1 Introduction

Research on extreme solar and solar-terrestrial activity dates to the notable event of 1859 (Carrington 1859; Hodgson 1859; Stewart 1861), but it is only within the last twenty years that extreme events, as a separate class, have been examined in detail. The threat posed by extreme space weather events to Earth’s technological infrastructure provided much of the impetus for this development. Detailed studies of the impact of a severe space weather event on modern society have been conducted by the US National Research Council (NRC; 2008), Lloyds of London (2010), JASON (2011), and the UK Royal Academy of Engineering (2013), among others.Footnote 1 The NRC report contains an estimate for the economic costs of such a storm of “$1 trillion to $2 trillion during the first year alone … with recovery times of 4–10 years”. From the Lloyds’ report: “Sustained loss of power could mean that society reverts to nineteenth century practices. Severe space weather events that could cause such a major impact may be rare, but they are nonetheless a risk and cannot be completely discounted.”

The investigation of extreme space weather has broadened as new windows—historical cosmogenic nuclide events (Miyake et al. 2012) and Kepler observations of flares on Sun-like stars (Maehara et al. 2012)—were opened. In this review, we trace the evolution of research on extreme solar activity and review work on the limits of the various types of extreme space weather and their occurrence probabilities.

1.1 1859: in the beginning

Richard Carrington, the accomplished nineteenth century English astronomer (Cliver and Keer 2012), described his discovery of the first solar flare—on 1 September 1859—in a brief Monthly Notices paper that is a mixture of scientific rigor (e.g., avoiding any correspondence with Hodgson, who also observed the flare, to maintain the independence of their accounts), excitement (“being somewhat flurried by the surprise, I hastily ran to call someone to witness the exhibition with me, and on returning within 60 s, was mortified to find that it [the flare] was already much changed and enfeebled”), and caution (“While the contemporary occurrence [of solar flare and geomagnetic disturbance] may deserve noting, he [Carrington] would not have it supposed that he even leans toward hastily connecting them. ‘One swallow does not make a summer.’”). From his detailed account of the "singular appearance seen in the Sun", it seems clear that Carrington knew that the transient bright emission patches he observed in the large spot group near central meridian (Fig. 1) represented a new and important solar phenomenon. What he could not know was that the “sudden conflagration” he observed on the Sun and the ensuing geomagnetic storm were historically large—possibly the largest that have been directly observed (Cliver and Dietrich 2013). It was recognized at the time that the 1859 magnetic storm was strong, but just how strong was hard to say. It was accompanied by a widespread aurora (Loomis 1859, 1860, 1861) and the magnetometers at Greenwich (east of London) and Kew (west), both near to Carrington’s observatory in Reigate (south of London) and Hodgson’s in Highgate (north), were driven off-scale, but systematic magnetic records only dated from the 1830s (Chapman and Bartels 1940) and there was little basis for comparison.

Fig. 1
figure 1

Image reproduced with permission from Cliver and Keer (2012), copyright by Springer; the enlarged portion of RGO 67/266 in (B) is reproduced by kind permission of the Syndics of Cambridge University Library

a Carrington’s (1859) carefully executed drawing of his sunspot group 520 on 1 September 1859, the first visual record of a solar flare. The initial (A and B) and final (C and D) positions of the white-light emission are indicated. Solar east is to the right. b Enlargement of region 520 from an early solar photograph made at Kew Observatory by Warren De la Rue on 31 August 1859.

1.2 2003–2004: surge in interest in extreme events

The establishment of the extreme strength of the Carrington storm awaited the study by Tsurutani et al. (2003) who analyzed long-buried geomagnetic observations of the event from Colaba Observatory where the automatic recording that led to off-scale readings at Greenwich and Kew had not yet been instituted. The manual observations at ~ 10-min intervals at Bombay indicated a storm three times as intense as any that has been observed since. While the three-times assessment is likely an overestimate (Akasofu and Kamide 2005; Siscoe et al. 2006; Cliver and Dietrich 2013), the 1859 storm remains among the strongest events ever observed (Hayakawa et al. 2020c, 2022). The year 2003 was ripe for a renewed appreciation of the intensity of the Carrington storm. The discipline of space weather was becoming established and interest in long-term space weather, sparked by Eddy’s (1976) rediscovery of the Maunder Minimum, was fueled by the concern of global warming. The journal Space Weather began publication in 2003 (Lanzerotti 2003) and a series of Space Climate Symposia (Mursula et al. 2004) was inaugurated in 2004. In this context, the Tsurutani et al. (2003) paper became a seminal paper for research on extreme solar activity.

The NASA Astrophysics Data System (ADS) annual citation rate for Carrington's Monthly Notices paper on the 1859 event (Fig. 2) shows the recent growth of interest in the first recognized extreme solar event and in extreme events generally. The citation spike in 2006 reflects the publication in Advances in Space Research of the proceedings of a workshop on the Carrington event at the University of Michigan in late 2003 (Clauer and Siscoe 2006).

Fig. 2
figure 2

Recent growth (through 2020) in citations to Carrington's (1859) Monthly Notices paper on the discovery of the 1859 flare

In 2004, Cliver and Svalgaard presented a paper at the first Space Climate Symposium that listed the observed/inferred extreme values of: the peak intensity of solar flares (based on soft X-ray observations and magnetic crochet amplitudes); Sun-Earth disturbance transit time (a proxy for the average speed of coronal mass ejections (CMEs)); solar proton event amplitude, inferred—erroneously, it now appears (Wolff et al. 2012; Sukhodolov et al. 2017)—from nitrate concentrations in ice cores; geomagnetic storm intensity; and low-latitude auroral extent. The limiting cases of the various types of events they considered are analogous to the 100-year floods or hurricanes of terrestrial climate. Note the inclusion of the CME (Webb and Howard 2012), intermediary to flare and storm, in this list of space weather phenomena. CMEs are a relatively new aspect of solar activity first directly observed in the early 1970s (Koomen et al. 1974, and references therein; Gopalswamy 2016) that were pointedly brought to the forefront in solar-terrestrial physics by Gosling (1993) with an important precursor in Kahler (1992).

1.3 2012: new windows

The year 2012 witnessed significant advances, based on disparate data sets, in the study of extreme solar activity. Miyake et al. (2012) obtained high-time-resolution measurements of the 14C concentration in the tree rings of two Japanese cedar trees that showed a transient increase of ~ 12‰ from 774 to 775 AD (Fig. 3). The origin of the 774–775 AD 14C enhancement was immediately a matter of debate. Was it caused by a very large solar energetic proton (SEP) event (or a closely-spaced cluster of high-energy proton events such as occurred from August to October 1989) (Usoskin et al. 2013) or by a galactic gamma-ray burst (GRB; Hambaryan and Neuhäuser 2013; Pavlov et al. 2013a, b)? Since 2012, numerous studies, including a multi-isotope investigation (Mekhaldi et al. 2015) of the 774–775 AD cosmogenic-nuclide event and a similar event in 993–994 AD discovered by Miyake et al. (2013), have provided strong evidence for the solar scenario (Miyake et al. 2020a).

Fig. 3
figure 3

Image reproduced with permission from Miyake et al. (2012), copyright by Macmillan

Measured 14C content at 1–2-year time resolution for two Japanese cedar trees showing the cosmogenic nuclide event of 774–775 AD.

Potential evidence far from the Sun bearing on the limits of the strength of solar flares was provided by the precise, long-term, and continuous photometry of stars by the Kepler satellite (Koch et al. 2010) that had the search for extra-solar planets as its primary task. For the first 120 days of the Kepler mission, Maehara et al. (2012) reported the observations of 14 superflares (with energy > 1034 erg) on 14,000 Sun-like stars slowly-rotating G-type main-sequence stars with surface temperatures of 5600–6000 K). They calculated that a flare with bolometric energy > 1034 erg could occur on a Sun-like star once every 800 years. The question of whether the Sun in its present state is capable of producing a flare of this size remains unsettled (Aulanier et al. 2013; Shibata et al. 2013; Tschernitz et al. 2018; Schmieder 2018; Notsu et al. 2019; Okamoto et al. 2021). In a detailed survey of 38 large eruptive (i.e., CME-associated) solar flares from 2002 to 2006, Emslie et al. (2012) found that the X28 soft X-ray flare on 4 November 2003 had the largest radiative energy (4.3 × 1032 erg), while the X17 flare on 28 October 2003 had the largest total (radiative plus CME kinetic energy) energy (1.6 × 1033 erg). The corresponding inferred values for the Carrington flare are essentially identical to these estimates: ~ 5 × 1032 erg (bolometric) and ~ 2 × 1033 erg (total) (Cliver and Dietrich 2013).

There was a further important development in 2012. On 23 July of that year a major backside eruption on the Sun was observed both remotely and in situ by the STEREO spacecraft (Kaiser et al. 2008). Work by several teams of investigators (Russell et al. 2013; Baker et al. 2013; Liu et al. 2014a, b; Riley et al. 2016; cf. Ngwira et al. 2013) indicated that had the eruption occurred on the front side of the Sun, it might have produced a magnetic storm greater than that inferred for the Carrington event.

1.4 Subsequent developments

Hayakawa et al. (2017) presented evidence for a space weather event in 1770 that rivaled or exceeded aspects of the 1859 event. Aurorae from 16 to 18 September 1770 were observed at geomagnetic latitudes as low as ~ 20° in both the southern and northern hemispheres, comparable to those following the 1859 event, and the estimated area of the likely associated sunspot region (from a contemporary drawing) was ~ 6000 millionths of a solar hemisphere, approximately three times that of the mean area for the source region of the Carrington flare (Newton 1943; Jones 1955). Similarly, Love et al. (2019c) deduced a minimum Dst intensity for the 15 May 1921 geomagnetic storm that equaled (within uncertainties) that inferred for the Carrington storm. Other developments include the Knipp et al. (2016) review of the notable May 1967 space weather event which drew attention to an aspect of extreme space weather that deserves increased attention—extreme radio bursts that pose a threat to radar operations and radio communications (e.g., Cerruti et al. 2008)—and the discovery and verification of a third historical cosmogenic nuclide event in ~ 660 BC (Park et al. 2017; O’Hare et al. 2019) that was comparable to the 774–775 AD event. More recently, Cliver et al. (2020b) inferred a bolometric energy of ~ 2 × 1033 erg for the flare associated with the 774 AD proton event and Reinhold et al. (2020) presented evidence suggesting that the Sun is currently in a state of subdued activity relative to its stellar peers.

1.5 Related work

Previous reviews on extreme events have been published by Riley (2012), Schrijver et al. (2012), Hudson (2015, 2021), Riley et al. (2018), Gopalswamy (2018) and Hapgood et al. (2021). Here, in addition to the phenomena of solar flares, CMEs, geomagnetic storms, and low-energy proton events, we consider sunspot groups, flares on Sun-like stars, solar radio bursts, fast transit interplanetary coronal mass ejections (ICMEs), low-latitude aurorae, and high-energy proton events that give rise to cosmogenic nuclide enhancements—topics that were not included or were more lightly treated in the reviews of extreme events listed above. We do not, however, consider the effects of extreme solar events on the ionosphere (e.g., sudden ionospheric disturbances and polar cap absorption events), atmosphere (e.g., ozone depletion) and lithosphere (e.g., geomagnetically induced currents), all of which are addressed by Riley et al. (2018). The emphasis in Hapgood et al. (2021) is on the terrestrial impacts of extreme solar activity. Recent reviews by Tsurutani et al. (2020) and Temmer (2021) focuses on space weather generally. Although the rarity of extreme events makes their footing less certain, there is evidence for certain of the phenomena we consider that the physics of extreme space weather events can differ from that in events which are merely large.

Our focus will be on the largest directly observed, inferred, and theoretically derived values of size/intensity measures of the various types of solar emissions and of geomagnetic storms. For solar flares, geomagnetic storms, and solar proton events, long-term indirect observations are provided by magnetograms (via magnetic crochets), auroral records, and cosmogenic nuclide data, respectively. We compile lists of the largest observed events in each category for comparison with future events and tabulate estimates of 100- and 1000-year events based on occurrence probability distribution functions.

In Sects. 27 we consider the different types of extreme activity, in turn, and in Sect. 8 we present and discuss a summary table of extreme events.

2 Sunspot groups

2.1 100- and 1000-year spot groups

The primary data base used to estimate the areas of 100-year and 1000-year solar spot groups is that compiled at the Royal Greenwich Observatory from 1874 to 1976 (RGO; Willis et al. 2013a, b; Erwin et al. 2013) and extended to the present using data from the US Air Force’s Solar Observing Optical Network (SOON) and other observatories (Muñoz-Jaramillo et al. 2015; Giersch et al. 2018; Mandal et al. 2020).Footnote 2 Such estimates begin with histograms of projection-corrected spot group areas (measured in millionths of a solar hemisphere (μsh; 1 μsh = 3.0 × 106 km2)). Various functional forms have been used to fit the size distribution of spot groups, dependent in part on the data sets and time intervals considered.

Bogdan et al. (1988), Baumann and Solanki (2005), and Leuzzi et al. (2018) used lognormal functions to fit distributions of group spot areas. Baumann and Solanki considered the RGO data set from 1874 to 1976, restricting their analysis to groups within ± 30° from central meridian, to minimize the errors resulting from visibility corrections. They only considered groups with umbral areas > 15 μsh and total areas > 60 μsh in their fits because of intrinsic measurement errors and distortion due to seeing for small groups. Their lognormal number density functions for the maximum (peak value observed during disk passage) and instantaneous (all daily observations), umbral and total (umbral plus penumbral), group areas are shown in Fig. 4. In Fig. 4b, the curves for maximum and snapshot total spot areas are essentially identical.

Fig. 4
figure 4

Image reproduced with permission from Baumann and Solanki (2005), copyright by ESO

a Size distribution of maximum (circles) and instantaneous (crosses) sunspot group umbral areas. b Same as a for total group areas. The log-normal fits are over-plotted. The vertical lines indicate the smallest areas considered for the fits.

Double or bi-lognormal functions have also been used to fit distributions of spot group areas (Kuklin 1980; Nagovitsyn et al. 2012, 2017; Nagovitsyn and Pevtsov et al. 2021). Nagovitsyn and Pevtsov considered the maximum total areas of groups observed by Greenwich from 1874 to 1976 and from Kislovodsk Mountain Astronomical Station (KMAS) from 1977 to 2018. The crossover point for the two lognormal distributions of Nagovitsyn and Pevtsov corresponds to the 60 μsh lower limit of the area range that Baumann and Solanki considered for their single lognormal distribution. From analysis of instantaneous group areas from multiple data sets including RGO and KMAS, Muñoz-Jaramillo et al. (2015) concluded that while the larger spot groups had a lognormal distribution, the smaller groups were better represented by a Weibull (1951) function (cf. Nagovitsyn and Pevtsov 2021).

Figure 5 shows a comparison made by Muñoz-Jaramillo et al. (2015) of fits to the instantaneous total group sunspot number area from RGO for four different functions: lognormal, power law, exponential, and Weibull. For the five data sets Muñoz-Jaramillo et al. considered (RGO, KMAS, SOON, Pulkovo Observatory (Nagovitsyn et al. 2008), and the Heliospheric and Magnetic Imager (HMI; Scherrer et al. 2012) on the Solar Dynamics Observatory (SDO; Pesnell et al. 2012), the best fit was provided by the Weibull function distribution in each case although it only passed the Kolomogorov-Smirnov test for the HMI data set.

Fig. 5
figure 5

Image reproduced with permission from Muñoz-Jaramillo et al. (2015), copyright by AAS

Distribution of the instantaneous total group sunspot area from RGO, fitted by four different functions. The dark-gray shaded area indicates the range over which the fits were made.

Figure 6 shows a downward cumulative distribution of spot group areas (left hand axis) from Gopalswamy et al. (2018) for a data set consisting of observed daily whole sunspot areas for spot groups from 1874 to 2016 (based on RGO data from 1874 to 1976 and SOON data after 1976). The blue curve is a modified exponential function to the annualized distribution on the right hand axis

$$ Y = a \left(1 - {\text{exp}}\left[ { - \left( {\frac{ - X + b}{c}} \right)} \right] \right ) $$
Fig. 6
figure 6

Downward cumulative distribution (left hand axis) of the number of solar spot groups from 1874 to 2016 with instantaneously-measured total areas greater than a given value A (black circle data points). This annualized distribution (right hand axis) is fitted with a modified exponential function (solid blue line) and power-law (dashed red line; for the tail of the distribution) to give the occurrence frequency distribution (OFD). The fit parameters to Eq. (1; exponential) and Eq. (2; power law) are given in the figure. The intersections of the dashed horizontal lines with the fitted curves give the areas of the 100-year and 1000-year spot groups. Adapted from Gopalswamy (2018)

with a normalization factor (a), in addition to location (b) and scale (c) parameters, that gives the occurrence frequency distribution (OFD).Footnote 3 The scale factor (c) reflects the spread of the distribution in the X-parameter. The data are fitted to this function by making an initial guess of the three parameters and using an IDL routine called MPFIT (e.g., https://pages.physics.wisc.edu/~craigm/idl/fitting.html) (Gopalswamy, personal communication, 2021) to determine the best-fit values.

Gopalswamy (2018) also obtained a power-law fit (Newman 2005) for the tail of the distribution (dashed red line),

$$ Y = \alpha - \beta X $$

with the minimum X-value determined by the maximum-likelihood estimator (MLE; Clauset et al. 2009). The parameters for both the exponential and power-law fits are given in the figure. In both Eqs. (1) and (2), X is the log of the spot group area and Y is the log of the number of number of groups per year for this area.

Equations (1) and (2) define occurrence frequency distributions (OFDs) that give the probability of a spot group with area A ≥ a given value occurring during one year. The intersection of the exponential and power-law fits with the dashed horizontal lines drawn from 10−2 and 10−3 on the right-hand y-axis indicate the spot areas of 100-year and 1000-year spot groups. The exponential fit indicates a maximum 100-year spot area of 5780 μsh sunspot that is comparable to the largest (corrected for foreshortening) whole sunspot area of 6132 μsh for Greenwich sunspot group 14886 on 8 April 1947 (Newton 1955). The corresponding estimate for a 1000-year sunspot is 8200 μsh. The power-law fit yields 100-year and 1000-year estimates for maximum group areas of 7100 and 13,600 μsh.

From the above it is clear that the choice of a function to fit a size distribution is not straightforward and the form used will affect the 100-year and 1000-year estimates obtained. For this reason we adopt a conservative empirical approach that favors the modified exponential function of Gopalswamy (2018) for several of the phenomena considered below; with its three free parameters, this function generally fits distributions well over their full parameter range, as in Fig. 6. Because power-law functions are commonly used for flare X-ray and radio distributions (e.g., Aschwanden 2014), we consider them as well, comparing exponential and power-law 100-year and 1000-year event estimates when both functions are available.

2.2 Sunspot groups versus active regions

To date, sunspot group area is the parameter most commonly used to make estimates of the largest possible solar flare (as discussed in Sect. 3.1.7 below). This is likely due to the ready availability of the digital RGO record but there is no guarantee that group spot area is the optimum parameter from which to determine the limiting energy of extreme flares. The entire magnetic active region (AR) determines field configuration, although spots dominate where the (free) energy can be stored. Harvey and Zwaan (1993) considered AR size at maximum development and found a power law with slope close to − 2 for the distribution, without a clear indication of turnover at the largest sizes. It may well be significant that sunspot groups (the strong core fields of active regions) have a lognormal distribution while AR sizes that include more dispersed peripheries have a power law.

3 Flares on the sun and sun-like stars

3.1 Solar flares

3.1.1 Solar flare soft X-ray burst classification

The current standard measure for solar flare intensity is the widely-used Geostationary Environmental Satellite System (GOES) ABCMX SXR classification system which is defined as follows: SXR classes A1-9 through X1-9 correspond to flare peak 1–8 Å fluxes of 1–9 × 10−n W m−2 where n = 8, 7, 6, 5, and 4, for classes A, B, C, M, and X, respectively. Occasionally, flares are observed which peak intensities ≥ 10−3 W m−2. Rather than being assigned a separate letter designation, such flares are referred to as X10 events and above.Footnote 4

Approximately 20 flares of class X10 or higher have been observed during the last ~ 40 years.Footnote 5 The largest GOES SXR flare yet recorded occurred on 4 November 2003. The 1–8 Å detector saturated at a level of X18.4, with an estimated SXR class of X35 ± 5 (Cliver and Dietrich 2013) based on consideration of values given in Kiplinger and Garcia (2004), Thomson et al. (2004, 2005), Brodrick et al. (2005), and Tranquille et al. (2009). The bulk of the SXR class estimates for this flare were obtained via comparisons of flare SXR intensities and the amplitudes of flare-associated sudden ionospheric disturbances (SIDs; Mitra 1974; Prölls 2004; Tsurutani et al. 2009) caused by energetic flare photons leading to increased electrical conductivity in the day-side ionosphere, e.g., the magnetic crochet observed for the 1859 event. Cliver and Dietrich (2013) obtained an estimate for the Carrington flare of X45 ± 5 based on the magnetic crochet (a type of SID recorded by ground-based magnetometers) observed for this event (Cliver and Svalgaard 2004; Boteler 2006; Clarke et al. 2010). The November 2003 and September 1859 flares provide the current benchmarks for extreme flare activity. Less certain is an estimate of X285 ± 140 (Cliver et al. 2020b: Sect. 7.8 below) for the flare associated with the inferred SEP event of 774 AD (Miyake et al. 2012; Usoskin et al. 2013). Because flares such as those observed/inferred in 1859 and 774 AD are rare, we need to look at the ensemble of lesser flares observed by the GOES system since 1976 to estimate the occurrence frequency of extreme flares of this size and larger.

3.1.2 Solar flare frequency distributions

The GOES 1−8 Å soft X-ray measurements provide the longest, uninterrupted, and uniformly calibrated data set available for solar flares. Thus these records provide the basis for statistical studies that seek to establish the frequency of occurrence of flares as a function of flare magnitude. The most often used magnitude metric is the peak intensity of the flare as measured in that passband (expressed in W m−2), i.e., the ABCMX GOES classification.

Flares of classes A and B are under-reported in the GOES records because during active phases of the solar cycle such faint flares are often hard to separate from (or detect above) the supposedly quiescent background and are judged to be of low interest. As a further complication, flare magnitude is often not corrected for background emission, so that the weak flares that are reported are intrinsically overestimated in their strength. The least observationally biased records are those for flares of classes C, M, and X.

Many studies over the past decades have established that, at least for the larger flares, the frequency distribution for peak brightness of flares is well approximated by a power law (a thorough review in the literature of power-law fits to flare strength parameters can be found in Aschwanden et al. (2016) who also discuss waiting time distributions). The first such distribution listed by Aschwanden is that of Akabane (1956) who presented a distribution for burst peak radio intensities at 3.0 GHz. Subsequently, Hudson et al. (1969) showed a frequency histogram based on solar hard X-rays as observed by the Orbiting Solar Observatory (OSO-3) satellite that, with hindsight, hints at a power-law distribution for the more intense flares, although it lacks an explicit power-law fit to the data. Such an approximation was made by Drake (1971) based on soft X-ray measurements by the Explorer 33 and 35 spacecraft. Similar power-law distributions were first reported a few years earlier for UV Ceti type flare stars (very cool main sequence stars that exhibit frequent flaring; e.g., Kunkel 1968; Gershberg 1972; Lacy et al. 1976; Kowalski et al. 2013).

Rosner and Vaiana (1978) suggested that one way to create such frequency distributions would be to have a system in which exponential growth in stored energy is interrupted at random times by a flare-like event in which a substantial amount of the stored energy is removed from the system. But if such an energy build-up would occur in regions on the solar surface, a correlation between flare brightness and waiting time would be expected. The absence of such a correlation in both solar observations (Aschwanden et al. 1998) and stellar data (as already noted by, e.g., Lacy et al. 1976) led to other ideas, including that of self-organized criticality (SOC; Bak et al. 1987, 1988; applied to solar flares by Lu and Hamilton 1991). The SOC model for solar flares is based on the assumption that flares occur as a time-varying, or non-stationary, Poisson process. We point the reader to Aschwanden et al. (1998, 2016) for descriptions of the developments of such interpretations, focusing here on the distribution functions rather than the processes behind them.

In an extensive study of ~ 50,000 GOES SXR bursts observed over a 25-year period from 1976 through 2000–38,000 of which were in classes C-X, Veronig et al. (2002a) obtained well-defined fluence and peak flux distributions that could be fitted by power laws with similar exponents: − 2.03 ± 0.09 and − 2.11 ± 0.13, respectively. These power-law approximations to the frequency distribution hold for over 2.5 orders of magnitude, with no significant indication of a change in behavior up to the largest flares observed. At a flare magnitude bin at ~ X15 (based on only two flares), the histogram simply aborts at the end of a straight power law (Fig. 7).

Fig. 7
figure 7

Image reproduced with permission from Veronig et al. (2002a), copyright by ESO

Frequency distribution of flare 1–8 Å peak flux with least squares fit for GOES SXR flares from 1976 to 2000.

The work by Gopalswamy (2018) includes a comparable analysis of flare magnitudes over a 47-year period from 1969 through 2016, including 55,285 flares of class C1 or larger (it did not consider the ~ 11,500 B class flares included in the Veronig et al. 2002a, sample). This larger sample was obtained over a time base that almost doubles that of Veronig et al. (2002a) by including more recent data and also extending the GOES records with Solar Radiation (SOLRAD) satellite data for the period 1969–1975. The downward-cumulative representation of the frequency distribution for that data set (Fig. 8) exhibits a deviation from a power-law distribution (see also, e.g., Riley 2012). In a log–log downward cumulative representation, an approximation by power laws suggests a break towards less-frequent larger flares somewhere around X4, defined by a total of about 100 flares at that magnitude or larger. Such a downward break is consistent with the conclusions reached by Schrijver et al. (2012).

Fig. 8
figure 8

Downward cumulative distribution (left hand axis) of the number of flares from 1969 to 2016 with peak SXR fluxes greater than a given value F (blue diamond and red circle data points). This annualized distribution (right hand axis) is fitted with a modified exponential function (solid blue line) and power law (dashed red line; for the tail of the distribution (red circle data points)) to give the occurrence frequency distribution. The fit equations and parameters are given in the figure. Adapted from Gopalswamy (2018)

The century- and millennium-level flares based on the modified exponential function fit to the SXR peak values in Fig. 8 are X44 and X101, respectively. The corresponding bolometric energy values from Gopalswamy (2018) are ~ 4 × 1032 erg and ~ 1033 erg. The 100- and 1000-year estimates based on power laws are comparable (X42 and X115; ~ 4 × 1032 erg and ~ 1.2 × 1033 erg). Both functional forms fit the tail of the distribution well. For a 10,000-year flare, the exponential function yields ~ X200 versus ~ X310 for the power-law fit.

While the radiative energy of the strongest observed solar flare is often given variously as 1032 erg, about 1032 erg, or ~ 1032 erg (e.g., Candelaresi et al. 2014; Peter et al. 2014; Osten and Wolk 2015; Hudson 2016; Maehara et al. 2017; Notsu et al. 2019; Brasseur et al. 2019), the measured value is a half-decade larger. Emslie et al. (2012) reported that the inferred/observed radiative energies of 12 events from 2002 to 2006 were above the 1032 erg threshold. For the three largest such events (4 November 2003, 4.3 × 1032 erg, X35 ± 5; 28 October 2003, 3.6 × 1032 erg, X17; 7 September 2005, 3.2 × 1032 erg, X17), the bolometric energy was measured directly by the Total Irradiance Monitor (TIM; Kopp and Lawrence 2005) on the Solar Radiation and Climate Experiment (SORCE; Rottman 2005). Thus, the largest observed (November 2003) and inferred (September 1859; ~ 5 × 1032 erg, ensemble estimate of X45 ± 5; Cliver and Dietrich 2013; based on electromagnetic emissions) flares are comparable to the 100-year flare given by Fig. 8.

3.1.3 White-light flares

The flare Carrington and Hodgson independently recorded on 1 September 1859 was observed in integrated light. Thus their Monthly Notices papers were the first reports of what came to be known as “white light flares” (WLFs). A compilation of reports of such events from (1859–1982) is given in Neidig and Cliver (1983a). At that time, Neidig and Cliver (1983b) reckoned that such flares were relatively rare, with an occurrence frequency of ~ 15 per year based on a ~ 2.5-year period from June 1980 to December 1982 following the maximum of solar cycle 21. They determined that a flare with SXR class ≥ X2 originating in a large (≥ 500 μsh), magnetically complex, sunspot group was a sufficient condition for a white light flare. Two decades later, Hudson et al. (2006) used Transition Region and Coronal Explorer (TRACE; Handy et al. 1999) images, which encompassed a wavelength range from 1500 to 4000 Å, and identified 11 white-light flares that had SXR classes ranging from C1.6 to M9.1 (median = M1.2) to “support the conclusion of Neidig (1989) that white-light continuum occurs in essentially all flares.” Subsequently—based on observations of many flares with the Variability of Solar Irradiance and Gravity Oscillations (VIRGO: Fröhlich et al. 1995) experiment on the Solar and Heliospheric Observatory (SOHO; Domingo et al. 1995)—Kretzschmar (2011) estimated that white-light emission might typically account for ~ 70% of the total radiated flare energy, following an earlier estimate by Neidig (1989) of ≲ 90% based on analysis of the 24 April 1981 white-light flare (Neidig 1983).

White-light flares are of particular importance in studies of extreme flares on Sun-like stars because of the observations of such flares by the high-precision (better than 0.01% for moderately bright stars) visible wavelength (i.e., white-light, 4000–9000 Å) photometer on the Kepler spacecraft (Koch et al. 2010), which operated from 2009 to 2018. The durations of stellar white-light flares can exceed several hours (Maehara et al. 2012) versus typical durations of ≲ 10 min (Neidig and Cliver 1983a) for their largest solar counterparts, although some of this discrepancy may be due to the lower sensitivity of early WLF observations (see Fig. 13 below). Stellar flares detected by Kepler have bolometric energies exceeding 1033 erg (Maehara et al. 2012; Shibayama et al. 2013), exceeding those of the largest solar flares (~ 4 × 1032 erg; Emslie et al. 2012; cf. Kane et al. 2005) by about a factor of three. Whereas the Kepler photometer detects essentially all superflares with amplitudes > 1% of the average stellar flux on solar-type (G-type dwarfs, 5100 K < Teff < 6000 K and log g > 4.0) stars, corresponding to a flare bolometric energy of ~ 5 × 1034 erg (Shibayama et al. 2013), the estimated detection completeness for ~ 1034 and 1033 erg flares are ~ 0.1 and 0.001, respectively (Maehara et al. 2012).

It is well-accepted that the paradigm of the solar flare also holds also for the large (1034–35 erg) stellar flares (Gershberg 2005). In the standard picture of solar flares, a rapid conversion occurs, via magnetic reconnection, of energy stored in the magnetic field to energies distributed over bulk kinetic energy and thermal and non-thermal particle distributions. A significant fraction of this converted energy is eventually released as a pulse in the visible radiation, the observable diagnostic of the Kepler superflares on Sun-like stars. Like solar flares, stellar flares: (1) occur on single “solar-type” stars, viz., G-type main sequence stars with Teff and gravity similar to that of the Sun and rotation periods > 10 days and without hot Jupiters (Maehara et al. 2012; Shibayama et al. 2013; Okamoto et al. 2021; Sect. 3.2.2); (2) show evidence of non-thermal particle populations and of temperatures ≳ 107 K (e.g., Osten et al. 2007; Benz and Güdel 2010); (3) exhibit a characteristic fast rise and slower, near-exponential, decay in X-rays (Kahler et al. 1982; Haisch et al. 1983); and (4) have lower-energy emissions that often scale with the time integral of their high-energy emissions (equivalent to the Neupert effect for solar flares; Hawley et al. 1995; Güdel 2002; Osten et al. 2004; see Sect. 3.1.4). The above evidence for the commonality of stellar flare characteristics and processes with those that occur during smaller solar flares implies that the statistics of energetic flares on a multitude of Sun-like stars (see Sect. 3.2) can provide information on extreme solar flares that may occur only once per millennium or even less frequently (Notsu et al. 2019; Okamoto et al. 2021).

3.1.4 Impulsive and gradual phases of flares: The Neupert effect

The separation of flare emission into a fast-rise impulsive phase followed by a slowly decaying gradual phase has long been noted in multiple wavelengths (Fletcher et al. 2011). The impulsive phase of flares (Dennis and Schwartz 1989; Benz 2017), characterized by non-thermal hard X-ray and radio bursts, marks the time of the early principal energy release in a flare. See Fletcher et al. (2011) and Benz (2017) for detailed reviews of flare observations.

In a prescient six-page paper based on early flare soft X-ray data for three flares, Neupert (1968) found that the integral of impulsive phase microwave emission in each flare resembled the rise of the SXR light curve. He hypothesized that collisional losses by the energetic electrons responsible for the microwave burst heated the chromosphere to sufficient temperatures to eject plasma into the low corona where its cooling was manifested by the slow decay of thermal SXR emission during the flare gradual phase.

The support structure for Neupert’s conjecture on the conversion of non-thermal energy to thermal energy in a solar flare lay in the future. It consisted of: the thick target model of hard X-ray emission (Brown 1971; Kane and Donnelly 1971), the CSHKP model for eruptive solar flares (Carmichael 1964; Sturrock 1968; Hirayama 1974; Kopp and Pneuman 1976; Hudson 2021), and the establishment of chromospheric evaporation (Antonucci et al. 1984; Fisher et al. 1985). In addition, evidence that white light emission is powered by energetic electrons accelerated during the flare impulsive phase accumulated (Hudson 1972; Machado and Rust 1974; Rust and Hegwer 1975; Neidig 1989; Hudson et al. 1992; Neidig and Kane, 1993). Figure 9 (taken from Hayes et al. 2016) shows the inverse aspect of the Neupert effect—hard X-ray emission is the derivative of the SXR time profile (Dennis and Zarro 1993)—using modern data.

Fig. 9
figure 9

Image reproduced with permission from Hayes et al. (2016), copyright by AAS

Illustration of the Neupert (1968) effect in which the time derivative of flare soft X-ray (1–8 Å) emission during the impulsive phase of a flare mimics the flare hard X-ray time profile (Dennis and Zarro 1993). (Top panel) Normalized light curves for different wavelengths/radio frequencies/energies for a flare on 28 October 2013 flare. (Bottom panel) Normalized derivatives of the four SXR channels in the top panel. The red vertical lines bracket the flare impulsive phase.

Figure 10 shows a CSHKP schematic for an eruptive flare that illustrates various elements of the Neupert effect: acceleration of electrons and protons at a neutral current sheet, propagation of electrons to the low atmosphere giving rise to hard X-ray and microwave emission and heating of the chromosphere via particle bombardment, evaporation of the heated plasma to fill SXR emitting loops, and subsequent retraction and cooling of these loops as manifested by the appearance of post-flare Hα loops (Švestka et al. 1987).

Fig. 10
figure 10

CSHKP schematic showing the global structure of an eruptive solar flare and the major energy conversion via reconnection viewed in cross section along the magnetic neutral line. Adapted from Martens and Kuin (1989); with input from Anzer and Pneuman (1982), Howard et al. (2017), and Veronig et al. (2018)

According to the Neupert effect, the impulsive phase of flares can be interpreted as the time interval during which magnetic reconnection occurs. In the two-dimensional schematic in Fig. 10, this phase would end when the last closed loop of the CME is disconnected from the pinched off lower loop system. In theory, the SXR intensity at this time would be at it or near its maximum. In reality, Veronig et al. (2002b) found that this coincidence or near-coincidence of the end of the impulsive phase with SXR maximum was observed for only about 50% of their sample of ~ 1100 C2-X4 flares. In ~ 25% of the cases, the SXR emission peak occurred after the impulsive phase hard X-ray emission, and another ~ 25% of cases were unclear. For a much smaller subset of 66 large impulsive hard X-ray flares, Dennis and Zarro (1993) found that 80% of the events were consistent with the Neupert effect, suggesting that a relationship between SXR emission and white light emission can be valid for extreme events.

The significance of the Neupert effect for this review is two-fold: (1) it provides a link between the widely used (and more readily available) SXR intensity as a measure of flare size and the physically more significant measure of flare white-light energy which dominates the radiative energy budget of flares; and (2) as will be seen in Sect. 3.2, it allows flares on Sun-like stars to be directly compared with solar flares in terms of the standard CMX SXR classification.

The classic Neupert effect suggests that magnetic reconnection, particle acceleration, and post-flare loop formation, are confined to the flare impulsive phase, a useful generalization. Subsequently, Zhang et al. (2001) demonstrated that the SXR rise phase corresponded to the interval of CME acceleration. That said, magnetic reconnection, particle acceleration, and loop formation can continue for up to ≳ 10 h (e.g., Bruzek 1964; Kahler 1977; Akimov et al. 1996; Gallagher et al. 2002; Tripathi et al. 2004), well beyond the impulsive phase, as can particle acceleration by CME-driven shocks. Moreover, these late phase phenomena can give rise to strong, and even extreme (e.g., Frost and Dennis 1971; Chupp et al. 1987; Gary 2008; Hudson 2018) solar phenomena. Time profiles of flare emission at various frequencies/energies are given in Fig. 11. The hard and soft X-ray traces during the impulsive phase in panels (b) and (c) display the classic Neupert effect, which applies for the majority of flares, particularly the smaller confined events, but also for eruptive events with late phase reconnection too weak (or too high in the corona) to significantly affect the SXR time profile in panel (c)—consistent with the radiative energy domination of the impulsive phase. As shown in panels (b), (d), and (e), and discussed in Sect. 4, electrons accelerated in certain of these eruptive events can result in gradual hard X-ray and microwave bursts as well as intense decimetric bursts at ~ 1 GHz, with peak flux densities as large as 105–106 solar flux units.

Fig. 11
figure 11

Schematic of time profiles of flare emissions at various wavelengths and energy ranges. The meter wavelength range (m–λ; 30–300 MHz in panel a) exhibits various spectral types of which fast drift type III bursts are a characteristic emission of the impulsive phase and slow-drift type II bursts are the defining emission of the second phase of fully developed radio events (McLean and Labrum 1985). The various time profiles shown have been simplified to emphasize the separation of the delayed non-thermal emissions both from the impulsive phase and from each other. The delayed electron-generated emissions in panels b, d, and e are attributed to late-phase reconnection in post flare loop systems while the delayed > 100 MeV γ-ray emission in panel f is attributed to shock-accelerated high-energy protons that precipitate to the photosphere. These extended phase emissions have little effect on the flare SXR time profile (panel c)

Late phase particle acceleration in eruptive flares can also manifest itself in remotely-sensed high-energy solar γ-ray emission. When first observed in the 1980s and early 1990s, prolonged 100 MeV γ-ray emission, attributed to the decay of neutral pions which require acceleration of protons to ≳ 300 MeV energies—the highest that can be inferred from γ-ray observations—for their production, contrasted sharply with the Neupert effect in which flare energy degrades over time from non-thermal X-ray and radio emissions to thermal soft X-rays. Rather than flare electromagnetic emission becoming progressively less energetic after the impulsive phase, 100 MeV γ-ray emission characteristically occurs during the late phase of flares (Share et al. 2018). The most economical (Occam’s razor) explanation for acceleration of the energetic protons responsible for such emission is that they are accelerated by the same coronal shocks (manifested by the slow-drift metric type II burst in panel (a)) that give rise to the solar energetic particles detected by spacecraft near Earth (Sect. 7).

3.1.5 Energetics of flares and CMEs

1−8 Å soft X-ray and broad-band visible light emissions are but two of the channels into which the processes related to solar flares deposit energy. Magnetic energy is converted not only into photon emission in other channels, but also into kinetic energy of non-thermal particle populations, and the bulk kinetic energy associated with the motion of material ejected from the flaring active region. Specific electromagnetic, plasma, and particle aspects of solar eruptions will be considered in following sections of this review.

Figure 12 (taken from Emslie et al. 2012) shows the various pathways by which energy is converted and released during eruptive solar flares. Emslie et al. analyzed wide-ranging data for a sample of 38 M- and, mostly, X-class flares. Figure 12 summarizes their findings for six well-studied X-class flares. All of the listed phenomena in the figure derive their energy ultimately from the free (or excess) magnetic energy of an active region (Efm), defined to be the non-potential energy beyond (or in excess of) that obtained from a potential field model. Emslie et al. (2012) estimated that, on average, the amount of Efm of an active region was equal to 30% of the modelled potential energy determined from line-of-sight magnetograms. For the six events upon which Fig. 12 is based, with an average flare magnitude of X6, the median Efm was ~ 15 × 1032 erg. Emslie et al. estimated that ~ 30% of the free energy in an active region was released in an eruptive event. This, in turn, implies that the energy released in large flares amounts to ~ 10% of the magnetic energy in an active region (see also Shibata et al. 2013). Thus, it is not surprising that magnetic field changes between before and after a flare are hard to spot unambiguously in the overall, always-evolving, multi-thermal coronal configuration. In contrast, rapid changes associated with flares are detected in the photospheric field, most readily in association with SXR flares of class M or X (e.g., Wang et al. 2002; Liu et al. 2005; Sudol and Harvey 2005; Castellanos Durán et al. 2018). The search for such changes was lengthy, however, beginning with Carrington (1859) who looked unsuccessfully following the 1 September 1859 flare (Fig. 1) for any changes in the sunspot group based on the sketch he had made prior to the flare.

Fig. 12
figure 12

Image reproduced with permission from Emslie et al. (2012), copyright by AAS

Bar chart showing the free magnetic energy (denoted by the bottom bar) contained in various energy sinks (with their standard deviations in logarithmic values) for six major eruptive flares (X2.5, X3.0, X3.8, X7.1, X8.3, X10; averaging to X5.8). “Magnetic energy” is the assumed free energy, taken to be 30% of the estimated energy in a potential field over the active region based on observed line-of-sight magnetograms. Energy release is dominated by CMEs and flare electromagnetic radiation.

In keeping with the increased emphasis on CMEs versus flares per se for space weather during the last ~ 25 years, Emslie et al. (2012) find that, on average—for the two dominant terms of energy release (Fig. 12)—CME energy (kinetic plus gravitational potential) is ~ 3 times larger than the flare bolometric energy. Aschwanden et al. (2017) performed an analogous analysis to that of Emslie et al. (2012) for a sample of 157 M- and X-class eruptive flares observed from 2002–2006 and found that the ratio of CME to bolometric energy was ~ 1:1 versus ~ 3:1 determined by Emslie et al. (2012). These different samples preclude a straightforward direct comparison. X-class flares constituted 76% of the Emslie et al. (2012) sample with a median of X2 versus < 10% of that for Aschwanden et al. (2017; median < M2). As this review focuses on the largest, most energetic events, we use the Emslie et al. result.

This ~ 3:1 ratio CME to flare energy obtained by Emslie et al. is presumably an underestimate because the magnetic energy of the CME (Webb et al. 1980) is not taken into account in the determination. In addition, the CME masses and speeds in the SOHO Large Angle Spectroscopic Coronagraph (LASCO; Brueckner et al. 1995) CME data base (https://cdaw.gsfc.nasa.gov/CME_list/; Yashiro et al. 2004; Gopalswamy et al. 2009) used by Emslie et al. are underestimated because of projection effects (Vourlidas et al. 2002, 2010; Vršnak et al. 2007; Paouris et al. 2021). Emslie et al. (2012) write that the mass is underestimated by about a factor of two for CMEs that are associated with flares ≲ 40° from the limb—approximately half of their sample—with greater underestimates for those closer to disk center. Vršnak et al. find that average velocities of non-halo limb-CMEs are 1.5–2 times higher than for such CMEs originating near disk center. This underestimation of CME parameters impacts estimates of the largest possible solar flare (Sect. 3.1.7) that are based on active region energy and flare energy released by reducing the radiative energy budget. For a plausible 9:1 apportionment (vs. 3:1), only 10% of the released energy would go into the flare.

In reference to extreme solar flares, we can safely conclude that they are eruptive. Schrijver (2009) reviewed studies on flares with and without CMEs, including Andrews (2003), Yashiro et al. (2005), and Wang and Zhang (2007), that showed whereas ~ 20% of low to mid C-class flares have associated CMEs, ~ 90% of X-class flares are eruptive. The largest reported flares that lacked CMEs were an X3.1 flare on 24 October 2014 from NOAA spot group 12192 that produced several confined X-class flares (e.g., Thalmann et al. 2015; Liu et al. 2016; Green et al. 2018; Gopalswamy 2018), an X3.6 flare on 16 July 2004 (Wang and Zhang 2007), and an X4.0 flare on 9 March 1989 (Feynman and Hundhausen (1994). For flares on active stars, it has been argued (Drake et al. 2016; Alvarado-Gómez et al. 2018; Moschou et al. 2019; Li et al. 2020, 2021) that much stronger magnetic fields than observed on the Sun would not allow free energy to be released via eruption below some threshold. Based on observations of solar CMEs, Li et al. (2021) speculate that for an active region with unsigned flux of 1024 Mx on a solar- type star, the CME association rate for X100 flares would be < 50%. To date, the most intense solar flares yet inferred (viz., 1 September 1859 and 774 AD (Sect. 7.8)) are eruptive by virtue of their terrestrial effects—geomagnetic storm and proton event, respectively—which both imply CME association (Kahler 1992; Gosling 1993).

3.1.6 Empirical relationship between flare total solar irradiance and SXR class

Even though most of the solar flare electromagnetic radiation, or bolometric output, occurs in the visible range, the contrast with the quiescent photosphere is low. Only the larger white-light flares stand out clearly against the photospheric background in high-resolution images while only the most extreme flares such as 28 October or 4 November 2003 can be recognized as spikes in the record of total solar irradiance (TSI; Woods et al. 2004, 2006; Kretzschmar 2011). As shown in Fig. 13 (from Kopp 2016), the X17 flare on 28 October in the Halloween sequence in 2003 reached a peak brightness in the visible wavelength range of only 0.028% of TSI.

Fig. 13
figure 13

Image reproduced with permission from Kopp (2016), copyright by the author

Comparison of the total solar irradiance (TSI) signal from the X17 flare on 28 October 2003 with a bolometric energy of 3.6 × 1032 erg, and the corresponding scaled GOES 1−8 Å signal.

However, when averaging observations over multiple flares, the signal-to-noise ratio increases. Kretzschmar et al. (2010) and Kretzschmar (2011) used this ensemble technique in a superposed-epoch analysis of observations made with the VIRGO experiment on SOHO to correlate the GOES measured soft X-ray energy (\(\mathcal{F}\) GOES) and bolometric energy of over 2100 solar flares (with GOES classes from C4 up to X17) over an 11-year period. Kretzschmar (2011) presented his results averaged over fairly wide energy ranges in order to have sufficient signal-to-noise. In their analysis of solar and stellar flares, Schrijver et al. (2012) fitted a power-law relationship to these SOHO and GOES measurements to obtain

$$ {\mathcal{F}}_{{{\text{TSI}}}} = 2.4 \times 10^{12} {\mathcal{F}}_{{{\text{GOES}}}}^{0.65 \pm 0.05} , $$

for the conversion from 1 to 8 Å GOES radiated energy to bolometric energy (both expressed in erg) based on flares with bolometric energies ranging from 3.6 × 1030 erg to 5.9 × 1031 erg. The GOES 1−8 Å channel, commonly used to characterize flares and their frequency distribution, captures < 2% of the total radiation emitted by large flares (Woods et al. 2006; Kretzschmar 2011; Emslie et al. 2012).

To go from soft X-ray radiated energy to peak brightness (which sets the GOES flare class) we can use observations of the almost 50,000 flares that were analyzed by Veronig et al. (2002a). They find that the GOES 1−8 Å radiated energy,

$$ {\mathcal{F}}_{{{\text{GOES}}}} = 2.8 \times 10^{29} { }\left( {\frac{{{\mathcal{C}}_{{{\text{GOES}}}} }}{{{\mathcal{C}}_{{{\text{GOES}},{\text{ X}}1}} }}} \right)^{1.10} { ,} $$

with their SXR fluence measurement converted to energy by multiplying by 4π × (1 AU)2 and peak SXR flux scaled to equal unity for an X1 class flare (10−4 W m−2), where 1 AU = 1 astronomical unit (1.5 × 108 km).

Combining Eqs. (3) and (4) leads to a relationship between total radiated energy, \({\mathcal{F}}_{{{\text{TSI}}}}\) (in erg), and the scaled GOES flare class:

$$ {\mathcal{F}}_{{{\text{TSI}}}} = 0.33 \times 10^{32} { }\left( {\frac{{{\mathcal{C}}_{{{\text{GOES}}}} }}{{{\mathcal{C}}_{{{\text{GOES}},{\text{ X}}1}} }}} \right)^{0.72} . $$

Thus values of 1032 (1033) erg correspond to a ~ X5 (~ X115) flare, with uncertainties in the mean relationships as well as flare-to-flare differences resulting in overall uncertainties of easily half an order of magnitude (e.g., Emslie et al. 2012; Benz 2017; Gopalswamy 2018). Because TSI is dominated by white-light emission in the impulsive phase of flares, Eq. (5) can be considered a corollary of the Neupert effect.

3.1.7 Estimates of the largest possible solar flare based on the largest observed sunspot group

(a) Estimates based on reconnection flux

Large flares require large active regions with substantial amounts of magnetic flux near a high-gradient polarity separation line (Schrijver 2007). As noted in Sect. 2.1, the largest sunspot group recorded since systematic area measurements began at RGO in 1874 (Greenwich group 14886) occurred on 8 April 1947 (Fig. 14), with a corrected total spot area of 6132 μsh. Estimates of the total unsigned magnetic flux of such an active region range from 2 to 6 × 1023 Mx (Toriumi et al. 2017; Schrijver et al. 2012). Figure 15 (adapted from Toriumi et al.) shows a plot of unsigned magnetic flux (from spots and plage) for active regions associated with 51 flares located within 45° of solar central meridian from 2011–2015 (black diamonds). The red circle data points with fitted dashed line are based on unsigned flux values from Kazachenko et al. (2017). Linear fits to both data sets give a value of ~ 2 × 1023 Mx for the April 1947 spot group. At the same time, the largest spot groups in this figure (all but one from spot group 12192 in October 2014) hint at a non-linear fit line in the log–log plot that could reach the 6 × 1023 Mx value used by Tschernitz et al. The large black circle in Fig. 15 represents a compromise average flux value of 4 × 1023 Mx.

Fig. 14
figure 14

Image reproduced with permission from Aulanier et al. (2013), copyright by ESO

The largest sunspot group reported since 1874, as observed on 5 April 1947 in Ca II K1v (left) and Hα (right) by the Meudon spectroheliograph. The largest area of this group (Greenwich 14886), as reported by the Royal Greenwich Observatory (RGO) was 6132 μsh on 8 April.

Fig. 15
figure 15

Scatter plot of total unsigned active region (AR) magnetic flux versus sunspot area (black diamonds) for 51 ≥ M5.0 flares (from 29 active regions) located within 45° of disk center from 2011 to 2015 (Toriumi et al. 2017). The location of the black circle point for the April 1947 spot group (Greenwich 14886) is based on an average of the estimated total unsigned flux from Schrijver et al. (2012; 6 × 1023 Mx) and Toriumi et al. (2017; 2 × 1023 Mx). The black straight line is the result of a linear fit to data in the log–log plot. The red circle data points with fitted dashed line are based on the unsigned flux values for these events from Kazachenko et al. (2017). Adapted from Toriumi et al. (2017)

The Schrijver et al. estimate of ~ 6 × 1023 Mx assumes a uniform field strength of ~ 3000 G for the spotted area of group. While ~ 3000 G is not unreasonable for the peak field strength of a large single spot, such a value is reached < 10% of the time in sunspot umbrae (Pevtsov et al. 2014). Moreover, the corrected umbral area of Greenwich 14886, is only 739 μsh. Schrijver et al. (2012) used an overall sunspot group area of 6000 μsh but did not consider plage. This neglect of the plage contribution offsets the assumption of the high uniform field strength in the sunspots.

Cycle18 (1944–1954), which included the great spot group of April 1947 is known as the cycle of “giant” sunspots (Dodson et al. 1974). The five spot groups since 1874 with observed areas > 4500 μsh all occurred in this cycle. (The next largest group, with an area of 3716 μsh, occurred in January 1926.) While large spot areas are indicative of the potential for extreme events, they are not sufficient for their occurrence. Of these five spot groups, two (February 1946 (− 220 nT; http://dcx.oulu.fi/dldatadefinite.php) and July 1946 (− 268 nT) were associated with “great”, but not “outstanding” magnetic storms, the designation given the May 1921 event (Jones 1955). As we shall see in Sects. 6 and 7, the intensities of geomagnetic storms and solar energetic proton events, respectively, can be affected by other factors than the size of the parent sunspot group.

Tschernitz et al. (2018) find a strong correlation between the GOES class of a flare and the rate of magnetic reconnection for a sample of 51 flares (SXR class B3-X17). They estimate the reconnection rate by mapping the Hα ribbon evolution in time onto the region’s magnetogram (Fig. 16a). Among the most pronounced correlations (r = 0.92) they find a scaling between GOES SXR class and the total reconnected flux \({\Phi }_{\mathrm{r}}\) (in Mx; 1 Mx = 10−8 Wb) (Fig. 16b). Tschernitz et al. (2018) find that the largest events involve about half of the magnetic flux contained in the active region. For the April 1947 spot group with a total magnetic flux of 6 × 1023 Mx (after Schrijver et al. 2012), Tschernitz et al. inferred a largest possible SXR flare of class ~ X500 (with confidence bounds from X200-X1000) as shown in Fig. 16b, with a corresponding bolometric energy of ~ 3 × 1033 erg. However, because the correlation of Tschernitz et al. used the average of the positive and negative magnetic flux rather than the total unsigned flux as estimated by Schrijver et al. (2012), the reconnection flux for the 1947 region should be reduced by another factor of two from 3.0 × 1023 Mx to 1.5 × 1023 Mx (A. Veronig, personal communication, 2021), reducing the SXR class estimate to X180 (− 100, + 300) (1.4 (− 0.6, + 1.4) × 1033 erg) via Eq. (5). For the compromise total (average) unsigned flux of 4(2) × 1023 Mx for the April 1947 active region in Fig. 15 (with 50% flux involvement), Fig. 16b yields a SXR class of X80 (− 40, + 120) (with bolometric energy of 0.8 (− 0.3, + 0.7) × 1033 erg). For reasons given in 3.1.7(c) below, our preferred estimate based on Fig. 16b is the X180(− 100, + 300) value for an unsigned flux of 6 × 1023 Mx. Figure 17 gives an impression of the areas of sunspot groups that appear to be required to power flares of different magnitudes. The inset shows the large 1947 active region for comparison with a modelled spot group in the upper right of the figure that, based on the above calculation, could produce a 1033 erg flare.

Fig. 16
figure 16

Image reproduced with permission from Tschernitz et al. (2018), copyright by AAS

a Determination of reconnection flux φ for an eruptive M1.1 flare on 2011 November 9. The cumulated pixels swept over by the flare ribbons are superimposed on the HMI line of sight magnetogram (scaled to ± 500 G). Red (blue) areas indicate negative (positive) magnetic polarity. b Total flare reconnection flux \({\Phi }_{\mathrm{r}}\) (defined as the mean of the absolute values of the reconnection fluxes in both polarity regions) versus GOES 1–8 Å SXR peak flux FSXR. Blue squares indicate confined events, and red triangles indicate eruptive events. The linear regression line derived in log–log space for all events (thick line) is plotted along with the 95% confidence intervals (thin lines).

Fig. 17
figure 17

A cartoon depicting estimated areas of sunspot groups (exclusive of any surrounding facular areas) needed to power flares with different energy budgets. These estimates were originally developed by, and depicted in, Aulanier et al. (2013) for the parts of the sunspot groups involved in the flaring, which is assumed to be up to a third of the overall group size. In this modified version, Schmieder (2018) tripled the areas of the groups for the bipole involved in flaring (with red outlines) to show the sizes of spot groups required to reach up to a maximum value of a large stellar flare of around 1036 erg. In the Sun and stars, the flux in two thirds of the region may not be fully contained in spots, of course, but could also be distributed over extended facular regions. In a modification of the figure from Schmieder (2018), we overlaid, in the solar image on the left, the largest observed sunspot group since 1874 (shown in Ca II K1v; full disk image in Fig. 14) for comparison with the group in the upper right quadrant which has an estimated peak flare energy of 1033 erg. The two symbols in the right-hand image that resemble tires are sunspot drawings of uncertain scale by John of Worcester from 1128 AD December 8 (https://sunearthday.nasa.gov/2006/locations/firstdrawing.php)

Kazachenko et al. (2017) performed a similar study to that of Tschernitz et al. based on over 3000 flare events (C1.0-X5.4) by measuring the ribbons observed in the 1600 Å channel of SDO/AIA (Atmospheric Imaging Assembly; Lemen et al. 2012) instead of Hα spectroheliograms, and mapping these to SDO/HMI magnetograms. They find

$$ \Phi_{{\text{r}}} = 10^{22.13} { }\left( {\frac{{{\mathcal{C}}_{GOES} }}{{{\mathcal{C}}_{{GOES,{\text{ X}}1}} }}} \right)^{0.67} $$

Equation (6) was derived as a linear reduced major axis fit to logarithmic variables (Isobe et al. 1990). The slope of 0.58 derived by Tschernitz et al. (2018) for the fit line in Fig. 16b and that of Kazachenko et al. in Eq. (6) are statistically consistent. Toriumi et al. (2017) obtain a slope of 0.28 ± 0.10 for a sample of 51 events but for a small (M5.0-X5.4) range. For an active region with total unsigned flux of 4 × 1023 Mx and 50% flux involvement, Eq. (6) yields an ~ X55 SXR flare with radiative energy of ~ 6 × 1032 erg from Eq. (5), comparable to the ~ X45 and 5 × 1032 erg values inferred for the 1 September 1859 event.

(b) Estimates based on active region energy and released flare energy

As noted in Sect. 3.1.5, solar flares and eruptions are powered by some fraction of the excess or free (i.e., non-potential) magnetic energy (Efm) of an active region. Efm is distributed throughout the magnetic field of an active region. Sometimes, authors define a “free energy density” as if that would provide information on where the primary contributions to free energy would be located. However, despite the fact that the difference of the integral of the field energy for non-potential and potential fields can be mathematically written as the integral over the difference of what looks like a local quantity, namely

$$ E_{{\text{fm }}} = { }\mathop \int \limits_{V} \frac{{B_{{{\text{non}} - {\text{pot}}}}^{2} }}{8\pi }dV - { }\mathop \int \limits_{V} \frac{{B_{{{\text{pot}}}}^{2} }}{8\pi }dV = { }\mathop \int \limits_{V} \left( {\frac{{B_{{{\text{non}} - {\text{pot}}}}^{2} }}{8\pi } - \frac{{B_{{{\text{pot}}}}^{2} }}{8\pi }} \right)dV $$

this should not be interpreted to mean that the physical quantity of the resulting integrand maps to, say, the electrical currents that most contribute to the free energy. Efm is intrinsically a large-scale quantity.

Estimation of Efm is complicated by the facts that we cannot directly measure the coronal magnetic field and that models of that field based on the measurements of polarized photospheric light are, at best, uncertain at present. Nevertheless, as summarized by Emslie et al. (2012): “Numerous efforts have been undertaken to estimate nonpotential magnetic energies in active regions near disk center. The methods include: (1) using the magnetic virial theorem estimates from chromospheric vector magnetograms (Metcalf et al. 1995, 2005), (2) semi-empirical flux-rope modelling using Hα and EUV images with MDI LOS magnetograms (Bobra et al. 2008), and (3) MHD modelling (Metcalf et al. 1995; Jiao et al. 1997) and non-potential field extrapolation based upon photospheric vector magnetograms (Guo et al. 2008; Schrijver et al. 2008; Thalmann and Wiegelmann 2008; Thalmann et al. 2008). These methods are labor intensive, and uncertainties in their energy estimates are large. For example, error bars on virial free-energy estimates can exceed the potential magnetic energy. Also, there is considerable scatter in estimates from studies that employ several methods to analyze the same data (e.g., Schrijver et al. 2008). A couple of generalizations, however, can be made. Free energies determined by virial methods matched or exceeded the potential field energy, while free energies estimated using other techniques typically amounted to a few tens of percent of the potential field energy. Published values for free energies in analytic (Schrijver et al. 2006) and semi-empirical (Metcalf et al. 2008) fields meant to model solar fields also hover around a few tens of percent of the potential field energy.” Thus, as noted above, Emslie et al. (2012) adopted 30% of the modelled potential energy of an active region as their estimate of the free energy available for eruptive flares.

It may well be that a value of ~ 30% of the total magnetic energy of an active region represents a maximum value of the free energy beyond which active-region coronal fields cannot be stable. A model example of this can be found in the magnetohydrodynamic experiment performed by Aulanier et al. (2010, 2012) and analyzed in light of extreme flaring by Aulanier et al. (2013). In their bipolar region, one polarity is subjected to a rotational shear, which builds up until an instability in the field develops, manifested in the eruption of a flux rope mimicking a CME. Aulanier et al. (2013) note that the “CME itself was triggered by the ideal loss-of-equilibrium of a weakly twisted coronal flux rope …, corresponding to the torus instability (Kliem & Török 2006; Démoulin and Aulanier 2010)”. During the eruption, the field above their modelled bipolar region released 19% of the total magnetic energy, with most of that energy available for the thermal evolution of the corona (i.e., a flare) and the remaining 5% going into the bulk kinetic energy of the CME. While the fraction of total energy released in this numerical experiment approaches 30%, the ~ 20:1 apportionment between flare and CME is opposite to that of Emslie et al. (2012).

Aulanier et al. (2013) obtained relationships between active region length scale, typical field strength, flux, and energy content. Reformulation of their results yields the following scaling between the total energy E(AR) (erg) contained in the active region magnetic field, the region’s total unsigned flux \(\Phi \) (Mx), and the core field strength of the simulated spot pair, BC (Mx cm−2):

$$ E\left( {AR} \right) = 0.14 \left( {\frac{{B_{C} }}{{1000 {\text{G}}}}} \right)^{1/2} \Phi^{3/2} . $$

This equation permits an independent determination of the largest possible flare from that determined in Sect. 3.1.7(a). With the scaling of Eq. (8), a maximum field strength of BC = 3.5 kG (Aulanier et al. 2013) and an active region flux of Φmax = 4 × 1023 Mx yields a total active region energy E of 6.6 × 1034 erg. Apportioning per Sect. 3.1.5, the released energy (approximately one-tenth of active region magnetic potential energy) is 6.6 × 1033 erg, with bolometric energy (~ one-fourth of released energy) of ~ 1.7 × 1033 erg and a SXR class of ~ 240 via Eq. (5)Footnote 6.

In another estimate of this type, Toriumi et al. (2017) calculated the total energy released in a flare based on the integral of the flux contained in the flare ribbons over time. For an estimated total magnetic flux of 2 × 1023 Mx for the April 1947 active region (Fig. 15) and under the assumptions that the ratio of ribbon area to spot area was 0.85 and that two-thirds of the computed energy would be released in a flare, they obtained a released energy of ~ 1 × 1034 erg, which after Emslie et al. (2012) would translate to a bolometric energy ~ 2.5 × 1033 erg, and a ~ X410 SXR flare from Eq. (5).

The likely underestimation of CME energies noted in Sect. 3.1.5 will reduce the SXR class estimates based on the work of Aulanier et al. (2013) and Toriumi et al. (2017). Assuming a 6:1 (vs. 3:1) apportionment of released free energy between CMEs and flares, respectively, results in a SXR class estimate of X105 (X185) for Aulanier et al. (Toriumi et al.).

(c) Composite estimate

Altogether, the above estimates have a surprisingly low range of values, all lying within the X80 (− 40, + 120) range for the estimate from Fig. 16b, based on a total unsigned flux of 4 × 1023 Mx from Fig. 15. The lower limit SXR class and radiative energy estimates of Kazachenko et al. (X55; 6 × 1032 erg) and Tschernitz et al. (X40; 5 × 1032 erg) are comparable to the parameters for the 1 September 1859 and 4 November 2003 flares, which originated in spot groups less than half as large as April 1947. Because of this and the apparent upward curvature of magnetic flux with increasing group spot area in Fig. 15, we adopt: (1) the reconnection flux based estimate of X180 (− 100, + 300) given by Fig. 16b from Tschernitz et al. as our preferred estimate of the largest possible flare, and (2) the total unsigned flux value of 6 × 1023 Mx they used as the best estimate of the unsigned flux for the April 1947 active region. An unsigned flux of 6 × 1023 Mx would increase the SXR flare class (radiative energy) estimate of Kazachenko et al. to X105 (9.4 × 1032 erg), and those of Aulanier et al. and Toriumi et al. to ~ X240 (1.7 × 1033 erg) and ~ X4000 (1.3 × 1034 erg), respectively. We reject the ~ X4000 estimate of Toriumi et al. as a clear outlier—an overestimate apparently due to assumptions regarding spot:ribbon area ratio and the fraction of the computed energy released. From Fig. 8, a 10,000-year flare would have a bolometric energy of ~ 2 × 1033 erg versus the 1.3 × 1034 erg value calculated from Eq. (5) for the ~ X4000 estimate from Toriumi et al. Nominal SXR class and energy values of X180 and 1.4 × 1033 erg are approximately five and three times as large as the respective values (~ X40; ~ 5 × 1033 erg) for the 1859 and 2003 flares. Our preferred estimate of X180 (− 100, + 300), indicated by bold/italic font in Table 1, can be considered a composite because it encompasses the ~ X105 and ~ X240 estimates of Kazachenko et al. and Aulanier et al., respectively, with all three estimates based on an unsigned magnetic flux of 6 × 1023 Mx. The lower end of the X180 (− 100, + 300) estimate for the largest possible flare based on the April 1947 active region exceeds that of the largest flares observed since 1859 (from smaller spot groups) and the upper limit encompasses the X285 (± 140) SXR class estimated for the inferred flare for the 774–775 AD SEP event (Sect. 7.8).

Table 1 Estimates of the largest possible present-era solar flare based on an unsigned magnetic flux of 6 × 1023 Mx for the active region of April 1947 (peak daily area = 6132 µsh), and a 6:1 CME:flare energy apportionment

3.2 Flares on Sun-like stars

3.2.1 Stellar flare research before Kepler

The first generally recognized observations of stellar flares (on the faint dwarf star L 726-8; also designated UV Ceti) occurred on 25 September 1948 (Joy and Humason 1949) and 7 December 1948 (Luyten 1949a, b). Luyten (1949a) noted that Edwin Carpenter’s photographic plates, taken on 7 December with the 36-in. telescope at Steward Observatory in Tucson (Fig. 18), give “the first observation of the extreme rapidity of the [flare brightness] change—to twelve times the original luminosity [of the star] in less than 3 min …” Precursor observations of stellar flares were made by Hertzsprung in 1924 (Bastian 1990; Gershberg 2005), Luyten (1926) and Van Maanen (1940, 1945). Subsequently, radio (Lovell 1963; Slee et al. 1963a, b; Orchiston 2004) and X- ray emissions (Heise et al. 1975; Güdel 2004) were found to accompany optical stellar flares.

Fig. 18
figure 18

Image reproduced with permission from https://speccoll.library.arizona.edu/, copyright by University of Arizona

Edwin Carpenter (center) shows University of Arizona students the 36-in. telescope of the Steward Observatory in 1957. Carpenter was the first to record the rapidity with which some stellar flares can unfold.

Joy and Humason (1949) and Luyten (1949a, b) used the terms “flare-up” or “flare” when describing the stellar phenomenon but neither paper makes an analogy to solar flares. Luyten (1949b) suggested that the cause might be “the same as that which produces a nova, or an SS Cygni star.” In the preface of his monograph on solar activity in main sequence stars, Gershberg (2005) places the establishment of the similarity of solar and stellar flares in the mid-1960s, following the first high-time resolution spectra of flares on UV Ceti variables. As noted in Sect. 3.1.3, several lines of evidence support the close relationship of physical processes observed in stellar and solar flares.

3.2.2 Discovery of superflares on solar-type (vs. Sun-like) stars

X-ray observations show that young stars can have very large flares, termed “superflares” (Schaefer et al. 2000), defined as flares with radiative energy more than 1033 erg (~ 2–3 times more energetic than the largest directly observed solar flares; Emslie et al. 2012). These young stars rotate very fast with rotational periods of a few days, much less than solar sidereal rotation period of ~ 25 days at the equator. Generally, such fast rotating stars are strong X-ray sources even in their quiescent phase, indicating the presence of strong magnetic field and often with evidence that large fractions of the stellar surface are covered by starspots (Pevtsov et al. 2003). Consequently, it has been assumed that superflares would never occur on the present Sun, since the Sun is old and is slowly rotating.

However, by analyzing previously existing astronomical data, Schaefer et al. (2000) identified nine superflares with energies ~ 1033–38 erg in ordinary solar-type stars (i.e., G type main sequence stars with rotational velocities less than 10 km s−1; rotation period > 5 days). Concurrently, Rubenstein and Schaefer (2000) argued from analogy with RS CVn binary star systems that superflares on solar-type stars were caused by a “hot Jupiter” (Mayor and Queloz 1995; Marcy and Butler 1998; Schilling 1996), a Jovian-type exoplanet orbiting close-in to these stars. Rubenstein and Schaefer thus implied that—because of the lack of a hot Jupiter—the Sun is not a candidate to produce superflares. Cuntz et al. (2000) laid the framework for how a close-in giant planet could cause stellar activity via gravitational and magnetic interaction. Ip et al. (2004) used a numerical MHD simulation to show that a star-planet magnetic interconnection could lead to energy release comparable to that of a typical solar flare and Lanza (2008) explained the phase relation between stellar hot spots and planetary location in terms of such a linkage. Direct observational evidence between stars with close-in Jovian planets and superflares was lacking, however (e.g., Saar et al. 2004).

One of the authors of this review, Kazunari Shibata, questioned the hot-Jupiter hypothesis of Rubenstein and Schaefer (2000) because the RS CVn stars on which the hypothesis is based have short rotation periods due to tidal locking. Hence in 2010 Shibata started to encourage young researchers and students to search for superflares on solar-type stars (G type dwarfs) observed by Kepler. As a result, Maehara et al. (2012) reported 365 superflares (with radiative energy > 1033 erg) on 148 “solar-type” stars (defined as G-type main sequence stars with effective temperature of 5100–6000 K and log (surface gravity (cm/s2) ≥ 4.0) during 120 days of Kepler observations. For more restrictive criteria, they identified [9] superflares on 5 “Sun-like” stars in the sample with effective temperatures of 5600–6000 K and rotational periods between 11.0 d and 17.1 d (Maehara et al.; Supplementary Information). As a reflection of Shibata’s encouragement, the Nature paper reporting these results included five undergraduate researchers as co-authors (T. Shibayama, S. Notsu, Y. Notsu, T. Nagao, and S. Kusaba).

Later, Shibayama et al. (2013) extended the survey and confirmed the work of Maehara et al. by finding 1547 superflares (including those reported by Maehara et al. 2012) on 279 G-type dwarf stars during 500 days (from 2009 May to 2010 September) of Kepler observations. In all, they found 44 superflares on 19 Sun-like stars. Of these 19 Sun-like stars, three had rotational periods longer than the solar rotation period of ~ 25 days, suggesting that slowly rotating stars like the Sun could exhibit superflares.

Maehara et al. (2012) and Shibayama et al. (2013) argued that the superflares they discovered were not due to hot Jupiters orbiting the superflare stars. Quoting from Shibayama et al., “According to the Kepler candidate planet data explorer (Batalha et al. 2013), 2321 planets have been found in 1790 stars among 156,453 stars. Hence, the probability of finding exoplanets orbiting stars is about 1%. Howard et al. (2012) showed that the probability of finding a hot Jupiter was 0.5%. However, none of our superflare stars (279 G-type dwarfs [of which 69 had rotation periods > 10 d]) have a hot Jupiter according to the data explorer. For a G-type dwarf with a hot Jupiter, the probability of [detecting] a transit of the planet across the star is about 10% averaged over all possible orbital inclinations (Kane and von Braun 2008). If all of our 279 [69] superflare stars are caused by a hot Jupiter as suggested by Rubenstein and Schaefer (2000), Kepler should detect 28 [7] of them from transits” versus the zero that were found. Recently, for a sample of 265 solar-type super flare stars (G-type main-sequence; effective temperature of 5100–6000 K; 139 with rotation periods > 10 days), with 43 in common with the 279 G-type dwarfs of Shibayama et al. 2013), Okamoto et al. (2021) found only three stars with candidate exoplanets in the NASA Exoplanet Archive (https://exoplanetarchive.ipac.caltech.edu/).

Figure 19a shows a typical example of a superflare observed by Kepler, with a spike-like increase peaking at 1.5% of the stellar brightness for a few hours. Flares on Sun-like (solar-type) stars have brightness variations in the range of ~ 0.1 to ~ 20% of the stellar luminosity (Notsu et al. 2013a). In contrast, one of the largest solar flares in the past 20 years (X17 SXR flare on 28 October 2003 shown in Fig. 13; Woods et al. 2004; Kopp 2016) showed only a ~ 0.03% solar brightness increase for ~ 20 min (FWHM). The estimated bolometric energy of the superflare in Fig. 19a is ~ 1035 erg, ~ 200 times larger than that for the largest solar flare at ~ 5 × 1032 erg.

Fig. 19
figure 19

a (left) Typical example of a superflare (shown on an expanded time scale in the inset) on a solar-type star. b (right) An artist's conception of a superflare and big starspots on a solar-type star based on Kepler observations. Images reproduced with permission from [left] Maehara et al. (2012), copyright by Macmillan; and [right] courtesy of H. Magara

In Fig. 19a the stellar brightness itself shows significant rotational variation with an amplitude of ~ 1% at a characteristic time of 10–15 days. Almost all superflare stars show such a variation ranging from 0.1 to 1% (Maehara et al. 2012) versus ~ 0.1% for the 11-year solar cycle TSI variation (Willson and Hudson 1991; Kopp 2016). This stellar brightness variation can be interpreted in terms of the rotation of a star with a substantial coverage by unevenly distributed starspots (Strassmeier et al. 2002, and references therein; Strassmeier 2009; Notsu et al. 2013a; Namekata et al. 2019), such as depicted in Fig. 19b. The interpretation of Kepler brightness variations in terms of rotation of large area starspot groups can be used to measure the rotation period of stars and to infer the fractional coverage of starspots (technically, one can only infer the maximum difference between opposing hemispheres of total spot coverage; see below), or total magnetic flux, assuming the average magnetic flux density is the same as that of a large sunspot. The derivation of superflare star, and superflare, parameters from Kepler light curves will be discussed in Sect. 3.2.4 below.

Before proceeding further, we need to add a caveat. Up to this point, we have been using the terms radiative energy, bolometric energy, and TSI interchangeably, taking it to mean flare-radiated energy time-integrated over all wavelengths. For Kepler white light flare observations, the term “bolometric flare energy” has a different, more limited, definition. The “bolometric energy” as referenced for Kepler here is the flare-radiated energy in the blackbody continuum, assuming the flare to have an effective temperature of 10,000 K to convert the emission that occurs in the Kepler 4000–9000 Å bandpass into a bolometic energy (e.g., Notsu et al. 2013a) based on a solar study by Kretzschmar (2011). We further note that there is no consideration of line versus continuum emission in the photosphere and chromosphere or allowance for coronal emission. Thus far such computations have not been done for solar-like stars other than the Sun. Consequently, the Kepler flare bolometric energies could be underestimated by up to a factor of ten in comparison with solar values based on actual TSI measurements (Osten and Wolk 2015).

3.2.3 Verifying the Sun-like nature of superflare stars

(a) Initial verification

It is long known that close binary systems such as the RS CVn-type (Hall 1976) can produce superflares with energies ~ 1035 erg or more (e.g., Doyle et al. 1991). Doppler observations can be used to eliminate such binary systems (except for systems seen (nearly) pole-on), as well as single stars rotating much faster than the Sun, from the Kepler sample of nominally Sun-like superflare stars. Other spectroscopic observations can be used to substantiate the Sun-like character of superflare stars. These include measurements of: Hα and Ca II 8542 fluxes (gauges of chromospheric activity; Linsky et al. 1979; Herbig 1985; Foing et al. 1989; Soderblom et al. 1993), Ca II K flux (indicator of the stellar mean magnetic field strength; Skumanich et al. 1975; Schrijver et al. 1989); Fe/H ratio (metallicity) and Li abundance (both indicators of stellar age; Edvardsson et al. 1993; Skumanich 1972; Soderblom 2010), and equivalent widths of Fe I and Fe II lines (effective temperature and surface gravity; Takeda et al. 2002, 2005; Valenti and Fischer 2005). Early spectroscopic studies of superflare stars were conducted by Notsu et al. (2013b; one star), Nogami et al. (2014; two stars), and Wichmann et al. (2014; 11 stars). The detailed discussion of the 11 stars examined in Wichmann et al. illustrates the complexity of the task, with all indicators of “Sun-likeness” seldom pointing in the same direction.

Notsu et al. (2015a, b) reported on their spectroscopic investigation of 34 non-binary solar-type (G-type main sequence) superflare stars observed with the High Dispersion Spectrograph (HDS; Noguchi et al. 2002) on the Japanese 8.2-m Subaru telescope in Hawaii. As a check on light-curve based stellar rotation periods (and inferred rotation rates), Notsu et al. (2015a), used HDS data to determine the projected rotation rates (v sin i; where i is the inclination angle between the stellar rotational axis and the observer’s line of sight, measured from the pole; e.g., for i = 0°, the stellar rotation axis points directly along the line-of-sight) of these 34 starsFootnote 7 and parameters such as metallicity, effective temperature, and surface gravity (all via Fe I and Fe II line widths) to isolate a sample of seven Sun-like stars that met the following criteria: 5600 ≤ Teff ≤ 6000 K, log g ≥ 4.0, and rotation periods (based on brightness variation) > 10 days.

The seven Sun-like stars identified by Notsu et al. (2015a) are listed in the first seven rows of Table 2 along with their effective temperature (Teff), surface gravity (g), amplitude of brightness variation (BV), metallicity (Fe/H), mean magnetic flux density (fB), spot area (Aspot), rotational velocity (determined both photometrically (vlc (lc = light curve)) and spectroscopically (v sin(i)), rotation period (Prot), and bolometric energy of the largest observed flare (Max flare). Recently an additional Sun-like star (KIC 11253827) has been identified by Notsu et al. (2019). Its parameters are given in row 8. Corresponding parameters for the Sun are given in the bottom row.

Table 2 Sun-like superflare stars identified by Kepler (from Nogami et al. 2014; Notsu et al. 2015a, b, 2019)

Light curves for the two stars in Table 2 that were analyzed in detail by Nogami et al. (2014) are shown in Fig. 20. Those authors reported photometrically-based rotation periods of 21.8 days and 25.3 days for KIC 9766237 and KIC 9944137, respectively. Subsequently, these two periods were reduced in Notsu et al. (2015a, see their Appendix A1 (Supplementary data)) to the 14.2 day and 12.6 day values shown in Table 2. The Nogami et al. (2014) periods were calculated by Shibayama et al. (2013) from Kepler Quarters 0–6 data while those from Notsu et al. (2015b) were based on updated Quarter 2–16 data.

Fig. 20
figure 20

Image reproduced with permission from Nogami et al. (2014), copyright by the authors

a Typical light curve of KIC 9766237 from the long cadence Kepler data. The y-axis represents the relative flux normalized by the average flux: (F − Fav)/Fav. Quasi-periodic modulations with a timescale of about 14 days (see text) are seen. An arrow points out the superflare. The inset figure shows an enlarged light curve around the superflare. The amplitude is about 0.17%, and the duration is about 0.1 days. Though many “spikes” other than that of the superflare are seen, they consist of only one point. b Same as a, but for KIC 9944137. The amplitude of the superflare in the inset figure is about 0.28%, and the duration exceeds 0.2 days.

Witzke et al. (2020) document the difficulty of obtaining photometry-based light curves for “solar-like” stars by analyzing a set of synthesized light curves for stars that are identical to the Sun except for their metallicity. They find a minimum in the detection rate of the rotation period for stars having a metallicity value close to that (0.01) of the Sun because of the close balance of facular and spot contributions in our star. Witzke et al. substantiate this result by considering a sample of Kepler solar-like stars with a range of metallicities from − 0.35 to 0.35 and near-solar effective temperatures and photometric variabilities. The underlying assumption of long-term coherence, i.e., of long-lived spots comparable to the stellar rotation period, is not tenable for the Sun (in its present state). As of 2019, KIC 9766237 and KIC 9944137 appeared to be the most Sun-like superflare stars in terms of their brightness variation (BV) and rotation velocities (Y. Notsu, personal communication, 2019), but as noted below, even the reduced rotation periods from Notsu et al. (2015a) are now in question. The two associated superflares (one per star) had energies of ~ 1034 erg, at the lower end of the Kepler superflare range and had associated Sun-like spot areas of ≲ 1000. Such spot areas are at or below the limit of Kepler detection (Nogami et al. 2014), but see Sect. 3.2.4 below for a different interpretation of these inferred values.

Maehara et al. (2017) identified two other candidates for Table 2, both with photometrically-determined rotation periods > 20 days. KIC 6347656 and KIC 10011070 have spot area brightness-variation-based rotation periods (spot areas, Teff) of 28.4 days (3600 µsh, 5623 K) and 24.0 days (3400 µsh, 5669 K), respectively. Each of these stars had one superflare with bolometric energy of 3.1 × 1034 erg for KIC 6347656 and 2.6 × 1034 erg for KIC 10011070. The two stars have yet to be examined spectroscopically (Y. Notsu, personal communication, 2019).

In Table 2, the mean magnetic flux densities (fB) for the Sun-like stars cannot be distinguished from that of the Sun. The eight listed stars in Table 2 have a median flux density of 4(± 24) G versus ~ 10 G for the Sun. The values of < fB > are derived from a calibration against the intensity of the stellar Ca II infrared triplet relative to its local continuum (Notsu et al. 2015b). However, this calibration assumes observations of an area around a single active region. The authors do not discuss the consequences of using such observations to calibrate the signal against the disk-integrated signal. Furthermore, the uncertainties that they list are characteristic of the spread of points around the mean relationship, not the uncertainty in the mean itself. Finally, we note that that relationship when applied to the Sun yields a mean magnetic flux density of 0.2G, where observations show values of ~ 10 G, i.e., ~ 1.5 orders of magnitude larger. In addition, Notsu et al. (2015a, b) make no mention of center-to-limb effects for the transformation from their single spot group measurement to a disk-integrated signals for stars (cf. Namekata et al. 2017).

(b) A subsequent challenge

Subsequent development and refinement of the Kepler data base challenged the case for superflares on Sun-like stars. Quoting from Notsu et al. (2019): “The recent Gaia-DR2 stellar radius data (Berger et al. 2018; Lindegren et al. 2007, 2018) have suggested the possibility of severe contaminations of subgiant stars in the classification of Kepler solar type (G-type main-sequence) stars used for the statistical studies of superflares … The classification of solar-type stars in our previous studies … was based on Teff and log g values from the Kepler Input Catalog … and there can be large differences between the real and catalog values. For example, Brown et al. (2011) reported that uncertainties of Teff and log g in the initial KIC are ± 200 K and 0.4 dex [decimal exponent], respectively. … In a strict sense, we cannot even deny an extreme possibility that all of the slowly rotating Sun-like superflare stars [we considered] were the results of contaminations of subgiant stars. In addition, the Kepler solar-type superflare stars discussed in our previous studies can include some number of binary stars [excepting the events in Table 2 that were confirmed to be single, and Sun-like, by spectroscopic examination]. This is a problem, especially for investigating whether even truly Sun-like stars can have large super flares or not.“

To address the above concern, Notsu et al. (2019) revisited earlier analyses (e.g., Maehara et al. 2012, 2015, 2017; Shibayama et al. 2013) by using the Gaia-DR2 stellar radius estimates (Berger et al. 2018) and Teff values updated in DR25-KSPC (Mathur et al. 2017; Pinsonneault et al. 2012). In so doing, they found that of the 245 (of 279) stars in the Shibayama et al. (2013) sample of solar-type (G-type dwarfs) superflare stars (from 2009 May to 2010 September) for which Berger et al. (2018) determined a radius, 108 (~ 45%) were subgiants (with one red giant) and, of the remaining 136 identified as main-sequence stars, only 106 had Teff,DR25 from 5100–6000 K. Spectroscopic analysis (3200–10,000 Å) was then conducted using the Apache Point Observatory 3.5 m telescope for 18 main-sequence superflare stars observed by Kepler at 1-min cadence for which Gaia-DR2 (Release 2) stellar radius and Teff,DR25 values were available; 13 of these 18 stars were found to be single.

(c) Further development

In a further development based on the entire Kepler primary mission data set (Kepler Data Release 25 (Thompson et al. 2016; Mathur et al. 2017); 2009 May to 2013 May) and the Gaia-DR2 catalog, Okamoto et al. (2021) used a flare detection method that employed high-pass filtering to remove low-frequency rotational variations due to starspots to identify superflares on 15 Sun-like stars (based on Teff from 5600 to 6000 K, a more stringent Prot requirement of > 20 d (with Prot taken from McQuillan, et al. 2014), and age of 4.6 Gyr). The parameters for these 15 Sun-like superflare stars are given in Table 3 (for 15 vs. the 16 in the table, see Flag 2). This list includes the two Sun-like stars with rotation periods > 20 days identified by Maehara et al. (2017) noted above. The range of spot areas from ~ 1500 to 13,000 µsh for the superflare stars in Table 3 are comparable to those in Table 2 while, as noted by Okamoto et al. (2021), the faster rotating stars in Table 2 are capable of producing stronger superflares (by up to an order of magnitude) for the listed events. At present the case for superflares on Sun-like stars is based largely on Table 3.

Table 3 Parameters of Sun-like superflare stars

In a reappraisal of KIC 9766237 and KIC 9944137 (from Table 2) for which light curves are shown in Fig. 20 above, Okamoto et al. (2021) assess them to be “Sun-like star candidates”, rather than Sun-like stars, because “the amplitude of the rotational brightness variation is so small that [its] value is not reported in McQuillan et al. (2014), and the accuracy of the Prot value is considered to be low.” In contrast, as discussed in Sect. 8.3.1 below, the inability to photometrically detect a rotation period may be a signature of a Sun-like star.

3.2.4 Inferring superflare star properties from brightness variations

From the Kepler photometric observations, it is possible to estimate the stellar rotation period, starspot area, and flare energy. We will consider each of these in turn.

Maehara et al. (2012) and Shibayama et al. (2013) calculated the rotational periods of superflare stars by discrete Fourier analysis of the stellar brightness variations for the first 500 days of the Kepler mission. They chose the largest peak that exceeded the error level as the stellar rotation period. Alternatively, McQuillan et al. (2014) used an automated autocorrelation function (ACF) approach to determine rotation periods based on the entire Kepler data set.

The lower limit of the spot area of superflare stars (Aspot) can be inferred from the ΔFrot (defined as the full amplitude of the rotational brightness variation normalized by the average brightness) as follows:

$$ \Delta F_{{{\text{rot}}}} \approx { }\left[ {1 - \left( {\frac{{T_{{{\text{spot}}}} }}{{T_{{{\text{star}}}} }}} \right)^{4} } \right]\frac{{A_{{{\text{spot}}}} }}{{A_{{{\text{star}}}} }} $$

where Tspot is the spot temperature, Tstar is the effective temperature of the unspotted photosphere of the star, and Astar is the apparent area of the star (Maehara et al. 2012, 2017). The spot area in this equation is a lower limit because it is assumed that there is a reference hemisphere that has no (or negligible) starspot coverage (e.g., Fig. 3b in Notsu et al. 2013a). Notsu et al. (2019) used a relationship between star temperature and spot temperature (Maehara et al. 2017) for which Tstar (in the range from 5600 to 6000 K) was taken from the Kepler catalog and Tspot was determined spectroscopically from intensities of the photospheric lines of sunspots.

Figure 21 (taken from Maehara et al. 2017) is a downward cumulative frequency distribution (solid line) showing the occurrence rate of active regions of a given spot area for solar-type stars (defined to be early G- and late F-type main-sequence stars that meet both of the following criteria: (1) 5600 K < Teff < 6300 K and log g > 4.0 in Huber et al. (2014); and (2) 5100 K < Teff < 6000 K and log g > 4.0 in Brown et al. (2011)). The corresponding solar distribution (dotted line) is shown for ~ 140 years of solar data. The continuation of the extrapolated power-law fit to the distribution for solar-type stars (dashed line) through that of solar active regions suggests that both sunspots and larger starspots might be produced by the same physical process and that the Sun is capable of producing spot groups much larger than observed in modern times. Maehara et al. (2017) suggest that the steep drop-off of the solar distribution is due to “the lack of a ‘super-active’ phase on our Sun during the last 140 years”. This seems problematic, however, because the four solar cycle interval of solar activity from ~ 1945 to 1995, termed the modern grand maximum, is considered to be one of the strongest in the past 2000–4000 years (Solanki et al. 2004; Usoskin et al. 2006a; Muscheler et al. 2006; Usoskin 2017). Figure 21 indicates that a ~ 30,000 µsh spot region, region ~ 5 times larger than that of April 1947, would be expected to be observed once every ~ 1000 years, with no evidence of a region approaching this size over the ~ 400 years of telescopic observation of the Sun. That said, the time interval for which we have detailed knowledge of the Sun's variability is minuscule compared to the time-scale of stellar evolution. As discussed in Sect. 8.3.1 below, recent analysis of Kepler photometric and Gaia astrometric data by Reinhold et al. (2020) raises the possibility that our Sun is currently in a state of subdued activity in comparison with the bulk of Sun-like stars.

Fig. 21
figure 21

Image reproduced with permission from Maehara et al. (2017), copyright by the authors

Cumulative frequency distribution of starspots on slowly rotating solar-type stars (solid line) and that of sunspot groups (dotted line). The thin dashed line represents the power-law fit (with index =  − 2.3 ± 0.1) to the frequency distribution of starspots in the spot area range of 10−2.5 to 10−1.0.

After Shibata et al. (2013), the energy (in erg) of a stellar flare (Eflare) in a sunspot group spot with magnetic flux density B, scale-length L, and area Aspot has an upper limit determined by the total magnetic energy stored in a volume A3/2 near the spot, i.e.,

$$ E_{{\text{flare }}} \approx fE_{{{\text{mag}}}} \approx f\frac{{B^{2} L^{3} }}{8\pi } \approx f\frac{{B^{2} }}{8\pi }{ }A_{{{\text{spot}}}}^{3/2} \approx 7 \times 10^{32} { }\left( {\frac{f}{0.1}} \right)\left( {\frac{B}{{10^{3} G}}} \right)^{2} \left( {\frac{{A_{{{\text{spot}}}} }}{{3 \times 10^{19} {\text{ cm}}^{2} }}} \right)^{3/2} { } \approx 7 \times 10^{32} {\text {(erg)}} { }\left( {\frac{f}{0.1}} \right)\left( {\frac{B}{{10^{3} G}}} \right)^{2} \left( {\frac{{A_{{{\text{spot}}}} /\left( {2\pi R^{2}_{ \odot } } \right) }}{0.001}} \right)^{3/2} $$

where f (0.1; Sect. 3.1.5) is the fraction of magnetic energy that is released during the flare and R is the solar radius (to which the stellar radius is scaled). To scale the flare energy determined by Eq. (10) to SXR class, Shibata et al. (2013) assumed linear proportionality between these parameters from consideration of previous observational estimates of energies of typical solar flares (e.g., Benz 2008; also used by Gopalswamy et al. 2018). The linear scaling of Shibata et al. (2013) differs from that of Schrijver et al. (2012; Eq. (5) in this paper with an exponent of 0.72) as well as the similar scaling in Tschernitz et al. (2018; 0.79), both of which are based on the data in Kretzschmar (2011). Namekata et al. (2017) show that an exponent of 1.0 lies between those for an ordinary least squares (OLS) fit (0.84 ± 0.04) and a linear regression bisector fit (1.18 ± 0.04; Isobe et al. 1990) to a scatter plot of the logs of HMI white-light flare energy versus SXR peak flux for a sample of M- and X-class flares. It is not clear that the bisector fit is more appropriate in this case than the OLS fits in log–log space used by Schrijver et al., Tschernitz et al., and Namekata et al., which yielded slopes from 0.72 to 0.84. The Schrijver et al., Tschernitz et al., and Shibata et al. scalings are shown in Fig. 22 along with the Kretzschmar data points. The Tschernitz et al. and Schrijver et al. fits differ slightly because Schrijver did not correlate bolometric energy and GOES SXR class directly but first related TSI to SXR fluence and then converted SXR fluence to SXR class after Veronig et al. (2002a). The three scalings in Fig. 22 agree reasonably well for events in the X10-X1000 range of interest here, although Shibayama et al. (2013; their Appendix B) note an apparent disagreement between the observed frequency of nanoflares and the expected rate based on Kretzschmar's relation (Eq. (3) above) between bolometric energy and SXR energy.

Fig. 22
figure 22

Three separate scalings of flare bolometric energy to SXR class: Schrijver et al. (2012), Tschernitz et al. (2018), and Shibata et al. (2013). The similar scalings of Schrijver et al. and Tschernitz et al. are both based on the data (“X” data points) from Kretzschmar (2011)

Figure 23 from Okamoto et al. (2021) is a combined scatter plot of (1) the SXR classes of solar flares (right-hand y-axis) and (2) the bolometric energies of Kepler superflares on the 15 solar-type stars in Table 3 (left-hand axis) versus their spot areas. For both sets of flares, the energy to flare class scaling of Shibata et al. (2013) in Fig. 22 is used to infer the missing parameter—bolometric energy for solar flares and SXR class for stellar superflares. The SXR flare versus spot area comparison was patterned after Sammis et al. (2000).Footnote 8 For the stellar flare data points, spot areas were inferred from Eq. (9) with bolometric energy given by Kepler photometry. The solid and dashed slanting lines given by Eq. (10) represent the released flare energy (with f = 0.1) for a given de-projected spot area (inclination angle i assumed to be 90°) on the x-axis and spot field strengths of 3 kG and 1 kG, respectively.

Fig. 23
figure 23

Image reproduced with permission from Okamoto (2021), copyright by AAS

Scatter plot of flare energy versus sunspot area for solar flares (black dots) and superflares on solar-type stars (red squares) observed by Kepler (Table 3). The black solid and dashed lines correspond to the relationship in Eq. (10) between flare energy and spot area for B = 3000 and 1000 G, respectively. For the Kepler superflares, the inclination angle (i) between the stellar rotational axis and the line-of-sight is assumed to be 90° (e.g., as is the case for the Sun viewed from Earth at the equinoxes). The measured SXR peak intensities for solar flares and calculated values are given on the right-hand y-axis. The values on the horizontal axis at the top show the total magnetic flux of a spot corresponding to the area on the horizontal axis at the bottom when B = 3 kG.

The apparent correlation in the peak SXR flare flux versus solar spot group area in Fig. 23 has been discounted by Hudson (2021) because of “a strong selection bias, resulting from the under-reporting of weaker GOES events due to higher background levels during active times (Wheatland 2010)”. This selection effect will also apply to the flares observed by Kepler, for which the detection completeness of 1033 erg flares is 0.001 (Maehara et al. 2012). That said, the increase in the largest observed flare energies of stellar flares relative to those of solar flares in Fig. 23, in correspondence with the larger maximum spot areas inferred for stellar flares, is in accord with expectations—particularly because the starspot areas are likely underestimated due to the “null hemisphere” hypothesis. From this figure, Okamoto et al. (2021) conclude that “stellar superflares [represent] the magnetic energy release stored around the starspots, and the process is the same as that of solar flares”.

3.2.5 Occurrence frequency of superflares on Sun-like stars

Figure 24 (adapted from Okamoto et al. 2021; see also Schrijver et al. 2012; Maehara et al. 2015; Lin et al. 2019; Notsu et al. 2019) shows the unified figure of occurrence frequency of flares as a function of flare energy, for solar flares and superflares on Sun-like stars (based on short time cadence Kepler data). There is an apparent smooth distribution from the smallest EUV solar flares all the way to the largest bolometric flares on Sun-like stars. It is remarkable that superflare frequency is roughly on the same line as that for solar flares, microflares, and nanoflares, dN/dE ~ E−1.8, suggesting the same physical mechanism for both solar and stellar flares (see Sect. 3.1.3). In an early result for a sample of solar-type stars observed by the Transiting Exoplanet Survey Satellite (TESS; Ricker 2015), Tu et al. (2020) reported a slightly steeper (− 2.16 ± 0.10) slope, but within the uncertainties of the Kepler results. Recently, Aschwanden and Güdel (2021) obtained a slope of − 1.82 ± 0.005 for the fluence distribution of 162,262 flares on stars of types A-M and giants observed by Kepler at optical wavelengths (Yang and Liu 2019) veraus a slope of − 1.98 ± 0.11 for the (background- subtracted) peak fluxes of 338,661 GOES SXR flares (Aschwanden and Freeland 2012).

Fig. 24
figure 24

Comparison between the frequency distribution of superflares on Sun-like stars and solar flares. The red filled squares, blue filled square, blue dashed line, and blue open squares indicate the occurrence frequency distributions of superflares on Sun-like stars (slowly rotating solar-type stars with Teff = 5600–6000 K). The red filled circles correspond to the updated frequency values of superflares on Sun-like stars with Prot = 20–40 days, which are calculated with gyrochronology and sensitivity correction. Image adapted from Okamoto et al. (2021)

A power-law slope of ~  − 1.8 pertains to a wide variety of solar emissions (Aschwanden et al. 2016) and is in theoretical agreement, within uncertainties, with the − 1.67 value for hard X-ray (> 10 keV) peak fluxes obtained by Aschwanden (2012) from the SOC model for solar flares (Lu and Hamilton 1991). The similarity of slopes in these various studies may reflect the universality of the reconnection process in converting magnetic energy to radiative energy via particle acceleration in different types of settings (the Sun and other stars; Benz and Güdel 2010; Aschwanden and Güdel 2021) as well as in different types of flares (e.g., confined and eruptive; Harra et al. 2016; Tschernitz et al. 2018; cf., Cliver and D'Huys 2018). The Neupert effect implies a power-law slope of − 2.0 for solar SXR flares (Aschwanden and Freeland 2012).

In their earlier analysis of a power-law distribution of Sun-like superflare stars (defined as G-type main sequence stars with 5800 ≤ Teff < 6300 and rotation period > 10 days) observed during the first 500 days of the Kepler mission, Maehara et al. (2015) determined that the occurrence frequency of 1033 erg solar flares would be once every 500 years. Subsequently, Notsu et al. deduced that flares with energy of > 1034 to ≲ 5 × 1034 erg occur on old (~ 4.6 Gyr), slowly rotating (Prot ∼ 25 days) Sun-like stars approximately once every 2000–3000 years. For a larger sample size than that of Notsu et al. and using a gyrochronology correction, Okamoto et al. (2021) inferred that solar superflares with energies of 1034 erg (SXR class = X1000) could occur on slowly rotating Sun-like stars once every ~ 6000 years. This rate is preliminary, pending spectroscopic analysis of metallicity, surface gravity, binarity, and rotation period of the 15 (presumably) Sun-like stars in Table 3 on which it is based (see Appendix B in Okamoto et al.). In addition, the possible underestimation of the energy of Kepler superflares (Osten and Wolk 2015) noted in Sect. 3.2.2 will need to be addressed.

Five of the 15 Sun-like stars in Table 3 produced more than one such superflare during the ~ 4 years of Kepler observations that were considered. Statistically, this seems incongruent with ~ 1034 erg superflares happening only once in six millennia. This might suggest clustering of superflares on Sun-like stars (as witnessed for intense solar flares; e.g., Cliver et al. 2020b) but it also might mean that the sample of candidate Sun-like stars in Table 3 is contaminated by faster rotators (a possibility noted for five stars with flag = 1 in Table 3; see Appendix A in Okamoto et al.), underscoring the need for spectroscopic analysis. Until that is done, it is uncertain whether the stars in Table 3 are truly Sun-like. As Okamoto et al. (2021) explicitly note, “the number of Sun-like superflare stars (Teff = 5600–6000 K, Prot ∼ 25 days, and t ∼ 4.6 Gyr) that have been investigated spectroscopically and confirmed to be “single” Sun-like stars, is now zero …” Thus the question posed in the title of Notsu et al. (2019)—Do Kepler Superflare Stars Really Include Slowly Rotating [with rotation periods ~ 25 days] Sun-like Stars?—remains unanswered. Until it is answered affirmatively, the Kepler observations of superflares (≥ 1033 erg) can only hint at the possibility that the Sun is capable of producing such flares.

While further spectroscopic work is required to substantiate the implications of superflares on Sun-like stars for extreme solar activity, observations of solar flares with bolometric energies within a half-decade of 1033 erg (Sect. 3.1.2 above) suggest that the Sun is capable of producing a threshold level superflare over time, while analyses of historical solar proton events based on cosmogenic radionuclides (Sect. 7 below) provide evidence that it has already done so.

4 Extreme solar radio bursts

Radio bursts from the Sun were discovered in late February 1942 when Hey (1946, with delayed publication due to wartime restrictions) correctly identified as solar radio emission what at first appeared to be enemy jamming of army radar receivers. More recently, radio frequency interference due to a great decimetric (range from 300 to 3000 MHz) burst on 6 December 2006 severely degraded the performance of Global Navigation Satellite Systems (including the U.S. Global Positioning System (GPS)), signaling a modern hazard of solar radio bursts (Gary 2008; Cerruti et al. 2008; Kintner et al. 2009; Carrano et al. 2009). In this section, after reviewing the climatology of solar radio bursts, we will focus on extreme radio bursts in the frequency range from 1.0 to 1.6 GHz range that includes the 1.58 GHz (L1 band) and 1.23 GHz (L2 band) operational frequencies of GPS.

4.1 Climatology of solar radio bursts

Following the pioneering work of Akabane (1956) on the peak flux density distribution of 3 GHz solar radio bursts observed at Ottawa, Toyokawa, and Tokyo, the most extensive studies of radio burst occurrence statistics have been conducted by Nita and colleagues (Nita et al. 2002, 2004a, b) and Giersch et al. (2017). Nita et al. (2002) analyzed a 40-year (1960–1999) compilationFootnote 9 by the National Geophysical Data Center (NGDC) of solar radio burst intensities to obtain occurrence frequency distributions for six discrete frequency groupings in the range from 0.1 to 37 GHz. Giersch et al. (2017) performed a similar analysis based on the updated NGDC compilation, but using only data from the U.S. Air Force (USAF) Radio Solar Telescope Network (RSTN) for the 1966–2010 interval for the eight fixed frequencies of the RSTN patrol.

Both Nita et al. (2002) and Giersch et al. (2017) stress that the data sets they used are incomplete, with nearly half of all events being missed. Nita et al. (2002) attribute the missing events to uneven geographical coverage while Giersch et al. suggest that breakdowns in "the process of report transmission [from the RSTN observatories] to [the National Geophysical Data Center (NGDC) or its National Center for Environmental Information (NCEI) successor] is likely to account for the ‘missing’ burst data.” The power-law slopes of the peak flux density distributions obtained by Nita et al. (2002) for the six frequency ranges they considered range from ~  − 1.7 to − 1.85 versus ~  − 1.8 to − 2 for the eight RSTN frequencies examined by Giersch et al. For a shorter interval from 1994 to 2005, Song et al. (2012) obtain a range of slopes from ~  − 1.75 to − 1.85 for data from Nobeyama (http://solar.nro.nao.ac.jp/) for six frequencies from 1.0 to 35.0 GHz. Power-law slopes of ~  − 1.8 are similar to those found for the hard X-ray burst count rate (Dennis 1985; Crosby et al. 1993). For bursts in the ~ 1.2–1.6 GHz frequency range employed by GPS, the slopes found by the various investigators are consistent with values of − 1.83 ± 0.01 for 1.0–1.7 GHz (Nita et al. 2002), − 1.815 at 1.415 GHz (Giersch et al. 2017), and − 1.87 ± 0.12 at 1.0 GHz and − 1.83 ± 0.07 at 2.0 GHz (Song et al. 2012). The peak flux density distribution obtained by Giersch et al. (2017) for 1.415 GHz is given in Fig. 25. From this distribution, Giersch et al. estimated that a solar 1.4 GHz burst of peak flux density 3.2 × 106 solar flux units (sfu; 1 sfu = 10−22 W m−2 Hz−1) would occur about once per century while a burst of 6.1 × 107 sfu would occur once in a thousand years. The corresponding peak intensity values for 100-year and 1000-year bursts calculated from the 1.0–1.7 GHz distribution of Nita et al. (2002) are 1.2 × 107 sfu and 2 × 108 sfu, respectively (G. Nita, personal communication, 2018). Both sets of authors emphasized the uncertainty of the 1000-year events because of the large extrapolations involved (personal communication: Nita 2018; Giersch 2019).

Fig. 25
figure 25

Image reproduced with permission from Giersch et al. (2017), copyright by AGU

Peak flux density distribution for solar ~ 1.4 GHz (L-band) bursts from 1996 to 2010.

4.2 Observations of extreme decimetric radio bursts

The great decimetric solar radio burst of 6 December 2006 that disrupted GPS operations was the largest burst yet observed from 1.0 to 1.6 GHz, with a peak flux density of ~ 1 × 106 sfu recorded by the Frequency-Agile Solar Radiotelescope (FASR) Subsystem Testbed (FST; Liu et al. 2007) at Owens Valley (Gary 2008). A search by Klobuchar et al. (1999) for ~ 1.4 GHz bursts with peak flux densities > 4 × 104 sfu observed by RSTN (and predecessor sites in Athens and Manila) that might affect the GPS identified 14 such events from 1967 to 1998 with peak flux densities ranging up to 8.8 × 104 sfu. The extreme burst observed at Owens Valley on 6 December 2006 prompted the following questions from Kintner et al. (2009): "Was the 6 December [solar radio burst; SRB] a simple outlier on a well-behaved statistical distribution such as, for example, a 100-year flood? … Or has the method of monitoring SRBs been inadequate so that the power of previous intense SRBs was underestimated?" In reference to the second question, Giersch et al. (2017) deduced a nominal saturation level of 105 sfu at 1.415 GHz for the RSTN observatories (Gary 2008), with a station-to-station uncertainty on this value of over 50%.

In Figs. 26, 27 and 28, we show intensity versus time records of solar bursts at several frequencies that had intense (> 80,000 sfu) emission at 1.415 GHz. Figures 26 and 27 give time profiles for events on 28 May 1967 and 29 April 1973, respectively. Figure 28 contains the records of bursts on: (a) 6 December 2006, (b) 13 December 2006, (c) 14 December 2006, and (d) 21 April 2002. For the 6 December 2006 event (Fig. 28a), the Sagamore Hill 1.415 GHz radiometer saturated at a level of ~ 1.4 × 105 sfu, well below the ~ 1 × 106 sfu peak flux recorded at Owens Valley. For both the 13 December 2006 event and the 21 April 2002 event (Fig. 28b, d), the RSTN 1.4 GHz receivers at Palehua and Learmonth saturated at ~ 105 sfu.Footnote 10

Fig. 26
figure 26

Sagamore Hill record at five frequencies (with peak fluxes given) for the great solar radio burst of 23 May 1967. Image adapted from Castelli et al. (1968)

Fig. 27
figure 27

Sagamore Hill record at four selected frequencies (with peak fluxes given) for the great solar radio burst of 29 April 1973. Image adapted from Barron et al. (1980)

Fig. 28
figure 28

Image reproduced with permission from Cliver et al. (2011), copright by AAS

Time-intensity profiles of great (≳105 sfu) decimetric 1.4 or 1.0 GHz bursts (red traces) along with bursts at other frequencies from ~ 1–17 GHz for: a 6 December 2006 (Sagamore Hill); b 13 December 2006 (Nobeyama); c 14 December 2006 (Learmonth); and d 21 April 2002 (Nobeyama). The dashed gray line in each of the panels is the time-intensity profile of the associated 1–8 Å SXR burst.

As shown in Table 4 below, only 14 1.0–1.6 GHz bursts with peak fluxes ≥ 80,000 sfu were recorded from 1966 to 2015. The rarity of these large decimetric bursts and the fact that NOAA active region 10930 in December 2006 produced three such bursts (Fig. 28a–c) with peak fluxes ≳ 105 sfu (Gary 2008) suggests that a special circumstance is required for the occurrence of extreme ~ 1.5 GHz bursts. Giersch and Kennewell (2013) investigated the sunspot group and magnetic features of region 10930 to determine if any distinguishing features (e.g., spatial locations of minimum and maximum magnetic fields, magnetic field gradient between these locations, small-scale mixing of positive and negative polarities) were present. From a comparison of NOAA 10930 with a set of active regions from 2002 to 2008, they found no features that were unique to 10930, thus making such bursts “inherently unpredictable.”

Table 4 1.0–1.6 GHz radio bursts with peak fluxes ≥ 80,000 sfu (1966–2015).

While region 10930 may not have distinguished itself from the 2002–2008 comparison set of active regions, inspection of the great decimetric bursts themselves reveals a pattern that may provide insight into the conditions under which these extreme bursts arise. As noted by Cliver et al. (2011), all three of the events from region 10930 exhibited intense (≳ 105 sfu) L-band (defined here to be ~ 1.0–1.5 GHz) emission some tens of minutes after the impulsive phase of these flares (Fig. 28a–c, although the strongest such emission for the 13 December 2006 flare occurred during the impulsive phase (Fig. 28b). Similar delayed decimetric emission is also seen in Figs. 26, 27, and 28d. Delayed microwave bursts with spectral maxima at ~ 3 GHz (Tanaka and Kakinuma 1962) and gradual or extended hard X-ray bursts with hardening spectra (Tsuneta et al. 1984; Dennis 1985; Tanaka 1987) that often accompany them, have both been interpreted (Cliver 1983; Cliver et al. 1986) in terms of electrons accelerated via reconnection and trapped on coronal loops in the standard CSHKP model for eruptive solar flares (Hudson and Cliver 2001; Shibata and Magara 2011). The late phase of such flares is characterized by the growth of post-flare loop systems (Bruzek 1964; Kahler 1977; Švestka et al. 1982) as loops are formed at successively greater heights in the wake of a CME (e.g., Forbes 2000; Lin and Forbes 2000; Hudson 2011; Benz 2017).Footnote 11 Because of their similar origins in delayed microwave bursts and gradual hard X-ray events, Cliver et al. (2011) suggested that the great decimetric bursts in December 2006 could also be interpreted in terms of magnetic reconnection and electron trapping in the CSHKP model. Tanaka and Kakinuma (1962) were the first to note the concurrence of delayed microwave and intense decimetric bursts.

The list of great (peak intensity ≥ 80,000 sfu) 1.0–1.6 GHz bursts in Table 4 is based on searches of the NGDC website by Nita et al. (2002; for 1960–1999) and Giersch et al. (2017; 1966–2010) as well as our own search for large 1 GHz bursts on the Nobeyama website (1988–2015). Four of the eight additional cases (i.e., those not shown in Figs. 26, 27, 28) listed in Table 4 (4 July 1974, 16 February 1979, 15 April 1990, and 5 March 2012) also appear to follow the characteristic (delayed) pattern seen in Figs. 26, 27 and 28, while in the other four cases (29 October 1968, 2 November 1992, 8 February 1993, 11 July 2005), intense decimetric peaks occurred relatively close (within 10 min) to the impulsive phase with no strong late phase emission.Footnote 12 Setting aside the 13 December 2006 radio burst in which strong (105 sfu) L-band emission occurred both during and after the impulsive phase, in 9 of the 13 other cases in Table 4, commensurately intense decimetric emission avoided the flare impulsive phase—widely considered to be the time of primary energy release in flares (e.g., Fletcher et al. 2011). Recently, Marqué et al. (2018) presented another example of such behavior in a radio burst on 4 November 2015 (Fig. 29) and listed another event not in Table 4 with a large (> 100,000 sfu) delayed decimetric burst on 24 September 2011 (Shakhovskaya et al. 2019). The spike burst at 2.65 GHz on 11 April 1978 reported by Slottje (1978) shares this timing characteristic. The frequently observed delay of the largest decimetric bursts relative to the flare impulsive phase suggests that a special circumstance is required for their generation.

Fig. 29
figure 29

Image reproduced with permission from Marqué et al. (2018), copyright by the authors

Decimetric emission at several discrete frequencies for a radio burst on 4 November 2015 observed by the Humain Solar Radio Spectrometer (HSRS) in Humain, Belgium and the Observations Radio pour FEDOME et l’Étude des Éruptions Solaires (ORFEES) spectrograph in Nançay, France).

4.3 Are out-sized decimetric spike bursts due to electron cyclotron maser emission?

The evolution of the decimetric burst emission in Fig. 28(c; at 1.4 GHz), (d; 1.0 GHz) is instructive. In both cases the decimetric emission more or less tracks that at centimeter wavelengths (cm-λ; 3–30 GHz) through the impulsive phase. After ~ 22:40UT in 26(c) and 01:40 UT in 26(d), however, the ~ 1 GHz emission dwarfs that at ~ 4–5 GHz by 1–2 orders of magnitude, before dropping abruptly to a level that closely tracks the higher frequency emissions. This behavior suggests an additional component of the decimetric emission, specifically a coherent emission contribution (Melrose 2017) because of the high peak fluxes (brightness temperatures), either plasma emission or electron cyclotron maser (ECM; see Treumann 2006; Fleishman 2006, for reviews) emission, riding atop an incoherent gyrosynchrotron (GS) component (Nindos 2020) that is dominant at centimeter wavelengths. This behavior is pronounced in the ~ 1000 MHz emission in the event on 4 November 2015 in Fig. 29 which intermittently rapidly increases and decays by up to three orders of magnitude, returning to levels characteristic of encompassing frequencies at ~ 600 MHz and ~ 1400 MHz.

The cumulative density function for 1 GHz in Fig. 30 from Song et al. (2012), which is unaffected by receiver saturation, is consistent with a transition from gyrosynchrotron emission to a dominant coherent process for events with peak fluxes > 104 sfu which lie above the extension of the power-law distribution to higher intensities, in contrast to the behavior at higher frequencies up to 35 GHz where the data points fall below the power-law fit for peak fluxes < 104 sfu. Thus extreme decimetric bursts suggest a different emission mechanism than that responsible for merely large events. The high Xmin value determined by the maximum likelihood estimator method (Clauset et al. 2009) for the 1 GHz bursts in Fig. 30 results in the power-law fit being based on only 53 events versus 139 events for 2 GHz and a larger uncertainty in the slope of the fit at 1 GHz. That said, the largest 1 GHz peak flux density in the figure is more than an order of magnitude larger than the corresponding value at 2 GHz. The GS spectrum for large cm-λ bursts characteristically has its maximum at ~ 5–10 GHz (e.g., Castelli et al. 1967; Guidice and Castelli 1975; Stähli et al. 1989), tapering down to a minimum in the decimetric range. The markedly larger maximum flux value at 1 GHz relative to that at 2 GHz in Fig. 30 supports the picture of an additional non-GS (coherent) emission mechanism for the largest bursts at 1 GHz.

Fig. 30
figure 30

Cumulative density functions for solar radio bursts recorded at Nobeyama Observatory from 1994 to 2005 at six separate frequencies from 1 to 35 GHz. The red oval in the 1 GHz panel highlights the unusual behavior relative to the other frequencies on the tail of the distribution. Xmin is the smallest value used for the power-law fit. Image adapted from Song et al. (2012), copyright by AAS

Following Gary (2008), Cliver et al. (2011) interpreted the delayed decimetric peak in Fig. 28a in terms of ECM emission. Figure 31 contains a ~ 40 s time sample (with gaps) of the Owens Valley FST record from 1.0–1.5 GHz for the 6 December 2006 burst showing a proliferation of intense narrow-band (3–4 MHz) spikes with durations less than 20 ms that are a signature of coherent radio emission. Wang et al. (2008) reported spike emission at 2.6–3.8 GHz for the 13 December 2006 radio burst and also attributed it to electron cyclotron maser emission. ECM emission driven by a loss cone instability was proposed by Wu and Lee (1979) to account for terrestrial auroral kilometric radiation (AKR) and subsequently developed by Holman et al. (1980) and Melrose and Dulk (1982) to explain solar millisecond spike bursts (Dröge 1977; Slottje 1978) such as seen in Fig. 31.

Fig. 31
figure 31

Image reproduced with permission from Cliver et al. (2011), copyright by AAS

Detail of Owens Valley FST observation of the extreme decimetric burst on 6 December 2006. The figure shows four-second scans of burst intensity in right circular polarization, with 4 s gaps between them (not shown). Time labels on the bottom axis are for the center of each 4 s scan. The emission consists of a proliferation (many hundreds per second) of narrow-band (3–4 MHz) spike bursts with durations less than 20 ms.

ECM emission is based on direct amplification of free-space electromagnetic waves in a plasma with a non-thermal electron population.Footnote 13 The amplification results from a linear plasma instability that can occur when the local electron cyclotron frequency (fce) exceeds the local plasma frequency (fpe), i.e., fce ≳ fpe, where fce (MHz) = eB/2πme and fpe (MHz) = (e/2π)(ne/meε0)1/2. However, the effective condition, viz., fce >> fpe, for fundamental ECM emission to escape the Sun's atmosphere is more stringent because of resonant gyromagnetic absorption at the second harmonic layer in the weaker fields of the overlying thermal plasma (Treumann 2006; Holman et al. 1980; Melrose 2017). Calculations of such gyromagnetic absorption (Melrose and Dulk 1982; McKean et al. 1989) indicate that it can effectively suppress ECM emission. Various suggestions have been proposed to overcome this difficulty, with none yet generally accepted (Melrose 1999, 2009; Treumann 2006; Ning et al. 2021a, b).

Beginning with the assumption that ECM emission arises from an unstable “horseshoe” distribution (Ergun et al. 2000), rather than a loss-cone-driven instability, Melrose and Wheatland (2016) argued that the fce >> fpe condition can be satisfied if deep density cavities (Calvert 1981; Hilgers 1992; Alm et al. 2015) driven by parallel electric fields (Temerin et al. 1982; Ergun et al. 2001) which make ECM emission possible in terrestrial aurora, can exist in the coronal arcades of eruptive flares. Cliver et al. (2011) made a similar suggestion. Quoting from Melrose and Wheatland (2016), “The formation of a density cavity … suggests a new possibility for allowing a fraction of the ECME to escape through the second-harmonic layer. This possibility arises if the density cavity extends to above the second-harmonic absorption layer. This is the case when the flux tube in which the acceleration (and associated density depletion) occurs extends to a height where B has decreased by a factor of two from its value at the emission point of the ECME. The radiation then passes through the second-harmonic layer in the low-density cavity. The gyromagnetic absorption coefficient is proportional to the density of thermal electrons and hence would be anomalously weak in an anomalously low-density region. If the density in the cavity is orders of magnitude lower inside the flux tube than outside it, as is the case of AKR, then gyromagnetic absorption at the second harmonic would be unimportant. The fraction of the ECME that escapes would then be the fraction that is ducted along the low-density flux tube to above the second-harmonic layer.” The vertical extension of a density cavity suggested by Melrose and Wheatland as a pre-requisite for the escape of ECM emission is in keeping with the evolution of loop arcades in the standard CSHKP model of eruptive flares. Such loop systems can reach heights ~ 105 km (Kahler 1977; Švestka et al. 1982). The outward motion of the post-eruption arcade associated with the great decimetric burst on 21 April 2002 (Fig. 28d), as derived from TRACE 195 Å images and X-ray source centroids from the Ramaty High-Energy Solar Spectroscopic Imager (RHESSI), is shown in Fig. 32 (adapted from Gallagher et al. 2002).

Fig. 32
figure 32

Radial growth of the post-eruption loop system for the flare associated with the great decimetric (DCIM) burst on 21 April 2002 based on TRACE 195 Å and RHESSI images. X-ray source centroids at 3–6, 6–12, and 12–25 keV are indicated by diamonds, squares, and triangles, respectively. The red horizontal bar indicates the interval of peak decimetric emission (~ 01:45–02:30 UT). Image adapted from Gallagher et al. (2002)

The pattern of higher-energy emissions from the most recently formed loops (e.g., Švestka et al. 1987; Anzer and Pneuman 1982) is a standard feature of the CSHKP model (Fig. 10). Our speculative working hypothesis to account for delayed out-sized decimetric burst such as shown in Figs. 26, 27, 28 and 29, based on analogy with AKR in Earth’s magnetosphere (where such characteristics of the emission region as a strong electric field, density cavity, and horseshoe electron distribution, can be observed in situ), is given in schematic form in Fig. 33, taken from Cliver et al. (2011).

Fig. 33
figure 33

Image reproduced with permission from Cliver et al. (2011), copyright by AAS

Schematic showing how ECM decimetric emission might arise in post-eruption loops as a result of strong late-phase reconnection and electron acceleration in a field-aligned potential drop (and density depletion) in conjunction with delayed coronal hard X-ray and microwave bursts.

The decimetric spike emission in Fig. 31 could also be produced by plasma emission (Zheleznyakov and Zaitsev 1975; Kuijpers et al. 1981; Bárta et al. 2011a, b; Karlický and Bárta 2011; Feng et al. 2018; Karlický et al. 2021), the accepted mechanism for solar metric type II and type III bursts as well as for fine structure in Type IV bursts (e.g., Kaneda et al. 2017). Marqué et al. (2018) write, “[The ECM emission] mechanism is usually expected to not operate in the solar corona, because of the high electron plasma frequency. The exceptional occurrence of the strong radio burst near 1000 MHz [in reference to an event on 4 November 2015 they analyzed (Fig. 29)] could of course be explained by an exceptional situation of plasma parameters in the corona, so that electron cyclotron maser emission might arise. The argument was put forward by Régnier (2015) and Cliver et al. (2011). However, the identification of well-known fine structures, like fiber bursts and zebra patterns, that are observed in type IV bursts, but usually at lower frequencies where it is still more unlikely that the cyclotron frequency exceeds the plasma frequency, makes such an exceptional situation improbable. Instead, it rather argues for coherent plasma emission.”

For either interpretation (plasma emission or ECM emission), an exceptional situation of plasma parameters seems a necessity to explain the rarity of outsized decimetric spike bursts in the context of the standard CSHKP model for eruptive flares. The vast majority of such flares are not accompanied by such emission. Such a special condition, e.g., the formation of a density cavity permitting the fce >> fpe inequality required for ECM emission generation and escape, can also account for the rapid switch on/off behavior of the intense delayed component of certain ~ 1 GHz bursts. The observed/inferred late superimposed spike emission in great decimetric bursts (Figs. 26, 27, 28, 29) implies that in certain cases extreme events are not achieved merely by scaling up an event—adding more of the same (in this case incoherent gyrosynchrotron emission)—but by crossing a physical threshold after which out-sized growth of the event is based on a different emission mechanism from that associated with smaller events.

Such events are referred to as Dragon Kings (Sornette and Quillon 2012), a double metaphor indicating their extreme size and peculiar nature. Dragon Kings have the following characteristics: (1) they do no not belong to the same population as other such events as manifested by the size distributions for 1 GHz bursts in Fig. 30; and (2) they appear as a result of amplifying mechanisms (in this case coherent radio spike emission; Fig. 31) that are not fully active for the rest of the population. The intense delayed ~ 1 GHz solar radio bursts satisfy both criteria.

5 Coronal mass ejections (CMEs)

As shown in Fig. 12 in Sect. 3.1.5, the kinetic energy of CMEs dominates the energy budget of large eruptive flares, accounting for ~ 75% (or more, Sect. 3.1.5) of the total energy released (Emslie et al. 2012; cf. Aschwanden et al. 2017). Not coincidentally, CMEs are the drivers for the two major space weather phenomena—solar particle events and geomagnetic storms (Kahler 1992; Gosling 1993; Green et al. 2018; Schrijver and Siscoe 2012). In fact, Gopalswamy et al. (2018) have shown that CME speed can be used to organize the full range of heliospheric and terrestrial effects of solar eruptions (Fig. 34). One of the geo-effective CMEs that occurred during the sequence of strong solar activity in October–November 2003 is shown in Fig. 35.

Fig. 34
figure 34

Image reproduced with permission from Gopalswamy (2018), copyright by Elsevier

Cumulative frequency distribution function for plane-of-sky speeds of SOHO LASCO CMEs showing the organization of interplanetary and terrestrial phenomena by this parameter. The average speeds of CME populations responsible for various coronal and interplanetary phenomena are indicated by vertical lines. Abbreviations refer to CMEs that are associated with: metric type II bursts (m2), magnetic clouds (MC; Klein and Burlaga 1982), interplanetary CMEs (ICMEs; Russell 2001; cf. Burlaga 2001) without flux rope structure (EJ, for ejecta), shocks detected in situ in the solar wind (S), non-recurrent geomagnetic storms (GM), decametric-hectometric type II bursts (DH), NOAA class S1 SEP events (with peak > 10 MeV fluxes > 10 proton flux units), and ground level enhancement (GLE; > 500 MeV) proton events detected by ground-based neutron monitors.

Fig. 35
figure 35

Image reproduced with permission from https://sci.esa.int/web/soho/-/47806-lasco-c2-image-of-a-cme, copyright by ESA & NASA

CME observed on 28 October 2003 during the Halloween storms (Gopalswamy et al. 2005a, b; Webb and Allen 2004). The measured linear speed was 2,459 km s−1 versus an average CME speed of ~ 450 km s−1. The CME was associated with both an extreme geomagnetic storm and a severe solar proton event (G5 and S4, respectively on the NOAA Space Weather Scales (http://www.swpc.noaa.gov/noaa-scales-explanation). (LASCO CME image and Extreme-ultraviolet Imaging Telescope (EIT; Delaboudinière et al. 1995) image of solar disk.

Webb and Howard (2012) reviewed observations of CMEs and Temmer (2021) recently discussed them from a space weather perspective. See Vršnak (2021) for a recent review of the origins and interplanetary propagation of CMEs.

5.1 CME climatology

Gopalswamy (2018) constructed a CME speed distribution for ~ 27,000 coronal transients observed by SOHO LASCO (Brueckner et al. 1995) from January 1996 to March 2016. As was the case for sunspot group areas (Fig. 6) and flares (Fig. 8), Gopalswamy found that a modified exponential function provided a better fit to the data than a power law over the full range of observations. Riley (2012) had previously noted a well-defined “knee” in the CME speed distribution.

Exponential and power-law fits to CME speed and kinetic energy distributions are shown in Fig. 36 (Gopalswamy 2018). From the speed distribution (left panel), Gopalswamy obtained 100-year (1000-year) CME speeds of 3800 km s−1 (4700 km s−1) based on the exponential fit. For the power-law fit the corresponding values are 4500 km s−1 and 6600 km s−1. The 100-year exponential-based speed is only ~ 10% higher than that of the fastest CME observed by LASCO, 3387 km s−1 (mass = 9.5 × 1015 g) for an event on 10 November 2004 that was associated with a W49 flare. The CME kinetic energy values (right panel) for 100-year (1000-year) events are 4.4 × 1033 erg (1034 erg) based on the modified exponential function. The most energetic LASCO CME occurred on 9 September 2005 (4.2 × 1033 erg; mass = 1.6 × 1017 g; speed = 2257 km s−1; W67).

Fig. 36
figure 36

a Downward cumulative distribution (left hand axis) of the number of CMEs from January 1996 to March 2016 with speeds greater than a given value V (black diamond and red circle data points). This annualized distribution (right hand axis) is fitted with a modified exponential function (solid blue line) and power-law (dashed red line; for the tail of the distribution (red circle data points) to give the corresponding annual occurrence frequency distribution (OFD). The fit equations and parameters are given in the figure panel. b Same as a for CME kinetic energies (E) from 1996 to 2015. Image adapted from Gopalswamy (2018)

Gopalswamy (2011, 2018) obtained an estimate of the fastest possible CME of 6700 km s−1—comparable to the 1000-year estimate from Fig. 36 using a power-law fit—based on a spot group area (6000 μsh) similar to that of April 1947 with a uniform field strength of 6.1 × 103 G (Livingston et al. 2006; cf. Sect. 3.1.7(a)) to yield a total magnetic energy of ~ 4 × 1036 erg. Using the rules of thumb from Emslie et al. (2012; Sect. 3.1.5), this would imply a flare with bolometric energy ~ 1035 erg and a SXR class of ~ X70,000 via Eq. (5). If we assume a 6:1 ratio of bolometric to CME energy, then the corresponding flare would still have a SXR class of ~ X30,000 (~ 5.7 × 1034 erg). For the largest possible SXR flare (X180) flare from Sect. 3.1.7 based on the April 1947 spot group, the corresponding CME would have a total energy (kinetic plus potential) of 8.4 × 1033 erg for a 6:1 CME to flare apportionment.

The search for stellar CMEs is a rapidly developing field (e.g., Odert et al. 2017; Moschou et al. 2017, 2019; Argiroffi et al. 2019; Koller et al. 2021; Namekata et al. 2021). Studies by Harra et al. (2016) and Veronig et al. (2021) suggest that the transient dimming signatures on the Sun left by CMEs (Hudson et al. 1996; Sterling and Hudson 1997; Dissauer et al. 2019; Jin et al. 2020) may provide the best evidence of stellar eruptions. Notably, Veronig et al. (2021) used EUV and X-ray observations from the Extreme Ultraviolet Explorer (Bowyer and Malina 1991), Chandra (Weisskopf et al. 2000) and XMM-Newton (Jansen et al. 2001) to identify 21 CME candidates via dimmings on 13 different Sun-like and late-type flaring stars.Approximately half of the dimmings were found on three stars—the young and rapidly rotating K0V star AB Dor (five events), the young M0Ve star AU Mic (three) and Proxima Centauri (two)—with the others on G- to M-type pre-main-sequence and main-sequence stars. The 21 dimmings exceed the total number of stellar CME detections previously reported.

5.2 Fast transit interplanetary coronal mass ejections (ICMEs)

Before CMEs were discovered (in 1971), their existence was inferred, if not fully appreciated, by the timing delay between solar flares and magnetic storms at Earth (first hinted at by the 1859 event; Stewart, 1861) and spectroscopically via Doppler shifts or direct off-limb (Fényi 1892) observations of eruptive prominences (Švestka and Cliver 1992; Cliver 1995, and references therein). In terms of extreme events, the short timing delays (~ 15–30 h) between sudden commencements of great historical geomagnetic storms (caused when CME-driven shocks strike the magnetosphere) and their associated flares are important because they provide a significant extension of the data base for major CMEs. The shortest ICME transit time yet recorded, 14.6 h (Cliver et al. 1990b), was that for an eruptive flare on 4 August 1972. The flare was preceded by four hours by two closely-spaced sudden commencements at Earth signifying ICMEs (that can be linked to large flares from the same region on 2 August; Pomerantz and Duggal 1974) that presumably “pre-conditioned” the interplanetary medium (e.g., Lugaz et al. 2017), resulting in the record low transit time. From the empirical shock arrival time model of Gopalswamy et al. (2005b, c), a 4700 km s−1 CME (100-year event; exponential fit in Fig. 36a) would have a transit time of 11.8 h versus an asymptotic time in the model of 11.6 h.

Gopalswamy et al. (2005b) compiled a list of fast transit ICMEs (defined as events with Sun-Earth transit times ≲ 30 h) that is reproduced here as Table 5. The only such event that we are aware of that has occurred since is the backside event on 22–23 July 2012, where the travel time from the Sun to ~ 1 AU was 18.6 h (Russell et al. 2013; Baker et al. 2013; Liu et al. 2014a, b; Gopalswamy et al. 2016). The short transit time in this event has been attributed to a preceding CME that reduced the drag force (Vršnak et al. 2013) due to the interaction of the ICME and the ambient solar wind by reducing the density and increasing the flow speed in the ambient medium (Liu et al. 2014b; Temmer and Nitta 2015). Including the September 1859 eruption (transit time ≤ 17.1 h; W12), 19 of the 31 fast transit events in Table 5 originated within 30° of solar central meridian.

Table 5 Historical fast transit events

Active region magnetic fields are the key determinant of peak CME speed in the corona, prior to deceleration. Vršnak (2021) writes, “Statistically, fast and impulsively-accelerated CMEs originate from strong-field regions, and start to accelerate at low heights (Vršnak 2001; Vršnak et al. 2007; Bein et al. 2011) [See also Dere et al. 1997]. This is consistent with the hypothesis that stronger CME accelerations are driven by stronger magnetic fields, as the Lorentz force is the main driver of the eruption. … the kinetic energy of the eruption comes from the free energy stored in the magnetic field, and … it can be concluded that ρv2/2 < B2/2μ0 [where ρ = the CME plasma mass-density and μ0 = the magnetic permeability of free space], i.e., that the CME kinetic energy density cannot exceed the total magnetic energy density, implying vCME < vA, where vA represents the Alfvén speed within the CME body (for details see Vršnak 2008, and Sect. 2.2.3 in Green et al. 2018). Thus, in stronger fields an eruptive structure can basically achieve a higher speed.”

6 Geomagnetic storms and aurorae

While geomagnetic storms are not solar events per se, extreme storms are Earth’s natural detection system for powerful ICMEs with strong embedded southward-pointing magnetic fields. Systematic geomagnetic observations originated nearly 200 years ago (Cawood 1979; Chapman and Bartels 1940) and detailed auroral observations are available for a few centuries before that, providing a long-term indirect record of extreme solar activity.

In this section, we will use the minimum hourly Dst index (Sugiura 1964; Sugiura and Kamei 1991) during a storm as the measure of storm intensity. Because of the threat of geomagnetically induced currents (GICs) to the power grid (Pirjola 2000; Molinski 2002; Schrijver et al. 2014; 2015), dB/dt (where B is the ground magnetic field) is increasingly used as a measure of storm strength (Kataoka and Ngwira 2016; Pulkkinen et al. 2017). Thomson et al. (2011) used extreme value statistics (Coles 2001; Beirlant et al. 2004) to calculate that a magnetic storm with a dH/dt variation (where H is the horizontal component of B) of 1000–4000 (1000–6000) nT/minute at 55–60 north geomagnetic latitude could be expected once every 100 (200) years. Storm strength and GIC amplitude, which is strongly dependent on local ground conductivity (e.g., Love et al. 2019b; Lucas et al. 2020), are not closely related (Pulkkinen et al. 2012), however. For example, Huttunen et al. (2008) write that for GIC measurements in Finland, “The largest GIC of the solar cycle 23 (57.0 A on 29 October 2003) [recorded on the Finnish natural gas pipeline network] took place when Dst was barely at the intense storm level while the largest Dst storm of the solar cycle 23 (on 20 November 2003 with Dst minimum of − 422 nT) was associated with much lower-amplitude GIC (23.8 A).” In Sect. 6.3, we show that the extreme Dst storms considered here are characteristically accompanied by auroral effects—indicative of rapid geomagnetic field variations—extending to the low magnetic latitudes of the four stations (Hermanus, Kakioka, Honolulu, San Juan) used to determine Dst.

6.1 Climatology of extreme geomagnetic storms

Riley and Love (2017) classified storms with minimum Dst values in the range from − 600 nT < Dst <  − 250 nT as severe (see also Tsurutani et al. 1992; Gonzalez et al. 1994; Lakhina et al. 2012) and those with minimum Dst <  − 600 nT as extreme. Several determinations of waiting times have been made for storms with minimum Dst values of ~  − 600 nT, corresponding to the lowest Dst value of − 589 nT observed in modern times (on 14 March 1989 storm; Allen et al. 1989), and − 850 nT, the minimum hourly Dst value inferred by Siscoe et al. (2006) for the 2 September 1859 storm. Results of these studies are given in Table 6. The calculated waiting times in the table for a storm with minimum Dst ~  − 600 nT range from 25 to 60 years with a median value of 55 years depending on the data interval considered, the assumed form of the Dst peak intensity distribution, and the analysis techniques employed. The corresponding values for a storm with minimum Dst of − 850 nT are a range from 49 to 333 years with a median of 93 years. Estimates for the minimum Dst value of 100-year storms range from − 542 nT (Love 2020) based on a lognormal distribution to − 1100 nT (Kataoka and Ngwira 2016; power-law distribution). Gopalswamy’s (2018) estimates for a 100-year storm are − 603 nT (modified exponential distribution) and − 774 nT (power-law distribution). For a 1000-year event he obtained − 845 nT (exponential function) and − 1470 nT (power law). As we will see below, there is evidence that the 1000-year estimate of − 845 nT has been exceeded three times within the last ~ 160 years, so in this case the power law may be more appropriate.

Table 6 Waiting time studies for extreme geomagnetic storms with Dst ≲ − 600 nT

Gonzalez et al. (2011) and Baker et al. (2013) have presented evidence based on the “fast transit” ICMEs of 4 August 1972 and 22–23 July 2012 (see Sect. 5.2 and Table 5) that suggest the possibility of far larger storms than one with a minimum Dst of ~  − 850 nT. For the 4 August 1972 event, Gonzalez et al. (2011) estimated that if the ICME magnetic field had been southward, the associated geomagnetic storm would have had a minimum Dst of ~  − 1400 nT because of its measured/inferred high CME speed (Zastenkar et al. 1978; Cliver et al. 1990a). For the July 2012 backside ICME, a model for the 22–23 July 2012 storm based on solar wind observations that assumed optimal solar wind magnetosphere coupling conditions (based on seasonal and time-of-day orientation of Earth's magnetic dipole; Russell and McPherron 1973; Cliver et al. 2000, 2002; Temerin and Li 2002) yielded a minimum Dst value of − 1182 nT (Baker et al. 2013). As we will see below, the low-latitude aurorae of September 1859 and February 1872 suggest minimum hourly Dst values from ~ 1200–1250 nT. Vasyliũnas (2011) has argued that the maximum strength of a geomagnetic storm will be limited to ~  − 2500 nT by the inability of Earth's dipole field to balance the mechanical stresses on magnetospheric plasma beyond a certain point. Recently, from a consideration of an “ICME in a sheath” as a storm driver, Liu et al. (2020) obtained a comparable limiting Dst value of ~  − 2000 nT for an extreme geomagnetic storm.

6.2 Identified geomagnetic storms (1500-present) with Dst <  − 500 nT

Until recently, there were only two well-documented cases of geomagnetic storms comparable to or more intense than the 14 March 1989 event (Allen et al. 1989; Yokoyama et al. 1998; Kappenman