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A&A
Volume 697, May 2025
Article Number A80
Number of page(s) 7
Section Astrophysical processes
DOI https://doi.org/10.1051/0004-6361/202452999
Published online 07 May 2025

© The Authors 2025

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

With an age of approximately 4.6 Grs and a rotation rate of about 25 days, our Sun is an old and slowly rotating main sequence G-type star. However, it remains magnetically active, often producing eruptive events such as flares and coronal mass ejections. These events vary by several orders of magnitude in energy and occur on a wide range of timescales. The upper threshold of such events is not well constrained and, until recently, had been limited to direct instrumental observations from the past ∼50 years. Strong solar energetic particle events, such as ground-level enhancements, have been observed to occur on average once per year (e.g., Papaioannou 2023). Likewise, with the help of cosmogenic radionuclide records (i.e., of 10Be, 14C, and 36Cl), much more extreme solar energetic particle events were identified on a multi-millennial timescale (e.g., Miyake et al. 2012; Mekhaldi et al. 2015).

Missions such as Kepler (e.g., Koch et al. 2010) and the Transiting Exoplanet Survey Satellite (TESS, e.g., Ricker et al. 2015) have observed flares on other stars. In line with what we know about solar flares (e.g., Shibata & Magara 2011), it is commonly accepted that these stellar flares (e.g., Gershberg 2005; Reid & Hawley 2005; Cliver et al. 2022) result from the sudden discharge of magnetic energy that accumulates around starspots. Accordingly, the observations show that (i) young stars with rotation rates of only a few days, (ii) binary stars, and (iii) cool M stars are even more magnetically active and tend to produce so-called superflares (e.g., Benz & Güdel 2010; Tristan et al. 2023) that exceed solar bolometric flare energies by a factor of 10 to 106 (i.e., ∼1033−1038 erg; Schaefer et al. 2000).

Using data from the space-borne Kepler mission, Maehara et al. (2012) analyzed a sample of 83 000 solar-type stars – specifically G-type main-sequence stars with effective temperatures (Teff) between 5100 K and 6000 K and surface gravity log (g) values of at least 4.0. They identified 365 superflares occurring on 148 solar-type stars that had bolometric energies at least an order of magnitude higher than those of typical solar flares (∼1032 erg; Emslie et al. 2012).

By investigating more extended Kepler periods, on the order of 500 days, Shibayama et al. (2013) found 1547 flares on 279 solar-type stars and determined that the bolometric flare energies of solar-type stars are 10–104 orders of magnitude stronger (i.e., 1033−1036 erg). However, the Gaia Data Release (DR) 2 stellar radius data released in 2018 indicated a potential contamination of subgiants in the Kepler sample of solar-type stars. Accordingly, Notsu et al. (2019) reinvestigated the Shibayama et al. (2013) sample and found that 40% of its solar-type stars were subgiants, which thus needed to be removed from the statistical sample. This drastically reduced the number of stars in the stellar solar-type sample and thus in the sample of Sun-like stars, a subset of solar-type stars with effective temperatures ranging between 5600 K and 6000 K, log(g)≥4.0, and rotation periods > 20 days (i.e., slow rotators; see, e.g., Table 1 in Okamoto et al. 2021).

Taking into account data from the four-year Kepler mission, covering about 1500 days of observations, as well as results from Gaia DR2 and using updated methods to detect flares in the sample, Okamoto et al. (2021) conducted a new statistical analysis. This led to an increase in the number of known solar-type stars and, in particular, Sun-like stars compared to the sample from Notsu et al. (2019). This new sample allowed for a more dedicated statistical analysis, revealing 2341 superflares on 265 solar-type stars and 26 superflares on 15 Sun-like stars. In addition, Okamoto et al. (2021) show that (i) flare energies of up to 1036 erg occurred on young (a few hundred megayear-old) and fast-rotating (rotation period, Prot, of around a few days) solar-type stars; (ii) in the case of solar-type stars, the flare energy decreased with an increasing rotation period; and (iii) superflares with energies of up to 4×1034 erg occurred on Sun-like stars. Statistically speaking, Sun-like stars show superflares with energies of around 7×1033 erg (1034 erg) on average every 3000 (6000) years (e.g., Hayakawa et al. 2023; Usoskin et al. 2023). We note that their methods can only be applied to stars with known stellar rotation periods. However, when it comes to flaring stars, a crucial stellar question was not addressed in any of these statistical studies: whether these Sun-like stars are spot-dominated or – like the current Sun – faculae-dominated (and therefore directly comparable to the Sun).

Bright magnetic features such as faculae, plages, and networks as well as dark magnetic features like spots have been extensively differentiated and analyzed in the Sun. These differences are attributed to the relative thickness of emerging magnetic flux tubes (e.g., Fligge & Solanki 2000; Solanki & Fligge 2002) with specific references to their structural dimensions. However, in the context of other stars, obtaining data of comparable quantity and quality is currently unrealistic. The lack of continuous observations and unresolved surface details makes distinguishing faculae from spots particularly challenging.

The center-to-limb contrast modulates the brightness distribution of stellar disks and manifests as a limb-darkening effect in the continuum. This optical phenomenon causes the central region of the disk to appear brighter than its edges, especially at optical wavelengths. This effect arises because the disk's central region is optically thicker than the limb. This causes most of the light to be emitted from the central regions, making them appear brighter. In contrast, the limb region appears darker as it consists of upper and thinner layers that emit less light. Consequently, dark features (such as spots) and bright features (such as faculae, plages, and the network) imprint distinctive signatures in the time series of stellar brightness variations. Shapiro et al. (2020) and Amazo-Gómez et al. (2020a) find that – by characterizing the particular shapes generated by faculae (an M-like shape) or spot (a V-like shape) transits recorded in the total solar irradiance and comparing them with the Michelson Doppler Imager observations – it is possible to infer whether facular or spot regions dominated the stellar surface of solar analogs (see also Amazo-Gómez et al. 2020b). This method involves analyzing the gradient of the power spectrum (GPS) from photometric time series, which are available for thousands of stars.

Overall, the current Sun is a faculae-dominated star. Thus, to directly compare solar-type stars to the Sun, it is mandatory to distinguish between spot and facular domination (e.g., Shapiro et al. 2020; Amazo-Gómez et al. 2020a). Consequently, the Sun should only be compared to Sun-like stars of comparable masses, radii, effective temperatures, and rotation periods that show faculae-dominated characteristics.

Solar brightness fluctuates over periods ranging from minutes to decades. Notably, these variations in observed brightness can directly be linked to the Sun's rotation period. Understanding this relationship will help us make inferences about other stars. Despite successful stellar surveys by photometric missions like Kepler, the CHaracterising ExOPlanets Satellite (CHEOPS; e.g., Benz et al. 2021), and TESS, a lack of photometric data on rotation periods for stellar solar analogs remains. The main challenges in accurately determining rotation periods for the Sun and its analogs are the nonperiodic light-curve (LC) profiles, the low modulation amplitudes resulting from the random appearance of magnetic features, and the short lifespan of these features relative to the rotation period (Shapiro et al. 2017, 2020).

Solar LCs analyzed with regular periodograms such as Lomb-Scargle, autocorrelation functions, or power spectra do not show a realistic rotation period value equal to (or close to) the expected 26-day period (see Aigrain et al. 2015; Amazo-Gómez et al. 2020a). This suggests that for stars with similar brightness variations as the Sun, similar issues in detecting their rotation periods might occur, which therefore implies that only a small fraction of solar-type systems have been thoroughly analyzed (see Basri et al. 2010; Reinhold et al. 2020).

Despite efforts to construct extensive stellar rotation rate catalogs, we still lack reliable information on rotation periods (e.g., they are unknown for about 87% of the 530 506 stars observed by the Kepler telescope1). With regard to solar twins, this is especially challenging. Even after successfully deriving extensive stellar survey catalogs (e.g., McQuillan et al. 2014; Reinhold et al. 2023, who published rotation periods of about 34030 and 67163 Kepler stars, respectively), we have not adequately studied or characterized many solar twin systems (see Amazo-Gómez et al. 2020b). Thus, a detectability bias toward active stars with clear, sinusoidal, and stable magnetic feature modulation prevents us from finding analog systems with comparable magnetic and activity environments where similar planetary habitability conditions could exist.

The newly developed method discussed in Amazo-Gómez (2021) allows us to infer and quantify the degree of spot or facular contribution on stellar surfaces based on observed LCs and revealed that Sun-like stars are distributed between three regimes: spot- and faculae-dominated stars or those transitioning between the two branches. The analysis of several Kepler Sun-like stars with the GPS method showed that their photospheres display a smooth transition between being dominated by spots or faculae, with the Sun sitting roughly in the middle (Amazo-Gómez et al. 2020a). We quantify this transition with the α factor, which corresponds to the instantaneous area ratio between dark or bright features (Sf/Ss) detected on the stellar surface. We note that in the case of the Sun, the α factor has a value of α⊙ = 0.158, which corresponds to a Sfac/Sspot ratio of ∼3. This directly indicates that the current Sun, on a rotation timescale, is transitioning to a regime dominated by faculae.

Surface manifestations of magnetic flux rely not only on parameters such as effective temperature and rotation period, but also on the strength of the magnetic field, which, in turn, depends on the underlying dynamo. Small differences in the generation of internal magnetic fields will influence surface features and their development into higher atmospheric layers, such as the chromosphere and the corona. Furthermore, these differences might impact the mechanism responsible for driving stellar winds and the frequency of transient events such as flares and coronal mass ejections. These factors are crucial for understanding the evolution of stellar rotation and magnetic activity. They are also critical for characterizing the environments in which exoplanets are embedded as well as for estimating the survival and conditions of their possible exoatmospheres.

2. Magnetic energy stored around starspots

In theory, the magnetic energy accumulated near starspots can explain observed flare intensities. To put stellar and solar observations into context, often a scatter plot of the flare energy as a function of the starspot area is discussed. The upper panel of Fig. 1 shows the solar flare intensities observed by the Geostationary Operational Environmental Satellites (GOES; black dots), the superflare regime based on Kepler 1-minute-cadence observations (red squares; see Maehara et al. 2015), and the sample obtained by Okamoto et al. (2021) (black squares). To generate the observed superflares, the existence of large starspots is an almost mandatory condition (see the discussion in Notsu et al. 2019; Okamoto et al. 2021), which is reflected in the much bigger spot group areas of the superflare regime. This panel also shows that the low-energy superflares of the short-cadence Kepler sample from Maehara et al. (2015) are on the order of the maximum flare energies observed on the current Sun.

thumbnail Fig. 1.

Flare energy as a function of spot group area for solar flares (black dots) and superflares of G-type stars from the Kepler sample by Okamoto et al. (2021). Left panel: Original Sun-like sample from Okamoto et al. (2021). Right panel: Updated solar observations based on the rescaling discussed in Hudson et al. (2024) (purple dots) and updated stellar sample utilizing the updated spot group area function from Herbst et al. (2021) (blue dots and associated errors).

As discussed in Maehara et al. (2012), Shibayama et al. (2013), and Notsu et al. (2013, 2019), the stellar area covered by a starspot can be described as a function of the stellar brightness variation and the ratio between the spot temperature and the stellar effective temperature. However, Herbst et al. (2021) show that the analytically derived stellar spot group areas are erroneous because the ratio between the spot and effective temperature that had previously been proposed in Berdyugina (2005) needed to be updated. Furthermore, Hudson et al. (2024) show that the solar flare energies measured by GOES-1 through GOES-15 (1975–2016) needed to be scaled up by a factor of 1/0.7. An updated version of the plot incorporating the aforementioned corrections is shown in the lower panel of Fig. 1. These updates further support the idea that superflares may occur on the Sun, as these corrections shift the solar observations well within the flare energy range of the 1-minute-cadence Kepler observations.

In fact, first hints of the occurrence of solar superflares within the past 25 000 years have been revealed with the help of cosmogenic radionuclides like 10Be, 14C, and 36Cl (e.g., Beer et al. 2012). As of now, five extreme solar events (AD993, AD774/775, 660 BC, 5259 BC, and 7176 BC) have been identified with extreme spike-like increases in all three archives (see, e.g., Miyake et al. 2013; Mekhaldi et al. 2015; O’Hare et al. 2019; Brehm et al. 2022, respectively). We note that three more events around 5410 BC, 1052 AD, and 1279 AD (see Miyake et al. 2021; Brehm et al. 2021, respectively) have been found in the radionuclide records and that more may be detected in the future (e.g., Herbst & Papaioannou 2025). However, since these events have not yet been confirmed in all three records, we did not consider them in this study. Nevertheless, one of the strongest (and most famous) is the AD774/775 event (e.g., Papaioannou et al. 2023, 2024). Assuming AD774/775 was a single, extreme solar event, and based on the GOES recalibration by Hudson et al. (2024), Cliver et al. (2022) and Papaioannou et al. (2023) derived an associated X-ray flare intensity of 2×1033−6×1033 erg (see the purple horizontal band in the lower panel of Fig. 1). This brought the upper solar flare energies much closer to the upper limit of Sun-like stars based on the Kepler sample (here from Okamoto et al. 2021) and further supported the consistency between solar- and Sun-like star observations. In the following, the stellar sample by Okamoto et al. (2021) is reanalyzed with a focus on spot or facular domination of Sun-like stars.

3. Stellar sample used in this study

In this study, as a first step, we utilized the Sun-like sample from Okamoto et al. (2021) to check the reported Sun-like stars for facular domination. Utilizing the Gaia DR3 temperatures of the initial 265 solar-type main-sequence stars, we found 119 to be within the effective temperature regime of 5600 K < Teff < 6000 K. These 119 stars form the background of our study.

4. Facular versus spot dominance

In this study, we analyzed these 119 stars in the field of view (FOV) observed during a 90-day quarter by the Kepler telescope and considered LCs acquired in the long-cadence mode (i.e., with a cadence of 29.42 min). We find that 48 stars showed a definitive signal of magnetic features transiting the stellar disk.

However, we note that high-quality LCs were required to improve our ability to distinguish between surface features – spots or faculae – when applying the GPS method. A highly periodic stellar signal improves the definitive detection of a certain feature, an enhancement normally observed in stars that are more active than the Sun. This behavior is evident in the entire sample, where most of the analyzed stars show predominant periodic activity (e.g., with high bright contrast, longer than a single rotation lifetime evolution, and a non-stochastic emergence of magnetic features). This results in a periodic LC with a detectable M or V shape, which is characteristic of faculae or spots, respectively (see Fig. 2a for a periodic facular transit LC profile).

thumbnail Fig. 2.

Analysis of the rotation period based on the Kepler LC of KIC 11599385 (a), generalized Lomb-Scargle (GLS) periodogram (b), autocorrelation function (ACF; c), power spectrum (d), and GPS (d). Each panel displays the most prominent periods detected with each method.

However, such ideal periodic conditions have only been observed on the Sun on isolated occasions (see Kopp 2016, and Figs. 1 and 2 in Amazo-Gómez et al. 2020a). It is more common to observe a superposition of faculae and spots (i.e., the two phenomena coexisting). Thereby, a solar LC is more disorganized compared to regular LCs of more active stars. Nevertheless, on rare occasions, it is possible to observe a clear transit of isolated, stable (long-lasting), and periodic dark or bright features dominating the solar LC within a rotational timescale. We note a similar behavior in the LC of KIC 11599385 shown in Fig. 2a. Since a higher contribution of bright features is observed in the total solar irradiance, stronger faculae and network area components are typically detected. This phenomenology results from less stable and thinner emerging magnetic flux tubes, making the presence of bright features more evident (Fligge & Solanki 2000).

Within the time frame of a solar magnetic cycle, faculae dominate. This explains why a brighter Sun is observed even during solar maximum conditions. We note that the brightness of the Sun, in general, varies only slightly with solar activity. Reconstructions of cosmogenic radionuclide production rates further indicate that the Sun is somewhat brighter today than it was in the past 8,000 years (e.g., Solanki et al. 2004; Steinhilber et al. 2009). This suggests that the Sun likely had less facular contribution over the last few millennia. Thus, understanding the transition of stellar activity investigated in Amazo-Gómez et al. (2020b) is crucial to forming a clear picture of the past, recent, and future changes of our Sun, and also to gain further insight into the emergence and evolution of life in the Solar System and beyond.

This solar perspective gives a basic approach to the challenge of finding solar twins among Sun-like stars and highlights the need for a careful comparison of the Sun to more periodic, more active, and, ultimately, easier to detect Sun-like stars. Further studies of low-periodicity and faculae-dominated stars are necessary, and the scarcity of such stars in current observations may be due to observational challenges like low signal-to-noise ratios, long rotation periods, and the superposition of observable features.

In order to recover a characteristic signature from the surface features, we calculated the GPS as the ratio between the power spectral density, P(ν) (see Fig. 2d) at two adjacent frequency grid points (GPS≡P(νk+1)/P(νk)) as follows:

GPS = 1 + d ln P ( ν k ) d ln ν · ( Δ ν ) k ν k , $$ {\rm GPS} = 1+\frac {{\rm d} \ln P(\nu _k)}{{\rm d} \ln \nu } \cdot \frac {{(\Delta \nu )}_k}{\nu _k}, $$(1)

where Δν represents the spacing of the frequency grid. Here, the power spectrum gradient is calculated on an equidistant logarithmic scale grid, with Δν/ν assumed to be constant. After deriving the GPS profile (see Fig. 2e), we located the highest amplitude peak that corresponds to the inflection points (IPi) of the power spectrum profile. Therefore, the IPi values represent a GPS plotted on a log-log scale. As discussed in Torrence & Compo (1998), Shapiro et al. (2020), and Amazo-Gómez (2021), the GPS depends on the chosen frequency grid. The location of the inflection points is proportional to the stellar-rotation period via the calibration factor, defined as αi=IPi/Prot, where IPi is the period corresponding to the (i) inflection point. In this case, we only used the high-frequency, high-intensity inflection point and its corresponding α proportionality factor. We disregarded other inflection point peaks with lower frequency and intensity values. Furthermore, we computed the uncertainty of the α value as the standard deviation between the positions of the inflection points.

Thus, the relative presence of faculae (M-like shapes) or spots (V-like shapes) in the LC profile results in a characteristic inflection point location in the GPS. This allows for the estimation of a proportionality factor that quantitatively describes the presence of specific magnetic features on the stellar surface. We estimated the relative dependence of the facular-to-spot area ratio, Sfac/Sspot, at the time of maximum area from the observed inflection point position and from the corresponding α values, according to the models discussed in Shapiro et al. (2020) and the observed stars in Amazo-Gómez et al. (2020b).

Figure 3 shows the α factor of the 48 Kepler stars in our sample (blue and green dots) compared with the values obtained for 400 modeled LCs with different Sfac/Sspot ratios (adapted from Shapiro et al. 2020; Amazo-Gómez et al. 2020b). As can be seen, the Sun (α⊙ = 0.158 and Sfac/Sspot∼3; yellow dot) is located in the transition region between the branches of spot-dominated surfaces (left) and faculae-dominated LCs (right). This indicates that the Sun is transitioning to a regimen dominated by faculae. As shown, only 4 of the 48 stars analyzed with the GPS (highlighted in green; i.e., KIC 3853938, KIC 8424356, KIC 11599385, and KIC 12266582) are transitioning between spot- and faculae-domination (i.e., GPS α<0.18, as indicated by the dashed black line). Furthermore, two of these four show α values in agreement with that of the Sun (i.e., KIC 11599385, and KIC 12266582) within estimated errors. However, within a rotational timescale, all the stars in the sample show a faster, periodic, and sinusoidal modulation in the LC. This implies that our selected stars deviate strongly from the typical solar brightness behavior. The derived parameters of the 48 stars in our sample are listed in Table A.1.

thumbnail Fig. 3.

Diagram of the α factor (parameter proportional to the instantaneous surface ratio between bright and dark features, Sfac/Sspot) for the 48 Kepler stars compared with values obtained for 400 modeled LCs with different Sfac/Sspot (adapted from Shapiro et al. 2020; Amazo-Gómez et al. 2020b). The Sun (yellow circle) is located in the transition region between the branches of spot-dominated surfaces (left) and faculae-dominated LCs (right). Blue circles represent the 48 stars analyzed with the GPS in this work, 4 of which are highlighted in green (i.e., KIC 3853938, KIC 8424356, KIC 11599385, and KIC 12266582) since they are transitioning between spot- and faculae-domination (i.e., GPS α<0.18, as indicated by the dashed black line). The dotted black lines correspond to solar values.

Note that the predominantly high variability of the Sun-like stars in our sample does not necessarily imply that the Sun is an outlier. One possible unbiased interpretation is that the solar levels of photometric variability are typical for stars with near-solar fundamental parameters but with difficult to measure (or even unknown) rotation periods, which we see by comparing the solar variability and of the stellar sample.

5. Conclusions and future prospects

Of the 48 stars analyzed by the GPS, only 4 (i.e., KIC 3853938, KIC 8424356, KIC 11599385, and KIC 12266582) were found to be transitioning between spot- and faculae-domination (i.e., GPS α<0.18) and thus are comparable to the current Sun (α∼0.158). Only three of the four have effective temperatures, radii, and masses roughly similar to the solar values (i.e., KIC 3853938, KIC 8424356, and KIC 11599385).

However, only one star of the entire sample, namely KIC 11599385, with a rotation period of Prot = 23.94 days, further matches the solar rotation period. Thus, in the given sample, KIC 11599385 is the only true current Sun analog. During its observation, Kepler detected one flare that had a bolometric energy of 5.5×1033 erg. As shown in Fig. 4, this flare energy falls right within the rescaled flare intensities of the AD774/775 event detected in the cosmogenic radionuclide records.

thumbnail Fig. 4.

Updated solar and stellar samples as shown in the lower panel of Fig. 1. Upper panel: Flaring of the 48 stars used in our analysis (highlighted in blue). Lower panel: Flaring activity of the 4 Sun-like stars transitioning between spot- and faculae-dominated stars. The flare energy of the only star in the sample that also shows a comparable rotation period (i.e., KIC 11599385) is highlighted by the dashed red line, which falls right within the reconstructions of the AD774/775 event.

Of course, 48 stars are not enough to fairly compare the Sun with its analogs, and follow-up studies are needed. Most recently, with the help of the Kepler archive2 and the Gaia DR3 release3, Vasilyev et al. (2024) derived a new sample of main-sequence stars with effective temperatures between 5000 K < Teff<6500 K and absolute magnitudes between 4 mag < MG < 6 mag. They identified 2889 superflares on 2527 Sun-like stars. This new statistical analysis suggests that superflares are much more frequent on Sun-like stars than previously thought: for instance, superflares with energies >1034 erg on average likely occur roughly every 1000 years. The study further suggests that the resulting flare-frequency distribution fits an extrapolation of the solar distribution indicative of the same physical mechanisms. This was also investigated by Yang et al. (2025) who conclude that fast rotators have the same dynamo process as the Sun and that a solar-type dynamo can be assumed at all evolutionary stages of a star. Therefore, in a next step, we will utilize the newly derived Kepler sample from Vasilyev et al. (2024) and add available TESS data to redo the analysis presented above.

However, even though we searched for similar Teff, Prot, stellar ages, and faculae-dominated stars (or those transitioning to facular domination), a difference in stellar metallicity would change the picture. As discussed in Witzke et al. (2018), for example, even a slight variation in metallicity or effective temperature within the observational error range substantially impacts the photometric brightness change in relation to the Sun. Thus, it is essential to precisely ascertain the fundamental stellar properties to fully understand variations in stellar brightness.

The Kepler mission has provided us with long-term time series for thousands of Sun-like stars. However, the specific location of its relatively faint FOV has hindered a proper ground-based spectroscopic follow-up, limiting the spectral information for many targets. The lack of complementary information on the Kepler field prevents a robust characterization of the solar analogs. Spectral follow-up was possible for the extensive TESS survey. However, the 27-day averageg short LCs represent an issue in obtaining the rotational modulation for closer solar analogs with a slower rotation period compared to the Sun.

In the coming years, the support activities for the PLAnetary Transits and Oscillations of stars mission (PLATO, see Rauer et al. 2014, 2024) are expected to provide long-term follow-up observations of stellar activity, particularly that of G stars, and will improve, for example, the sampling of targets with simultaneous or contemporaneous spectral information for the upcoming PLATO high-cadence and long-term photometric time series. Measuring stellar surface rotation, photometric activity, and long-term modulations is an integral part of the PLATO pipeline (Breton et al. 2024). In addition, the mission will be able to deliver more accurate and precise stellar characterization for a larger sample of targets than Kepler (Goupil et al. 2024), hence improving our insight into the science topics discussed in this paper.

Acknowledgments

KH acknowledges the support of the DFG priority program SPP 1992 “Exploring the Diversity of Extrasolar Planets (HE 8392/1-1)”. EMAG was partially supported by HST GO-15299 and GO-15512 grants and acknowledges support from the German Leibniz-Gemeinschaft under project number P67/2018. AP acknowledges the support from NASA/LWS project NNH19ZDA001N-LWS. KH and AP acknowledge the International Space Science Institute and the supported International Team 464 (ETERNAL). This work presents results from the European Space Agency (ESA) space mission PLATO. The PLATO payload, the PLATO Ground Segment and PLATO data processing are joint developments of ESA and the PLATO Mission Consortium (PMC). Funding for the PMC is provided at national levels, in particular by countries participating in the PLATO Multilateral Agreement (Austria, Belgium, Czech Republic, Denmark, France, Germany, Italy, Netherlands, Portugal, Spain, Sweden, Switzerland, Norway, and United Kingdom) and institutions from Brazil. Members of the PLATO Consortium can be found at https://platomission.com/. The ESA PLATO mission website is https://www.cosmos.esa.int/plato. We thank the teams working for PLATO for all their work. This project further has received funding from the Research Council of Norway through the Centres of Excellence funding scheme, project number 332523 (PHAB) and was supported by the European Union (ERC, PastSolarStorms, grant agreement no. 101142677). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

4

Stars in the transition between being spot- and faculae-dominated stars are highlighted in blue.

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Appendix A: Stellar sample

Table A.1.

Characteristics of the stellar sample and corresponding parameters derived in this study.

All Tables

Table A.1.

Characteristics of the stellar sample and corresponding parameters derived in this study.

In the text

All Figures

thumbnail Fig. 1.

Flare energy as a function of spot group area for solar flares (black dots) and superflares of G-type stars from the Kepler sample by Okamoto et al. (2021). Left panel: Original Sun-like sample from Okamoto et al. (2021). Right panel: Updated solar observations based on the rescaling discussed in Hudson et al. (2024) (purple dots) and updated stellar sample utilizing the updated spot group area function from Herbst et al. (2021) (blue dots and associated errors).
In the text
thumbnail Fig. 2.

Analysis of the rotation period based on the Kepler LC of KIC 11599385 (a), generalized Lomb-Scargle (GLS) periodogram (b), autocorrelation function (ACF; c), power spectrum (d), and GPS (d). Each panel displays the most prominent periods detected with each method.
In the text
thumbnail Fig. 3.

Diagram of the α factor (parameter proportional to the instantaneous surface ratio between bright and dark features, Sfac/Sspot) for the 48 Kepler stars compared with values obtained for 400 modeled LCs with different Sfac/Sspot (adapted from Shapiro et al. 2020; Amazo-Gómez et al. 2020b). The Sun (yellow circle) is located in the transition region between the branches of spot-dominated surfaces (left) and faculae-dominated LCs (right). Blue circles represent the 48 stars analyzed with the GPS in this work, 4 of which are highlighted in green (i.e., KIC 3853938, KIC 8424356, KIC 11599385, and KIC 12266582) since they are transitioning between spot- and faculae-domination (i.e., GPS α<0.18, as indicated by the dashed black line). The dotted black lines correspond to solar values.
In the text
thumbnail Fig. 4.

Updated solar and stellar samples as shown in the lower panel of Fig. 1. Upper panel: Flaring of the 48 stars used in our analysis (highlighted in blue). Lower panel: Flaring activity of the 4 Sun-like stars transitioning between spot- and faculae-dominated stars. The flare energy of the only star in the sample that also shows a comparable rotation period (i.e., KIC 11599385) is highlighted by the dashed red line, which falls right within the reconstructions of the AD774/775 event.
In the text

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