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We report progress on the ongoing recalibration of the Wolf sunspot number (SN) and Group sunspot number (GN) following the release of version 2.0 of SN in 2015. This report constitutes both an update of the efforts reported in the 2016 Topical Issue of Solar Physics and a summary of work by the International Space Science Institute (ISSI) International Team formed in 2017 to develop optimal SN and GN re-construction methods while continuing to expand the historical sunspot number database. Significant progress has been made on the database side while more work is needed to bring the various proposed SN and (primarily) GN reconstruction methods closer to maturity, after which the new reconstructions (or combinations thereof) can be compared with (a) ``benchmark'' expectations for any normalization scheme (e.g., a general increase in observer normalization factors going back in time), and (b) independent proxy data series such as F10.7 and the daily range of variations of Earth's undisturbed magnetic field. New versions of the underlying databases for SN and GN will shortly become available for years through 2022 and we anticipate the release of next versions of these two time series in 2024., Comment: 21 figures, 4 tables. To be published in Solar Physics

Aims. The first relativistic solar proton event of solar cycle 25 (SC25) was detected on 28 October 2021 by neutron monitors (NMs) on the ground and particle detectors onboard spacecraft in the near-Earth space. This is the first ground level enhancement (GLE) of the current cycle. A detailed reconstruction of the NM response together with the identification of the solar eruption that generated these particles is investigated based on in-situ and remote-sensing measurements. Methods. In-situ proton observations from a few MeV to $\sim$500 MeV were combined with the detection of a solar flare in soft X-rays (SXRs), a coronal mass ejection (CME), radio bursts and extreme ultraviolet (EUV) observations to identify the solar origin of the GLE. Timing analysis was performed and a relation to the solar sources was outlined. Results. GLE73 reached a maximum particle rigidity of $\sim$2.4 GV and is associated with type III, type II, type IV radio bursts and an EUV wave. A diversity of time profiles recorded by NMs was observed. This points to an anisotropic nature of the event. The peak flux at E$>$10 MeV was only $\sim$30 pfu and remained at this level for several days. The release time of $\geq$1 GV particles was found to be $\sim$15:40 UT. GLE73 had a moderately hard rigidity spectrum at very high energies ($\gamma$ $\sim$5.5). Comparison of GLE73 to previous GLEs with similar solar drivers is performed, Comment: accepted for publication in A&A Let

Solar activity in all its varied manifestations is driven by the magnetic field. Particularly important for many purposes are two global quantities, the Sun's total and open magnetic flux, which can be computed from sunspot number records using models. Such sunspot-driven models, however, do not take into account the presence of magnetic flux during grand minima, such as the Maunder minimum. Here we present a major update of a widely used simple model, which now takes into account the observation that the distribution of all magnetic features on the Sun follows a single power law. The exponent of the power law changes over the solar cycle. This allows for the emergence of small-scale magnetic flux even when no sunspots are present for multiple decades and leads to non-zero total and open magnetic flux also in the deepest grand minima, such as the Maunder minimum, thus overcoming a major shortcoming of the earlier models. The results of the updated model compare well with the available observations and reconstructions of the solar total and open magnetic flux. This opens up the possibility of improved reconstructions of sunspot number from time series of cosmogenic isotope production rate.

The 11-year solar cycle is the dominant pattern of solar activity reflecting the oscillatory dynamo mechanism in the Sun. Solar cycles were directly observed since 1700, while indirect proxies suggest their existence over a much longer period of time but generally without resolving individual cycles and their continuity. Here we reconstruct individual cycles for the last millennium using recent 14C data and state-of-the-art models. Starting with the 14C production rate determined from the so far most precise measurements of radiocarbon content in tree rings, solar activity is reconstructed in three physics-based steps: (1) Correction of the 14C production rate for the changing geomagnetic field; (2) Computation of the open solar magnetic flux; and (3) Conversion into sunspot numbers outside of grand minima. Solar activity is reconstructed for the period 971-1900 (85 individual cycles). This more than doubles the number of solar cycles known from direct solar observations. We found that lengths and strengths of well-defined cycles outside grand minima are consistent with those obtained from the direct sunspot observations after 1750. The validity of the Waldmeier rule is confirmed at a highly significant level. Solar activity is found to be in a deep grand minimum when the activity is mostly below the sunspot formation threshold, during about 250 years. Therefore, although considerable cyclic variability in 14C is seen even during grand minima, individual solar cycles can hardly be reliably resolved therein. Three potential solar particle events, ca. 994, 1052 and 1279 AD, are shown. A new about 1000-year long solar activity reconstruction, in the form of annual (pseudo) sunspot numbers with full assessment of uncertainties, is presented based on new high-precision 14C measurements and state-of-the-art models, more than doubling the number of individually resolved solar cycles., Comment: Accepted for publication in Astronomy and Astrophysics

A new model of cosmogenic tritium ($^3$H) production in the atmosphere is presented. The model belongs to the CRAC (Cosmic-Ray Atmospheric Cascade) family and is named as CRAC:3H. It is based on a full Monte-Carlo simulation of the cosmic-ray induced atmospheric cascade using the Geant4 toolkit. The CRAC:3H model is able, for the first time, to compute tritium production at any location and time, for any given energy spectrum of the primary incident cosmic ray particles, explicitly treating, also for the first time, particles heavier than protons. This model provides a useful tool for the use of $^3$H as a tracer of atmospheric and hydrological circulation. A numerical recipe for practical use of the model is appended.

Numerous analyses suggest the existence of various quasi-periodicities in solar activity. The power spectrum of solar activity recorded in sunspot data is dominated by the $\sim$11-year quasi-periodicity, known as the Schwabe cycle. In the mid-term range (1 month -- 11 years) a pronounced variability known as a quasi-biennial oscillation (QBO) is widely discussed. In the shorter time scale a pronounced peak, corresponding to the synodic solar rotation period ($\sim$ 27 days) is observed. Here we revisited the mid-term solar variability in terms of statistical dynamic of fully turbulent systems, where solid arguments are required to accept an isolated dominant frequency in a continuous (smooth) spectrum. For that, we first undertook an unbiased analysis of the standard solar data, sunspot numbers and the F10.7 solar radioflux index, by applying a wavelet tool, which allows one to perform a frequency-time analysis of the signal. Considering the spectral dynamics of solar activity cycle by cycle, we showed that no single periodicity can be separated, in a statistically significant manner, in the specified range of periods. We examine whether a model of solar dynamo can reproduce the mid-term oscillation pattern observed in solar data. We found that a realistically observed spectrum can be explained if small spatial (but not temporal) scales are effectively smoothed. This result is important because solar activity is a it global feature, although monitored via small-scale tracers like sunspots., Comment: accepted in MNRAS

As a result of intense solar activity during the first ten days of September, a ground level enhancement occurred on September 10, 2017. Here we computed the effective dose rates in the polar region at several altitudes during the event using the derived rigidity spectra of the energetic solar protons. The contribution of different populations of energetic particles viz. galactic cosmic rays and solar protons, to the exposure is explicitly considered and compared. We also assessed the exposure of a crew members/passengers to radiation at different locations and at several cruise flight altitudes and calculated the received doses for two typical intercontinental flights. The estimated received dose during a high-latitude, 40 kft, $\sim$ 10 h flight is $\sim$ 100 $\mu$Sv., Comment: 16 pages, 6 figures

Changes in solar irradiance and in its spectral distribution are among the main natural drivers of the climate on Earth. However, irradiance measurements are only available for less than four decades, while assessment of solar influence on Earth requires much longer records. The aim of this work is to provide the most up-to-date physics-based reconstruction of the solar total and spectral irradiance (TSI/SSI) over the last nine millennia. The concentrations of the cosmogenic isotopes 14C and 10Be in natural archives have been converted to decadally averaged sunspot numbers through a chain of physics-based models. TSI and SSI are reconstructed with an updated SATIRE model. Reconstructions are carried out for each isotope record separately, as well as for their composite. We present the first ever SSI reconstruction over the last 9000 years from the individual 14C and 10Be records as well as from their newest composite. The reconstruction employs physics-based models to describe the involved processes at each step of the procedure. Irradiance reconstructions based on two different cosmogenic isotope records, those of 14C and 10Be, agree well with each other in their long-term trends despite their different geochemical paths in the atmosphere of Earth. Over the last 9000 years, the reconstructed secular variability in TSI is of the order of 0.11%, or 1.5 W/m2. After the Maunder minimum, the reconstruction from the cosmogenic isotopes is consistent with that from the direct sunspot number observation. Furthermore, over the nineteenth century, the agreement of irradiance reconstructions using isotope records with the reconstruction from the sunspot number by Chatzistergos et al. (2017) is better than that with the reconstruction from the WDC-SILSO series (Clette et al. 2014), with a lower chi-square-value.

Using neutron monitor and space-borne data we performed an analysis of the second ground level enhancement of solar cycle 24, namely the event of 10 September 2017 (GLE 72) and derive the spectral and angular characteristics of GLE particles. We employ new neutron monitor yield function and a recently proposed model based on optimization procedure. The method consists of simulation of particle propagation in a model magnetosphere in order to derive the cut-off rigidity and neutron monitor asymptotic directions. Subsequently the rigidity spectrum and anisotropy of GLE particles are obtained in their dynamical evolution during the event on the basis of inverse problem solution. The derived angular distribution and spectra are briefly discussed., Comment: 21 pages, 10 figures, 1 table

Aims. Lunar soil and rocks are not protected by a magnetic field or an atmosphere and are continuously irradiated by energetic particles that can produce cosmogenic radioisotopes directly inside rocks at different depths depending on the particle's energy. This allows the mean fluxes of solar and galactic cosmic rays to be assessed on the very long timescales of millions of years. Methods. Here we show that lunar rocks can serve as a very good particle integral spectrometer in the energy range 20-80 MeV. We have developed a new method based on precise modeling, that is applied to measurements of $^{26}$Al (half-life ~0.7 megayears) in lunar samples from the Apollo mission, and present the first direct reconstruction (i.e., without any a priori assumptions) of the mean energy spectrum of solar and galactic energetic particles over a million of years. Results. We show that the reconstructed spectrum of solar energetic particles is totally consistent with that over the last decades, despite the very different levels of solar modulation of galactic cosmic rays ($\phi=496\pm 40$ MV over a million years versus $\phi= 660\pm 20$ MV for the modern epoch). We also estimated the occurrence probability of extreme solar events and argue that no events with the F(>30 MeV) fluence exceeding $5*10^{10}$ and $10^{11}$ cm$^2$ are expected on timescales of a thousand and million years, respectively. Conclusions. We conclude that the mean flux of solar energetic particles hardly depends on the level of solar activity, in contrast to the solar modulation of galactic cosmic rays. This puts new observational constraints on solar physics and becomes important for assessing radiation hazards for the planned space missions., Comment: Astron. Astrophys., in press

Solar activity in the past millennia can only be reconstructed from cosmogenic radionuclide records in terrestrial archives. However, because of the diversity of the proxy archives, it is difficult to build a homogeneous reconstruction. Here we provide a new consistent multiproxy reconstruction of the solar activity over the last 9000 years, using available long-span datasets of 10Be and 14C in terrestrial archives. A new method, based on a Bayesian approach, was applied for the first time to solar activity reconstruction. A Monte Carlo search for the most probable value of the modulation potential was performed to match data from different datasets for a given time. We used six 10Be series from Greenland and Antarctica, and the global 14C production series. The 10Be series were resampled to match wiggles related to the grand minima in the 14C reference dataset. The GRIP and the EDML 10Be series diverge from each other during the second half of the Holocene, while the 14C series lies between them. A likely reason for this is the insufficiently precise beryllium transport and deposition model for Greenland. A slow 6-millennia variability with lows at ca. 5500 BC and 1500 AD of solar activity is found. Two components of solar activity can be statistically distinguished: the main 'normal' component and a component corresponding to grand minima. A possible existence of a component representing grand maxima is indicated, but it cannot be separated from the main component in a statistically significant manner. A new consistent reconstruction of solar activity over the last nine millennia is presented with the most probable values of decadal sunspot numbers and their realistic uncertainties. Independent components of solar activity corresponding to the main moderate activity and the grand-minimum state are identified; they may be related to different operation modes of the dynamo., Comment: A&A, in print (the abstract has been slightly modified w/r to the original paper to match the ArXiv length limitations)

Context. There is no consensus on the amplitude of the historical solar forcing. The estimated magnitude of the total solar irradiance difference between Maunder minimum and present time ranges from 0.1 to 6 W/m2 making uncertain the simulation of the past and future climate. One reason for this disagreement is the applied evolution of the quiet Sun brightness in the solar irradiance reconstruction models. This work addresses the role of the quiet Sun model choice and updated solar magnetic activity proxies on the solar forcing reconstruction. Aims. We aim to establish a plausible range of the solar irradiance variability on decadal to millennial time scales. Methods. The spectral solar irradiance (SSI) is calculated as a weighted sum of the contributions from sunspot umbra/penumbra, fac- ulae and quiet Sun, which are pre-calculated with the spectral synthesis code NESSY. We introduce activity belts of the contributions from sunspots and faculae and a new structure model for the quietest state of the Sun. We assume that the brightness of the quiet Sun varies in time proportionally to the secular (22-year smoothed) variation of the solar modulation potential. Results. A new reconstruction of the TSI and SSI covering the period 6000 BCE - 2015 CE is presented. The model simulates solar irradiance variability during the satellite era well. The TSI change between the Maunder and recent minima ranges between 3.7 and 4.5 W/m2 depending on the applied solar modulation potential. The implementation of a new quietest Sun model reduces, by approximately a factor of two, the relative solar forcing compared to the largest previous estimation, while the application of updated solar modulation potential increases the forcing difference between Maunder minimum and the present by 25-40 %., Comment: Accepted for publication in Astronomy & Astrophysics, 10 pages, 10 figures

In this work, we use two decadal sunspot number series reconstructed from cosmogenic radionuclide data (14C in tree trunks, SN-14C and 10Be in polar ice, SN-10Be) and the Extreme Value Theory to study variability of solar activity during the last 9 millennia. The peaks-over-threshold technique was used to compute, in particular, the shape parameter of the generalized Pareto distribution for different thresholds. Its negative value implies an upper bound of the extreme SN-10Be and SN-14C time series. The return level for 1000 and 10000 years were estimated leading to values lower than the maximum observed values, expected for the 1000-year, but not for the 10000-year return levels, for both series. A comparison of these results with those obtained using the observed sunspot numbers from telescopic observations during the last four centuries suggest that the main characteristics of solar activity have already been recorded in the telescopic period (from 1610 to nowadays) which covers the full range of solar variability from a Grand minimum to a Grand maximum., Comment: 15 pages, 5 figures, 5 tables, accepted for publication in ApJ

The conventional definition of ground-level enhancement (GLE) events requires a detection of solar energetic particles (SEP) by at least two differently located neutron monitors. Some places are exceptionally well suitable for ground-based detection of SEP - high-elevation polar regions with negligible geomagnetic and reduced atmospheric energy/rigidity cutoffs. At present, there are two neutron-monitor stations in such locations on the Antarctic plateau: SOPO/SOPB (at Amundsen-Scott station, 2835 m elevation), and DOMC/DOMB (at Concordia station, 3233 m elevation). Since 2015, when the DOMC/DOMB station started continuous operation, a relatively weak SEP event that was not detected by sea-level neutron-monitor stations was registered by both SOPO/SOPB and DOMC/DOMB, and it was accordingly classified as a GLE. This would lead to a distortion of the homogeneity of the historic GLE list and the corresponding statistics. To address this issue, we propose to modify the GLE definition so that it maintains the homogeneity: A GLE event is registered when there are near-time coincident and statistically significant enhancements of the count rates of at least two differently located neutron monitors, including at least one neutron monitor near sea level and a corresponding enhancement in the proton flux measured by a space-borne instrument(s). Relatively weak SEP events registered only by high-altitude polar neutron monitors, but with no response from cosmic-ray stations at sea level, can be classified as sub-GLEs.

Recent data from the Kepler mission has revealed the occurrence of superflares in sun-like stars which exceed by far any observed solar flares in release of energy. A natural idea is that the dynamo mechanism in superflaring stars differs in some respect from that in the Sun. We search for a difference in the dynamo-related parameters between superflaring stars and the Sun to suggest a dynamo-mechanism as close as possible to the conventional solar/stellar dynamo but capable of providing much higher magnetic energy. Dynamo based on joint action of differential rotation and mirror asymmetric motions can in principle result in excitation of two types of magnetic fields. First of all, it is well-known in solar physics dynamo waves. The point is that another magnetic configuration with initial growth and further stabilisation is also possible for excitation. For comparable conditions, magnetic field strength of second configuration is much larger rather of the first one just because dynamo do not spend its efforts for periodic magnetic field inversions but use its for magnetic field growth. We analysed available data from the Kepler mission concerning the superflaring stars in order to find tracers of anomalous magnetic activity. Starting from the recent paper, we find that anti-solar differential rotation or anti-solar sign of the mirror-asymmetry of stellar convection can provide the desired strong magnetic field in dynamo models. We confirm this concept by numerical models of stellar dynamos with corresponding governing parameters. We conclude that the proposed mechanism can plausibly explain the superflaring events at least for some cool stars, including binaries, subgiants and, possibly, low-mass stars and young rapid rotators., Comment: 13 pages, 3 figures and 1 table

An important topic in the field of space weather is the precise assessment of the contribution of galactic cosmic rays and solar energetic particles on air crew exposure, specifically during eruptive events on the Sun. Here we present a model, a full chain analysis based on ground based measurements of cosmic rays with neutron monitors, subsequent derivation of particle spectral and angular characteristics and computation of dose rate. The model uses method for ground level enhancement analysis and newly numerically computed yield functions for conversion of secondary particle fluence to effective dose and/or the ambient dose equivalent. The precise an adequate information about the solar energetic particle spectra (SEPs) is the basis of the model. Since SEPs possess an essential isotropic part, specifically during the event onset, the angular characteristics should be also derived with good precision. This can be achieved using neutron monitor data during a special class of SEP events the ground level enhancements (GLEs). On the basis of the method representing a sequence of consecutive steps: computation of the NM asymptotic cones, NM rigidity cut-off and application of convenient optimization procedure, we derive the rigidity spectra and anisotropy characteristics of GLE particles. For the computation we use newly computed yield function of the standard sea-level 6NM64 neutron monitor for primary proton and alpha CR nuclei as well as 6NM64 yield function at altitudes ranging from the sea level up to 5000 m above the sea level. We derive the SEP spectra and pitch angle distributions in their dynamical development throughout the event. Subsequently on the basis of the derived spectra and angular characteristics and previously computed yield functions we calculate the effective dose and/or ambient dose equivalent during the GLE. Several examples are shown., Comment: "XXV ECRS 2016 Proceedings - eConf C16-09-04.3", highlight talk

It has been recently claimed (Zolotova and Ponyavin, Solar Phys., 291, 2869, 2016, ZP16 henceforth) that a mid-latitude optical phenomenon, which took place over the city of Astrakhan in July 1670, according to Russian chronicles, was a strong aurora borealis. If this was true, it would imply a very strong or even severe geomagnetic storm during the quietest part of the Maunder minimum. However, as we argue in this article, this conclusion is erroneous and caused by a misinterpretation of the chronicle record. As a result of a thorough analysis of the chronicle text, we show that the described phenomenon occurred during the daylight period of the day ("the last morning hour"), in the south direction ("towards noon"), and its description does not match that of an aurora. The date of the event was also incorrectly interpreted. We conclude that this phenomenon was not a mid-latitude aurora but an atmospheric phenomenon, the so-called sundog (or parhelion) which is a particular type of solar halo. Accordingly, the claim about a strong mid-latitude aurora during the deep Maunder minimum is not correct and should be dismissed., Comment: accepted to Solar Physics

A ground-based neutron monitor is a standard tool to measure cosmic ray variability near Earth, and it is crucially important to know its yield function for primary cosmic rays. Although there are several earlier theoretically calculated yield functions, none of them agrees with experimental data of latitude surveys of sea-level neutron monitors, thus suggesting for an inconsistency. A newly computed yield function of the standard sea-level 6NM64 neutron monitor is presented here separately for primary cosmic ray protons and $\alpha-$particles, the latter representing also heavier species of cosmic rays. The computations have been done using the GEANT-4 Planetocosmics Monte-Carlo tool and a realistic curved atmospheric model. For the first time, an effect of the geometrical correction of the neutron monitor effective area, related to the finite lateral expansion of the cosmic ray induced atmospheric cascade, is considered, that was neglected in the previous studies. This correction slightly enhances the relative impact of higher-energy cosmic rays (energy above 5--10 GeV/nucleon) in neutron monitor count rate. The new computation finally resolves the long-standing problem of disagreement between the theoretically calculated spatial variability of cosmic rays over the globe and experimental latitude surveys. The newly calculated yield function, corrected for this geometrical factor, appears fully consistent with the experimental latitude surveys of neutron monitors performed during three consecutive solar minima in 1976--77, 1986--87 and 1996--97. Thus, we provide a new yield function of the standard sea-level neutron monitor 6NM64 that is validated against experimental data., Comment: Includes also Erratum (submitted to JGR) to Table 1

Nearly 70 Ground Level Enhancements (GLEs) of cosmic rays have been recorded by the worldwide neutron monitor network since the 1950s depicting a big variety of energy spectra of solar energetic particles (SEP). Here we studied a statistical relation between the event-integrated intensity of GLEs (calculated as count-rate relative excess, averaged over all available polar neutron monitors, and expressed in percent-hours) and the hardness of the solar particle energy spectra. For each event the integral omnidirectional event-integrated fluences of particles with energy above 30 MeV ($F_{30}$) and above 200 MeV ($F_{200}$) were computed using the reconstructed spectra, and the ratio between the two fluences was considered as a simple index of the event's hardness. We also provided a justification of the spectrum estimate in the form of the Band-function, using direct PAMELA data for GLE 71 (17-May-2012). We found that, while there is no clear relation between the intensity and the hardness for weak events, all strong events with the intensity greater 100 \%*hr are characterized by a very hard spectrum. This implies that a hard spectrum can be securely assumed for all extreme GLE events, e.g., those studied using cosmogenic isotope data in the past., Comment: Advances in Space Research (in press)

The distribution of the sunspot group size (area) and its dependence on the level of solar activity is studied. It is shown that the fraction of small groups is not constant but decreases with the level of solar activity so that high solar activity is largely defined by big groups. We study the possible influence of solar activity on the ability of a realistic observer to see and report the daily number of sunspot groups. It is shown that the relation between the number of sunspot groups as seen by different observers with different observational acuity thresholds is strongly non-linear and cannot be approximated by the traditionally used linear scaling ($k-$factors). The observational acuity threshold [$A_{\rm th}$] is considered to quantify the quality of each observer, instead of the traditional relative $k-$factor. A nonlinear $c-$factor based on $A_{\rm th}$ is proposed, which can be used to correct each observer to the reference conditions. The method is tested on a pair of principal solar observers, Wolf and Wolfer, and it is shown that the traditional linear correction, with the constant $k-$factor of 1.66 to scale Wolf to Wolfer, leads to an overestimate of solar activity around solar maxima., Comment: Solar Physics, accepted

Cosmogenic isotopes provide the only quantitative proxy for analyzing the long-term solar variability over a centennial timescale. While essential progress has been achieved in both measurements and modeling of the cosmogenic proxy, uncertainties still remain in the determination of the geomagnetic dipole moment evolution. Here we improve the reconstruction of solar activity over the past nine millennia using a multi-proxy approach. We used records of the 14C and 10Be cosmogenic isotopes, current numerical models of the isotope production and transport in Earth's atmosphere, and available geomagnetic field reconstructions, including a new reconstruction relying on an updated archeo-/paleointensity database. The obtained series were analyzed using the singular spectrum analysis (SSA) method to study the millennial-scale trends. A new reconstruction of the geomagnetic dipole field moment, GMAG.9k, is built for the last nine millennia. New reconstructions of solar activity covering the last nine millennia, quantified in sunspot numbers, are presented and analyzed. A conservative list of grand minima and maxima is provided. The primary components of the reconstructed solar activity, as determined using the SSA method, are different for the series based on 14C and 10Be. These primary components can only be ascribed to long-term changes in the terrestrial system and not to the Sun. They have been removed from the reconstructed series. In contrast, the secondary SSA components of the reconstructed solar activity are found to be dominated by a common ~2400-yr quasi-periodicity, the so-called Hallstatt cycle, in both the 14C and 10Be based series. This Hallstatt cycle thus appears to be related to solar activity. Finally, we show that the grand minima and maxima occurred intermittently over the studied period, with clustering near highs and lows of the Hallstatt cycle, respectively., Comment: In press in Astronomy & Astrophysics, doi: 10.1051/0004-6361/201527295

A new model CRAC:EPII (Cosmic Ray Atmospheric Cascade: Electron Precipitation Induced Ionization) is presented. The CRAC:EPII is based on Monte Carlo simulation of precipitating electrons propagation and interaction with matter in the Earth atmosphere. It explicitly considers energy deposit: ionization, pair production, Compton scattering, generation of Bremsstrahlung high energy photons, photo-ionization and annihilation of positrons, multiple scattering as physical processes accordingly. The propagation of precipitating electrons and their interactions with atmospheric molecules is carried out with the GEANT4 simulation tool PLANETOCOSMICS code using NRLMSISE 00 atmospheric model. The ionization yields is compared with an analytical parametrization for various energies of incident precipitating electron, using a flux of mono-energetic particles. A good agreement between the two models is achieved. Subsequently, on the basis of balloon-born measured spectra of precipitating electrons at 30.10.2002 and 07.01.2004, the ion production rate in the middle and upper atmosphere is estimated using the CRAC:EPII model, Comment: 13 pages, 4 figures. Under review in JASTP

Although the sunspot-number series have existed since the mid-19th century, they are still the subject of intense debate, with the largest uncertainty being related to the "calibration" of the visual acuity of individual observers in the past. Daisy-chain regression methods are applied to inter-calibrate the observers which may lead to significant bias and error accumulation. Here we present a novel method to calibrate the visual acuity of the key observers to the reference data set of Royal Greenwich Observatory sunspot groups for the period 1900-1976, using the statistics of the active-day fraction. For each observer we independently evaluate their observational thresholds [S_S] defined such that the observer is assumed to miss all of the groups with an area smaller than S_S and report all the groups larger than S_S. Next, using a Monte-Carlo method we construct, from the reference data set, a correction matrix for each observer. The correction matrices are significantly non-linear and cannot be approximated by a linear regression or proportionality. We emphasize that corrections based on a linear proportionality between annually averaged data lead to serious biases and distortions of the data. The correction matrices are applied to the original sunspot group records for each day, and finally the composite corrected series is produced for the period since 1748. The corrected series displays secular minima around 1800 (Dalton minimum) and 1900 (Gleissberg minimum), as well as the Modern grand maximum of activity in the second half of the 20th century. The uniqueness of the grand maximum is confirmed for the last 250 years. It is shown that the adoption of a linear relationship between the data of Wolf and Wolfer results in grossly inflated group numbers in the 18th and 19th centuries in some reconstructions., Comment: Accepted for publication at Solar Physics

A two-wave dynamo model was recently proposed by Zharkova et al. (2015, Zh15 henceforth), which aims at long-term predictions of solar activity for millennia ahead and backwards. Here we confront the backward model predictions for the last 800 years with known variability of solar activity, using both direct sunspot observations since 1610 and reconstructions based on cosmogenic radionuclide data. We show that the Zh15 model fails to reproduce the well-established features of the solar activity evolution during the last millennium. This means that the predictive part for the future is not reliable either., Comment: 8 pages, 1 figure, an extended version of a comment sent to Sci. Rep

We use sunspot group observations from the Royal Greenwich Observatory (RGO) to investigate the effects of intercalibrating data from observers with different visual acuities. The tests are made by counting the number of groups $R_B$ above a variable cut-off threshold of observed total whole-spot area (uncorrected for foreshortening) to simulate what a lower acuity observer would have seen. The synthesised annual means of $R_B$ are then re-scaled to the observed RGO group number $R_A$ using a variety of regression techniques. It is found that a very high correlation between $R_A$ and $R_B$ ($r_{AB}$ > 0.98) does not prevent large errors in the intercalibration (e.g. sunspot maximum values can be over 30% too large even for such levels of $r_{AB}$). In generating the backbone sunspot number, Svalgaard and Schatten [2015] force regression fits to pass through the scatter plot origin which generates unreliable fits (the residuals do not form a normal distribution) and causes sunspot cycle amplitudes to be exaggerated in the intercalibrated data. It is demonstrated that the use of Quantile-Quantile (Q-Q) plots to test for a normal distribution is a useful indicator of erroneous and misleading regression fits. Ordinary least squares linear fits, not forced to pass through the origin, are sometimes reliable (although the optimum method used is shown to be different when matching peak and average sunspot group numbers). However other fits are only reliable if non-linear regression is used. From these results it is entirely possible that the inflation of solar cycle amplitudes in the backbone group sunspot number as one goes back in time, relative to related solar-terrestrial parameters, is entirely caused by the use of inappropriate and non-robust regression techniques to calibrate the sunspot data., Comment: 14 pages, 1 Table, 11 figures

Long-term records of sunspot number and concentrations of cosmogenic radionuclides (10Be and 14C) on the Earth reveal the variation of the Sun's magnetic activity over hundreds and thousands of years. We identify several clear periods in sunspot, 10Be, and 14C data as 1000, 500, 350, 200 and 100 years. We found that the periods of the first five spherical harmonics of the slow magnetic Rossby mode in the presence of a steady toroidal magnetic field of 1200-1300 G in the lower tachocline are in perfect agreement with the time scales of observed variations. The steady toroidal magnetic field can be generated in the lower tachocline either due to the steady dynamo magnetic field for low magnetic diffusivity or due to the action of the latitudinal differential rotation on the weak poloidal primordial magnetic field, which penetrates from the radiative interior. The slow magnetic Rossby waves lead to variations of the steady toroidal magnetic field in the lower tachocline, which modulate the dynamo magnetic field and consequently the solar cycle strength. This result constitutes a key point for long-term prediction of the cycle strength. According to our model, the next deep minimum in solar activity is expected during the first half of this century., Comment: 4 pages, 4 figures, accepted in ApJL

The Maunder minimum (MM) of greatly reduced solar activity took place in 1645-1715, but the exact level of sunspot activity is uncertain as based, to a large extent, on historical generic statements of the absence of spots on the Sun. Here we aim, using a conservative approach, to assess the level and length of solar cycle during the Maunder minimum, on the basis of direct historical records by astronomers of that time. A database of the active and inactive days (days with and without recorded sunspots on the solar disc respectively) is constructed for three models of different levels of conservatism (loose ML, optimum MO and strict MS models) regarding generic no-spot records. We have used the active day fraction to estimate the group sunspot number during the MM. A clear cyclic variability is found throughout the MM with peaks at around 1655--1657, 1675, 1684 and 1705, and possibly 1666, with the active day fraction not exceeding 0.2, 0.3 or 0.4 during the core MM, for the three models. Estimated sunspot numbers are found very low in accordance with a grand minimum of solar activity. We have found, for the core MM (1650-1700), that: (1) A large fraction of no-spot records, corresponding to the solar meridian observations, may be unreliable in the conventional database. (2) The active day fraction remained low (below 0.3-0.4) throughout the MM, indicating the low level of sunspot activity. (3) The solar cycle appears clearly during the core MM. (4) The length of the solar cycle during the core MM appears $9\pm 1$ years, but there is an uncertainty in that. (5) The magnitude of the sunspot cycle during MM is assessed to be below 5-10 in sunspot numbers; A hypothesis of the high solar cycles during the MM is not confirmed., Comment: Accepted to Astron. Astrophys

Context. The relation between solar surface rotation and sunspot activity still remains open. Sunspot activity has dramatically reduced in solar cycle 24 and several solar activity indices and flux measurements experienced unprecedentedly low levels during the last solar minimum. Aims. We aim to reveal the momentary variation of solar surface rotation, especially during the recent years of reducing solar activity. Methods. We used a dynamic, differentially rotating reference system to determine the best-fit annual values of the differential rotation parameters of active longitudes of solar X-ray flares and sunspots in 1977-2012. Results. The evolution of rotation of solar active longitudes obtained with X-ray flares and with sunspots is very similar. Both hemispheres speed up since the late 1990s, with the southern hemisphere rotating slightly faster than the north. Earlier, in 1980s, rotation in the northern hemisphere was considerably faster, but experienced a major decrease in the early 1990s. On the other hand, little change was found in the southern rotation during these decades. This led to a positive asymmetry in north-south rotation rate in the early part of the time interval studied. Conclusions. The rotation of both hemispheres has been speeding up at roughly the same rate since late 1990s, with the southern hemisphere rotating slightly faster than the north. This period coincides with the start of dramatic weakening of solar activity, as observed in sunspots and several other solar, interplanetary and geomagnetic parameters., Comment: Astron. Astrophys. Lett. (accepted)

Firestone (2014) asserted evidence for numerous (23) nearby (d<300 pc) supernovae within the Middle and Late Pleistocene. If true, this would have strong implications for the irradiation of the Earth; at this rate, mass extinction level events due to supernovae would be more frequent than 100 Myr. However, there are numerous errors in the application of past research. The paper overestimates likely nitrate and 14C production from moderately nearby supernovae by about four orders of magnitude. Moreover, the results are based on wrongly selected (obsolete) nitrate and 14C datasets. The use of correct and up-to-date datasets does not confirm the claimed results. The claims in the paper are invalidated., Comment: 8 pages, 1 figure. to be published in the Interenational Journal of Astrobiology

Aims. The Sun shows strong variability in its magnetic activity, from Grand minima to Grand maxima, but the nature of the variability is not fully understood, mostly because of the insufficient length of the directly observed solar activity records and of uncertainties related to long-term reconstructions. Here we present a new adjustment-free reconstruction of solar activity over three millennia and study its different modes. Methods. We present a new adjustment-free, physical reconstruction of solar activity over the past three millennia, using the latest verified carbon cycle, 14C production, and archeomagnetic field models. This great improvement allowed us to study different modes of solar activity at an unprecedented level of details. Results. The distribution of solar activity is clearly bi-modal, implying the existence of distinct modes of activity. The main regular activity mode corresponds to moderate activity that varies in a relatively narrow band between sunspot numbers about 20 and 67. The existence of a separate Grand minimum mode with reduced solar activity, which cannot be explained by random fluctuations of the regular mode, is confirmed at a high confidence level. The possible existence of a separate Grand maximum mode is also suggested, but the statistics is too low to reach a confident conclusion. Conclusions. The Sun is shown to operate in distinct modes - a main general mode, a Grand minimum mode corresponding to an inactive Sun, and a possible Grand maximum mode corresponding to an unusually active Sun. These results provide important constraints for both dynamo models of Sun-like stars and investigations of possible solar influence on Earth's climate., Comment: Astron. Astrophys. Letters (in press)

Aims. Sunspot number is a benchmark series in many studies, but may still contain inhomogeneities and inconsistencies. In particular, an essential discrepancy exists between the two main sunspot number series, Wolf (WSN) and group (GSN) sunspot numbers, before 1848. The source of this discrepancy has so far remained unresolved. However, the recently digitized series of solar observations in 1825-1867 by Samuel Heinrich Schwabe, who was the primary observer of the WSN before 1848, makes such an assessment possible. Methods. We construct sunspot series, similar to WSN and GSN, but using only Schwabe's data. These series, called WSN-S and GSN-S, respectively, were compared with the original WSN and GSN series for the period 1835-1867 to look for possible inhomogeneities. Results. We show that: (1) The GSN series is homogeneous and consistent with the Schwabe data throughout the entire studied period; (2) The WSN series decreases by roughly ~20% around 1848 caused by the change of the primary observer from Schwabe to Wolf and an inappropriate individual correction factor used for Schwabe in the WSN; (3) This implies a major inhomogeneity in the WSN, which needs to be corrected by reducing its values by 20% before 1848; (4) The corrected WSN series is in good agreement with the GSN series. This study supports the earlier conclusions that the GSN series is more consistent and homogeneous in the earlier part than the WSN series., Comment: Published as: Leussu, R., I.G. Usoskin, R. Arlt and K. Mursula, Inconsistency of the Wolf sunspot number series around 1848, Astron. Astrophys., 559, A28, 2013

Samuel Heinrich Schwabe made 8486 drawings of the solar disk with sunspots in the period from November 5, 1825 to December 29, 1867. We have measured sunspot sizes and heliographic positions on digitized images of these drawings. A total of about 135,000 measurements of individual sunspots are available in a data base. Positions are accurate to about 5% of the solar radius or to about three degrees in heliographic coordinates in the solar disk center. Sizes were given in 12 classes as estimated visually with circular cursor shapes on the screen. Most of the drawings show a coordinate grid aligned with the celestial coordinate system. A subset of 1168 drawings have no indication of their orientation. We have used a Bayesian estimator to infer the orientations of the drawings as well as the average heliographic spot positions from a chain of drawings of several days, using the rotation profile of the present Sun. The data base also includes all information available from Schwabe on spotless days., Comment: 8 pages, 4 figures, Mon. Not. R. Astron. Soc., accepted

Miyake et al. (henceforth M12) recently reported, based on 14C data, an extreme cosmic event ca. AD775. Using a simple model, M12 claimed that the event was too strong to be caused by a solar flare within the standard theory. This implied a new paradigm of either an impossibly strong solar flare or a very strong cosmic ray event of unknown origin occurred ca. AD775. We show that the strength of the event was significantly overestimated by M12. Several subsequent works have attempted to find a possible exotic source for such an event, but they are all based on incorrect estimates by M12. We revisit this event with analysis of new datasets and consistent theoretical modelling. We verified the experimental result for the AD775 event using independent datasets including 10Be series and newly measured 14C annual data. We surveyed available historical chronicles for astronomical observations for the AD770s to identify potential sightings of aurorae or supernovae. We interpreted the 14C measurements using an appropriate carbon cycle model. We show that: (1) The reality of the AD775 event is confirmed by new measurements of 14C; (2) by using an inappropriate carbon cycle model, M12 strongly overestimated the event's strength; (3) The revised magnitude of the event is consistent with different independent datasets (14C, 10Be, 36Cl) and can be associated with a strong, but not inexplicably strong, SEP event (or a sequence of events), and provides the first evidence for an event of this magnitude (the fluence >30 MeV was about 4.5*10^{10} /cm2) in multiple datasets; (4) This is in agreement with increased auroral activity identified in historical chronicles. This point to the likely solar origin of the event, which is the greatest solar event on a multi-millennial time scale, placing a strong observational constraint on the theory of explosive energy releases on the Sun and cool stars., Comment: Published by Astron. Astrophys. Lett

We report on the 2012 May 17 Ground Level Enhancement (GLE) event, which is the first of its kind in Solar Cycle 24. This is the first GLE event to be fully observed close to the surface by the Solar Terrestrial Relations Observatory (STEREO) mission. We determine the coronal mass ejection (CME) height at the start of the associated metric type II radio burst (i.e., shock formation height) as 1.38 Rs (from the Sun center). The CME height at the time of GLE particle release was directly measured from a STEREO image as 2.32 Rs, which agrees well with the estimation from CME kinematics. These heights are consistent with those obtained for cycle-23 GLEs using back-extrapolation. By contrasting the 2012 May 17 GLE with six other non-GLE eruptions from well-connected regions with similar or larger flare size and CME speed, we find that the latitudinal distance from the ecliptic is rather large for the non-GLE events due to a combination of non-radial CME motion and unfavorable solar B0 angle, making the connectivity to Earth poorer. We also find that the coronal environment may play a role in deciding the shock strength., Comment: 16 pages, 4 figures, 1 table

The long-term variability of the sunspot cycle, as recorded by the Wolf numbers, are imprinted in different kinds of statistical relations which relate the cycle amplitudes, duration and shapes. This subject always gets a special attention because it is important for the solar activity forecast. We discuss statistical properties of the mean-field dynamo model with the fluctuating \alpha-effect. Also, we estimate dynamical properties of the model for the long and short time-scale and compare it with the dynamics of the sunspot numbers data sets., Comment: 2 pages, IAU 294 Proceedings

The probability of occurrence of extreme solar particle events (SPEs) with the fluence of (>30 MeV) protons F30>10^{10} cm^{-2} is evaluated based on data of cosmogenic isotopes 14C and 10Be in terrestrial archives centennial-millennial time scales. Four potential candidates with F30=(1-1.5)x10^{10} cm^{-2} and no events with F30>2x10^{10} cm^{-2} are identified since 1400 AD in the annually resolved 10Be data. A strong SPE related to the Carrington flare of 1859 AD is not supported by the data. For the last 11400 years, 19 SPE candidates with F30=(1-3)x10^{10} cm^{-2} are found and clearly no event with F30>5x10^{10} cm^{-2} (50-fold the SPE of 23-Feb-1956) occurring. This values serve as an observational upper limit for the strength of SPE on the time scale of tens of millennia. Two events, ca. 780 and 1460 AD, appear in different data series making them strong candidates to extreme SPEs. We built a distribution of the occurrence probability of extreme SPEs, providing a new strict observational constraint. Practical limits can be set as F30~1x, 2-3x, and 5x10^{10} cm^{-2} for the occurrence probability ~10^{-2}, 10^{-3} and 10^{-4} year^{-1}, respectively. Because of uncertainties, our results should be interpreted as a conservative upper limit of the SPE occurrence near Earth. The mean SEP flux is evaluated as ~40 (cm2 sec)^{-1} in agreement with estimates from the lunar rocks. On average, extreme SPEs contribute about 10% to the total SEP fluence., Comment: accepted to Astrophys. J

We present the results of full new calculation of radiocarbon 14C production in the Earth atmosphere, using a numerical Monte-Carlo model. We provide, for the first time, a tabulated 14C yield function for the energy of primary cosmic ray particles ranging from 0.1 to 1000 GeV/nucleon. We have calculated the global production rate of 14C, which is 1.64 and 1.88 atoms/cm2/s for the modern time and for the pre-industrial epoch, respectively. This is close to the values obtained from the carbon cycle reservoir inventory. We argue that earlier models overestimated the global 14C production rate because of outdated spectra of cosmic ray heavier nuclei. The mean contribution of solar energetic particles to the global 14C is calculated as about 0.25% for the modern epoch. Our model provides a new tool to calculate the 14C production in the Earth's atmosphere, which can be applied, e.g., to reconstructions of solar activity in the past., Comment: Published in EPSL, 337, 114, 2012

The most powerful explosions on the Sun [...] drive the most severe space-weather storms. Proxy records of flare energies based on SEPs in principle may offer the longest time base to study infrequent large events. We conclude that one suggested proxy, nitrate concentrations in polar ice cores, does not map reliably to SEP events. Concentrations of select radionuclides measured in natural archives may prove useful in extending the time interval of direct observations up to ten millennia, but as their calibration to solar flare fluences depends on multiple poorly known properties and processes, these proxies cannot presently be used to help determine the flare energy frequency distribution. Being thus limited to the use of direct flare observations, we evaluate the probabilities of large-energy solar explosions by combining solar flare observations with an ensemble of stellar flare observations. We conclude that solar flare energies form a relatively smooth distribution from small events to large flares, while flares on magnetically-active, young Sun-like stars have energies and frequencies markedly in excess of strong solar flares, even after an empirical scaling with the mean activity level of these stars. In order to empirically quantify the frequency of uncommonly large solar flares extensive surveys of stars of near-solar age need to be obtained, such as is feasible with the Kepler satellite. Because the likelihood of flares larger than approximately X30 remains empirically unconstrained, we present indirect arguments, based on records of sunspots and on statistical arguments, that solar flares in the past four centuries have likely not substantially exceeded the level of the largest flares observed in the space era, and that there is at most about a 10% chance of a flare larger than about X30 in the next 30 years., Comment: 14 pages, 3 figures (in press as of 2012/06/18); Journal of Geophysical Research (Space Physics), 2012

We present an overview of the observed properties of the GLEs and those of the associated flares and CMEs. The solar eruptions are very intense involving X-class flares and extreme CME speeds (average ~2000 km/s). The active regions in which the GLE events originate are generally large: 1290 msh (median 1010 msh) compared to 934 msh (median: 790 msh) for SEP-producing active regions. The initial acceleration of GLE-associated CMEs is much larger (by a factor of 2) than that of ordinary CMEs (2.3 km/s2 vs.1 km/s2). The GLE particle release is delayed with respect to the onset of all electromagnetic signatures of the eruptions: type II bursts, low frequency type III bursts, soft X-ray flares and CMEs. The presence of metric type II radio bursts some 17 min (median: 16 min; range: 3 to 48 min) before the GLE onset indicates shock formation well before the particle release. The release of GLE particles occurs when the CMEs reach an average height of ~3.09 Rs for well-connected events. For poorly connected events, the average CME height at GLE particle release is ~66% larger (mean: 5.18 Rs). The longitudinal dependence is consistent with shock accelerations because the shocks from poorly connected events need to expand more to cross the field lines connecting to an Earth observer. The CME height at metric type II burst onset is in the narrow range 1.29 to 1.8 Rs, with A mean of 1.53 Rs. The CME heights at metric type II burst onset and GLE particle release correspond to the minimum and maximum in the Alfven speed profile. The CME heights at GLE particle release are in good agreement with those obtained from the velocity dispersion analysis (Reames, 2009a,b) including the source longitude dependence. We also discuss the implications of the delay of GLE particle release with respect to complex type III bursts and hard X-ray emission., Comment: 17 figures, 56 pages, based on two LWS/CDAWs on GLE events, part of Topical Issue on GLE events in Space Science Reviews

Solar cycles vary in their amplitude and shape. There are several empirical relations between various parameters linking cycle's shape and amplitude, in particular the Waldmeier relations. As solar cycle is believed to be a result of the solar dynamo action, these relations require explanation in the framework of this theory.Here we aim to present a possible explanation of such kind. We relate the cycle-to-cycle variability of solar activity to fluctuations of solar dynamo drivers and primarily to fluctuations in the parameter responsible for recovery of the poloidal magnetic field from the toroidal one. To be specific, we develop such a model in the framework of the mean-field dynamo based on the differential rotation and $\alpha$-effect. We demonstrate that the mean-field dynamo based on a realistic rotation curve and nonlinearity associated with the magnetic helicity balance reproduces both qualitatively and quantitatively the Waldmeier relations observed in sunspot data since 1750 (SIDC data). The model also reproduces more or less successfully other relations between the parameters under discussion, in particular, the link between odd and even cycles (Gnevyshev-Ohl rule). We conclude that the contemporary solar dynamo theory provides a way to explain the cycle-to-cycle variability of solar activity as recorded in sunspots. We discuss the importance of the model for stellar activity cycles which, as known from the data of HK project, demonstrate the cycle-to-cycle variability similar to solar cycles., Comment: 12 pages, 9 figures, to appear in A&A

Maunder Minimum forms an archetype for the Grand minima, and detailed knowledge of its temporal development has important consequences for the solar dynamo theory dealing with long-term solar activity evolution. Here we reconsider the current paradigm of the Grand minimum general scenario by using newly recovered sunspot observations by G. Marcgraf and revising some earlier uncertain data for the period 1636--1642, i.e., one solar cycle before the beginning of the Maunder Minimum. The new and revised data dramatically change the magnitude of the sunspot cycle just before the Maunder Minimum, from 60--70 down to about 20, implying a possibly gradual onset of the Minimum with reduced activity started two cycles before it. This revised scenario of the Maunder Minimum changes, through the paradigm for Grand solar/stellar activity minima, the observational constraint on the solar/stellar dynamo theories focused on long-term studies and occurrence of Grand minima., Comment: 8 pages, 2 figures, accepted for publication in ApJ Letters

We consider to what extent the long-term dynamics of cyclic solar activity in the form of Grand Minima can be associated with random fluctuations of the parameters governing the solar dynamo. We consider fluctuations of the alpha-coefficient in the conventional Parker migratory dynamo, and also in slightly more sophisticated dynamo models, and demonstrate that they can mimic the gross features of the phenomenon of the occurrence of Grand Minima over a suitable parameter range. The temporal distribution of these Grand Minima appears chaotic, with a more or less exponential waiting time distribution, typical of Poisson processes. In contrast however, the available reconstruction of Grand Minima statistics based on cosmogenic isotope data demonstrates substantial deviations from this exponential law. We were unable to reproduce the non-Poissonic tail of the waiting time distribution either in the framework of a simple alpha-quenched Parker model, or in its straightforward generalization, nor in simple models with feedback on the differential rotation. We suggest that the disagreement may only be apparent and is plausibly related to the limited observational data, and that the observations and results of numerical modeling can be consistent and represent physically similar dynamo regimes., Comment: Solar Physics, in print

Using a reconstruction of sunspot numbers stretching over multiple millennia, we analyze the statistics of the occurrence of grand minima and maxima and set new observational constraints on long-term solar and stellar dynamo models. We present an updated reconstruction of sunspot number over multiple millennia, from $^{14}$C data by means of a physics-based model, using an updated model of the evolution of the solar open magnetic flux. A list of grand minima and maxima of solar activity is presented for the Holocene (since 9500 BC) and the statistics of both the length of individual events as well as the waiting time between them are analyzed. The occurrence of grand minima/maxima is driven not by long-term cyclic variability, but by a stochastic/chaotic process. The waiting time distribution of the occurrence of grand minima/maxima deviates from an exponential distribution, implying that these events tend to cluster together with long event-free periods between the clusters. Two different types of grand minima are observed: short (30--90 years) minima of Maunder type and long ($>$110 years) minima of Sp\"orer type, implying that a deterministic behaviour of the dynamo during a grand minimum defines its length. The duration of grand maxima follows an exponential distribution, suggesting that the duration of a grand maximum is determined by a random process. These results set new observational constraints upon the long-term behaviour of the solar dynamo., Comment: 10 Figures