1]Department of Astrophysical and Planetary Sciences, University of Colorado,
2000 Colorado Ave, Boulder, 80305, CO, United States

2]National Solar Observatory, University of Colorado, 3665 Discovery Drive,
Boulder, 80303, CO, United States

3]Laboratory for Atmospheric and Space Sciences, University of Colorado, 3665
Discovery Drive, Boulder, 80303, CO, United States

# Stellar flares

Adam F. Kowalski<EMAIL_ADDRESS>[  [  [

###### Abstract

Magnetic storms on stars manifest as remarkable, randomly occurring changes of
the luminosity over durations that are tiny in comparison to the normal
evolution of stars. These stellar flares are bursts of electromagnetic
radiation from X-ray to radio wavelengths, and they occur on most stars with
outer convection zones. They are analogous to the events on the Sun known as
solar flares, which impact our everyday life and modern technological society.
Stellar flares, however, can attain much greater energies than those on the
Sun. Despite this, we think that these phenomena are rather similar in origin
to solar flares, which result from a catastrophic conversion of latent
magnetic field energy into atmospheric heating within a region that is
relatively small in comparison to normal stellar sizes. We review the last
several decades of stellar flare research. We summarize multi-wavelength
observational results and the associated thermal and nonthermal processes in
flaring stellar atmospheres. Static and hydrodynamic models are reviewed with
an emphasis on recent progress in radiation-hydrodynamics and the physical
diagnostics in flare spectra. Thanks to their effects on the space weather of
exoplanetary systems (and thus in our search for life elsewhere in the
universe) and their preponderance in _Kepler_ mission data, white-light
stellar flares have re-emerged in the last decade as a widely-impactful area
of study within astrophysics. Yet, there is still much we do not understand,
both empirically and theoretically, about the spectrum of flare radiation, its
origin, and its time evolution. We conclude with several big-picture questions
that are fundamental in our pursuit toward a greater understanding of these
enigmatic stellar phonemena and, by extension, those on the Sun.

###### keywords:

Solar-stellar connection

###### Contents

  1. 1 Introduction
    1. 1.1 Overview of this review
  2. 2 Our flare star neighbor: Proxima Centauri
  3. 3 Solar Flares and the Standard Flare Model
  4. 4  A Survey of Flare Stars
  5. 5 Flare Rates and Flare Frequency Distributions
    1. 5.1 Basic Methods and Early Studies
    2. 5.2 White-Light Flare Rates & _Kepler_
    3. 5.3 Timing on Long Scales: Flare Rates and Stellar Age
  6. 6 Light Curve Analyses
    1. 6.1 Classification of Flares by Optical Broadband Evolution
    2. 6.2 Quasi-Periodic Pulsations
  7. 7 Multi-Wavelength Spectral Observations
    1. 7.1 NUV and Optical: The Thermal Radiative Response of the Footpoints to Impulsive Nonthermal Heating from Above
      1. 7.1.1 Overview of Emission Line Properties
    2. 7.2 Spectral Properties of the Optical and NUV Continuum Radiation
      1. 7.2.1 Rise and Peak Phase
      2. 7.2.2 The Decay Phase and The Evolution of Optical Spectral Properties
      3. 7.2.3 Empirically-determined, Broadband Radiated Energy Budgets Relative to the $U$ band
      4. 7.2.4 Optical Broadband Flare Colorimetry
    3. 7.3 FUV Observations: The Rapidly Evolving Flare Transition Region (TR)
    4. 7.4 Centimeter (Microwave/Radio) Observations: Nonthermal Gyrosynchrotron Radiation from Mildly Relativistic Electrons
    5. 7.5 Millimeter Observations from ALMA: Nonthermal Radiation with Linear Polarization
    6. 7.6 Soft X-rays and EUV Observations: Loops Filled with Evaporated, Previously-Chromospheric/TR, $T>10^{7}$ K Thermal Plasma
    7. 7.7 The Neupert Effect in Stellar Flares
    8. 7.8 Abundance Changes
  8. 8 Atmosphere Modeling of Stellar Flares: Overview of Results
    1. 8.1 Slab Models
    2. 8.2 Atmospheric Models
  9. 9 Atmosphere Modeling of Stellar Flares: Overview of Physics and Methods
    1. 9.1 An Introduction to Radiative-Hydrodynamic Flare Modeling
    2. 9.2 The Flare Heating Term
  10. 10 Models of Chromospheric & Transition Region Flare Emission Line Broadening (& Asymmetries)
    1. 10.1 An Overview of Broadening Sources
    2. 10.2 Symmetric Wing Broadening of H Lines
      1. 10.2.1 Recombination Edge Modeling
  11. 11 A Comprehensive Multi-Wavelength Stellar Flare Dataset
    1. 11.1 Inferring Nonthermal Electron Properties from Stellar Data
  12. 12 Stellar Flare Geometries
    1. 12.1 A Dominant Loop or an Arcade?
    2. 12.2 Stellar Core-Halo
  13. 13 Conclusions and Future Outlook
  14. A Optical Filter Reference
  15. B A Summary of Supplemental Model Constraints in the Optical
  16. C Slab Model Reference
    1. C.1 Blackbody Fitting to Stellar Flare Optical Data
    2. C.2 Hydrogen Continuum Emissivity Models
  17. D Pressure Broadening of Hydrogen Lines in Flares

##  1 Introduction

Stars produce energetic bursts of electromagnetic radiation that follows a
sudden magnetic energy release into their atmospheres. These electromagnetic
bursts are called flares, which occur on a very wide range of timescales, from
seconds to days. Stellar flares are observed in all spectral windows, from the
high-energy X-rays through the long wavelength radio waves. However, all
wavelength regimes do not respond equally in energy and simultaneously in
time. Some wavelengths (e.g., microwave) are dominated by nonthermal
radiation, while others (optical, low-energy X-rays) are thought to result
from primarily thermal radiative processes. The flare energy – both thermal
and nonthermal – ultimately originates in the energy that is released from
magnetic fields in stellar coronae. These magnetic fields, in turn, originate
from turbulent convective energy transport and shear in rotating stellar
envelopes, beneath the visible photosphere. The Sun is the best studied flare
star due to its proximity to Earth and a fleet of multi-messenger instruments
that continuously provide spatially resolved observations.

From Earth, solar and stellar flares are not detectable by the unaided human
eye. Stars like the Sun, and other active stars like Algol that are visible to
the naked eye, have too much background glare for our eyes to sense the flare
brightness increases. In all-sky surveys that are sensitive to the apparent
magnitudes of low-mass stars (fainter than a Johnson $V$-band magnitude of
$\sim 9$), however, stellar flares are the most dramatic source of
variability. By _most dramatic_ , we mean that flares produce the largest
changes in their apparent magnitudes per unit time when compared against
nearly all other astrophysical phenomena111Perspective provided by Z. Ivezic
(priv. communication, 2010)..

In data from the upcoming Vera C. Rubin Observatory’s ten-year Legacy Survey
of Space and Time (LSST), stellar flares will constitute a major source of
variability [305]. Further, many of the largest events that are observed
serendipitously (and those from very faint but populous low-mass stars) will
be bona-fide transients, whereby the changes in brightness occur from sources
that have not been detected in quiescence. Stellar flares are much shorter in
duration and much less luminous than extra-galactic transient phenomena, such
as supernovae, tidal disruption events, and optical counterparts to neutron
star mergers. Most stellar flares can in principle be readily distinguished
from these events given enough empirical information, despite some striking
spectral similarities [124] in the optical spectra of kilonovae [669]. Due to
their proximity to the solar system, the largest stellar flare events can
trigger gamma ray observatories; these events are known as “superflares” in
X-rays. Stellar superflares are factors of $\approx 10^{2}-10^{4}$ more
energetic and luminous in comparison to the largest solar flares, which have
bolometric energies of $E_{\rm{bol}}\approx 3-6\times 10^{32}$ erg [778, 779,
136, 306]. Such superflares were likely common when the Sun was very young and
rotating much more rapidly than today [472]. Understanding the physical
processes in stellar flares thus provides insight into the heliospheric
conditions in the early history of our solar system [625].

It is timely for a review of recent observations and models of stellar flares.
The next sunspot cycle maximum is approaching in late 2024 [727]222See also
the recent assessments in Gopalswamy et al [272] and McIntosh et al [493] and
the continuously updated web resource,
https://helioforecast.space/solarcycle., and with it a deluge of flares and
eruptions from the Sun. New and upcoming solar observatories, such as the
Daniel K. Inouye Solar Observatory [628], the Expanded Owens Solar Valley
Array [252], and the Interface Region Imaging Spectrograph [164], will provide
new avenues for solar-stellar comparisons. The immaculate precision of
_Kepler_ [396], K2 [336], and the Transiting Exoplanet Survey Satellite [TESS;
627] observations has recently facilitated a resurgence in the study of
optical broadband stellar flares. These missions provide access to stellar
magnetic activity over long time baselines previously not feasible to observe
from ground-based campaigns, and over a much larger variety of stars. Models
of energy transport, atmospheric response, and emission line broadening have
increased in accuracy and sophistication over a large range of heating
parameters. As the community looks to next-generation modeling paths, analysis
methods, and observational capabilities (such as new space missions and
ground-based instruments), synthesizing recent findings and outstanding
problems could help to steer these into productive directions. Additionally,
stellar flare radiation is now considered an important factor in assessing
exoplanet habitability and photochemistry [676, 666], and general interest
within the solar and astrophysics communities has grown over the last decade.

###  1.1 Overview of this review

There have been several reviews on stellar flares prior to _ca._ 1990 [588,
106, 291]. The current review supplements these with results from the past
three decades. This review features many results from flare studies of low-
mass, M-dwarf (dM / MV) stars. Because of their low background glare from non-
flaring regions, large contributions to Galactic populations, and inherently
high flare rates, the M dwarfs tend to be the most commonly studied flare
stars. Flares from active binary systems have also been very well-observed
across the electromagnetic spectrum. Results from studies of post main-
sequence single stars and young solar-type stars are covered as well.
Surprisingly, certain wavelength regimes in M-dwarf flares have been more
thoroughly observed than solar flares. Solar flares are not reviewed in any
detail here, but a general overview is provided, since the interpretation of
and modeling approach to stellar flares are dependent on knowledge of the
particle acceleration, magnetic fields, and spatial resolution from the study
of the Sun.

The primary purpose of this review is to serve as a compendium of references
and to facilitate research in multi-wavelength observations and models of
stellar flares. The target audience consists of beginning PhD students or
interested scientists in other areas of astrophysics or solar physics. Every
current topic in the study of stellar flares is not included here: for
example, I leave all results from the TESS mission to the next iteration of
this _Living Review_. Unfortunately, many important results and references
cannot be discussed extensively in order to keep this review a reasonable
length, while also allowing growth with future addenda. For the same reasons,
every important result from the references cited herein regrettably cannot be
included at this time. There are several comprehensive introductions to sub-
topics in the stellar flare literature; these are indicated throughout instead
of being repeated. Broader applications of flare research beyond stellar
astrophysics (e.g., exoplanet habitability and exoplanet atmospheric
photochemistry) are outside the scope of this review and are not covered. The
study of stellar coronal mass ejections and their associated energetic
particles is a vast and newly emerging field; these topics are only very
briefly alluded to. _This review focuses on the electromagnetic response of
flaring stellar atmospheres and detailed modeling of the associated physical
processes._

This review is organized as follows. We begin with a brief tour of the flare
star next door, Proxima Centauri, which is about the same age as the Sun but
is otherwise a much different star (Sect. 2). Then we take a detour for a
brief introduction to solar flare terminology and phenomenology (Sect. 3). A
general overview of all known types of stars that tend to flare is summarized
in Sect. 4. The bulk of the review contains a synthesis of stellar flare
observations across the electromagnetic spectrum with a focus on the near-
ultraviolet (NUV) and optical response, which has not been the subject of most
other reviews. Observational studies can be separated into two general topics:
flare rates, which are primarily diagnosed through single-band photometry
(Sect. 5), and multi-wavelength analyses, which are primarily accomplished
through spectroscopy (Sect. 7). The final third of the review summarizes
stellar flare modeling approaches (slab, semi-empirical, and radiative-
hydrodynamic), beginning in Sect. 8. We go into detail in several areas within
stellar flare modeling, focusing on key elements of radiation-hydrodynamics
(Sect. 9) and the interpretation of chromospheric spectral line broadening in
flares (Sect. 10). A comprehensive analysis of an “ideal” multi-wavelength
data set is discussed in Sect. 11, and inferences of the stellar flare
geometries are reviewed in Sect. 12. We conclude in Sect. 13 with six
questions that we think are currently central in stellar flare research
(excluding questions directly related to exoplanets). Throughout, we use the
_cgs_ (Gaussian) units system except for wavelengths in Å or in cases that are
otherwise noted. In the appendices, we include a visual guide to common
filters used in optical studies of stellar flares (Appendix A), some
additional observational results from the literature are merged (Appendix B),
several common slab modeling practices and assumptions are reviewed (Appendix
C), and the basic microphysical processes that symmetrically broaden hydrogen
lines in flares is summarized (Appendix D).

##  2 Our flare star neighbor: Proxima Centauri

The nearest star outside the solar system is Proxima Centauri, which is
$\approx$1.3 parsecs, or 268,000 astronomical units (au), from the Earth, and
is a low-mass ($\approx 0.12$ MSun) M dwarf star of spectral type M5.5Ve
(dM5.5e) [299]. The hydrogen Balmer $\alpha$ line (6562.81 Å) is in emission
(e)333A spectral line “in emission” means that it rises above the local
continuum. A spectral line “in absorption” means that the line spectrum
decreases below the local continuum or pseudo-continuum flux.  in quiescence
outside of detectable flaring events, which occur very frequently. This star
is a relatively old ($\approx 5-6$ Gyr) main sequence star, but it is still
very magnetically active, consistent with the long magnetic activity lifetimes
of low mass stars of its spectral type [769]. We describe some of the flare
properties of this star from a night of monitoring during an observing run at
La Silla Observatory in 2010, while also introducing basic point-source
photometry measurements and terminology that are common to almost all analyses
of flare star data.

A NUV light curve of Prox Cen [416] is shown in Fig. 1. The photometry was
measured through a custom filter with a full-width-at-half-maximum (FWHM) of
100 Å centered at $\lambda_{\rm{cen}}=3500$ Å. This is a wavelength range over
which the star has a very faint luminosity during non-flaring (“quiescent”,
denoted by $q$) intervals. In less than seven hours of continuous 3 s
exposures (and 1.5 s exposures at the end of the night), the star produced at
least 16 flares at $>$3$\sigma$ (1$\sigma\approx 6$%) above the mean, which is
indicated by a flux enhancement equal to 1.0 in the light curve. In
quiescence, the star is very “red” since it has a very cool photospheric
temperature ($<3000$ K), but the flare light causes the star’s total flux at
Earth to become much “bluer”. In addition to the time-integrated flare energy
(fluence), the peak flux enhancement is sometimes used to characterize the
size of a flare. The peak magnitude change of a flare is related to the peak
flux enhancement by

$\Delta\mathrm{mag}_{\mathrm{peak}}=-2.5\ \mathrm{log}_{10}\
\left(C_{\mathrm{rel}}(t_{\mathrm{peak}})/C_{\mathrm{rel},q}\right)=-2.5\
\mathrm{log}_{10}\big{(}I_{f}(t_{\rm{peak}})+1\big{)}$ (1)

where $C_{\rm{rel}}$ represents the measured counts per exposure through a
bandpass and is calculated relative to the counts in that same exposure from a
nearby comparison (non-variable) star or set of stars. The count flux
enhancement, $C_{\rm{rel}}(t)/C_{\rm{rel},q}$, is normalized to a time
interval over which the flare star is not clearly varying, usually taken right
before a flare. The count flux enhancement has traditionally been denoted as
$I_{f}+1$ [254]. Because the star is spatially unresolved, the values of
$I_{f}(t)+1$ during a flare consist of the flux from the non-flaring regions
on a flare star in addition to the flaring source; if the flare source has an
optical thickness $\tau\gtrsim 1$ at wavelengths in this bandpass, then some
or all of the pre-flare flux from the flare area may be diminished at time $t$
[298].

Figure 1: A light curve of Proxima Centauri in a narrowband (filter FWHM of
100 Å) centered in the near-ultraviolet, $U$-band wavelength regime. These
data are from [416] and can be obtained from the Zenodo repository at
https://doi.org/10.5281/zenodo.45878. The horizontal dashed lines indicate the
mean flux enhancement (1.0) and $3\sigma$ above the mean (where $\sigma$ is
the standard deviation outside of flaring). The inset shows the flare peaking
at 2010 May 25 01:31:02 UT, which is referred to as the “IF10” event in [416].
The inset shows time relative to peak.

None of the flares in Fig. 1 are all that energetic. The flare with the
largest peak flux enhancement of five would correspond to an energy of
$3\times 10^{29}$ erg in the Johnson $U$ bandpass, which is traditionally the
optimal broadband filter [$\lambda_{\rm{peak}}\sim 3700$ Å, a full-width-at-
half maximum of $\sim 700$ Å; 506, 67]444Hereafter, following the convention
established in [420], we refer to the wavelength regime from the atmospheric
cutoff at $\approx 3200$ Å to the violet at $4000$ Å as the $U$ band, in order
to distinguish it from the near-ultraviolet (NUV) band at $\lesssim 3200$ Å,
which can only be observed from space. for stellar flare observations at
ground-based observatories. This energy is nearly four times larger than the
average $U$-band flare energy from Proxima Centauri [754] and a factor of 100
smaller than the largest event on the Sun that has been observed at similar
wavelengths [539]. Multi-wavelength scaling relations [297, 422, 663, 562,
526, 136] facilitate comparison to the standard classification of solar flares
according to their peak flux at Earth over the $\lambda=1-8$ Å band, as
measured by the X-ray Sensor (XRS) on the Geostationary Operational
Environmental Satellite (GOES). The A1.0, B1.0, C1.0, M1.0, and X1.0-class
flares correspond to a range of peak X-ray irradiances logarithmically
spanning $10^{-8}$ to $10^{-4}$ W m-2, while those at $\geq 10^{-3}$ W m-2 are
sometimes clipped [339] and are given designations beginning with X10.0. The
scaling relations imply that the large event on Proxima Centauri would
correspond to approximately a GOES $1-8$ Å C-class flare if it had occurred on
the Sun. However, Prox Cen occasionally erupts with energies that are
comparable to, or even much greater than, typical GOES X-class flares from the
Sun [283, 332]. Prox Cen also hosts a near Earth-mass planet at $\sim 0.05$ au
[17], and unlike our relatively safe distance from solar flares at 1 au,
equivalent flare energies from this star would bathe the surrounding planet in
$400x$ the flux of high-energy radiations. Higher-mass, M3-M4 stars are known
to emit even higher-energy flares than Prox Cen. For example, the most
luminous event known resulted in a remarkably fast $\Delta t\approx 35$ s rise
to a peak flux change of $\Delta V=-5$ mags ($L_{V,\rm{peak}}\approx 1.7\times
10^{32}$ erg s-1) from a young M4$+$M4 binary DG CVn [108], for which the
estimated values of the $U$-band energy and GOES class are $4.7\times 10^{34}$
erg and X600,000, respectively [571, 791]. [589] describes a remarkable EV Lac
flare with a peak magnitude change in the $U$-band of $-7.2$ mags
($L_{U,\rm{peak}}=4.6\times 10^{31}$ erg s-1) and a $U$-band energy of
$7.23\times 10^{33}$ erg. [333] highlight several flares with extreme
amplitudes ($\Delta g^{\prime}\leq-3$ mag) and energies ($E\approx
10^{35}-10^{36}$ erg) on M1-M4 stars in their optical survey.

The most common qualitative description of a flare light curve is a “FRED”: a
fast-rise, exponential-decay. In Fig. 1, the flares exhibit a simple FRED
shape at low time-resolution; however, at high-time resolution, much more
variation in the temporal morphology is apparent, including two periods of the
rise phase (1a, 1b), an extended peak (1c), two intervals of fast decay (2a,
2c), a stall between these two intervals (2b), and a gradual decay phase (3).
These features are clear in some other events too [e.g., Figure 2 of 414,
which is reproduced in the middle, right of Fig. 8 here], though the
respective phases may have different durations, relative amplitudes, and
integrated energies. High-time resolution observations of stellar flares were
actually routine in the 1970s and 1980s using photometers, which have been
superseded by high efficiency CCDs, culminating in the unprecedented precision
from _Kepler_ , K2, and TESS. The short timescale variations in stellar flares
have actually long been recognized. The seminal study of [78] aptly summarized
their findings as follows: “As the time resolution of observations has
improved, the great complexity of the flare phenomenon has been revealed. The
classical definition of a flare (i.e., rapid rise to maximum followed by a
slower quasi-exponential decay) appears to be a gross oversimplification of
the complex structures observed”.

##  3 Solar Flares and the Standard Flare Model

Only the very largest solar flare events could be detected in the optical if
the Sun was at a similar distance as other stars. Recently, Kepler has
provided the precision to detect a signal of $0.0170$%$-0.0270$% in optical
enhancements that have been observed in Sun-as-a-star data [778, 422, 510] and
from slowly rotating G dwarfs [472, 551, 555]. Nonetheless, it is generally
thought that flares from other stars originate from the same or similar
processes as solar flares. This is supported by the empirical Neupert effect
[540], which was first reported in stellar flares in [298] and [280] (Sect.
7.7). In this section, we briefly review the standard solar flare model
paradigm, whose phenomenology (e.g., footpoints, loops) and fundamental
physical processes are widely adopted in the interpretation and analyses of
stellar flares – either through direct application or through some sort of
physical scaling to higher densities, magnetic fields, accelerated particle
fluxes, etc… This section synthesizes decades of observations and theoretical
work, and it presents our own (rather highly simplified) viewpoint of the
entire process. Thus we give a non-exhaustive reference list. For more
extensive reviews of solar flares, see [704], [342], [343], [62], [344], the
entries within the _Space Science Reviews_ Volume [166], the books by [23] and
[706], and the review by [670]. For modern, comprehensive reviews of the
observations and modeling of solar flares, see [232], [613], [387], and [388].

In the standard flare paradigm, magnetic potential energy is transferred to
the atmosphere, which responds by radiating away this energy as the flare.
There are generally two categories of solar flares [711, 385]. Mass and
magnetic field are ejected away from the Sun during eruptive flares. In
contrast, magnetized plasma is not ejected in confined, or compact, flares or
the eruption may be undetectable. The collective process of the mass eruption
and the electromagnetic flare is called a “solar eruptive event” (SEE). The
left panel of Figure 2 illustrates the geometry of eruptive magnetic field
above compact flare loops, with reconnection of magnetic fields in between
[673]. This framework underlies most modern generalizations of flare-
productive magnetic field topologies [e.g., Figure 1 of 386], which we now
describe in further detail.

An SEE begins in active regions where there are colliding, non-
conjugate555Meaning that they are not N and S poles that are connected by the
same field lines. sunspots of opposite polarity [128, 718, 623]. The
demarcation that separates the bulk of the north and south polarities in an
active region is termed the polarity inversion line (PIL). At the PIL, mixed
negative and positive magnetic polarities appear as “salt-and-pepper” patterns
in magnetograms where there is newly emerging magnetic flux. Ongoing magnetic
flux emergence adds to a twisted, cool filament (flux rope) that is parallel
to the PIL, and it is thus driven to an instability [449]. The start of the
SEE occurs when this filament becomes unstable and erupts through overarching
magnetic field into interplanetary space, developing into a coronal mass
ejection (CME). The CME produces a shock that accelerates a large number of
solar energetic particles [SEPs; 609] away from the Sun; these particles may
reach Earth within $\approx 10$ minutes of the X-ray flare, peak about
$\approx 6-12$ hours after, and persist for days. Another class of SEPs are
prompt or impulsive-type SEPs, which originate from the flare site. The
arrival of the CME itself within $\approx 18-50$ hours of the flare disturbs
the terrestrial magnetic field (if the orientation of the CME magnetic field
is opposite that of the Earth’s magnetic field upon arrival), induces DC
electric fields and currents in the ground, and increases the particle flux
into the poles and radiation belts. The ground-induced currents can damage
transformers if power is not diverted away from high-load regions.

Returning to the Sun, the solar magnetic field lines that initially arch over
a filament current channel [which is relatively low-lying in the corona; 702,
623] are “stretched” during the eruption. Oppositely-directed magnetic field
lines pinch together in a “current sheet” in the wake of the erupting filament
(a simplified analogy to magnetic tension and retraction is often made to a
rubber band building up elastic tension and releasing it by snapping back).
The fields undergo magnetic reconnection at many so-called X-points, and the
rapid speed of this process is thought to be facilitated by the tearing
mode/plasmoid instability and MHD turbulence. The magnetic potential energy in
the stretched field line is released as they shorten and retract [Section 4 of
462].

The fundamental physics of the conversion of magnetic energy into particle
kinetic energy is described in the overview by [146]. As the retraction of
field proceeds to a more relaxed state, an Alfven-ish-speed outflow
(“jet”/“exhaust”) is directed from the reconnection region in the direction
toward the stellar surface. Slow-mode MHD (“Petschek”) shocks [e.g., 461] and
parallel electric fields [198] form in the just-reconnected, bent field lines,
enhancing the temperature and density in the reconnection outflow. Thus, the
magnetic potential energy initially stored in the contorted magnetic field is
converted into kinetic energy of bulk flows, the thermal energy of the plasma,
and the kinetic energy of particles out of the ambient/thermal distribution.

Figure 2: (_Left_) Illustration of several salient features and processes in
the standard model of solar flares, reproduced from Shibata et al [673] with
permission. (_Right)_ Generalization of the chromospheric evaporation and
condensation phenomena in radiative-hydrodynamic models of high-flux electron-
beam heating at the footpoints of newly-reconnected flare loops [adapted from
410]. Here, we also include an illustration of the radiative backwarming of
the photosphere, which is predicted by some 1D models [9] of flare
chromospheres that are optically thin at Balmer and Paschen continuum
wavelengths [see 225, for the expected geometry of 3D radiative backwarming of
the photosphere].

In large solar flares, a sizable fraction of the magnetic energy that is
released seems to go into accelerating ambient particles (electrons, protons,
ions) to mildly relativistic energies [452, 204, 759, 760]. The nonthermal
(“beam”) distribution is a power-law extending from $\sim 10$ keV to tens of
MeV for electrons and $\sim 1-1000$ MeV for protons. Recent observations have
shown that the Sun is a prolific particle accelerator: $\approx 1-100$% of the
pre-flare coronal electrons are accelerated [771, 427, 424, 554, 423, 227,
402]. The inferred nonthermal particle flux density into the lower atmosphere
can attain a correspondingly extreme range, $\sim 10^{12}-10^{13}$ erg cm-2
s-1, at some locations in solar flares [492, 111, 785, 537, 425].

Determining how and where the bulk of nonthermal particles is accelerated in
the solar atmosphere is actually a very active research topic [for an
overview, see 796]. Generally speaking, there are three classes of particle
acceleration that may operate in the solar atmosphere [23, and see also Knuth
and Glesener 395 and Chen et al 126 for succinct overviews of the more modern
classifications]: first-order Fermi and shock (including shock-drift and
diffusive shock), stochastic (“AC”, or resonant interaction with a spectrum of
magnetic turbulence), and electric field (“DC”) acceleration. Many theoretical
frameworks roughly fall under one of these types in one way or another, and
all may operate to some degree over the volume and time-evolution of a flare.

In and around the volume of reconnection and its outflow jets, there have been
exciting advances in producing power-laws from multiple first-order Fermi
reflections (through curvature-drift motions of particles) among volume-
filling plasmoids, which are contracting and coalescing magnetic “islands”
[671, 187, 553, 378, 188, 147, 286]. These models have recently been extended
to 3D [148] and expanded to much larger physical sizes [189, 21] than
particle-in-cell simulations, demonstrating efficient acceleration of high-
energy electrons into power-law tails that extend over many orders of
magnitude. The Alfvenic-speed reconnection exhaust may develop turbulence, and
stochastic acceleration within turbulence [500, 586, 585, 70] in the presence
of Coulomb collisions [72] is thought to be one of the possible acceleration
mechanisms666Stochastic acceleration in gyro-resonant interaction with
Whistler-modes of an inertial Alfven-wave turbulent spectrum in the lower
atmosphere has also been suggested [230].. Sub-Driecer [e.g., 317, 60, 7]
electric fields have also been investigated in accelerating a small fraction
of the ambient particles, and super-Driecer electric fields (accelerating the
entire ambient population) that result from the observed decay of magnetic
field over a large loop-like volume have also been discussed [227], possibly
related to a type of hybrid particle acceleration involving electric fields
and turbulent reconnection [752]. Despite the apparent lack of consensus among
these impressive modeling and theoretical directions, a commonality is that a
significant amount of energy is transferred to the atmosphere through
shortening of magnetic field lines (facilitated by reconnection), and
particles are efficiently accelerated over a large volume. For reviews of
magnetic reconnection and particle acceleration, see also [729, 119, 597, 120,
596, 19, 439, 542, 366].

We should mention that acceleration at a coronal termination shock where the
reconnection outflow/exhaust collides with the lower-lying (previously
reconnected) magnetic loops could also be important [e.g., 690], and plasmoid
interaction with a fast-mode MHD shock at the collision interface could
contribute to the accelerated particle flux into the lower atmosphere [671].
Acceleration based on terrestrial aurorae or magnetotail sub-storm processes
are also considered [75, 290], and some models argue for a mechanism where the
bulk of electron acceleration takes place close to [680, 724] or within [230]
the chromosphere. Whatever the case or cases may be, acceleration models
should attempt to reconcile one way [e.g., 98, and see also Section 3.6 in
Aschwanden et al 32] or another [127] with energy-dependent time delays of
hard X-rays [31, 33, 32, 22, 602, 13, 395] and the general agreement between
inferred path lengths and direct imaging measurements of flare loop sizes.
Other critical tests are matching the properties of the white-light continuum
radiation [231], reproducing the energy content in accelerated protons [e.g.,
$E\approx 7\times 10^{32}$ erg above a proton kinetic energy of 30 MeV; 524],
and generating strong nonthermal electron radio signatures at coronal heights
[126].

In order to consider the rest of the flare process, let us assume that the
nonthermal particle beams are injected into a region around the apex of a
just-reconnected-and-retracted coronal loop. Initially, the particles are
injected in relatively low-lying magnetic fields ($\lesssim 10$ Mm), resulting
in the onset of the flare impulsive phase. Over time, larger (and weaker)
magnetic fields reconnect and energize particle beams. The particle beams are
injected into the newly reconnected loops with a distribution of pitch angles
with respect to the magnetic field orientation. The $E\gtrsim 100$ keV
electrons in the beams produce gyrosynchrotron radiation as they spiral (with
Larmor radii of $\sim$ cm) in the magnetic fields. If the magnetic flux
density [G] increases along the beam path (e.g., if the cross-sectional area
of the loop correspondingly converges into the lower atmosphere), then the
particles with small pitch angles with respect to the magnetic field direction
will freely stream (“precipitate”) into the chromosphere. Those with larger
initial pitch angles will gradually and adiabatically increase their pitch
angle, which is a result of Faraday’s Law of Induction in the frame of the
gyrating electrons. These particles can eventually reflect, or “mirror”, at
the footpoints (which could occur in the transition region, high chromosphere,
low chromosphere, or low corona)777The particles can also mirror as the loops
retract after they reconnect; see [24], and they may also exhibit non-
adiabatic/chaotic orbits [198].. After some time, the mirrored particles too
will scatter off enough ambient particles in the corona and precipitate out of
the magnetic “trap” into the chromosphere. This is the so-called
“trap$+$precipitation” paradigm [498]. The highest energy particles take the
longest to scatter and thus remain trapped for a time proportional to
$\frac{E^{1.5}}{n_{e}}$ where $E$ is the kinetic energy of the nonthermal
electrons and $n_{e}$ is the ambient/thermal electron density that the beam
heats (under the dilute beam assumption: $n_{\rm{e}}\gg n_{\rm{e-beam}}$).
These times are rather short ($<10$ s) for typical conditions and energies. In
solar flares, proton/ion beams can certainly contribute to atmospheric heating
[598, 11], nuclear excitation, a gamma-ray continuum spectrum, the 2.2 MeV
neutron-capture Deuterium-formation line, the 511 keV positron-annihilation
line, and neutrons that are directly detected at Earth [524, 751]. For the
sake of brevity, we further consider the effects of only accelerated
electrons.

When the electron beam particles stream into the chromosphere, they encounter
a wall of dense, partially ionized gas to which they rapidly lose their energy
through Coulomb collisions with ambient electrons. At the same time, free-free
radiative transitions in collisions with ambient protons occur in the
chromosphere, producing hard X-ray ($E\gtrsim 25$ keV), nonthermal
bremsstrahlung radiation that exhibits a power-law distribution. Hard X-ray
light curves define the impulsive phase in solar flares. The standard model of
hard X-ray emissivity that accounts for Coulomb energy loss as the electrons
radiate bremsstrahlung is called the collisional (cold) thick target model
[CTTM; 96, for reviews, see Brown et al 99, Holman et al 320, Kontar et al
400]. Electron beam power-laws that are inferred from the CTTM and injected
into radiative-hydrodynamic simulations are typically characterized by power-
law indices of $\delta\approx 5$, a low-energy cutoff of $E_{c}\approx 15-25$
keV, and energy flux densities in the range of $\approx 10^{10}$ to $\approx
10^{11}$ erg cm-2 s-1 [117]. Often, the hard X-ray light curves consist of
gradual variations with superimposed short bursts of $\mathcal{O}(0.5-5)$ s.
In some interpretations, these timescales are sensibly related to the
processes of prompt and delayed precipitation (described in the previous
paragraph) of nonthermal electrons.

The relative displacement of the accelerated electrons from the slower ions in
the corona results in a “return current electric field” [e.g., 730, 684]; this
electric field drains energy from the beam and transfers it as a drifting
velocity component to the ambient Maxwellian distribution of electrons. The
drift is towards the loop apex. In steady state, the flux densities ($J$; el
s-1 cm2) of the beam and return current drifting electrons are equal and
opposite, thus preventing pulsar-strength, $B\approx 10^{9}$ G, magnetic
fields from forming in the solar atmosphere. The resistivity of the background
results in “Joule heating” of the coronal plasma [$\approx\eta
e^{2}J_{\rm{beam}}^{2}$; 318, 11, 6]. For non-dilute beams, the density of the
electron beam is comparable to the ambient electron density, and interactions
of plasma wave turbulence and parallel electric fields (e.g., Langmuir waves,
electrostatic double layers) with the beam are additionally expected.

The nonthermal electron beam kinetic energy is thermalized in the low corona
and mid-to-upper chromosphere; see right side of Figure 2. Initially hydrogen
primarily provides the radiative cooling that balances the chromospheric
temperature increase. As the chromosphere increases from $T\approx 8000$ K to
$T\approx 30,000$ K, hydrogen becomes fully ionized, and then helium I and
helium II take over the radiative cooling in the range of $T\approx
30,000-80,000$ K. Thus, it is thought that non-equilibrium rates of helium and
detailed radiative transfer are important for calculating accurate
evolutionary states of the atmosphere. After helium is fully ionized, oxygen,
carbon, and neon regulate the cooling at $T\approx 10^{5}$ K, which is the
peak of the optically thin cooling curve [e.g., 139, 639, 168, 174]. If enough
heat is deposited to fully ionize these elements, then a temperature runaway,
or explosion, ensues because the optically thin cooling decreases as a
function of temperature up to $T\approx 20$ MK. Thermal conduction transports
energy away from the region of maximum beam deposition. Thus, the temperature
“bubble” expands, resembling a one-dimensional blast wave (which is more or
less confined to the magnetic field direction) in an exponential atmosphere
with pervasive beam heating. The increased pressures and their gradients
result in a multitude of mass motions. Additionally, an increase in
temperature of the very low corona (through thermal heat conduction) can also
drive mass motions on top of the beam-generated mass motions. The sources and
sinks of the equation for energy conservation are discussed further in Sect.
9.

The impulsively-generated upflows and downflows have been studied numerically
as part of the standard solar flare paradigm for many decades [458, 224, 223,
222, 220]. They are respectively known as explosive chromospheric evaporation
and chromospheric condensation (Figure 2, right)888The terms “condensation”
and “evaporation” are used even though they are misnomers because there are no
better words in the English language for the phase transitions from neutral
gas to plasma and vice-versa. “Ionization” and “recombination” could be
appropriate but these are reserved for the element-dependent, microscopic
atomic processes rather than the macroscopic flows and density changes. . The
condensations and evaporations manifest as redshifts and blueshifts in cool
and hot lines, respectively, with a transition temperature that has been
constrained in solar flares [501]. The high-speed upflows fill the loops with
$T\approx 10-30$ MK plasma, while the higher density downflows accrue/accrete
mass over time and radiatively cool through $T\approx 10,000$ K (i.e., this is
a non-adiabatic process). The chromosphere is compressed as in a snow plow
effect while also experiencing continuous energy deposition directly by the
beam. The condensations are a result of shock phenomena because they increase
in density up to 10x the ambient density (whereas sound waves would ostensibly
smooth out the density perturbations). The closest physical process to the
temperature bubble and condensation that we have found [e.g., 421] is that of
a thermal wave [Section 6.6, pp 671-672 of 793, and see also Atzeni and Meyer-
ter Vehn 37 for similarities to supersonic ablative heating waves in
laboratory implosion experiments], which are described as a “second-kind
temperature wave” in Livshits et al [458].

The representative properties of the condensation and evaporation flows at any
time in the evolution follow from a simple momentum balance equation:

$\left(v\rho\Delta z\right)_{\rm{cond}}\approx-\left(v\rho\Delta
z\right)_{\rm{evap}},$ (2)

where $\Delta z$ is a physical depth range over which gas with a mass density
of $\rho$ has a bulk velocity of $v$.

Here, we ignore the downward momentum of proton and electron beam particles
[349, 10]. Typical values of the condensation are $\Delta z\approx 30$ km
[which scales inversely with surface gravity; 410] and $v\approx-50$ km s-1.
Typical evaporation flows are fully ionized and have $n_{e}\approx 10^{11}$
cm-3 ($\rho\approx 3\times 10^{-13}$ g cm-3) with flow speeds of $v\approx
100-500$ km s-1. The density of the condensation increases over time, which is
compensated by an increase in the $\Delta z$ of the evaporation as
chromospheric/transition region mass ablates into and fills the newly-
reconnected magnetic loop. After the flows fill up a $\Delta z\approx 10$ Mm
loop, then Equation 2 implies a gas density of $10^{-9}$ g cm-3 in the
condensation, in agreement with numerical simulations999In reality, the
densities and velocities have large gradients in both the condensation and
evaporation, so these representative, average values are used only to elicit
some basic insight.. The coronal loops thus emit in soft X-rays, while the
footpoints emit in optical and UV radiation. The condensation manifests as
red-shifts in broad, chromospheric lines (Fe II, Mg II, H$\alpha$, Si II) that
exhibit a red-wing asymmetry (RWA). The RWA evolves in velocity and intensity
over $\approx 30$ s [349, 792, 785, 274].

[220] calculated the analytic relationship between the beam energy flux
deposited above the flare transition region ($F_{\rm{evap}}$), the peak
downflow (condensation) speed ($v_{\rm{peak}}$) just below the flare
transition region, and the pre-flare chromospheric mass density
($\rho_{\rm{chrom}}$) at which the flare transition region forms:

$v_{\rm{peak}}\approx
0.6\big{(}\frac{F_{\rm{evap}}}{\rho_{\rm{chrom}}}\big{)}^{1/3}$ (3)

Typical upflows of $100-500$ km s-1 imply durations up to $\approx 30-90$s to
fill a flare loop, at which time the confined, field-aligned flows from
conjugate footpoints collide. Superhot (lower volume and lower density)
$T\approx 30-50$ MK sources can be produced above the looptops of the cooler,
$T\approx 20$ MK, plasma that is thought to be due to chromospheric
evaporation; these are thought to be due to heating from slow-mode (Petschek)
shocks in reconnection, as demonstrated in recent multi-dimensional MHD
simulations [e.g., 461]. However, some of such sources that are consistent
with ultrahot $100-200$ MK thermal spectra that appear early on are generally
more consistent with nonthermal power-law spectra.

In the impulsive phase, bright sources are observed from the footpoints
(Figure 2, right) of flare loops in cooler emission lines and in the UV,
optical, and infrared continuum radiation. A variety of source morphologies
appear and could be affected by temporal and spatial resolution. However, one
can typically separate the geometry into elongated “ribbons” and compact,
brighter circular “kernels” [see, for example, the remarkable H$\alpha$ images
in 384]. Within a relatively short time, the footpoints light up quasi-
sequentially along the PIL, forming flare ribbons [e.g., 603, 386]. One ribbon
typically develops in plage101010Plage is typically considered non-spot, but
highly magnetized, locations of the non-flaring active region Sun that is
bright in the core of H$\alpha$, and another ribbon develops in a penumbral or
umbral region of sunspot. As the flare progresses, the distance between the
two ribbons increases, which is the so-called perpendicular apparent motion.
This is thought to be a signature of the reconnection occurring higher and
higher in the corona, thus releasing energy from larger and larger loops,
which is a hallmark of the famous “CSHKP” model of two-ribbon flares [118,
701, 316, 403]. As a result, the brightest emission is generally seen at the
“leading bright edge” of the ribbons as they spread apart (Figure 2 left). It
is thought that excitation from above, e.g. through particle beams, changes
location rather than through cross-field thermal heat transport / diffusion in
the low atmosphere. However, radiative backwarming from the X-ray, EUV, and
FUV lines may play an important role in heating the surrounding atmosphere
away from the brightest kernels [such as in the so-called “core-halo”
morphology; 537, 9, 357, 225, 528]. After the excitation front passes, the
impulsively-generated chromospheric condensation continues to propagate even
in the absence of electron beam heating, but it eventually reaches pressure
equilibrium with the lower atmosphere. The loops cool from tens of MK to 1 MK,
first primarily through thermal conduction, then primarily through radiation,
resulting in the appearance of the famous “post-flare” loops as seen in
filtergrams such as AIA 171 Å (which are, really, post-impulsive phase loops;
see Fig. 3 and Liu et al 456). In cooler lines such as H$\alpha$ or He II 304,
the very late phase of each flare loop is sometimes associated with the
phenomena of coronal rain, which is a second phase of “condensation”. In this
phase, dense material that was chromospheric before the flare drains out of
the corona. New flare loops are continuously formed through the gradual decay
phase of GOES soft X-ray solar flares [762], but the details of the partition
among various possible heating sources in the low atmosphere (thermal heat
flux111111Thermal conductivity during the flare cools the corona and heats the
top of the chromosphere., electron beams, proton beams, direct heating by
waves) remain poorly understood, especially during stellar flares. We note
that the origin of the long cooling times of post-flare loops on the Sun is
still a very active area of research [e.g., 645, 457, 600, 71, 798, 389, 612,
12, 35].

Several notable and well-studied solar flares are the SOL1992-Jan-13T17:25
M2.0 flare [“the Masuda flare”; 481], the SOL2000-Jul-14T10:00 X5.7 flare
[“the Bastille Day flare”121212Referred to colloquially as “the slinky flare”
within the stellar community., e.g., 601], the SOL2003-Oct-28T11:30 X17 flare
[one among the “the Halloween storms”, e.g., 778], the SOL2002-Jul-23T00:30
X4.8 [ “double power-law”; 319] flare, the SOL2014-Mar-29T17:48 X1.0 flare
[NASA’s “best observed X-class flare”; e.g., 392], the SOL2014-Sep-10T17:45
X1.6 flare [e.g., 273], and the SOL2017-Sep-10T16:06 X8.2 flare [e.g., 126].

Figure 3: SDO/AIA images around the EUV wavelength of $171$ Å [442], obtained
from Helioviewer.org, showing thermal emission from flare loops at two times
during a large X-class flare on the solar limb. The top panel shows the
compact, low-lying thermal flare arcade (indicated by the arrow) during what
is usually the peak of the nonthermal hard X-ray (not shown) luminosity from
the footpoints. Later in the flare, the bright arcade volume exhibits an
expansion horizontally and vertically – the expansion is, however, apparent in
the sense that newly reconnected, larger magnetic loops participate in the
flare as time progresses. The large arcades have traditionally been called
“post-flare” loops, which shine brightly much longer after the end of the main
hard X-ray impulsive phase (they are perhaps more accurately called “post-
nonthermal-X-ray-footpoint” loops). It is generally thought that the flare
loops in these images are the result of magnetic structures that have cooled
down from much higher temperatures to $T\approx 1$ MK [e.g. 28]. Note that a
cusp-like (fork-shaped) dark region just above the bright loops in the top
panel is where many models and observations suggest that the bulk of magnetic
reconnection and particle acceleration occur [see 462, 126]. For illustrative
comparisons of flare and active region loops, see [276].

##  4  A Survey of Flare Stars

The characteristics that lead to flaring in stellar atmospheres are generally
thought to be some combination of the following: rapid rotation, an outer
convective zone, and disorganized surface magnetic fields. The ratio of
stellar rotation to the convective turnover time131313This is derived from
models or theory [e.g., see 475, and references therein]. One qualitative
reasoning for M stars being such prolific flare stars early in their lives
($\lesssim 500$ Myr) is that for stars born with a large amount of angular
momentum and rapid rotation, the low-luminosity of these stars facilities slow
energy escape through convection and thus long convective turnover times. Very
low-luminosity main sequence stars, like Proxima Centauri, thus, may maintain
high rates of flare activity at relatively long rotation periods (and old
ages) because this ratio stays at a small value for a long time.  is called
the Rossby number, or the inverse Coriolis number, which may be indicative of
how much internal shear, differential rotation, and turbulent amplification of
kinetic energy into magnetic energy contribute to the dynamo. The dynamo may
operate in the shear flows around the tachocline interface, and/or throughout
the turbulent convection zone, and/or in the near-surface shear layers. Stars
with small Rossby numbers are found in the so-called “saturated regime” of
quiescent magnetic activity (outside of large flares). The saturated regime is
empirically characterized by nearly constant luminosity ratios of
$L_{X}/L_{\rm{bol}}\approx 10^{-3}$ (Figure 4) and $L_{H\alpha}/L_{bol}\approx
10^{-4}$ [e.g., 594, 781, 621, 770, 780, 541, 102, 782], where $L_{\rm{bol}}$
is the quiescent bolometric luminosity, and the numerator is a luminosity
[e.g., the soft X-ray luminosity, $L_{X}$; 659, 781] that is a proxy for
magnetic heating in the corona or chromosphere141414Saturated regimes are also
seen in many other quiescent magnetic activity indicators, such as Ca II [36,
81] and the photospheric magnetic field [747, 622].. There is a rather large
spread about the saturated value from star to star, and the Sun varies from
$L_{X}/L_{\rm{bol}}\approx 5\times 10^{-8}$ to $\approx 5\times 10^{-7}$ over
its 11 year sunspot cycle [369, 42]. For comparison, the least active M dwarfs
have ratios of $10^{-6}$ [780, 235, 94, 205]. At very small Rossby numbers, a
super-saturated regime may occur, while stellar activity proxies follow a
power-law with increasing Rossby numbers $\gtrsim 0.1$, relatively independent
of spectral type151515The Sun’s Rossby number is often discussed as being
around a value of two, but it is not widely accepted whether it is even
meaningful to characterize the Sun with a single number; further, the Rossby
number itself is a model-dependent parameter.  For an introduction to dynamo
theory in the stars and the Sun, see [84], [85], [125], [102], and [664]. For
a review of the literature, see [102] and [373]. Much recent progress has been
made in modeling stellar dynamos in low-mass stars just below [73, 74] and
above [101, 95, 372] the fully convective transition, and with especially
high-resolution simulations of the solar magnetic field and differential
rotation [325].

Figure 4:  Quiescent $R_{X}=L_{X}/L_{\rm{bol}}$ _vs._ the Rossby number for a
large sample of stars, reproduced from Wright et al [781] with permission.
$R_{X}$ is “saturated” at $\approx 10^{-3}$ for stars with Rossby numbers
$\lesssim 0.1$, and $R_{X}$ decreases according to a power-law for lower-
activity stars.

Magnetic fields buoyantly rise through the stellar photosphere; once in the
corona, they undergo destabilization and reconnection to release energy into
the atmosphere to produce flares. Flaring occurs at different times through
stellar evolution as convective envelopes come and go with changes in internal
structure, core fusion processes, and external factors such as tidal locking
and synchronous rotation. An overview of the myriad of flare stars is best
summarized in a color-magnitude diagram (CMD). Prolific and/or representative
flare stars are shown on a Gaia CMD in Fig. 5. The data were obtained from
Gaia Data Release 2 [246], except for a few cases that are only available
through the Early Data Release 3 or through common filter transformations
[737] for some of the brightest stars. The background data are the Gaia
photometry for a collection of stars compiled for the Palomar/Michigan State
University spectroscopic survey [PMSU; 618, 299, 264, 619, 617], downloaded
from Neill Reid’s website161616https://www.stsci.edu/~inr/cmd.html. Many of
these stars are discussed in the recent 10 pc Gaia sample [624] and the CNS5
catalog [268]. Isochrones from the PARSEC v1.2S [89] stellar evolution models
are shown for solar age and metallicity and the age of the $\beta$ Pic moving
group [478]. Characteristics171717We have tried to update properties with some
“standard” values as best as possible within the time frame of writing this
review but we have certainly missed some notable flare stars. Spectral types
of low-mass stars are determined to only about $\pm$0.5 spectral types using
broadband flux spectral-typing facilities [138]. For example, [162] quote a
spectral type of M4.0Ve for YZ CMi, whereas spectral typing with the Hammer
facility can result in determinations closer to M5 (and similar differences
result for the M0/M1 star AU Mic). [624] discuss the evolutionary status of YZ
CMi, a notorious flare star, as a candidate pre-main sequence star with an age
of around 24 Myr. Gyrochronology suggests an age range of $810\pm^{60}_{56}$
Myr [205]; the large intrinsic age spread at a given rotation period,
especially at young ages for low-mass stars, is widely-recognized [e.g., 356,
206]. There is a correspondingly large range of radius estimates for this star
found in the literature [514, 54]. Aside from coordinates, parallaxes, and
proper motions, one should use caution for some fundamental stellar parameters
(binarity, surface gravity, evolutionary status) obtained from large catalog
databases such as SIMBAD and Gaia. For example, the Gaia DR3 archive returns
an inaccurate surface gravity of log $g$/(cm s${}^{-2})=4.0$ for the notorious
flare star AD Leo (as discussed by [300] and [162], surface gravities
determined from spectra of active low mass stars are unreliable).  of selected
flare stars in the CMD are summarized in Tables 1 \- 6, and in some cases a
reference to an example of a flare study featuring the respective star.

Figure 5: Gaia color-magnitude diagram of absolute magnitude vs. color for
notable flare stars (outside of flaring). Note that corrections for extinction
and reddening from interstellar dust are not applied because most of these
stars are in close proximity (within $\sim 20$ pc) of Earth. The linecolors
and linestyles for the annotations are arbitrary. Open circles signify that
the data were transformed from Hipparcos magnitudes and/or parallaxes. Several
solar-metallicity isochrones from the PARSEC stellar evolutionary models are
shown for reference.

Table 1: Table of common and notable flare stars.

Name | SpT | Class | $d$ | Gaia $G$ | $U$ | RA, Dec | Ref
---|---|---|---|---|---|---|---
|  |  | (pc) | (app mag) | (app mag) | (J2000) |
dMe / UV Cet |  |  |  |  |  |  |
YY Gem (Gl 278C) | M0+M0 | BY Dra/EB | 15.0984 | 8.3 | 11.6 | 07 34 37, +31 52 10 |  [180],[696]
DT Vir (Gl 494) | M1 | dMe | 11.5132 | 8.91 | 12.3 | 13 00 46, +12 22 32 |
AU Mic (Gl 803)a | M0/M1 | (d)Me | 9.7248 | 7.84 | 11.3 | 20 45 09, -31 20 27 |  [145], [722]
CR Dra (GJ 9552)b | M1 | dMe | 20.2626 | 9.06 | 12.0 | 16 17 05, +55 16 08 | [767]
SDSS J211827.00 | M1 | dMe/S82 | 711.9443 | 17.01 | … | 21 18 27, +00 02 48 | [411]
+000248.5c |  |  |  |  |  |  |
SDSS J005903.60- | M1 | dMe/S82 | 123.2745 | 13.89 | … | 00 59 03, -01 13 31 | [411]
011331.1 |  |  |  |  |  |  |
Gl 411 | M2 | dMe | 2.55 | … | 10.1 | 11 03 20, +35 58 11 | [39]
VV Lyn (Gl 277A) | M2.5 | dMe | 11.86 | … | … | 07 31 57, +36 13 09 |
AD Leo (Gl 388) | M3 | dMe | 4.966 | 8.21 | 11.9 | 10 19 36, +19 52 12 | [297]
GQ And (Gl 15B) | M3.5 | dMe | 3.5614 | 9.68 | … | 00 18 25, +44 01 38 |
EQ Peg A (Gl 896A) | M3.5 | dMe | 6.2614 | 9.04 | 13.2 | 23 31 52, +19 56 14 |  [447], [416]
EV Lac (Gl 873) | M3.5 | dMe | 5.0502 | 9.00 | 13.0 | 22 46 49, +44 20 02 | [567]
GJ 3685 | M4 | dMe | 18.7988 | 11.98 | … | 11 47 40, +00 15 20 | [632]
GJ 1243 | M4 | dMe | 11.9787 | 11.55 | 15.5 | 19 51 09, +46 29 00 | [304]
DG CVn (GJ 3789) | M4+M4 | (d)Me | 18.2857 | 10.61 | … | 13 31 46, +29 16 36 | [571]
BL Lyn (Gl 277B) | M4 | dMe | 12.001 | 10.47 | … | 07 31 57, +36 13 47 |
V374 Peg (GJ 4247) | M4 | dMe | 9.1041 | 10.63 | … | 22 01 13, +28 18 24 | [741]
YZ CMi (Gl 285) | M4.5 | dMe | 5.9874 | 9.68 | 13.8 | 07 44 40, +03 33 08 | [412]
EQ Peg B (Gl 896B) | M4.5 | dMe | 6.2477 | 10.82 |  $\approx$15 | 23 31 52, +19 56 14 |  [635], [487]
AT Mic A | M4.5 | (d)Me | 9.8816 | 9.56 | … | 20 41 51, -32 26 06 | [251]
AT Mic B | M4.5 | (d)Me | 9.8312 | 9.59 | … | 20 41 51, -32 26 10 | [251]

1717footnotetext: The Gaia $G$ and Johnson/Bessel $U$-band magnitudes are
apparent magnitudes, and the Gaia $G$ mags are not adjusted for galactic dust
extinction. Note that many of the stars have large proper motions, and
truncated J2000 coordinates are given here. “EB” indicates an eclipsing
binary. “S82” refers to a SDSS 82 flare star with very large magnitude
enhancements during the observed events. aThe reported $U-$band range in the
literature is 11.2–11.4 [441, 722]. bCR Dra has a combined spectral type of
M1, but the system consists of components with orbital parameters consistent
with a M0 and an M3 with angular separations between
$0.\mbox{${}^{\prime\prime}$}06-0.\mbox{${}^{\prime\prime}$}16$ [705].
cProduced the largest flare from an early-type M dwarf in the SDSS Stripe 82
flare star sample. Other notable early/mid-type dMe flare stars are the
$U=11.7$ M1.5e star, FK Aqr [107, 379] and the $U=11.6$ M3e spectroscopic
binary, Gl 644AB (V1054 Oph) [178, 483, 142].

Table 2: Continued …

Name | SpT | Class | $d$ | Gaia $G$ | $U$ | RA, Dec | Ref
---|---|---|---|---|---|---|---
|  |  | (pc) | (app mag) | (app mag) | (J2000) |
V639 Her (Gl 669B) | M4.5 | dMe | 10.7566 | 11.46 | … | 17 19 52, +26 30 02 |
AF PSc (GJ 1285) | M4.5 | dMe | 34.957 | 12.81 | … | 23 31 44, -02 44 39 | [767]
CM Dra (Gl 630.1) | M4.5+M4.5 | dMe/EB | 14.85 | 11.5 | … | 16 34 20, +57 09 44 | [511, 432]
GJ 1245A | M5 | dMe | 4.6919 | 11.47 | … | 19 53 54, +44 24 51 | [466]
GJ 1245B | M5 | dMe | 4.6614 | 11.92 | … | 19 53 55, +44 24 54 | [466]
FL Vir (Wolf 424 AB/Gl 473AB) | M5+M7 | dMe | 4.40 | 11.24 | 15.4 | 12 33 17, +09 01 15 |
CU Cncd (GJ 2069) | M3.5+M3.5 | dMe/EB | 16.6012 | 10.55 | … | 08 31 37, +19 23 39 |
SDSS J025951.71 | M5 | dMe/S82 | 106.8607 | 15.87 | … | 02 59 51, +00 46 19 | [411]
+004619.1e |  |  |  |  |  |  |
Prox Cen (Gl 551) | M5.5 | dMe | 1.3012 | 8.95 | 14.3 | 14 29 42, -62 40 46 | [283]
CX Cnc (GJ 1111) | M6 | dMe | 3.5805 | 12.19 | 19.0 | 08 29 49, +26 46 33 | [313]
UV Cetif (Gl 65B) | M6 | dMe | 2.687 | 10.81 | … | 01 39 01, -17 57 01 | [197]
CN Leo (Gl 406) | M5.5-6 | dMe | 2.4086 | 11.04 | 17.03 | 10 56 28, +07 00 52 | [661]
SDSS J001309.33 | M6 | dMe/S82 | 48.2787 | 15.93 | … | 00 13 09, -00 25 51 | [411]
$-$002552.0g |  |  |  |  |  |  |

1717footnotetext: dStellar parameters extensively reviewed and discussed in
[214]; eMost flares in the SDSS Stripe 82 flare sample. f2.′′3 from Gl 65A (BL
Ceti), which is a dM5.5e BY Dra system. “EB” indicates an eclipsing binary.
“S82” refers to a SDSS 82 flare star with very large magnitude enhancements
during the observed events. gLargest magnitude enhancement in the SDSS Stripe
82 flare sample.

Table 3: Continued …

Name | SpT | Class | $d$ | Gaia $G$ | $U$ | RA, Dec | Ref
---|---|---|---|---|---|---|---
|  |  | (pc) | (app mag) | (app mag) | (J2000) |
Ultracool dMe / VLMh |  |  |  |  |  |  |
VB8 (Gl 644C) | M7 | dMe | 6.5014 | 13.84 | … | 16 55 35, -08 23 40 |
VB10 (Gl 752B) | M8 | dMe | 5.9185 | 14.32 | … | 19 16 57, +05 09 01 | [371]
TRAPPIST-1 | M8 | dMe | 12.4299 | 15.65 | … | 23 06 29, -05 02 29 | [742]
Gaia DR2 87125610- | M8 | … | 75.5971 | 19.28 | … | 02 21 16, +19 40 20 | [655]
622839424 |  |  |  |  |  |  |
WISEP J190648.47 | L1 | … | 16.7867 | 17.84 | … | 19 06 48, +40 11 08 | [265]
+401106.8 |  |  |  |  |  |  |
Optically inactivei dM |  |  |  |  |  |  |
Gl 832j | M1.5 | dM | 4.9651 | 7.74 | 11.5 | 21 33 33, -49 00 32 |
Gl 674 | M2.5 | dM | 4.5496 | 8.33 | 12.0 | 17 28 39, -46 53 42 | [237]
Barnard’s star (Gl 699) | M4 | dM | 1.8266 | 8.2 | 12.5 | 17 57 48, +04 41 36 |  [235],[579]
IL Aqr (Gl 876) | M4/5 | dM | 4.6758 | 8.88 | 12.9 | 22 53 16, -14 15 49 | [234]
YSO’s and Pre-Main Sequence |  |  |  |  |  |  |
GSC 8056-0482k | M2 | … | 38.8466 | 11.06 | … | 02 36 51, -52 03 03 | [465]
V773 Tau | … | T Tau | 128.1264 | 9.98 | … | 04 14 12, +28 12 12 | [480]
DQ Taul | … | T Tau | 197.4475 | 12.49 | $\approx 15.5$ | 04 46 53, +17 00 00 | [258, 715]
COUP 1462 | … | YSO | 376.405 | 12.9 | … | 05 35 31, -05 15 33 | [257]

1717footnotetext: hOther notable VLMs/brown-dwarfs that have produced
spectacular flares are 2MASSW J0149 [446], LP 412-3 [653], LP 944-20 [643],
and V780Tau [587]. iThe optically inactive dM stars were mostly taken from the
MUSCLES [234, 464] and Mega-MUSCLES [774, 238] HST Treasury programs; see also
[94]. jQuiescent chromosphere recently modeled in [233]. kColloquially
nicknamed the “Hazflare” star. lSee [715]: this PMS star exhibits a large
amount of $U$-band variability due to accretion phenomena.

Table 4: Continued …

Name | SpT | Class | $d$ | Gaia $G$ | $U$ | RA, Dec | Ref
---|---|---|---|---|---|---|---
|  |  | (pc) | (app mag) | (app mag) | (J2000) |
Active Binary |  |  |  |  |  |  |
HR 1099 (V711 Tau) | G5IV/K1IV | RS CVn | 29.6272 | 5.6 | … | 03 36 47, +00 35 15 | [566]
II Peg | K2-3V-IV/?? | RS CVn | 39.363 | 7.1 | … | 23 55 04, +28 38 01 | [568]
$\sigma$ Gem | K1III/?? | RS CVn | 37.0194 | 3.86 | … | 07 43 18, +28 53 00 | [100]
ER Vul | G0V+G5V | RS CVn | 50.7822 | 7.18 | … | 21 02 25, +27 48 26 |
$\sigma^{2}$ CrB | F6V+G0V | RS CVn | 22.658 | 5.41 | … | 16 14 40, +33 51 31 | [565]
V824 Ara | G5IV+K0V-IV | RS CVn | 30.5063 | 6.45 | … | 17 17 25, -66 57 03 |
V815 Her | G5V+M1-2V | RS CVn | 32.396 | 7.32 | … | 18 08 16, +29 41 28 |
EI Eri | G5IV/?? | RS CVn | 55.068 | 6.95 | … | 04 09 40, -07 53 34 |
AR Lac | G2IV | RS CVn | 42.6744 | 5.89 | … | 22 08 40, +45 44 32 |
BH CVn | F2IV/K2IV | RS CVn | 46.1482 | 4.73 | … | 13 34 47, +37 10 56 |
CF Tuc | G0V/K4IV | RS CVn | 87.9555 | 7.45 | … | 00 53 07, -74 39 05 |
UX Ari | K3-4V-IV/?? | RS CVn | 50.4723 | 6.33 | … | 03 26 35, +28 42 54 | [282]
VY Ari | K3-4-IV/?? | RS CVn | 41.2999 | 6.61 | … | 02 48 43, +31 06 54 |
AR Psc | K2V/?? | RS CVn | 45.9586 | 7.03 | … | 01 22 56, +07 25 09 |
Lambda And | G8IV-III/?? | RS CVn | 24.1695 | 3.26 | … | 23 37 33, +46 27 29 |
SZ PSc | K1IV+F8V | RS CVn/EB | 90.1098 | 7.08 | … | 23 13 23, +02 40 31 | [376]
CC Eri | K7V+M4V | RS CVn/BY Dra | 11.5373 | 8.17 | … | 02 34 22, -43 47 46 | [374]
ER Vul | G0V+G5V | RS CVn | 50.7822 | 7.18 | … | 21 02 25, +27 48 26 |
EI Eri | G5IV+dM | RS CVn | 55.068 | 6.95 | … | 04 09 40, -07 53 34 |
KIC 11551430 Am | … | … | 324.4975 | 10.7 | … | 19 09 40, +49 30 05 | [473]
KIC 11551430 B | … | … | 321.2161 | 12.62 | … | 19 09 40, +49 30 05 | [473]

1717footnotetext: mAB system colloquially referred to as “Pearl” (D.
Soderblom, priv. communication 2012). The spectral types for the RS CVn
systems were mostly reproduced from the tables in [561].

Table 5: Continued …

Name | SpT | Class | $d$ | Gaia $G$ | $U$ | RA, Dec | Ref
---|---|---|---|---|---|---|---
|  |  | (pc) | (app mag) | (app mag) | (J2000) |
Single Evolved / Active Giants |  |  |  |  |  |  |
HR 9024 | G1III | Gap | 139.485 | 5.62 | … | 23 49 40, +36 25 31 | [710]
FK Com | G5III | Gap | 216.9119 | 7.91 | … | 13 30 46, +24 13 57 |
v Peg | F8III | Gap | 2705.5292 | 9.84 | … | 22 01 02, +06 07 11 |
31 Comae | G0III | Gap | 87.0072 | 4.67 | … | 12 51 41, +27 32 26 |
EK Eri | G8IV | Gap | 64.1692 | 6.07 | … | 04 20 38, -06 14 45 |
$\beta$ Ceti | K0III | Clump | 29.53 | $\approx 1.77$ | 3.90 | 00 43 35, -17 59 11 | [47]
Gamma Tau | G9.5III | Clump | 44.202 | 3.29 | … | 04 19 47, +15 37 39 |
$\theta^{1}$ Tau | G9III | Clump | 46.689 | 3.49 | … | 04 28 34, +15 57 43 |
$\beta$ Boo/ Nekkar | G8 III | Pre-RG?n | 62.7702 | 3.14 | 5.2 | 15 01 56, +40 23 26 | [347]
YY Men | rapid | FK Com | 216.6981 | 7.92 | … | 04 58 17, -75 16 37 |
$\mu$ Vel | G6III(+ dF) | Pre-RG | 35.7812 | $\approx 2.35$ | 4.16 | 10 46 46, -49 25 12 | [46]
Dwarf K starso |  |  |  |  |  |  |
AB Dor | K0V | Ke | 15.3093 | 6.67 | … | 05 28 44, -65 26 55 | [476]
BY Dra | Ke+Ke | BY Dra | 16.5108 | 7.59 | … | 18 33 55, +51 43 08 |
$\epsilon$ Eri | K2V | … | 3.2029 | 3.37 | … | 03 32 55, -09 27 29 |

1717footnotetext: Pre-RG are pre-red giants, i.e., sub-giants about to ascend
the red giant branch; nKönig et al [399], Jim Kaler’s star database
http://stars.astro.illinois.edu/sow/sowlist.html. oOther notable active K (Ke)
stars are [617] PZ Mon (K2e), DK Leo (K7e), EQ Vir (K5e), and 61 Cyg AB
(K5e$+$K7e).

Table 6: Continued …

Name | SpT | Class | $d$ | Gaia $G$ | $U$ | RA, Dec | Ref
---|---|---|---|---|---|---|---
|  |  | (pc) | (app mag) | (app mag) | (J2000) |
Young Solarp |  |  |  |  |  |  |
EK Dra | G1.5V | solar | 34.4502 | 7.5 | … | 14 39 00, +64 17 29 | [43]
KIC 11610797 | G | solar | 279.3635 | 11.46 | … | 19 27 36, +49 40 14 | [473]
KIC 11253827 | G | solar | 225.1926 | 11.83 | … | 19 44 31, +48 58 38 | [473]
KIC 10422252 | G | solar |  |  |  |  | [675]
KIC 12023498 | … | … | 530.1268 | 12.12 | … | 19 47 02, +50 27 18 | [87]
KIC 8164653 | … | … | 3816.1876 | 14.32 | … | 19 25 36, +44 05 10 | [87]
KIC 8803106 | … | … | 867.4893 | 13.79 | … | 18 57 37, +45 05 13 | [87]

1717footnotetext: pSeveral other well-known young G-type flare stars are
$\kappa$ Ceti and 47 Cas [39].

Stellar classification according to binarity, evolutionary status, and
spectral type is important for understanding the origin and release of
magnetic energy. We describe general classifications of the types of stars
that flare across the CMD (Fig. 5). In the following “early” spectral types
refer to those of hotter effective temperatures (e.g., M0-M2), and “later”
spectral types refer to the cooler effective temperatures (e.g., M7-M9); this
jargon has no direct correspondence to time or age. Mid-type M dwarfs span the
fully convective transition and roughly correspond to stars with spectral
types $\approx$ M3-M6, though there is over an order of magnitude range of
bolometric luminosities just within this sub-grouping.

  * •

RS CVn. Though rapid rotation is often associated with stellar youth, tidal
locking of binaries can lead to prolonged magnetic activity and enhanced flare
rates for billions of years [570] in systems that would not normally produce
energetic events. RS CVn systems are synchronously rotating, detached binaries
consisting of an early-type main sequence star and an evolved, cooler G/K sub-
giant or giant companion. In RS CVn systems, the flares are thought to
originate from the cooler component. The cool star component has a large
convection zone, which combined with very rapid rotation for its size, leads
to large flares with rise and decay times on the order of days. The flares of
RS CVns are among the highest energy ($10^{36}-10^{37}$ erg) and longest-
lasting stellar flares. Flare rates from the EUV and X-rays indicate about one
flare per day [561], but optical enhancements are relatively rare due to the
enormous background glare from the non-flaring source; peak $U$-band changes
(Equation 1) are typically much less than a magnitude for energies of
$E_{U}>10^{33}$ erg [rates of $\approx 0.17$ hour-1; 484]; the largest flares
increase the $U$-band flux by a factor of $\approx 2$ [181]. The best-studied
RS CVn system is arguably HR 1099 (V711 Tau), which consists of a primary K1
IV (sub-giant) and a G5 V (main sequence) star with orbital separation of
$\approx 3R_{\rm{Sun}}$ and a 2.8 day synchronized orbital and rotation period
[217]. The most extensive multi-wavelength study of the flaring and quiescence
of RS CVns is presented in [561, 563, 564, 565, 566]. A large catalog of the
non-flaring properties of these stars is contained in [700].

  * •

Algol-type flare stars are semi-detached binaries with a history of
significant mass transfer from a cool evolved sub-giant onto a main sequence
B-type star. Algol (B8V + G8III/K1IV/K2III) consists of eclipsing binary (EB)
stars separated by 14.14$R_{\rm{Sun}}$, whose components have radii of 2.9 and
3.5$R_{\rm{Sun}}$ with a 2.87-day orbital period. Algol produces giant flares
on occasion [732, 212], and serendipitous eclipses of the flares constrain the
geometries and locations of the events on the cooler, less-massive subgiant
component, Algol B [658, 660], which is also the source of the coronal radio
emission [445]. Algol B is one of the few stars besides the Sun whose
magnetosphere has been spatially resolved in radio observations [583].
Detailed comparisons of Algol-type and RS CVn systems have been presented in
[682, 650, 190]. They are otherwise quite similar, except the total optical
(non-flaring) light from the system is dominated by the main sequence B star
in Algol-like systems, whereas RS CVns consist of two cool stars (spectral
type G or later) with one subgiant or giant star that dominates the total
system light181818In Algols, the components have evolved such that the
subgiant is no longer the more massive star due to mass transfer, whereas in
RS CVn systems, the more evolved star is the more massive component.. The
different locations on the CMD (Fig. 5) are clear.

  * •

BY Dra-type stars are, here, informally referred to as closely orbiting (EB or
non-EB), detached binaries consisting of two main-sequence, typically late K
or early M, stars. Arguably, the best known BY Dra flare star is the eclipsing
0.8 day period binary YY Gem, which consists of nearly identical, and probably
rotationally synchronized, dM0e stars with radii of $\approx 0.6R_{\rm{Sun}}$
and masses of $\approx 0.6M_{\rm{Sun}}$ separated by 3.83 $R_{\rm{Sun}}$ [720,
214, 105, 397]. Other eclipsing BY Dra flare star systems are CM Dra and CU
Cnc. BY Dra itself is composed of two active main-sequence K stars with semi-
major axes of $7.4-8.4\times R_{\rm{Sun}}$ and pseudo-synchronous rotation
with a period of 5.9 d [310]. Formally, BY Dra is a term that describes the
class of magnetically active spotted stars that exhibit periodic variation in
their optical light curves, due to starspots and rotational modulation
[General Catalogue of Variable Stars: Version GCVS 5.1;
649]191919http://www.sai.msu.su/gcvs/gcvs/vartype.htm); however, the BY Dra
classification more uniquely describes [e.g., 617] detached active binary
systems that consist of main sequence late-type (K or M) stars that orbit very
closely together202020In contrast to UV Ceti-type binary flare stars like EQ
Peg AB whose components orbit each other at much larger separations of $4-5$
au. At these orbital separations, the active M dwarf stars are considered
“non-interacting” binaries because it is not that plausible that interacting
magnetospheres, given their radio sizes [717], are the source of the flares.,
like BY Dra itself.

  * •

Pre-Main Sequence (PMS) stars, such as Young stellar objects (YSO’s) and T
Tauri stars exhibit flaring emission [e.g., 226] in addition to the emission
from accretion: the flow, shocks, and hotspots at the stellar surface all
generate optical, UV, IR, and X-ray continuum and emission line radiation
[e.g. 312] that can be both steady-state and transient. The optical, NUV, and
FUV flares on T Tauri stars tend to be very energetic, $\gtrsim 10^{35}$ erg
[715, 315, 259]. [715] conducted a large optical photometric monitoring
campaign on the T Tauri binary star DQ Tau and compared the color differences
between accretion and flare variations. [258] performed an extensive study of
X-ray giant flares in the DQ Tau system. The Chandra Orion Ultradeep Project
[COUP; 256] detected many X-ray superflares from YSOs; the flares produced
X-ray hardness ratios consistent with coronal temperatures of $T>10^{8}$ K
[257]. M-type stars take longer to contract onto the main sequence after
losing their gaseous disks and are thus not actively accreting, but may
exhibit inflated radii for tens of Myr (e.g., AU Mic) as they are still
contracting. These are still PMS stars but are also often considered as dwarf,
UV Ceti-type stars.

  * •

UV Ceti-type stars are low-mass, M-type main sequence (dwarf) flare stars that
exhibit frequent flaring and H$\alpha$ in emission in quiescence [255]. There
is a high correspondence between the dMe status and high flare rates [411].
These stars are traditionally denoted as “dMe” or “MVe” stars, they exhibit
optical rotational modulation (with periods typically in the $0.2-5$ day
range, but see West et al 770 and Newton et al 541), and many but not all are
in the saturated regime ($L_{x}/L_{\rm{bol}}\approx 10^{-3}$) of quiescent
X-ray luminosity. Many of the UV Cet/dMe-type stars are in the early stages of
their main sequence evolution and are possibly only several hundred Myr or
less in age. UV Ceti itself is a triple star system with two dM5.5e stars (A
and B components) separated by 5.3 au [63]; UV Ceti B is a binary BY Dra-type
system (using our adopted definition above). This category includes some non-
accreting PMS stars (AU Mic, AT Mic). A catalog of UV Ceti stars and their
quiescent properties was compiled by [255] and is provided on
VizieR212121http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=J/A+AS/139/555.
Though the M-dwarfs are the most populous stars in the Galaxy [e.g., 617],
none are visible to the naked eye ($V\lesssim 6$) from Earth due to their low
quiescent luminosities.

  * •

Very Low-Mass (VLM) Stars, Ultracool dwarfs, Brown dwarfs. The spectral types
M7 through early L span the edge of the core-hydrogen burning regime (i.e.,
the star/brown-dwarf boundary). The exoplanet host star TRAPPIST-1 is now a
well-known, M8 flare star, due to its system of planets in or around the
traditional habitable zone [263, 5]. VB 10 is another prolific VLM flare star
[311, 454, 229, 371]. The G’́udel-Benz relationship [278] is a correlation
between the quiescent radio and soft X-ray fluxes that holds over many orders
of magnitudes but breaks down in the VLM regime (with the VLMs being radio
overluminous compared to the X-rays). Nonetheless, flares from VLMs and L-type
stars have shown dramatic optical continuum enhancements and broad and bright
H$\alpha$ lines [446, 653, 265] that otherwise resemble the very energetic
flares from the earlier M dwarf spectral types, as noted by [617]. White-light
flare rates and magnitude variations have been rather well-characterized
recently in time-domain surveys [e.g., ASAS-SN, Kepler, K2, NGTS; 655, 656,
266, 742, 575, 576, 359, 577]. The L-dwarf flare luminosities are remarkably
super-bolometric and exceed magnitude changes of $-9$ in the $V$-band [655].
X-ray superflares have been reported from L dwarfs [e.g., 163] as well.

  * •

Weakly active and “inactive” M dwarfs (dM or MV) stars are “optically
inactive” stars without H$\alpha$ in emission during quiescence. The inactive
dM’s also flare on occasion [579, 411]. These are thought to be descendants of
MVe stars, which are inferred to hold on to their high levels of activity for
several billion years [301, 769, 391]. After several Gyr on the main sequence,
the M-dwarf stars maintain a lower level of chromospheric emission in Ca II H
& K (with sometimes H$\alpha$ showing a deeper absorption profile), Mg II h &
k emission lines, transition region emission (e.g., C II), and faint X-rays.
One notable example of a low-activity M dwarf is the $\approx 10$ Gyr-old,
slowly rotating ($P\approx 150$ d) star Gl 699 [Barnard’s star; 755, 233, 716,
235]. Optical flares are rare [313, 304] on optically inactive stars, which do
not exhibit much if any rotational modulation in _Kepler_. The visibility
(contrast) of flares at shorter wavelengths is much greater, however, and thus
the FUV and X-ray flares are rather prolific [463, 234, 464, 237, 235, 94].

  * •

Solar-type stars Flares from solar-type, G-dwarf stars have been observed
rather infrequently, and mostly through serendipitous means in the FUV [44],
X-ray [257, 599], and optical [651, 472]. The detectable flares above the
spatially integrated, background luminosity are very energetic, and in almost
all cases they exceed broadband optical energies of $E>10^{33}$ erg. Detailed
flare rates in the EUV have been studied for the rapidly rotating, young,
single G-type stars, EK Dra, 47 Cas, and $\kappa$ Ceti [39].

The _Kepler_ and K2 missions have transformed our knowledge about the white-
light flare rates of G-type stars. The literature has been summarized in
detail by [136] and [307], and we refer the reader to these works. Here, we
provide only a brief synopsis. After the discovery of G-dwarf white-light
superflares in Kepler [472], [675] and [548] performed a comprehensive flare
rate analysis of the 30-minute cadence data (with some comparisons to the
1-minute cadence data), and [109] extended analyses to lower mass K and M
stars [see also 756, for analyses of 30-minute cadence data of M star flares].
[789] also compares short and long cadence _Kepler_ data. [547], [544], and
[377] began the detailed spectroscopic characterization of the chromospheric
and photospheric properties of the solar-type superflare stars reported in
[472] and [675]. [549] and [550] presented spectroscopic follow-up of 46 of
the flare stars with 30-minute cadence data in Shibayama et al [675]. [473]
analyzed 1-minute cadence data of 23 solar-type superflare stars. The most
prolific flare star (KIC 11551430; Table 4) was confirmed as a visual and
spectroscopic binary in [551], who presented spectroscopic follow-up of 18 of
these 23 stars, leveraging Gaia DR2 data as well to constrain evolutionary
statuses on the subgiant branch or main sequence. [555] re-analyzed all solar-
type _Kepler_ data and presented up-to-date flare rates. As a result of the
group’s detailed follow-up, the calculated superflare occurrence rates on Sun-
like stars ($T_{\rm{eff}}=5600-6000$ K, $P_{\rm{rot}}=20-40$ d) has decreased
by $\approx$ an order of magnitude since [675]. These refined constraints
provide interesting comparisons to extrapolated, multi-wavelength scaling
relations from smaller solar flare energies and from large solar energetic
particle events [see also, e.g., Figure 33 of 728]. [783] presented flare
rates and power-law indices for 77 individual G stars that produce
superflares, including KIC 10422252 which is the prolific flare star that is
discussed in [675]. [155], [734], and [788] present _Kepler_ flare catalogs,
where the latter includes all spectral types including A-type stars (see
below) and only long-cadence data. [438] discussed source contamination in the
[155] catalog in the higher mass stellar range. Recently, [30] analyzed the
power-law distributions of the _Kepler_ long-cadence data flare catalog of
[788] and tested against self-organized criticality theory [26].

  * •

Single Active Giants [681] are classified into one of two general categories
[see 45]. The first type is helium core burning (post Helium flash) red clump
active giants like $\beta$ Ceti. Giant flare events with rise times of $\sim
1$ day are observed from these stars: an impressive EUV light curve of $\beta$
Ceti is shown in Fig. 6 from [47]. The active clump stars are rotating slower
than evolutionary models predict, yet they have detectable magnetic fields
(unlike the rest of this stellar population). The origin of the magnetic
activity in these stars is not well understood, but it has been hypothesized
that swallowing a companion planet or brown dwarf increases the shear in the
stellar interior [678], resulting in “magnetic rejuvenation” in old age. An
alternative idea is that these stars, like some other active giant stars
[698], are descendants of Ap stars with strong fossil fields [725]. The second
type of active giants occur in the Hertzsprung Gap, which is a short-lived
phase of stellar evolution after rapidly rotating A and B-type stars leave the
main sequence before rotationally breaking down and ascending the red giant
branch. White-light flares have been widely reported in _Kepler_ data of
subgiants and giants [551, 555, 382, 390, 557], and some exceed $10^{38}$ erg
from KIC 2852961, whose binary status is not yet clear.

  * •

B5-F5 single stars have convective cores and radiative exteriors and are not
expected to produce complex surface magnetic fields and flares. B stars and Ap
stars are magnetic but the magnetism is thought to be a remnant of fossil
fields, which are not disorganized enough to facilitate reconnection and
impulsive energy release. The F2 star HR 120 [520] is a surprisingly early-
type flare star that has a detailed EUV flare rate presented in [39]. White-
light superflares from A-type stars have been reported in _Kepler_ data [48,
49], and spectroscopic follow-up by [581] found evidence for binarity in most
of these sources. If originating from the companion, [48] argues that the
energy requirements are yet rather large and unreasonable [see also 519]. The
A7 star Altair is magnetically active, and its dynamo may operate locally in
an equatorial zone that is cool enough for convection to occur due to the
star’s very fast rotation [634]. But to our knowledge, this “backyard star” is
not known to produce white-light superflares. Algol-like systems consist of a
B- or A-type star, but the flares originate from the cooler, evolved star.

Figure 6: A small fraction of single, red clump giant stars like $\beta$ Ceti
exhibit continuous variability that consists of week-long, giant EUV ($60-180$
Å) flares. A flare on $\mu$ Velorum (also shown) is even more unexpected in
its evolutionary stage. Figure reproduced from [47] with permission.

##  5 Flare Rates and Flare Frequency Distributions

The rates at which stars flare have been measured through continuous ground-
based observations, continuous space-based observations, and coarse sampling
with serendipitous detections. The coarse sampling may result in just one data
point per flare; thus an accurate characterization of the flaring source
(Sect. 4) is often desired through follow-up spectroscopy. The study of flare
rates is important for a wide variety of wider applications, including coronal
heating physics [341], characterizing serendipitous variability in time-domain
and exoplanet surveys [58, 768, 314, 65, 260, 305, 244, 507, 333, 124, 362,
636, 398, 764, 784], and studying the effects on exoplanet environments [e.g.,
332, 713]. Much knowledge of stellar flares from low-mass stars is a result of
monitoring a handful of nearby dMe stars, which reliably flare at a high rate.
Large etendue222222Defined as a survey telescope’s primary mirror diameter
multiplied by the field of view [358]. time-domain capabilities of wide-field
surveys such as the Sloan Digital Sky Survey, 2MASS, and ASAS-SN have been
leveraged to characterize flare rates of inactive and early-type M dwarfs
[411], at infrared wavelengths [156], and of ultracool dwarfs [655, 656, 657].

###  5.1 Basic Methods and Early Studies

[314] distinguish among _flare rates_ , _flare duty cycles_ (flaring
fractions), and _flare frequency distributions_. Here, we focus on flare
frequency distributions (FFDs), which refer to the energy dependence of the
flare rates from continuous monitoring. Flaring fractions in sparsely sampled
data are discussed in Section 5.3.

The seminal study of [433] presented an extensive flare rate analysis of
several nearby active M dwarf (dMe) stars from hundreds of hours of ground-
based monitoring in typical Johnson bandpasses. The high-time resolution data
and observational details, along with some fundamental measurements such as
equivalent durations, timescales, and peak amplitudes, can be found in [506].
In this section, we first present several basic measurements that are derived
from flare light curves. The most fundamental calculation is the equivalent
duration of a flare through a bandpass. The equivalent duration of a flare is
the time that a star must spend in quiescence to produce the same energy as in
the flare. It is a proxy of the flare energy, which is spectral type and
luminosity class dependent, but can be readily obtained from relative
photometry. The equivalent duration is [254]:

$ED=\int_{t_{\rm{start}}}^{t_{\rm{end}}}{\frac{I(t)-I_{q}}{I_{q}}}\,dt=\int_{t_{\rm{start}}}^{t_{\rm{end}}}{I_{f}(t)}\,dt$
(4)

which has units of time and requires establishing flare start and end times. A
small equivalent duration can result from a small flare energy and/or a large
quiescent count flux ($I_{q}$) at Earth. Note that the symbol for intensity,
$I$, is traditionally used, though this quantity is actually the spatially
unresolved instrumental ($I$) count flux from the star232323Some recent
studies use $\delta$ instead of “ED” to refer to equivalent duration; later in
this review, we follow the standard convention and use $\delta$ to refer to
the power-law index of accelerated particles. The equivalent duration is
referred to as the _photometric equivalent width_ in several studies of flare
rates from _Kepler_ data. . The equivalent duration is multiplied by quiescent
luminosity through this bandpass to obtain the absolute flare energy.242424The
quiescent luminosity can be obtained from the literature for a handful of
flare stars, it can be calculated from an apparent magnitude of the star
assuming a bandpass zeropoint flux, or it can be synthesized from
spectrophotometry of a quiescent spectrum – all assuming that the quiescent
flux of the star does not largely vary among observing dates. Alternatively,
$L_{q}$ can be calculated directly using standard star fields during the
observations; this is a time-consuming process that requires taking time to
point to the standard star fields over the course of the night, potentially
missing awesome flares! Only with _Kepler_ and other high-precision ground-
based datasets [e.g. 308] have we seen that the values of $I_{q}$ for typical
flare stars can vary by several percent due to gradual stellar rotational
modulation.

The integrand of the equivalent duration is known as the “flare visibility” or
“flare contrast” and is denoted as $I_{f}(t)$. The flux enhancement relative
to the flare star’s quiescence is known as $I_{f}(t)+1$ (Sect. 2). The
numerator in the integrand is the “flare excess”, which can be denoted using a
prime symbol to indicate a change. If the average flux is calculated from a
continuum or pseudo-continuum region of a spectrum, then the flare-only flux
is denoted as C$\lambda_{c}^{\prime}$, where $\lambda_{c}$ is a representative
wavelength within the spectral window. The $I$ quantity is formally the count
flux of the flare star normalized to the count flux from a nearby non-variable
star ($I(t)=C_{\mathrm{rel}}(t)$ in Eq. 1) in the same exposure and same
aperture radius. For a photon-counting instrument, such as a CCD, the
theoretical value of $I$ is

$I(t)=\frac{\int{T(\lambda)f_{\lambda}(\lambda,t)\lambda\,d\lambda}}{\int{T(\lambda)f_{\lambda,0}(\lambda)\lambda\,d\lambda}}$
(5)

where $T(\lambda)$ is the total system response or effective area, including
filter, atmospheric transmission (if applicable), and the detector gain;
$f_{\lambda,0}$ is the flux spectrum of a comparison (non-variable) star or
stars, and $f_{\lambda}$ is the flux spectrum of the target star in
units252525Throughout the review, we refer to $f_{\lambda}$ as the “flux”,
following traditional shorthand in the observational astronomy community.
Formally, this quantity is the specific/monochromatic/spectral flux density at
Earth. in erg cm-2 s-1 Å-1. The ED is then multiplied by the bandpass ($T$)
quiescent luminosity ($L_{q,T}$) to find the bandpass-integrated flare energy,
$E_{T}=ED\times L_{q,T}$. For known and well-characterized bandpasses, these
can be determined using zeropoint fluxes, $f_{\rm{ZP}}$ [773] and published
apparent magnitudes262626Using the results of others who have done the
painstaking work of observing standard star fields and deriving atmospheric
extinction corrections. , $m_{T}$, at low-levels of flare activity:

$m_{T}=-2.5\,\log_{10}\frac{\langle
f_{q,\lambda}\rangle_{T}}{f_{\mathrm{ZP},T}}$ (6)

where the zeropoint flux in the Vega mag system is determined by the numerical
integration of a bandpass over the spectrum of the A0 V spectrophotometric
standard star, Vega. If a flux-calibrated quiescent spectrum of the flare star
is available, numerical integration over the total system response,
$T(\lambda)$, including bandpass, alternatively yields the following:

$L_{q,T}=\langle f_{q,\lambda}\rangle_{T}\,\Delta\lambda\,4\pi
d^{2}=\frac{\int{T(\lambda)f_{q,\lambda}(\lambda)\lambda}\,d\lambda}{\int{T(\lambda)\lambda}\,d\lambda}\Delta\lambda\,4\pi
d^{2},$ (7)

where $\langle{f_{q,\lambda}}\rangle_{T}$ is the system-weighted flux [683],
$f_{q,\lambda}(\lambda$) is the quiescent stellar spectrum at Earth in units
of erg cm-2 s-1 Å-1, and $d$ is the distance to the star. The width of the
bandpass, $\Delta\lambda$, is usually taken to be the FWHM or the effective
width, which vary from $\Delta\lambda\approx 600$ Å to $4000$ Å, for the SDSS
$u$-band and _Kepler_ bands, respectively. In recent years, quasi272727Usually
not accounting for very short wavelength radiation at EUV and X-ray
energies.-bolometric conversions from white-light filters have assumed a
blackbody function for all flares, and all phases of flares [e.g., 675]. Other
studies assume constant spectral energy distributions, with energies reported
over the limits of the bandpass [e.g., 304]. Several quiescent luminosities
used for M dwarfs are given in Table 16 by [506], but it should be noted that
stars, especially active M dwarfs, have a rather large spread of $U$-band
magnitudes reported in the literature (e.g., reported quiescent apparent
$U$-band magnitudes for the flare star YZ CMi range between $13.70-13.85$).
Total system response characteristics through $U$ bandpasses are also
inherently uncertain [e.g. 477] and to some extent, intractable. Thus, non-
negligible systematic uncertainties in absolute energies are to be expected
from a combination of these various issues [see 313, for empirical comparisons
of flare energies in SDSS $u$ and Johnson $U$ from two different telescopes].
In Appendix A, we show several of the broadband and narrowband filters that
are commonly employed in optical stellar flare studies.

FFDs are presented as either downward cumulative, $Q(>E)$, or differential,
$n(E)$, distributions in log10-log10 space [see 39, for comparisons of these
methods]. The most common practice is to fit a power-law (pareto) model to the
unbinned, downward cumulative FFD:

$Q(>E)=N\big{(}\frac{E}{E_{0}}\big{)}^{\beta}$ (8)

with power-law index, $\beta$, and $N$ total flares greater than or equal to a
fiducial low-energy limit, $E_{0}$ (e.g., the detection completeness limit)
observed within a monitoring duration of $\Delta t$ (hr). The usual convention
is that $\beta$ is negative. The differential FFD (# of flares per unit
energy) is then

$n(E)=-\frac{dQ}{dE}=N\frac{\alpha-1}{E_{0}}\big{(}\frac{E}{E_{0}}\big{)}^{-\alpha}$
(9)

where $\beta=1-\alpha$ (again, the convention is that $\beta$ is negative and
$\alpha$ is positive). Note that if $A\int_{E_{0}}^{\infty}n(E)dE=1$ is solved
for $A$, the converted differential distribution $A\ n(E)$ is a probability
density function (PDF): given a flare occurred, $A\ n(E)dE$ is the probability
for a flare to have an energy between $E$ and $E+dE$.

Convenient analytic forms exist for the power-law index and its statistical
uncertainty from a direct (i.e., unbinned) maximum likelihood (ML) analysis
[757, 133, Appendix B]:

$\hat{\beta}_{\text{ML}}=\frac{N}{\sum_{i=1}^{N}ln\frac{E_{i}}{E_{0}}}$ (10)

from which the result follows that
$\sigma_{\hat{\beta}_{\text{ML}}}\approx\hat{\beta}_{\text{ML}}/\sqrt{N}$.

Alternatively, weighted least-squares fits of the power-law index (slope) and
intercept of a line in log-log space has been sometimes employed in stellar
flare studies. For example, [433] express the power-law for each flare star in
their sample of active M dwarfs as

$\log_{10}\nu(>E)=a+\beta\,\log_{10}\,E$ (11)

where $\nu(>E)=Q(>E)/\Delta t$ is the number of flares per hour greater than
or equal to energy $E$ (expressed in erg) through an optical bandpass $T$. (To
increase statistical confidence, flares that are observed in several
bandpasses were multiply-counted using empirical bandpass energy conversions
with the effective observing time adjusted accordingly.) Of course, the counts
in even well-populated cumulative distributions have statistical uncertainties
that are asymmetric in logarithmic space, and the values in cumulative
distributions are non-independent. Nevertheless, this method is employed as a
sufficient approximation in some cases given the statistics that are possible
with the relatively low number of flares in ground-based observing campaigns.

Power-laws are typically fit to FFDs over an intermediate range of energy that
is carefully chosen [e.g., 679] to exclude the high-energy and low-energy
ends, which may have incomplete sampling that can affect the fits. Note that
[133] present more sophisticated methods for uncertainty estimation for power-
laws and have been employed in some more recent flare rate studies [496]. The
Markov Chain Monte Carlo approach is described in [159], and [160] use an
appropriate uncertainty analysis for the counts [253]. Modifications to the
basic power-law form (Eq. 11) and completeness functions have been used to
make adjustments to the the low-energy end [638, 27, 156, 496, 555] and high-
energy tail [30] of FFDs. For reference, solar flare frequency distributions
[136] exhibit power-law indices in (nonthermal) hard X-ray energy of $\approx
1.5$ and peak luminosity of $\approx 1.7$ [143], whereas the thermal soft
X-ray peak flux exhibits a larger power-law index of 2.0 [738, 29, 339].

We summarize the general range of values of $\alpha$ calculated from FFDs in
the X-ray and the optical regimes. Ground-based optical and space-based XEUV
($0.01-10$ keV, which corresponds to $1.24-1240$ Å, and where only EUV data,
such as DS count rates, are available, a two-temperature, $T=6+23$ MK,
optically thin free-free model extrapolates to the full XEUV range to
calculate energies) flare rates have generally given values from
$\alpha\approx 1.4-2.2$ with statistical uncertainties of $\gtrsim 0.1$ [39].
RS CVn systems exhibit power-law indices in the EUV on the lower end of this
distribution [561]. [783] summarize the power-laws from 77 G-type stars. [304]
and [351] summarize values of $\alpha$ for low-mass stars in the optical, and
[464] present power-laws for low-mass stars of different activity levels in
the FUV. A useful document that summarizes flare rates from low mass M stars
of various spectral types is Osten
(2016)282828https://www.stsci.edu/files/live/sites/www/files/home/hst/instrumentation/stis/documentation/instrument-
science-reports/_documents/2017_02.pdf.. Among the active M dwarfs, the higher
luminosity stars (in quiescence) exhibit flatter power-laws [590], and thus
the sum of the energy in all flares is dominated by the occurrence of the
highest-energy flares [433]. Alternatives to the power-law functional form (Eq
9) are summarized in [25], [638], and [646].

A few stars have had power-law indices measured independently several times;
these stars show some interesting systematic differences in the reported
power-law properties. In the M dwarf study of [433], the famous flare star AD
Leo appeared to be an outlier from the others. Due to the relatively short
observing time in the [433], its FFD was recalculated after more data were
collected. [590] found a value of $\alpha=1.62\pm 0.09$ from 85 $U$-band flare
events (with 115 individual flare peaks) in 111 hr of monitoring. The most
energetic flare was $10^{33}$ erg. The $U$-band FFD of Proxima Centauri has
been quantified by [754] for flare energies between $5\times 10^{27}$ erg and
$10^{30}$ erg, giving a similar value of $\alpha\approx 1.7$ to AD Leo but
flatter than other stars of the same quiescent luminosity, such as CN Leo.
Recently, [159] doubled the number of flares in the statistics for Prox Cen
using white-light optical data from the MOST satellite and calculated a
similar power-law slope to [754]. [332] found a steeper power-law for Proxima
Centauri from the Evryscope survey and investigated the effect on the power-
law index with and without an extreme superflare outlier.

[433] and [590] found trends in power-law indices with effective temperature
or quiescent $U$-band luminosity: larger luminosity (earlier-type) main-
sequence stars exhibiting flatter power-laws (i.e., smaller $\alpha$,
$\mid\beta\mid$). However, the study of [590] did not include the results of
[754] for the interesting case of Proxima Centauri, which shows a flatter
power-law than expected; evidence for and against Proxima Centauri being less
magnetically active than others of the same sub-type, such as CN Leo, are
summarized in [620]. [39] suggested that $\alpha$ derived from the XEUV may
exhibit an opposite trend with spectral type, though this is reported with low
confidence. They find an increasing rate of $E>10^{32}$ erg flares corresponds
with greater quiescent X-ray luminosity.

Other interesting quantities that have been calculated from flare star
monitoring are the energy per flare (average flare energy) and the radiated
flare energy per unit time (average luminosity due to flaring). These
quantities are usually calculated through the $U$ band. They correlate with
the non-flaring, bandpass quiescent luminosity [433] or non-flaring,
bolometric luminosity of the star over several orders of magnitude. It should
be noted that the number of detectable flares is anti-correlated with the
quiescent luminosity, an effect largely due to brighter stellar background
that raises the detection limit [e.g., 156]. Nonetheless, including many
small, unobserved flares in the total energy calculation does not change the
trends. From modern space-based data, it seems that these relationships
generally extend to active binaries, which comprise much higher luminosity
stellar components [570]. An extrapolation of these relationships to the Sun,
however, dramatically over-predicts the flare energy release rate in optical
solar flares by many orders of magnitude, as first noted by [433]. From
_Kepler_ data (Sect. 5.2), a somewhat similar quantity of $L_{f}$ vs.
$L_{\rm{Kp}}$ has been constructed by [466] and is making for powerful
comparisons for normalized flaring efficiencies among stars of different
spectral types [e.g., 160]. Davenport [155] investigates evidence for
saturation in the relative flare luminosity given by $L_{f}/L_{\rm{Kp}}$ at
small Rossby numbers, analogous to the quiescent $L_{X}/L_{\rm{bol}}$
saturation (Section 4).

There has been a severe paucity of low-activity M dwarf flare rates until
[304] presented white-light 1-minute cadence data from _Kepler_ and some
previously unpublished ground-based $U$-band data from the PhD work within
[313]. These studies revealed that stars that are optically inactive in
quiescence (Sect. 4) produce broadband optical flares, a result that had been
quantified in sparsely sampled data from the Sloan Digital Sky Survey [411]
and from serendipitous spectral detections [579]. [464] compared FUV flare
energies among stars with comparable quiescent bolometric luminosities and
found that FFDs are shifted according to the quiescent flux in the FUV.

The different apparent behaviors at the high-energy ends of stellar flare FFDs
are still largely unexplained. Does the power-law that is well-fit to a middle
range of energies continue on indefinitely, or does it turn over at the high-
energy end and follow a steeper power-law index [176, 570] or another type of
roll-over [e.g., 566, 149, 30, 136, 646] for some stars? What physical
parameters determine the upper limits of flare energies (and peak
luminosities) that are possible on a given star? Extending the power-law that
is fit to the FFD of YZ CMi predicts a $U$-band flare to occur about once per
month with an energy that is a factor of $5000-10,000$ larger than its mean
flare energy [433, 412]. Dal and Evren [152] discuss an upper limit to the
energy released in dMe flares is approached as the flares become longer in
total duration. Shibata et al [674] and Aulanier et al [41] present
theoretical grounds and semi-empirical relations for upper limits of flare
energies. Empirical constraints from very long monitoring times have only been
possible recently using Kepler data.

###  5.2 White-Light Flare Rates & _Kepler_

The release of _Kepler_ data [80] has transformed knowledge of flare rate
statistics in several important ways. First, the high precision and long
monitoring times facilitate detection of relatively rare white-light flares
from older, less active low-mass stars, from much larger luminosity stars such
as solar-type stars, and from higher mass K and early-type M dwarf stars, and
evolved stars. Second, it provided many more hours of monitoring per star:
ground-based monitoring efforts resulted in $\approx 30-100$ hours per star,
while the nominal baseline for Kepler was two contiguous months of
observations per star. The flare statistics correspondingly improved, but the
high precision also revealed comparable amounts of background flux variation
due to starspots and rotation. Nonetheless, the _Kepler_ field began as a
relatively poorly known star field (due to its relatively faint magnitude
limits), and systematic contamination of the flare sources from binaries,
subgiants, and other non-flare sources was an issue until targeted, ground-
based spectroscopic (and asteroseismic) analyses caught up and Gaia DR2
parallaxes became available. Additionally, the vast majority of data from the
_Kepler_ mission is at 30-minute cadence, and most superflares span just two
to four data points. The 1-minute cadence data were made possible for select
stars through Guest Observer programs [e.g., 304].

Figure 7:  Flare frequency distributions of several main-sequence M stars in
the Kepler band (top) and in the $U$-band (bottom). Figure reproduced from
[304] with permission. Note the spread in flare rates in the bottom panel
among the very active and less active stars of the same spectral types.

The dM4e star GJ 1243 has been observed for 11 months at 1-minute cadence with
_Kepler_ , making it the longest observed flare star in the sky besides the
Sun. The FFDs and flare properties of this star have been studied extensively
by [604, 304, 157, 155, 679]. From the database of over 6000 flares, [679]
obtained a value of $\alpha=2.008\pm 0.002$, and [161] inferred
$\alpha=1.942\pm 0.001$. From the first two months of data, [304] found
$\alpha=2.01$. They also noted that the turnover in the FFD at the low-energy
end, which has often been attributed to a detection limit effect, may be
astrophysical at some level since flares of these energies should be readily
detectable above the precision of _Kepler_. The FFD of GJ 1243 is shown in
Fig. 7(top) in comparison to several other M dwarfs spanning different
spectral types and levels of quiescent magnetic activity (dM v. dMe).
Comparable diagrams with compilations of FFDs of many individual stars are
shown in [668], [604], and [149]. [466] presented a detailed analysis of the
GJ 1243 AB system, where they separated the _Kepler_ light into the two
stellar components and found values of $\alpha\approx 2$ for both stars, but a
higher flare rate occurs on the slower rotator. The bottom panel of Fig. 7
shows the _Kepler_ energies converted to energies in the $U$-bandpass. The
flare rates of “less-active” M3–M5 stars are closer to the FFD of GJ 1243,
whereas the very-active M3-M5 stars include the dMe stars from [433], such as
YZ CMi, EV Lac, and AD Leo. A continuum of flare rates is clearly evident as
the vertical offsets of the power-laws, while different power-law slopes may
be a result of vast differences in statistics from ground and space-based
monitoring [304].

The spread in flare rates among active M-dwarfs of similar spectral types
(Fig. 7) contrasts with the findings of [411], who found that the average
M4Ve-M5Ve flare frequency distribution from sparsely sampled data of field
stars in the Sloan Digital Sky Survey (SDSS) Stripe 82 is sensibly in line
with that of the well-known flare star YZ CMi (which is one of those among the
“more active” stars in Fig. 7). [313] began to investigate the chance
alignment of FFDs in [411] using Monte Carlo sampling of simulated flares at
the SDSS $riuzg$ cadence. The resolution of this issue (which is a non-trivial
adjustment to the statistical rate calculation; E. Hilton, priv. communication
2013) will be important for accurately mapping flare rates of the Galaxy using
sparsely sampled data from the LSST [358].

The immaculate flare statistics of _Kepler_ facilitated new timing analyses of
flare occurrence and correlation. The flare occurrence of GJ 1243 was
investigated as a function of the phase of the longer-timescale, flux
modulation that exhibits a $\approx 1$% amplitude and is attributed to stellar
rotation. No significant correlation was found at either maximum or minimum
flux levels. The relationship between flare occurrence and phase was further
investigated in detail with larger samples of flares on GJ 1243 [679] and for
a large sample of M dwarfs [466, 185]. Morris et al [515] reported a larger
flare occurrence when the very low-mass (M8V) star TRAPPIST-1 is slightly
brighter than average and when it goes through the increasing flux phase in
the Kepler light curve. [151] investigated the occurrence of different types
of flares with phase from a large sample of ground-based observations of
flares.

[304] finds that the intervals between successive flares (“waiting times”) on
an active M dwarf (GJ 1243) over many rotation periods is consistent with a
single exponential probability distribution, $p(\Delta
t_{\rm{next}};\tau_{0})$ $\propto\frac{1}{\tau_{0}}e^{-\Delta
t_{\rm{next}}/\tau_{0}}$ (where $p$ is the probability density and $\tau_{0}$
is the average interval between consecutive flares). Equivalently, the number
of flares occurring in fixed time intervals, $\Delta t$, is Poisson-
distributed, $p(N_{\rm{flares}}$; $r,\Delta t)=(r\Delta
t)^{N_{\rm{flares}}}e^{-r\Delta t}/(N_{\rm{flares}}!)$ [e.g., 433]. Stellar
flares from active stars thus occur independently and continuously at a
constant rate, $r=1/\tau_{0}$ (note that some studies use $\lambda$ instead of
$r$ for the rate), except possibly on some short time-intervals ($\Delta
t\lesssim 30$ min) within complex events [304, 157]. Correlations between
waiting time and flare energy, as may be expected from magnetic energy buildup
and release [e.g., 638] in a single region on the star, are not clearly
consistent with the stellar data [304]. Efforts to understand the flare
occurrence as a function of light-curve phase are ongoing in the era of TESS
data [350].

[157] constructed a canonical flare template from the normalized, classical
(single-peaked) flares of GJ 1243 in short-cadence _Kepler_ data. They modeled
the decay phase with a sum of two exponentials and the rise phase with a
polynomial. The template reflects a remarkable self-similarity in many flares
on one-minute integrations. Further analysis in this work demonstrated that
many complex flares can be decomposed into linear superpositions of several
templates with adjustable parameters. They then injected synthetic flares over
a long time baseline and simulated flare statistics, concluding that the
number of complex flares observed in the data is not fully consistent with a
chance superposition of many uncorrelated classical flares. This is
tantalizing, quantitative evidence for a solar-like “sympathetic” [e.g., 467]
flaring property of stars.

The rates of flares in the high-energy regime have been investigated in detail
with Kepler data. Silverberg et al [679] discuss a deviation from a single
power-law between $E_{\rm{Kp}}=10^{32.5}-10^{34}$ erg flares in the GJ 1243
sample. (In the following, flare energies now refer to the energies integrated
over a $T=9000$ K blackbody function, as reported in the respective studies,
rather than an energy over a particular bandpass). The upper limit flare
energy for white-light flares has been pretty well established from the long
monitoring times from _Kepler_ to be $E_{\rm{max}}\approx 10^{37}$ erg for
subgiants, $E_{\rm{max}}\approx 10^{36}$ erg for rapidly rotating G-dwarfs,
and $E_{\rm{max}}\approx 10^{35}$ erg for slower rotating G dwarfs [551, 555].
For G, K, and M stars in the Kepler field, Candelaresi et al [109] discuss a
correlation between a star’s superflare ($E>5\times 10^{34}$ erg) rate and its
light-curve amplitude modulation due to starspots. The ground-based Evryscope
network has also provided new insights into $g^{\prime}$-band superflare
rates: Howard et al [333] find that the rate of $E\geq 10^{33}$ erg
superflares (by extrapolating the best-fit power-laws down over $0.5-1.5$
orders of magnitude in energy for the K5-M2 stars) decrease from the averaged
active K5-K7 stars to the averaged active M4 stars. Similar to many previous
studies (Section 5.1), they report a larger average flare energy for earlier-
type stars, which they attribute to the “size of the stellar convective
region”. Pettersen [588] review and analyze the time-averaged luminosity due
to flaring (and quiescent magnetic activity proxies) in a sample of dKe and
dMe stars, and they correlate it with the convective envelope volume. Recent
stellar evolution models of dM flare stars in eclipsing binary systems [214,
215] and dK/dG stars [488] also show that early type stars, which have larger
mean flare energies, have larger convective zone volumes.

###  5.3 Timing on Long Scales: Flare Rates and Stellar Age

By characterizing the FFDs across many types of stars in different
evolutionary stages, astronomers begin to piece together how flare rates
change over billions of years. With age-dating techniques such as open cluster
membership / association and gyrochronology, quantitative evolutionary tracks
in the flare history of stars are possible. At the youngest stellar ages, high
flare rates and energies have been reported from PMS stars in nearby open
clusters, such as Orion, in the X-ray and optical [256, 361]. Signatures of
accretion (e.g., in T Tauri systems) indicate a very young age range for some
flare stars [e.g., 715]. Membership among moving groups, such as the well-
known $\beta$ Pic moving group (with the flare star members AU Mic and AT Mic
AB) probe ages around $\sim$ 20 Myr for stars that are no longer actively
accreting gaseous material from a circumstellar disk. At older ages ($\gtrsim
100$ Myr), flare rates and properties have been reported in open clusters,
such as the Pleiades [695, 352]. At the oldest ages, globular cluster and the
Galactic bulge membership facilitates calibration of the flare-rate–age
relationship, but such stars are located at very large distances. As stars
spin down due to angular momentum loss over their main sequence evolution,
flare rates are expected to correspondingly decrease. This correlation is
supported by detections of stellar superflares from main sequence, rapidly
rotating (periods $<$ 5 d) G-type stars. However, binarity plays an important
role in maintaining flaring among a population stars to old ages, compared to
the Galactic disk populations. This may occur through a co-evolutionary
process in M dwarf-white dwarf binaries [513] or rotational tidal
synchronization in RS CVn-like systems [570]. Binarity may also directly
affect flare occurrences and energies [e.g., 176, 250, 4, and see references
in Section 7.5], but it is in general a difficult property to assess without
adaptive optics [e.g., 132] and high-resolution spectroscopy [e.g., 547],
which are observational techniques that are limited to nearby stars.

Following the statistical inference of quiescent magnetic activity lifetime
analyses of [769], the vertical distances above or below the Galactic plane
have been used as proxies of stellar age for M dwarf stars in the field. The
basic idea is that over time, stars experience repeated gravitational
scatterings in the midplane of the Galaxy and are consequently more likely to
be found at larger vertical displacements. Large time-domain surveys, such as
the SDSS and soon the LSST, allow characterization of flare rates in multiple
populations of stars in sparsely sampled data sets. The stellar flaring
fraction and the flare fraction (duty cycle) are two quantities that are
useful in mapping the flare rate of a galaxy and the star’s vertical distance
from the galactic plane. A _stellar flaring fraction_ is the fraction of stars
that produce some number of flares, while _flaring fraction_ , or duty cycle,
is the fraction of sparsely sampled epochs that flare. UV and optical surveys
have been leveraged to map these quantities with increasing vertical distances
[768, 411, 314, 756]. There is suggestive evidence that flaring fraction from
sparsely sampled data indicates that flaring occurrence among a population
decreases faster with stellar age than other measures of quiescent magnetic
activity [411, 314]. One may be tempted to speculate that this is one possible
source for the spread in flare rates among the mid-type dMe stars in Fig. 7,
but more investigation is needed. The effects of solar-like activity cycles
[143] might also sensibly contribute to the variation of flare rates within a
population of stars of the same spectral type [but see 590, 161].

Most recently, the improved methods and data samples for flare statistics,
asteroseismology, gyrochronology, and open cluster calibrations using _Kepler_
and K2 data have facilitated much more quantitative characterization of the
flare rate as a function of stellar age [e.g., 351, 352]. [160] used _Kepler_
field stars and gyrochronology [479] to present the first stellar flaring
fraction as a function of age and $g-i$ color, which maps to spectral type
[138]. [160] used MCMC non-linear least squares to constrain $\alpha(t)$ in an
equation similar to Eq. 11 over a range of stellar masses, from G to M dwarfs,
and ages older than $t\geq 10$ Myr. They found that $\alpha(t)$ remains
approximately constant, flare rates indeed decrease over time for the M
dwarfs, but that the flare rate decreases much less than for the G dwarfs.
[551] and [555] characterized the evolution of differential flare frequency of
G-type stars in the _Kepler_ field by splitting up into different rotational
period bins from $<5$ d to $20-40$ d. They further divided the $<5$ d sample
into smaller bins and found that flare frequency is approximately constant for
periods $<$ 3 d, for which the corresponding ages are $\lesssim 500$ Myr but
are not as well constrained from gyrochronology calibrations. Johnstone et al
[368] combine several state-of-the-art techniques to calculate the quiescent
$L_{X}$ as a function of age and mass, and then they leverage a scaling
relation (discussed in Sect. 5.1) from Audard et al [39] to map the rate of
high-energy XEUV flares on GKM stars over ages spanning 2 Myr to 5 Gyr.

##  6 Light Curve Analyses

In addition to integrated flare energy in a particular bandpass (Sect. 5), a
number of other strictly empirical quantities can be calculated from single-
bandpass data with a moderately fast cadence. A useful decay timescale
measurement is the time it takes for a light curve to reach $1/e$ of its
maximum flux [e.g., 473, 526], which prevents ambiguities in determining the
bona-fide end of a flare. This measure also avoids model-dependent
uncertainties in fitting an exponential function to flares with events in the
decay phase, which may also show lengthening timescales [567, 157]. Here, we
summarize a few relationships between decay times and other quantities. The
total rise and total decay times of flares are not well-correlated [506, 433,
304], which possibly suggests some loss of “memory” about the details of the
rise phase after the peak. Integrated energy and total duration exhibit a
tighter correlation. Larger peak-flare amplitudes typically correspond to
longer decay times and larger energy flares, but there is significant scatter.
[526] find that the $1/e$ decay duration scales with total white-light energy
to the $1/3$ power and magnetic field to the $-5/3$ power for flares on the
Sun, rapidly rotating G stars [see also 473], and M stars [see also 333].
Kővári et al [390] found that the $1/3$ power scaling between duration and
energy also extends to a late-type giant. For similar-duration ($\approx 5-10$
min) solar flares and white-light flares on solar-type stars, the energies of
the latter are several orders of magnitude larger [_cf_ Figures 8-9 of 526].

###  6.1 Classification of Flares by Optical Broadband Evolution

Several classification schemes of flares have been developed based on optical
broadband durations and light curve shapes (“morphologies”). [78] and [506]
established some of the basic descriptive terminology of flare classification,
such as _complex_ , _spike_ , _slow_ , _typical_ , or _multipeak_ (a type of
complex flare). More recently, a condensed scheme denotes flares as either
_classical_ or _complex_ [157]; several classical and complex events are
evident throughout a small section of the Kepler data of GJ 1243 in Figure
8(top) panel. [420] argued that flares with several events that comprise the
rise phase can appear as relatively simple, “classical” flares if degraded to
1-minute time-integrations. A classification of solar flares into “gradual”,
“impulsive”, and “secondary” was proposed by [134] based on extremely high-
time resolution, hard X-ray data. Hard X-ray data is rarely available for
stellar flares, and impulsive and gradual classifications have thus been
determined from broadband $U$\- and optical light curves.

A common classification utilizes a measure of the impulsiveness (also referred
to as “impulse” or “impulsivity”), of a flare. [297] quantified the
impulsiveness as the fraction of energy emitted in the impulsive phase, which
ends after the fast-decay phase(s) in broadband optical radiation. They found
that about 70% of the energy was emitted in the impulsive phase during
two292929The light curve of the smaller event is shown in the review by Byrne
[106]. flares that are separated by $\sim 2$ orders of magnitude in total
energy. These were both considered very impulsive events. [413] classified
three events as “gradual”, “impulsive”, and “traditional” in ultra-high-
cadence (0.16 s) photometry data. [414] calculated the impulsiveness index
($\mathcal{I}$) as the peak value of $I_{f}$ divided by the FWHM ($t_{1/2}$)
of the broadband ($U$) light curve to classify events into “impulsive flares
(IF)”, “hybrid flares” (HF), and “gradual flares” (GF). Examples of high-
energy, $E_{U}>10^{32}$ erg, GF and IF events are shown in Figure 8(middle).
Lower-energy and lower-amplitude IF, HF, and GF events are shown on the same
axes in the bottom panel of Figure 8. [150] divide a large sample of dMe
flares into fast and slow flares and suggest the differences are related to
the position on the stellar disk. This hypothesis could be verified with
spectral properties that are thought to change with position on the solar disk
[538].

The Kowalski et al [414] IF/HF/GF impulsiveness index is primarily a
diagnostic of the broadband evolution during the impulsive _phase_ of a flare.
Thus, it is smaller for flares with a gradual decay phase that starts at a
flux level that is a significant fraction of the peak flux. Among the IF-type
events, there is a large range, $30-70$%, in the percentage of the total
energy that is emitted in the impulsive phase. Several spectral quantities
(Balmer jump ratios, Balmer line-to-continuum ratios; Sect. 7.2) at the time
of peak broadband flux correlate with this index; one possible interpretation
is that the average impulsive-phase heating rate varies among events. If the
heating rate is attributed to the flux of electron beams, then the inter-flare
variation of these optical spectral quantities suggests [416] an important
connection between large-scale flare evolution and electron acceleration,
which is sometimes discussed in the context of nonthermal hard X-ray
properties of solar flares [492, 320]. This expands upon the foundational
paradigm that stellar flare scaling relations (e.g., H$\gamma$ _v._ $U$-band
energies), which extend over many orders of magnitude, mean that differences
from flare to flare primarily result from the flaring surface area [e.g., 104,
297]. The empirical differences in IF/HF/GF spectra thus eventually motivated
the semi-empirical modeling work in Kowalski [408], which suggests that
differences among flares further result from scaling the relative areas of
higher and lower concentrations of particle beams in the stellar atmosphere.
Recent analyses of H$\alpha$ line profiles from different locations within
solar flare ribbons have begun to quantitatively explore the implications for
spatially unresolved stellar data [528].

The fraction of flares within each of the IF, HF, and GF types is not yet
well-established. The reported numbers of flares that are IF/HF/GF [414] are
biased toward the IF type because the larger signal-to-noise at peak
facilitates spectral characterization. Note that rigorous distinctions cannot
be made between an HF and GF classification for certain events [420]. It is
also not clear how many IF events produce larger Balmer jumps [416] than
expected according to the original [414] IF/HF/GF classification scheme. GF
events in the decay phase of highly-impulsive events also have shown some of
the smallest Balmer jumps (in absorption).

Figure 8:  Representative high-cadence optical light curves of M Ve flares.
(Top) A randomly chosen section of the 1-minute cadence Kepler light curve of
GJ 1243 from [157] showing a variety of classical and complex flares
superimposed on the modulation, which is due to starspots [158]. The data were
obtained from https://github.com/jradavenport/GJ1243-Flares. Note, there was a
single flare event (not shown) that was observed simultaneously in both Kepler
and in the $U$-band [304]. For reference, this flare had a relative peak
contrast of $\Delta F/F_{\rm{median}}=0.014$ in the Kepler band, a peak
contrast of $I_{f}=0.9$ in the $U$ band, and a total $U$-band energy of
$10^{31}$ erg. (Middle left) The impulsive and gradual decay _phases_ of a
high-energy gradual flare (GF) event on EV Lac; reproduced with permission
from [414]. The impulsive phase and gradual decay phases are indicated. At the
peak (vertical black dashed line), GF-type events like this one have large
Balmer jumps, which are still much smaller than predicted by optically thin
hydrogen recombination theory alone (e.g., Sect. 7.2.1). The $t_{1/2}=13$ min
is the FWHM of the light curve, and the $U$-band impulsiveness index,
$\mathcal{I}=I_{f}/t_{1/2}$, is 0.5 for this flare. (Middle right) An example
high-energy IF event ($t_{1/2}=2.5$ min, $\mathcal{I}=8.3$) on EQ Peg A,
reproduced with permission from Kowalski et al [414]. (Bottom) Examples of
lower energy flares within the flare IF/HF/GF classification scheme based on
the impulsiveness in the $U$-band or a narrow-band filter (NBF) with
$\lambda_{c}=3500$Å. The HF and IF events have similar peak contrasts
($I_{f}=2.7-3.7$) but differ significantly in the values of $t_{1/2}$ (14 s
and 120 s). The IF and GF events are referred to as IF4 and GF2 in Kowalski et
al [416]; the HF event is referred as the HST-2 flare in Kowalski et al [420].
Note, the flare-only impulsive-phase spectrum of the IF event is shown in
comparison to common photometry filters in Appendix A.

Nonetheless, it is rather clear that the most gradual-type events in the
light-curve evolution exhibit the largest Balmer jumps ratios and ratios of
H$\gamma$ / C4170′ at the peak time in broadband optical flux, whereas the
most impulsive-type events show the smallest Balmer jumps in emission and the
largest fraction of optical energy that is radiated in the continuum (Sect.
7.1). This is possibly an interesting connection to the Type I and Type II
white-light flare classification on the Sun (Sect. 7.2), but the stellar
events that would, ostensibly, be most analogous to Type II white-light flares
in the spectral properties are actually the stellar events that are most
impulsive.

We should clarify that an impulsive-type flare exhibits an impulsive phase and
a distinct gradual decay phase, while a gradual-type flare can show a gradual
decay phase that is clearly distinct from its impulsive phase. Figure 8