille et al., 2008). This has been shown as
valid in Terzo et al. (2011) and Jess et al. (2014) alongside Paper I and
Paper II, and allows for sensitivity in the detection of any residual
intensity deviations within the expected intensity ranges of a normal
distribution. The number of macroscopic flares removed were used to calculate
approximate flare rates for the M stars, which are displayed in Table A1. To
ensure consistency with previous stellar nanoflare investigations, the
filtering steps employed were identical to those used in Paper II, with the
filtered lightcurves subsequently cropped to 97 060 datapoints each to match
the number statistics from the previous study. This allows a direct comparison
to be made with the work of Paper II, since the previously published nanoflare
simulations can be re-used due to identical number statistics, filtering
techniques, desired $\alpha$ (power-law index) and $\tau$ ($e$-folding time)
ranges, in addition to specific NGTS-modeled noise characteristics.

##  4 Analysis and Discussion

To investigate the possible changing nanoflare properties with spectral type,
we utilized the statistical and Fourier analysis techniques outlined in
Section 2. As outlined, nanoflares give rise to 2 distinct statistical
signatures, which can be used to diagnose stellar nanoflare activity. We
present two example histograms of intensity fluctuations in Figure 2 for stars
NGTS J052346.3-361114 (M0V spectral type; top panel) and NGTS J050423.8-373021
(M4V spectral type; lower panel). From Figure 2 it is clear that opposite ends
of the included spectral types, which lie on either side of the predicted
fully-convective boundary, demonstrate distinctly different statistical
signatures. The M0V star exhibits weak nanoflare signatures, with a marginal
negative median offset and no elevated intensity fluctuations at $\sim
2~{}\sigma_{N}$. On the contrary, the M4V star has a clear excess of $\sim
2~{}\sigma_{N}$ intensity fluctuations in addition to a prominent negative
median offset. The signatures of the M4V star shown in Figure 2 are consistent
with previous positive stellar nanoflare identifications in Paper II. The
distinct increase of visible nanoflare signatures within the expected regime
of full convection indicates that the enhanced nanoflare rates are related to
the underlying convective nature of the star.

Figure 3: The bootstrap-averaged statistical properties of the intensity
fluctuation histograms for each stellar classification. Beyond the convective
boundary, at approximately M2.5V and later, sub-types begin to exhibit
statistical signatures that are consistent with the presence of nanoflare
activity, including larger median offsets (top panel), increasing levels of
kurtosis (second panel from top), and higher Fisher skewness values (second
panel from bottom). The $\zeta$ (FW$\frac{1}{8}$M-to-FWHM ratio) values do not
vary significantly as a function of stellar classification. However, this is
likely due to the interplay between the power-law index of the nanoflares and
the duration of the $e$-folding timescales, which are able to counteract the
statistical effects of one another.

These examples illustrated in Figure 2 clearly identify the vastly different
nanoflare signatures present at either end of the investigated range of
spectral sub-types. To better examine the change in nanoflare activity across
the given spectral range (M0V – M4V), the derived properties were averaged
according to their specific spectral type following the bootstrap method
documented by Efron et al. (1979). Straightforward averaging of features that
are dependent on the underlying stellar plasma conditions from multiple stars
is challenging due to the uncertain behavior of the standard errors of the
given parameters. Hence, bootstrapping techniques are used extensively
throughout the physical sciences to better calculate confidence intervals for
data following non-standard or unknown distributions (Simpson & Mayer-
Hasselwander, 1986; Desmars et al., 2009; Yao et al., 2017).

Figure 3 shows the change in the median offset, kurtosis, Fisher skewness, and
$\zeta$ values, respectively, as a function of spectral sub-type. The results
are also tabulated in Table 1. From Figure 3, we find a distinct change in the
nanoflare statistical signatures as a function of spectral sub-type,
suggesting the convective boundary may play an important role in the
generation of efficient nanoflare conditions. We find that M2.5V (and beyond)
stars exhibit distinct nanoflare statistical signatures that are consistent
with those put forward by Paper II. Specifically, the average median offset
for the pre-M2.5V stars exhibits a large spread around a weakly offset value
(upper panel of Figure 3), while the post-M2.5V stars demonstrate a larger
consistent offset magnitude (with less uncertainty) of approximately
$-0.05\sigma_{N}$.

The Fisher skewness value is effectively zero for pre-M2.5V stars (second
panel from bottom in Figure 3), suggesting no, or very weak, nanoflare
activity. From M2.5V onward, there is a clear increasing trend in the Fisher
skewness value of the fluctuation distribution, with the M4V sub-type
displaying a Fisher skewness equal to $0.051\pm 0.014$, providing strong
evidence for the presence of nanoflares.

Figure 4: The Fourier power spectral densities (PSDs) for example M0V (upper
panel) and M4V (lower panel) stellar sources, displayed in normalized units of
$\sigma_{N}^{2}/\text{mHz}$. The crosses in each panel depict the individual
power values as a function of frequency, while the solid red line reveals a
trendline calculated over $\pm 6$ frequency elements ($\pm 0.478$ mHz). It can
be seen that the PSD for the M0V star is relatively flat, with small-amplitude
power enhancements in the range $3-10$ mHz, which is consistent with typical
$p$-mode oscillations. On the contrary, the PSD for the M4V star exhibits a
clear enhancement of spectral energy at lower frequencies, resulting in a
spectral slope of $\beta=-0.57\pm 0.05$ that begins at $0.32\pm 0.04$ mHz,
followed by numerous power peaks in the range of $1-10$ mHz, which is
consistent with the presence of both nanoflare activity and $p$-mode
oscillations.

In the additional distribution diagnostics, the relationship is less clear.
Regarding the kurtosis (second panel from top in Figure 3) there appears to be
a trend in that statistical kurtosis increases across the full sample, between
M0V and M4V. However, the exact nature of this relationship is obscured by the
large uncertainty in the kurtosis for spectral types around the dynamo mode
transition. In particular, the M3V sub-type appears to exhibit a decrease in
kurtosis, though this may be due to the large uncertainty in M2V & M2.5V
producing abnormally large values.

There is no clear trend visible in the corresponding $\zeta$ values (lower
panel of Figure 3) as a function of spectral sub-type. It must be remembered
that the $\zeta$ value is a measure of the deviation away from a standard
Gaussian distribution, which has a value of $\zeta=1.73$. As discussed in
Paper I, increased nanoflare decay timescales (i.e., larger $\tau$ values)
result in broader tails of the intensity fluctuation distributions, hence
giving rise to $\zeta>1.73$. On the contrary, large power-law indices help
reduce the widths of the tails in the intensity fluctuation distributions due
to the superposition of positive intensity fluctuations (e.g., new nanoflares)
superimposed on top of decaying (i.e., negative) intensity fluctuations, which
result in $\zeta<1.73$. As such, the interplay between the power-law index and
the nanoflare $e$-folding time produces the specific value of $\zeta$
measured. As such, the relatively consistent values of $\zeta$ found across
the spectral range M0V – M4V may result from an increased nanoflare rate
expected for M4V stars being negated by an increase in the associated decay
timescales of the resulting nanoflares, i.e., a larger $\alpha$ term being
coupled with longer $\tau$ values.

As described in Section 2 (and in detail in Paper I), the products of the four
distribution diagnostics can be used to derive the power-law index, $\alpha$,
and the nanoflare decay timescale, $\tau$, for each observation. Through the
calculation of median offset, Fisher skewness, kurtosis, and $\zeta$ for each
of the $6100$ synthesized lightcurves, these can be compared to the values
seen derived in Table 1. This was achieved by considering each diagnostic
individually. The values of the statistical diagnostic within a range of $\pm
1\sigma_{N}$ are directly compared to the corresponding simulated signatures,
to determine which $\alpha$ and $\tau$ values match. The values of these
nanoflare paramaters which equal all four of the observed diagnostics are the
derived $\alpha$ and $\tau$ values. For pre-M2.5V stars, it was not possible
to establish values for the power-law index and $e$-folding time that were
self-consistent with the Monte Carlo models provided by Paper II. For example,
it was possible to find self-similarity between the observational and model
power-law indices, but this resulted in decay timescales that were
incompatible and inconsistent. As a result, we are unable to define nanoflare
characteristics for pre-M2.5V stars, suggesting that nanoflare activity may be
very weak (or not present) on these specific stellar sub-types.

Table 2: Nanoflare parameters per spectral type, derived from statistical properties of Monte-Carlo modeled nanoflare timeseries. The approximately symmetrical distribution of statistical properties leads to an ambiguity in the derived power-law indices, hence $\alpha_{1}$ and $\alpha_{2}$.  Spectral type | $\alpha_{1}$ | $\alpha_{2}$ | $\tau(s)$
---|---|---|---
M2.5V | $2.25\pm 0.25$ | $3.00\pm 0.25$ | $200\pm 100$
M3V | $2.25\pm 0.20$ | $3.00\pm 0.20$ | $200\pm 100$
M4V | $2.30\pm 0.20$ | $3.10\pm 0.20$ | $450\pm 50$
Figure 5: The bootstrap-averaged properties of the Fourier power spectral
densities (PSDs) across each spectral type. The upper panel displays the peak
frequency values (in mHz), which are found to reside within the range of
approximately $1-4$ mHz, which is consistent with both nanoflare activity and
$p$-mode oscillations, and therefore cannot be used as an indicator of
nanoflare activity by itself. The middle and lower panels display the turning
point frequencies (in mHz) and subsequent spectral slopes, respectively, as a
function of stellar classification. When compared to the Monte Carlo nanoflare
simulation outputs depicted in Figure 6, the distinct jump in turning point
frequency and spectral gradient at the convective boundary (M2.5V) provides
clear evidence of prominent nanoflare activity in M2.5V – M4V stellar sources.

The statistical parameters for the M2.5V, M3V and M4V stars, which are
believed to be beyond the convective boundary and therefore best described as
‘fully convective’, exhibit values consistent with the power-law indices of
$\alpha=2.25\pm 0.25$ or $\alpha=3.00\pm 0.25$, $\alpha=2.25\pm 0.20$ or
$\alpha=3.00\pm 0.20$, and $\alpha=2.30\pm 0.20$ or $\alpha=3.10\pm 0.20$,
alongside the $e$-folding timescales of $\tau=200\pm 100$ s, $\tau=200\pm 100$
s, and $\tau=450\pm 50$ s, respectively (see Table 2). As highlighted by Paper
II, the approximate symmetry of the statistical distributions about their peak
values leads to ambiguity in the derived power-law indices (see, e.g., the
bands of similar values shown in each panel in Figure 5 of Paper II ). As a
result, it is possible to map each sub-type onto two distinct solutions for
the power-law index. Irrespective of this ambiguity, both sets of possible
nanoflare conditions are highly active (i.e., $\alpha>2$), in stark contrast
to the effectively zero statistical nanoflare signals observed in the
pre-M2.5V spectral sub-types. The larger uncertainty in the M2.5V power-law
indices are due to the larger uncertainty associated with the kurtosis value
for these spectral sub-types. M2.5V stars are at the boundary of predicted
full convection, so a larger spread in their nanoflare properties would be
expected if full convection is the cause of the spectral ‘break’ in associated
power-law indices.

Table 3: Average characteristics of the Fourier PSD properties by each spectral type.  Spectral type | Gradient | Turning point (mHz) | Peak frequency (mHz)
---|---|---|---
M0V | $+0.044\pm 0.023$ | $0.000\pm 0.000$ | $3.952\pm 0.320$
M1V | $-0.051\pm 0.019$ | $0.000\pm 0.000$ | $2.695\pm 0.278$
M2V | $-0.035\pm 0.022$ | $0.000\pm 0.000$ | $1.956\pm 0.611$
M2.5V | $-0.326\pm 0.044$ | $0.762\pm 0.105$ | $1.821\pm 0.338$
M3V | $-0.330\pm 0.022$ | $0.684\pm 0.063$ | $3.276\pm 0.557$
M4V | $-0.518\pm 0.018$ | $0.467\pm 0.103$ | $1.757\pm 0.822$

Interestingly, the M4V stars display evidence for longer $e$-folding
timescales when compared to their M2.5V and M3V counterparts. This may imply
that the power-law index is marginally greater than for the earlier spectral
classes. As previously discussed, constant $\zeta$ values are seen throughout
the spectral sub-type range, and are thought to be due to the statistical
effects of larger power-law indices being negated by the slower decay
timescales associated with those stars (see Paper I for a more thorough
discussion of this interplay). The specific values for the $e$-folding
timescales for the M2.5V and M3V stars of $\tau=200\pm 100$ s, are consistent
with the previous work of Paper II, who studied similar stellar types.

Overall, the changes in the statistical parameters indicate that post-dynamo
mode transition M-dwarf stars (i.e., M2.5V and later and fully convective)
exhibit greatly enhanced stellar nanoflare activity when compared to the
partially convective pre-dynamo mode transition M-dwarfs that show little-to-
no evidence for nanoflare activity.

As highlighted in Paper II, the examination of Fourier signatures, which are
derived directly from the stellar lightcurves, can help disambiguate any
derived nanoflare characteristics and further substantiate the evidence for
specific activity levels. Following the methods documented by Welch (1961) and
Vaughan (2012), power spectral densities (PSDs) were derived from the stellar
time series. The longest continuous time series (i.e., the longest
uninterrupted series of frames) common to all stars was $2095$ datapoints,
slightly shorter than the $2316$ consecutive frames employed by Paper II. This
resulted in the frequency resolution being slightly reduced from
$\Delta{f}=0.0356$ mHz to $\Delta{f}=0.0398$ mHz in the present study. In
order to readily compare the observational PSDs to those calculated from the
Monte Carlo nanoflare models of Paper II, the Fourier signatures needed to be
re-calculated adhering to the new frequency resolution. Hence, utilizing the
new frequency resolution, we re-computed the PSDs and corresponding ‘heat map’
of the simulated Fourier properties (c.f., Figure 7 of Dillon et al., 2020) as
a function of both the nanoflare power-law and $e$-folding time. The
recalculated heat map is displayed in Figure 6. Due to the change in frequency
resolution being a relatively small value ($0.0042$ mHz), no noticeable
deviations from Figure 6 and the original distribution (Figure 7 of Dillon et
al., 2020) can be seen, with the trends identical in both studies. This is
likely a result of the Fourier nanoflare trends being relatively broad in
frequency with peak values sufficiently away from the lowest and highest
(i.e., Nyquist) frequencies of the time series, and therefore are not
significantly affected by very small changes in the underlying frequency
resolution.

As with the statistical signatures shown in Figure 3, there are dramatic
differences in the Fourier properties between M0V and M4V stars. As seen in
Figure 4, the M0V has an effectively flat power spectrum (suggesting no
nanoflare signal is present; Dillon et al., 2020), which is contrasted by the
M4V star that demonstrates a spectral slope of ${\beta=-0.57\pm 0.05}$ between
the frequencies ${\sim 0.3-6.0}$ mHz. In Figure 4, the black crosses represent
the individual frequency-dependent power measurements, while the solid red
line depicts a trendline established over $\pm 6$ frequency elements ($\pm
0.478$ mHz). In the lower panel of Figure 4, a PSD slope is consistent with
enhanced rates of stellar nanoflare activity, which begins at the ‘turning
point’ of $0.32\pm 0.04$ mHz. As defined by Paper II, the turning point is
defined as the initial peak before the gradual reduction in Fourier power with
increasing frequency. It must be noted that both PSD plots shown in Figure 4
(i.e., for M0V and M4V spectral types) exhibit numerous power peaks in the
range of $1-10$ mHz, consistent with both stellar nanoflare signatures (Dillon
et al., 2020) and the presence of $p$-mode oscillations generated in the
convective layers of M-dwarf stellar sources (M-dwarf stars are believed to
exhibit solar-like oscillations, hence $p$-modes synonymous with the typical
solar frequency range; Rodríguez-López et al., 2014; Rodríguez et al., 2016).
As the entire range of spectral types included in this study (M0V – M4V) are
expected to exhibit $p$-mode oscillations, the peak frequencies within this
interval are not conclusive evidence alone of nanoflare activity.

The averaged (following bootstrap procedures) Fourier properties per spectral
type are shown in Figure 5, and tabulated in Table 3. As with the averaged
statistical signatures shown in Figure 3, there is a marked change in Fourier
features consistent with nanoflare activity for spectral classifications M2.5V
and later. Evidence for this is shown in the averaged PSD spectral gradient
(lower panel of Figure 5), where pre-M2.5V stars have relatively flat spectral
slopes ($\beta\sim 0$), yet stellar sources past the convective boundary at
M2.5V and later demonstrate increased magnitude spectral slopes in the range
of $-0.6\leq\beta\leq-0.3$. Note that the peak frequency values (upper panel
of Figure 5) are relatively consistent across all M-dwarf stellar sources,
approximately in the range of $2-4$ mHz. As discussed above, this alone does
not constitute evidence of nanoflare activity since all of these sources are
expected to demonstrate $p$-mode oscillations spanning that particular
frequency interval (Guenther et al., 2008; Rodríguez-López et al., 2014).

The corresponding ‘turning point’, where the spectral slopes are observed to
begin, is, of course, equal to zero for the pre-M2.5V stars since they do not
exhibit any associated spectral slopes (middle panel of Figure 5). However,
for spectral classifications beyond M2.5V, where the stars are believed to be
fully convective, a relatively constant value (when errors are included) in
the range of ${0.3\leq f\leq 0.9}$ mHz is found, which is consistent with the
previous work of Paper II. In simulated nanoflare lightcurves documented by
Paper II, an increased flare decay rate (i.e., a longer $\tau$ value) gave
rise to a decreased frequency of the Fourier turning point. Examination of the
middle panel of Figure 5 shows that while the turning point frequencies are
distinctly different from the pre-M2.5V stars, there does seem to be tentative
evidence that the average turning point frequency decreases across the M2.5V,
M3V, and M4V spectral types. This is further evidenced in Table 3, where the
turning points of the M2.5V, M3V, and M4V stars are computed as $0.762\pm
0.105$ mHz, $0.684\pm 0.063$ mHz, and $0.467\pm 0.103$ mHz, respectively. The
evidence suggests that the $e$-folding timescales associated with the M4V
stars are longer than their M2.5V counterparts, which is consistent with the
intensity fluctuation statistical signatures discussed above.

Table 4: Nanoflare parameters per spectral type, derived from Fourier properties of Monte-Carlo modeled nanoflare timeseries. There is no ambiguity in the derived power-law indices.  Spectral type | $\alpha$ | $\tau(s)$
---|---|---
M2.5V | $3.00\pm 0.15$ | $200\pm 100$
M3V | $3.00\pm 0.15$ | $250\pm 100$
M4V | $3.10\pm 0.15$ | $450\pm 50$

Comparing the derived Fourier properties to the heat maps shown in Figure 6,
it is possible to estimate the power-law indices and decay timescales for each
of the M2.5V, M3V, and M4V stellar types that shown clear evidence for
nanoflare activity. We find power-law indices of $\alpha=3.00\pm 0.15$,
$\alpha=3.00\pm 0.15$, and $\alpha=3.10\pm 0.15$, alongside nanoflare
$e$-folding timescales of $\tau=200\pm 100$ s, $\tau=250\pm 100$ s, and
$\tau=450\pm 50$ s, for the M2.5V, M3V, and M4V spectral types, respectively
(see Table 4). Importantly, these values are consistent with the statistical
analyses, with the Fourier techniques providing additional benchmarks to
validate the nanoflare properties extracted from the observational time series
and resolve the ambiguity in power-law index arising from the statistical
analysis. In contrast to the statistical mapping, the derived Fourier
parameters of the M3V stars are consistent with a marginal $e$-folding time
enhancement compared to the M2.5V classifications. This is likely related to
the same physical processes that caused enhanced $e$-folding timescales in the
M4V star. However, this is difficult to ascertain due to the relatively large
errors in determining the plasma decay rate over the entire stellar surface.

Combining the Fourier and statistical analyses (see Table 5), we find that the
fully convective M2.5V and M3V sub-types exhibit nanoflare power-law indices
of $\alpha=3.00\pm 0.20$ and $\alpha=3.00\pm 0.18$, respectively. The M2.5V
sub-types are consistent with a decay timescale of $\tau=200\pm 100$ s,
whereas the M3V stars display tentative evidence for a slightly enhanced
$e$-folding timescale of $\tau=225\pm 100$ s. These $e$-folding timescales and
power-law indices are values consistent with similar M-dwarf spectral types
studied by Paper II, whereas M4V stars exhibit elevated power-law indices of
$\alpha=3.10\pm 0.18$, with an increased decay timescale of $\tau=450\pm 50$
s. With these properties confirmed, the behavior of these flares in comparison
to M-dwarf flare samples as-a-whole can be inferred. It has been established
that a general relationship between the flare duration, $t$, and energy holds
for observable flare populations, namely that $t\propto E^{x}$, where
$x\approx 0.33$ for solar and G-type stellar flares (Veronig et al., 2002;
Maehara et al., 2015), whereas it drops to $x\approx 0.2$ for solar
microflares (Christe et al., 2008). In Chang et al. (2015), a directly
comparable relationship between the $e$-folding time (in minutes) and the
flare energy was defined from a sample of 420 energetic M-dwarf flares
($E\simeq 10^{31}-10^{34}$ erg). The log-log fit of the data was established
at a high statistical significance as,

$log~{}\tau=(0.57\pm 0.05)~{}log~{}E-(15.61\pm 1.57)\ .$ (3)

Taking the peak energies for nanoflares as $E=10^{25}$ erg, Equation 3
produces $e$-folding times of $1\leq\tau\leq 258$ s. From inspection, the
derived $e$-folding times of the M2.5V and M3V populations are consistent with
the upper boundary of predicted values, whereas the M4V values lie outwith the
derived relationship. This is not necessarily unexpected, since a
complementary study by Howard et al. (2019) found a broken power-law index
relationship between the $e$-folding time and flare energy, where at $E\leq
10^{33}$ erg, $\tau$ remained approximately constant instead of following the
trend associated with Equation 3. The authors attributed this to the
limitations of flare characterizations around the detection limit, but the
flares in this study suggest that the effect may be physical. Equation 3 was
derived from flare energies orders of magnitude above those under
consideration in this study, thus the disparity between the predicted
$e$-folding time and that seen in M4V stars may be indicative of a transition
from large-scale Petschek reconnection to the Sweet-Parker process. As was
proposed by Tsuneta & Katsukawa (2004), small-scale pico/nano-scale flares
occur more favorably via Sweet-Parker than Petschek reconnection. As Paper II
suggest, this would explain a discontinuity in the power-law relationship
between nanoflares and their larger scale counterparts, which remain driven by
Petschek-like reconnection (Loureiro & Uzdensky, 2016). The Sweet-Parker
reconnection process is inversely proportional to the square root of the
plasma Lundquist number, which is itself inversely proportional to the plasma
resistivity. As such, Sweet-Parker reconnection is more favorable in poorly
conducting plasmas. The increased decay timescale of $\tau=450\pm 50$ s,
alongside the associated increased power-law index of $\alpha=3.10\pm 0.18$,
found for the M4V sub-type may be related to increased plasma resistivity,
which matches expectations for mid-to-late M-dwarfs (Mohanty et al., 2002).
Caution is required however ; these increased $\alpha$ values are within 1
$\sigma_{N}$, and the $\tau$ values within 3 $\sigma_{N}$ of the uncertainties
of the less enhanced M2.5 and M3V stars, so this trend cannot yet be
considered statistically significant. Future investigation of M5V and later
sub-types is required to determine if there is a statistically significant
trend exceeding 3 $\sigma_{N}$ confidence in the observed properties. This
could be complimented by multi-color observations that would allow for lower
uncertainty in the $\tau$ value at each color band due to the reliance on
underlying plasma properties, which are naturally more separated across color
bands due to their associated temperature sensitivities.

In contrast to the fully convective sub-types, pre-dynamo mode transition M0V
– M2V stars exhibited weak (if any) nanoflare signals, suggesting that fully
convective stellar atmospheres lead to a large enhancement of nanoflare
activity.

While the observed trend of fully convective stars exhibiting enhanced
nanoflare activity is clear, the exact mechanism leading to this is still a
matter of debate. While the Sweet-Parker hypothesis is plausible, there is
also a potential issue. If enhanced nanoflare activity occurs in the corona,
it would lead to enhanced heating of that plasma. Consequently, this would
lower the resistivity and hence lower the rate of Sweet-Parker reconnection.
This ‘feedback loop’ behavior may reach some natural and stable equilibrium,
but it may be necessary to incorporate additional theory to ensure the
stability of this mechanism. Referring to the original nanoflare mechanism
theorized by Parker (1988) may provide this. In that paper, Parker suggested
random convective motion in the photosphere causes ‘shuffling’ and subsequent
deformation and braiding of the photospheric footpoints of the coronal
magnetic fields and consequently the generation of free energy. The coupling
of the magnetic field lines between the photosphere and corona provides the
framework to allow this free energy to flow into the corona. This energy is
then dissipated in coronal current sheets, leading to small-scale
reconnection. As such, enhanced heating leading to decreased resistivity would
improve the magnetic coupling between these footpoints and the corona,
consequently enhancing the flow of free energy available for nanoflare
activity. One can imagine a combination of these scenarios, wherein the
sympathetic transfer of hot plasma and free energy through these coupled
fields regulates the resistivity and drives a stable rate of Sweet-Parker
reconnection.

To uncover the source of this enhanced activity, it’s vital to obtain two sets
of observations; multi-band photometry, and observations of later MV star
types. The multi-color observation of these stars will allow us to make a
limited analysis of the change in nanoflaring properties across different
wavelengths and consequently the contribution at different atmospheric
heights. Comparing relative photospheric and coronal signatures could diagnose
the underlying mechanism powering this enhanced nanoflare activity. The multi-
color analysis should also provide a lower uncertainty in the $\tau$ values.
Secondly, sourcing M5V and beyond stars would allow the continuation of the
trend in flare decay rate (if any) to be investigated. If later MV stars
continue to exhibit enhanced activity it would support the Sweet-Parker
reconnection theory, as it would suggest the enhanced resistivity is key.
Ultimately, observations of later MV stars, and across multiple photometry
bands will need to be coupled with detailed physical modeling to try and
uncover what changes in these stars are driving their nanoflare behavior.

Regardless of the specific physical mechanism causing this enhancement across
the convective boundary, it is there. The observational evidence points to
nanoflare contributions increasing significantly in the fully convective M2.5V
and later stars. This novel result is independent of the modeled nanoflare
lightcurves, which serve only to diagnose the parameters of the nanoflare
signatures within observed lightcurves. It is also independent of the range of
stellar luminosities present in the sample. It is established that greater
macroscopic flare rates in later-type stars, as seen in Table A1, are
influenced by the reduced luminosity threshold of these stars. It is therefore
of interest to investigate whether the reduced flare detection threshold may
influence the nanoflare study presented here. Average luminosities for the M0
and M4 stars, representing the largest range of spectral classes under
consideration, are $L_{M0}=0.068L_{\odot}$ and $L_{M4}=0.014L_{\odot}$.
Therefore, the energy rates associated with these luminosities can be
estimated as $E_{M0}=2.6\times 10^{32}$ erg s-1 and $E_{M4}=5.5\times 10^{31}$
erg s-1, respectively. This order-of-magnitude drop in the energy rate
associated with the fundamental stellar brightness agrees with previously
detected flare energies (e.g., Rodríguez Martínez et al., 2020). In the
present study, a $1\sigma_{N}$ deviation is modeled in the simulation as
$5\times 10^{24}$ erg (Jess et al., 2019; Dillon et al., 2020), and by
projecting an order-of-magnitude energy threshold decrease between M0 and M4
classifications to nanoflare conditions, $1\sigma_{N}$ deviations in M4V stars
equates to flares with energies $\sim 10^{23}$ erg, still well within the
typical nanoflare regime. This differential in flare energy sampling is also
not sufficient to explain the lack of nanoflare enhancement in early-type MV
stars. The flare frequency, $dN/dE$ from Equation 2, associated with the M2.5V
nanoflares calculated from the $\alpha$ and $\tau$ values seen in Table 5
reveals a two order-of-magnitude increase in nanoflare frequency between M0
and M4 ($10^{-46}-10^{-44}$ erg-1 cm-2 s-1), consistent with previous solar
studies (Purkhart & Veronig, 2022). Therefore, in a scenario where early-type
M-dwarfs had the same flaring profile as their later-type counterparts, there
would still be an ample frequency of 1$\sigma_{N}$ signatures present in their
histograms and Fourier spectra. Given that there is no such behavior seen, we
posit this as evidence that the nanoflare frequency spectrum detected in this
work exists only in the fully-convective sample. Finding the source of this
convective divide should be a key focus of future studies.

Table 5: Nanoflare parameters per spectral type, derived from combined statistical and Fourier properties of Monte-Carlo modeled nanoflare timeseries.  Spectral type | $\alpha$ | $\tau(s)$
---|---|---
M2.5V | $3.00\pm 0.20$ | $200\pm 100$
M3V | $3.00\pm 0.18$ | $225\pm 100$
M4V | $3.10\pm 0.18$ | $450\pm 50$

The enhanced small-scale flare rates in fully convective stars holds profound
implications for the energy budgets of those stellar sources. The energy
output of rapid and continuous nanoflares may be a major component of the
overall stellar energy budget, yet are hidden within the noise envelope of the
observations and can only be extracted through use of large-scale statistical
and Fourier analyses. The question of whether the enhanced flaring visible in
post-dynamo mode transition M2.5V – M4V stars is due to the helical dynamo or
altered plasma Lundquist conditions in these stars is an avenue to explore in
future work. Furthermore, our work reveals tentative evidence that M4V stars
are linked to nanoflare events that have inherently longer decay timescales
(i.e., larger $\tau$ values) as well as larger power-law indices. Importantly,
mid M-dwarf sub-types should have decreased optical depths, alongside
increased plasma resistivities, a trend which continues to late M9 sub-types
(Mohanty et al., 2002). If the nanoflare $e$-folding times continue to
increase with increasing M-dwarf sub-type, it would support the scenario of
increased plasma resistivity leading to increased small-scale flaring via
Sweet-Parker reconnection. This would appear to support the findings of Wright
& Drake (2016); Wright et al. (2018), that solar and stellar dynamos operate
independent of a tachocline. As a result, it is of paramount importance to
source sufficient late M-type stellar time series for follow-up analyses.

##  5 Conclusions

Evidence for stellar nanoflares has been observed on a further 15 post-dynamo
mode transition (M2.5V, M3V, and M4V classification) stars, with nanoflare
power-law indices and $e$-folding times consistent with the enhanced rates of
nanoflare activity put forward by Paper II. The marked increase in nanoflare
activity is coincident with M2.5V and later sub-types, suggesting that the
change from partial to fully convective atmospheres may be responsible. The
post-dynamo mode transition stars exhibit nanoflare rates that are enhanced
from those seen at larger energies in other stars and the Sun, with power-law
indices found to be in the region of $\alpha=3.00\pm 0.20$ for M2.5V and M3V
sub-types, with slightly larger values of $\alpha=3.10\pm 0.18$ for M4V sub-
types. Given the relation between power-law index and low-energy flare
frequency, it is clear that the atmospheres of late MV stars have an energy
budget dominated by small-scale flaring. Whereas observational evidence of
nanoflares being the dominant heating mechanism in the solar atmosphere
remains elusive, their energy output in fully-convective stars may well be
sufficient to produce bulk heating. The decay timescales for M2.5V and M3V
stars were found to be on the order of $\tau=200\pm 100$ s, while evidence was
presented for increased plasma $e$-folding times of ${\tau=450\pm 50}$ s in
the M4V stars, suggesting the presence of Sweet-Parker reconnection processes.
It must be noted these enhanced values for the M4V star remain within
$1-3\sigma_{N}$ of the M2.5 and M3V stars, so we cannot yet consider these to
be fully distinct.

On the contrary, pre-dynamo mode transition M-dwarf (M0V, M1V, and M2V
classification) stars exhibit marginal statistical or Fourier-based nanoflare
signals, indicating that the large power-law index for MV stars reported in
Paper II is not uniform across all spectral types. Instead, it is implied that
a fully-convective interior is necessary to exhibit the $\alpha\geq 3$ that
distinguish them from other stellar candidates. Additionally, the underlying
reason why fully convective atmospheres lead to enhanced nanoflare activity
should be explored, i.e., is this due to an altered dynamo, or due to other
plasma changes such as modification of the corresponding Lundquist number? One
avenue of exploration would be examining M5V (and later) stellar types, to
investigate if there is a continuing and more statistically significant trend
in the flare decay rate and associated power-law index, which could be linked
to increasing plasma resistivity, and thus increased Sweet-Parker reconnection
rates. It is likely such observations would need to be coupled to detailed
theoretical and modeling efforts using well-developed numerical simulations
(e.g., Takahashi et al., 2011; Tenerani et al., 2015; Shi et al., 2018; Papini
et al., 2019).

Additionally, sampling late-type M dwarfs may reveal the traditional
observational signatures of nanoflares. Since the advent of M-dwarf flare
studies following the seminal works of Gershberg (1972) and Lacy et al.
(1976), it has been clear that the higher luminosity of early-type MV stars
can skew flare population studies in favor of less luminous later-type stars
due to a lower intensity threshold for flare detection. This has led to
subsequent studies constraining the population of stars under consideration,
or studying a single star in detail (e.g., Hawley et al., 2014b; Davenport et
al., 2014). This is the case in our present study, where we limited the
population of MV stars to those with comparable magnitudes. However, utilizing
the reduction in surface temperature in late-type MV stars would allow for the
intensity threshold of a macroscopic flare detection to be reduced. A sample
of MV stars with varying magnitudes would allow for the $2\sigma_{N}$
nanoflare intensity excursions found in this study to potentially lie above
the minimum observable detection threshold in cooler late-type MV stars. This
could confirm the temporal morphology and occurrence frequencies of flares at
these energies and provide further insight into the validity of previously
defined relationships, such as Equation 3. Additionally, the statistical
techniques employed here could provide the first signatures of even smaller
flares, such as the proposed pico-flare energy regime (Katsukawa & Tsuneta,
2001; Katsukawa, 2003).

It goes without saying that enhanced small-scale reconnection in fully
convective stars may mean that nanoflare activity could be a significant
component of their overall energy budget. Large-scale multi-year studies of
stellar nanoflare rates in fully convective M-dwarfs would further our
understanding of nanoflare behavior across different activity cycles, which
would further shine light on the ubiquity and role nanoflares play in these
dynamic host stars. This can be achieved through further use of large-scale
sky surveys (like the NGTS) and space-based observations from the likes of the
Transiting Exoplanet Survey Satellite (TESS; Ricker et al., 2014), alongside
targeted campaigns using high-cadence observational platforms, such as
HiPERCAM (Dhillon et al., 2016), or multi-band photometry such as the Rapid
Eye Mount (REM) telescope (Antonelli, 2005) to investigate the nanoflare
signature across layers of the Stellar atmosphere.

We thank the anonymous referee for their valuable input in improving this
manuscript. S.D.T.G, D.B.J., and C.J.D. wish to thank Invest NI and Randox
Laboratories Ltd. for the award of a Research and Development Grant
(059RDEN-1) that allowed the computational techniques employed to be
developed. D.B.J. and S.D.T.G. also acknowledge support from the UK Space
Agency for a National Space Technology Programme (NSTP) Technology for Space
Science award (SSc 009). S.D.T.G., D.B.J., and M.M. would like to thank the UK
Science and Technology Facilities Council (STFC) for the consolidated grants
ST/T00021X/1 and ST/X000923/1. D.B.J. would like to thank the STFC for an
Ernest Rutherford Fellowship (ST/K004220/1), in addition to a dedicated
standard grant (ST/L002744/1) that allowed this project to be started. D.B.J.
also wishes to thank The Leverhulme Trust for grant RPG-2019-371. C.A.W.
acknowledges support from STFC consolidated grant ST/X00094X/1. J.A.G.J.
acknowledges support from grant HST-GO-15955.004-A from the Space Telescope
Science Institute, which is operated by the Association of Universities for
Research in Astronomy, Inc., under NASA contract NAS 5-26555. S.L.C would like
to thank STFC for an Ernest Rutherford Fellowship (ST/R003726/1). P.J.W.,
D.R.A., and R.G.W. acknowledge support from STFC consolidated grants
ST/L000733/1 and ST/P000495/1. This project is based on data collected under
the NGTS project at the ESO La Silla Paranal Observatory. The NGTS facility is
operated by the consortium institutes with support from the UK STFC under
projects ST/M001962/1 and ST/S002642/1. This research has made use of data
obtained from the 4XMM XMM-Newton Serendipitous Source Catalog compiled by the
10 institutes of the XMM-Newton Survey Science Centre selected by ESA. J.I.V.
acknowledges support of CONICYT-PFCHA/Doctorado Nacional-21191829. Finally,
S.D.T.G. and D.B.J. wish to acknowledge scientific discussions with the Waves
in the Lower Solar Atmosphere (WaLSA; https://www.WaLSA.team) team, which has
been supported by the Research Council of Norway (project no. 262622), The
Royal Society (award no. Hooke18b/SCTM; Jess et al., 2021), and the
International Space Science Institute (ISSI Team 502).

##  Appendix A Stellar Parameters

Additional stellar parameters, including the RA and Dec for each star are
described in Table A1.

Sp Type | NGTS ID | GAIA ID | TIC ID | RA | Dec | Mass ($M_{\odot}$) | Radius ($R_{\odot}$) | Luminosity ($L_{\odot}$) | Distance (pc) | Approximate Flare Rate per Hour | Magnitude
---|---|---|---|---|---|---|---|---|---|---|---
M0V | NGTS J233315.1-385757 | 6538313140873424640 | 224245757 | 353.312913 | -38.965817 | 0.487596 | 0.489513 | 0.04772778 | 100.33 | 0.0123635 | 13.10
M0V | NGTS J045221.8-312424 | 4874911889552910000 | 1310695 | 73.090834 | -31.406834 | 0.597557 | 0.611732 | 0.07879962 | 137.421 | 0.0154544 | 13.26
M0V | NGTS J052346.3-361114 | 4822374303400198144 | 167745038 | 80.94287 | -36.187338 | 0.566358 | 0.574022 | 0.07396496 | 123.332 | 0.00618174 | 13.04
M0V | NGTS J061346.1-362248 | 2885025813007881728 | 267248553 | 93.442125 | -36.380098 | 0.585147 | 0.596407 | 0.07490093 | 137.363 | 0.0123635 | 13.24
M0V | NGTS J061054.6-370701 | 2884885281677800448 | 300200809 | 92.72739 | -37.116954 | 0.575812 | 0.585163 | 0.06398756 | 112.406 | 0.0401813 | 12.99
M1V | NGTS J233248.3-382456 | 6538532356004046592 | 224244565 | 353.201262 | -38.415564 | 0.542372 | 0.546805 | 0.05344092 | 90.7535 | 0.0123635 | 12.97
M1V | NGTS J051250.6-361938 | 4821058497219315328 | 14173066 | 78.210928 | -36.327354 | 0.596268 | 0.61012 | 0.06084004 | 201.14 | 0.0123635 | 14.16
M1V | NGTS J052652.1-373123 | 4821222942926810752 | 192785958 | 81.717213 | -37.523125 | 0.567364 | 0.575197 | 0.05580399 |  | 0.00309087 | 13.57
M1V | NGTS J235034.7-373312 | 2310510165491596672 | 183536494 | 357.644544 | -37.553375 | 0.506923 | 0.509045 | 0.04646323 | 83.9291 | 0.00927262 | 12.84
M1V | NGTS J111257.7-331216 | 5403344977522967424 | 23438898 | 168.240279 | -33.20455 | 0.460039 | 0.462724 | 0.03412572 | 73.1095 | 0.0185452 | 12.92
M2V | NGTS J045136.3-321720 | 4874656837214833664 | 1309522 | 72.901424 | -32.288803 | 0.56 | 0.806538 | 0.1124733 | 247.955 | 0.0278178 | 14.32
M2V | NGTS J050254.6-352000 | 4825066629419253632 | 1526841 | 75.72729 | -35.333409 | 0.469279 | 0.471581 | 0.03436859 | 69.7222 | 0.0185452 | 12.72
M2V | NGTS J051926.5-253444 | 2957763042671388416 | 30960826 | 79.860208 | -25.578811 | 0.460925 | 0.463568 | 0.03302738 | 90.6288 | 0.00927262 | 13.49
M2V | NGTS J053614.4-353309 | 4821870486556489216 | 24612475 | 84.059908 | -35.552443 | 0.501857 | 0.503859 | 0.03897486 | 90.6101 | 0.0154544 | 13.2
M2V | NGTS J062005.7-372555 | 5575203489668007936 | 393481864 | 95.023721 | -37.431899 | 0.400268 | 0.407808 | 0.02305492 | 87.7167 | 0 | 13.83
M2.5V | NGTS J045008.8-362401 | 4818804257863710336 | 77369893 | 72.536742 | -36.400372 | 0.545633 | 0.550421 | 0.05525498 | 118.868 | 0.0494539 | 13.07
M2.5V | NGTS J050359.5-305327 | 4876285488813663232 | 1439071 | 74.755373 | -30.999599 | 0.456069 | 0.458954 | 0.02748212 | 54.6863 | 0.00618174 | 12.8
M2.5V | NGTS J045901.2-305958 | 4875598534564509312 | 1535810 | 75.997888 | -30.890807 | 0.380031 | 0.389898 | 0.01965184 | 68.7496 | 0.0710901 | 13.65
M2.5V | NGTS J050810.8-371850 | 4823535318959536256 | 14084620 | 77.044884 | -37.313753 | 0.514811 | 0.517217 | 0.03755042 | 59.1582 | 0.0309087 | 12.29
M2.5V | NGTS J061516.4-360818 | 2885223381503521536 | 267327257 | 93.818249 | -36.138454 | 0.496041 | 0.497965 | 0.03437556 | 118.666 | 0.0278178 | 13.99
M3V | NGTS J035219.1-311459 | 4886786408973741568 | 166804322 | 58.079759 | -31.249846 | 0.544884 | 0.549588 | 0.03602405 | 123.566 | 0.0216361 | 13.63
M3V | NGTS J050230.0-355301 | 4824660428592359552 | 13982951 | 75.624229 | -35.883643 | 0.308218 | 0.327511 | 0.01185441 | 41.1162 | 0.00927262 | 13.1
M3V | NGTS J051925.5-235535 | 2958246827787260032 | 30961390 | 79.856241 | -23.926311 | 0.564934 | 0.572365 | 0.04099549 | 131.705 | 0.0865444 | 13.94
M3V | NGTS J052116.1-322429 | 4826608831916384640 | 78053729 | 80.317094 | -32.357958 | 0.401901 | 0.409266 | 0.01826787 | 39.3307 | 0.0494539 | 12.52
M3V | NGTS J000722.8-293528 | 2320750123439437184 | 12418184 | 1.845141 | -29.5912 | 0.22872 | 0.257223 | 0.007069197 | 40.1093 | 0.0216361 | 13.44
M4V | NGTS J035624.7-311140 | 4886831592030178944 | 166869904 | 59.102836 | -31.194413 | 0.402508 | 0.409808 | 0.01474621 | 37.7653 | 0.10509 | 12.7
M4V | NGTS J044312.0-322643 | 4874430475258301184 | 170882537 | 70.800028 | -32.445142 | 0.512283 | 0.514585 | 0.02822354 |  | 0.0834535 | 12.6
M4V | NGTS J045519.0-321222 | 4873878176823736192 | 1357792 | 73.829291 | -32.206129 | 0.170431 | 0.201449 | 0.003005163 | 21.8796 | 0.0865444 | 13.72
M4V | NGTS J050423.8-373021 | 4823476460727785728 | 14001734 | 76.099199 | -37.505698 | 0.434448 | 0.438763 | 0.02066878 |  | 0.148362 | 12.42
M4V | NGTS J2341092-363819 | 2311548448064869120 | 224276435 | 355.288136 | -36.638609 | 0.253768 | 0.279818 | 0.005977345 | 38.7917 | 0.0649083 | 13.79

Table A1: The Spectral type, NGTS identifier, Gaia source ID, Tess Input
Catalog (TIC) ID, RA, Dec, Stellar Mass (in Solar mass units), Stellar Radius
(in Solar radi units), Distance (in parsecs), Stellar Luminosity (in Solar
luminosity units), and the Macroscopic Flare Rate (per hour) for the stars
used in the analysis. The Stellar masses, radi. and luminosity data is from
the Tess Input Catalog release V8. (Stassun et al., 2018)

##  Appendix B Re-Calculated Fourier Simulation Heat Map

Figure 6 shows a ‘heat map’ of the simulated PSDs (c.f., Figure 7 of Dillon et
al., 2020), which has been recalculated for the $2095$ element long time
series employed in the present study. While the frequency resolution is
slightly reduced ($\Delta{f}=0.0398$ mHz) than that utilized by Dillon et al.
(2020, $\Delta{f}=0.0356$ mHz), the overall trends and evolution remain
consistent across the power-law index and $e$-folding timescale values.

Figure 6: A reproduction of Figure 7 from Paper II, with the constituent PSDs
re-calculated for $2095$ datapoints to match the longest continuous time
series used in the present study. The primary peak frequencies (lower-left),
spectral slopes (upper-left), dominant frequencies following detrending
(upper-right), and the percentage of nanoflare power above the noise floor in
the range of $1-5$ mHz (lower-right), is displayed as a function of the power-
law index, $\alpha$, and the decay timescale, $\tau$, used to generate the
synthetic time series. While a few individual values differ, the overall
trends and the magnitude of the derived signals are consistent with the PSD
properties generated from $2316$ datapoints and reported by Paper II.

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